Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Georges Bastin and Jean-Michel Coron

Transcription

1 Stability and Boundary Stabilization of 1-D Hyperbolic Systems Georges Bastin and Jean-Michel Coron April 18, 216

2 2 c G. Bastin and J-M. Coron, 216.

3 Contents Preface 9 1 Hyperbolic Systems of Balance Laws Definitions and notations Riemann coordinates and characteristic form Steady state and linearization Riemann coordinates around the steady state Conservation laws and Riemann invariants Stability, boundary stabilization and the associated Cauchy problem Systems in Riemann coordinates General hyperbolic systems Telegrapher equations Raman amplifiers Saint-Venant equations for open channels Boundary conditions Steady state and linearization The general model Saint-Venant-Exner equations Rigid pipes and heat exchangers The shower control problem Isothermal model: the water hammer problem Heat exchangers Plug flow chemical reactors Euler equations for gas pipes Isentropic equations Steady state and linearization Musical wind instruments Fluid flow in elastic tubes Aw-Rascle equations for road traffic Ramp metering Kac-Goldstein equations for chemotaxis Age-dependent SIR epidemiologic equations Steady state Chromatography SMB chromatography Scalar conservation laws Physical networks of hyperbolic systems Networks of electrical lines Chains of density-velocity systems

4 4 Contents Gas pipe lines Navigable rivers and irrigation channels Genetic regulatory networks References and further reading Systems of Two Linear Conservation Laws Stability conditions Exponential convergence Lyapunov stability and convergence in L 2 -norm A note on the proofs of stability in L 2 -norms Frequency domain stability Example: stability of a lossless electrical line Boundary control of density-flow systems Feedback stabilization with two controls Dead-beat control Feedback-feedforward stabilization with a single control Proportional-integral control Stability analysis in the frequency domain Lyapunov stability analysis The non uniform case Conclusions Systems of Linear Conservation Laws Exponential stability for the L 2 -norm Dissipative boundary conditions Exponential stability for the C -norm: analysis in the frequency domain. 88 A simple illustrative example Robust stability Comparison of the two stability conditions The rate of convergence Application to a system of two conservation laws Differential linear boundary conditions Frequency domain Lyapunov approach Example: a lossless electrical line Example: a network of density-flow systems under PI control Example: stability of genetic regulatory networks The non uniform case Switching linear conservation laws The example of SMB chromatography A simulation experiment References and further reading Systems of Nonlinear Conservation Laws Dissipative boundary conditions for the C 1 -norm Control of networks of scalar conservation laws Application: Ramp-metering control in road traffic networks Interlude: Solutions without shocks

5 Contents Dissipative boundary conditions for the H 2 -norm Proof of Theorem Stability of general systems of nonlinear conservation laws in quasi-linear form Stability condition for the C 1 -norm Stability condition for the C p -norm Stability condition for the H p -norm References and further reading Systems of Linear Balance Laws Lyapunov exponential stability Example: feedback control of an exothermic PFR Linear systems with uniform coefficients Application to linearized Saint-Venant-Exner model Steady state and linearization Riemann coordinates Lyapunov stability Existence of a basic quadratic control Lyapunov function Application to the control of an open channel Boundary control of density-flow systems Transfer functions Boundary feedback stabilization with two local controls Feedback-feedforward stabilization with a single control Proportional-Integral control Proportional-Integral control in navigable rivers Dissipative stability condition Control error propagation Limit of stabilizability References and further reading Quasi-Linear Hyperbolic Systems Stability of systems with uniform steady states Stability of general quasi-linear hyperbolic systems Stability condition for the H 2 -norm Stability condition for the H p -norm References and further reading Backstepping Control Motivation and problem statement Full-state feedback Observer design and output feedback Backstepping control of systems of two balance laws References and further reading

6 6 Contents 8 Case Study: Control of Navigable Rivers Geographic and technical data Modelling and simulation Control implementation Local or nonlocal control? Steady state and set-point selection Choice of the time step for digital control Control tuning and performance References and further reading A Well-posedness of the Cauchy problem for linear hyperbolic systems 223 B Well-posedness of the Cauchy problem for quasi-linear hyperbolic systems 233 C Properties and comparisons of the functions ρ, ρ 2 and ρ 237 C.1 Properties of the function ρ 2 K) C.2 Proof of Theorem C.3 Proof of Proposition D Proof of Lemma 4.12 b) and c) 255 E Proof of Theorem F Notations 265 Index 267

7 Preface THE TRANSPORT of electrical energy, the flow of fluids in open channels or in gas pipelines, the light propagation in optical fibers, the motion of chemicals in plug flow reactors, the blood flow in the vessels of mammalians, the road traffic, the propagation of age-dependent epidemics or the chromatography are typical examples of processes that may be represented by hyperbolic partial differential equations PDEs). In all these applications, described in Chapter 1, the dynamics are usefully represented by one-dimensional hyperbolic balance laws although the natural dynamics are three-dimensional, because the dominant phenomena evolve in one privileged coordinate dimension, while the phenomena in the other directions are negligible. From an engineering perspective, for hyperbolic systems as well as for all dynamical systems, the stability of the steady states is a fundamental issue. This book is therefore entirely devoted to the exponential) stability of the steady states of one-dimensional systems of conservation and balance laws considered over a finite space interval, i.e. where the spatial domain of the PDE is an interval of the real line. The definition of exponential stability is intuitively simple: starting from an arbitrary initial condition, the system time-trajectory has to exponentially converge in spatial-norm to the steady state globally for linear systems and locally for nonlinear systems). Behind the apparent simplicity of this definition, the stability analysis is however quite challenging. First because this definition is not so easily translated into practical tests of stability. Secondly, because the various function norms that can be used to measure the deviation with respect to the steady state are not necessarily equivalent and may therefore give rise to different stability tests. As a matter of fact, the exponential stability of steady states closely depends on the so-called dissipativity of the boundary conditions which, in many instances, is a natural physical property of the system. In this book, one of the main tasks is therefore to derive simple practical tests for checking if the boundary conditions are dissipative. Linear systems of conservation laws are the simplest case. They are considered in Chapters 2 and 3. For those systems, as for systems of linear ordinary differential equations, a necessary and sufficient) test is to verify that the poles i.e. the roots of the characteristic equation) have negative real parts. Unfortunately, this test is not very practical and, in addition, not very useful because it is not robust with respect to small variations of the system dynamics. In Chapter 3, we show how a robust necessary and sufficient) dissipativity test can be derived by using a Lyapunov stability approach, which guarantees the existence of globally exponentially converging solutions for any L p -norm. The situation is much more intricate for nonlinear systems of conservation laws which are considered in Chapter 4. Indeed for those systems, it is well known that the trajectories of the system may become discontinuous in finite time even for smooth initial conditions that are close to the steady state. Fortunately, if the boundary conditions are dissipative and if the smooth initial conditions are sufficiently close to the steady state, it is shown in 7

8 8 Preface this chapter that the system trajectories are guaranteed to remain smooth for all time and that they exponentially converge locally to the steady state. Surprisingly enough, due to the nonlinearity of the system, even for smooth solutions, it appears that the exponential stability strongly depends on the considered norm. In particular, using again a Lyapunov approach, it is shown that the dissipativity test of linear systems holds also in the nonlinear case for the H 2 -norm, while it is necessary to use a more conservative test for the exponential stability for the C 1 -norm. In Chapters 5 and 6, we move to hyperbolic systems of linear and nonlinear balance laws. The presence of the source terms in the equations brings a big additional difficulty for the stability analysis. In fact the tests for dissipative boundary conditions of conservation laws are directly extendable to balance laws only if the source terms themselves have appropriate dissipativity properties. Otherwise, as it shown in Chapter 5, it is only known through the special case of systems of two balance laws) that there are intrinsic limitations to the system stabilizability with local controls. There are also many engineering applications where the dissipativity of the boundary conditions, and consequently the stability, is obtained by using boundary feedback control with actuators and sensors located at the boundaries. The control may be implemented with the goal of stabilization when the system is physically unstable, or simply because boundary feedback control is required to achieve an efficient regulation with disturbance attenuation. Obviously, the challenge in that case is to design the boundary control devices in order to have a good control performance with dissipative boundary conditions. This issue is illustrated in Chapters 2 and 5 by investigating in detail the boundary proportionalintegral output feedback control of so-called density-flow systems. Moreover Chapter 7 addresses the boundary stabilization of hyperbolic systems of balance laws by full-state feedback and by dynamic output feedback in observer-controller form, using the backstepping method. Numerous other practical examples of boundary feedback control are also presented in the other chapters. Finally, in the last chapter Chapter 8), we present a detailed case-study devoted to the control of navigable rivers when the river flow is described by hyperbolic Saint-Venant shallow water equations. The goal is to emphasize the main technological features that may occur in real life applications of boundary feedback control of hyperbolic systems of balance laws. The issue is presented through the specific application of the control of the Meuse river in Wallonia south of Belgium). In our opinion, the book could have a dual audience. In one hand, mathematicians interested in applications of control of 1-D hyperbolic PDEs may find the book a valuable resource to learn about applications and state-of-the-art control design. On the other hand, engineers including graduate and post-graduate students) who want to learn the theory behind 1-D hyperbolic equations may also find the book an interesting resource. The book requires a certain level of mathematics background which may be slightly intimidating. There is however no need to read the book in a linear fashion from the front cover to the back. For example, people concerned primarily with applications may skip the very first Section 1.1 on first reading and start directly with their favorite examples in Chapter 1, referring to the definitions of Section 1.1 only when necessary. Chapter 2 is basic to an understanding of a large part of the remainder of the book, but many practical or theoretical sections in the subsequent chapters can be omitted on first reading without problem. The book presents many examples that serve to clarify the theory and to emphasize the

9 9 practical applicability of the results. Many examples are continuation of earlier examples so that a specific problem may be developed over several chapters of the book. Although many references are quoted in the book our bibliography is certainly not complete. The fact that a particular publication is mentioned simply means that it has been used by us as a source material or that related material can be found in it. Acknowledgements. The material of this book has been developed over the last fifteen years. We want to thank all those who, in one way or another, contributed to this work. We are especially grateful to Fatiha Alabau, Fabio Ancona, Brigitte d Andrea- Novel, Alexandre Bayen, Gildas Besançon, Michel Dehaen, Michel De Wan, Ababacar Diagne, Philippe Dierickx, Malik Drici, Sylvain Ervedoza, Didier Georges, Olivier Glass, Martin Gugat, Jonathan de Halleux, Laurie Haustenne, Bertrand Haut, Michael Herty, Thierry Horsin, Long Hu, Miroslav Krstic, Pierre-Olivier Lamare, Günter Leugering, Xavier Litrico, Luc Moens, Hoai-Minh Nguyen, Guillaume Olive, Vincent Perrollaz, Benedetto Piccoli, Christophe Prieur, Valérie Dos Santos Martins, Catherine Simon, Paul Suvarov, Simona Oana Tamasoiu, Ying Tang, Alain Vande Wouwer, Paul Van Dooren, Rafael Vazquez, Zhiqiang Wang and Joseph Winkin. During the preparation of this book we have benefited from the support of the ERC advanced grant CPDENL, European 7th Research Framework Programme FP7)) and of the Belgian Programme on Inter-university Attraction Poles IAP VII/19) which are also gratefully acknowledged. The implementation of the Meuse regulation reported in Chapter 8 is carried out by the Walloon Region, Siemens and the University of Louvain. Louvain-la-Neuve, Paris, February 216 Georges Bastin Jean-Michel Coron

10 1 Preface

11 Chapter 1 Hyperbolic Systems of Balance Laws IN THIS CHAPTER we provide an introduction to the modeling of balance laws by hyperbolic partial differential equations PDEs). A balance law is the mathematical expression of the physical principle that the variation of the amount of some extensive quantity over a bounded domain is balanced by its flux through the boundaries of the domain and its production/consumption inside the domain. Balance laws are therefore used to represent the fundamental dynamics of many physical open conservative systems. In the first section, we give the basic definitions and properties that will be used throughout the book. We successively address the characteristic form, the Riemann coordinates, the steady states, the linearization and the boundary stabilization problem. The remaining of the chapter is then devoted to a presentation of typical examples of hyperbolic systems of balance laws for a wide range of physical engineering applications, with a view to allow the readers to understand the concepts in their most familiar setting. With these examples we also illustrate how the control boundary conditions may be defined for the most commonly used control devices Definitions and notations In this section we give the basic definition of one-dimensional systems of balance laws as they are used throughout the book. Let Y be a non-empty connected open subset of R n. A one-dimensional hyperbolic system of n nonlinear balance laws over a finite space interval is a system of PDEs of the form 1 1.1) t eyt, x)) + x fyt, x)) + gyt, x)) =, t [, + ), x [, L], where t and x are the two independent variables: a time variable t [, + ) and a space variable x [, L] over a finite interval; Y : [, + ) [, L] Y is the vector of state variables; e C 2 Y; R n ) is the vector of the densities of the balanced quantities; the map e is a diffeomorphism on Y; f C 2 Y; R n ) is the vector of the corresponding flux densities; 1 The partial derivatives of a function f with respect to the variables x and t are indifferently denoted xf and tf or f x and f t. 11

12 12 Chapter 1. Hyperbolic Systems of Balance Laws g C 1 Y; R n ) is the vector of source terms representing production or consumption of the balanced quantities inside the system. Under these conditions, system 1.1) can be written in the form of a quasi-linear system 1.2) Y t + F Y)Y x + GY) =, t [, + ), x [, L], with F : Y M n,n R) and G : Y R n are of class C 1 and defined as F Y) e/ Y) 1 f/ Y), GY) e/ Y) 1 gy). As usual, M n,n R) denotes the set of n n real matrices. Also in 1.2), and often in the rest of the book, we drop the argument t, x) when it does not lead to confusion. We assume that system 1.2) is hyperbolic i.e. that F Y) has n real eigenvalues called characteristic velocities) for all Y Y. In this book, it will be also always assumed that these eigenvalues do not vanish in Y. It follows that the number m of positive eigenvalues counting multiplicity) is independent of Y. Except otherwise stated, we will always use the following notations for the m positive and the n m negative eigenvalues: λ 1 Y),..., λ m Y), λ m+1 Y),..., λ n Y), λ i Y) > Y Y, i. In the particular case where F is constant i.e. does not depend on Y), the system 1.2) is called semi-linear. Obviously, in that case, the system has constant characteristic velocities denoted: λ 1,..., λ m, λ m+1,..., λ n, λ i > i. Remark that, in contrast with most publications on quasi-linear hyperbolic systems, we use here the notation λ i Y) to designate the absolute value of the characteristic velocities. The reason for using such an heterodox notation is that it simplifies the mathematical writings when the sign of the characteristic velocities matters in the boundary stability analysis which is one of the main concerns of this book. Riemann coordinates and characteristic form In this book we shall often focus on the class of hyperbolic systems of balance laws that can be transformed into a characteristic form by defining a set of n so-called Riemann coordinates see for instance Dafermos, 2, Chapter 7, Section 7.3) ). The characteristic form is obtained through a change of coordinates R = ψy) having the following properties: The function ψ : Y R R n is a diffeomorphism: R = ψy) Y = ψ 1 R), with Jacobian matrix ΨY) ψ/ Y. The Jacobian matrix ΨY) diagonalizes the matrix F Y): ΨY)F Y) = DY)ΨY), Y Y, with DY) = diag λ 1 Y),..., λ m Y), λ m+1 Y),..., λ n Y) ).

13 1.1. Definitions and notations 13 The system 1.2) is then equivalent for C 1 solutions to the following system in characteristic form expressed in the Riemann coordinates: 1.3) R t + ΛR)R x + CR) =, t [, + ), x [, L], with ΛR) Dψ 1 R)) and CR) Ψψ 1 R))Gψ 1 R)). Clearly, this change of coordinates exists for any system of balance laws with linear flux densities i.e. with fy) = AY, A M n,n R) constant) when the matrix A is diagonalizable, in particular when the characteristic velocities are distinct. For systems with nonlinear flux densities, finding the change of coordinates R = ψy) requires to find a solution of the first order partial differential equation ΨY)F Y) = DY)ΨY). As it is shown in Lax, 1973, pages 34 35), this partial differential equation can always be solved, at least locally, for systems of size n = 2 with distinct characteristic velocities see also Li, 1994, p.3) ). By contrast, for systems of size n 3, the change of coordinates exists only in non-generic specific cases. However we shall see in this chapter that there is a multitude of interesting physical models for engineering which have size n 3 and can nevertheless be written in characteristic form. Steady state and linearization A steady state or equilibrium) is a time-invariant space-varying solution Yt, x) = Y x) t [, + ) of the system 1.2). It satisfies the ordinary differential equation 1.4) F Y )Y x + GY ) =, x [, L]. The linearization of the system about the steady state is then 1.5) Y t + Ax)Y x + Bx)Y =, t [, + ), x [, L], where Ax) F Y x)) and Bx) [ )] F Y)Y Y x + GY). Y=Y x) In the special case where there is a solution to the algebraic equation GY ) =, the system has a constant steady state independent of both t and x) and the linearization is 1.6) Y t + AY x + BY =, t [, + ), x [, L], where A and B are constant matrices. In this special case where Y is constant, the nonlinear system 1.2) is said to have a uniform steady state. In the general case where the steady state Y x) is space varying, the nonlinear system 1.2) is said to have a nonuniform steady state. Riemann coordinates around the steady state By definition, the steady state of system 1.3) is R x) = ψy x)) such that ΛR )R x + CR ) =.

14 14 Chapter 1. Hyperbolic Systems of Balance Laws Then, alternatively, Riemann coordinates may also be defined around this steady state as R ψy) ψy ). With these coordinates the system is now written in characteristic form as 1.7) R t + ΛR, x)r x + CR, x) =, t [, + ), x [, L], with and ΛRt, x), x) Dψ 1 Rt, x) + ψy x))) CRt, x), x) D ψ 1 Rt, x) + ψy x)) ) ψ x Y x)) + Ψ ψ 1 Rt, x) + ψy x)) ) G ψ 1 Rt, x) + ψy x)) ). The linearization of the system 1.7) gives: with R t + Λx)R x + Mx)R =, t [, + ), x [, L], [ ] CR, x) Λx) DY x)) and Mx) R. R= Remark that this linear model is also the linearization of system 1.3) around the steady state and that it could be obtained as well by transforming directly the linear system 1.5) into Riemann coordinates. In other words the operations of linearization and Riemann coordinate transformation can be inverted. Conservation laws and Riemann invariants In the special case where there are no source terms i.e. GY) =, Y Y), system 1.1) or 1.2) reduces to 1.8) t ey) + x fy) = or Y t + F Y)Y x =, t [, + ), x [, L], A system of this form is a hyperbolic system of conservation laws, representing a process where the balanced quantity is conserved since it can change only by the flux through the boundaries. In that case, it is clear that any constant value Y can be a steady state, independently of the value of the coefficient matrix F Y). Thus such systems have uniform steady states by definition. After transformation in Riemann coordinates if possible), a system of conservation laws is written in the form t R i + λ i R) x R i =, i = 1,..., m, t R i λ i R) x R i =, i = m + 1,..., n. The left-hand sides of these equations are the total time derivatives dr i dt t R i + dx dt xr i

15 1.1. Definitions and notations 15 t R i t, x) =R i,x o ) i i x o L Fig.1.1: Characteristic curves x of the Riemann coordinates along the characteristic curves which are the integral curves of the ordinary differential equations dx dt = λ irt, x)), i = 1,..., m, dx dt = λ irt, x)), i = m + 1,..., n, in the plane t, x) as illustrated in Fig.1.1. Since dr i /dt =, it follows that the Riemann coordinates R i t, x) are constant along the characteristic curves and are therefore called Riemann invariants for systems of conservation laws. Stability, boundary stabilization and the associated Cauchy problem In order to have a unique well defined solution to a quasi-linear hyperbolic system 1.2) over the interval [, L], initial and boundary conditions must obviously be specified. In this book, we address the specific issue of identifying and characterizing dissipative boundary conditions which guarantee bounded smooth solutions converging to an equilibrium. Of special interest is the feedback control problem when the manipulated control input, the controlled outputs and the measured outputs are physically located at the boundaries. Formally, this means that we consider the system 1.2) under n boundary conditions having the general form 1.9) B Yt, ), Yt, L), Ut) ) = with the map B C 1 Y Y R q, R n ). The dependence of the map B on Yt, ), Yt, L)) refers to natural physical constraints on the system. The function Ut) R q represents a set of q exogenous control inputs that can be used for stabilization, output tracking or disturbance rejection. In the case of static feedback control laws UYt, ), Yt, L)), one of our main concerns is to analyze the asymptotic convergence of the solutions of the Cauchy problem: System Y t + F Y)Y x + GY) =, t [, + ), x [, L], B. C. BYt, ), Yt, L), UYt, ), Yt, L))) =, t [, + ), I. C. Y, x) = Y o x) x [, L].

16 16 Chapter 1. Hyperbolic Systems of Balance Laws Additional constraints on the initial condition I.C.) and the boundary conditions B.C.) are needed to have a well-posed Cauchy problem. We examine this issue first in the case when the system can be transformed into characteristic form and then in the general case. The Cauchy problem in Riemann coordinates As we shall see later in this chapter, for many physical systems described by hyperbolic equations written in characteristic form 1.3) R t + ΛR)R x + CR) =, t [, + ), x [, L], it is a basic property that at each boundary point the incoming information R in is determined by the outgoing information R out Russell, 1978, Section 3), with the definitions ) R + t, ) ) R + t, L) 1.1) R in t) R t, L) and R out t) R t, ), where R + and R are defined as follows 2 : R + = R 1,..., R m ) T, R = R m+1,..., R n ) T. This means that the system 1.3) is subject to boundary conditions having the nominal form 1.11) R in t) = H R out t) ), where the map H C 1 R n ; R n ). Moreover, the initial condition 1.12) R, x) = R o x) x [, L] must be specified. Hence, in Riemann coordinates, the Cauchy problem is formulated as follows: System R t + ΛR)R x + CR) =, t [, + ), x [, L], B. C. R in t) = H R out t) ), t [, + ), I. C. R, x) = R o x) x [, L]. The well-posedness of this Cauchy problem may require that the initial condition 1.12) be compatible with the boundary condition 1.11). The compatibility conditions which are necessary for the well-posedness of the Cauchy problem depend on the functional space to which the solutions belong. In this book, we will be mainly concerned with solutions Rt,.) that may be of class C or L 2 for linear systems and of class C 1 or H 2 for quasilinear systems. For each case, the required compatibility conditions will be presented at the most suitable place see also Appendices A and B). It is also interesting to remark that the hyperbolic system 1.3) under the boundary condition 1.11) can be regarded as the closed loop interconnection of two causal inputoutput systems as represented in Fig In this section and everywhere in the book the notation M T denotes the transpose of the matrix M.

17 1.1. Definitions and notations 17 R in R t + R)R x + CR) = R out System S 1 H.) System S 2 Fig.1.2: A quasi-linear hyperbolic systems with boundary conditions in nominal form is a closed loop interconnection of two causal inputoutput systems The well-posedness of the general Cauchy problem for strictly hyperbolic systems Let us now consider the case of a general quasi-linear hyperbolic system Y t + F Y)Y x + GY) =, t [, + ), x [, L], BYt, ), Yt, L)) =, t [, + ), which cannot be transformed into characteristic form. We assume that the system is strictly hyperbolic which means that for each Y Y, the matrix F Y) has non zero distinct eigenvalues. Therefore, for all x [, L], the matrix F Y x)), where Y x) is the steady state as in 1.4), can be diagonalized, i.e. there exists a map N : x [, L] Nx) M n,n R) of class C 1 such that Nx) is invertible for all x [, L], Nx)F Y x)) = Λx)Nx), with Λx) diag{λ 1 x),..., λ m x), λ m+1 x),..., λ n x)}. We define the following change of coordinates: Zt, x) Nx) Yt, x) Y x)), Z = Z 1,..., Z n ) T. In the coordinates Z, the system is rewritten Z t + AZ, x)z x + BZ, x) =, B N) 1 Zt, ) + Y ), NL) 1 Zt, L) + Y L) ) =, with AZ, x) Nx)F Nx) 1 Z + Y x))nx) 1 with A, x) = Λx), BZ, x) Nx) [ F Nx) 1 Z + Y x))y xx) Nx) 1 N x)nx) 1 Z) + GNx) 1 Z + Y x)) ].

18 18 Chapter 1. Hyperbolic Systems of Balance Laws Let us now define the incoming and outgoing boundary signals: ) Z + t, ) ) Z + t, L) Z in t) Z t, L) and Z out t) Z t, ), where Z + and Z are as follows: Z + = Z 1,..., Z m ) T, Z = Z m+1,..., Z n ) T. Obviously there exists a map B C 1 R n R n ; R n ) such that 1.13) BN) 1 Zt, ), NL) 1 Zt, L)) = BZ in t), Z out t)). The requirement that, at each boundary point, the incoming information should be determined by the outgoing information imposes that 1.13) can be solved for Z in : Z in t) = H Z out t) ). Then, provided the system is strictly hyperbolic and the initial condition is compatible with the boundary condition, the well-posed Cauchy problem is formulated as follows: 1.14) System Z t + AZ, x)z x + BZ, x) =, t [, + ), x [, L], B. C. Z in t) = H Z out t) ), t [, + ), I. C. Z, x) = Z o x), x [, L], with appropriate compatibility conditions for the initial state Z o. The rest of this chapter is now devoted to presenting typical examples of hyperbolic systems of balance laws for various physical engineering applications. We shall see that in many examples, the system can indeed be transformed into Riemann coordinates. With these examples we also illustrate how the control boundary conditions may be defined for the most commonly used control devices Telegrapher equations First published by Heaviside 1892), page 123, the telegrapher equations describe the propagation of current and voltage along electrical transmission lines see Fig.1.3). It is a system of two linear hyperbolic balance laws of the following form: 1.15) t L l I) + x V + R l I =, t C l V ) + x I + G l V =, where It, x) is the current intensity, V t, x) is the voltage, L l is the line self-inductance per unit length, C l is the line capacitance per unit length, R l is the resistance of the two

19 1.2. Telegrapher equations 19 Power supply Ut) It, ) It, L) Transmission line R V t, ) L x V t, L) Load R L Fig.1.3: Transmission line connecting a power supply to a resistive load R L ; the power supply is represented by a Thevenin equivalent with electromotive force Ut) and internal resistance R. conductors per unit length and G l is the admittance per unit length of the dielectric material separating the conductors. For the circuit represented in Fig.1.3, the line model 1.15) is to be considered under the following boundary conditions: 1.16) V t, ) + R It, ) = Ut), V t, L) R L It, L) =, where R is the internal resistance of the power supply and R L is the load. The telegrapher equations 1.15) coupled with these boundary conditions constitute therefore a boundary control system with the voltage Ut) as control input. A steady state I x), V x) of system 1.15) is a solution of the differential equation ) ) V Rl I 1.17) x I + G l V =. From equations 1.15) and 1.17), we can write the model, around a steady state, in the general linear form 1.18) Y t + AY x + BY = with ) It, x) I x) Yt, x) V t, x) V, A x) ) ) L 1 l Rl L 1 l C 1, B l G l C 1. l Here, because the physical system 1.15) is linear, we observe that the linear system 1.18) has uniform coefficients although the steady state may be non-uniform. The system has two characteristic velocities which are the eigenvalues of the matrix A), one positive and one negative: λ 1 = 1 Ll C l, λ 2 = 1 Ll C l.

20 2 Chapter 1. Hyperbolic Systems of Balance Laws Riemann coordinates can be defined as R 1 t, x) = V t, x) V x) ) + It, x) I x) ) L l C l, R 2 t, x) = V t, x) V x) ) It, x) I x) ) L l C l, with the inverse coordinate transformation It, x) I x) = R 1t, x) R 2 t, x) 2 V t, x) V x) = R 1t, x) + R 2 t, x). 2 Cl L l, With these coordinates, the system 1.15), 1.16) is written as follows in characteristic form: 1.19) 1.2) with t R 1 + λ x R 1 + γr 1 + δr 2 =, t R 2 λ x R 2 + δr 1 + γr 2 =, R 1 t, ) = [ 1 + λr C l )R 2 t, ) + Ut) U ] 1 + λr C l ) 1, R 2 t, L) = [ 1 + λr C l )R 1 t, L) ] 1 + λr C l ) 1, λ 1, γ 1 [ Gl + R ] l, δ 1 [ Gl R ] l. Ll C l 2 C l L l 2 C l L l 1.3. Raman amplifiers Raman amplifiers are electro-optical devices that are used for compensating the natural power attenuation of laser signals transmitted along optical fibres in long distance communications. Their operation is based on the Raman effect which was discovered by Raman and Krishnan 1928). The simplest implementation of Raman amplification in optical telecommunications is depicted in Fig.1.4. The transmitted information is encoded by pump beam input signal L x output signal Fig.1.4: Optical communication with Raman amplification

21 1.4. Saint-Venant equations for open channels 21 amplitude modulation of a laser signal with wavelength ω s. The signal is provided by an optical source at the channel input and received by a photo-detector at the output. A pump laser beam with wavelength ω p is injected backward in the optical fibre. If the wavelengths are appropriately selected, the energy of the pump is transferred to the signal and produces an amplification that counteracts the natural attenuation. The dynamics of the signal and pump powers along the fibre are represented by the following system of two balance laws Dower and Farrel 26) ): ) t S + λ s x S + α s S β s SP =, 1.21) ) t P λ p x P α p P β p P S =, where St, x) is the power of the transmitted signal, P t, x) is the power of the pump laser beam, λ s and λ p are the propagation group velocities of the signal and pump waves respectively, α s and α p are the attenuation coefficients per unit length, β s and β p are the amplification gains per unit length. All these positive constant parameters α s and α p, β s and β p, λ s and λ p are characteristic of the fibre material and dependent of the wavelengths ω s and ω p. Here, the physical model 1.21) is directly given in characteristic form 1.3). The Riemann coordinates are the powers R 1 = St, x) and R 2 = P t, x). The system is hyperbolic with characteristic velocities λ s > > λ p. As the input signal power and the launch pump power can be exogenously imposed, the boundary conditions are 1.22) St, ) = U t), P t, L) = U L t). Then the system 1.21) coupled to the boundary conditions 1.22) is a boundary control system with the boundary control inputs U and U L Saint-Venant equations for open channels First proposed by Barré de Saint-Venant 1871), the Saint-Venant equations also called shallow water equations) describe the propagation of water in open channels see Fig.1.5). In the simple standard case of a channel with a constant slope, a rectangular cross section and a unit width, the Saint-Venant model is a system of two nonlinear balance laws of the form 1.23) t H + x HV ) =, V 2 ) t V + x 2 + gh + C V 2 ) H gs b =, where Ht, x) is the water depth and V t, x) is the horizontal water velocity. More precisely, V t, x) denotes the horizontal velocity averaged across a vertical column of water. S b is the constant bottom slope, g is the constant gravity acceleration and C is a constant friction coefficient. The first equation is a mass balance and the second equation is a momentum balance.

22 22 Chapter 1. Hyperbolic Systems of Balance Laws Ht, x) V t, x) L x Fig.1.5: Lateral view of a pool of an open channel with constant bottom slope and rectangular cross section. This model is written in the general quasi-linear form Y t + F Y)Y x + GY) = with ) ) ) H V H Y, F Y), GY) V g V CV 2 H 1. gs b The eigenvalues of the matrix F Y) are V + gh and V gh. The flow is said to be subcritical or fluvial) if the so-called Froude s number F r = V t, x) ght, x) < 1. Under this condition, the system is hyperbolic with characteristic velocities λ 1 Y) = V + gh > > λ 2 Y) = V gh. Riemann coordinates may be defined as R 1 = V + 2 gh, R 2 = V 2 gh and are inverted as H = R 1 R 2 ) 2 /16g, V = R 1 + R 2 )/2. With these coordinates, the system is written in characteristic form R t + ΛR)R x + CR) = with ) 3R 1 + R 2 λ1 R) ΛR) = 4 λ 2 R) R 1 + 3R 2 4 and ) ) ) 2 R1 + R 2 1 CR) 4gC gs b. R 1 R 2 1

23 1.4. Saint-Venant equations for open channels 23 Boundary conditions When the flow is subcritical, two boundary conditions at both ends of the interval [, L] are needed to close the Saint-Venant equations. These conditions are imposed by physical devices located at the ends of the pool, as for instance the two spillways of the channel in Fig.1.5. A very simple situation is when the pool is closed but endowed with pumps that impose the discharges at x = and x = L. In that case, the boundary conditions are 1.24) Ht, )V t, ) = U t), Ht, L)V t, L) = U L t). Then the system of the Saint-Venant equations 1.23) coupled to the boundary conditions 1.24) is a boundary control system with the two boundary flow rates U and U L as command signals. Another interesting case is when the boundary conditions are assigned by tunable hydraulic gates as in irrigation canals and navigable rivers, see Fig.1.6. Z U U L Ht, L) Z Ht, ) U Ht, L) U L Z L Fig.1.6: Hydraulic gates at the input and the output of a pool: above) overflow gates, below) underflow gates. Standard hydraulic models give the boundary conditions for overflow gates or mobile spillways): 1.25) Ht, )V t, ) = k G 2g ) [ Z t) U t) ] 3, Ht, L)V t, L) = k G 2g ) [ Ht, L) UL t) ] 3,

24 24 Chapter 1. Hyperbolic Systems of Balance Laws and for underflow or sluice) gates: 1.26) Ht, )V t, ) = k G 2g ) U t) Z t) Ht, ), Ht, L)V t, L) = k G 2g ) UL t) Ht, L) Z L t). In these expressions, Ht, ) and Ht, L) denote the water depth at the boundaries inside the pool, Z t) and Z L t) are the water levels on the other side of the gates, k G is a constant adimensional discharge coefficient, U t) and U L t) represent either the weir elevation for overflow gates or the height of the aperture for underflow gates. Again the Saint-Venant equations 1.23) coupled to these boundary conditions constitute a boundary control system with U and U L as command signals, and Z and Z L as disturbance inputs. Steady state and linearization A steady state H x), V x) is a solution of the differential equations x H V ) =, x V gh ) + These equations may also be written as C V 2 ) gs b =. H V x H = H x V = H V gs b CV 2 /H ) gh V 2. In order to linearize the model, we define the deviations of the states Ht, x) and V t, x) with respect to the steady states H x) and V x) : ht, x) Ht, x) H x), vt, x) V t, x) V x). Then the linearized Saint-Venant equations around the steady state are : 1.27) t h + V x h + H x v + x V )h + x H )v =, t v + g x h + V x v CV 2 /H 2 )h + [ x V + 2CV /H )] v =. The Riemann coordinates for the linearized system 1.27) are defined as follows: g R 1 t, x) = vt, x) + ht, x) H x), R 2 t, x) = vt, x) ht, x) with the inverse coordinate transformation ht, x) = R 1t, x) R 2 t, x) 2 vt, x) = R 1t, x) + R 2 t, x). 2 g H x), H x), g

25 1.5. Saint-Venant-Exner equations 25 With these definitions and notations, the linearized Saint-Venant equations are written in characteristic form: 1.28) t R 1 t, x) + λ 1 x) x R 1 t, x) + γ 1 x)r 1 t, x) + δ 1 x)r 2 t, x) = t R 2 t, x) λ 2 x) x R 2 t, x) + γ 2 x)r 1 t, x) + δ 2 x)r 2 t, x) = with the characteristic velocities λ 1 x) = V x) + gh x), λ 2 x) = V x) gh x) and, using the relation H x V ) = V x H ), the coefficients γ 1 = δ 1 = γ 2 = δ 2 = CV 2 H CV 2 H CV 2 H CV 2 H The general model [ 3 4 gh + V ) + 1 V 1 2 gh [ 1 4 gh + V ) + 1 V gh [ 1 4 gh V ) + 1 V 1 2 gh [ 3 4 gh V ) + 1 V gh ] + ] + ] ] 3gS b 4 gh + V ), gs b 4 gh + V ), gs b 4 gh V ), 3gS b 4 gh V ). To conclude this section, we give a more general version of the Saint-Venant equations which holds for channels with non-constant slopes and cross-sections. The equations are as follows: 1.29) t A + x Q =, t Q + x Q 2 A + ga[ x H S b + S f ] = where At, x) is the cross-sectional area of the water in the channel, Qt, x) is the flow rate or discharge), Ht, x) is the water depth, S f is the friction term, S b x) is the bottom slope and g is the constant gravity acceleration. The friction term S f is usually assumed to be proportional to V 2 = Q 2 /A 2 and to the perimeter P of the cross-sectional area. Clearly it is natural to assume that both the water depth HA) and the perimeter P A) are monotonic increasing functions of A Saint-Venant-Exner equations First proposed by Exner 192) see also Exner 1925) ), the Exner equation is a conservation law that represents the transport of sediments in a water flow in the case where the sediment moves predominantly as bedload. A common modeling of the dynamics of open

26 26 Chapter 1. Hyperbolic Systems of Balance Laws Bt, x) Ht, x) V t, x) x Fig.1.7: Lateral view of an open channel with a sediment bed. channels with fluctuating bathymetry is therefore achieved by the coupling of the Exner equation to the Saint-Venant equations. The state variables of the model see Fig.1.7) are the water depth Ht, x) and the average horizontal water velocity V t, x) as for Saint-Venant equations, and the bathymetry Bt, x) which is the elevation of the sediment bed above a fixed reference datum. For an horizontal channel with a rectangular cross-section and a unit width, the equations are written as follows see e.g. Hudson and Sweby 23) ): t H + x HV ) =, 1.3) ) V 2 t V + x + gh + B) + C V 2 2 H =, t B + x a V 3 ) =. 3 In these equations, g is the gravity acceleration constant, C is a friction coefficient and a is a constant parameter that encompasses porosity and viscosity effects on the sediment dynamics. The first two equations are the Saint-Venant equations and the third one is the Exner equation. This model is in the general quasi-linear form Y t + F Y)Y x + GY) = with H Y V, B V H F Y) g V g, av 2 The characteristic polynomial of the matrix F Y) is λ 3 2V λ 2 + V 2 gav 2 + H))λ + agv 3. GY) C V 2 H. From this polynomial, analytic expressions of the eigenvalues of F Y) are not easily derived. However, as shown by Hudson and Sweby 23), good approximations can be obtained for small values of the parameter a under the subcritical flow condition V 2 < gh. As a, the eigenvalues of F Y) tend to λ 1 V + gh, λ 2, λ 3 V gh.

27 1.6. Rigid pipes and heat exchangers 27 The determinant of F Y) is 1.31) detf Y)) = λ 1 λ 2 λ 3 = agv 3. Then, for small values of a, we have the following realistic approximations 1.32) [ λ1 V + gh ] [ λ 2 agv 3 gh V 2 ] > > [ λ3 V gh ] where the value of λ 2 is obtained by substituting the values of λ 1 and λ 3 in 1.31). Here λ 1 and λ 3 are the characteristic velocities of the water flow and λ 2 the characteristic velocity of the sediment motion. Obviously the sediment motion is much slower than the water flow. Thus, the Saint-Venant-Exner model 1.3) is a hyperbolic system of three balance laws with characteristic velocities approximately given by 1.32) Rigid pipes and heat exchangers The management of hydro-electric plants, the design of water supply networks with water hammer prevention or the temperature control in heat exchangers are typical engineering issues that require dynamic models of water flow in pipes. Under the assumptions of axisymmetric flow and negligible radial fluid velocity, a standard model for the motion of water in a rigid cylindrical pipe is given by the following system of three balance laws: )) )) gh gh t exp + x V exp =, 1.33) c 2 t V + x gh + V 2 ) + C V V =, 2 2d t T + x V T ) + k o T e T ) =, where Ht, x) is the piezometric head, V t, x) is the water velocity, T t, x) is the water temperature, c is the sound velocity in water, C is a constant friction coefficient, g is the gravity acceleration, d is the pipe diameter and T e is the external atmospheric temperature. The first equation is a mass conservation law, the second equation is a momentum balance and the third equation is a heat balance. The piezometric head H is defined as Ht, x) = Zx) + c 2 P t, x), ρg where Zx) is the elevation of the pipe, P t, x) is the pressure and ρ is the density. For an horizontal pipe, the piezometric head is just proportional to the pressure. The constant parameter k o is defined as k o α c p ρa,

28 28 Chapter 1. Hyperbolic Systems of Balance Laws where α is the thermal conductance of the pipe wall, c p is the water specific heat and A = πd 2 /4 is the cross-section area of the pipe. This kind of model based on one-dimensional mass, momentum or heat balances was already present in the engineering scientific literature by the late nineteenth century see e.g. the paper by Allievi 193) and also other references quoted in the survey paper by Ghidaoui et al. 25) ). The model 1.33) is written in the general quasi-linear form Y t +F Y)Y x +GY) = with H Y V T, V c 2 /g F Y) g V, T V The characteristic polynomial of the matrix F Y) is V λ)v λ) 2 c 2 ). GY) CV V /2d. k o T e T ) The roots of this polynomial are V, V + c and V c. In practice, the sound velocity is about 14 m/s and the flow velocity is much lower. In that case, the system is hyperbolic with characteristic velocities which are the eigenvalues of the matrix F Y)): Riemann coordinates may then be defined as and are inverted as λ 1 = V + c > λ 2 = V > > λ 3 = V c. R 1 = V + g c H, R 2 = g c H c ln T, R 3 = V g c H, H = c g R 1 R 3, V = R 1 + R ln T = R 1 2R 2 R 3. 2c With these coordinates, the system is written in characteristic form R t + ΛR)R x + CR) = with λ 1 R) ΛR) λ 2 R) = R 1 c 1 + R λ 3 R) 1 c and CR) = k o c CR 1 + R 3 ) R 1 + R 3 8d )) R1 + 2R 2 + R 3 2c. CR 1 + R 3 ) R 1 + R 3 8d 1 T e exp

29 1.6. Rigid pipes and heat exchangers 29 hot cold Fig.1.8: The shower control problem. The shower control problem Everybody knows the shower control problem which is the problem of simultaneously regulating the temperature and the flow rate of a shower by manipulating the two valves of hot and cold water as illustrated in Fig.1.8. The system is described by the model 1.33) with L being the length of the pipe between the valves and the shower outlet. This control problem may be analysed under the following boundary conditions: 1.34) AV t, ) = Q c t) + Q h t), P t, L) = P a ρg, T t, ) = Q ct)t c t) + Q h t)t h t). Q c t) + Q h t) The first condition represents the flow conservation at the junction of the valves, with Q c t) and Q h t) the cold and hot flow rates assigned by the two valves respectively. The second condition is that the atmospheric pressure P a is imposed at the outlet. The third condition expresses that the inlet temperature is an average of the cold T c and hot T h temperatures. Then the system of the shower equations 1.33) with the boundary conditions 1.34) is a boundary control system with the flow rates Q c and Q h as command signals. The water hammer problem The device of Fig.1.9 is a typical example of a system that may have a water hammer problem if the valve is closed too quickly or the pump is started up too quickly, see e.g. Van Pham et al. 214). Such a problem can be analyzed with the the first two equations of 1.33) and appropriate boundary conditions imposed by the pump and the valve respectively, see e.g. Luskin and Temple 1982). For instance, the pump may be regarded as a device which is able to deliver a desired pressure drop no matter the flow rate: 1.35) H in t) Ht, ) = Ut).

30 3 Chapter 1. Hyperbolic Systems of Balance Laws pump valve L x Fig.1.9: A pipe connecting a pump and a valve Moreover, the valve is typically modeled by a quadratic relationship between the pressure drop and the velocity: 1.36) Ht, L) H out t) = k v V t, L) V t, L), where k v is a constant characteristic parameter. The pipe equation 1.33) coupled with these boundary conditions form a boundary control system with the pump command signal Ut) as control input and the external piezometric heads H in t) and H out t) as disturbance inputs. Heat exchangers A simple tubular heat exchanger is depicted in Fig.1.1. It is made up of two counter current concentric pipes. Clearly a dynamical model is obtained by duplicating the basic warm inflow T 2 t, x) T 1 t, x) cold inflow heated outflow x cooled outflow Fig.1.1: A tubular heat exchanger balance equations 1.33) supplemented with appropriate interconnection terms as follows: t H 1 + V 1 x H 1 + c2 g xv 1 =, t V 1 + x gh 1 + V ) + C 2d V 1 V 1 =,

31 1.7. Plug flow chemical reactors 31 t T 1 + x V 1 T 1 ) k 1 T 1 T 2 ) k o T 1 T e ) =, t H 2 + V 2 x H 2 + c2 g xv 2 =, t V 2 + x gh 2 + V ) + C 2d V 2 V 2 =, t T 2 + x V 2 T 2 ) + k 2 T 1 T 2 ) =, where T 1 t, x), T 2 t, x) are the water temperatures, V 1 t, x), V 2 t, x) the water velocities and H 1 t, x), H 2 t, x) the piezometric heads in the heating and heated tubes respectively and T e is the external atmospheric temperature. The constant parameters k o, k 1 and k 2 are defined as k o α 1, k 1 α 2, k 2 α 2, c p ρa 1 c p ρa 1 c p ρa 2 where α i i = 1, 2) are the thermal conductivities of the tube walls and A i i = 1, 2) are the effective cross-sections of the tubes. The system 1.1) is hyperbolic with the characteristic velocities λ 1 = V 1 + c, λ 2 = V 1, λ 3 = V 1 c, λ 4 = V 2 + c, λ 5 = V 2 λ 6 = V 2 c, and the corresponding Riemann coordinates R 1 = V 1 + g c H 1, R 2 = g c H 1 c ln T 1, R 3 = V 1 g c H 1, R 4 = V 2 + g c H 2, R 5 = g c H 2 c ln T 2, R 6 = V 2 g c H Plug flow chemical reactors A plug flow chemical reactor PFR) is a tubular reactor where a liquid reaction mixture circulates. The reaction proceeds as the reactants travel through the reactor. Here, we consider the case of an horizontal PFR where a simple mono-molecular reaction takes place: A B. A is the reactant species and B is the desired product. The reaction is supposed to be exothermic and a jacket is used to cool the reactor. The cooling fluid flows around the wall of the tubular reactor. Therefore, the dynamics of the system are naturally described by the model 1.33) of the flow in a heat exchanger supplemented with the mass balance equations for the concerned chemical species. However it is usual to assume, for simplicity, that the dynamics of velocity and pressure in the reactor and the jacket are negligible. The dynamics of the PFR are then described by the following semi-linear system of balance

32 32 Chapter 1. Hyperbolic Systems of Balance Laws laws: t T c V c x T c k o T c T r ) =, 1.37) t T r + V r x T r + k o T c T r ) k 1 rt r, C A, C B ) =, t C A + V r x C A + rt r, C A, C B ) =, t C B + V r x C B rt r, C A, C B ) =, where V c t) is the coolant velocity in the jacket, V r t) is the reactive fluid velocity in the reactor, T c t, x) is the coolant temperature, T r t, x) is the reactor temperature. The variables C A t, x) and C B t, x) denote the concentrations of the chemicals in the reaction medium. The function rt r, C A, C B ) represents the reaction rate. A typical form of this function is: rt r, C A, C B ) = ac A bc B ) exp E ), RT r where a and b are rate constants, E is the activation energy and R is the Boltzmann constant. This model is in the general quasi-linear form Y t + F Y)Y x + GY) = with Y T c T r C A C B, F Y) V c V r V r, V r k o T c T r ) k GY) o T c T r ) k 1 rt r, C A, C B ) rt r, C A, C B ). rt r, C A, C B ) It is a hyperbolic system of four balance laws with characteristic velocities V c and V r. This system is not strictly hyperbolic because it has three identical characteristic velocities. It is nevertheless endowed with Riemann coordinates because F Y) is diagonal Euler equations for gas pipes The motion of an inviscid ideal gas in a rigid cylindrical pipe with a unit cross-section is most often described by the classical Euler 1755) equations which have the form of a system of three balance laws: 1.38a) 1.38b) 1.38c) t ϱ + x ϱv ) =, t ϱv ) + x ϱv 2 + P ) + CϱV V =, [ t ϱ V P ] [ + x V ϱ V 2 γ γp )] γ 1 =,

33 1.8. Euler equations for gas pipes 33 where ϱt, x) is the gas density, V t, x) is the gas velocity, P t, x) is the gas pressure, C is a constant friction coefficient and γ > 1 is the constant heat capacity ratio. The first equation is a mass balance, the second equation is a momentum balance and the third equation is an energy balance, with the total energy defined as E ϱ V P γ 1. Using ϱ, V and P as state variables, the model 1.38) is equivalent to t ϱ + x ϱv ) =, 1.39) t V + x V ϱ xp + CV V =, t P + V x P + γp x V γ 1)Cϱ V 3 =. This model is in the general quasi-linear form Y t + F Y)Y x + GY) = with ϱ Y V, P V ϱ F Y) V 1/ϱ, γp V The characteristic polynomial of the matrix F Y) is GY) V λ)v λ) 2 c 2 ) with c γp ϱ. CV V γ 1)Cϱ V 3 The quantity c is the sound velocity in the concerned medium. In subsonic conditions i.e. V < c) the system is hyperbolic with characteristic velocities which are the roots of the characteristic polynomial): λ 1 = V + γp ϱ, λ γp 2 = V, λ 3 = V ϱ. The Euler equations 1.38) are a typical example of a hyperbolic system of size > 2 which cannot be transformed into an equivalent system expressed in Riemann coordinates.. Isentropic equations For the modeling and the analysis of gas pipeline networks, a common model is made of the isentropic equations which correspond to the special case where the dynamics of the energy balance are neglected. The model reduces to the first two equations of 1.39): 1.4) t ϱ + x ϱv ) =, t V + x V ϱ xp ϱ) + CV V =,

34 34 Chapter 1. Hyperbolic Systems of Balance Laws where ϱ, V, C are defined as above. The gas pressure P ϱ) is a monotonically increasing function of the gas density P ϱ) > ). This model is equivalent to the general quasi-linear form Y t + F Y)Y x + GY) = with ) ) ) ϱ V ϱ Y V, F Y) c 2 ϱ)/ϱ V, GY) CV V, where cϱ) P ϱ) is the sound velocity. Under subsonic conditions i.e. V 2 < c 2 ϱ)), the system is hyperbolic with characteristic velocities λ 1 Y) = V + cϱ) > > λ 2 Y) = V cϱ). Riemann coordinates may then be defined as R 1 = V + φϱ), R 2 = V φϱ), and inverted as ϱ = φ 1 R 1 R 2 )/2, V = R 1 + R 2 )/2, where φϱ) is a primitive of cϱ)/ϱ, i.e. a function such that φ ϱ) cϱ) ϱ. Steady state and linearization A steady state ϱ x), V x) is a solution of the differential equations x ϱ V ) =, x V ϱ xp ϱ ) + CV V =. In order to linearize the model, we define the deviations of the states ϱt, x) and V t, x) with respect to the steady states ϱ x) and V x): ρt, x) ϱt, x) ϱ x), vt, x) V t, x) V x). Then the linearized isentropic equations around the steady state are t ρ + V x ρ + ϱ x v + x V )ρ + x ϱ )v = 1.41) t v + φ ϱ )cϱ ) x ρ + V x v, + [2φ ϱ )c ϱ ) φ ϱ )) 2] x ϱ ) ρ + x V + 2C V )v =. The Riemann coordinates for the linearized system 1.41) are defined as follows: R 1 t, x) = vt, x) + ρt, x)φ ϱ x)), R 2 t, x) = vt, x) ρt, x)φ ϱ x)),

35 1.8. Euler equations for gas pipes 35 with the inverse coordinate transformation ρt, x) = R 1t, x) R 2 t, x) 2φ ϱ, x)) vt, x) = R 1t, x) + R 2 t, x). 2 With these definitions and notations, the linearized isentropic Euler equations are written in characteristic form: t R 1 t, x) + λ 1 x) x R 1 t, x) + γ 1 x)r 1 t, x) + δ 1 x)r 2 t, x)] = t R 2 t, x) λ 2 x) x R 2 t, x) + γ 2 x)r 1 t, x) + δ 2 x)r 2 t, x)] = with the characteristic velocities λ 1 x) = V x) + cϱ x)), λ 2 x) = V x) cϱ x)) and the coefficients γ 1 x) c ϱ x)) x ϱ x) + x V x) + C V x), ) δ 1 x) c ϱ x)) + φ ϱ x)) x ϱ x) + C V x), γ 2 x) ) + c ϱ x)) φ ϱ x)) x ϱ x) + C V x), δ 2 x) c ϱ x)) x ϱ x) + x V x) + C V x). Musical wind instruments Musical wind instruments flute, trumpet, organ, etc.) are a nice example of devices that use air motion in pipes to produce a pleasant entertainment. Here, as a matter of illustration, we consider the case of a simple slide flute as described by d Andréa-Novel et al. 21). A slide flute Fig.1.11) is a recorder without finger holes which is ended by a piston mechanism D) to modify the length of the resonator chamber and consequently the sound pitch. The air motion dynamics in the tube are assumed to be described by the isentropic equations 1.4). D Fig.1.11: Cross section of a slide flute adapted from en.wikipedia.org /Recorder). L x

36 36 Chapter 1. Hyperbolic Systems of Balance Laws The player s breath is transformed into a linear airstream in the channel B) called the windway, through the mouthpiece A) of the instrument. Exiting from the windway, the airflow is directed against a sharp edge C), called the labium. This structure is a nonlinear oscillator which, amplified by the acoustic resonator, produces a stationary wave at the desired frequency determined by the pipe length L. The pitch of a note may therefore be adjusted by moving the piston at the back of the instrument. The boundary condition at x = is given by a second order non-linear differential relation 1.42) d 2 V t, ) dv t, ) dt 2 + c 1 + c 2 V t, ) V t, ) + c 3 P ϱt, )) + c 4 P f =, dt where c i i = 1,..., 4) are constant parameters. This equation is motivated in d Andréa- Novel et al., 21, Section 5.3.3) and describes the resonance phenomenon that occurs in the labium under the external pressure P f imposed by the player s breath. At x = L, a moving boundary condition is determined by the piston motion represented by the expression 1.43) m d2 L dt 2 = F s pp ϱt, L)), where F is the force exerted on the piston, m is the piston mass and s p is the pipe section. Then the isentropic equations 1.4) coupled to the boundary conditions 1.42) and 1.43) constitute a boundary control system with P f and F as command signals Fluid flow in elastic tubes The laminar flow of an incompressible fluid in an elastic tube is of special interest because of its relevance to the dynamics of blood flow in arteries. Most often, one-dimensional hyperbolic balance law models are adopted under the assumption of cylindrical tubes with axisymmetric flow and negligible radial fluid velocity. The general form of such models e.g. Barnard et al. 1966) and Li and Canic 29) ) is as follows: 1.44) t A + x AV ) =, t AV ) + x αav 2 ) + A δ xp A) + CV =, where At, x) is the cross-sectional area, V t, x) is the average fluid velocity, P A) is the pressure, C is a constant friction coefficient, δ is the fluid density and α > 1 is a constant depending on the shape of the axial velocity profile. The first equation is a mass balance and the second equation is a momentum balance. The pressure function P A) is a monotonically increasing function of the area P A) > ) such that P A o ) = with A o denoting the cross-sectional area at rest. Using A and AV as state variables, the model 1.44) is equivalent to the general quasilinear form Y t + F Y)Y x + GY) = with ) ) ) A 1 Y, F Y), GY), AV c 2 A) αv 2 2αV CV

37 1.1. Aw-Rascle equations for road traffic 37 where the function ca) AP A)δ 1 has the dimension of a velocity. The system is hyperbolic with characteristic velocities which are the eigenvalues of F Y)) λ 1 Y) = αv + α 2 α)v 2 + c 2 A), λ 2 Y) = αv α 2 α)v 2 + c 2 A) that have opposite signs for numerical values corresponding to human arteries. For α 1) sufficiently small, the Riemann coordinates may be approximated as follows: R 1,2 V ± [ A A o ) ] ca) V da + α 1)V a 2 ca) Aw-Rascle equations for road traffic In the fluid paradigm for road traffic modeling, the traffic is described in terms of two basic macroscopic state variables: the density ϱt, x) and the speed V t, x) of the vehicles at position x along the road at time t. The following dynamical model for road traffic was proposed by Aw and Rascle 2): 1.45) t ϱ + x ϱv ) =, t + V x )V + t + V x )P ϱ) + σv V o ϱ)) =. In this model the first equation is a continuity equation representing the conservation of the number of vehicles on the road. The second equation is a phenomenological model describing the speed variations induced by the drivers behaviour. The function V o ϱ) represents the monotonically decreasing relation between the average speed of the vehicles and the density: the larger the density the smaller the average speed. A typical experimental example of this function is shown in Fig The parameter σ is a relaxation constant. The function P ϱ) is a monotonically increasing function of the density P ϱ) > ), called traffic pressure, which is selected such that the term t + V p x )V + t + V x )P ϱ) represents the dynamics of the transient speed variations around the average V o ϱ) when the density is changing. The use of the Lagrangian derivative t + V x allows to account for the density variations that are really perceived by the drivers in front of them. Now, multiplying the first equation of 1.45) by V + P ϱ) and the second equation by ϱ, and adding the two, we obtain a system of two nonlinear balance laws of the form 1.46) t ϱ + x ϱv ) =, t ϱv + ϱp ϱ) ) + x ϱv 2 + ρv P ϱ) ) + σϱv V o ϱ)) =. The model can also be written in the general quasi-linear form Y t + F Y)Y x + GY) = with ) ) ) ϱ V ϱ Y, F Y), GY). V V ϱp ϱ) σv V o ϱ))

38 38 Chapter 1. Hyperbolic Systems of Balance Laws average speed km/h) V o ) 1 2 density #veh/km) Fig.1.12: Relation between average speed and density on a three lane highway. The experimental data have been recorded on the E411 highway Belgium) at ten measurement stations between Namur and Brussels from October 1 to October 31, 23 Data provided by the Service Public de Wallonie). Therefore, for a positive density ϱ >, the system is hyperbolic with characteristic velocities λ 1 Y) = V > λ 2 Y) = V ϱp ϱ), which are the eigenvalues of F Y). Moreover Riemann coordinates may be defined as with the inverse coordinate transformation: R 1 = V + P ϱ), R 2 = V, V = R 2, ϱ = P 1 R 1 R 2 ). Q in L x Q out Fig.1.13: Ramp metering on a stretch of a motorway.

39 1.11. Kac-Goldstein equations for chemotaxis 39 Ramp metering Ramp metering is a strategy that uses traffic lights to regulate the flow of traffic entering freeways according to measured traffic conditions as illustrated in Fig For the stretch of motorway represented in this figure, the boundary conditions are ϱt, )V t, ) = Q in t) + Ut), ϱt, L)V t, L) = Q out t), where Ut) is the inflow rate controlled by the traffic lights. The Aw-Rascle equations 1.46) coupled to these boundary conditions form a boundary control system with Ut) as the command signal. In a feedback implementation of the ramp metering strategy, Ut) may be a function of the measured disturbances Q int t) or Q out t) that are imposed by the traffic conditions Kac-Goldstein equations for chemotaxis Chemotaxis refers to the motion of certain living micro-organisms bacteria, slime molds, leukocytes...) in response to the concentrations of chemicals. A simple model for onedimensional chemotaxis, known as the Kac-Goldstein model, has been proposed by Goldstein 1951) in order to explain the spatial pattern formations in chemosensitive populations. Revisited by Kac 1956), this model, in its simplest form, is a semi-linear hyperbolic system of two balance laws of the form: 1.47) t ϱ + + γ x ϱ + + µϱ +, ϱ )ϱ ϱ + ) =, t ϱ γ x ϱ + µϱ +, ϱ )ϱ + ϱ ) =, where ϱ + denotes the density of right-moving cells and ϱ the density of left-moving cells. The function µϱ +, ϱ ) is called the turning function. The constant parameter γ is the velocity of the cell motion. With the change of coordinates ϱ ϱ + + ϱ, q γϱ + ϱ ), we have the following alternative equivalent model: 1.48a) 1.48b) t ϱ + x q =, t q + γ 2 x ϱ 2µ ϱ 2 + q 2γ, ϱ 2 q ) q =, 2γ where ϱ is the total density and q is a flux proportional to the difference of densities of right and left moving cells. Remark that we have q = ϱv where V γ ϱ+ ϱ ϱ + + ϱ can be interpreted as the average group velocity of the moving cells. Various possible turning functions are reviewed in Lutscher and Stevens 22). A typical example is µϱ +, ϱ ) = αϱ + ϱ µ o, where α and µ o are positive constants. It is an evidence that the system 1.47) is directly written in characteristic form 1.3) with the coordinates R 1 ϱ + and R 2 ϱ as Riemann coordinates. The system is hyperbolic with characteristic velocities λ 1 = γ and λ 2 = γ.

40 4 Chapter 1. Hyperbolic Systems of Balance Laws A special case of interest see e.g. Lutscher 22) ) is when the cells are confined in the domain [, L]. This situation may be represented by no-flow boundary conditions of the form: 1.49) qt, ) = γ ϱ + t, ) ϱ t, ) ) =, qt, L) = γ ϱ + t, L) ϱ t, L) ) =. These boundary conditions can be written in the nominal form ϱ + ) ) t, ) 1 ϱ + ) t, L) 1.5) ϱ = t, L) 1 ϱ, t, ) which is required for the well-posedness of the associated Cauchy problem see page 16). We remark that, under these boundary conditions, the total amount of cells in the domain [, L] is constant since, from 1.48a) and 1.49), we have 1.51) t ϱt, x)dx = qt, ) qt, L) =. Let us now consider the chemotaxis system 1.47) and 1.5) under an initial condition 1.52) ϱ +, x) ϱ + o x), ϱ, x) ϱ o x). Then, using 1.51), it can be verified that the Cauchy problem 1.47), 1.5) and 1.52) has a single equilibrium state ϱ + = ϱ 1 2 ϱ + o x) + ϱ o x) ) dx. In this example, we observe that the steady state depends on the initial condition. This is in contrast with most other examples of this book where the values of the steady states are generally independent of the initial conditions Age-dependent SIR epidemiologic equations In the field of epidemiology, mathematical models are currently used to explain epidemic phenomena and to assess vaccination strategies. For infectious diseases where individuals are infected by pathogen micro-organisms like viruses or bacteria), a first fundamental model was formulated by Kermack and McKendrick 1927). In this model, the population is classified into three groups : i) the group of individuals who are uninfected and susceptible S) of catching the disease, ii) the group of individuals who are infected I) by the concerned pathogen, iii) the group of recovered R) individuals who have acquired a permanent immunity to the disease. The propagation of the disease is represented by a compartmental diagram shown in Fig The model is derived under three main assumptions : i) a closed population without immigration or emigration, ii) spatial homogeneity, iii) disease transmission by contact

41 1.12. Age-dependent SIR epidemiologic equations 41 S I R Fig.1.14: Compartmental diagram of disease propagation between susceptible and infected individuals. In the case where the age of patients is an important factor to be taken into account, St, a) represents the age distribution of the population of susceptible individuals at time t. This means that 1.53) a2 a 1 St, a)da is the number of susceptible individuals with ages between a 1 and a 2. Similar definitions are introduced for the age distributions It, a) of infected individuals and Rt, a) of recovered individuals. The dynamics of the disease propagation in the population are then described by the following set of hyperbolic partial integro-differential equations, e.g. Hethcote, 2, Section 5), Thieme, 23, Chapter 22), Perthame, 27, Chapter 1): 1.54) t St, a) + a St, a) + µa)st, a) + βa)st, a) It, b)db =, t It, a) + a It, a) + γa)it, a) + µa)it, a) βa)st, a) t Rt, a) + a Rt, a) γa)it, a) + µa)it, a) =, It, b)db =, under the boundary conditions St, ) = Bt), It, ) =, Rt, ) =. In these equations, µa) > denotes the natural age-dependent per capita death rate in the population. In the first and second equations, the term βa)st, a) It, b)db represents the disease transmission rate by contact between susceptible and infected individuals which is assumed to be proportional to the sizes of both groups with βa) > the age-dependent transmission coefficient between all infected individuals and susceptibles having age a. In the second and third equations, the parameter γa) > is the age-dependent rate at which infected individuals recover from the disease. Obviously, L denotes here the maximal life duration in the considered population. In the boundary conditions, Bt) > is the inflow of newborn individuals in the susceptible part of the population at time t. The system 1.54) is a semi-linear system of balance laws in Riemann coordinates R t + R x + CR) = with R T S, I, R) T. In contrast with all other models presented in this chapter, CR) is here a so-called non-local source term because it depends on the

42 42 Chapter 1. Hyperbolic Systems of Balance Laws spatial integral of the state variable It, x) over the interval [, L]. Moreover, it can be also observed that, in accordance with the physical reality, the system 1.54) is positive, which means that, if the initial state is non negative, i.e. S, a), I, a), R, a), then the solution is non negative, i.e. St, a), It, a), Rt, a) for all t [, + ). Steady state For a constant birth rate Bt) = B for all t, a steady state S a), I a), R a) is a solution of the system of integro-differential equations 1.55a) 1.55b) 1.55c) a S a) + µa)s a) + βa)s a) I b)db =, a I a) + γa)i a) + µa)i a) βa)s a) a R a) γa)i a) + µa)r a) =, I b)db =, with the initial conditions S ) = B, I ) =, R ) =. There is one trivial disease free steady state: S a) = B exp a ) µb)db, I a) =, R a) =. In order to determine a non trivial endemic steady state, let us integrate equations 1.55a) and 1.55b): a ) S a) = B exp µb) + βb)ψ )db with ψa) exp I a) = ψa)ψ a a βb)s b) db, ψb) ) µc) + γc))dc, ψ Substituting for the expression of I into ψ, we get ψ = ψa)ψ a βb)s ) b) db da. ψb) I a)da. Clearly, an endemic steady state may exists only if ψ > which implies that 1.56) 1 = a ψa) βb)s ) b) db da. ψb) Let us now substitute for the expression of S into 1.56): L a ) ) βb) b 1.57) 1 = ψa) ψb) B exp µc) + βc)ψ )dc db da Rψ ).

43 1.13. Chromatography 43 Clearly Rψ ) is a positive exponentially decreasing function of ψ such that R+ ) =. It follows that the equality 1.57) may be satisfied if and only if R) 1 and therefore that there exists an endemic steady state if and only if R) 1. Furthermore, it can be shown that this endemic equilibrium, when it exists, is uniquely determined see e.g. Thieme, 23, Theorem 22.1) ) Chromatography In chromatography, a fluid carrying dissolved chemical species flows through a porous solid fixed bed. The carried species are partially adsorbed on the bed. The fluid flow is supposed to have a constant velocity V. Let us first consider the case of a single carried species with concentration C f in the fluid and concentration C s deposited on the solid. Then we have the conservation equation 1.58) t C f + C s ) + V x C f =. A standard model for the net exchange rate between the fluid and the solid is k 1 C f 1 C s C max ) k 2 C s where k 1, k 2 are positive kinetic constants. The first term represents the deposition from the fluid to the solid at a rate proportional to the amount in the fluid, but limited by the amount already on the solid up to capacity C max. The second term is the reverse transfer from the solid to the fluid. Here we assume quasi steady state conditions such that the net exchange rate vanishes and, consequently, such that C s is a function of C f : C s = LC f ) = hc f 1 + bc f where h k 1 /k 2 is the so-called Henry coefficient and b k 1 /k 2 C max ) is the adsorption equilibrium coefficient. This function was proposed by Langmuir 1916) and is known under the name of Langmuir isotherm. Using this expression in 1.58), we obtain the following scalar hyperbolic conservation law: 1.59) t C f + The characteristic velocity λ such that V 1 + L C f ) xc f =, with L h C f ) = ) bcf < λ = V 1 + L C f ) < V is the propagation speed of the carried species. It is slower for species with larger affinity for the solid material which is measured by L C f ). Let us now suppose that a pulse of a mixture of species with different affinities is injected in the carrying fluid at the entrance of the process as illustrated in Fig Clearly,

44 44 Chapter 1. Hyperbolic Systems of Balance Laws injection separation elution of blue solute elution of red solute chromatogram Fig.1.15: Principle of chromatography the various substances will travel at different propagation speeds and will ultimately be separated in different bands. Obviously, the scalar conservation law 1.59) for a single component applies only after separation. Before the separation, the dynamics of the mixture are described by a system of coupled conservation laws that generalizes the scalar case in the following way: 1.6) t C i + h i C i 1 + n j b jc j ) + V x C i = i = 1,..., n. where C i i = 1,..., n) denote the densities of the n carried species. SMB chromatography Simulated moving bed SMB) chromatography is a technology where several interconnected chromatographic columns are switched periodically against the fluid flow. This allows for a continuous separation with a better performance than the discontinuous singlecolumn chromatography. A standard SMB chromatography process Suvarov et al. 212)) is represented in Fig The input flows feed mixture and solvent) and output flows extract and raffinate) divide the system in four zones each containing one chromatographic column. The four mobile) columns are labeled 1, 2, 3, 4. The four operating zones are

45 1.13. Chromatography 45 Fig.1.16: SMB chromatography. adapted from de/research/projects/1119/1127/dsmbc) labeled I, II, III, IV. Pumps connected at each port determine the liquid phase flow rates in all the zones. The feed mixture composed of species A and B is injected between zone I and IV. The adsorbent is chosen in such a way that the two components are adsorbed at different rates, allowing them to travel with different velocities. The less adsorbed component A) is collected at the raffinate port and the more adsorbed one B) at the extract port. The separation of the two components is performed in zone I and IV, whereas zones II and III are dedicated to adsorbent regeneration and solvent recycling. A liquid-solid countercurrent movement is obtained by a periodic circular switching of the columns containing in the opposite direction to the liquid phase flow as shown in Fig We introduce the following notations: The switching time period is T and the column length is L; C l i t, x), x L, t, is the density of species l {A, B} in the column i {1, 2, 3, 4}; V I is the fluid velocity in the columns located in zones I and III; V II is the fluid velocity in the columns located in zones II and IV with V I > V II > ; V F is the fluid velocity and C l F is the density of species l {A, B} in the feeding stream; h A and h B are the Henry coefficients, b A and b B are the adsorption equilibrium coefficients. We consider the operation of the SMB process in the standard conditions where V I, V II, V F and C l F are all constant. We first state the dynamical model during the first time

46 46 Chapter 1. Hyperbolic Systems of Balance Laws period, assuming that column 1 is in zone I, column 2 in zone II, column 3 in zone III and column 4 in zone IV. The process dynamics are represented by the following set of conservation laws of the form 1.6): For t < T, t [C 1 A h A C A ] V I x C1 A =, with the boundary conditions: 1 + b A C A 1 + b BC B 1 t [C 2 A h A C2 A b A C2 A + b BC2 B t [C 3 A h A C3 A b A C3 A + b BC3 B t [C 4 A h A C4 A b A C4 A + b BC4 B t [C 1 B h B C1 B b A C1 A + b BC1 B t [C 2 B h B C2 B b A C2 A + b BC2 B t [C 3 B h B C3 B b A C3 A + b BC3 B t [C 4 B h B C4 B b A C4 A + b BC4 B ] + V II x C A 2 =, ] + V I x C A 3 =, ] + V II x C A 4 =, ] + V I x C B 1 =, ] + V II x C B 2 =, ] + V I x C B 3 =, ] + V II x C B 4 =. V I C A 1 t, ) = V II C A 4 t, L) + V F C A F, C A 2 t, ) = C A 1 t, L), V I C A 3 t, ) = V II C A 2 t, L), C A 4 t, ) = C A 3 t, L). V I C B 1 t, ) = V II C B 4 t, L) + V F C B F, C B 2 t, ) = C B 1 t, L), V I C B 3 t, ) = V II C B 2 t, L), C B 4 t, ) = C B 3 t, L). We introduce the following vector and matrix notations: C l t, x) C l 1t, x), C l 2t, x), C l 3t, x), C l 4t, x)) T, l { A, B}, U l V F C l F,,, ) T,

47 1.13. Chromatography 47 ) C A t, x) Ct, x) = C B t, x), DC) diag{1 + b A Ci A + b B Ci B ) 1 ; i = 1, 2, 3, 4}, Υ diag{v I, V II, V I, V II }, V II /V I 1 K V II /V I. 1 With these notations, the model equations are written in compact form as follows: t < T, l {A, B}, t I + h l DC))C l) + Υ x C l =, C l t, ) = KC l t, L) + U l. We now consider the second time period when the columns have been shifted by one position such that column 1 is now located in zone IV, column 2 in zone I, etc... To take the shifting process into account in a systematic way, we introduce the following permutation matrix: 1 1 P = 1. 1 Then, noticing also that P 1 = P T, it can be checked that the model equations during the second period become T t < 2T, l {A, B}, t I + h l DC))C l) + P ΥP T x C l =, C l t, ) = P KP T C l t, L) + P U l. It is then clear that, by iteration, we have the following general form for the hyperbolic system of conservation laws describing the periodic SMB chromatography process: mt t < m + 1)T, m =, 1, 2, 3, 4, 5,...,, l {A, B}, 1.61) t I + h l DC))C l) + P m )ΥP m ) T x C l =, C l t, ) = P m )KP m ) T C l t, L) + P m )U l. This system 1.61) has a periodic solution C C A, C B ) such that C t, x) = C t + 4T, x)), x [, L], t.

48 48 Chapter 1. Hyperbolic Systems of Balance Laws Scalar conservation laws In several examples of hyperbolic systems presented above, the first equation represents a mass balance with the form 1.62) t ϱ + x ϱv ) =. The variable ϱ is a density and the variable V is a velocity such that q = ϱv is a flux density. For instance, it is the case for the Saint Venant equations 1.23) with ϱ = H), for the Euler equations 1.38), for the elastic tube model 1.44) with ϱ = A), for the Aw- Rascle traffic model 1.45) or for the chemotaxis model 1.48). Equation 1.62) is the basis for many physical models of interest in engineering where momentum and energy balances are assumed to be quasi balanced. The assumption that the mass balance is the dominant effect then relies on the quasi steady state simplification of considering the velocity V as a static function of the density ϱ: 1.63) V = V o ϱ) = t ϱ + x qϱ) = t ϱ + x ϱv o ϱ)) =. Obviously, this is just a scalar conservation law since it involves only one dependent variable ϱ. This scalar conservation law 1.63) can also be written in a quasi-linear form: t ϱ + λϱ) x ϱ =, λϱ) = q ϱ) = V o ϱ) + ϱv o ϱ), where λϱ) is the characteristic velocity of the system. As a first example, in road traffic modelling, we all have experienced that the larger is the density and the smaller is the speed. The simplest model is to assume that the drivers instantaneously adapt their speed to the local traffic density such that the relation V o ϱ) between the vehicle speed and the traffic density is linearly decreasing see e.g. Garavello and Piccoli, 26, Chapter 3) ): 1.64) V o ϱ) = V max 1 ϱ ). ϱ max Here V o ) = V max is the maximal velocity of the vehicles when the road is almost) empty while V o ϱ max ) = means that the velocity drops to zero when the density is maximal and the traffic is totally congested. This model was first proposed by Greenshields 1935) and gives rise to the well known scalar LWR model Lighthill and Whitham 1955) ) for traffic flow: )) t ϱ + x V max ϱ ϱ2 =. ϱ max A direct improvement of this model is to adopt the experimental nonlinear function of Fig.1.12 for V o ϱ) instead of the affine function 1.64). Non-monotonic variants of the function V o ϱ) are also suggested for pedestrian flow modelling e.g. Chalons et al. 213) ). Another classical example is the scalar model for open channels. Let us consider again the Saint-Venant equations 1.23). Let us assume that the momentum dynamics are negligible so that the momentum equation reduces to the static source term which links the water velocity to the slope S b and the viscous friction C: g [C V 2 ] H S b =.

49 1.15. Physical networks of hyperbolic systems 49 From this equation, we recover the Torricelli s formula between the velocity V and the water depth H: 1.65) V = V o H) = k Sb gh with k = C. By substituting this expression into 1.63), we get t H + x kh gh) =. which is the basic scalar equation for open channels. A further example of the model 1.63) is the Buckley and Leverett 1942) equation: ) ϱ ϱ 2 ) 1.66) V = V o ϱ) = ϱ ϱ) 2 = t ϱ + x ϱ ϱ) 2 = for modelling a two phase e.g. oil and water) fluid flow in a porous medium. Here ϱ 1 represents the saturation of water: ϱ = is a flow of pure oil while ϱ = 1 is a flow of pure water. This kind of model has applications in oil reservoir studies. Let us now observe that, in all these examples, the function V o is monotone: decreasing for road models 1.64) and increasing for open channels 1.65) and two-phase fluid flow 1.66). Hence the function V o ϱ) may be supposed to be invertible as ϱ = ϱ o V ) in such a way that the system may also be written in the inverse form 1.67) t V + cv ) x V =, with cv ) = V o ϱ o V ))λϱ o V ))ϱ ov ). We see that, depending on the application, ϱ or V can be equally taken as the state variable. In both cases the system is a scalar conservation law with either λϱ) or cv ) as the characteristic velocity. The simplest physical model which is usually given in the form 1.67) is the well known equation of Burgers 1939) with cv ) = V such that 1.68) t V + V x V =. This equation may be considered as a simplified model of isentropic gas motion because it can be derived from the momentum balance Euler equation 1.38b) by assuming that the friction is negligible C ) and that the pressure is almost constant with respect to x x P ϱ) ) Physical networks of hyperbolic systems The operation of many physical networks having an engineering relevance can be represented by hyperbolic systems of balance laws in one space dimension. Typical examples

50 5 Chapter 1. Hyperbolic Systems of Balance Laws are hydraulic networks for water supply, irrigation or navigation), road traffic networks, electrical line networks, gas transportation networks, networks of heat exchangers, communication networks, blood flow networks etc. Such physical networks can generally be schematized by using a graph representation. The edges of the network represent the physical links for instance the pipes, the canals, the roads, the electrical lines, etc...) that are governed by hyperbolic systems of balance laws. Without loss of generality and for simplicity, it can always be assumed that, by an appropriate scaling, all the links have exactly the same length L. Typically, the links carry some kind of flow and the network has only a few nodes where flows enter or leave the network. The other nodes of the network represent the physical junctions between the links. The mechanisms that occur at the junctions are described by junction models under the form of algebraic or differential relations that determine the boundary conditions of the PDEs. Networks of electrical lines Let us consider an electrical line network in which several lines meet at a given junction as shown in Fig In this Figure, the directions of the lines represented by arrows are arbitrary. For convenience, each arrow denotes both the direction of the increasing space coordinate x and the direction of a positive current It, x) on the concerned line. According to the Kirchhoff s law, we know that the following conditions hold at the junction: I j t, L) = j k I k t, ) and V j t, L) = V k t, ) j, k) where j and k index the incoming and outgoing lines respectively. In words, these conditions mean that the voltage is identical on all lines at the junction and that the currents balance. These expressions are the boundary conditions for the hyperbolic system of balance laws that describes the network. We observe that the number of independent boundary conditions at each junction is equal to the number of incident lines at the junction. Then Fig.1.17: An electrical line junction.

51 1.15. Physical networks of hyperbolic systems 51 Power supply Line 1 x Transformer Line 2 x Load Fig.1.18: Two electrical lines interconnected with a transformer it is clear that, if the whole network has the structure of a simply connected graph, the hyperbolic system is well-posed. Now, it is evident that we may also have more complex devices at the junctions, such as transformers or amplifiers. In such cases, we have to use the appropriate equations of lumped electrical circuits as boundary conditions. For example, let us consider two electrical lines connected by a transformer as shown in Fig This system is described by two electrical line equations: t L 1 I 1 ) + x V 1 + R 1 I 1 =, t L 2 I 2 ) + x V 2 + R 2 I 2 =, t C 1 V 1 ) + x I 1 + G 1 V 1 =, t C 2 V 2 ) + x I 2 + G 2 V 2 =. coupled by the boundary conditions V 1 t, ) + R g I 1 t, ) = Ut), V 2 t, L) R l I 2 t, L) =, V 1 t, L) = L 1s di 1 t, L)/dt) + MdI 2 t, )/dt), V 2 t, ) = L 2s di 2 t, )/dt) + MdI 1 t, L)/dt), where L 1s and L 2s denote the self-inductance of the transformer coils and M is the mutual inductance see Section 1.2 for the meaning of the other notations). Remark that this is a typical example of a situation where some of the boundary conditions have the form of ordinary differential equations. Chains of density-velocity systems We consider a chain of two-by-two hyperbolic density-velocity systems of the form 1.69) t ϱ j + x ϱv j ) =, t V j + x fϱ j, V j ) + gϱ j, V j ) =, j = 1,..., n. The system is subject to n 1 physical boundary conditions 1.7) ϱ j t, L)V j t, L) = ϱ j+1 t, )V j+1 t, ) j = 1,..., n 1 which are the expression of the natural constraint of flow conservation between two successive elements of the chain. The form of the remaining n+1 boundary conditions then depends on the particular application. We illustrate the issue with two typical examples: gas pipe lines and navigable rivers.

52 52 Chapter 1. Hyperbolic Systems of Balance Laws pipe j 1 compressor compressor pipe j pipe j +1 L x Fig.1.19: The structure of a gas pipe line. Gas pipe lines In gas pipes, the pressure decreases along the pipes due to friction. Compressors are therefore used from place to place in order to amplify the pressure. A gas pipe line is thus a chain of n pipes separated by compressor stations Gugat and Herty 211), Dick et al. 21) ) as illustrated in Fig The gas flow in the pipes is described by isentropic Euler equations: 1.71) t ϱ j + x ϱ j V j ) =, Vj 2 t V j + x x P ϱ j ) + CV j V j =, ϱ j j = 1,..., n. In addition to the flow conservation conditions 1.7), the compressors impose the following n 1 boundary conditions: ) m P ϱj+1 t, )) kϱ j+1 t, )V j+1 t, ) 1) = U j t), j = 1,..., n 1. P ϱ j t, L)) where k and m are positive constant parameters. This is a common static model of the pressure amplification which is achieved with a centrifugal adiabatic compressor under a power supply U j t) which can be considered as a control input. Two additional boundary conditions are needed to close the system. For instance: P ϱ 1 t, )) = P o t) and ϱ n t, L)V n t, L) = Q L t), where P o t) is the input pressure imposed by the producer and Q L t) is an outflow rate constraint at the consumer side. Navigable rivers and irrigation channels In navigable rivers or irrigation canals see e.g. Litrico et al. 25), Cantoni et al. 27), Bastin et al. 29) ) the water is transported along the channel under the power of gravity through successive pools separated by automated gates that are used to regulate the water flow, as illustrated in Figures 1.2 and Here, we consider a channel with n pools the dynamics of which are described by Saint-Venant equations 1.23) ) ) ) + x 1.72) t Hi V i H i V i 1 2 V i 2 + gh i + g[c i V 2 i H 1 i S i ] =, i = 1..., n.

53 1.15. Physical networks of hyperbolic systems 53 Pool i Hi Vi x Pool i + 1 ui Hi+1 x Vi+1 ui+1 Fig.1.2: Lateral view of successive pools of a navigable river with overflow gates. Fig.1.21: Automated control gates in the Sambre river Belgium). The left gate is in operation. The right gate is lifted for maintenance. c L.Moens)

54 54 Chapter 1. Hyperbolic Systems of Balance Laws In this model, for simplicity, we assume that all the pools have a rectangular section with the same width W. The system 1.72) is subject to a set of 2n boundary conditions that are distributed into three subsets: 1) A first subset of n 1 conditions expresses the flow-rate conservation at the junction of two successive pools the flow that exits pool i is equal to the flow that enters pool i + 1): 1.73) H i t, L)V i t, L) = H i+1 t, )V i+1 t, ), i = 1,..., n 1. 2) A second subset of n boundary conditions is made up of the equations that describe the gate operations. A standard gate model is given by the algebraic relation 1.74) H i t, L)V i t, L) = k G 2g ) [ Hi t, L) u i t) ] 3, i = 1,..., n. where k G is a positive constant coefficient and u i t) denotes the weir elevation which is a control input see Fig.1.2). 3) The last boundary condition imposes the value of the canal inflow rate Q t): 1.75) Q t) = W H 1 t, )V 1 t, ). Depending on the application, Q t) may be viewed as a control input in irrigation channels) or as a disturbance input in navigable rivers). A steady state or equilibrium) is a constant state Hi, Vi i = 1,..., n) that satisfies the relations S i Hi = CVi ) 2, i = 1,..., n. The subcritical flow condition is 1.76) gh i V i ) 2 >, i = 1,..., n. The characteristic velocities are λ i = V i + gh i >, λ n+i = V i gh i <, i = 1,..., n. Genetic regulatory networks Systems of nonlinear delay-differential equations constitute an obvious special case of quasilinear hyperbolic systems which are represented by a set of scalar conservation laws transport equations) coupled by nonlinear ordinary differential equations. An early pioneering reference on this topic is the fundamental book Bellman and Cooke 1963). A typical and important example of this class of systems is given by the continuous time models of genetic regulatory networks when time delays are included to allow for the time required for transcription, translation, and transport Smolen et al. 2) ). For a genetic regulatory network which involves n genes interconnected through activator or repressor proteins, the expression of the i-th gene in the network i = 1,..., n)

55 1.15. Physical networks of hyperbolic systems 55 is represented by the following standard delay-differential system see e.g. Bernot et al. 213) ): 1.77) dm i t) dt dp i t) dt = b i + h i P k t τ k )) δ i M i t), = α i M i t τ n+i ) β i P i t), where, at time t, M i t) is the density of mrna molecules, P i and P k are the densities of proteins expressed by the i-th and k-th genes respectively. Here, for simplicity, the model is restricted to the case where the i-th gene is controlled by only one protein expressed by the k-th gene k depends on i), see Fig.1.22 for an illustration. The constants b i and α i denote activation or repression transcription translation gene mrna protein Fig.1.22: Scheme of genetic transcription and translation with activation or repression. respectively the basal transcription rate and the specific translation rate. The constants β i and δ i are the natural degradation rate coefficients. The constant delays τ i and τ n+i are the times needed for transcription and translation respectively. The function h i P k ) describes how the transcription of the i-th gene is activated or repressed by the density P k of the protein expressed by the k-th. It may therefore be either an activation Hill function of the form h i P k ) = or an inhibition Hill function of the form h i P k ) = k ci i k ci i υ ip ci k + P ci k υ ik ci i + P ci k where υ i, k i and c i are positive constant parameters with υ i the maximal transcription rate, k i the half-saturation coefficient and c i the so-called Hill coefficient. A steady state of the system is a constant solution M i, P i, i = 1,..., n, of the dynamical system 1.77) i.e. a solution of the algebraic system b i + h i P k ) δ i M i =, α i M i β i P i =.,,

56 56 Chapter 1. Hyperbolic Systems of Balance Laws It is well known that these systems may have multiple steady states or equilibria. A critical issue is to determine the stability of these equilibria. This issue will be adressed in Section 3.4. Let us define the deviations of P i and M i with respect to a steady state M i, P i : m i t) = M i t) M i, p i t) = P i t) P i. With these coordinates, the model 1.77) is alternatively written under the form dm i t) dt dp i t) dt where the function g i is defined such that = g i p k t τ k ))p k t τ k ) δ i m i t), = α i m i t τ n+i ) β i p i t), g i p)p = h i P + p) h i P ). This system is then a special case of a general hyperbolic system of linear conservation laws in Riemann coordinates 1.78) R t + Λ R R x =, S t + Λ S S x =, t [, + ), x [, 1], with the maps R : [, + ) [, 1] R n and S : [, + ) [, 1] R n such that Rt, ) p 1 t),..., p n t) ) T and St, ) m 1 t),..., m n t) ) T, and the maps Λ R : R n D + and Λ S : R n D + such that { 1 Λ R diag,..., 1 } { 1, Λ S diag,..., 1 }. τ 1 τ n τ n+1 τ n The system 1.78) is subject to nonlinear differential boundary conditions of the form 1.79) drt, ) dt dst, ) dt = ASt, 1) BRt, ), = GRt, 1))Rt, 1) DSt, ) with the notations A diag{α 1,..., α n }, B diag{β 1,..., β n }, D diag{δ 1,..., δ n } and GR) M n,n R) the matrix with entry [GR)] ik g i R k ) if k i and zero otherwise. The notation k i means that the protein expressed by the k-th gene is an activator or a repressor of the i-th gene transcription. Here, for simplicity, the model has been restricted to the case of networks where each gene can be controlled by one protein at most. The model 1.78), 1.79) is obviously also valid for situations where a gene can be controlled by several proteins simultaneously provided the definition of the matrix GR) is adequately extended.

57 1.16. References and further reading References and further reading In this chapter, we have presented various examples of physical systems or engineering problems that are usefully represented by one-dimensional hyperbolic systems of conservation and balance laws. Among many other examples which can be found in the literature, let us mention the following interesting additional references. Multi-layer Saint-Venant equations with interfacial exchanges of mass for the modelling of suspended matters transported by shallow-water streams, see e.g. Audusse et al. 211). This kind of model is applied for example in the design of so-called raceway processes used for the cultivation of micro-algae where the water flows around a circular channel, driven by a paddlewheel, see e.g. Bernard et al. 213). Alternative models of conservation laws for bedload sediment transport in shallow waters are presented and compared in Castro Diaz et al. 28). Isentropic gas flow models generalized to the case of flow through porous media where the velocity obeys the Darcy s law, see e.g. Dafermos and Pan 29). Relativistic isentropic Euler equations for the description of one-dimensional gas flow at speeds where the relativistic effects become significant as in high energy particle beams for instance, see e.g. Smoller and Temple 1993), Chen and Li 24) and LeFloch and Yamazaki 27). For general gas transportations networks, the determination of steady states may be a complicated task. Gugat et al. 215) present a method for the computation of steady states in certain pipeline networks that involve cycles and are represented by isentropic Euler equations. Balance law models with non-local source terms for the description of crystal growth or multilane road traffic with non-local flux density, see e.g. Colombo et al. 27), Lee and Liu 215). Aw-Rascle traffic flow models with phase transitions, see e.g. Goatin 26). Conservation laws similar to the models of car traffic for the description of packets flow on telecommunication networks with routing algorithms at the nodes, e.g. D Apice et al. 26). Conservation law models for supply chains and other highly re-entrant manufacturing systems as encountered, for instance, in semi-conductor production. These models may have non-local characteristic velocities, see e.g. Armbruster et al. 23), Armbruster et al. 26), Shang and Wang 211), Coron et al. 21), Armbruster et al. 211), Coron and Wang 213). Semi-linear hyperbolic systems with nonlinear reaction interactions of Lotka-Volterra type, e.g. Pavel 213). In addition to age-dependent epidemiologic models, Perthame 27) describes also other interesting examples of biological and ecological systems represented by hyperbolic equations, such as intra-host virus or sea phytoplankton dynamics see also Calvez et al. 21), Calvez et al. 212) ).

58 58 Chapter 1. Hyperbolic Systems of Balance Laws Balance law models for extrusion processes, e.g. Dos Santos Martins 213), Diagne et al. 215a), Diagne et al. 215b). Balance law models for oil well drilling processes, e.g. Aamo 213), Hasan and Imsland 214). Balance law models for incompressible two-phase flow with an interfacial pressure and a drag force as the coupling terms between the two phases, e.g. Djordjevic et al. 21) and Djordjevic et al. 211). Quasi-linear hyperbolic models for the onset of bubbling in fluidized bed chemical reactors, e.g. Hsiao and Marcati 1988). Finally, readers interested in numerical simulations of one-dimensional hyperbolic equations may refer to the classical textbooks of LeVeque 1992) and Godlewski and Raviart 1996).

59 Chapter 2 Systems of Two Linear Conservation Laws IN THIS CHAPTER, we start the analysis of the stability and the boundary stabilization design with the simple case of systems of two linear conservation laws. There are two good reasons for beginning in this way. The first reason is that a system of two linear conservation laws is simple enough to allow an explicit and complete mathematical analysis of many aspects of the stability issue. It is therefore an excellent pedagogical starting point for more complex studies on general systems of conservation and balance laws. The first section of this chapter is devoted to a thorough presentation of the necessary and sufficient stability conditions for systems of two linear conservation laws under local boundary conditions, both in the time and the frequency domain. A second reason is that, in many instances as we have illustrated with numerous examples in Chapter 1, a system of two linear conservation laws may be a valid approximation of a physical system having a direct engineering interest. In particular, in the second section of this chapter, we shall develop in great details the issue of the boundary feedback stabilization of so-called density-flow systems. Finally, in the third section, we examine how the stability conditions are extended to the case of nonuniform characteristic velocities Stability conditions We consider a hyperbolic system of two linear conservation laws in Riemann coordinates: 2.1) t R 1 + λ 1 x R 1 =, t R 2 λ 2 x R 2 =, t [, + ), x [, L], λ 1 > > λ 2, with constant characteristic velocities λ 1 > and λ 2 <. If the solutions R 1 and R 2 are differentiable with respect to t and x, it is immediate to check that the system 2.1) is equivalent to a pair of scalar delay equations such that 2.2) R 1 t + x x, x ) = R 1 t, x), λ 1 R 2 t + x x λ 2, x) = R 2 t, x ), t, x, x ) such that x < x L. 59

60 6 Chapter 2. Systems of Two Linear Conservation Laws The behaviour of the solutions is illustrated in Fig. 2.1 where they are seen as waves moving to the right or to the left without change of shape at the constant velocities λ 1 and λ 2. R 1 t, x) R 2 t, x) t 2 t 1 t 1 t 2 x x Fig.2.1: The motion of the solutions of linear conservation laws in Riemann coordinates t 2 > t 1 ). It is however evident that the delay equations 2.2) make sense also if the solutions are not differentiable and even not continuous with respect to t and x. In such case, the dynamics of the system may still be represented by the hyperbolic PDEs 2.1) albeit with appropriate definitions of weak partial derivatives for t and x. Let us define linear boundary conditions of the form 2.3) R 1 t, ) = k 1 R 2 t, ), R 2 t, L) = k 2 R 1 t, L), where k 1 and k 2 are constant real coefficients. Observe that this is a special case of the general nominal boundary conditions 1.11) that we have introduced in Chapter 1. The mechanism described by these boundary conditions can be interpreted as a reflection of the waves at the boundaries with amplifications or attenuations) of size k 1 and k 2, as illustrated in Fig R 1 t 1,x) R 2 t 2,x) L x Fig.2.2: The reflection mechanism at the boundaries with t 2 > t 1 and k 2 =.5. When k 1 k 2 1, the unique equilibrium solution of the system 2.1), 2.3) is R 1 =, R 2 = which can be stable or instable depending on the values of k 1 and k 2. When k 1 k 2 = 1, the system has infinitely many non-isolated equilibria: R 1 = k 1 R 2 for every R 2 R which are therefore not asymptotically stable. Our concern, in this section is to show that the equilibrium solution is exponentially stable if and only if k 1 k 2 < 1.

61 2.1. Stability conditions 61 Exponential stability for the L norm In this subsection, we use the method of characteristics to analyze the exponential convergence of the system solutions in the L and C spaces. We consider the system 2.1) under an initial condition for t =, 2.4) R 1, x) = R 1o x), R 2, x) = R 2o x), x [, L], and, for t >, under the boundary conditions 2.5) R 1 t, ) = k 1 R 2 t, ), R 2 t, L) = k 2 R 1 t, L), t, + ). For a function ϕ = ϕ 1, ϕ 2 ) T L, L); R 2 ), we define the L norm: ) ϕ L,L);R 2 ) max ϕ 1 L,L);R), ϕ 2 L,L);R) < +. We assume that the functions 1 R 1o : [, L] R and R 2o : [, L] R are bounded, and therefore that R 1o, R 2o ) T L, L); R 2 ). Theorem 2.1. There exist positive constants C, ν such that, for any bounded functions R 1o and R 2o, the solution of the Cauchy problem 2.1), 2.4), 2.5) defined by the delay equations 2.2) satisfies 2.6) R 1 t,.), R 2 t,.)) T L,L);R 2 ) Ce νt R 1o, R 2o ) T L,L);R 2 ), t [, + ), if and only if k 1 k 2 < 1. Proof. The mechanism of the proof is illustrated in Fig As shown in this figure, since the system 2.1) is linear, the two characteristic curves C 1 and C 2 are straight lines with slopes λ 1 and λ 2 respectively. By using twice alternatively the formula 2.2) and the boundary condition 2.5), for a given x [, L], we have R 1 τ, x) = R 1 t 2, ) = k 1 R 2 t 2, ) with τ L λ 1 + L λ 2 and t 2 τ x λ 1 = k 1 R 2 t 1, L) = k 1 k 2 R 1 t 1, L) = k 1 k 2 R 1, x) with t 1 = L x λ 1 = k 1 k 2 R 1o x). By continuing this recurrence for increasing time delays nτ, we get 2.7) R 1 nτ, x) = k 1 k 2 ) n R 1o x) n = 2, 3, 4,.... From this expression, we conclude readily that R 1 t, x) cannot converge to zero as t if k 1 k 2 1. A similar argument can obviously be developed for R 2 t, x). 1 Here, and in the rest of the book except otherwise stated, we always assume that these functions are measurable.

62 62 Chapter 2. Systems of Two Linear Conservation Laws R 1,x)=k 1 R 2 t 2, ) R 2 t 2, ) = R 2 t 1,L) t 2 t 1 C 2 R 2 t 1,L)=k 2 R 1 t 1,L) = k 2 R 1,x) C 1 x Fig.2.3: Illustration of the proof of Theorem 2.1. On the other hand, using again 2.2) and 2.5) appropriately, the following upper bounds can be established: x [, L] t s.t. nτ t x λ 1 < n + 1)τ : R 1 t, x) 1 + k 1 ) k 1 k 2 ) n R max o, x [, L] t s.t. nτ t L x λ 2 < n + 1)τ : R 2 t, x) 1 + k 2 ) k 1 k 2 ) n R max o, with Ro max R 1o, R 2o ) T L,L);R 2 ). From these expressions, if < k 1 k 2 < 1, it follows readily that inequality 2.6) is satisfied with C 2 + k 1 + k 2 )R max o and ν 1 τ ln 1 k 1 k 2 and therefore that the solutions exponentially converge to zero for the L norm. In the special case where k 1 k 2 =, R 1 t, x) = and R 2 t, x) = for all t, x) [τ, + ) [, L] see also page 71). This concludes the proof of Theorem 2.1. Let us now consider the special case when the initial condition is not only bounded but also continuous, i.e. when R 1o, R 2o ) T C [, L]; R 2 ), and when it is desired to have a so-called C solution, i.e. a solution R 1 t,.), R 2 t,.)) T C [, L]; R 2 ) which is continuous with respect to x for all t [, + ). In that case, to avoid discontinuities with respect to x at the initial time instant and to get a C solution, it is clearly necessary that the initial condition 2.4) be compatible with the boundary condition 2.5) as follows: 2.8) R 1o ) = k 1 R 2o ), R 2o L) = k 2 R 1o L). Under this condition, it follows readily not only that the solution is unique, but also that it inherits the regularity of the initial condition, i.e. that it is a C solution. Then, since by definition Ro max is also the C norm of R 1o, R 2o ) T, we have the following corollary of Theorem 2.1. )

63 2.1. Stability conditions 63 Corollary 2.2. For every R 1o, R 2o ) T C [, L]; R 2 ) satisfying the compatibility condition 2.8), the Cauchy problem 2.1), 2.4), 2.5) has a unique C solution. Furthermore, the solutions exponentially converge to zero for the C norm if and only if k 1 k 2 < 1. Exponential Lyapunov stability for the L 2 -norm In this subsection, using a Lyapunov stability approach, we shall now examine how the convergence to zero of the solutions of the system 2.1), 2.5) can be analyzed in the L 2 space. We consider again the system 2.1), 2.5) under a non-zero initial condition 2.9) R 1, x) = R 1o x), R 2, x) = R 2o x), but we assume now that the function R 1o, R 2o ) T L 2, L); R 2 ) with a L 2 norm ) 1/2 2.1) R 1o, R 2o ) T L 2,L);R 2 ) R1ox) 2 + R2ox) dx) 2 < +. Thus here, provided it is in L 2, the initial condition may be unbounded in contrast with the assumption of Theorem 2.1) and does not need to satisfy any compatibility condition in contrast with the assumption of Corollary 2.2). We will now give the definition of a solution to the Cauchy problem 2.1), 2.5), 2.9) in L 2, L); R 2 ). In order to motivate this definition let us multiply 2.1) on the left by ϕ 1, ϕ 2 ) C 1 [, T ] [, L]; R 2 ) where T is given. We get the equation ϕ 1 t R 1 + λ 1 x R 1 ) + ϕ 2 t R 2 λ 2 x R 2 ) =. Let us now integrate this equation on, T ), L). Assuming for a while that the solution R 1, R 2 is of class C 1 with respect to both t and x, we have, using integrations by parts and 2.9): = = T ) ϕ 1 t R 1 + λ 1 x R 1 ) + ϕ 2 t R 2 λ 2 x R 2 ) dtdx ) ϕ 1 T, x)r 1 T, x) + ϕ 2 T, x)r 2 T, x) dx ) ϕ 1, x)r 1o x) + ϕ 2, x)r 2o x) dx T ) + λ 1 ϕ 1 t, L)R 1 t, L) λ 2 ϕ 2 t, L)R 2 t, L) dt T ) λ 1 ϕ 1 t, )R 1 t, ) λ 2 ϕ 2 t, )R 2 t, ) dt T ) t ϕ 1 + λ 1 x ϕ 1 )R 1 + t ϕ 2 λ 2 x ϕ 2 )R 2 dtdx. Then, using the boundary condition 2.5), we get = ) ϕ 1 T, x)r 1 T, x) + ϕ 2 T, x)r 2 T, x) dx

64 64 Chapter 2. Systems of Two Linear Conservation Laws ) ϕ 1, x)r 1o x) + ϕ 2, x)r 2o x) dx T ) + λ 1 ϕ 1 t, L) k 2 λ 2 ϕ 2 t, L) R 1 t, L)dt T ) k 1 λ 1 ϕ 1 t, ) λ 2 ϕ 2 t, ) R 2 t, )dt T ) t ϕ 1 + λ 1 x ϕ 1 )R 1 + t ϕ 2 λ 2 x ϕ 2 )R 2 dtdx. Now, if the functions ϕ 1 and ϕ 2 are selected such that 2.11) we get 2.12) = T k 1 λ 1 ϕ 1 t, ) λ 2 ϕ 2 t, ) = λ 1 ϕ 1 t, L) k 2 λ 2 ϕ 2 t, L) =, ) ϕ 1 T, x)r 1 T, x) + ϕ 2 T, x)r 2 T, x) dx ) ϕ 1, x)r 1o x) + ϕ 2, x)r 2o x) dx t ϕ 1 + λ 1 x ϕ 1 )R 1 + t ϕ 2 λ 2 x ϕ 2 )R 2 ) dtdx. The key point here is that, although this latter equation has been derived under the assumption that the functions R 1 and R 2 are of class C 1 with respect to t and x, it appears that it makes sense also even if the functions R 1 and R 2 are not differentiable and can therefore be considered as weak solutions of the system. The L 2 solutions are then defined as the functions R 1, R 2 ) which satisfy 2.12) for all ϕ 1, ϕ 2 ) verifying 2.11), when the initial conditions are in L 2. The technical definition is as follows. Definition 2.3. Let R 1o, R 2o ) L 2, L); R 2 ). A map R 1, R 2 ) : [, + ), L) R 2 is a L 2 solution of the Cauchy problem 2.1), 2.5), 2.9) if R 1, R 2 ) C [, + ); L 2, L); R 2 )) is such that 2.12) is satisfied for every T [, + ) and for every ϕ 1, ϕ 2 ) C 1 [, T ]) [, L]; R 2 ) satisfying 2.11). We then have the following stability theorem. Theorem 2.4. For any function R 1o, R 2o ) L 2, L); R 2 ), the Cauchy problem 2.1), 2.5), 2.9) has one and only one solution. Furthermore, there exist positive constants C, ν such that R 1 t,.), R 2 t,.)) T L 2,L);R 2 ) Ce νt R 1o, R 2o ) T L 2,L);R 2 ), t [, + ), if and only if k 1 k 2 < 1.

65 2.1. Stability conditions 65 Proof. The existence of a unique L 2 solution follows, as a special case, from Theorem A.4 in Appendix A. For the convergence analysis, the following candidate Lyapunov function is introduced: 2.13) Vt) [ p1 R λ 1t, 2 x) exp µx ) + p 2 R 1 λ 1 λ 2t, 2 x) exp+ µx ] ) dx 2 λ 2 with positive constant coefficients p 1, p 2 and µ. The time derivative of Vt) along the C 1 solutions of the Cauchy problem 2.1), 2.5), 2.9) is dvt) dt = = = = µ [ 2 p 1 R 1 t R 1 ) exp µx ) + 2 p 2 R 2 t R 2 ) exp+ µx ] ) dx λ 1 λ 1 λ 2 λ 2 [ 2p 1 R 1 x R 1 ) exp µx ) + 2p 2 R 2 x R 2 ) exp+ µx ] ) dx λ 1 λ 2 [ p 1 x R 2 1) exp µx λ 1 ) + p 2 x R 2 2) exp+ µx p 1 λ 1 R 2 1 exp µx λ 1 ) + p 2 λ 2 R 2 2 exp+ µx [ p 1 R1 2 exp µx ) λ 1 ] L + [ p 2 R 2 2 exp+ µx λ 2 ) λ 2 ) ) dx ] L [ = µvt) p 1 R1t, 2 L) exp µl ] ) R 2 λ 1t, ) 1 [ p 2 R2t, 2 ) R2t, 2 L) exp+ µl ] ) λ 2 Using the boundary condition 2.5), we have: dvt) dt = µvt) [ p 1 exp µl λ 1 ) p 2 k 2 2 exp+ µl λ 2 ) If k 1 k 2 < 1, we can select µ > such that Then, we can select p 1 and p 2 such that which implies that exp µl λ 1 + µl λ 2 )k 2 1k 2 2 < 1. exp µl λ 1 + µl λ 2 )k 2 2 < p 1 p 2 < 1 k 2 1 ]. ] ) dx λ 2 [ ] R1t, 2 L) p 2 p 1 k1 2 R2t, 2 ). p 1 exp µl λ 1 ) p 2 k 2 2 exp+ µl λ 2 ) > and p 2 p 1 k 2 1 >.

66 66 Chapter 2. Systems of Two Linear Conservation Laws Hence, we see that dv/dt µv along the trajectories of the system 2.1), 2.5) which are of class C 1. By density see the explanation given in the next section), this inequality also holds in the distribution sense for every solution of 2.1), 2.5) which is in C [, + ); L 2, L)). Then, since there exists γ > such that 2.14) 1 γ R 2 1 t, x) + R2t, 2 x) ) dx Vt) γ R 2 1 t, x) + R2t, 2 x) ) dx, we get that R 1 t,.), R 2 t,.) ) T L 2,L);R 2 ) γe µt/2 R 1o, R 2o ) T L 2,L);R 2 ), t [, + ). Consequently, if k 1 k 2 < 1, the solutions converge in L 2 -norm and the equilibrium is exponentially stable. We thus have proved the sufficient condition. In order to prove the necessary condition, we assume that k 1 k 2 1 and, as Xu and Sallet 214), we use the same Lyapunov function as above albeit with µ = : Wt) = [ p1 R λ 1t, 2 x) + p ] 2 R 2 1 λ 2t, x) dx. 2 Along the C 1 solutions of the system 2.1), 2.5), the time derivative of W is dwt) dt = [ p 2 k 2 2 p 1 ] R 2 1 t, L) + [ p 1 k 2 1 p 2 ] R 2 2 t, ). Since k 1 k 2 1, we can select p 1 and p 2 such that k1 2 p 1 1 p 2 k2 2, which implies that dw/dt along the system trajectories which, therefore, cannot exponentially converge to zero. This completes the proof of Theorem 2.4. From this proof it follows also that the maximum admissible value of the parameter µ 2.15) µ max = 2 ) 1 τ ln k 1 k 2 is an estimate of the fastest possible decay rate of V. We observe that it is identical to the convergence rate ν which was obtained in Theorem 2.1 with the method of characteristics. Remark 2.5. The weights e µx/λ1 and e µx/λ2 in 2.13) are essential to get a strict Lyapunov function in Theorem 2.4. The use of such terms in a quadratic Lyapunov function was originally introduced in Coron 1999) for the stabilization of the Euler equation of incompressible fluids.

67 2.1. Stability conditions 67 A note on the proofs of stability in L 2 -norm In the course of the proof of Theorem 2.4, we have seen that the Lyapunov stability analysis was technically derived for the C 1 solutions of the system. At the end of the proof, it was then stated that the analysis is actually also valid for L 2 solutions. Roughly speaking, the reason is that the C 1 solutions are dense in the set of L 2 solutions of the system. Indeed, since the set of C 1 functions vanishing at and L is dense in L 2, L); R 2 ), there exists a sequence of initial conditions R1o, k R2o) k T of class C 1 vanishing at and L and therefore satisfying the compatibility condition 2.8)) which converge to R 1o, R 2o ) T in L 2, L); R 2 ). Let R1, k R2) k T : [, + ), L) R 2 be the L 2 solution of the Cauchy problem 2.1), 2.5), for the initial condition R1o, k R2o) k T. By Theorem A.1 in Appendix A, we have R1, k R2) k T C 1 [, + ); L 2, L) 2 ) C [, + ); H 1, L) 2 ). This regularity is sufficient to get the inequality dv k /dt µv k with V k t) [ p1 R λ 1) k 2 t, x) exp µx ) + p 2 R 1 λ 1 λ 2) k 2 t, x) exp+ µx ] ) dx. 2 λ 2 Then letting k +, we get dv/dt µv in the sense of distributions. See also Comment 4.6 in Chapter 4 for an explanation of this assertion in a more general case). In the sequel of the book, each time we will establish stability conditions for the L 2 norm, we will follow the same line of deriving the Lyapunov analysis for smooth solutions but, somewhat abusively, we will generally not explicitly mention that the analysis is, obviously, also valid for general weak solutions. Frequency domain stability Taking the two-sided) Laplace transform of equations 2.2), we have the following representation in the frequency domain with s σ + jω the Laplace complex variable): 2.16) R 1 s, x ) = exp x x s)r 1 s, x), λ 1 R 2 s, x) = exp x x λ 2 s)r 2 s, x ), x, x ) such that x < x L. For x = and x = L, the system 2.1) of linear conservation laws endowed with the linear boundary conditions 2.5) can be represented under the form of a feedback loop of two scalar delay systems as shown in Fig.2.4 with the notations τ 1 L/λ 1 and τ 2 L/λ 2. The poles σ n + jω n of the system of Fig.2.4 are the roots of the characteristic equation: 2.17) e sτ k 1 k 2 =, with τ τ 1 + τ 2 ). There is a countable infinity of poles lying on a vertical line in the complex plane. Their values depend on the sign of k 1 k 2 as follows: k 1 k 2 > σ n = 1 1 ) τ ln, ω n = ± 2nπ, n =, 1, 2,... k 1 k 2 τ k 1 k 2 < σ n = 1 τ ln 1 ) 2n + 1)π, ω n = ±. k 1 k 2 τ

68 68 Chapter 2. Systems of Two Linear Conservation Laws R 1 s, ) e s 1 R 1 s, L) k 1 k 2 R 2 s, ) e s 2 R 2 s, L) Fig.2.4: Representation of the hyperbolic system of two linear conservation laws 2.1), 2.5) as a feedback delay system. The following stability theorem follows immediately. Theorem 2.6. The poles of the system 2.1), 2.5) have a strictly negative real part σ n < if and only if k 1 k 2 < 1. From Theorems 2.1, 2.4 and 2.6, we see that having a pole spectrum strictly located in the complex left-half plane is equivalent to the global exponential stability of the equilibrium for the system 2.1), 2.5), whatever the considered norm L or L 2. Moreover, the rate of exponential convergence of the solutions given by the absolute real part σ n of the poles is again identical to the previous estimates obtained with the method of characteristics and the Lyapunov approach. Example: stability of a lossless electrical line Let us come back to the example of an electrical line that we have already presented in Section 1.2 and which is illustrated in Fig.2.5. Assuming a lossless line i.e. with zero re- Power supply Ut) It, ) It, L) Transmission line R V t, ) L x V t, L) Load R L Fig.2.5: Transmission line connecting a power supply to a resistive load. sistance R l and zero conductance G l ), the dynamics are described by the following system of two conservation laws: 2.18) t I + 1 L l x V =, t V + 1 C l x I =,

69 2.2. Boundary control of density-flow systems 69 with the boundary conditions 2.19) R It, ) + V t, ) = Ut), V t, L) = R L It, L). For a given constant input voltage Ut) = U, the system has a unique constant steady state I U =, V = R LU. R + R L R + R L For this system, as we have seen in Section 1.2, the Riemann coordinates around the steady state are defined as R 1 V V ) + I I ) L l /C l, and the characteristic velocities are R 2 V V ) I I ) L l /C l, λ 1 = 1, λ 2 = 1. Ll C l Ll C l Then, expressing the boundary conditions 2.19) in Riemann coordinates, we have: R 1 t, ) = k 1 R 2 t, ), R 2 t, L) = k 2 R 1 t, L), with k 1 = R Cl L l R Cl +, L l k 2 = R L Cl L l R L Cl +. L l It is easy to see that k 1 k 2 < 1 for any positive) value of R, R L, C l and L l. Consequently, the equilibrium I, V is exponentially stable in C -norm according to Theorem 2.1, in L 2 -norm according to Theorem 2.4 and in the frequency domain according to Theorem 2.6. This is obviously a natural property for the device of Fig.2.5 since it is a passive electrical circuit. However, the analysis provides here the accurate value of the exponential decay-rate Boundary control of density-flow systems In this section, we shall now give a first example of the use of boundary feedback control for the stabilization of an hyperbolic system. We consider a system of two linear conservation laws of the general form: 2.2) t H + x Q = t Q + λ 1 λ 2 x H + λ 1 λ 2 ) x Q = where λ 1 and λ 2 are two real positive constants. The first equation can be interpreted as a mass conservation law with H the density and Q the flow density. The second equation can then be interpreted as a momentum conservation law. The model 2.2) may be used to represent many physical systems. For instance, it may be used as a valid approximate linearized model for the motion of liquid fluids in pipes interconnected by pumps where H is the piezometric head and Q is the flow rate, while

70 7 Chapter 2. Systems of Two Linear Conservation Laws λ 1 = λ 2 = c is the sound velocity. A detailed justification of this model is nicely presented in Nicolet, 27, Chapter 2). In such fluid distribution networks, it may be relevant to provide the system with feedback controllers that regulate the piezometric head at certain places in order, for instance, to prevent water hammer phenomena. We are concerned with the solutions of the Cauchy problem for the system 2.2) under an initial condition: H, x) = H o x), Q, x) = Q o x) x [, L] and two boundary conditions of the form: 2.21) Qt, ) = Q t), Qt, L) = Q L t), t [, + ). Any pair of constant states H, Q is a potential steady state of the system. We assume that one of them has been selected as the desired steady state or set point. The Riemann coordinates are defined around the set point by the following change of coordinates: R 1 = Q Q + λ 2 H H ), R 2 = Q Q λ 1 H H ), with the inverse change of coordinates: H = H + R 1 R 2 λ 1 + λ 2, Q = Q + λ 1R 1 + λ 2 R 2 λ 1 + λ 2. With these coordinates, the system 2.2) is written in characteristic form: 2.22) t R 1 + λ 1 x R 1 =, t R 2 λ 2 x R 2 =. Then, assuming a constant flow rate Q t) = Q L t) = Q and expressing the boundary conditions 2.21) in Riemann coordinates, we have: R 1 t, ) = k 1 R 2 t, ), R 2 t, L) = k 2 R 1 t, L) with k 1 = λ 2 λ 1, k 2 = λ 1 λ 2. Consequently k 1 k 2 = 1 and the equilibrium H, Q ) is not asymptotically stable. In fact, all the system poles are located on the imaginary axis as we have seen in the previous section see also Litrico and Fromion, 29, Section 3.2) ). It is therefore relevant to study the boundary feedback stabilization of the control system 2.2), 2.21). It will be the main concern of this section. More precisely, we are looking for controls Q t) and Q L t) that are functions of the state variables at the boundaries and that guarantee the asymptotic stability of the steady state H, Q ). Feedback stabilization with two local controls From the stability analysis of section 2.1, a simple and natural solution is to select control laws that realize boundary conditions of the form R 1 t, ) = k 1 R 2 t, ), R 2 t, L) = k 2 R 1 t, L)

71 2.2. Boundary control of density-flow systems 71 in Riemann coordinates. This is easily achieved by defining the following feedback control laws: 2.23) Q t) Q + k H Ht, )), k λ 1k 1 + λ 2 1 k 1, Q L t) Q k L H Ht, L)), k L λ 2k 2 + λ 1 1 k 2. Then, the steady state H, Q ) of the closed loop system 2.2), 2.21),2.23) is exponentially stable if and only if the control tuning parameters k and k L are chosen such that k 1 k 2 = k λ ) 2 kl λ ) 1 < 1. k + λ 1 k L + λ 2 H + Ht, ) Ht, L) + + k Q o t) System Q L t) k L + Q + + Fig.2.6: Block diagram of the control system with two local controllers. A block diagram representation of the control system is given in Fig.2.6. It can be seen that each control Q has the form of a feedback of the density H at the same boundary: the implementation of Q t) requires only the on-line measurement of Ht, ) and the implementation of Q L t) requires only the on-line measurement of Ht, L). For this reason, the controls are said to be local. Dead-beat control An interesting very special choice of the control tuning parameters is k = λ 2 and k L = λ 1. Indeed with this choice we have k 1 k 2 = and, therefore, a so-called dead-beat control because, starting from an arbitrary initial condition, the steady state is reached as fast as possible in a finite time τ given by τ L λ 1 + L λ 2. It must however be pointed out that dead-beat control may have the severe drawback of producing excessively big transients and feedback control actions that are too strong to be achieved with the available physical actuators.

72 72 Chapter 2. Systems of Two Linear Conservation Laws Feedback-feedforward stabilization with a single control We now consider the situation where there is only one boundary control input, say Q t), available for feedback stabilization. The other boundary flow Q L t) perturbs the system in an unpredictable manner and cannot be manipulated. But we assume that this disturbance can be measured on-line. Then, from the previous section, a natural candidate control law is: 2.24) Q t) Q L t) + k P H Ht, )), k P λ 1k 1 + λ 2 1 k 1, where k P is a tuning parameter and H is the density set point. This control law involves a feedforward term Q L t) which compensates for the measured disturbance) and a proportional feedback term k P H Ht, )) for the density regulation. The control system is illustrated in Fig.2.7. H + Ht, ) k P + Q o t) System Q L t) + Fig.2.7: Block diagram of the closed-loop system with a feedback-feedforward control. Assuming a constant disturbance Q L t) = Q, if k P the closed-loop system has a unique steady state H, Q ) and can be written in Riemann coordinates with boundary conditions R 1 t, ) = k 1 R 2 t, ), k 1 = k P λ 2 k P + λ 1, R 2 t, L) = k 2 R 1 t, L), k 2 = λ 1 λ 2. Then, the steady state H, Q ) of the closed loop system 2.2), 2.21), 2.24) is exponentially stable if and only if the control tuning parameter k P is selected such that k 1 k 2 = kp λ ) 2 λ1 k P + λ 1 < 1. Proportional-integral control In the previous section we have used a controller that involves a feedforward action when the flow Q L t) is measurable. But, obviously, it could arise that this flow is a so-called load λ 2

73 2.2. Boundary control of density-flow systems 73 disturbance which cannot be measured and cannot therefore be directly compensated in the control. In such case it is useful to implement an integral action in order to eliminate offsets and to attenuate the incidence of the load disturbance. A so-called Proportional- Integral control law may be of the following form: t 2.25) Q t) Q R + k P H Ht, )) + k I H Hs, ))ds, The first term Q R is a constant reference value for the flow which is arbitrary and freely chosen by the designer. The second term is the proportional correction action with the tuning parameter k P. The last term is the integral action with the tuning parameter k I. In case of a constant unknown) disturbance Q L t) = Q, the closed-loop system has a unique steady state H, Q ). The control system is illustrated in Fig.2.8. As it is explained H Q R + k P + k I Z + + Ht, ) Q o t) System Q L t) Fig.2.8: Block diagram of the closed-loop system with a Proportional- Integral control. in details in Chapter 11 of the textbook Feedback Systems Åström and Murray 29) ), PI control is by far the most popular way of using feedback in engineering systems because it is the simplest way to cancel offset errors and to attenuate load disturbances in a robust way. The integral gain k I is a measure of the disturbance attenuation but a too large value of k I may lead to instability in some instances. It is therefore of interest to characterize the range of values of k I for which the closed-loop system is guaranteed to be stable. In Riemann coordinates, the control law 2.25) gives a first boundary condition at x = : 2.26) R 1 t, ) = k 1 R 2 t, ) + k 3 Xt) and Xt) Q R Q with k 1 k P λ 2 k P + λ 1, k 3 k I λ 1 + λ 2 ) + t k I k P + λ 1 R 2 τ, ) R 1 τ, ))dτ. The constant disturbance Q L t) = Q gives the second boundary condition at x = L: 2.27) R 2 t, L) = k 2 R 1 t, L) with k 2 = λ 1 λ 2.

74 74 Chapter 2. Systems of Two Linear Conservation Laws From 2.27), since R 1 t, x) and R 2 t, x) are constant along their respective characteristic lines, we have that 2.28) R 2 t + τ, ) = k 2 R 1 t, ) with τ L λ 1 + L λ 2 and therefore that 2.29) dr 2 t + τ, ) dt dr 1 t, ) = k 2. dt Moreover, by differentiating 2.26) with respect to time, the first boundary condition is rewritten as: 2.3) dr 1 t, ) dt = k 1 dr 2 t, ) dt + k 3 R2 t, ) R 1 t, ) ). Then, by eliminating R 1 t, ) and dr 1 t, )/dt between 2.28), 2.29) and 2.3), we get that R 2 t, ) is the solution of the following delay-differential equation of neutral type: 2.31) dr 2 t + τ, ) dt k 1 k 2 dr 2 t, ) dt ) + k 3 R 2 t + τ, ) k 2 R 2 t, ) =. The Laplace transform of this equation is: [ ] 2.32) e sτ k 1 k 2 )s + k 3 e sτ k 2 ) R 2 s, ) =. This is a so-called neutral delay-differential equation. The roots of the characteristic equation 2.33) e sτ k 1 k 2 )s + k 3 e sτ k 2 ) = are called the poles of the system 2.22), 2.26), 2.27). Stability analysis in the frequency domain In the next theorem, we give necessary and sufficient conditions to have stable poles, i.e. poles located in the left-half complex plane and bounded away from the imaginary axis. The stability of the poles is equivalent, in the time domain, to the exponential stability of the equilibrium when the disturbance Q L t) = Q is constant) and therefore the inputto-state stability when the disturbance Q L t) is bounded time-varying) for the C -norm see e.g. Hale and Verduyn-Lunel, 22, Section 9.3) and Michiels and Niculescu, 27, Section 1.2)). In the proof of the theorem we use a variant of the Walton-Marshall procedure see Walton and Marshall 1987) and Silva et al., 25, Section 5.6)). Theorem 2.7. There exist δ > such that the poles of the system 2.32) are in the half plane, δ] R if and only if when λ 1 λ 2 i.e. 1 k 2 < ), k 1 k 2 < 1 and < k 3 ;

75 2.2. Boundary control of density-flow systems 75 when λ 1 > λ 2 i.e. k 2 < 1), k 1 k 2 < 1 and < k 3 < ω sin ω τ k 21 + k 1 ) 1 k 2 2 where ω is the smallest positive ω such that cosωτ) = 1 + k 1k 2 2 k k 1 ). Proof. For s +, the equation 2.33) is approximated by e sτ k 1 k 2 =, from which it follows that k 1 k 2 < 1 is a necessary condition to have stable poles i.e. Rs) < δ, see e.g. Hale and Verduyn-Lunel 22) and Michiels and Vyhlidal 25) ). From now on, we assume that k 1 k 2 < 1. It is also easily checked that, for every k 1 and k 2, for every η > ln k 1 k 2 ) and for every C >, there exists C 1 > such that 2.34) { } k 3 C, s C 1 and 2.33) { } Rs) η. Indeed the existence of C 1 results from rewriting 2.33) under the form 2.35) e sτ = k 1k 2 s + k 3 k 2 s + k 3 which implies τrs) = ln k 1 k 2 s + k 3 k 2 s + k 3 s ln k 1 k 2 where the convergence is uniform for k C. With the notation s σ + iω, the poles satisfy the following equation: 2.36) k 3 = esτ k 1 k 2 )s e sτ k 2 [ωaσ, ω) σbσ, ω)] i[σaσ, ω) + ωbσ, ω)] = e 2στ + k2 2 2k 2e στ cosωτ) with 2.37) 2.38) aσ, ω) k 2 e στ k 1 1) sinωτ) and bσ, ω) e 2στ k k 1 )e στ cosωτ) + k 1 k 2 2. Since the left-hand side of equation 2.36) is real, it follows that the imaginary part of the right-hand side must be zero. Therefore we are looking for the values of σ and ω such that 2.39) σaσ, ω) + ωbσ, ω) =. Let us now consider the poles with non-positive real parts, i.e. σ. If k 3 =, we see that the poles are roots of e sτ k 1 k 2 )s =. This means that there is a pole s = at the origin and the other poles are stable if and only if k 1 k 2 < 1. Now for small non-zero k 3, we have: 1 k 1 k 2 )s + k 3 1 k 2 ), that is s k 3 1 k 2 1 k 1 k 2.

76 76 Chapter 2. Systems of Two Linear Conservation Laws This approximation can be justified by using the implicit function theorem applied to the map s, k 3 ) C R F s, k 3 ) = e sτ k 1 k 2 )s + k 3 e sτ k 2 ) since F/ s, ) = 1 k 1 k 2. Then, since k 1 k 2 < 1 and k 2 = λ 1 /λ 2 <, it follows that for small k 3 > the pole at zero moves inside the negative half-plane while the other poles stay inside the negative half-plane. Moreover, for small k 3 <, the pole at zero moves inside the right half plane. As k 3 decreases, this simple pole cannot come back on the imaginary axis since k 2 ) and therefore it remains in the right half plane for all k 3 <. Now, in order to analyze what happens when k 3 > becomes larger, we consider the conditions for having poles on the imaginary axis i.e. σ =. Since k 3, the case σ =, ω = is excluded. Therefore σ = implies b = from 2.39), which together with 2.38) gives: cosωτ) = 1 + k 1k 2 2 k k 1 ). In this case, it can be readily verified that, since k 1 k 2 < 1, λ 1 < λ 2 k 2 < 1 1 k 2 2 > 1 k 2 2)1 k 2 1k 2 2) > 1 + k1k k 1 k2 2 > k k1) 2 + 2k 1 k k 1 k2 2 k k 1 ) > 1. which implies that there is no eigenvalue on the imaginary axis. Then, using also 2.34), we can conclude, using a standard deformation argument on k 3, that, when k 2 < 1 and k 1 k 2 < 1, the poles remain stable for every k 3 >. Let us now consider the case where λ 1 > λ 2 i.e. k 2 < 1 the case λ 1 = λ 2 is discussed later). In this case, it can be readily verified that 1 + k 1 k2 2 k k 1 ) < 1. Therefore, from 2.36) and 2.38) with σ =, there is a pair of poles ±iω on the imaginary axis for any positive value of ω such that: 2.4) cos±ωτ) = 1 + k 1k 2 2 k k 1 ) and ω sinωτ) = k 3k 2 2 1) k k 1 ). Let ω be the smallest value of ω such that 2.4) is satisfied. Now, if iω is a pole on the imaginary axis, the corresponding value of k 3 computed from 2.4) ω = ω is as follows: k 3 = ω sinω τ) k 21 + k 1 ) 1 k 2 2 Then, using again 2.34), we can conclude, using a standard deformation argument on k 3, that the poles are stable for any k 3 such that < k 3 < k 3. In order to determine the motion >.

77 2.2. Boundary control of density-flow systems 77 of the pole on the imaginary axis for small variations of k 3 around k 3, we consider the root s of the characteristic equation as an explicit function of k 3. Then, by differentiating the characteristic equation 2.33), we have the following expression for the derivative of s with respect to k 3 : 2.41) s ds k 2 e sτ = dk 3 e sτ. 1 + τs + k 3 )) k 1 k 2 We now evaluate this expression at iω: s = k 2 e iωτ e iωτ 1 + τiω + k 3 )) k 1 k 2. Using 2.4), after some calculations, we obtain that the real part of s at iω is given by: Rs τk 3 k2 2 1) ) = e iωτ τiω + k 3 )) k 1 k 2 Hence, since k 2 2 > 1 and k 3 > by assumption, Rs ) is a positive number. It follows that any pole reaching the imaginary axis from the left when k 3 is increasing will cross the imaginary axis from left to right. This readily implies that, as soon as k 3 > k 3, there is necessarily at least one pole in the right half plane. Let us finally consider the case where λ 1 = λ 2 i.e. k 2 = 1). In that case, it follows directly from 2.4) that cosωτ) = 1 and sinωτ) = for any pole iω on the imaginary axis. Therefore the characteristic equation 2.33) reduces to k 1 1)iω = which is impossible if ω because the conditions k 2 = 1 and k 1 k 2 < 1 imply that k 1 < 1. Hence there is no imaginary pole when λ 1 = λ 2. This completes the proof of Theorem 2.7. As a matter of illustration, a sketch of the root locus for fixed values of k 1 and k 2 and increasing values of k 3 from to + is given in Fig.2.9. In the previous Theorem, for the clarity of the proof, we have carried out the analysis in terms of the parameters k 1, k 2 and k 3. However, from a practical viewpoint, it is clearly more relevant and more interesting to express the stability conditions in terms of the control tuning parameters k P and k I. Replacing k 1, k 2 and k 3 by their expressions in function of k P, k I, λ 1 and λ 2 as given in 2.26), 2.27), the conditions of Theorem 2.7 are translated as follows. Theorem 2.8. There exist δ > such that the poles of the closed-loop system 2.2),2.25) are in the half plane, δ] R if and only if the control tuning parameters k P, k I are selected such that: when λ 1 < λ 2, k P > and k I > or k P < 2λ 1λ 2 λ 2 λ 1 and k I < ;

78 78 Chapter 2. Systems of Two Linear Conservation Laws x x 1 > 2 x x x x x x x 1 > Fig.2.9: Sketch of the root locus for fixed values of k 1 and k 2 and increasing values of k 3 from to +. when λ 1 = λ 2, k P > and k I > ; when λ 1 > λ 2, < k P < 2λ 1λ 2 2k P + λ 1 λ 2 )λ 1 λ 2 and < k I < ω λ 1 λ 2 λ 2 1 sinω τ) λ2 2 where ω is the smallest positive ω such that cosωτ) = λ2 2k P + λ 1 ) + λ 2 1k P λ 2 ). λ 1 λ 2 λ 2 λ 1 2k P ) Lyapunov stability analysis Up to now, in this subsection, we have taken the frequency domain viewpoint for the stability analysis of PI control of density-flow systems. We now move to the Lyapunov approach. In the next two theorems, we show how a quadratic Lyapunov function can be used to establish sufficient stability conditions which are similar but slightly more restrictive.

79 2.2. Boundary control of density-flow systems 79 Theorem 2.9. The solution R 1 t, x), R 2 t, x) of the system 2.1), 2.26), 2.27) exponentially converges to zero for the L 2 -norm if 2.42) k 2 < 1, k 1 k 2 < 1, k 3 >. Proof. We define the candidate Lyapunov function Vt) = [ p1 R λ 1t, 2 x) exp µx ) + p 2 R 1 λ 1 λ 2t, 2 x) exp+ µx ] ) dx + bx 2 t) 2 λ 2 with positive constant coefficients p 1, p 2, b and µ. The time derivative of Vt) along the system trajectories is dv dt = µ [ p1 R1 2 exp µx ) + p 2 R2 2 exp+ µx ] ) dx λ 1 λ 1 λ 2 λ 2 [ p 1 R1 2 exp µx ] L [ ) + p 2 R2 2 exp+ µx ] L ) + 2bXX, λ 1 λ 2 Using the boundary conditions 2.26), 2.27), we have: dv dt = µv + W 1 + W 2 with 2.43) 2.44) W 1 W 2 [ λ 2 1 p 2 λ 2 exp+ µl ) p 1 exp µl ] ) R 2 λ 2 λ 1t, 2 L) 1 ] [ ] [p 1 k1 2 p 2 R2t, 2 ) + p 1 k3 2 2bk 3 + µb X 2 [ ] + 2p 1 k 1 k 3 + 2b1 k 1 ) XR 2 t, ). We have to prove that there exist positive µ >, p 1 >, p 2 >, b > such that 2.43) and 2.44) are negative definite ND) quadratic forms in R 1 t, L) and in R 2 t, ) and X respectively. We set p 2 = 1. Then, it follows that there exists µ > such that 2.43) is ND if and only if 2.45) p 1 > λ1 λ 2 ) 2 = k 2 2. We now consider the special case where µ =. Then the right-hand side of 2.44) is: [ ] ] [ ] p 1 k1 2 1 R2t, 2 ) + [p 1 k3 2 2bk 3 X 2 + 2p 1 k 1 k 3 + 2b1 k 1 ) XR 2 t, ). This is a ND quadratic form if and only if 2.46) p 1 < 1 k 2 1

80 8 Chapter 2. Systems of Two Linear Conservation Laws and 2.47) P 1 k 1 ) 2 b 2 + 2k 3 p 1 k 1 1)b + p 1 k3 2 <. We consider P as a degree-2 polynomial in b. The discriminant of P is = 4k3p 2 1 k1 2 1)p 1 1), which, if 2.46) holds, is positive if 2.48) p 1 < 1. In this case, the two roots of P which have the sign) are positive if 2.49) p 1 k 1 < 1. Since k 1 k 2 < 1 and k 2 < 1, there exists p 1 > such that 2.45), 2.48) and 2.49) hold. We choose such a p 1. Then, it follows from our analysis that there exists b > such that 2.43), 2.44) are ND quadratic forms when µ =. By continuity with respect to µ, under the same conditions, there exist µ >, p 1 >, p 2 >, b > such that 2.43), 2.44) are ND quadratic forms. Theorem 2.1. The solution R 1 t, x), R 2 t, x) of the system 2.1), 2.26), 2.27) exponentially converges to zero for the L 2 -norm if k 1 k 2 < 1, and k 3 > is sufficiently small. Proof. We introduce the following change of coordinates: S 1 R 1 In these coordinates, the system 2.22) is rewritten k 3 1 k 1 k 2 X, S 2 R 2 k 2k 3 1 k 1 k 2 X. k 3 t S 1 + λ 1 x S 1 + X =, 1 k 1 k 2 t S 2 λ 2 x S 2 + k 2k 3 X = 1 k 1 k 2 and the boundary conditions 2.26), 2.27) are respectively S 1 t, ) = k 1 S 2 t, ), S 2 t, L) = k 2 S 1 t, L). Under these boundary conditions, we have 2.5) X = 1 k 1 )S 2 t, ) 1 k 2 ) X. 1 k 1 k 2 k 3 We define the candidate Lyapunov function Vt) [ p1 S λ 1t, 2 x) exp µx ) + p 2 S 1 λ 1 λ 2t, 2 x) exp+ µx ] ) dx + bx 2 t). 2 λ 2

81 2.2. Boundary control of density-flow systems 81 with positive constant coefficients p 1, p 2, b and µ. The time derivative of Vt) along the system trajectories is dv dt = µ [ p1 S1 2 exp µx ) + p 2 S2 2 exp+ µx ] ) dx λ 1 λ 1 λ 2 λ 2 [ p 1 exp µl ) p 2 k2 2 exp+ µl ] ) S λ 1 λ 1t, 2 L) [p 2 p 1 k1]s 2 2t, 2 ) 2 + 2bX X [ ) ] p 1 S 1 exp µx ) + 2 p 2 S 2 k 2 exp+ µx k 3 ) X dx. λ 1 λ 1 λ 2 λ 2 1 k 1 k 2 Since k 1 k 2 < 1, we can select µ > such that Then, we can select p 1 and p 2 such that which implies that exp µl λ 1 + µl λ 2 )k 2 1k 2 2 < 1. exp µl λ 1 + µl λ 2 )k 2 2 < p 1 p 2 < 1 k 2 1 p 1 exp µl λ 1 ) p 2 k 2 2 exp+ µl λ 2 ) > and p 2 p 1 k 2 1 >. Then, using also 2.5), there are δ > and κ > such that dvt) dt δ We introduce the following notations: Then dvt) dt S S 2 2) dx δ S 2 1 t, L) + S 2 2t, ) ) 2bk 3 1 k 2 1 k 1 k 2 X 2 + κb S 2 t, ) X [ ]1 + κk 3 S S2) 2 2 dx S2 t, ) + k 3 X ). A 1 k 2 1 k 1 k 2 >, ϑ [ ]1 S S2) 2 2 dx. δs1t, 2 L) X S 2 t, ) ϑ ) 2bk 3 A bκ/2 κk 2 3/2 X bκ/2 δ κk 3 /2 S 2 t, ). κk3/2 2 } κk 3 /2 {{ δ } Q ϑ

82 82 Chapter 2. Systems of Two Linear Conservation Laws We set b = k3 2 and we compute the three principal minors of the matrix Q: Aκ 2k3A, 3 2Aδ)k3 3 κ 4 /4)k3, 4 2Aδ 2 )k3 2 δκ 2 k3/ κ3 4 ) k 5 3. Clearly, we have that the three minors are strictly positive if k 3 > is chosen sufficiently small and therefore that Q is positive definite. We conclude that, along the system trajectories, there exists ν > such that dvt) dt νv t. Consequently the equilibrium is asymptotically stable and the solutions exponentially converge to zero for the L 2 -norm The non uniform case In this section, we examine how the previous results can be extended to the non uniform case where the characteristic velocities λ i are function of the space coordinate x. More precisely, we now consider the following system: 2.51) t R 1 + λ 1 x) x R 1 =, t R 2 λ 2 x) x R 2 =, λ 1 x) > > λ 2 x), x [, L], under boundary conditions in canonical form 2.52) R 1 t, ) = k 1 R 2 t, ), R 2 t, L) = k 2 R 1 t, L), and an initial condition 2.53) R 1, x) = R 1o x) L 2 [, L], R), R 2, x) = R 2o x) L 2 [, L], R). The well-posedness of the Cauchy problem 2.51), 2.52), 2.53) in L 2 follows, as a special case, from Theorem A.4 in Appendix A. In order to analyze the exponential stability of the system we introduce the following tentative Lyapunov function: Vt) = q1 x)r 2 1t, x) + q 2 x)r 2 2t, x) ) dx where q 1 C 1 [, L];, + )) and q 2 C 1 [, L];, + )) have to be determined. The time derivative of V along the trajectories of 2.51), 2.52), 2.53) is dv dt = = ) L ) 2q1 R 1 t R 1 + 2q 2 R 2 t R 2 dx = 2q1 R 1 λ 1 x R 1 2q 2 R 2 λ 2 x R 2 dx [ ] λ 1 q 1 ) x )R1 2 λ 2 q 2 ) x )R2 2 dx [ λ 1 L)q 1 L) λ 2 L)q 2 L)k2] 2 R 2 1 t, L) [ λ 2 )q 2 ) λ 1 )q 1 )k 2 1] R 2 2 t, ).

83 2.3. The non uniform case 83 It follows that V is a Lyapunov function if q 1 and q 2 are such that λ 1 q 1 ) x >, λ 2 q 2 ) x <, x [, L], and k 2 1 λ 2)q 2 ) λ 1 )q 1 ), k2 2 λ 1L)q 1 L) λ 2 L)q 2 L). On the basis of our previous results in this chapter, a natural and convenient choice for the functions q 1 and q 2 is: q 1 x) = p x ) 1 λ 1 x) exp µ λ 1 σ) dσ, 2.54) q 2 x) = p 2 λ 2 x) exp + x µ λ 2 σ) dσ where p 1, p 2, µ are positive constants. With this choice, we have the following stability theorem. Theorem The solution R 1 t, x), R 2 t, x) of the Cauchy problem 2.51), 2.52), 2.53) exponentially converges to zero for the L 2 -norm if k 1 k 2 < 1. Proof. With the functions 2.54), the derivative of the Lyapunov function becomes dv dt = µv [ p 1 exp [ p 2 p 1 k 2 1] R 2 2 t, ). µ ) λ 1 σ) dσ p 2 k2 2 exp + Since k 1 k 2 < 1, we can select µ sufficiently small such that [ k1k exp µ Then, we can select p 1 and p 2 such that [ k2 2 exp µ which implies that [ p 1 exp 1 λ 1 σ) + 1 λ 2 σ) 1 λ 1 σ) + 1 λ 2 σ) ) ] ) dσ < 1. ] ) dσ < p 1 < 1 p 2 k1 2 µ ) λ 1 σ) dσ p 2 k2 2 exp + ] µ ) λ 2 σ) dσ R1t, 2 L) ] µ ) λ 2 σ) dσ > and [ p2 p 1 k1] 2 >. Hence, we have dv µv. dt Therefore V is a strict Lyapunov function and the solutions of the system 2.51), 2.52), 2.53) exponentially converge to zero for the L 2 -norm.

84 84 Chapter 2. Systems of Two Linear Conservation Laws 2.4. Conclusions In Section 2.1, this chapter was first devoted to a comprehensive treatment of the exponential stability of a system of two linear conservation laws under local linear static boundary conditions. A necessary and sufficient stability condition has been given. This condition ensures that the L 2 norm is an exponentially decaying Lyapunov function and, equivalently, it guarantees that the poles of the system are located in the left-half complex plane and bounded away from the imaginary axis. In the subsequent chapters, the main concern will be to examine how such static boundary conditions can be generalized for the exponential stability analysis of hyperbolic systems of conservation and balance laws. Then, Section 2.2 was devoted the case when the stability boundary conditions are obtained by using boundary feedback control with actuators and sensors located at the boundaries. The stabilization of density-flow systems which are open-loop unstable and subject to unknown disturbances was investigated in detail. Using Proportional-Integral PI) control the stabilization of the two conservation laws is achieved under dynamic boundary conditions. A necessary and sufficient condition on the values of the control tuning parameters is given see also Bastin et al. 215) ). Let us also mention that, in this case, adding a derivative action to the PI controller is known to always produce an unstable closed-loop see e.g. Coron and Tamasoiu 215) ).

85 Chapter 3 Systems of Linear Conservation Laws THIS CHAPTER mainly deals with the stability of general systems of linear conservation laws under static linear boundary conditions. Depending on whether the issue is examined in the time or in the frequency domain, different stability criteria emerge and are compared, namely from the viewpoint of robustness against uncertainties in the characteristic velocities. The chapter ends with the study of the stability of linear conservation laws under more general boundary conditions that may be dynamic, nonlinear or switching. We consider hyperbolic systems of linear conservation laws in Riemann coordinates 3.1) R t + ΛR x =, t [, + ), x [, L], where R : [, + ) [, L] R n. As we have already explained in Chapter 1, the matrix Λ is diagonal and defined as 3.2) Λ Λ + With the notations R + = Λ R 1. R m ) the system 3.1) is also written with and R = { Λ + = diag{λ 1,..., λ m }, Λ = diag{λ m+1,..., λ n }, R m+1. R n such that R = ) ) ) R + Λ + R + 3.3) t + R Λ x =. R λ i > i. ) R +, Our concern is to analyze the exponential stability of this system under linear boundary conditions in canonical form ) ) ) R + t, ) R + t, L) K K 1 3.4) = K with K, t [, + ), R t, L) R t, ) K 1 K 11 and an initial condition 3.5) R, x) = R o x), x, L). R 85

86 86 Chapter 3. Systems of Linear Conservation Laws 3.1. Exponential stability for the L 2 -norm In this section, using a Lyapunov approach, we give an explicit condition on the matrix K under which the steady state Rt, x) of the system 3.3), 3.4) is globally exponentially stable for the L 2 norm according to the following definition. Definition 3.1. The system 3.3), 3.4) is exponentially stable for the L 2 norm if there exist ν > and C > such that, for every R o L 2, L); R n ), the L 2 solution of the Cauchy problem 3.3), 3.4), 3.5) satisfies Rt,.) L2,L);R n ) Ce νt R o L2,L);R n ), t [, + ). The definition of the L 2 solutions and the well-posedness of the Cauchy problem 3.3), 3.4), 3.5) are given in Appendix A, see Definition A.3 and Theorem A.4. In order to state the stability condition, we first introduce the functions ρ p : M n,n R) R defined by 3.6) ρ p M) inf { M 1 p, D n + }, 1 p, where D + n denotes the set of diagonal n n real matrices with strictly positive diagonal entries and for ξ ξ 1,..., ξ n ) T R n, ξ p [ n i=1 for M M n,n R), M p max ξ p=1 Mξ p. We have the following stability theorem. ξ i p ] 1 p, ξ max{ ξ 1,..., ξ n }, Theorem 3.2. The system 3.3), 3.4) is exponentially stable for the L 2 norm if ρ 2 K) < 1. Proof. We introduce the following candidate Lyapunov function, which is a direct extension of the function used in Chapter 2: 3.7) with 3.8) 3.9) V = = [ m i=1 p i λ i R 2 i t, x) exp µx λ i ) + n i=m+1 ] p i Ri 2 t, x) exp+ µx ) dx λ i λ i [ R +T Λ + ) 1 P + µx)r + ) + R T Λ ) 1 P µx)r ) ] dx { P + µx) diag p 1 exp µx } ) λ m ),..., p m exp µx λ 1 { P µx) diag p m+1 exp+ µx ),..., p n exp+ µx ) λ m+1 λ n, p i >, }, p i >.

87 3.1. Exponential stability for the L 2 -norm 87 The time derivative of V along the C 1 solutions of 3.3), 3.4) is dv dt = µv + W with W [ R +T P + µx)r +] L + [ R T P µx)r ] L. First we will show that the parameters p i and µ can be selected such that, under the condition ρ 2 K) < 1, W is a negative definite quadratic form in R t, ) and R + t, L). For this analysis, we introduce the following notations: R t) R t, ) R + L t) R+ t, L). Using the boundary condition 3.4), we have 3.1) W = [ R +T P + µx)r +] L + [ R T P µx)r ] L = R +T L P + µl)r + L + R T P )R ) + R +T L KT + R T KT 1) P + ) K R + L + K 1R + R +T L KT 1 + R T KT 11) P µl) K 1 R + L + K 11R ) ). Since ρ 2 K) < 1 by assumption, there exist D D + m, D 1 D + n m and diag{d, D 1 } such that 3.11) K 1 < 1. The parameters p i are selected such that P + ) = D 2 and P ) = D1. 2 With these definitions, regarding W as a function of µ, we have Wµ) = R +T L D R T D ) D R + ) L 1 Ωµ) D 1 R with Ωµ) P + µl)d 2 I ) ) D K D 1 D K 1 D1 1 T and, for µ =, D 1 K 1 D 1 D 1 K 11 D1 1 W) = R +T L D D K D 1 D K 1 D1 1 P µl)d1 1 K 1D 1 P µl)d1 1 K 11D1 1 R T D 1) ) D I K 1 ) T K 1 R + L ) D 1 R Since K 1 < 1, it follows that W) is a strictly negative definite quadratic form in R + L and R. Then, by continuity, Wµ) remains a strictly negative definite quadratic form for µ > sufficiently small. ). )

88 88 Chapter 3. Systems of Linear Conservation Laws Hence, we have dv dt = µv + W µv. along the system trajectories. Therefore V is a strict Lyapunov function and the solutions of the system 3.3), 3.4), 3.5) exponentially converge to zero for the L 2 norm. Dissipative boundary conditions It is remarkable that the stability condition ρ 2 K) < 1 depends on the value of K but not on the values of the characteristic velocities λ i. In other words, the stability condition is independent of the system dynamics 3.3) and depends only on the boundary conditions 3.4). When the matrix K satisfies such a stability condition, the boundary conditions are said to be dissipative and the stability is guaranteed whatever the length L and the time required for solutions to cross the system. Intuitively, this is understood as follows: the solutions, which are moving back and forth between the two boundaries, remain constant along the characteristic lines and are exponentially damped at the boundaries only. This can be also understood using a small gain principle. Indeed we have observed in Chapter 1, Section 1, that the hyperbolic systems 1.3) under the boundary conditions 1.11) can be regarded as a closed loop interconnection of two causal input-output systems as represented in Fig.3.1. It is therefore natural that the stability requires a small gain of the feedback loop. It is not surprising that the condition relies only on the gain ρ 2 K) of the system S 2 since the system S 1 has a unit gain by definition. R in R t + R x = R out System S 1 K System S 2 Fig.3.1: The linear hyperbolic system 3.3), 3.4) viewed as a closed loop interconnection of two causal input-output systems 3.2. Exponential stability for the C -norm: analysis in the frequency domain In this section, we now take the frequency domain viewpoint to analyze the exponential stability of the system 3.3), 3.4) for the C norm according to the following definition.

89 3.2. Exponential stability for the C -norm: analysis in the frequency domain 89 Definition 3.3. The system 3.3), 3.4) is exponentially stable for the C norm if there exist ν > and C > such that, for every R o C [, L], R n ) satisfying the compatibility condition ) ) R + o ) R + o L) 3.12) = K, Ro L) Ro ) the solution of the Cauchy problem 3.3), 3.4), 3.5) satisfies Rt,.) C,L);R n ) Ce νt R o C,L);R), t [, + ). As we have already emphasized in Chapter 2, the system 3.3) can be regarded as a set of scalar delay systems R i t, L) = R i t τ i, ) i = 1,..., m, R j t, ) = R j t τ j, L) j = m + 1,..., n, τ k L λ k, k = 1,..., n, which are interconnected by the boundary conditions 3.4). Taking the Laplace transform, it follows that the characteristic function of the system 3.3), 3.4) is: 3.13) det [ I n diag { e sτ1,..., e sτn} K ], where I n is the identity matrix of M n,n R). The roots of this function are called the poles of the system. Definition 3.4. The poles of the system 3.3), 3.4) are stable if there exists δ > such that the poles are located in the half plane, δ] R. A fundamental property is given in the following theorem. Theorem 3.5. The system 3.3), 3.4) is exponentially stable for the C norm if and only if the poles of the system are stable. Proof. See Hale and Verduyn-Lunel, 1993, Chapter 9, Theorem 3.5). Remark 3.6. Theorem 3.5 deals with the C norm. However, it must be pointed out that the proof, as it is given by Hale and Verduyn-Lunel, also works for the L p norm for every p [1, + ]. Hence the stability analysis does not require to know the actual location of the poles. It is sufficient to know that they have a negative real part which is bounded away from zero. From the viewpoint of boundary control design, it is obviously of major interest to predict the stability, and therefore the sign of the real parts of the poles, directly from the coefficients of the matrix K. Two stability conditions are presented below. The first one is the same as in the previous section. Theorem 3.7. The poles of the system 3.3), 3.4) are stable if ρ 2 K) < 1. Proof. If ρ 2 K) < 1 there exists η, 1) and D n such that 3.14) K 1 η.

90 9 Chapter 3. Systems of Linear Conservation Laws Let us assume that s is a pole of the system. Then det [ I n diag { e sτ1,..., e sτn} K 1] = det [ I n diag { e sτ1,..., e sτn )K } 1] = det [ I n diag { e sτ1,..., e sτn} K) ] =, which implies that 3.15) diag { e sτ1,..., e sτn} K 1 1. Since diag { e sτ1,..., e sτn} K 1 diag { e sτ1,..., e sτn} K 1 exp min{τ 1 Rs),..., τ n Rs)}) K 1 where Rs) denotes the real part of the pole s, we have, using also 3.14) and 3.15), 3.16) exp min{τ 1 Rs),..., τ n Rs)})η 1. Inequality 3.16) implies that Rs) δ lnη) max{τ 1,..., τ n } <. Another stability condition is stated in the following Theorem by Silkowski 1976) which relies on the Kronecker density theorem e.g. Bridges and Schuster 26) ). Theorem 3.8. Let 3.17) ρk) max{ρdiag { e iθ1,..., e iθn} K); θ 1,..., θ n ) T R n } where ρm) denotes the spectral radius of the matrix M. If the time delays τ 1,..., τ n ) are rationally independent, the poles of the system 3.3), 3.4) are stable if and only if ρk) < 1. Proof. See Hale and Verduyn-Lunel, 1993, Chapter 9, Theorem 6.1). The statement of this theorem includes the rather unexpected feature that the time delays have to be rationally independent which is a generic property. In fact, when the τ i s are rationally dependent the condition ρk) < 1 is no longer necessary and can be violated while keeping the exponential stability as we shall illustrate with a simple example below. In Michiels et al. 21), it is explained how when approaching rational dependence of the delays, the supremum of the real parts of the poles can have a discontinuity... ) compatible with the continuous movement of individual roots in the complex plane.

91 3.2. Exponential stability for the C -norm: analysis in the frequency domain 91 A simple illustrative example Let us now present an example that illustrates the conditions of Theorems 3.7 and 3.8. We consider the most simple case of a system of two linear conservation laws with a full matrix K. More precisely, we have the system ) ) ) R1 λ1 R1 3.18) t + x =, λ 2 < < λ 1, R 2 λ 2 R 2 with the boundary condition K ) {}} ){ ) R1 t, ) k k 1 R1 t, L) 3.19) =. R 2 t, L) k 2 k 3 R 2 t, ) Taking the Laplace transform of system 3.18), 3.19), the characteristic equation is 3.2) e sτ1 k )e sτ2 k 3 ) k 1 k 2 =. Let us consider the very special case τ 1 = 1, τ 2 = 2 which allows a simple and explicit computation of the poles. In this case, the characteristic equation is 3.21) e 3s k e 2s k 3 e s + k k 3 k 1 k 2 =. Defining z e s, we get the third-order polynomial equation 3.22) z 3 k z 2 k 3 z + k k 3 k 1 k 2 =. Let z l l = 1, 2, 3) denote the three roots of this polynomial. Then, for each z l, there is an infinity of system poles s n = σ n + jω n lying on a vertical line in the complex plane: 3.23) σ n = ln z i, ω n = 2πn + argz i ), n =, ±1, ±2,.... The poles are stable if and only if z l < 1, l = 1, 2, 3. For simplicity, let us now address the special case where k k 3 = k 1 k 2. In that case, it can be shown after a few calculations that the stability condition of Theorems 3.7 and 3.8 is 3.24) ρk) = ρ 2 K) = k + k 3 < 1. The region of stability corresponding to this condition is thus the square represented in Fig.3.2. From Theorem 3.8 we know that the condition is necessary and sufficient when τ 1 /τ 2 is an irrational number. But, when τ 1 /τ 2 is rational, the stability region may be larger as we shall now illustrate by computing the poles of the system. Using the condition k k 3 = k 1 k 2 the polynomial equation 3.22) becomes 3.25) zz 2 k z k 3 ) = and we can compute the roots explicitly 3.26) z 1 =, z 2,3 = k ± k 2 + 4k 3. 2

92 92 Chapter 3. Systems of Linear Conservation Laws k k -1 Fig.3.2 Remark that e s = z 1 = has no solution. The system poles corresponding to z 2 and z 3 are stable if and only if the parameters k and k 3 satisfy one of the following two conditions: 3.27) k 2 + 4k 3 and k ± k 2 + 4k 3 < 2 or 3.28) k 2 + 4k 3 < and k 2 + k2 + 4k 3 < 2. The region of stability in the k, k 3 ) plane is the triangular region shown in Fig.3.2. It can be easily checked that the stability conditions 3.27), 3.28) can be in fact formulated in the following simpler way: 3.29) k 3 > 1, k + k 3 < 1, k k 3 > 1. A numerical illustration is given in Fig.3.3a) for parameter values k = k 1 = k 2 = k 3 =.6 which satisfy inequalities 3.29) but not inequality 3.24). As expected from the above analysis, the poles are located on a vertical line in the left half complex plane. Robust stability From this simple example, it appears that the strict negativity of the pole real parts is not a robust stability condition as far as robustness with respect to small variations of the characteristic velocities is concerned. A numerical illustration of this fact is given in Fig.3.3b) where the τ 2 value of the previous example is slightly perturbed: τ 2 is set to 2.1 instead of 2. The consequence of this small variation of τ 2 is that half of the poles are progressively shifted to the right up to instability. In contrast, we may observe that the stability condition ρk) < 1 is robust with respect to small changes on K since ρk) depends continuously on K. We then introduce the following definition for the robustness with respect to the characteristic velocities. Definition 3.9. The system 3.3), 3.4) is robustly exponentially stable with respect to the

93 3.2. Exponential stability for the C -norm: analysis in the frequency domain 93 a) 4 b) Fig.3.3: Pole configuration for the characteristic equation 3.2) with k = k 1 = k 2 = k 3 =.6: a) τ 1 = 1, τ 2 = 2, b) τ 1 = 1, τ 2 = 2.1 characteristic velocities if there exists ε > such that the the perturbed system ) ) ) R + Λ+ R + t + R x =. Λ R is exponentially stable for every Λ such that λ i λ i ε i 1,..., n. It is then evident that the following robust stability condition follows as a simple corollary of Theorem 3.8. Corollary 3.1. The system 3.3), 3.4) is robustly exponentially stable with respect to the characteristic velocities if and only if ρk) < 1. Comparison of the two stability conditions Another interesting observation is that we have ρk) = ρ 2 K) in the above simple example. This observation suggests to further investigate the comparison between ρ and ρ 2 and to determine to which extent they can be equal. This is done in the following theorems. Theorem For every integer n and for every real n n matrix K, ρk) ρ 2 K).

94 94 Chapter 3. Systems of Linear Conservation Laws Proof. For every θ 1,..., θ n ) T R n and for every D D n ρdiag { e ιθ1,..., e ιθn} K) = ρd diag { e ιθ1,..., e ιθn} KD 1 ) = ρdiag { e ιθ1,..., e ιθn} DKD 1 ) diag { e ιθ1,..., e ιθn} DKD 1 diag { e ιθ1,..., e ιθn} DKD 1 = DKD 1. Theorem a) For every n {1, 2, 3, 4, 5} and for every real n n matrix K, ρk) = ρ 2 K). b) For every integer n > 5, there exist a real n n matrix K such that ρk) < ρ 2 K). The proof of this theorem can be found in Appendix C. The following corollary follows trivially. Corollary If there exist a permutation matrix P such that the matrix K = P KP 1 is a block diagonal matrix { K = diag K1, K 2,..., K } p where each block K i is a real n i n i matrix with n i {1, 2, 3, 4, 5}, then ρk) = ρ 2 K) The rate of convergence In the previous two sections, we have given explicit conditions on the matrix K that guarantee the exponential convergence to zero of the solutions of the system 3.3), 3.4). From a control design viewpoint it is also of major interest to be able to quantify the rate of convergence. For that purpose, we define the following change of state variables: ) P + µx) 3.3) St, x) e µt P µx)rt, x), < µ R, P µx), P µx) where P + and P are defined in 3.8) and 3.9) respectively. The dynamics of these new coordinates can be shown to be governed by the same hyperbolic system as is R: S t + ΛS x =, with adequately adapted boundary conditions: ) S + t, ) ) S + t, L) S t, L) = Kµ) S t, ), ) D 2 K P + µl)d 4 DK 2 1 D1 2 Kµ). P µl)k 1 P + µl)d 4 P µl)k 11 D1 2

95 3.4. Differential linear boundary conditions 95 It follows that, for a given K such that ρk) < 1, the robust convergence is guaranteed with any µ such that ρ Kµ)) < 1. Let us define µ c sup{µ : ρ Kµ)) < 1}. It follows that, for any ν, µ c ), there exists C > such that, for every solution of 3.3), 3.4), Rt,.) L2,L);R n ) Ce νt R o L2,L);R n ), t [, + ). Application to a system of two conservation laws Let us consider again the system of two conservation laws ) ) ) R1 λ1 R1 3.31) t + x =, λ 1 >, λ 2 >, R 2 λ 2 R 2 under the boundary condition ) ) ) R1 t, ) k k 1 R1 t, L) 3.32) =. R 2 t, L) k 2 k 3 R 2 t, ) In this case, the change of coordinates 3.3) is and the matrix Kµ) is Kµ) S 1 = p 1 e µt e µx/λ1 R 1, S 2 = p 2 e µt e µx/λ2 R 2, D 2 k e µτ1 k 1 D1 2 k 2 e µτ1+τ2) D2 1 D 2, with τ i L, i = 1, 2. λ k 3 e µτ2 i In the special case of local boundary conditions where k = k 3 = and k 1 k 2 < 1, the value of µ c is explicitly given by µ c = sup{µ : ρ Kµ)) = 1} = 1 ) 1 τ ln k 1 k 2 which is, as expected, identical to the convergence rate that we have found in Chapter Differential linear boundary conditions Up to now, in this chapter, we have discussed the stability of linear hyperbolic systems under static linear boundary conditions. In this section, we examine how the previous results can be generalized to the case of boundary conditions that are dynamic and represented by linear differential equations. More precisely, we consider the linear hyperbolic system of

96 96 Chapter 3. Systems of Linear Conservation Laws conservation laws in Riemann coordinates 3.3) under linear differential boundary conditions of the following form: 3.33) X = AX + BR out t), R in t) = CX + KR out t), where A M l,l R), B M l,n R), C M n,l R), K M n,n R), X R l, l n. The notations R in and R out were introduced in Section 1.1 and stand for ) ) R + t, ) R + t, L) R in t) R, R out t) t, L) R. t, ) The well-posedness of the Cauchy problem associated to this system is addressed in Appendix A, see Theorem A.6. Frequency domain Using the Laplace transform, the system 3.3), 3.33) is written in the frequency domain as R out s) = Ds)R in s) with Ds) diag{e sτ1,..., e sτn }, τ i = L/λ i, si A)Xs) = BR out s), R in s) = CXs) + KR out s). Hence the poles of the system are the roots of the characteristic equation [ det I Ds) CsI A) 1 B + K )] =. Theorem The steady state Rt, x) of the system 3.3), 3.4) is exponentially stable for the L norm if and only if the poles of the system are stable i.e. have strictly negative real parts and are bounded away from zero). Proof. See Hale and Verduyn-Lunel, 22, Section 3) and Michiels and Niculescu, 27, Section 1.2)). Lyapunov approach In the line of the previous results of this chapter, we may also introduce the following Lyapunov function candidate: 3.34) V = [ m i=1 p i Ri 2 t, x) exp µx ) + λ i λ i n i=m+1 ] p i Ri 2 t, x) exp+ µx ) dx + λ i λ i with X X 1,..., X l ) T, p i > i = 1,..., n), q j > j = 1,..., l). The time derivative of this function along the C 1 solutions of 3.3), 3.33) is ) V = µv + Rout, T X T Rout )Mµ), X l q j Xj 2 j=1

97 3.4. Differential linear boundary conditions 97 with the matrix ) K T P 1 µ)k P 2 µ) K T P 1 µ)c + B T Q Mµ), C T P 1 µ)k + QB µq + C T P 1 µ)c + A T Q + QA) and { P 1 µ) diag p 1,..., p m, p m+1 exp µl ),..., p n exp µl } ) λ m+1 λ n { P 2 µ) diag p 1 exp µl ),..., p m exp µl } ), p m+1,..., p n λ 1 λ m Q diag{q 1,..., q l }. Exponential stability holds if there exist p i > and q j > such that the matrix M) is negative definite see Castillo et al. 212) for a related reference). A simple example of this Lyapunov approach can be found in Theorem 2.9 for the stability analysis of a density-flow system under Proportionnal-Integral control. Example: a lossless electrical line connecting an inductive power supply to a capacitive load Let us come back to the example of a lossless electrical line that we have presented in Section 2.1. We now consider the case where the line connects an inductive power supply to a capacitive load as shown in Fig.3.4. Power supply Ut) R Load Transmission line R L x L C L L Fig.3.4: Transmission line connecting an inductive power supply to a capacitive load. The dynamics of the line are described by the following system of two conservation laws: 3.35) t I + 1 L l x V =, t V + 1 C l x I =, with the differential boundary conditions: 3.36) L dit, ) dt + R It, ) + V t, ) = Ut), C L dv t, L) dt + V t, L) R L = It, L).

98 98 Chapter 3. Systems of Linear Conservation Laws For a given constant input voltage Ut) = U, the system has a unique constant steady state I U = V = R lu. R + R l R + R l The Riemann coordinates are defined as with the inverse coordinates R 1 V V ) + I I ) L l /C l, R 2 V V ) I I ) L l /C l, I = I + R 1 R 2 Ll /C l, 2 V = V + R 1 + R 2. 2 Then, expressing the dynamics 3.35) and the boundary conditions 3.36) in Riemann coordinates, we have: t R 1 + λ 1 x R 1 =, t R 2 λ 2 x R 2 =, λ 1 = λ 2 1 Ll C l, with ) X α1 1 = X 2 α 2 }{{} A ) ) R1 t, ) 1 = R 2 t, L) 1 }{{} C X1 X 2 X1 X 2 ) ) ) β1 R1 t, L) +, β 2 R 2 t, ) }{{} B ) ) ) 1 R1 t, L) +, 1 R 2 t, ) }{{} K The characteristic equation is α 1 = 1 L Cl L l + R L, α 2 = 1 β 1 = 2 L Cl L l, β 2 = 2 C L C L Ll C l. Ll C l + 1 R L C L, s + α 1 )s + α 2 ) + s + α 1 β 1 )s + α 2 β 2 ) e sτ =, τ 2L L l C l. }{{}}{{} ds) ns) In order to analyse the stability of the poles in function of the length L of the line, we follow the Walton and Marshall procedure as it is described in Silva et al., 25, Section 5.6).

99 3.4. Differential linear boundary conditions 99 The first step is to examine the stability when L = i.e. τ = ) where the characteristic equation reduces to the following second order polynomial with positive coefficients: τ = = s 2 R ) s R ) =. L R L C L L C L R L Obviously, in that case the two poles are stable. In the second step, we compute the following polynomial in ω 2 : W ω 2 ) djω)d jω) njω)n jω) = ω 2 + α 1 + α 2 )jω + α 1 α 2 ) ω 2 α 1 + α 2 )jω + α 1 α 2 ) ω 2 + α 1 + α 2 β 1 β 2 )jω + α 1 β 1 )α 2 β 2 )) ω 2 α 1 + α 2 β 1 β 2 )jω + α 1 β 1 )α 2 β 2 )) = α 1 α 2 ω 2 ) 2 + α 1 + α 2 ) 2 ω 2 α 1 β 1 )α 2 β 2 ) ω 2) 2 After a few computations, we get α 1 + α 2 β 1 β 2 ) 2 ω ) W ω 2 ) = γ 1 + γ 2 )ω 2 γ 1 γ 2 + γ 1 α γ 2 α 2 1, with γ 1 α1 2 α 1 β 1 ) 2 = 4R Cl L 2, γ 2 α2 2 α 2 β 2 ) 2 = 4 Ll L l R L CL 2. C l It follows that the sign of W ω 2 ) for large ω is positive. This means that all the system poles have strictly negative real parts for sufficiently small non-zero values of L. In the third step, we observe that the polynomial 3.37) has a single root: ω 2 = γ 1γ 2 γ 1 α 2 2 γ 2 α 2 1 γ 1 + γ 2 ) which is negative for all positive values of the physical parameters R, R L, L, L l, C L, C l. In accordance with the physical intuition, we conclude that, whatever the length of the line, the poles of the system are stable for any line length L. Example: a network of density-flow systems under PI control In Section 1.15, we have emphasized that many physical networks of interest are represented by hyperbolic systems of conservation or balance laws. In this section, we consider the special case of acyclic networks of density-flow conservation laws under PI control which is a typical example of a hyperbolic system of conservation laws with differential boundary conditions. We examine how the stability conditions of Section 2.2 can be extended. Depending on the concerned application, there are different ways of designing such networks. Here, as a matter of example, we consider a specific structure which leads to a natural generalization of Theorem 2.8. But other structures could be dealt with in a similar way, see e.g. Marigo 27) or Engel et al. 28) for relevant related references.

100 1 Chapter 3. Systems of Linear Conservation Laws The network has a compartmental structure illustrated in Fig.3.5. The nodes of the network are n storage compartments having the dynamics of density-flow systems e.g. the pipes of an hydraulic network): 3.38) { t H j + x Q j =, t Q j + λ j λ n+j x H j + λ j λ n+j ) x Q j =, j = 1,..., n. Without loss of generality and for simplicity, it can always be assumed that, by an appropriate scaling, all the systems have exactly the same length L. U 2 2 D 2 1 D 1 U 1 U 5 x 3 U 3 5 x D 5 D 3 U 4 4 x D 4 Fig.3.5: Physical network of density-flow systems The directed arcs i j of the network represent instantaneous transfer flows between the compartments. Additional input and output arcs represent interactions with the surroundings: either inflows injected from the outside into some compartments or outflows from some compartments to the outside. We assume that there is exactly one and only one control flow, denoted U i, at the input of each compartment. All the other flows are assumed to be disturbances and denoted D k k = 1,..., m). The set of 2n PDEs 3.38) is therefore subject to 2n boundary flow balance conditions of the form: 3.39) m Q i t, ) = U i t) + β ik D k t), i = 1,..., n, k=1 n m Q i t, L) = α ij U j t) + γ ik D k t), i = 1,..., n. j=1 k=1 In the summations, only the terms corresponding to actual links between adjacent compartments of the network are taken into account, i.e. the coefficients β ik and γ ik are equal to 1 for the existing links and for the others see Fig.3.5 for illustration). With the matrix notations H H 1. H n, Q Q 1. Q n, U U 1. U n, D D 1. D m, Λ + = diag{λ 1,..., λ n }, Λ = diag{λ n+1,..., λ 2n },

101 3.4. Differential linear boundary conditions 11 the system 3.38) is written 3.4) t H + x Q =, t Q + Λ + Λ x H + Λ + Λ ) x Q =. The boundary conditions 3.39) are written 3.41) Qt, ) = Ut) + B Dt), Qt, L) = A L Ut) + B L Dt), where A L, B and B L are the matrices with entries α ij, β jk and γ ik respectively. For example, in the network of Fig.3.5, we have A L 1, B 1, B 1 L Since the network is acyclic, the nodes of the network can be numbered such that the square matrix A L is strictly upper triangular. Therefore the matrix A L has the property that 3.42) A p L =, where p is the length of the longest path in the network. A steady state for the system 3.4), 3.41) is a quadruple which satisfies the boundary conditions: {H, Q, U, D } Q = U + B D, Q = A L U + B L D. The network has an infinity of positive steady states which are not asymptotically stable. In order to stabilize the network, each control input is endowed with a PI control law of the form: t 3.43) U i t) U R + k P i Hi H i t, )) + k Ii Hi H i τ, ))dτ, where U R is an arbitrary scaling constant, H i is the set point for the ith compartment, k P i and k Ii are the control tuning parameters. In matrix form, the set of control laws 3.43) is written ) t 3.44) U = U R + K P H Ht, ) + K I ) H Hτ, ) dτ,

102 12 Chapter 3. Systems of Linear Conservation Laws with K P diag{k P 1,... k P n } and K I diag{k I1,... k In }. We shall now examine how the stability analysis in the frequency domain can be generalized to the closed-loop system 3.4), 3.41), 3.44) for constant unknown disturbances D. The Riemann coordinates are defined as follows: { Ri Q i Q i + λ n+i H i Hi ), i = 1,..., n. R n+i Q i Q i λ i H i Hi ), Using this definition, we have the following system dynamics { t R i + λ i x R i =, i = 1,..., n, t R n+i λ n+i x R n+i =, and the following equalities hold at the boundaries: [ ] λ i + λ n+i )Q i t, ) Q i ) = λ i + λ n+i ) k Pi Hi H i t, )) + k Ii Z i t) = λ i R i t, ) + λ n+i R n+i t, ) = k Pi R n+i t, ) R i t, )) + λ i + λ n+i )k Ii Z i t), λ i + λ n+i )Q i t, L) Q i ) = λ i R i t, L) + λ n+i R n+i t, L), with Z i t) such that dz i dt = H i H i t, ) = R n+it, ) R i t, ) λ i + λ n+i. Since R i t, x) and R n+i t, x) are constant along their respective characteristic lines, we have that R i t + L λ i, L) = R i t, ) and R n+i t + L λ n+i, ) = R n+i t, L). Then, combining appropriately these equalities, it can be shown after some computations that, in the frequency domain, the transfer function between Q i t, L) Q i ) and Q i t, ) Q i ) is given by: G i s) Q is, ) Q i Q i s, L) Q i with the following notations: = 1 sλ i k i λ n+i ) + c i λ i λ n+i ) λ n+i e sτi k i k n+i )s + c i e sτi k n+i ) e sl λ i, k i k P i λ n+i k P i + λ i, k n+i λ i λ n+i, c i k Ii, τ i L + L. k P i + λ i λ i λ n+i It follows that the poles of each transfer function G i s) are the roots of the characteristic equation e sτi k i k n+i )s + c i e sτi k n+i ) =,

103 3.4. Differential linear boundary conditions 13 which is, as expected, identical to the characteristic equation of the simple case of Section 2.2. Let us now consider the closed-loop system 3.4), 3.41), 3.44) as an input-output dynamical system with input D and output U. Then, by iterating equations 3.41) p times and using property 3.42), it can be shown that the transfer matrix of the system is as follows: p 1 Hs) Gs)A L ) i Gs)B L B ), i= with Gs) diag{g 1 s),..., G n s)}. For example, in the network of Fig.3.5, since p = 3, we have Hs) = I + Gs)A L + Gs)A L Gs)A L )Gs)B L B ) G 1 G 1 G 3 G 1 G 3 G 1 G 4 G 1 G 3 G 5 1 G 2 = G 3 G 3 G 3 G 5 G 4. 1 G 5 As illustrated with this example, the poles of Hs) are given by the collection of the poles of the individual scalar transfer function G i s). Consequently, the system is stable if and only if the conditions of Theorem 2.8 hold for each PI controller of the network, i.e. if and only if for i = 1,..., n, when λ i < λ n+i, k P i > and k Ii > or k P i < 2λ iλ n+i λ n+i λ i and k Ii < ; when λ i = λ n+i, k P i > and k Ii > ; when λ i > λ n+i, < k Pi < 2λ iλ n+i 2k P + λ i λ n+i )λ i λ n+i and < k Ii < ω i λ i λ n+i λ 2 i sinω i τ i ) λ2 n+i where ω i is the smallest positive ω such that cosωτ i ) = λ2 n+i k P i + λ i ) + λ 2 i k P i λ n+i ). λ i λ n+i λ n+i λ i 2k P i ) Example: stability of genetic regulatory networks In Section 1.15 p.54), we have shown how the dynamics of genetic regulatory networks are represented by linear hyperbolic systems with nonlinear differential boundary conditions of the form 3.45) drt, ) dt R t + Λ R R x =, S t + Λ S S x =, t [, + ), x [, 1], = ASt, 1) BRt, ), dst, ) dt = GRt, 1))Rt, 1) DSt, )

104 14 Chapter 3. Systems of Linear Conservation Laws with Λ R D n +, Λ S D n +, A D n +, B D n +, D D n +, and G : R n + M n,n R) is a smooth bounded function of R representing genetic activations or repressions with R n + being the orthant {R R n ; R i > Pi, i = 1, n}. For this system, we see that the solutions are confined in the orthant R n + S n + with S n + {S R n ; S i > Mi, i = 1, n}. This means that, in accordance with the physical reality, if the initial conditions R, x) and S, x) are in the orthant R n + S n +, the solutions of the system are guaranteed to stay in the same orthant for all time. The stability of the system 3.45) is analyzed with the following Lyapunov function candidate: V = 1 [ ] R T t, x)q R T R E R µ, x)rt, x) + S T t, x)q S T S E S µ, x)st, x) dx RT t, 1)W R Rt, 1) ST t, 1)W S St, 1), Q R = diag{q 1,..., q n }, Q S = diag{q n+1,..., q 2n }, T R = diag{τ 1,..., τ n }, T S = diag{τ n+1,..., τ 2n }, E R µ, x) = diag { e µτ1x,..., e µτnx}, E S µ, x) = diag { e µτn+1x,..., e µτ2nx}, W R = diag{w 1,..., w n }, W S = diag{w n+1,..., w 2n }. Using integration by parts, it can be shown that the time derivative of V, along the system solutions, is Rt, ) dv dt = µv R T t, ) S T t, ) R T t, 1) S T t, 1) ) Mµ, R) St, ) Rt, 1), St, 1) with the matrix Mµ, R) defined as W R B Q R W R A Mµ, R) = We have the following proposition. W S D Q S W S GR) Q R E R µ, 1) Q S E S µ, 1) Proposition There exists µ > sufficiently small such that V is a strict exponentially decreasing Lyapunov function along the solutions of the system 3.45) if there exist q i >, q n+i >, w i >, w n+i > and δ > such that M, R) + M T, R) > δi n for all R R n +. Let us now use the toggle switch as an example of how this proposition can be used. A toggle switch is a system of two genes that repress each other see e.g. Smits et al. 28)

105 3.4. Differential linear boundary conditions 15 and the references therein). In the case of the toggle switch, the general model 3.45) is specialized as follows: t R 1 t, x) + 1 τ 1 x R 1 t, x) =, t R 2 t, x) + 1 τ 2 x R 2 t, x) =, t S 1 t, x) + 1 τ 3 x S 1 t, x) = t S 2 t, x) + 1 τ 4 x S 2 t, x) =, dr 1 t, ) dt dr 2 t, ) dt ds 1 t, ) dt ds 2 t, ) dt For this system, the matrix M, R) is = α 1 S 1 t, 1) β 1 R 1 t, ), = α 2 S 2 t, 1) β 2 R 2 t, ), = g 1 R 2 t, 1))R 2 t, 1) δ 1 S 1 t, ), = g 2 R 1 t, 1))R 1 t, 1) δ 2 S 2 t, ). ) M11 M 12 M, R) =, M 22 with M 11 = M 12 = w 1 β 1 q 1 w 2 β 2 q 2 w 3 δ 1 q 3, w 4 δ 2 q 4 w 1 α 1 w 2 α 2 w 3 g 1 R 2 ), w 4 g 2 R 1 ) q 1 M 22 = q 2 q 3. q 4 This matrix is positive definite if and only if the leading principal minors of the symmetric

106 16 Chapter 3. Systems of Linear Conservation Laws matrix M, R) + M T, R) are all positive. This leads to the following inequalities: < w 1 β 1 q 1 < w 2 β 2 q 2 < w 3 δ 1 q 3 < w 4 δ 2 q 4 < w 4 δ 2 q 4 )q 1 1/4)w 2 4g 2 2R 1 ) < w 3 δ 1 q 3 )q 2 1/4)w 2 3g 2 1R 2 ) < w 1 β 1 q 1 )q 3 1/4)w 2 1α 2 1 < w 2 β 2 q 2 )q 4 1/4)w 2 2α 2 2. Hence, the system is stable for any τ i if there exist positive values of q i and w i such that these inequalities are satisfied for all R 1, R 2 ) in R The non uniform case In this section, we explain how the previous results of this chapter are trivially extended to the non-uniform case where the characteristic velocities λ i x) depend of the spatial coordinate. We consider the linear hyperbolic system: 3.46) R t + Λx)R x = with the diagonal matrix Λx) diag { Λ + x), Λ x) } such that Λ + x) = diag{λ 1 x),..., λ m x)}, Λ x) = diag{λ m+1 x),..., λ n x)}, λ i x) > i, x [, L]. Our concern is to analyze the exponential stability of this system under linear boundary conditions in canonical form ) R + t, ) ) R + t, L) 3.47) R t, L) = K R t, ), t [, + ), and an initial condition 3.48) R, x) = R o x), x, L). The well-posedness of the Cauchy problem 3.46), 3.47), 3.48) in L 2 results, as a special case, from Theorem A.4 in Appendix A. We have the following stability theorem. Theorem The system 3.46), 3.47) is exponentially stable for the L 2 norm in the sense of Definition 3.1) if ρ 2 K) < 1. Proof. We use the following candidate Lyapunov function which is a direct extension of the function used in Section 2.3: V = [ R +T Λ + ) 1 P + µx)r + ) + R T Λ ) 1 P µx)r ) ] dx

107 3.6. Switching linear conservation laws 17 where P + µx) diag { p1 x ) µ λ 1 σ) dσ,..., λ 1 x) exp { x P pm+1 µx) diag λ m+1 x) exp µ λ 1 σ) dσ ),..., p x m λ m x) exp x p n λ n x) exp )} µ λ m σ) dσ, µ λ n σ) dσ )}, with positive coefficients µ and p i, i = 1,..., n. With this definition of the Lyapunov function, the proof of the theorem is a direct extension of the proof of Theorem 2.11 which can be written as a replicate of the proof of Theorem Switching linear conservation laws In certain practical applications, it is of interest to address situations where the system exhibits periodic time switching between various sets of boundary conditions. From the viewpoint of exponential stability analysis, a system of conservation laws with switching boundary conditions can be viewed as a hybrid system on an infinite dimensional state space. While hybrid systems based on ordinary differential equations are extensively considered in the literature e.g. Liberzon 23), De Schutter and Heemels 211), Shorten et al. 27) ), hybrid systems based on partial differential equations are relatively unexplored. In this section, through the specific example of SMB chromatography, our purpose is to illustrate how exponential stability in L 2 norm) can be established by switching between Lyapunov functions. The obtained stability conditions are direct generalizations of the corresponding results for the unswitched case. Other interesting references on the stability analysis for linear hyperbolic switching systems can be found in Sections 3.7 and 5.7. The example of SMB chromatography As described in Section 1.13, SMB chromatography is a technology where interconnected chromatographic columns are switched periodically. The SMB chromatography model is given by equations 1.61). It exhibits a periodic steady state denoted C t, x). Here we consider the linear case which is the special case where b A = and b B =. The linear system is therefore written: mt t < m + 1)T, m =, 1, 2, 3, 4, 5,...,, l {A, B}, 3.49) 1 + h l ) t C l + P m )ΥP m ) T x C l =, C l t, ) = P m )KP m ) T C l t, L) + P m )U l. In order to put the system in characteristic form, we define the Riemann coordinates: R A i = 1 + h A )C A i C A i ), R B i = 1 + h B )C B i C B i ), i = 1, 2, 3, 4. In these Riemann coordinates, the periodic linear system is written see Section 1.13) mt t < m + 1)T, m =, 1, 2, 3, 4, 5,...,, l {A, B},

108 18 Chapter 3. Systems of Linear Conservation Laws 3.5) t R l + Λ l m x R l =, R l t, ) = K m R l t, L), with the following notations: We have the following stability property. Λ l diag{λ l 1, λ l 2, λ l 1, λ l 2} with λ l 1 V I 1 + h l, λ l 2 V II 1 + h l, Λ l m P m )Λ l P m ) T, K m = P m )KP m ) T. Theorem The periodic solution C t, x) of the system 3.5) is exponentially stable if T > L λ l 2 L λ l, l {A, B}. 1 Proof. As advocated in Branicky 1998) for the analysis of hybrid systems, we follow a socalled multiple Lyapunov function approach with the two following candidate quadratic Lyapunov functions: V 1 V 2 l {A,B} l {A,B} { p1 λ l 1 { p2 [R l 1 t, x) ] 2 [ + R l 3 t, x) ] ) 2 exp µx ) λ l 1 + p 2 [R l λ l 2 t, x) ] 2 [ + R l 4 t, x) ] ) 2 exp µx 2 λ l 2 λ l 2 [R l 1 t, x) ] 2 [ + R l 3 t, x) ] ) 2 exp µx ) λ l 2 + p 1 [R l λ l 2 t, x) ] 2 [ + R l 4 t, x) ] ) 2 exp µx 1 λ l 1 )} dx, )} dx, with positive constant coefficients p 1, p 2 and µ. Now, until the end of the proof, we consider only even values of m: m {, 2, 4,... }. The time derivatives of V 1 and V 2 along the trajectories of the system 3.5) are For mt t < m + 1)T, dv 1 dt = µv 1 l {A,B} l {A,B} { { p 1 exp p 2 exp µl ) ) [R l λ l p 2 1 t, L) ] 2 [ + R l 3 t, L) ] )} 2 1 µl ) λ l 2 λ l 2 λ l 1 ) 2 ) [R l p 1 2t, L) ] 2 [ + R l 4 t, L) ] )} 2 For m + 1)T t < m + 2)T,

109 3.6. Switching linear conservation laws 19 dv 2 dt = µv 2 l {A,B} l {A,B} { { p 1 exp p 2 exp µl ) ) [R l λ l p 2 2 t, L) ] 2 [ + R l 4 t, L) ] )} 2 1 µl ) λ l 2 λ l 2 λ l 1 ) 2 ) [R l p 1 1t, L) ] 2 [ + R l 3 t, L) ] )} 2. Since V I > V II see Section 1.13), the parameters p 1 and p 2 can be selected such that 1 < p 1 p 2 < ) λ l 2 1 = λ l 2 VI V II ) 2, then µ > can be selected such that 3.51) p 1 exp µl ) p 2 λ l > 1, 1 p 2 p 1 λ l 1 λ l 2 ) 2 exp µl ) λ l > 1, 2 which imply mt t < m + 1)T, 3.52) dv 1 dt µv 1 and therefore V 1 m + 1)T ) V 1 mt )e µt, 3.53) m + 1)T t < m + 2)T, dv 2 dt Let us select p 1 /p 2 as follows: 3.54) µv 2 and therefore V 2 m + 2)T ) V 2 m + 1)T )e µt. p 1 p 2 = V I V II > 1. Let us define a parameter α > 1 selected such that 1 L 3.55) T λ l L ) 2 λ l < ln α 1 µt < 1. Using inequalities 3.54) and 3.55), we then have, for every x [, L], exp µl ) p 1 λ l 1 λ α exp µl ) l exp µx ) 1 λ l 1 p 2 λ l 2 λ l exp µx ) 1. 2 λ l 2 By combining this inequality with the definitions of V 1 and V 2, it can be checked that 3.56) From 3.52), 3.53) and 3.56), we then have 1 α V 2 V 1 αv 2, t, x. V 1 m + 2)T ) αv 2 m + 2)T ) αe µt V 2 m + 1)T )

110 11 Chapter 3. Systems of Linear Conservation Laws Mutatis mutandis, obviously we also have α 2 e µt V 1 m + 1)T ) αe µt ) 2 V1 mt ). V 2 m + 3)T ) αe µt ) 2 V2 m + 1)T ). Now, from 3.55) we have: αe µt < 1. Therefore, V 1 t) and V 2 t) exponentially converge to zero and the periodic time solution C is exponentially stable. A simulation experiment As a matter of illustration, we present a simulation experiment of a SMB process implemented under the operating conditions reported in Nobre et al. 213) for the separation of fructo-oligosaccharides. Fig.3.6: Time evolution of the concentrations inside column 1 : exponential convergence towards the periodic regime. The parameter values are L =.248 m, T = s, V I =.36 m/s, V II =.22 m/s, h A =.3954, h B =.251, CF A = 64 mg/ml, CF B = 85 mg/ml, V F = V I V II =.14 m/s. From these values, we verify that the stability condition of Theorem 3.17 is satisfied since T > L λ A 2 L λ B 1 56 s > L λ B 2 L λ B 1 44 s. We simulate the start-up of the process from zero initial conditions i.e. zero initial concentrations in the columns). The simulation results are shown in Fig.3.6 and Fig.3.7. We

111 3.7. References and further reading 111 Fig.3.7: Time evolution of the outlet concentrations dotted line: actual concentration, solid line: average concentration). see in Fig.3.6 that the steady state periodic regime is reached within about 1 column shifts i.e. 2.5 rounds). It may also be observed in Fig.3.7 that the separation between species A and B is effective but not perfect. This is an inherent limitation of SMB processes implemented with four columns as considered here for simplicity. In order to reach a total purity of the separation, industrial SMB processes are generally implemented with eight e.g. Suvarov et al. 212) ) or even twelve columns e.g. Lorenz et al. 21) ) References and further reading Tang et al. 215b) address the issue of singular perturbations in linear systems of conservation laws. For systems having a slow-fast behavior, they show that the stability in L 2 norm of the full system implies necessarily the stability for the L 2 norm of both the slow reduced system and the fast boundary-layer system, while the converse is not true. Furthermore, they establish a generalization of Tikhonov s theorem for this class of infinite-dimensional systems. An application to the control of gas transport systems described by Euler equations where the small parameter is the ratio between gas and sound velocities illustrates the theory. In Section 3.4, we have illustrated the stability of linear hyperbolic systems under differential boundary condition with the example of a power source connected to a load through a lossless electrical line. In Daafouz et al. 214), the authors address the nonlinear control of this system in the case where the source is a switched power converter. In Section 3.4, we have given the necessary and sufficient stability conditions for densityflow systems under PI control. Related references dealing with PI control of hyperbolic systems are Xu and Sallet 1999), Dos Santos et al. 28), Dos Santos Martins and Rodrigues 211) where sufficient stability conditions are given using respectively spectral,

112 112 Chapter 3. Systems of Linear Conservation Laws Lyapunov and LMI approaches. Experimental validations on a mini-channel set-up are also reported in Dos Santos et al. 28). In Section 3.6, we have presented a simple example of the stability analysis under switching boundary conditions. The exponential stability in L 2 norm for a class of switched linear systems of conservation laws is further investigated by Lamare et al. 213) and Lamare et al. 215b) in the case where the state equations and the boundary conditions are both subject to switching. The authors consider the problem of synthesizing stabilizing switching controllers. By means of Lyapunov techniques, three control strategies are developed based on steepest descent selection, possibly combined with a hysteresis and a low-pass filter.

113 Chapter 4 Systems of Nonlinear Conservation Laws THE PURPOSE of this chapter is to extend the exponential stability analysis to the case of systems of nonlinear conservation laws of the form 4.1) Y t + fy) x =, t [, + ), x [, L], where Y : [, + ) [, L] R n and f C 2 Y; R n ). Since the frequency domain is irrelevant for the nonlinear case, the main concern will be to examine how the Lyapunov approach is extended to nonlinear hyperbolic systems. For a nonlinear differential system, the well known Lyapunov s indirect method allows to deduce the local exponential stability of an equilibrium from the exponential stability of its linearization with the same Lyapunov function, see for instance Khalil, 1996, Chapter 3). In contrast, for hyperbolic systems, such an implication does not hold: the exponential stability of the linearization for a given norm does not directly induce the local exponential stability of the underlying nonlinear system for the same norm and with the same Lyapunov function. The analysis is more intricate as we shall see in the present chapter. An additional difficulty comes from the fact that, even for smooth initial conditions that are close to the steady-state, the trajectories of nonlinear conservation laws may become discontinuous in finite time generating jump discontinuities which propagate on as shocks. Fortunately, the main result of this chapter will be to show that, if the boundary conditions are dissipative and if the initial conditions are smooth enough and sufficiently close to the steady state, the system trajectories are guaranteed to remain smooth i.e. without schocks) for all time and that they exponentially converge locally to the steady state. Surprisingly enough, due to the nonlinearity of the system, even for smooth solutions, it will appear that the exponential stability strongly depends on the considered norm. In particular, using the Lyapunov approach, it is shown in Section 4.4 that the robust dissipativity test ρ 2 K) < 1 of linear systems holds also in the nonlinear case for the H 2 norm, while, in the next section, we shall see that a more conservative test is needed for solutions in C 1. For smooth solutions, the system of conservation laws 4.1) is equivalent to the quasilinear system 4.2) Y t + F Y)Y x =, t [, + ), x [, L], where F C 1 Y; M n,n R)) is the Jacobian matrix of f. Our main concern in this chapter is to address the exponential stability of the steady states of this quasi-linear system. For simplicity, we start the analysis with the special case of systems that can be written into 113

114 114 Chapter 4. Systems of Nonlinear Conservation Laws characteristic form. The case of generic systems 4.2) that cannot be transformed into characteristic form is treated at the end of the chapter in Section 4.5. The systems under consideration have the following form in Riemann coordinates: 4.3) R t + ΛR)R x =, t [, + ), x, L), where R : [, + ) [, L] R n. The map Λ is defined as: ) Λ + R) ΛR) Λ R) with { Λ + R) = diag{λ 1 R),..., λ m R)}, Λ R) = diag{λ m+1 R),..., λ n R)}, λ i R) >, i {1,..., n} and R Y. We assume that, for some σ >, the map Λ : B σ D n is of class C 1, with B σ an open ball of radius σ in R n. Hence there exist functions λ i C B σ ; R n ) such that 4.4) λ i R) λ i ) + λ i R)R. Our purpose is to analyze the exponential stability of the steady state Rt, x) under nonlinear boundary conditions in nominal form ) ) 4.5) R + t, ) R t, L) = H R + t, L) R t, ), t [, + ), where H C 1 B σ ; R n ), H) = and under an initial condition 4.6) R, x) = R o x), x [, L], which satisfies the following compatibility conditions: ) ) 4.7) R + o ) R o L) = H R + o L) R o ), 4.8) Λ + R o )) x R + o ) Λ R o L)) x R o L) ) ) R + = H o L) Λ + R o L)) x R + ) o L) Ro ) Λ R o )) x Ro ) where H denotes the Jacobian matrix of the map H Dissipative boundary conditions for the C 1 -norm In this section, our main result is to show how the exponential stability of systems of nonlinear conservation laws of the form 4.3), 4.5) can be established for the C 1 norm.

115 4.1. Dissipative boundary conditions for the C 1 -norm 115 Let us first introduce some norm notations that will be useful to state and prove the result. ξ ξ 1,..., ξ n ) T R n, ξ max{ ξ j ; j {1,..., n}}. For f C [, L]; R n ) resp. in C [T 1, T 2 ] [, L]; R n ), we denote 4.9) 4.1) f max{ fx) ; x [, L]}, resp. f max{ ft, x) ; t [T 1, T 2 ], x [, L]}). For f C 1 [, L]; R n ) resp. in C 1 [T 1, T 2 ] [, L]; R n )), we denote 4.11) 4.12) f 1 f + f, resp. f 1 f + t f + x f ). The well-posedness of the Cauchy problem and the existence of a unique C 1 solution result from the following theorem. Theorem 4.1. Let T >. There exist C 1 > and ε 1 > such that, for every R o C 1 [, L]; R n ) satisfying the compatibility conditions 4.7), 4.8) and such that 4.13) R o 1 ε 1, the Cauchy problem 4.3), 4.5) and 4.6) has one and only one solution Moreover, this solutions satisfies R C 1 [, T ] [, L]; R n ) 4.14) R 1 C 1 R o 1. Proof. A proof of this theorem was given by Li and Yu 1985) for the case of local boundary conditions of the form R + t, ) = H + R t, )), R t, L) = H R + t, L)). The general case can be reduced to this particular case by using the dummy doubling of the system size introduced in de Halleux et al. 23): see Li et al. 21) for further details See also Wang 26) for a generalization to the case of a nonautonomous coefficient matrix ΛR, t, x) depending explicitly on x and t. The definition of exponential stability is as follows. Definition 4.2. The steady state Rt, x) of the system 4.3), 4.5) is exponentially stable for the C 1 norm if there exist ε >, ν > and C > such that, for every R o in the set V {R C 1 [, L]; R n ) : R 1 < ε} and satisfying the compatibility conditions 4.7) and 4.8), the C 1 solution of the Cauchy problem 4.3), 4.5), 4.6) satisfies Rt,.) 1 Ce νt R o 1, t [, + ).

116 116 Chapter 4. Systems of Nonlinear Conservation Laws In order to state the next stability theorem, we define the matrix K as the linearization of the map H at the steady state: K H ) M n,n R) and we recall the definition of the function ρ see 3.6)) which can be stated as follows: where 4.15) R K) max{ ρ K) inf{r K 1 ); D + n } n k ij ; i {1,..., n}} j=1 and k ij denotes the i, j)th entry of the matrix K. Note that, by Li, 1994, Lemma 2.4, page 146), ρ K) is the spectral radius ρ of the matrix K [ k ij ]: 4.16) ρ K) = ρ K ). We have the following theorem. Theorem 4.3. If ρ K) < 1, the steady state Rt, x) of the system 4.3), 4.5) is exponentially stable for the C 1 norm. From now on, in this section, R : [, T ] [, L] R n denotes a C 1 solution of the system 4.3), 4.5). Let W 1 and W 2 be defined by 4.17) W ) W 2 [ m i=1 p p i R2p i e 2pµx + [ m p p i tr i ) 2p e 2pµx + i=1 n i=m+1 n i=m+1 p p i R2p 1 2p ] i e 2pµx dx ] p p i tr i ) 2p e 2pµx dx with p N + and p i > i {1,..., n}. The proof of Theorem 4.3 will be based on two preliminary lemmas. The first lemma provides an estimate of dw 1 /dt) along the solutions of the system 4.3), 4.5). Lemma 4.4. If ρ K) < 1, there exist p i > i {1,..., n}, α >, β 1 > and µ 1 > such that, for every µ, µ 1 ), for every p 1/µ 1, + ), for every solution of the system 4.3), 4.5) satisfying R < µ 1, 4.19) dw 1 dt µα 1 + β 1 R t ) W 1. Proof. Along the solutions the system 4.3), 4.5), the derivative of W 1 is:, 1 2p, dw 1 dt = 1 2p W1 2p 1 [ m i=1 p p i 2pR2p 1 i t R i e 2pµx

117 4.1. Dissipative boundary conditions for the C 1 -norm 117 Hence, using 4.4), dw 1 dt = 1 2p W1 2p 1 = 1 2p W1 2p 1 m i=1 [ [ m i=1 p p i 2pR2p 1 i + n i=m+1 m i=1 + n i=m+1 p p i 2pR2p 1 i t R i e 2pµx ] dx p p i 2pR2p 1 i λ i R) x R i )e 2pµx + n i=m+1 p p i λ i) x R 2p i )e 2pµx + λi R)R) x R i e 2pµx p p i 2pR2p 1 i Then, using integration by parts, we get p p i 2pR2p 1 i λ i R) x R i )e 2pµx ] dx. n i=m+1 λi R)R) x R i e 2pµx ]dx p p i λ i) x R 2p i )e 2pµx with [ T 1 W1 2p 1 2p m i=1 dw 1 dt = T 1 + T 2 + T 3 p p i λ i)r 2p i e 2pµx + n i=m+1 ] L p p i λ i)r 2p i e 2pµx, T 2 µw 1 2p 1 T 3 W 1 2p 1 [ m i=1 [ m i=1 p p i λ i)r 2p i e 2pµx + n i=m+1 p p i λ i R) 2pR2p 1 i λi R)R) t R i e 2pµx + n i=m+1 p p i λ i)r 2p i e 2pµx ]dx, p p i λ i R) 2pR2p 1 i λi R)R) t R i e ]dx. 2pµx Analysis of the first term T 1. Since ρ K) < 1, there exists diag{δ 1,..., δ n } D + n such that 4.2) R K 1 ) < 1. The parameters p i are selected such that 4.21) p p i λ i) = δ 2p i, i = 1,..., n.

118 118 Chapter 4. Systems of Nonlinear Conservation Laws Then, using the boundary condition 4.5), T 1 may be written [ T 1 = W1 2p m 1 δ 2p i R 2p i L)e 2pµL + 2p i=1 m i=1 δ 2p i n i=m+1 m k ij R j L) + j=1 n i=m+1 δ 2p i δ 2p i R 2p i ) n j=m+1 m k ij R j L) + j=1 k ij R j ) n j=m+1 ) 2p k ij R j ) ) 2p e 2pµL ] where we use the simplified notations R i ) R i t, ) and R i L) R i t, L). We define ξ i, i = 1,..., n such that ξ i δ i R i L) for i = 1,..., m and ξ i δ i R i ) for i = m + 1,..., n. Then, for µ =, the term between brackets in T 1 is written n i=1 ξ 2p i Let us now introduce ξ max such that n n ) 2p δ i k ij ξ j. δ j i=1 j=1 ξ 2 max = max{ξ 2 i, i = 1,..., n}. and therefore ξ 2p max n i=1 ξ 2p i nξ 2p max. Then, we have n j=1 and consequently, n i=1 ξ 2p i ) 2p δ i k ij ξ j δ j n n i=1 j=1 n j=1 k ij δ i δ j ξ j ) 2p k ij δ i ξ j δ j n j=1 k ij δ i δ j ) 2p ξ 2p max, ) 2p ξmax 2p 1 n R K 1 ) ) ) 2p. Now sign 1 n R K 1 ) ) ) ) 2p = sign 1 n 1/2p) R K 1 ). Then, using 4.2), we easily check by continuity that there exists µ 11, σ] such that µ, µ 11 ), p 1/µ 11, + ) and R, T 1 if R < µ 11.

119 4.1. Dissipative boundary conditions for the C 1 -norm 119 Analysis of the second term T 2. Defining 4.22) α 1 2 minλ 1),..., λ n )), there is µ 12, σ] such that µ, µ 12 ), p 1/µ 12, + ) and R, T 2 µαw 1 if R < µ 12. Analysis of the third term T 3. The integrand of T 3 is linear with respect to R t and of order 2p, at least, with respect to R. It follows that there exist β 1 > and µ 13, σ] such that µ, µ 13 ), p 1/µ 13, + ) and R T 3 β 1 R t W 1 if R < µ 13. Hence, with µ 1 min{µ 11, µ 12, µ 13 }, we conclude that dw 1 dt µαw 1 + β 1 R t W 1 for all µ, µ 1 ), for all p 1/µ 1, + ) and for all R such that R completes the proof of Lemma 4.4. < µ 1. This By time differentiation of the system equations 4.3), 4.5), R t can be shown to satisfy the following hyperbolic dynamics: 4.23) 4.24) R tt + ΛR)R tx + diag[λ R)R t ]R x =, ) [ )] R + t, ) R + t, L) t = R t H, t, L) R t, ) where Λ R) denotes the matrix with entries and [Λ R)] i,j λ i R j diag[λ R)R t ] diag{λ 1R)R t,..., λ mr)r t, λ m+1r)r t,..., λ nr)r t }. The next lemma provides an estimate of the functional dw 2 /dt) along the solutions of the system 4.3), 4.5), 4.23), 4.24). Lemma 4.5. If ρ K) < 1 and p i i = 1,..., n) are given by 4.21), there exist β 2 > and µ 2 > such that, for every µ, µ 2 ), for every p 1/µ 2, + ), for every solution of the system 4.3), 4.5), 4.23), 4.24), satisfying R < µ 2, 4.25) dw 2 dt with α defined by 4.22). µα + β 2 R t ) W 2, in the distribution sense on, T ),

120 12 Chapter 4. Systems of Nonlinear Conservation Laws Since R is only of class C 1, we remark that W 2 is only continuous and 4.25) has to be understood in the distribution sense on, T ), in contrast with inequality 4.19) that holds pointwise at every time t [, T ]. Proof. In order to prove this lemma, we temporarily assume that R is of class C 2 on [, T ] [, L]. The assumption will be relaxed at the end of the proof. Then along the solutions of the Cauchy problem 4.3) and 4.5), the derivative of W 2 can be computed as dw 2 dt = 1 2p W1 2p 2 [ m p p i 2p tr i ) 2p 1 tt R i e 2pµx i=1 + n i=m+1 p p i 2p tr i ) 2p 1 tt R i e 2pµx ] dx. Using 4.3) and 4.23), we have dw 2 dt = 1 2p W1 2p 2 + [ m p p i 2p tr i ) 2p 1 i=1 n i=m+1 p p i 2p tr i ) 2p 1 λ i R) t R i ) x + λ i R)R t λ i R) tr i λ i R) t R i ) x + λ i R)R t λ i R) tr i ) e 2pµx )e 2pµx ] dx. Integrating by parts, we obtain dw 2 dt = U 1 + U 2 + U 3 with [ U 1 1 2p W1 2p 2 U 2 µw 1 2p 2 U 3 W 1 2p 2 n p p i λ ir) t R i ) 2p e 2pµx + i=1 [ m p p i λ i) t R i ) 2p e 2pµx + i=1 n i=1 p p i λ ir) t R i ) 2p e 2pµx]L n i=m+1 [ m p p i tr i ) 2p λ ir)l i R)R t e 2pµx i=1 + n i=m+1 p p i λ i) t R i ) 2p e 2pµx ] dx, p p i tr i ) 2p λ ir)l i R)R t e 2pµx ]dx,, with L i R) 1 2p diag { 2pλ 1 i R) λ 1 1 R),..., 2pλ 1 i R) λ 1 n R) }.

121 4.1. Dissipative boundary conditions for the C 1 -norm 121 Analysis of the first term U 1. In a way similar to the analysis of T 1 in the proof of Lemma 4.4, it can be shown that, since ρ K) < 1, there exists µ 21, η] sufficiently small such that U 1 for all µ, µ 21 ) and for all p 1/µ 21, + ) if R < µ 21. Analysis of the second term U 2. There is µ 22, η] such that µ, µ 22 ), p 1/µ 22, + ), U 2 µαw 2. Analysis of the third term U 3. The integrand of U 3 is of order 2p + 1 with respect to R t and it is easily checked that there exist β 2 > and µ 23, η] such that µ, µ 23 ), p 1/µ 23, + ), U 3 β 2 R t W 2. if R < µ 23. Hence, with µ 2 min{µ 21, µ 22, µ 23 }, we conclude that 4.26) dw 2 dt µαw 2 + β 2 R t W 2 for all µ, µ 2 ), for all p 1/µ 2, + ) and for all R such that R < µ 2. The above estimate of dw 2 /dt has been obtained under the assumption that R is of class C 2. But the proof shows that the selection of µ, α and β 2 does not depend on the C 2 norm of R. Hence the estimate 4.26) remains valid, in the distribution sense, with R only of class C 1 as motivated in the following comment. Comment 4.6. In this comment we explain why the estimate 4.26) is valid, in the distribution sense, when R is only of class C 1. Let Λ k : R n D + n and H k : R n R n, k N, of class C 2 be such that 4.27) 4.28) H k Λ k Λ in C 1 B σ ; D n + ), k H in C 1 B σ ; R n ), H k ) =. k Let Ro k C 2 [, L]; R n ) k N be a sequence of functions which satisfy the boundary conditions of order 2 as they are defined in page 144, such that Ro k converges to R o in the C 1 norm when k. Let R k : [, T ] [, L] R n be the solutions of the Cauchy problem 4.29) 4.3) Rt k + Λ k R k )Rx k =, t [, + ), x, L), ) ) R k,+ t, ) R k,+ t, L) = H k, R k, t, L) R k, t, ) for initial data R k o. We know that, for k large enough, R k exists, is of class C 2 and that 4.31) R k R in C 1 [, T ] [, L]; R n ) as k.

122 122 Chapter 4. Systems of Nonlinear Conservation Laws Now, defining W 2,k [ m p p i tri k ) 2p e 2pµx + i=1 n i=m+1 we have, if R < µ 2 and for k sufficiently large, that 4.32) dw 2,k dt µαw 2,k + β 2 R k t W 2,k. Letting k in 4.32) and using 4.31), we get 1 2p ] p p i tri k ) 2p e 2pµx dx, dw 2 dt µαw 2 + β 2 R t W 2 This completes the proof of Lemma 4.5. in the sense of distributions. Proof of Theorem 4.3. Let us choose µ R such that 4.33) < µ < min {µ 1, µ 2 }, where µ 1 and µ 2 are as in Lemma 4.4 and Lemma 4.5 respectively. We define the functionals V 1 and V 2 by V 1 R) p 1 R 1 e µx,..., p m R m e µx, p m+1 R m+1 e µx,..., p n R n e µx, V 2 R t ) p 1 t R 1 e µx,..., p m t R m e µx, p m+1 t R m+1 e µx,..., p n t R n e µx, We consider the Lyapunov function candidate V defined by 4.34) VR, R t ) V 1 R) + V 2 R t ). There exists γ 1, + ) such that 4.35) 1 γ VR, R t) R + R t γvr, R t ). Let us select T > large enough so that 4.36) γ 2 e µαt/2 1 2 with α defined by 4.22). Let ε 2, + ) be such that 4.37) ε 2 < min { µ1, µ } 2, ε 1, C 1 C 1 where ε 1 and C 1 are as in Theorem 4.1 while µ 1 and µ 2 are as in Lemma 4.4 and Lemma 4.5 respectively. Let us now assume that the initial condition R,.) = R o.) be such that 4.38) R o 1 ε 2.

123 4.1. Dissipative boundary conditions for the C 1 -norm 123 Then, by Theorem 4.1, 4.37) and 4.38), the C 1 solutions of 4.3) and 4.5) satisfy 4.39) R R 1 < min{µ 1, µ 2 }. By the definition of W 1 and W 2, we have 4.4) 4.41) V 1 R) = lim p W 1R) and V 2 R t ) = lim p W 2R t ), t [, T ], M > such that W 1 R) + W 2 R t ) M R 1, p [1, + ), t [, T ]. Hence from Lemmas 4.4 and 4.5, 4.4) and 4.41), we have in the distribution sense in, T ), 4.42) dv 1 dt µαv 1 + β 1 R t V ) dv 2 dt µαv 2 + β 2 R t V 2 Summing 4.42) and 4.43), we get, in the distribution sense in, T ), 4.44) dv dt µαv + β R 1V, with β max {β 1, β 2 }. Let us impose on ε 2, besides 4.37), that 4.45) ε 2 µα 2βC 1. From 4.14), 4.38) and 4.45), we get that 4.46) β R 1 µα 2. From 4.44) and 4.46), we have, in the distribution sense in, T ), 4.47) which implies that dv dt µ 2 αv, 4.48) VT ) e αµt/2 V). From 4.35) and 4.48), we obtain that 4.49) RT,.) 1 γ 2 e αrt/2 R o 1, which, together with 4.36), implies that 4.5) RT,.) R o 1. Then, by repeating exactly the same argumentation, it can be iteratively shown that R is defined [, j + 1)T ] [, L] and that Rj + 1)T,.) RjT,.) 1, j =, 1, 2,.... This completes the proof of Theorem 4.3.

124 124 Chapter 4. Systems of Nonlinear Conservation Laws 4.2. Control of networks of scalar conservation laws The special case of nonlinear scalar conservation laws has been introduced in Section Here we consider a network of scalar laws as illustrated in Fig.4.1. The nodes of the network i.e. the rectangular boxes) represent physical devices called compartments ) with dynamics expressed by scalar conservation laws of the form 1.63): 4.51) t ρ j t, x) + x q j t, x) =, t, x, L), j = 1,..., n. We assume that each flux q j is a static monotone increasing function of the density ρ j : This relation is supposed to be invertible as q j = ϕ j ρ j ). ρ j = ϕ 1 j q j ), in such a way that the system is also written in the quasi-linear form 4.52) t q j + λ j q j ) x q j =, j = 1,..., n, with ) λ j q j ) [ ] >. ϕ 1 j q) q j ) q u 1 1 x u x y 5 4 x y 4 Fig.4.1: Network of scalar conservation laws The directed edges i j of the network represent instantaneous transfer flows between the compartments. The flow from the output of a compartment i to the input of a compartment j is denoted f ij t). Additional input and output arcs represent inflows u j t) injected from the outside into some compartments or outflows y j t) from some compartments to the outside. Hence, the set of PDEs 4.52) is subject to boundary conditions of the form: 4.54a) 4.54b) q j t, ) = i j f ij t) + u j t), q j t, L) = k j f jk t) + y j t), j = 1,..., n.

125 4.2. Control of networks of scalar conservation laws 125 In equations 4.54), only the terms corresponding to actual edges of the network are explicitly written. Otherwise stated, all the u j, y j and f ij for non existing edges do not appear in the equations. It is assumed here that the flows f ij and y i are fractions of the outgoing flow q i t, L) from compartment i: f ij t) a ij q i t, L), < a ij 1 and y i t) a i q i t, L), < a i 1. The conservation of flows then imposes the following obvious constraints: 4.55) n a ij = 1, i = 1,..., n. j= We introduce the following vector and matrix notations: q q 1, q 2,..., q n ) T, Λq) diag{λ 1 q 1 ),..., λ n q n )}, u = vector including only the actual input flows u j, y = vector including only the actual output flows y j, A = matrix with entries a ji j = 1,..., n; i = 1,..., n). With these notations, the system 4.52), 4.54) may be written in the following compact form: 4.56a) 4.56b) 4.56c) t q + Λq) x q =, qt, ) = Aqt, L) + But), yt) = Cqt, L). where the definition of B and C is obvious. The first equation 4.56a) is a system of hyperbolic quasi-linear PDEs that defines the system state dynamics. The second equation defines the boundary conditions of the system, some of them being assignable by the system input ut). The third equation can be interpreted as an output equation with system output yt) being the set of outflows. For any constant input u, a steady state or equilibrium state) of the system is defined as a constant state q which satisfies the state equation 4.56a) and the boundary condition 4.56b): A I)q + Bu =. Under the constraints 4.55) it is can be verified that the matrix A I is a full-rank compartmental matrix see e.g. Bastin and Guffens 26) for more details on compartmental systems). It follows that, for any positive u there exists a unique positive steady state: q = A I) 1 Bu. Our goal is to analyse the exponential stability of the steady state q of the control system 4.56) when the system is under a linear state feedback control of the form 4.57) ut) = u + G qt, L) q )

126 126 Chapter 4. Systems of Nonlinear Conservation Laws where the matrix G is the control gain. Defining the Riemann coordinates R = R 1,..., R n ) T q q, the Cauchy problem associated to the closed-loop control system 4.56), 4.57) is equivalently written as: 4.58a) 4.58b) 4.58c) R t + ΛR)R x =, Rt, ) = KRt, L), R, x) = R o x), with ΛR) Λq + R) and K A + BG. According to Theorem 4.3, the steady state q is exponentially stable for the C 1 norm if the control gain G is selected such that ρ A + BG) < 1. Example: Ramp-metering control in road traffic networks In the fluid paradigm for road traffic modelling, the traffic state is represented by a macroscopic variable ϱt, x) which represents the density of the vehicles # veh/km) at time t and at position x along the road. The traffic dynamics are represented by a conservation law t ϱt, x) + x qt, x)) =. which expresses the conservation of the number of vehicles on a road segment without entries nor exits. In this equation, qt, x) is the traffic flux representing the flow rate of the vehicles at t, x). By definition, we have qt, x) ϱt, x)vt, x) where vt, x) is the velocity of the vehicles at t, x). As we have seen in Section 1.14, the basic assumption of the so-called LWR model is that the drivers instantaneously adapt their speed to the local traffic density, which is expressed by a function vt, x) = V ϱt, x)). The LWR traffic model is therefore written as 4.59) t ϱt, x) + x ϱt, x)v ϱt, x)) =. In accordance with the physical observations, the velocity-density relation is a monotonic V m v q % m % c Fig.4.2: Velocity v and flux q viz. density ϱ % m % decreasing function dv/dϱ < ) on the interval [, ϱ m ] see Fig.4.2 and Fig.1.12, in Chapter 1, for an illustration of this function with experimental data) such that: 1. V ) = V m the maximal vehicle velocity when the road is almost) empty;

127 4.2. Control of networks of scalar conservation laws V ϱ m ) = : the velocity is zero when the density is maximal, the vehicles are stopped and the traffic is totally congested. Then the flux qϱ) = ρv ϱ) is a non-monotonic function with q) = and qϱ m ) = which is maximal at some critical value ϱ c which separates free-flow and traffic-congestion: the traffic is flowing freely when ϱ < ϱ c while the traffic is congested when ϱ > ϱ c see Fig.4.2). As a matter of example, let us now consider the network of interconnected one-way road segments as depicted in Fig.4.3. The network is made up of nine road segments with u 3 v 1 ¾ ½ u 2 u 4 Fig.4.3: A road network four entries and three exits. The densities and flows on the road segments are denoted ϱ j and q j, j = 1, 9. The flow rate v 1 is a disturbance input and the flow rates u 2, u 3, u 4 at the three other entries are control inputs. Our objective is to analyse the stability of this network under a feedback ramp-metering strategy which consists in using traffic lights for modulating the entry flows u i. The motivation behind such control strategy is that a temporary limitation of the flow entering a highway can prevent the appearance of traffic jams and improve the network efficiency possibly at the price of temporary queue formation at the ramps). The reader can refer to Reilly et al. 213) and the references therein for more fetails on ramp-metering. The traffic dynamics are described by a set of LWR models 4.59): 4.6) t ϱ j t, x) + x ϱ j t, x)v ϱ j t, x)) =, j = 1,..., 9. Under free-flow conditions, the flows q j ϱ j ) = ϱ j V ϱ j ) are monotonic increasing functions and the model for the network of Fig.4.3 is written as a set of kinematic wave equations 4.61) t q j t, x) + cq j t, x)) x q j t, x) =, cq j ) >, j = 1,..., 9 with the boundary conditions q 1 t, ) = v 1 t), q 4 t, ) = q 3 t, L) + u 2 t), q 7 t, ) = 1 α)q 1 t, L), q 2 t, ) = αq 1 t, L), q 5 t, ) = γq 4 t, L), q 8 t, ) = q 7 t, L) + u 4 t), q 3 t, ) = βq 2 t, L), q 6 t, ) = q 5 t, L) + u 3 t), q 9 t, ) = q 6 t, L) + q 8 t, L) where α, β, γ are traffic splitting factors at the diverging junction and the two exits of the network. Obviously the set-point for the feedback traffic regulation is selected as a freeflow steady state q 1, q 2,..., q 9 ) T. A linear feedback is then defined for the ramp-metering

128 128 Chapter 4. Systems of Nonlinear Conservation Laws of the three input flows: u 2 t) = ū 2 + k 2 q 6 t, L) q 6 ), u 3 t) = ū 3 + k 3 q 6 t, L) q 6 ), u 4 t) = ū 4 + k 4 q 8 t, L) q 8 ), where k 2, k 3, k 4 are tuning control parameters. Defining the Riemann coordinates R i = q i q i, the boundary conditions of the system under the ramp-metering control are: R 1 t, ) R 1 t, L) R 2 t, ) α R 2 t, L) R 3 t, ) β R 3 t, L) R 4 t, ) 1 k 2 R 4 t, L) R 5 t, ) = γ R 5 t, L). R 6 t, ) 1 k 3 R 6 t, L) R 7 t, ) 1 α R 7 t, L) R 8 t, ) 1 k 4 R 8 t, L) R 9 t, ) } {{ 1 1 } R 9 t, L) K In this example, with parameter values α =.8, β =.9, γ =.8, the stability condition ρ K) < 1 can be shown to be satisfied if and only if the control parameters are selected such that:.8 k 2 + k 3 < 1 k 4 < 1. These bounds on the values of the k i parameters are computed with an algorithm for the computation of the maximal stability region of nonnegative matrices which is described in Haut et al. 29) see also Haut 27) for further details on this example) Interlude: Solutions without shocks For systems of nonlinear conservation laws 4.1) which may be written in quasi-linear characteristic form 4.3), it is well known that, even for small smooth initial conditions, the trajectories of the system may become discontinuous in finite time generating jump discontinuities which propagate on as shocks. An important consequence of the analysis of Section 4.1, is that if the initial condition is of class C 1 and small enough, then the dissipativity condition ρ K) < 1 guarantees that shocks will not develop and that the solutions remain smooth i.e. of class C 1 ) for all t [, + ), while exponentially converging to zero for the C 1 norm. Moreover, in Chapter 3, we remember that, for linear systems of conservation laws, the steady state is exponentially stable for the L 2 norm under another dissipativity condition ρ 2 K) < 1 which appears to be weaker according to the following proposition. Proposition 4.7. For every K M n,n R), 4.62) ρ 2 K) ρ K). Proof. The proof is given in Appendix C.

129 4.4. Dissipative boundary conditions for the H 2 -norm 129 It is also worth to point out that there are matrices K for which the inequality 4.62) is strict. For example, for a >, we have ) a a K and ρ a a 2 K) = 2a < 2a = ρ K). So we have the following natural question: does the condition ρ 2 K) < 1 imply also the exponential convergence of smooth solutions for nonlinear conservation laws? Surprisingly enough, the answer is negative for the C 1 norm as stated in the next proposition. Proposition 4.8. For the C 1 norm the condition ρ 2 K) < 1 is not sufficient for the exponential stability of the steady state of systems of nonlinear conservation laws. Proof. This proposition is a special case of Theorem 2 in Coron and Nguyen 215). Hence, in contrast with the linear case, it appears that the considered norm matters when looking at dissipative boundary conditions for systems of nonlinear conservation laws. In the next section, we prove that the condition ρ 2 K) < 1 is in fact a dissipativity condition for the H 2 norm Dissipative boundary conditions for the H 2 -norm In this section, we consider again the system 4.3), 4.5), i.e. 4.63) 4.64) R t + ΛR)R x =, t [, + ), x, L), ) R + t, ) ) R + t, L) R t, L) = H R t, ), under the initial condition 4.6), i.e. 4.65) R, x) = R o x), x [, L], which satisfies the compatibility conditions 4.7) and 4.8). However, in contrast with the previous Section, we now assume that Λ and H are of class C 2 on the ball B η. Our main result is to show that the exponential stability of the steady state can be established for the H 2 norm under the condition ρ 2 K) < 1. For a function φ : x φx), φ H 2 a, b); R n ), the definition of the H 2 norm is 1/2 b 4.66) φ H 2 a,b);r n ) φ 2 + φ x 2 + φ xx )dx) 2. a The well-posedness of the Cauchy problem and the existence of a unique solution result from the following theorem. Theorem 4.9. There exists δ > such that, for every R o H 2, L), R n ) satisfying R o H 2,L),R n ) δ

130 13 Chapter 4. Systems of Nonlinear Conservation Laws and the compatibility conditions 4.7), 4.8), the Cauchy problem 4.63), 4.64) and 4.65) has a unique maximal H 2 solution with T, + ]. Moreover, if then T = +. Proof. See Appendix B. R C [, T ); H 2, L); R n )) C 1 [, T ) [, L]; R n ) Rt, ) H 2,L);R n ) δ, t [, T ), The definition of the exponential stability is as follows. Definition 4.1. The steady state Rt, x) of the system 4.63), 4.64) is exponentially stable for the H 2 norm if there exist δ >, ν > and C > such that, for every R o H 2, L); R n ) satisfying R o H 2,L);R n ) δ and the compatibility conditions 4.7), 4.8), the H 2 solution of the Cauchy problem 4.63), 4.64), 4.65) is defined on [, + ) [, L] and satisfies Rt,.) H2,L);R n ) Ce νt R o H2,L);R n ), t [, + ). We then have the following stability theorem. Theorem If ρ 2 K) < 1, the steady state Rt, x) of the system 4.63), 4.64) is exponentially stable for the H 2 norm. From now on in this section, R : [, T ] [, L] R n denotes a solution in C [, T ]; H 2, L); R n ) to the system 4.63), 4.64). In this case, as we shall see in the proof of the theorem, we need to expand the analysis up to the dynamics of R tt. Therefore, we introduce the candidate Lyapunov function defined by: 4.67) V R T P µx)rdx + } {{ } V 1 R T t P µx)r t dx + } {{ } V 2 R T ttp µx)r tt dx. } {{ } V 3 with P µx) diag { P + e µx, P e µx}, P + diag{p 1,..., p m }, P diag{p m+1,..., p n }. Let us remark that, if R C [, T ]; H 2, L); R n )), V is a continuous function of t. Proof of Theorem 4.11 In order to prove the theorem, we temporarily assume that R is of class C 3 on [, T ] [, L] and therefore that V is of class C 1 in [, T ]. This assumption that will be relaxed later on) allows, by time differentiation of the equations 4.23), 4.24), to express the dynamics of R tt as follows: 4.68) R ttt + ΛR)R ttx + 2diag[Λ R)R t ]R tx + diag[λ R)R t ] t R x =,

131 4.4. Dissipative boundary conditions for the H 2 -norm ) ) [ )] R + t, ) R + t, L) tt = R tt H, t, L) R t, ) and, consequently, to compute the time derivatives of V 1, V 2 and V 3 along the solutions of the system 4.3), 4.5), 4.23), 4.24), 4.68), 4.69) in the following way: 4.7) dv 1 dt = 2R T P µx)r t dx = 2R T P µx)λr)r x dx. 4.71) dv 2 dt = 2R T t P µx)r tt dx = ) 2R T t P µx) ΛR)R tx diagλ 1 R)R t dx. 4.72) dv 3 dt = 2R T ttp µx)r ttt dx = 2R T ttp µx) ΛR)R ttx + 2diag[Λ R)R t ]R tx diag[λ R)R t ] t Λ 1 R)R t )dx. Since ρ 2 K) < 1 by assumption, there exist D D + m, D 1 D + n m and diag{d, D 1 } such that K 1 < 1. The parameters p i are selected such that P + Λ + ) = D 2 and P Λ ) = D 2 1. The proof of the theorem is then based on the following lemma which provides estimates of the functionals V i and dv i /dt) i=1,2,3) along the system solutions. Lemma a) There exists µ 1 > such that, µ, µ 1 ), there exist positive real constants α 1, β 1, δ 1 independent of R, such that, if R < δ 1, 4.73) 4.74) 1 β 1 Rt, x) 2 dx V 1 t) β 1 Rt, x) 2 dx, dv 1 dt t) α 1V 1 t) + β 1 Rt, x) 2 R t t, x) dx. b) There exists µ 2 > such that, µ, µ 2 ), there exist positive real constants α 2, β 2, δ 2 independent of R, such that, if R < δ 2, 4.75) 4.76) 1 β 2 R t t, x) 2 dx V 2 t) β 2 R t t, x) 2 dx, dv 2 dt t) α 2V 2 t) + β 2 R t t, x) 3 dx.

132 132 Chapter 4. Systems of Nonlinear Conservation Laws c) There exists µ 3 > such that, µ, µ 3 ), there exist positive real constants α 3, β 3, δ 3 independent of R, such that, if R + R t < δ 3, 4.77) 1 β ) dv 3 dt t) α 3V 3 t) + β 3 R tt t, x) 2 dx V 3 t) β 3 R tt t, x) 2 dx, R t t, x) 2 R tt t, x) + R t t, x) R tt t, x) 2 )dx. Proof. Here, we give only the proof of item a) of Lemma The proofs of items b) and c) are given in Appendix D and are constructed in a similar way. Using integration by parts, from 4.7), we can decompose dv 1 /dt as follows: with 4.79) 4.8) 4.81) T 1 T 2 dv 1 dt T 3 µ = T 1 + T 2 + T 3, R T P µx)λr)r ) x dx, ) R T P µx)[λ R)R x ]R dx, ) R T P µx) ΛR) R dx. Analysis of the first term T 1. We have [ ] L [ ] L 4.82) T 1 = R T P µx)λr)r = R T P µx)λ)r + h.o.t. where h.o.t. stands for higher order terms. Let us introduce a notation in order to deal with estimates of these higher order terms. We denote by OX; Y ), with X and Y, quantities such that there exist C > and ε >, independent of R and R t such that Y ε) OX; Y ) CX). Hence, using P + Λ + ) = D 2 and P Λ ) = D 2 1, we have T 1 = ) D R + ) t, L) R +T t, L)D R T t, )D 1 Ωµ) with D 1 R t, ) + O R+ 3 ) t, L) R t, ) ; R+ t, L) R t, ),

133 4.4. Dissipative boundary conditions for the H 2 -norm 133 ) In e µl Ωµ) I ) D K D 1 D K 1 D1 1 T D K D 1 D K 1 D1 1 D 1 K 1 D 1 D 1 K 11 D1 1 D 1 K 1 D 1 eµl D 1 K 11 D 1 1 eµl Following the same argumentation as in the proof of Theorem 3.2, we know that there exists µ 1 > such that Ωµ) is positive definite for all µ, µ 1 ). Consequently, there exists δ 11 > such that T 11 for all µ, µ 1 ) if R +T t, L), R T t, ) δ 11. Analysis of the second term T 2. The absolute value of the integrand of T 2 is bounded by R 2 R t P µx)λ R)Λ 1 R). It follows that β 1 > can be taken sufficiently large such that inequalities 4.73) hold for every µ, µ 1 ), there exists δ 12 such that if R δ 12. T 2 β 1 R 2 R t dx ). Analysis of the third term T 3. By 4.73) there exist α 1 > and δ 13 > such that T 3 α 1 V 1. for every µ, µ 1 ) if R δ 13. Then we conclude that dv 1 dt = T 1 + T 2 + T 3 α 1 V 1 + β 1 R 2 R t dx if R δ 1 minδ 11, δ 12, δ 13 ). In the next lemma, we now show how it follows from the previous estimates that V = V 1 + V 2 + V 3 exponentially decreases along the system solutions. Lemma For every µ, min{µ 1, µ 2, µ 3 }), there exist positive real constants α, β and δ such that, for every R such that R + R t < δ, 4.83) 1 β 4.84) R 2 + R x 2 + R xx 2 )dx V β dv dt αv. R 2 + R x 2 + R xx 2 )dx,

134 134 Chapter 4. Systems of Nonlinear Conservation Laws Proof. Let β, δ be selected such that From 4.3) and 4.23), we know that β max{β 1, β 2, β 3 }, < δ min{δ 1, δ 2, δ 3 }. 4.85) 4.86) R t = ΛR)R x R tt = ΛR)ΛR)R x ) x [Λ R)ΛR)R x ]R x. Using these expressions, it readily follows from 4.73), 4.75) and 4.77) that, if R + R t < δ, then 4.83) holds if β is large enough. For every η >, we have ) n R t 2 R tt dx 4η R t 4 + η R tt 2 dx 4.87) 1 4η R t 2 R t 2 dx + η R tt 2 dx. In order to get 4.84), it is sufficient to combine inequalities 4.74), 4.76), 4.78) and 4.87) with η α 3 /2β 3 ) 2, and to point out that R 2 R t dx n R t R 2 dx, R t 3 dx n R t R t 2 dx, R t R tt 2 dx n R t R tt 2 dx. Proof of Theorem The estimates 4.83) and 4.84) have been obtained under the assumption that Λ and H of class C 3 and R is of class C 3. But the selection of α and β 2 does depend neither on the C 3 norms of Λ and H nor on the C 3 norm of R: they depend only on the C 2 norm of Λ and H and the C [, T ]; H 2, L); R n )) norm of R. Hence, using a density argument similar to Comment 4.6, the estimates 4.83) and 4.84) remain valid, in the distribution sense, with Λ, H, R only of class C 2. By the Sobolev inequality see, for instance, Brezis, 1983, Theorem VII, page 129) ), there exists C > such that, for every ϕ in the Sobolev space H 2, L); R n ), 4.88) ϕ + ϕ C ϕ H2,L);R n ). We choose µ, min{µ 1, µ 2, µ 3 }]. Let 4.89) ε min { δ 2C β, δ }. β Note that β 1 and therefore that δ δ. Using Lemma 4.13, 4.88) and 4.89), for every t [, T ]: 4.9) Rt,.) H 2,L);R n ) ε ) = Rt,.) + R x t,.) δ2 ) and Vt) βε2,

135 4.5. Stability of general systems of nonlinear conservation laws in quasi-linear form ) R + R x δ and V βε 2) = Rt,.) + Rt,.) t δ2 ) and Rt,.) H 2,L);R n) δ, 4.92) and R + R x δ) = ) dv dt in the distribution sense. Let R o H 2, L); R n ) satisfy the compatibility conditions 4.7),4.8) and R o H2,L);R n ) < ε. Let R C [, T ), H 2, L); R n )) be the maximal classical solution of the Cauchy problem 4.3), 4.5), 4.6). Using implications 4.9) to 4.92) for T [, T ), we get that 4.93) 4.94) Rt, ) H2,L);R n ) δ, t [, T ), Rt, ) + R t t, ) δ, t [, T ). Using 4.93) and Theorem 4.9, we have that T = +. Using Lemma 4.13 and 4.94), we finally obtain that Rt, ) 2 H 2,L);R n ) βvt) βv)e αt β 2 R o 2 H 2,L);R n ) e αt. This concludes the proof of Theorem Stability of general systems of nonlinear conservation laws in quasi-linear form In this section, we now consider systems having the general quasi-linear form 4.2): 4.95) Y t + F Y)Y x =, t [, + ), x [, L], where Y : [, + ) [, L] R n and F : Y M n,n R) is of class C 1. We do not assume that the system can be transformed into characteristic form with Riemann coordinates. We shall nevertheless be able to analyze the exponential stability of a steady state Yt, x) when the matrix F ) has n real non-zero eigenvalues denoted λ i, i = 1,..., n and is diagonalizable. In this section, for simplicity, we treat only the special case where all eigenvalues are positive: λ i >, i = 1,..., n. The generalization to the case where F ) has both positive and negative eigenvalues can be found in Coron and Bastin 215). The key point is that, by a linear change of coordinates, it is always possible to rewrite the system 4.95) in the equivalent form 4.96) Z t + AZ)Z x =, t [, + ), x [, L], where A C 1 Y; M n,n R)) and A) D + n is the positive diagonal matrix diag{λ 1,..., λ n }. Our concern is to analyze the exponential stability of the steady state Zt, x) under nonlinear boundary conditions in nominal form 4.97) Zt, ) = H Zt, L) ), t [, + ),

136 136 Chapter 4. Systems of Nonlinear Conservation Laws where H C 1 B σ, R n ), H) = and under an initial condition 4.98) Z, x) = Z o x), x [, L], which satisfies the compatibility conditions 4.99) Z o ) = H Z o L) ), 4.1) A Z o ) ) x Z o ) = H Z o L) ) A Z o L) ) x Z o ), where H denotes the Jacobian matrix of the map H. The matrix K is defined as the linearization of the map H at the steady state: K H ). In this section, we shall show that ρ K) < 1 and ρ 2 K) < 1 are dissipativity conditions for the C p norm and the H p norm respectively. In order to define appropriate Lyapunov functions for this analysis, we introduce the following assumption. Assumption Let DZ) be the diagonal matrix whose diagonal entries are the eigenvalues λ i Z), i = 1,..., n, of the matrix AZ). There exist a positive real number η and a map Q : B η M n,n R) of class C 2 such that 4.11) 4.12) QZ)AZ) = DZ)QZ), Z B η, Q) = Id n, where Id n is the identity matrix of M n,n R). Remark This assumption is the expression of the fact that the matrix AZ) can be diagonalized in a neighborhood of the origin. It is however important to realize that this does not imply that the quasi-linear hyperbolic system 4.96) itself can be written in characteristic form. Otherwise the present section would obviously be irrelevant! Remark It is worth noting that Assumption 4.14 is generic. Indeed it is satisfied as soon as the eigenvalues λ i ) are distinct. But, there are interesting cases where there is a matrix Q satisfying equality 4.11) even with non-distinct eigenvalues. For example, in the case where the matrix AZ) is block diagonal, it may be sufficient to have distinct eigenvalues in each block, but the different blocks may share identical eigenvalues. This situation will appear in particular in the proofs of Theorems 4.22 and 4.24 which deal with stability for C p norms with p > 1 and for H p norms with p > 2. Stability condition for the C 1 -norm Our purpose is to show that the dissipativity condition ρ K) < 1 of Section 4.1 for systems in Riemann coordinates can be extended to general systems 4.96), 4.97) in a similar fashion. The definition of exponential stability is as follows.

137 4.5. Stability of general systems of nonlinear conservation laws in quasi-linear form 137 Definition The steady state Zt, x) of the system 4.96), 4.97) is exponentially stable for the C 1 norm if there exist ε >, ν > and C > such that, for every Z o in the set V {Z C 1 [, L]; R n ) : Z 1 < ε} and satisfying the compatibility conditions 4.99) and 4.1), the C 1 solution of the Cauchy problem 4.96), 4.97) and 4.98) satisfies Zt,.) 1 Ce νt Z o 1, t [, + ). We have the following stability theorem. Theorem If ρ K) < 1, the steady state Zt, x) of the system 4.96), 4.97) is exponentially stable for the C 1 norm. In order to prove this theorem, we introduce the following generalizations of the functionals W 1 and W 2 that were used in Section 4.1: 4.13) W ) W 2 n i=1 n i=1 p p i p p i n ) 2pe q ij Z)Z 2pµx j dx j=1 1 2p n ) 2pe q ij Z) t Z 2pµx j dx j=1 with p N + and p i > i. In 4.13) and 4.14), q ij Z) denotes the i, j)th entry of the matrix QZ). The key point for the analysis is that Lemma 4.4 and Lemma 4.5 can be extended to these generalized forms of W 1 and W 2. Lemma If ρ K) < 1, there exist p i > i {1,..., n}, positive real constants α, β 1 and δ 1 such that, for every µ, δ 1 ), for every p 1/δ 1, + ), for every T >, for every C 1 solution Z : [, T ] [, L] R n of 4.96) and 4.97) satisfying Z < δ 1, we have dw 1 dt µα + β 1 Z x t) ) W 1. Proof. Let Z : [, T ] [, L] R n be a C 1 solution of 4.96) and 4.97). derivative of W 1 is: 4.15) dw 1 dt = 1 2p W1 2p 1 n n 2p p p i q ij Z)Z j i=1 j=1 [ n ] q ij Z) t Z j + j=1 2p 1 Using 4.96), the term between brackets can be written as n n n ) 4.16) q ij Z) t Z j = q ij Z) a jk Z) x Z k j=1 j=1 k=1, 1 2p, The time n t q ij Z) ) Z j e 2pµx dx. j=1

138 138 Chapter 4. Systems of Nonlinear Conservation Laws n n = q ij Z)a jk Z) x Z k, k=1 j=1 where a jk Z) is the j, k)th entry of the matrix AZ). Now, from 4.11), we have 4.17) n n q ij Z)a jk Z) x Z k = d ij Z)q jk Z) x Z k = λ i Z)q ik Z) x Z k, j=1 j=1 where d ij Z) is the i, j)th entry of the matrix DZ). From 4.16) and 4.17), we have 4.18) n n n q ij Z) t Z j = λ i Z) q ik Z) x Z k = λ i Z) q ij Z) x Z j. j=1 k=1 j=1 By substituting this expression for the term between brackets in 4.15), we get dw 1 dt = 1 n 2p W1 2p 1 2p p p i i=1 λ i Z) n q ij Z)Z j j=1 2p 1 n ) n q ij Z) x Z j + t q ij Z) ) ) ) Z j e 2pµx dx, j=1 j=1 which leads to 4.19) [ dw 1 = 1 L n n ) 2p) dt 2p W1 2p 1 p p i λ iz) q ij Z)Z j e 2pµx dx i=1 j=1 x L n n ) + 2p q ij Z)Z j 2p 1λ n ) i Z) x q ij Z))Z j i=1 p p i j=1 j=1 n + t q ij Z) ) ) ) ] Z j e 2pµx dx. j=1 Using integrations by parts, we now get 4.11) with 4.111) 4.112) T 1 W1 2p 1 2p i=1 T 2 µw 1 2p 1 dw 1 dt = T 1 + T 2 + T 3, n n ) 2p 1 p p i λ iz) q ij Z)Z j e µx, j=1 n n ) 2pe p p i λ iz) q ij Z)Z 2pµx j dx, i=1 j=1

139 4.5. Stability of general systems of nonlinear conservation laws in quasi-linear form ) T 3 W 1 2p 1 j=1 n i=1 p p i n ) 2p 1 n ) q ij Z)Z j λ i Z) x q ij Z))Z j j=1 n + t q ij Z) ) ) Z j + 1 2p n j=1 j=1 ) ) λi q ij Z)Z j Z Z) xz e 2pµx dx Analysis of the first term T 1. From 4.111), we have 4.114) T 1 = W1 2p 1 2p n n ) 2p p p i λ izt, L)) q ij Zt, L))Z j t, L)e µl i=1 j=1 n n ) 2p p p i λ izt, )) q ij Zt, ))Z j t, ). i=1 According to Li, 1994, Lemma 2.4, page 146), n 4.115) ρ K) = sup K ij δ i ; i {1,..., n}, δ j >, j {1,..., n} δ j. j=1 Since ρ K) < 1 and by 4.115), there exist δ i >, i {1,..., n}, such that 4.116) θ The parameters p i are selected such that n j=1 K ij δ i δ j < ) p p i λ i = δ 2p i, i = 1,..., n. We define ξ i : [, T ] R, i = 1,..., n, by 4.118) ξ i t) δ i Z i t, L), t [, T ]. From 4.114), 4.117) and 4.118), we have 4.119) T 1 = W1 2p 1 2p n i=1 j=1 n ) 2p λ i Zt, L)) q ij Zt, L)) δ i ξ j t)e µl Λ i δ j=1 j n n ) 2p λ i Zt, )) q ij Zt, ))δ i Z j t, ). Λ i i=1 j=1 Let t [, T ]. Without loss of generality, we may assume that 4.12) ξ 2 1t) = max{ξ 2 i t), i = 1,..., n}.

140 14 Chapter 4. Systems of Nonlinear Conservation Laws Let us denote by δ and C various positive constants which may vary from place to place but are independent of t [, T ], Z and p N +. From 4.12) and 4.12), we have, for Zt, L) δ, 4.121) n n λ i Zt, L)) q ij Zt, L)) δ i ξ j t)e µl δ j i=1 Λ i λ 1Zt, L)) Λ 1 j=1 n j=1 q 1j Zt, L)) δ 1 δ j ξ j t)e µl e 2pµ 1 C ξ 1 t) ) ξ 1 t) C ξ 1 t) 2) 2p ) 2p ) 2p = e 2pµ 1 C ξ 1 t) ) 2p+1 ξ 1 t)) 2p. From 4.97), 4.116), 4.118) and 4.12), we have, for Zt, ) δ, 4.122) n n ) 2p λ i Zt, )) q ij Zt, ))δ i Z j t, ) Λ i j=1 2p n n 1 + C ξ 1 t) ) C ξ 1 t) 2 + K ij δ i ξ j t) δ j i=1 i=1 j=1 n 1 + C ξ 1 t) ) θ ξ 1 t) + C ξ 1 t) 2) 2p. From 4.116), 4.119), 4.12) and 4.122), there exists δ 11, L), independent of Z, such that, for every µ, δ 11 ), for every p 1/δ 11, + ) N + and for every Z, we have 4.123) T 1 t) if Zt) < δ 11. Analysis of the second term T 2. Let 4.124) α minλ 1,..., Λ n )/2. From 4.112), 4.117) and 4.124) there is a δ 12, η] such that, for every µ, + ), for every p N + and for every Z, 4.125) T 2 µαw 1 if Z < δ 12. Analysis of the third term T 3. Using 4.96) and 4.113), we have 4.126) T 3 = W 1 2p 1 1 2p n i=1 p p i n ) 2p 1 q ij Z)Z j j=1 n ) λ i Z Z x q ij Z)Z j + j=1 n j=1 qij AZ) + λi Z) ) ) Z x Z j )e 2pµx dx. Z

141 4.5. Stability of general systems of nonlinear conservation laws in quasi-linear form 141 From 4.12) and 4.126) we get the existence of β 1 > and δ 13 > such that, for every µ, + ), for every p N + and for every Z, 4.127) T 3 β 1 Z x W 1 if Z < δ 13. Let δ 1 min{δ 11, δ 12, δ 13 }. From 4.11), 4.123), 4.125) and 4.127), we conclude that dw 1 µα 1 W 1 + β 1 Z x W 1 dt provided that Z is such that Z < δ 1, that p 1/δ 1, + ) N + and that µ, δ 1 ). Lemma 4.2. Let p i i = 1,..., n) be given by 4.117). If ρk) < 1, there exist β 2 and δ 2 such that, for every µ, δ 2 ), for every p 1/δ 2, + ) N + and for every C 1 solution Z : [, T ] [, L] R n of 4.96) and 4.97) such that Z < δ 2, we have, in the sense of distributions in, T ), with α defined by 4.124). dw 2 dt µα + β 2 Z x ) W 2 Let us remark that, since Z is only of class C 1, W 2 is only continuous and dw 2 /dt has to be understood in the distribution sense. Proof. In order to prove the lemma, we temporarily assume that Z is of class C 2 on [, T ] [, L]. The assumption will be relaxed at the end of the proof. By time differentiation of 4.96) and 4.97), we see that Z t satisfy the following hyperbolic dynamics for t [, T ] and x [, L]: 4.128) 4.129) Z t ) t + AZ)Z t ) x + A Z, Z t )Z x =, t Zt, ) = HZt, L)) t Zt, L), Zt, L) where A Z, Z t ) is a compact notation for the matrix whose entries are A Z, Z t ) i,j a ijz) Z Z t, i {1,..., n}, j {1,..., n}. Using 4.128) 4.129), we see that the time derivative of W 2 is: 4.13) j=1 dw 2 dt 2p 1 = 1 n n 2p W1 2p 2 2p p p i q ij Z) t Z j i=1 j=1 [ n ] n q ij Z) tt Z j + t q ij Z) ) t Z j e 2pµx dx. j=1 From 4.11), similarly as for 4.18), it can be shown that n n ) n n q ij Z) tt Z j = λ i Z) q ij Z) x t Z j + q ij Z) ã jk Z, Z t ) t Z k ), j=1 j=1 j=1 k=1

142 142 Chapter 4. Systems of Nonlinear Conservation Laws where ã ij Z, Z t ) is the i, j)th entry of the matrix ÃZ, Z t) A Z, Z t )A 1 Z). Then, by substituting this expression for the term between brackets in 4.13), we get dw 2 dt = 1 2p W1 2p 2 + n n ) 2p 1 2p p p i q ij Z) t Z j i=1 [ j=1 λ i Z) n ) q ij Z) x t Z j j=1 ] n n ) q ik Z)ã kj Z, Z t ) + t q ij Z) ) t Z j e 2pµx dx. j=1 k=1 Using integration by parts as in the proof of Lemma 4.19, we get dw 2 dt = U 1 + U 2 + U 3, with U 1 W1 2p 2 2p U 2 µw 1 2p 2 n p p i λ n iz) q ij Z) ) 2p 1 ) t Z j e µx, i=1 j=1 n p p i λ n iz) q ij Z) ) ) t Z j 2pe 2pµx dx, i=1 j=1 U 3 W 1 2p 2 k=1 n i=1 p p i n q ij Z) ) ) 2p 1[ n t Z j λ i Z) x q ij Z)) j=1 j=1 ) n ) t ) + q ik Z)ã kj Z, Z t ) + t q ij Z) Z j + 1 2p n j=1 ] ) λi q ij Z) t Z j Z Z)Z x e 2pµx dx. Analysis of the first term U 1. Using the boundary conditions 4.97) and 4.129), we have n U 1 = W1 2p 2 p p i 2p λ n izt, L)) q ij Zt, L)) t Z j t, L) )) 2p e 2pµ i=1 j=1 n n ) 2p p p i λ ihzt, L))) q ij HZt, L))) H jzt, L)) t Zt, L) Zt, L) i=1 j=1 Then, in a way similar to the analysis of T 1 in the proof of Lemma 4.19, we can show that, if ρ K) < 1, there exists δ 21, 1), such that, for every µ, δ 21 ), for every

143 4.5. Stability of general systems of nonlinear conservation laws in quasi-linear form 143 p 1/δ 21, + ) N + and for every Z, we have U 1 provided that Z < δ 21. Analysis of the second term U 2. Proceeding as in the proof of 4.125), we get the existence of δ 22, η] such that, for every µ, + ), for every p N + and for every Z, 4.131) U 2 µαw 2 if Z < δ 22, with α defined as in 4.124). Analysis of the third term U 3. Proceeding as in the proof of 4.127), we get the existence of β 2 > and δ 23 > such that, for every µ, + ), for every p N + and for every Z, 4.132) U 3 β 2 Z x W 2 if Z < δ 23. From the analysis of U 1, U 2 and U 3, we conclude that, with δ 2 min{δ 21, δ 22, δ 23 }, dw 2 dt µαw 2 + β 2 Z x W 2 for all Z such that Z < δ 2 provided that µ, δ 2 ) and that p 1/δ 2, + ) N +. The above estimate of dw 2 /dt has been obtained under the assumption that A, H and Z are of class C 2. But the selection of µα and β 2 does not depend on the C 2 norm of A, H and Z. Hence, using a density argument similar to Comment 4.6, the estimate 4.132) remains valid with A, H and Z only of class C 1. This completes the proof of Lemma 4.2. The proof of Theorem 4.18 then follows from Lemma 4.19, word-for-word as the proof of Theorem 4.3 follows from Lemmas 4.4 and 4.5. Stability condition for the C p -norm for any p N {} In the previous subsection, we have seen that ρ K) < 1 is a dissipativity condition with respect to the C 1 norm for general quasi-linear systems 4.133) Z t + AZ)Z x = 4.134) Zt, ) = H Zt, L) ) t [, + ), with initial conditions 4.135) Z, x) = Z o x), x [, L] satisfying the compatibility conditions 4.99), 4.1). Our purpose is now to emphasize that the exponential stability with respect to the C p norm holds in fact for any p N {} under the same dissipativity condition ρ K) < 1 when the maps A and H are of class C p. To establish this property it is first needed to generalize the definition of the compatibility of the initial conditions. For that, for a map G : C [, L]; R n ) C [, L]; R n ) of class C p, we introduce the sequence D j G : C j [, L]; R n ) C [, L]; R n ) defined by induction as follows: D G)Z) G Z C [, L]; R n ),

144 144 Chapter 4. Systems of Nonlinear Conservation Laws D j G)Z) ) D j 1 G) Z) AZ)Z x Z C j [, L]; R n ) j {1,..., p}. Let I be the identity map from C [, L]; R n ) into C [, L]; R n ) and let us define H : C [, L]; R n ) C [, L]; R n ) by ) Hϕ) x) = H ϕx) ) ϕ C [, L]; R n ) and x [, L]. Then, we say that the function Z o C p [, L]; R n ) satisfies the compatibility conditions of order p if ) ) 4.136) D j I)Z o ) ) = D j H)Z o ) L) for every j {, 1,..., p}. Less formally, let us explain how the explicit expression of the compatibility conditions of order 2 can be obtained. The first compatibility condition for j = is obtained by evaluating the boundary condition 4.134) at t = which, using the initial condition 4.135), gives: 4.137) Z o ) = H Z o L) ). Let us now differentiate the boundary condition 4.134) with respect to t. We get: Z t t, ) = H Zt, L))Z t t, L). Using the system equation 4.133), this latter relation is rewritten as 4.138) AZt, ))Z x t, ) = H Zt, L))AZt, L))Z x t, L). Then the second compatibility condition for j = 1 is obtained by evaluating 4.138) at t = which, using the initial condition 4.135), gives: 4.139) AZ o )) x Z o ) = H Z o L))AZ o L)) x Z o L). Let us now differentiate 4.138) with respect to t. We get: 4.14) AZt, ))Z xt t, ) + [A Zt, ))Z t t, )]Z x t, ) = H Zt, L))AZt, L))Z xt t, L) + { H Zt, L)) [ A Zt, L))Z t t, L) ] + [ H Zt, L))Z t t, L) ] AZt, L)) } Z x t, L). Differentiating the system equation 4.133) with respect to x, we have 4.141) Z tx + AZ)Z xx + [ A Z)Z x ] Zx =. Then, using 4.133) and 4.141), the relation 4.14) is rewritten as [ AZt, )) AZt, ))Z xx t, ) + [ A Zt, ))Z x t, ) ] ] Z x t, ) Z x t, ) + [A Zt, ))AZt, ))Z x t, )]Z x t, ) [ = H Zt, L))AZt, L)) AZt, L))Z xx t, L) + [ A Zt, L))Z x t, L) ] ] Z x t, L) + {H Zt, L)) [ A Zt, L))AZt, L))Z x t, L) ]

145 4.5. Stability of general systems of nonlinear conservation laws in quasi-linear form [ H Zt, L))AZt, L))Z x t, L) ] } AZt, L)) Z x t, L). Finally, the third compatibility condition for j = 2 is obtained by evaluating this latter relation at t = which, using the initial condition 4.135), gives: [ AZ o )) AZ o )) xx Z o ) + [ A Z o )) x Z o ) ] ] 4.142) x Z o ) + [A Z o ))AZ o )) x Z o )] x Z o ) [ = H Z o L))AZ o L)) AZ o L)) xx Z o L) + [ A Z o L)) x Z o L) ] ] x Z o L) + {H Z o L)) [ A Z o L))AZ o L)) x Z o L) ] + [ H Z o L))AZ o L)) x Z o L) ] } AZ o L)) x Z o L). So, the set of compatibility conditions of order p = 2 is given by equations 4.137), 4.139), 4.142). We then have the two following theorems, respectively for the well-posedness of the Cauchy problem associated to the system 4.133), 4.134) and for the exponential stability of the steady state. Theorem Let p N +. Let T >. There exist ε > and C > such that, for every Z o C p [, L]; R n ) satisfying the compatibility conditions of order p 4.136) and such that Z o C p [,L];R n ) ε, the Cauchy problem 4.133), 4.134), 4.135) has one and only one solution Z C p [, T ] [, L]; R n ). Furthermore, Z Cp [,T ] [,L];R n ) C Z o Cp [,L];R n ). Theorem If ρ K) < 1, there exist ε >, ν > and C > such that, for every Z o in the set V {Z C p [, L]; R n ) : Z Cp [,L];R n ) < ε} and satisfying the compatibility conditions of order p 4.136), the C p solution of the Cauchy problem 4.133), 4.134), 4.135) satisfies Zt,.) C p [,L];R n ) Ce νt Z o C p [,L];R n ), t [, + ). Here, for the sake of conciseness, we will not give the proofs of these theorems which are, indeed, straightforward but rather tedious extensions of the proofs of Theorems 4.1 and Theorem 4.18 respectively obtained, roughly speaking, by considering the augmented hyperbolic system of balance laws with state variables Z, t Z,..., p 1 t Z. These systems are systems of balance laws with a uniform zero steady state and a quadratic source term which is analyzed in Section 6.1, see Corollary 6.3. In this case, the augmented matrix AZ, t Z,..., p 1 t Z) is block diagonal and Remark 4.16 applies. Stability condition for the H p -norm for any p N {, 1} Let p N {, 1}. The definition of exponential stability for the H p norm is as follows. Definition The steady state Zt, x) of the system 4.133), 4.134) is exponentially stable for the H p norm if there exist δ >, ν > and C > such that, for every

146 146 Chapter 4. Systems of Nonlinear Conservation Laws Z o H p, L); R n ) satisfying Z o Hp,L);R n ) δ and the compatibility conditions of order p ), the solution Z of the Cauchy problem 4.133), 4.134), 4.135) is defined on [, + ) [, L] and satisfies Zt,.) Hp,L);R n ) Ce νt Z o Hp,L);R n ), t [, + ). We then have the following stability theorem. Theorem If ρ 2 K) < 1, the steady state Zt, x) of the system 4.133), 4.134) is exponentially stable for the H p norm. The proof of this theorem is an extension of Theorem 4.11 which deals with the case p = 2. Here also, similarly to the case of the C p norm see page 143), the proof is carried out by considering an augmented hyperbolic system of balance laws with state variables Z, t Z,..., p 2 t Z References and further reading For nonlinear systems of conservation laws, the issue of finding sufficient dissipative boundary conditions was addressed in the literature for more than thirty years. To our knowledge, first results were published by Slemrod 1983) and by Greenberg and Li 1984) for the special case of systems of size n = 2. A generalization to systems of size n was then progressively elaborated by the Li Ta-Tsien group, in particular by Qin 1985) and by Zhao 1986). All these contributions were dealing with the particular case of local boundary conditions having the specific form 4.143) R + t, ) = H + R t, )), R t, L) = H R + t, L)). With these boundary conditions, indeed, the analysis can be based on the method of characteristics which allows to exploit an explicit computation of the reflection of the solutions at the boundaries along the characteristic curves. This has given rise to the sufficient condition )) H + ) ) 4.144) ρ < 1, H ) ) for the dissipativity of the boundary conditions 4.143) for the C 1 norm. This result was first proved by Qin 1985) and Zhao 1986) using the method of characteristics and was also given by Li, 1994, Theorem 1.3, page 173) in his seminal book of 1994 on the stability of the classical solutions of quasi-linear hyperbolic systems. By using an appropriate dummy doubling of the system size, de Halleux et al., 23, Theorem 4) showed how the method of characteristics can be used to prove Theorem 4.3 for systems with the general non local boundary condition 4.5). This dummy doubling of the size of the system was also used by Li et al. 21) to prove the well-posedness of the Cauchy problem associated to 4.96) and 4.97) still in the framework of C 1 solutions. In contrast the proof of Theorem 4.3 given in Section 4.1 uses the Lyapunov method and was published much later in Coron and Bastin 215). Historically, the first approach for a Lyapunov stability analysis of 1-D hyperbolic systems was to use entropies as Lyapunov functions expressed in the physical coordinates.

147 4.6. References and further reading 147 This was done for instance by Coron et al. 1999) or by Leugering and Schmidt 22). The drawback of this approach was however that the time derivatives of such entropybased Lyapunov functions are necessarily only semi-definite negative. In such a case it is well known that the exponential convergence of the solutions may be proved with the LaSalle invariant set principle, see e.g. Khalil, 1996, Section 3.2) for nonlinear ODEs and Coron, 27, Chapter 13) for linear PDEs. Unfortunately, the LaSalle invariance principle requires the precompactness of the trajectories, a property which is quite difficult to get in the case of nonlinear partial differential equations. This difficulty is alleviated by using the strict Lyapunov functions with exponential weights that are repeatedly utilized in this book. Such functions were initially introduced by Coron 1999) for 2-D Euler equations of incompressible fluids, then by Xu and Sallet 22) for a class of symmetric linear hyperbolic systems, and finally by Coron et al. 27) and Coron et al. 28) for general systems of nonlinear conservation laws. Various interesting generalizations of the results presented in this chapter are worth mentioning. The stabilization of systems of two nonlinear conservation laws with integral actions is analyzed in Drici 211). The design of dead-beat controllers for the stabilization of systems of two nonlinear conservation laws is discussed in Perollaz and Rosier 213). The design of boundary observers for linear and quasi-linear hyperbolic systems of conservation laws is addressed by Castillo et al. 213). The analysis is illustrated with an application to flow control. The boundary feedback stabilization of gas pipelines modelled by isentropic Euler equations is studied by Dick et al. 21) and Gugat and Herty 211). The output feedback stabilization of a scalar conservation law with a nonlocal characteristic velocity is addressed in Coron and Wang 213). The stabilization of extrusion processes modelled by coupled conservation laws with moving boundaries is studied in Diagne et al. 215a). It is also interesting to notice that these boundary stabilization results of hyperbolic systems all assume that the characteristic velocities do not vanish. To our knowledge, the characterization of boundary conditions ensuring the exponential stability of systems with vanishing characteristic velocities remains an open question. Indeed in this case, the state components corresponding to vanishing characteristic velocities cannot be directly controlled from the boundary, but only indirectly through their coupling to other states. Let us however mention that controllability results for hyperbolic systems with vanishing characteristic velocities are already available and can be found for instance in Glass 27), Coron et al. 29), Glass 214), Hu and Wang 215).

148 148 Chapter 4. Systems of Nonlinear Conservation Laws

149 Chapter 5 Systems of Linear Balance Laws THE THREE PREVIOUS CHAPTERS have dealt with the stability and the boundary stabilization of systems of conservation laws. From now on, we shall move to the analysis of systems of balance laws. In this chapter, we begin with the case of linear balance laws. We consider the class of systems of linear hyperbolic balance laws with nonuniform coefficients of the form: 5.1) Y t + F x)y ) + Gx)Y = t [, + ), x, L), x where Y : [, + ) [, L] Y, F : [, L] M n,n R), G : [, L] M n,n R). The maps F and G are of class C 1. The matrix F x) has real non-zero real eigenvalues and is diagonalizable for all x [, L] through a change of variables which is of class C 1 with respect to x. Let us remark that any system of balance laws of the form 5.1) can also be written as the following linear hyperbolic system with nonuniform coefficients: 5.2) Y t + Ax)Y x + Bx)Y =, with Ax) F x) and Bx) Gx) + F x). Conversely, any linear system of the form 5.2) can be viewed as a system of balance laws of the form 5.1) with F x) Ax) and Gx) = Bx) A x). With respect to conservation laws, the presence of the source term Gx)Y brings a big additional difficulty for the stability analysis. In fact the tests for dissipative boundary conditions of conservation laws are directly extendable to balance laws only if the source terms themselves have appropriate dissipativity properties that may be expressed in terms of matrix inequalities. This is the topic of the next section. Otherwise, as shown in Sections 5.3 and 5.6, it is only known through the special case of systems of two balance laws) that there are intrinsic limitations to the system stabilizability with local controls. The other sections of the chapter are devoted to special cases and illustrating examples Lyapunov exponential stability As we have seen in Chapter 1, the system 5.1) can always be represented by an equivalent linear system having the following form in Riemann coordinates: 5.3) R t + Λx)R x + Mx)R = t [, + ), x, L). 149

150 15 Chapter 5. Systems of Linear Balance Laws where R : [, + ) [, L] R n, Λ : [, L] D n, M : [, L] M n,n R). The maps Λ and M are of class C 1. The matrix Λx) is diagonal and defined as ) Λ + x) Λx) = Λ x) with Λ + x) = diag{λ 1 x),..., λ m x)}, Λ x) = diag{λ m+1 x),..., λ n x)}, λ i x) > i, x [, L], The diagonal entries of Λx) are the eigenvalues of the matrix F x). As in the previous chapters, our concern is to analyze the exponential stability of this system under linear boundary conditions in canonical form ) R + t, ) ) R + t, L) 5.4) R t, L) = K R t, ), t [, + ), and an initial condition 5.5) R, x) = R o x), x, L). The well-posedness of the Cauchy problem 5.3), 5.4), 5.5) in L 2 is addressed in Appendix A, see Theorem A.4. For the stability analysis, we adopt a L 2 Lyapunov function candidate: with V = R T Qx)R dx, 5.6) Qx) diag { Q + x), Q x) }, Q + C 1 [, L]; D + m), Q C 1 [, L]; D + n m). The time derivative of this function along the solutions of the system 5.3), 5.4), 5.5) is 5.7) with W 1 [ R T Qx)Λx)R ] L = W 2 dv dt = W 1 + W 2 R + ) T t, L) Q + L)Λ + L) R t, ) R + t, L) R t, ) ) R + ) t, L) Q )Λ ) R t, ) ) T K T Q + )Λ + ) Q L)Λ L) R T ) Qx)Λx) )x Qx)Mx) MT x)qx) R dx. ) K R + ) t, L) R, t, ) From 5.7), we have the following straightforward stability result which was already sketched in the early papers by Rauch and Taylor 1974) and Russell 1978) ).

151 5.1. Lyapunov exponential stability 151 Proposition 5.1. The solution Rt, x) of the Cauchy problem 5.3), 5.4), 5.5) exponentially converges to for the L 2 norm if there exists a map Q satisfying 5.6) such that the following Matrix Inequalities hold: i) the matrix Q + L)Λ + ) L) Q )Λ K T Q + )Λ + ) ) ) Q L)Λ K L) is positive semi-definite; ii) the matrix is positive definite x [, L]. Qx)Λx) ) x + Qx)Mx) + MT x)qx) with In Chapter 3, for linear conservation laws, we have considered the special case where Qx) P µx) = diag { P + e µx, P e µx}, P + = diag{p 1,..., p m }, P diag{p m+1,..., p n }, p i > for i = 1,..., n. In that case, Proposition 5.1 is specialized as follows. Proposition 5.2. The solution Rt, x) of the Cauchy problem 5.3), 5.4), 5.5) exponentially converges to for the L 2 norm if there exists real µ and p i >, i = 1,..., n, such that the following Matrix Inequalities hold: i) the matrix ) P + Λ + L)e µl P P Λ K T + Λ + ) ) ) P Λ L)e µl K is positive semi-definite; ii) the matrix µp µx) Λx) P µx)λ x) + M T x)p µx) + P µx)mx) is positive definite x, L). For general linear systems of the form 5.3), it is rather clear that more explicit convergence conditions can be derived only when the internal structure and the numerical values of the involved matrices Λx), Mx) and K are at least partially specified. The special case of balance laws with uniform i.e. independent of x) Λ and M matrices, will be addressed in the next section. Another interesting special case is when K = and the characteristic velocities are all constant and positive, i.e. m = n and Λ = Λ + D n +. In that case, the following corollary of Proposition 5.2 states that the convergence is unconditionally guaranteed, in accordance with the physical intuition.

152 152 Chapter 5. Systems of Linear Balance Laws Corollary 5.3. If K = and Λ D n + is constant, the solution Rt, x) of the Cauchy problem 5.3), 5.4), 5.5) exponentially converges to zero in L 2 norm. Proof. If the matrix Λ D n + is constant and K =, the first stability condition of Proposition 5.2 reduces to i) P Λe µl is positive semi-definite, which is trivially satisfied for any P diag{p 1,..., p n } D + n. The second stability condition of Proposition 5.2 reduces to: ii) e µx ) µp Λ + M T x)p + P Mx) which is satisfied if µ > is taken sufficiently large. is positive definite, In practical applications, we may have K = either by the nature of the physical boundary conditions or by the choice of the boundary control laws as we illustrate with the example of a chemical plug flow reactor hereafter. Example: Feedback control of an exothermic plug flow reactor According to the description given in Section 1.7, the dynamics of a parallel plug flow reactor PFR) are represented by the following system of balance laws: 5.8) t T r + V r x T r + k o T c T r ) k 1 rt r, C A, C B ) =, t C A + V r x C A + rt r, C A, C B ) =, t C B + V r x C B rt r, C A, C B ) =, t T c + V c x T c k o T c T r ) =, where V r t) is the reactive fluid velocity in the reactor, V c t) is the coolant velocity in the jacket, T r t, x) is the reactor temperature, T c t, x) is the coolant temperature. The variables C A t, x) and C B t, x) denote the concentrations of the chemicals in the reaction mixture. The function rt r, C A, C B ) representing the reaction rate is defined as follows: rt r, C A, C B ) = ac A bc B ) exp E ), RT r where a and b are rate constants, E is the activation energy and R is the Boltzmann constant. Remark that here we consider a plug flow reactor where the chemical flow and the coolant flow are parallel while the model in Section 1.7 was for a countercurrent system. The characteristic velocities V r and V c are supposed to be constant and the system is subject to the following constant boundary conditions: 5.9) T r t, ) = T in r, C A t, ) = C in A, C B t, ) =, T c t, ) = T in c. From 5.8), by summing the second and third equations, it follows that the dynamics of the total concentration C T C A + C B are simply described by the delay equation t C T + V r x C T =.

153 5.1. Lyapunov exponential stability 153 Therefore, for the stability analysis, there is no loss of generality if we assume that C T = C in A is constant x and t. Then the reaction rate function may be redefined as rt r, C A ) = a + b)c A bca in ) exp E ) RT r and the third equation of the system 5.8) may be ignored. A steady state T r x), C Ax), T c x) is a solution, over the interval [, L], of the differential system V r x T r = k o T c T r ) + k 1 rt r, C A), under the boundary conditions V r x C A = rt r, C A), V c x T c = k o T c T r ), T r ) = T in r, C A) = C in A, T c ) = T in c. In order to linearize the model, we define the deviations of the states T c, T r and C A with respect to the steady states: R 1 = T r T r, R 2 = C A C A, R 3 = T c T c. Then the linearized system expressed in Riemann coordinates around the steady state is t R 1 R 2 V r + V r x R 1 R 2 k k 1 φ 1 x) k 1 φ x) k + φ 1 x) φ x) R 1 R 2, R 3 } {{ V c } Λ R 3 } k {{ k } Mx) R 3 with φ 1 x) E ) φ x) a + b) exp RTr, x) C Ax) b ) a + b Cin A E RT r x)) 2 φ x). Moreover, the boundary conditions 5.9) expressed in Riemann coordinates are: R 1 t, ) =, R 2 t, ) =, R 3 t, L) =. This implies trivially that we are in the special case where Λ is constant and belongs to D + n while K =. Therefore the stability is guaranteed according to Corollary 5.3. Although the reactor is stable, the use of feedback control is nevertheless required in practice because of the risk of peaks in the temperature profile hot spots) and the possibility of thermal runaway. Indeed, by quoting Karafyllis and Daoutidis 22), the occurrence of excessive temperatures can have detrimental consequences on the operation of the reactor, such as catalyst deactivation, undesired side reactions, and thermal decomposition of the products.

154 154 Chapter 5. Systems of Linear Balance Laws For simplicity, we consider here the case of a simple proportional control. The command signal is the inlet cooling temperature Ut) = Tc in t) and the regulated output variable is the exit reactor temperature T r t, L). The control law is defined as: Ut) = T ref + k P T sp T r t, L)). where T sp is the temperature set-point, k P is the tuning parameter of the controller and T ref is an arbitrary reference temperature freely chosen by the user. For the stability analysis, without loss of generality, we may assume that T ref = T c ) Then, in Riemann coordinates, the boundary conditions are: R 1 t, ) R 1 t, L) R 2 t, ) = R 2 t, L). R 3 t, ) k P }{{} K R 3 t, L) The two stability conditions of Proposition 5.2 may be expressed as follows: ii) i) k 2 p < e µl p 1 µv r 2k + k 1 φ 1 x))) p 1 k 1 φ x) + p 2 φ 1 x) p 1 + p 3 )k p 1 k 1 φ x) + p 2 φ 1 x) p 2 µv r + 2φ x)) p 1 + p 3 )k p 3 µv c 2k ) is positive semi-definite. It is then clear that, for any positive p 1, p 2, p 3, µ can be taken sufficiently large to satisfy condition ii), and that the stability is therefore guaranteed as long as the control tuning parameter k p satisfies condition i) Linear systems with uniform coefficients Let us now consider the special case of linear systems with uniform coefficients: 5.1) R t + ΛR x + MR =, t [, + ), x, L), where the matrices Λ and M are constant. For µ sufficiently small, it is clear that the condition ii) of Proposition 5.2 is satisfied if there exist p i > such that M T P )+P )M is positive definite. In such a case, the matrix M is said to be diagonally stable because it is stable and the associated Lyapunov equation is satisfied with a diagonal weighting matrix see e.g. Barker et al. 1978) and Shorten et al. 29) for more information on diagonally stable matrices). In the next theorem, we shall show, with an appropriate choice of the weighting matrix Qx) of the Lyapunov function, that condition ii) of Proposition 5.1 is satisfied even if there exists P D + n such that M T P + P M is only positive semi-definite.

155 5.2. Linear systems with uniform coefficients 155 Theorem 5.4. If there exists P D + n such that 5.11) M T P +P M is positive semi-definite. and 5.12) K 1 < 1 with P Λ, then the system 5.4), 5.1) is exponentially stable for the L 2 norm in the sense of Definition 3.3). Proof. We use the Lyapunov function with Qx) defined as follows: V = Qx) R T Qx)R dx, P + e φx) P e φx) ) φx) µx + 1) c, µ >, c >, where P + D + m, P D + n m are defined by P diag { P +, P }. With this definition, the matrix involved in the second stability condition ii) of Proposition 5.1 is P 2 µ) µcx + 1) c 1 Λ Qx) + M T Qx) + Qx)M. The Taylor expansion of P 2 µ) is as follows: P 2 µ) = M T P + P M) + Sc)µ + higher order terms in µ, with Sc) x + 1) c 1[ ] c Λ P x + 1)M T P + P M). Then, c > is selected sufficiently large such that the matrix Sc) is positive definite for all x [, L]. Hence, by 5.11), we know that for sufficiently small µ >, P 2 µ) is positive definite for all x [, L]. Moreover, the matrix of the first stability condition i) of Proposition 5.1 is ) P + Λ + e µl+1)c P + Λ + e µ ) P 1 µ) K T K. P Λ e µ P Λ e µl+1)c Then, using 5.12), we know that P 1 ) is positive definite same argument as for W) in the proof of Theorem 3.2) and therefore that µ > can be taken sufficiently small such that condition i) of Proposition 5.1 holds. In the next section, we illustrate this theorem with the example of a linearized Saint- Venant-Exner model.,

156 156 Chapter 5. Systems of Linear Balance Laws Q t) Ht, x) V t, x) control gate B Bt, x) Z L x Fig.5.1: A pool of a sloping channel with a moving bathymetry and a boundary control gate. Application to a linearized Saint-Venant-Exner model In this section adapted from Diagne et al. 212), we consider a pool of a prismatic sloping open channel with a rectangular cross-section, a unit width and a moving bathymetry because of sediment transportation), as described in Section 1.5 and represented in Fig.5.1. The state variables of the model are: the water depth Ht, x), the water velocity V t, x) and the bathymetry Bt, x) which is the depth of the sediment layer above the channel bottom. The dynamics of the system are described by the coupling of Saint-Venant and Exner equations: 5.13) t H + V x H + H x V =, t V + V x V + g x H + g x B gs b + C V 2 H =, t B + av 2 x V =. In these equations, g is the gravity constant, S b is the bottom slope of the channel, C is a friction coefficient and a is a parameter that encompasses porosity and viscosity effects on the sediment dynamics. Steady state and linearization Here we assume a uniform steady state H, V, B which satisfies the relation gs b H = CV 2. In order to linearize the model, we define the deviations of the state Ht, x), V t, x), Bt, x) with respect to the steady state: hx, t) = Hx, t) H, vx, t) = V x, t) V, bx, t) = Bx, t) B.

157 5.2. Linear systems with uniform coefficients 157 Then the Saint-Venant-Exner model 5.13) linearized around the constant steady state is 5.14) t h + V x h + H x v =, t v + V x v + g x h + g x b C V 2 t b + av 2 x v =. H 2 h + 2C V H v =, Riemann coordinates In matrix form, the linearized model 5.14) can be written as 5.15) Y t + AY x + BY = where h Y v, b V H A g V g, av 2 B C V 2 H 2 2C V H. As we have seen in Section 1.5 the eigenvalues of the matrix A can be approximated as 5.16) λ 1 V + gh, λ 2 agv 3 gh V 2, λ 3 V gh. Obviously exact, but rather more complicated, expressions of the eigenvalues of A could also be obtained by using the Cardano-Vieta method, see e.g. Hudson and Sweby 23). Remark also that here, exceptionally, we use the notation λ 3 instead of λ 3 ) for the negative eigenvalue, in order to keep the symetry of the ongoing mathematical developments. Once the eigenvalues λ i of the matrix A are obtained, the corresponding left eigenvectors can be computed as L k = 1 λ k λ i )λ k λ j ) k i j {1, 2, 3}. V λ i )V λ j ) + gh H λ k We multiply 5.15) by L k in order to write the model in terms of the Riemann coordinates R k k = 1, 2, 3). Then we obtain gh 5.17) t R k + λ k x R k + L k BY =, k = 1, 2, 3, where R k = 1 [ V λ i )V λ j ) + gh ) h + H λ k v + gh b ]. λ k λ i )λ k λ j ) Conversely, we can express h, u and b in terms of the Riemann coordinates: h = R 1 + R 2 + R 3, T,

158 158 Chapter 5. Systems of Linear Balance Laws v = 1 [ λ1 H V ) R 1 + λ 2 V ) R 2 + λ 3 V ) ] R 3, b = 1 [ λ1 gh V ) ) 2 gh R 1 λ2 + V ) ) 2 λ3 gh R 2 + V ) ) ] 2 gh R 3. Using the coordinates R k, the last term of 5.17) writes: L k BY = γ 1 l k 2h + γ 2 l k 2v 5.18) = 3 s=1 γ 1 + γ 2 λ s V H ) l k 2R s, where γ 1 = C V 2 H 2, γ 2 = 2C V H, and l k 2 is the second component of L k. Equation 5.18) can be rewritten as: L k BY = C V H λ k λ k λ i )λ k λ j ) k i j {1, 2, 3}. 3 l=1 3V 2λ l ) R l, For the sake of simplicity, we introduce the following notation θ k : Then equation 5.17) writes: 5.19) t S k + λ k x S k + θ k = C V H λ k λ k λ i )λ k λ j ). 3 2λ l 3V )θ l S l =, k = 1, 2, 3), l=1 where the characteristic coordinates are now redefined as S k = 1 θ k R k. From 5.19), the linearized model 5.17) in characteristic form may now be written as S t + ΛS x + MS = where M = S = S 1, S 2, S 3 ) T, Λ = diag{λ 1, λ 2, λ 3 }, α 1 α 2 α 3 α 1 α 2 α 3, with α k = α 1 α 2 α 3 2λ k 3V ) θ k.

159 5.2. Linear systems with uniform coefficients 159 From Section 1.5, we know that the three eigenvalues of the matrix A are such that 5.2) λ 1 λ 2 > > λ 3 with λ 1 and λ 3 the characteristic velocities of the water flow and λ 2 the characteristic velocity of the sediment motion. On the basis of 5.2), we are now going to determine the sign of the coefficients α k in M. For α 1, we have From 5.2), we have Using the trace of A, we have also α 1 = C V H 2λ 1 3V ) λ 1 λ 1 λ 2 )λ 1 λ 3 ). λ 1 >, λ 1 λ 3 > and λ 1 λ 2 >. 3V 2λ 1 = 2λ 3 + 2λ 2 V. Since λ 2 is small, 3V 2λ 1 has the same sign as 2λ 3 V. Since λ 3 < is negative, we obtain: 3V 2λ 1 < and consequently α 1 >. For α 2, we have α 2 = C V H 2λ 2 3V ) λ 2 λ 2 λ 1 )λ 2 λ 3 ). Since the sediment motion is much slower than the water flow, we may assume that 3V 2λ 2 >. Moreover from 5.2), we have also λ 2 >, λ 2 λ 3 > and λ 2 λ 1 <. From these inequalities, we conclude that α 2 >. Finally, for α 3, we have: α 3 = C V H 2λ 3 3V ) λ 3 λ 3 λ 2 )λ 3 λ 1 ). Since λ 3 <, we have 3V 2λ 3 >. Using 5.2), we infer that: λ 3 λ 2 < and λ 3 λ 1 <. From the above inequalities, we conclude that α 3 >. Hence all the coefficients α k in matrix M are strictly positive. Lyapunov stability We are now going to show how Theorem 5.4 may be applied to analyse the stability of an open channel represented by Saint-Venant-Exner equations 5.13). As seen in Fig.5.1 the channel is provided at the upstream boundary with a hydraulic device allowing to assign

160 16 Chapter 5. Systems of Linear Balance Laws the value of the flow-rate Q t). At the downstream boundary there is an underflow control gate. Therefore we have the following boundary conditions in physical coordinates: 5.21) Q t) = Ht, )V t, ), Q L t) = Ht, L)V t, L) = k G 2g ) Ut) Bt, L)) Ht, L) + Bt, L) ZL, Bt, ) = B, where k G is a constant positive parameter, Z L is the external water level and Ut) denotes the aperture of the control gate which is the command signal of the control system See Section 1.4 for more details). Remark that the aperture may be partially blocked by the sediment. For a given constant inflow rate Q, a uniform steady state of the system is a quadruple H, V, B, U which satisfies the four equations Q = H V, gs b H = CV 2, B = B, Q = k G 2g ) U B ) H + B Z L. It is evident that physical equilibria exist only for H > Z L B >. For any Q, there is a unique steady state with the value of H such that Q = θh gh, with θ Sb C 1, and the control gate position U given by U = B + θ H gh. k G 2g H + B Z L This function is represented in Fig.5.2. It is interesting to observe that there is a minimal aperture U of the control gate which is given by U min = B + )3 3 2 θ Z L B ), 2 k G Under this value, there is no uniform steady state but a nonuniform steady state may exist obviously). On the other hand, it can be seen that there are two distinct equilibria for each admissible position of the control gate. We shall now show how the stability of those equilibria can be analyzed using Theorem 5.4. Expressed in the S Riemann coordinates, linearized around a steady state, the boundary conditions 5.21) are as follows: S 1 t, ) π 1 S 2 t, ) = π 2 S 1t, L) S 2 t, L), S 3 t, L) χ 1 } χ 2 {{ } S 3 t, ) K

161 5.2. Linear systems with uniform coefficients 161 U B 1st equilibrium 2nd equilibrium U min B Z L B 2 H +B Z L Fig.5.2: Locus of steady states for the Saint-Venant-Exner system. χ 1 χ 2 λ1 λ 3 λ2 λ 3 π 1 π 2 ) λ2 λ 3 λ 1 λ 2 ) λ1 λ 3 λ 2 λ 1 λ3 λ 1 λ3 λ 2 ) gh V 2 + λ 2 λ 3 gh V 2 + λ 1 λ 2 ) gh V 2 + λ 1 λ 3 gh V 2 + λ 1 λ 2 ), ), ) λ1 φh ) + ψh )λ 1 V ) 2 gh ) ) λ 3 φh ) + ψh )λ 3 V ) 2 gh, ) ) λ2 φh ) + ψh )λ 2 V ) 2 gh ) ) λ 3 φh ) + ψh )λ 3 V ) 2 gh, ) with φh ) θ gh 2 H H, ψh gh + B Z L ) ) k G 2 + B Z L gh. In order to analyze the stability of the equilibria by invoking Theorem 5.4, we have 1) to find a matrix P = diag{p 1, p 2, p 3 } such that S T M T P + PM ) S is a positive semidefinite quadratic form, 2) to check that the dissipative boundary condition K 1 < 1 is satisfied. is For the matrix P, a natural choice is p i = α i i = 1,2,3) since then the quadratic form S T M T P + PM ) 3 ) 2 S = 2 α i S i. i=1

162 162 Chapter 5. Systems of Linear Balance Laws In order to check the dissipativity condition K 1 < 1, we have to compute the matrix P Λ. We know that P = diag{α 1, α 2, α 3 } and Λ = diag{λ 1, λ 2, λ 3 } by definition. Consequently: { = diag λ1 α 1, λ 2 α 2, } λ 3 α 3 and The inequality holds if and only if λ 1 α 1 π 1 λ 3 α 3 K 1 λ = 2 α 2 π 2. λ 3 α 3 λ 3 α 3 λ 3 α 3 χ 1 χ 2 λ 1 α 1 λ 2 α 2 K 1 < ) π1 2 λ 1 α 1 + π 2 λ 2 α 2 2 < 1 λ 3 α 3 λ 3 α 3 and 5.23) χ 2 λ 3 α χ 2 λ 3 α 3 2 < 1. λ 1 α 1 λ 2 α 2 Using the definitions of α i, we get χ 2 λ 3 α 1 1 = λ 1 α 1 χ 2 λ 3 α 1 2 = λ 2 α 2 π 2 1 π 2 2 ) λ 1 α 1 λ3 λ 2 2λ1 3V ) gh V 2 ) 2 + λ 2 λ 3 = λ 3 α 3 λ 1 λ 2 2λ 3 3V gh V 2, + λ 1 λ 2 ) λ 2 α 2 λ3 λ 1 2λ2 3V ) gh V 2 ) 2 + λ 1 λ 3 = λ 3 α 3 λ 2 λ 1 2λ 3 3V gh V 2, + λ 1 λ 2 ) λ3 λ 2 2λ3 3V ) λ1 φh ) + ψh )λ 1 V ) 2 gh ) 2 ) λ 1 λ 2 2λ 1 3V λ 3 φh ) + ψh )λ 3 V ) 2 gh, ) ) λ3 λ 1 2λ3 3V ) λ2 φh ) + ψh )λ 2 V ) 2 gh ) 2 ) 2λ 2 3V λ 3 φh ) + ψh )λ 3 V ) 2 gh, ) λ 2 λ 1 Let us now compute the values of these four expressions corresponding to the approximate values of the characteristic velocities given by 5.24) λ 1 V + gh, λ 2 agθ3 1 θ 2 gh, λ 3 V gh. We get π 2 1 λ 1 α 1 λ 3 α 3 1 θ 1 + θ ) ) 2 θ, 2 + θ

163 5.3. Existence of a basic quadratic control Lyapunov function 163 π 2 2 λ 2 α 2 = Oθ 3 ) as θ, λ 3 α 3 χ 2 λ 3 α 1 1 λ 1 α 1 1 θ 1 + θ ) 2 + θ 2 θ χ 2 λ 3 α 1 2 = Oθ 3 ) as θ. λ 2 α 2 ) 21 θ)h + B Z L ) θh ) θ)h + B Z L ) θh, Since < θ 1 and H > Z L B >, it follows immediately that inequalities 5.22) and 5.23) are satisfied and that all uniform steady states are exponentially stable Existence of a basic quadratic control Lyapunov function for a system of two linear balance laws In this section, we shall now focus on the special case of systems of two linear balance laws in characteristic form 5.25) t R 1 + λ 1 x) x R 1 + γ 1 x)r 1 + δ 1 x)r 2 =, t R 2 λ 2 x) x R 2 + γ 2 x)r 1 + δ 2 x)r 2 =, where the functions λ 1, λ 2 are in C 1 [, L];, + )) and the functions γ i, δ i are in C 1 [, L]; R). For this particular class of systems, the purpose is to generalize the previous results by giving explicit necessary and sufficient conditions for the existence of stabilizing boundary controls that can be derived from quadratic Lyapunov functions. This analysis is made possible because, for the system 5.25), there always exists a useful coordinate transformation which was emphasized in Krstic and Smyshlyaev, 28b, Chapter 9). In order to state this coordinate transformation, we introduce the notations 5.26) ϕ 1 x) exp and the new variables x γ 1 s) λ 1 s) ds), ϕ 2x) exp 5.27) S 1 t, x) = ϕ 1 x)r 1 t, x), S 2 t, x) = ϕ 2 x)r 2 t, x). x δ 2 s) λ 2 s) ds), ϕx) ϕ 1x) ϕ 2 x), Then the system 5.25) is transformed into the following system expressed in these new coordinates: 5.28) with t S 1 + λ 1 x) x S 1 + ax)s 2 =, t S 2 λ 2 x) x S 2 + bx)s 1 =, ax) = ϕx)δ 1 x), bx) = γ 2x) ϕx). Let us consider this system under boundary conditions of the form 5.29) S 1 t, ) = k 1 S 2 t, ), S 2 t, L) = k 2 S 1 t, L). For this system, we have the stability condition given in the following corollary of Theorem 5.1.

164 164 Chapter 5. Systems of Linear Balance Laws Corollary 5.5. The steady state solution solution S 1 t, x), S 2 t, x) of the system 5.28), 5.29) is exponentially stable for the L 2 norm if there exist positive parameters p 1 >, p 2 > and µ > such that i) k1 2 p 2λ 2 ) p 1 λ 1 ), k2 2 p 1λ 1 L) p 2 λ 2 L) e 2µL, ) µλ1 x) x λ 1 x))p 1 e µx ax)p 1 e µx + bx)p 2 e µx ii) The matrix ax)p 1 e µx + bx)p 2 e µx µλ 2 x) + x λ 2 x))p 2 e µx is positive definite x [, L]. Based on a Lyapunov function of the form V = p1 S 2 1t, x)e µx + p 2 S 2 2t, x)e µx) dx, the stability conditions of Corollary 5.5 are only sufficient, possibly conservative and given in a rather implicit way which is not easily checked. Our purpose, in the sequel of this section, is to show that explicit necessary and sufficient conditions can be given for the existence of basic quadratic control Lyapunov functions, as defined in Definition 5.6 below, that can be used to guarantee the stabilizability of system 5.28) with decentralized boundary feedback control laws. We now consider the system 5.28) under the boundary conditions 5.3) S 1 t, ) = u 1 t), S 2 t, L) = u 2 t) Equations 5.28) and 5.3) form a control system where, at time t, the state is St, ) = S 1 t, ), S 2 t, )) T L 2, L) 2 and the control is Ut) = u 1 t), u 2 t)) T R 2. We introduce the following control Lyapunov function candidate see e.g. Coron, 27, Section 12.1) for the classical concept of control Lyapunov function): 5.31) VS) q1 x)s 2 1t, x) + q 2 x)s 2 2t, x) ) dx, where q 1 C 1 [, L];, + )) and q 2 C 1 [, L];, + )) have to be determined. The time derivative of V along the trajectories of 5.28), 5.3) is 5.32) with 5.33) 5.34) dv S, U) = dt = = B 2q1 S 1 t S 1 + 2q 2 S 2 t S 2 ) dx 2q1 S 1 λ 1 x S 1 + as 2 ) + 2q 2 S 2 λ 2 x S 2 + bs 1 ) ) dx Idx, B λ 1 L)q 1 L)S 2 1t, L) λ 2 L)q 2 L)u 2 2 λ 1 )q 1 )u λ 2 )q 2 )S 2 2t, ), I λ 1 q 1 ) x )S q 2 b + q 1 a)s 1 S 2 + λ 2 q 2 ) x )S 2 2. We introduce the following definition.

165 5.3. Existence of a basic quadratic control Lyapunov function 165 Definition 5.6. A function VS) with given q 1 and q 2 is a basic control Lyapunov function for the control system 5.28), 5.3) if and only if 5.35) S H 1, L) 2, U R 2 such that dv S, U). dt It is a strict basic control Lyapunov function if and only if 5.36) S H 1, L) 2 \ {, ) T }, U R 2 such that dv S, U) <. dt We then have the following theorem. Theorem 5.7. There exists a basic quadratic strict control Lyapunov function for the control system 5.28), 5.3) if and only if the maximal solution η of the Cauchy problem η = a + b 5.37) η 2 λ 1 λ 2, η) =. is defined on [, L]. The proof of this theorem will be based on the following preliminary proposition. Proposition 5.8. Let L >, let α C [, L]) and β C [, L]). If there exist f C 1 [, L]) and g C 1 [, L]) such that 5.38) f > in [, L], 5.39) g > in [, L], 5.4) f in [, L], 5.41) g in [, L], 5.42) f g αf + βg) 2 in [, L], then the maximal solution η of the Cauchy problem 5.43) η = α + βη 2, η) =, is defined on [, L]. Conversely, if the maximal solution of the Cauchy problem 5.43) is defined on [, L], there exist f C 1 [, L]) and g C 1 [, L]) such that 5.38) and 5.39) hold while 5.4), 5.41) and 5.42) are strict inequalities. The proof of this proposition can be found in Bastin and Coron 211). Let us remark that the function x, s) [, L] R αx) + βx)s 2 R is continuous in [, L] R and locally Lipschitz with respect to s. Hence the Cauchy problem 5.43) has a unique maximal solution.

166 166 Chapter 5. Systems of Linear Balance Laws Proof of Theorem 5.7. a) Only if" condition. A necessary condition for VS) to be a strict) control Lyapunov function is that I is a strictly positive quadratic form with respect to S 1, S 2 ) for almost every x in [, L], i.e. 5.44) 5.45) 5.46) λ 1 q 1 ) x in [, L], λ 2 q 2 ) x in [, L], λ 1 q 1 ) x λ 2 q 2 ) x q 1 a + q 2 b) 2 in [, L]. We define the functions f C 1 [, L]) and g C 1 [, L]) such that 5.47) 5.48) fx) λ 1 x)q 1 x), x [, L], gx) λ 2 x)q 2 x), x [, L]. The quadratic form VS) is coercive with respect to S 1, S 2 ) T L 2, L) 2, i.e. σ > such that VS) σ S S 2 2)dx, if and only if 5.38) and 5.39) hold. Note that 5.44) is equivalent to 5.4) and that 5.45) is equivalent to 5.41) for almost every x in [, L]. Property 5.46) is equivalent to 5.42) for every x in [, L] with α and β defined by 5.49) αx) ax) bx), βx), x [, L]. λ 1 x) λ 2 x) Following Proposition 5.8, we consider the maximal solution η of the Cauchy problem η = a + b η 2 λ 1 λ 2, η) =. It follows from Proposition 5.8 that a necessary condition for the existence of a control Lyapunov function VS) of the form 5.31) is that η is defined on [, L]. b) If" condition. Let us assume that η is defined on [, L]. Then there is a strict control Lyapunov function V y) of the form 5.31). Indeed, by Proposition 5.8, there exist q 1 C 1 [, L];, + )) and q 2 C 1 [, L];, + )) such that 5.44), 5.45) and 5.46) are strict inequalities in [, L]. Let us define the following decentralized feedback control laws 5.5) u 1 t) k 1 S 2 t, ), u 2 t) k 2 S 1 t, L). Then for any constant k 1 and k 2 selected such that 5.51) we have 5.52) k 2 1 λ 2)q 2 ) λ 1 )q 1 ), k2 2 λ 1L)q 1 L) λ 2 L)q 2 L), dv dt θ S 1 2 L 2,L) + S 1 2 L 2,L) ), for some θ > independent of S 1, S 2 ). This leads to exponential stability with a rate depending on θ and σ, themselves depending on q 1 and q 2.

167 5.3. Existence of a basic quadratic control Lyapunov function 167 Remark 5.9. One could believe that more general stabilizability conditions could be obtained by considering a more general Lyapunov function with an additional cross-term) of the form 5.53) VS) q1 x)s q 2 x)s q 3 x)s 1 S 2 ) dx. In fact, this is not true because it can be shown that, for the control system 5.28)-5.3), if 5.53) is a control Lyapunov function then q 3 x) must be zero. The proof of this assertion can be found in Bastin and Coron 211). In the next section, the Theorem 5.7 is illustrated with an application to the control of an open channel represented by linearized Saint-Venant equations. Application to the control of an open channel We consider a pool of a prismatic horizontal open channel with a rectangular cross section and a unit width. The dynamics of the system are described by the Saint-Venant equations 5.54) t H + x HV ) =, ) V 2 t V + x 2 + gh + C V 2 H =, with the state variables Ht, x) = water depth and V t, x) = water velocity. C is a friction coefficient and g is the gravity acceleration. As illustrated in Fig.5.3, the channel is provided with hydraulic control devices pumps, valves, mobile spillways, sluice gates,...) which are located at the two extremities and allow to assign the values of the flow-rate on both sides: 5.55) Q t) = Ht, )V t, ), Q L t) = Ht, L)V t, L). The system 5.54), 5.55) is a control system with state Ht, x), V t, x) and controls Q t) and Q L t). Q t) Ht, x) V t, x) Q L t) L x Fig.5.3: Horizontal pool of an open channel controlled with boundary fluxes Q and Q L.

168 168 Chapter 5. Systems of Linear Balance Laws A steady state or equilibrium profile), corresponding to the set-point Q, is a couple of time-invariant nonuniform i.e. space-varying) state functions H x), V x) such that H x)v x) = Q which satisfy the differential equations CV ) V x H = H x V = gh V 2. For any constant inputs Q t) = Q L t) Q, the open-loop system has a continuum of non-isolated equilibria H x), V x) which are therefore not asymptotically stable. The objective is to design decentralized control laws, with Q t) function of Ht, ) and Q L t) function of Ht, L), in order to stabilize the system about a constant flow-rate set point Q and a constant level set-point H ). As we have seen in Section 1.4, using the definitions 5.57) g R 1 t, x) = V t, x) V x) + Ht, x) H x)) H x), g R 2 t, x) = V t, x) V x) Ht, x) H x)) H x), the linearized Saint-Venant equations around the steady state, written in characteristic form, are : t R 1 + λ 1 x) x R 1 + γ 1 x)r 1 + δ 1 x)r 2 =, with the characteristic velocities t R 2 λ 2 x) x R 2 + γ 2 x)r 1 + δ 2 x)r 2 =. λ 1 x) = V x) + gh x), λ 2 x) = V x) gh x), and the coefficients γ 1 x) = δ 1 x) = γ 2 x) = δ 2 x) = CV 2 H CV 2 H CV 2 H CV 2 H [ 3 4 gh + V ) + 1 V 1 2 gh [ 1 4 gh + V ) + 1 V gh [ 1 4 gh V ) + 1 V 1 2 gh [ 3 4 gh V ) + 1 V gh The steady state flow is subcritical or fluvial) if the following condition holds 5.58) gh x) V 2 x) > x [, L]. Under this condition, the system is strictly hyperbolic with λ 2 x) < < λ 1 x) x [, L]. ], ], ], ].

169 5.3. Existence of a basic quadratic control Lyapunov function 169 According to our analysis above, in order to check the condition for the existence of a basic quadratic control Lyapunov function, we need to solve the following third-order differential system on [, L]: dv dx = C V x)) 5 ) Q gq V x)) 3 dψ dx = γ 1x) λ 1 x) + δ 2x) ψ) =, λ 2 x) V ) = Q H ), dη dx = eψx) δ 1 x) + γ 2x) λ 1 x) e ψx) λ 2 x) η2 x) η) =. The first equation computes the steady state profile V x). It is obtained from 5.56) and Q = H x)v x). The solution ψ of the second equation is such that ϕ = expψ) involved in the computation of ax) and bx) see 5.28)). The third equation is the ODE 5.37) in the statement of Theorem 5.7. As a matter of illustration, we compute the solution of this system with the following parameter values : g = 9.81 m/s 2, C =.2, Q = 1 m 3 /s, H ) = 2 m. The function η exists over the interval [, L] with L km which is the maximal length for which the flow remains subcritical, i.e. V x) < gq ) 1/ m/sec. The functions V x) and ηx) are is shown in Fig m/sec V 5 # km Fig.5.4: Functions V x) and ηx) computed for the example of an openchannel. km Let us now impose a boundary condition of the form 5.59) S 1 t, ) = k 1 S 2 t, )

170 17 Chapter 5. Systems of Linear Balance Laws with k 2 1 λ 2)q 2 ) λ 1 )q 1 ) to the system 5.28). Then, using the definition 5.27) of the S i coordinates, the definition 5.57) of the R i coordinates and the physical boundary condition 5.55), it is a matter of few calculations to get the physical stabilizing control law which implements the boundary condition 5.59) Q t) = Ht, ) H ) [ Q ϕ ] 1) + k 1 ϕ 2 ) gh )Ht, ) H )) ϕ 1 ) k 1 ϕ 2 ) for the open channel represented by the Saint-Venant equations. We remark that this control law is a non-linear feedback function of the water depth Ht, ) although it is derived on the basis of a linearized model. Obviously, a similar derivation leads to a control law for Q L t) at the other side of the channel. In addition, it can also be emphasized that the implementation of the controls is particularly simple since only measurements of the levels Ht, ) and Ht, L) at the two boundaries are required. This means that the feedback implementation does not require neither level measurements inside the pool nor any velocity or flow rate measurements Boundary control of density-flow systems In Section 2.2 we have analyzed the boundary control of linear density-flow systems described by conservation laws. In this section, we generalize the analysis to the case of linear density-flow systems described by balance laws with uniform steady states. We consider a system of two linear balance laws of the general form: 5.6) t H + x Q = t Q + λ 1 λ 2 x H + λ 1 λ 2 ) x Q αh + βq = t [, + ), x [, L], where λ 1 R +, λ 2 R +, α R, β R. The first equation can be interpreted as a mass conservation law with H the density and Q the flow density. The second equation can then be interpreted as a momentum balance law. We are concerned with the solutions of the Cauchy problem for the system 5.6) under an initial condition: H, x) = H o x), Q, x) = Q o x) x [, L] and two boundary conditions of the form: 5.61) Qt, ) = Q t), Qt, L) = Q L t), t [, + ). Any pair of constant states H, Q such that αh = βq is a potential steady state of the system. We assume that one of them has been selected as the desired steady state or set point. The Riemann coordinates defined around the set point H, Q are: 5.62) R 1 = Q Q + λ 2 H H ), R 2 = Q Q λ 1 H H ),

171 5.4. Boundary control of density-flow systems 171 with the inverse change of coordinates: 5.63) H = H + R 1 R 2 λ 1 + λ 2, Q = Q + λ 1R 1 + λ 2 R 2 λ 1 + λ 2. With these coordinates, the system 5.6) is written in characteristic form: 5.64) t R 1 + λ 1 x R 1 + γr 1 + δr 2 =, t R 2 λ 2 x R 2 + γr 1 + δr 2 =, with γ λ 1β α λ 1 + λ 2, δ λ 2β + α λ 1 + λ 2. Transfer functions Using the Laplace transform, the transfer functions between the inputs Q, Q L and the outputs Ht, ), Ht, L) are computed as follows see e.g. Litrico and Fromion, 29, Section 3.3) ): ) ) ) Hs, ) P s) P = 1 s) Q s), Hs, L) P 1 s) P 11 s) Q L s) with P s) = σ 2s)e σ1s)l σ 1 s)e σ2s)l, se σ2s)l e σ1s)l ) P 1 s) = σ 1 s) σ 2 s) se σ2s)l e σ1s)l ), P 1 s) = σ 2s) σ 1 s))e σ1s)+σ2s))l, se σ2s)l e σ1s)l ) P 11 s) = σ 1s)e σ1s)l σ 2 s)e σ2s)l, se σ2s)l e σ1s)l ) σ 1 s) = λ 1 λ 2 )s + α ds) 2λ 1 λ 2, σ 2 s) = λ 1 λ 2 )s + α + ds) 2λ 1 λ 2, ds) = λ 1 + λ 2 ) 2 s 2 + 2[λ 1 λ 2 )α + 2λ 1 λ 2 β]s + α 2. The poles of the system are the roots of the characteristic equation se σ2s)l e σ1s)l ) =. There is a pole at zero p = since the system is an integrator of the difference of flows Q Q L. The other poles are given by Litrico and Fromion, 29, Equ. 3.51): p ±k = λ 1 γ + λ 2 δ) ± 2λ 1λ 2 γδ k2 π 2 λ 1 + λ 2 λ 1 λ 2 L 2, k Z \ {}.

172 172 Chapter 5. Systems of Linear Balance Laws In the special case where γ = δ = i.e. a system of two linear conservation laws), the poles are given by p ±k = ± 2jkπ λ 1 λ 2, k Z. L λ 1 + λ 2 Thus they are all located on the imaginary axis i.e. they have a zero real part) and we recover the property that the system is not asymptotically stable as in Section 2.2. In the general case where γ and/or δ, let k m be the greatest integer such that the radicand is positive. Then the poles obtained for < k k m are real, and those obtained for k > k m are complex conjugate with a real part equal to Rp ±k ) = λ 1 γ + λ 2 δ). These poles can be stable or unstable depending of the respective signs and the relative values of γ and δ. In any case, the system is never asymptotically stable since there is always a pole at zero and the issue of boundary feedback stabilization is worth be addressed. Boundary feedback stabilization with two local controls We consider the system 5.6) represented in Riemann coordinates by the model 5.64) with γ and/or δ the special case where γ = δ =, i.e. a system of two linear conservation laws, has been treated extensively in Section 2.2). In order to define local boundary feedback controls, we use the property, introduced in Section 5.3, that the model 5.64) is equivalent to 5.65) where t S 1 + λ 1 x S 1 + δe cx S 2 =, t S 2 λ 2 x S 2 + γe cx S 1 =, c 1 γ λ 1 c 2 δ λ 2, c c 1 + c 2, and the coordinates S 1 and S 2 are defined as 5.66) S 1 t, x) = e c1x R 1 t, x), S 2 t, x) = e c2x R 2 t, x). From Theorem 5.7, we know that there exists a basic quadratic strict control Lyapunov function for the system 5.65) under boundary control if and only if the Cauchy problem 5.67) η = ae cx + be cx η 2, a δ λ 1, b γ λ 2, η) =, has a solution on [, L]. Let us define θ e cx η. Then 5.67) becomes 5.68) θ + cθ = a + bθ 2, θ) =.

173 5.4. Boundary control of density-flow systems 173 The Cauchy problem 5.68) has a solution for all L if and only if cθ = a + bθ 2 for some θ, that is if and only if c and therefore if and only if 5.69) γ λ 1 + δ λ 2. Let us assume that 5.69) holds. Then, using the analysis of Section 5.3, we know that, for any L, there is a quadratic Lyapunov function V = q1 x)s 2 1t, x) + q 2 x)s 2 2t, x) ) dx, such that the stability of the system 5.65) is obtained under condition 5.69) with boundary conditions of the form 5.7) S 1 t, ) = k 1 S 2 t, ), k 2 1 λ 2 q 2 ) λ 1 q 1 ), S 2 t, L) = k 2 S 1 t, L), k 2 2 λ 1 q 1 L) λ 2 q 2 L). Using the definitions 5.62) and 5.66) of the coordinates R i and S i, we get the two local boundary feedback control laws that can be used to stabilize the density-flow system 5.6) at the set point H, Q : 5.71) Q t) Q + k H Ht, )), k λ 1k 1 + λ 2 1 k 1, Q L t) Q k L H Ht, L)), k L λ 2k 2 e cl + λ 1 1 k 2 e cl. It follows from our analysis above that, under conditions 5.69), the steady state H, Q ) of the closed loop system 5.6), 5.61), 5.71) is exponentially stable if the control tuning parameters k and k L are chosen such that inequalities 5.7) hold. Feedback-feedforward stabilization with a single control Let us now consider the situation where Q t) is the only control input while the other input Q L t) is a measurable disturbance. Then, from the previous section, a natural candidate control law is: 5.72) Q t) Q L t) + k P H Ht, )), where k P is a tuning parameter and H is the density set point. This control law involves a feedforward term Q L t) which compensates for the measured disturbance) and a proportional feedback term k P H Ht, )) for the stabilization. Assuming a constant disturbance Q L t) = Q, if k P the closed-loop system has a unique steady state H, Q ) and can be written in transformed Riemann coordinates 5.65) with boundary conditions S 1 t, ) = k 1 S 2 t, ), k 1 = k P λ 2 k P + λ 1, S 2 t, L) = λ 1 λ 2 e cl S 1 t, L).

174 174 Chapter 5. Systems of Linear Balance Laws Then, using again the analysis of Section 5.3, we know that the Cauchy problem 5.68) has a solution for all L if and only if inequality 5.69) holds. Using this inequality, it can be shown that the system 5.6) is exponentially stable with the control 5.72) for any L such that 5.73) ηl) e cl = θl) < λ 2 λ 1, provided the control tuning parameter k P is selected such that kp λ 2 k P + λ 1 ) 2 λ 2q 2 ) λ 1 q 1 ). The value of θl) can be computed explicitly in the special case where γ and δ are both positive. In such case, we have a >, b >, c > and 5.68) becomes 5.74) θ = a cθ + bθ 2, θ) =. The discriminant of equation bθ 2 cθ + a = is 5.75) = c 1 + c 2 ) 2 4c 1 c 2 = c 1 c 2 ) 2 Therefore the two zeroes of the polynomial bθ 2 cθ + a are θ 1 c 1 b = λ 2 λ 1, θ 2 c 2 b = δ γ, and 5.74) is equivalent to 1 θ 1 ) = θ 2 θ 1 ), θ) =. θ θ 2 θ θ 1 This gives the following function: e bθ2l e bθ1l θl) = θ 1 θ 2 θ 2 e bθ2l θ 1 e bθ1l. This function is monotonically increasing with lim θl) = min ) λ 2 θ 1, θ 2 = min, δ ). L + λ 1 γ Then, from 5.73) we see that the the system 5.6) is stabilizable for any L. Stabilization with Proportional-Integral control In the continuation of the previous paragraph, we now extend the stabilization analysis to the case where Q t) is provided with a Proportional-Integral PI) controller and Q L t) is a non-measurable disturbance see Section 2.2 for further details and motivations on the use of PI control in this case).

175 5.4. Boundary control of density-flow systems 175 The control law is defined as t 5.76) Q t) Q R + k P H Ht, )) + k I H Hτ, ))dτ, where Q R is a constant arbitrary reference value and H is the level set-point. For the closed-loop stability analysis there is no loss of generality in dealing with the special case Q L = Q R = Q = αh /β. In the Riemann coordinates 5.62), the closed-loop system dynamics are given by equations 5.64) that we recall here : 5.77) t R 1 + λ 1 x R 1 + γr 1 + δr 2 =, t R 2 λ 2 x R 2 + γr 1 + δr 2 =, Morover, the boundary conditions Qt, ) = Q t) and Qt, L) = Q are written, using the transformation 5.63): 5.78) R 1 t, ) = k 1 R 2 t, ) + k 3 Xt), k 1 k P λ 2 k I, k 3, k P + λ 1 k P + λ 1 R 2 t, L) = k 2 R 1 t, L), k 2 λ 1 λ 2, and Xt) t R2 τ, ) R 1 τ, ) ) dτ. For the stability analysis, we use the quadratic Lyapunov function V = p1 R λ 1e 2 µx/λ1 + p ) 2 R 2 1 λ 2e µx/λ2 dx + qx 2, 2 with positive constant coefficients p 1, p 2, q and µ. The time derivative of this function along the solutions of the system 5.77), 5.78) is dv dt = W 1 + W 2, where and ) ) R1 W 1 R1 R 2 Ω1 µ) dx, R 2 p 1 µ + 2γ) p e µx/λ1 1 δ e µx/λ1 + p 2γ e µx/λ2 Ω 1 µ) λ 1 λ 1 λ 2 p 1 δ e µx/λ1 + p 2γ p e µx/λ2 2 µ + 2δ), e µx/λ2 λ 1 λ 2 W 2 Xt) R 2 t, ) R 1 t, L) ) Xt) Ω 2 µ) R 2 t, ), R 1 t, L) λ 2

176 176 Chapter 5. Systems of Linear Balance Laws 2qk 3 p 1 k3 2 p 1 k 1 k 3 q1 k 1 ) Ω 2 µ) p 1 k 1 k 3 q1 k 1 ) p 2 p 1 k1 2. p 1 e µl/λ1 p 2 k2e 2 µl/λ2 We are looking for conditions under which the matrices Ω 1 and Ω 2 are positive definite for all x [, L]. We have the following proposition. Proposition 5.1. If 5.79) γ >, δ > and λ 1 λ 2 < γ δ < λ 2 λ 1, and if, the control tuning parameters k P and k I are selected such that 5.8) k 1 = k P λ 2 k P + λ 1 < δλ 2 k I, and k 3 = >, γλ 1 k P + λ 1 then there exist positive constant coefficients p 1, p 2, q and µ such that the matrices Ω 1 µ) and Ω 2 µ) are positive definite for all x [, L], and consequently, such that the PI control law 5.76) exponentially stabilizes the system 5.6). Proof. The determinant of the matrix Ω 1 µ) is given by Dµ) = p 1µ + 2γ) e p 2µ + 2δ) p1 δ µx/λ1 e µx/λ2 e µx/λ1 + p ) 2γ 2, e µx/λ2 λ 1 λ 1 λ 2 λ 2 = p 1p 2 µ 2 + 2µγ + δ)) p1 δ e µx/λ1 e µx/λ2 e µx/λ1 p ) 2γ 2. e µx/λ2 λ 1 λ 2 λ 1 λ 2 Let the coefficients p 1 > and p 2 > be selected such that 5.81) p 1 δ λ 1 = p 2γ λ 2. Then, under condition 5.79), the trace of Ω 1 µ) is positive and, under conditions 5.79) and 5.81), the determinant of Ω 1 µ) is positive for µ > sufficiently small since D) = and D ) = 2p 1p 2 γ + δ) λ 1 λ 2 >. The matrix Ω 2 µ) is positive definite if and only if the three following inequalities hold: i) p 1 e µl/λ1 p 2 k 2 2e µl/λ2 > ; ii) 2qk 3 p 1 k p 2 p 1 k 2 1 > ; iii) 2qk 3 p 1 k 2 3)p 2 p 1 k 2 1) p 1 k 1 k 3 + q1 k 1 ) ) 2 >. Using the definition of k 2 in 5.78) and the conditions 5.79) and 5.81), it is obvious that k2 2 = λ2 1 λ 2 < γλ 1 = p 1, 2 δλ 2 p 2

177 5.5. Proportional-Integral control in navigable rivers 177 and therefore that there exists µ > sufficiently small such that inequality i) is satisfied. Under conditions 5.8) and 5.81) we have 5.82) p 1 = γλ 1 < 1 p 2 δλ 2 k1 2, and therefore inequality ii) holds if 5.83) < k 3 < 2q p 1. The left hand side of iii) is a second order polynomial in k 3 : The discriminant of P k 3 ) is P k 3 ) = p 1 p 2 k qp 2 p 1 k 1 )k 3 q 2 1 k 1 ) 2. 4q 2 p 2 p 1 k 2 1)p 2 p 1 ). In view of 5.79) and 5.82), it is clear that >. Then, the polynomial P k 3 ) has two real roots given by k 3 ± = q [ p 2 p 1 k 1 ) ± ]. p 1 p 2 Under conditions 5.79) and 5.83), it can be checked that k + 3 > and k 3 < 2q p 1. It follows that, for any k 3 >, q > can be selected such that k 3, 2q/p 1 ) k 3, k+ 3 ) and therefore that conditions ii) and iii) are satisfied Proportional-Integral control in navigable rivers In Section 1.15 p.52), we have seen that a navigable river is a string of pools separated by hydraulic control gates as illustrated in Fig.1.2. An example of the control of a real life navigable river will be presented in detail in Chapter 8. In this section, for simplicity, we consider the special case of a channel having n rectangular pools with the same length L and the same width W. Using Saint-Venant equations 1.72), the dynamics of the system are described by the following set of balance laws: t Hi V i ) ) H i V i + x 1 2 V i gh i and the following set of boundary conditions: g[cv 2 i H 1 i S i ] ) =, i = 1..., n, 5.84a) 5.84b) H i t, L)V i t, L) = H i+1 t, )V i+1 t, ), i = 1,..., n 1, H i t, L)V i t, L) = k G 2g ) [ Hi t, L) U i t) ] 3, i = 1,..., n,

178 178 Chapter 5. Systems of Linear Balance Laws 5.84c) Q t) = W H 1 t, )V 1 t, ), where H i and V i denote the water level and the water velocity in the ith pool, U i is the position of the ith gate which is used as control action, S i is the constant slope of the ith pool, k G and C are constant shape and friction coefficients respectively, Q t) is the inflow rate considered as an external disturbance. For a constant inflow rate Q t) = Q, a steady state of the system is a constant state Hi, Vi i = 1,..., n) which satisfies the relations Q = W H i V i, S i H i = CV i ) 2. From Section 1.4, we know that the Riemann coordinates may be defined as 5.85) g R i = V i Vi ) + H i Hi ) H i g R n+i = V i Vi ) H i Hi ) Hi,, i = 1,..., n, and that the Saint-Venant equations, linearized about the steady state, are written as follows in these Riemann coordinates: 5.86) t R i + λ i x R i + γ i R i + δ i R n+i = t R n+i λ n+i x R n+i + γ i R i + δ i R n+i = i = 1,..., n, with the characteristic velocities and the parameters 1 γ i = gs i Vi λ i = V i + gh i, λ n+i = V i gh i, i = 1,..., n 1 2 gh i ), δ i = gs i 1 V i ) 1 + 2, i = 1,..., n ghi such that < λ n+i < λ i and < γ i < δ i. We assume that the control gates are provided with Proportional-Integral controllers that aim at regulating the water levels H i t, L) at the steady state set-point values H i. The control laws are as follows: t U i t) = U R k P i H i t, L) Hi ) k Ii H i τ, L) Hi )dτ. Then the linearization of the boundary conditions 5.84) gives the following relations in Riemann coordinates: 5.87a) R 1 t, ) = λ n+1 λ 1 R n+1 t, ), 5.87b) R i+1 t, ) = λ i + λ n+i )λ i + k i λ n+i ) R i t, L) λ n+i+1 R n+i+1 t, ) λ i+1 λ i+1 + λ n+i+1 ) λ i+1

179 5.5. Proportional-Integral control in navigable rivers λ n+iλ i + λ n+i )k n+i X i t), i = 1,..., n 1, λ i+1 λ i + λ n+i ) 5.87c) R n+i t, L) = k i R i t, L) + k n+i X i t), i = 1,..., n, with and X i t) k i t R i τ, L) R n+i τ, L)) dτ, i = 1,..., n, k P i λ i k P i + λ n+i, k n+i k Ii k P i + λ n+i. Dissipative boundary condition Theorem If the control tuning parameters k P i and k Ii are selected such that the dissipativity conditions γ i λ i k i <, k n+i > δ i λ n+i are satisfied, then the solutions of the system 5.86), 5.87) exponentially converge to zero for the L 2 norm. Proof. See Appendix E. Control error propagation In the decentralized control structure of a navigable river, the role of the PI controller in a given pool is to regulate the water level at its set-point while rejecting load disturbances. As we shall see in this paragraph, the decentralized structure where the PI controllers are tuned separately may lead to problems of global performance because of the spatial propagation of water level control errors and the possible amplification of the control actions in the downstream direction. To analyze that phenomenon, we consider a string of n pools with PI controllers. From Section 1.4, we know that the Saint-Venant equations, linearized about the steady state, have the structure of linear density-flow systems t h i + x q i =, t q i + λ i λ n+i x h i + λ i λ n+i ) x q i α i h i + β i q i =, i = 1,..., n, where h i H i H i, q i H i V i H i V i, α i CV i /Hi 2, β i 2CV i /H i. From Section 5.4, we deduce that the system dynamics are represented as follows in the frequency domain: 5.88) h i s) = P i s)q i s) + P n+i s)q i 1 s), i = 1,..., n, with the transfer functions P i s) σ n+is)e σn+is)l σ i s)e σis)l, se σn+is)l e σis)l )

180 18 Chapter 5. Systems of Linear Balance Laws P n+i s) σ n+is) σ i s))e σis)+σn+is))l, se σn+is)l e σis)l ) σ i s) λ i λ n+i )s + α i d i s) 2λ i λ n+i, σ n+i s) λ i λ n+i )s + α i + d i s) 2λ i λ n+i, d i s) λ i + λ n+i ) 2 s 2 + 2[λ i λ n+i )α i + 2λ i λ n+i β i ]s + α 2 i. Moreover, the PI controls are defined as 5.89) q i s) = C i s)h i s) with C i s) k P i + k Ii s, where k P i and k Ii are the proportional and integral gains respectively. Then, from 5.88) and 5.89), we see that the control errors propagate according to the transfer function 5.9) T i s) h is) h i 1 s) = P n+is)c i 1 s) 1 + P i s)c i s). In particular, it can be verified that the static gain of this transfer function is T i ) = k Ii 1) k Ii. Hence we see that T i ) = 1 if k Ii 1) = k Ii i.e. if the two integral gains are identically tuned), meaning that the level control error due to load disturbances will be propagated without attenuation nor amplification). An error propagation with a significant attenuation i.e. T i ) 1) requires that the two controllers be differently tuned with k Ii 1) k Ii. But, obviously this is necessarily detrimental for the performance quality in terms of set-point tracking in each pool Limit of stabilizability In this section we shall show that, for systems of balance laws, there is an intrinsic limit of stabilizability under local boundary control. We consider the simplest possible case of a control system with a single boundary control. The system is written in Riemann coordinates: t S 1 + x S 1 + cs 2 =, 5.91) t S 2 x S 2 + cs 1 =, t [, + ), x [, L], S 1 t, ) = ut), S 2 t, L) = S 1 t, L). This is a control system where the state is S 1, S 2 ) T L 2, L) 2 and the control is u R. This system is controllable see Russell 1978), and also Li 21) for the nonlinear case).

181 5.6. Limit of stabilizability 181 In the special case where c =, the system 5.91) is a control system of conservation laws and we know from Chapter 2 that it is stabilizable with a boundary feedback local control of the form ut) = ks 2 t, ) regardless of the value of L. In contrast, it is shown in Bastin and Coron, 211, Section 5) that for c > and for L large enough the control system 5.91) has no control Lyapunov of the form V q 1 x)s 1 x)) 2 + q 2 x)s 2 x)) 2 + q 3 x)s 1 x)s 2 x)dx. In the next proposition, we show that this limitation is intrinsic and that, in fact, the system 5.91) with c > can absolutely not be stabilized by means of linear boundary feedback laws of the form ut) = ks 2 t, ) if L is too large. Proposition If 5.92) L π c > there is no k R such that the equilibrium, ) T L 2, L) 2 is exponentially stable for the closed loop system t S 1 + x S 1 + cs 2 =, 5.93) t S 2 x S 2 + cs 1 =, t [, + ), x [, L], S 1 t, ) = ks 2 t, ), S 2 t, L) = S 1 t, L). Proof. Let us look at the real part of the eigenvalues of the generator associated to 5.93) as in finite dimension). Let σ C. We look for a solution S 1, S 2 ) T of 5.93) of the form S 1 t, x) = e σt fx), S 2 t, x) = e σt gx), t [, + ), x [, L]. Such a S 1, S 2 ) T is a solution of 5.93) if and only if 5.94) 5.95) 5.96) σf + f x + cg =, σg g x + cf =, fl) = gl), f) = kg). From 5.94), we have 5.97) g = 1 c σf + f x). From 5.95) and 5.97), we have 5.98) f xx + c 2 σ 2 )f =. From now on we assume that 5.99) σ 2 c 2.

182 182 Chapter 5. Systems of Linear Balance Laws Let ξ C be such that 5.1) ξ 2 = σ 2 c 2. From 5.98), we get the existence of A C and B C such that 5.11) f = Ae ξx + Be ξx. Then 5.97) gives 5.12) g = 1 c Aσ + ξ)e ξx + Bσ ξ)e ξx). From now on we assume that 5.11) and 5.12) hold. It is easily checked that 5.94) and 5.95) hold. Equation 5.96) is equivalent to 5.13) 5.14) A c + k σ + ξ)) + B c + k σ ξ)) =, A c + σ + ξ) e ξl + B c + σ ξ) e ξl =. There exists A, B) C C \ {, )} such that 5.13) and 5.14) hold if and only if 5.15) c + k σ + ξ)) c + σ ξ) = c + k σ ξ)) c + σ + ξ) e 2ξL. It follows from Lichtner 28), Neves et al. 1986) and Renardy 1993), that if the equilibrium, ) T L 2, L) 2 is exponentially stable for the closed loop system 5.93), then 5.16) From 5.15), we have and therefore k < 1. k = ξ e 2Lξ + 1 ) + σ + c) e 2Lξ 1 ) ξ e 2Lξ + 1) σ + c) e 2Lξ 1), k 1 = k + 1 = 2 σ + c) e 2Lξ 1 ) ξ e 2Lξ + 1) σ + c) e 2Lξ 1), 2ξ e 2Lξ + 1 ) ξ e 2Lξ + 1) σ + c) e 2Lξ 1), which imply 5.17) k 1 k σ + c) e 2Lξ 1 ) = 2ξ e 2Lξ = σ + c) + 1) sh Lξ) ξ 1 ch Lξ). Conversely, if 5.17) holds, then σ is an eigenvalue. We remark that the quantities sh Lξ), ch Lξ) ξ

183 5.7. References and further reading 183 are not changed if ξ is substituted by ξ. The functions sh Lξ) if σ c, σ C ξ L if σ = c, σ C ch Lξ), are holomorphic functions of σ. We now take σ, c). Then 5.17) is equivalent to 5.18) where F : [, c) R is defined by 5.19) F σ) := F σ) = k 1 k + 1, σ + c c σ tg L c 2 σ 2 ). We now assume that 5.92) holds. Let σ 1 [, c) and σ 2 σ 1, c) be defined by 5.11) σ 1 := c 2 π2 L 2, σ 2 := c 2 π2 4L 2. Then F is continuous on [σ 1, σ 2 ) and 5.111) 5.112) F σ 1 ) =, lim F σ) =. σ σ 2 From 5.16), we get that 5.113) k 1, ). k + 1 From 5.111), 5.112) and 5.113), there exists σ σ 1, σ 2 ) such that 5.18) holds. This concludes the proof of Proposition Remark Our proof of Proposition 5.12 shows that, if the inequality 5.92) is strict, then for every k R, the equilibrium, ) T L 2, L) 2 is exponentially unstable for the closed loop system 5.93) References and further reading Various interesting generalizations of the results presented in this chapter can be found in the following references. Tchousso et al. 29) show how the Lyapunov approach can be extended to linear systems of balance laws of higher spatial dimension. Prieur and Mazenc 212) show how time varying strict Lyapunov functions can be defined to get input-to-state stability for time varying linear systems of balance laws.

184 184 Chapter 5. Systems of Linear Balance Laws For a linearized Saint-Venant-Exner system, Diagne and Sène 213) use the Faedo- Galerkin method to prove the exponential convergence of the solutions with a Lyapunov function quadratic in the physical coordinates. Furthermore this reference presents simulations of the control system carried out with a finite volume method based on a Roe s scheme. The stability of switching systems of linear balance laws is addressed by Amin et al. 212) with the method of characteristics for the L norm, and by Lamare et al. 213) and Prieur et al. 214) with the Lyapunov method for the L 2 norm. The issue of singular perturbations in linear systems of balance laws is addressed in Tang et al. 215a). The propagation of control errors in strings of density-flow systems under non-local boundary control is discussed by Cantoni et al. 27) and Li and De Schutter 21).

185 Chapter 6 Quasi-Linear Hyperbolic Systems IN THIS CHAPTER, we continue to explore the use of Lyapunov functions for the stability analysis of quasi-linear hyperbolic systems under dissipative boundary conditions. We address the most general case of systems that cannot be transformed into Riemann coordinates. A balance law can be viewed as a conservation law which is perturbed by a so-called source term. In the first section, we shall see that, if the perturbation is not too big, the hyperbolic systems of balance laws with uniform steady states inherit of the stability properties of the corresponding hyperbolic systems of conservation laws. On the other hand, we remember that, in Chapter 5, linear hyperbolic systems are exponentially stable for the L 2 norm under the matrix inequalities of Proposition 5.1. In the second section, we shall see that, for quasi-linear systems, exactly the same matrix inequalities are sufficient to have the exponential stability of the steady state for the H 2 norm, in a way which is reminiscent to nonlinear conservation laws Stability of systems with uniform steady states Let us first consider the special case of a quasi-linear hyperbolic system with a uniform steady state see page13). The system is written: 6.1) 6.2) Y t + F Y)Y x + GY) =, t [, + ), x [, L], ) Y + t, ) ) Y + t, L) Y t, L) = H Y t, ), t [, + ), with Y : [, + ) [, L] R n and where F : Y M n,n R), G : Y R n and H : Y R n are of class C 1. We assume that the system is strictly hyperbolic which means that for each Y Y, the matrix F Y) has distinct real eigenvalues. Let us recall also that we assume that those eigenvalues do not vanish in Y. In this section, we assume furthermore that the system 6.1), 6.2) has a uniform steady state Y which is constant with respect to t and x and can therefore be assumed to be without loss of generality. In that case, we necessarily have G) = and H) =. As before, the matrix K is defined as the linearization of the map H at the steady state: K H ) M n,n R). The system 6.1), 6.2) is considered under an initial condition 6.3) Y, x) = Y o x), x [, L], 185

186 186 Chapter 6. Quasi-Linear Hyperbolic Systems which satisfies the following compatibility conditions of order 1 and are extensions of conditions 4.7), 4.8): ) ) Y + o ) Y + o L) 6.4) = H, Yo L) Yo ) 6.5) F + Y o )) x Y o ) + G + Y o )) = [ )] H + Y + o L) F + Y o L)) x Y o L) + G + Y o L))) Y + Yo ) [ H + + Y )] Y + o L) Yo F Y ) o )) x Y o ) + G Y o ))), 6.6) F Y o L)) x Y o L) + G Y o L)) = [ )] H Y + o L) F + Y o L)) x Y o L) + G + Y o L))) Y + Yo ) [ H + Y )] Y + o L) Yo F Y ) o )) x Y o ) + G Y o ))), where F + M m,n R), F M n m,n R), G + R m, G R n m, H + R m, H R n m are defined such that ) ) ) F + G + H + F, G, H. F Our purpose in this section is to show that the dissipativity conditions ρ 2 K) < 1 and ρ K) < 1 which are sufficient for the stability of systems of conservation laws see Chapter 4, Theorems 4.3, 4.11, 4.18, 4.22, 4.24), remain valid for the exponential stability of the steady state of general quasi-linear systems 6.1), 6.2) provided G ) is sufficiently small. We have the following stability theorem. Theorem 6.1. For given functions F and H, if ρ K) < 1, there exists ε >, ε 1 >, C 1 > and ν >, such that, for every function G of class C 1 such that G) = and G ) < ε, for every Y o C 1 [, L]; R n ) such that Y o 1 < ε 1 and satisfying the compatibility conditions 6.4), 6.5),6.6), the C 1 -solution of the Cauchy problem 6.1), 6.2), 6.3) satisfies Yt,.) 1 C 1 e νt Y o 1, t [, + ). Proof. The well-posedness of the Cauchy problem results from a straightforward extension of Theorem 4.1. The proof of the exponential stability is a direct extension of the proofs of Theorems 4.3 and It relies on the well known robustness of the Lyapunov analysis with respect to small perturbations of the system dynamics. An alternative proof using the method of characteristics can be found in Prieur et al. 28) see also Dos Santos Martins and Prieur 28) ). Remark 6.2. It is worth noting that the proof of this theorem gives an explicit estimate of ε which depends on F ) and K = H ). In the special case where G is quadratic with G H

187 6.2. Stability of general quasi-linear hyperbolic systems 187 respect to Y i.e. G ) = ), the stability becomes independent of F and K as stated in the next corollary. This property was repeatedly used in the stability proofs of Chapter 4 for systems of nonlinear conservation laws. Corollary 6.3. If ρ K) < 1, for every function G of class C 1 such that G) = and G ) =, the steady state Yt, x) of the system 6.1), 6.2) is exponentially stable for the C 1 -norm. Using the same approach as in Section 4.5 of Chapter 4, stability theorems similar to Theorem 6.1 can be stated for any C p norm, p N {}, under the dissipativity condition ρ K) < 1 and for any H p norm, p N {, 1}, under the dissipativity condition ρ 2 K) < 1, provided the initial condition satisfies compatibility conditions of order p and p 1 respectively Stability of general quasi-linear hyperbolic systems We now consider the case of a general quasi-linear hyperbolic system 6.7) 6.8) Y t + F Y)Y x + GY) =, t [, + ), x [, L], B Yt, ), Yt, L) ) =, t [, + ), with Y : [, + ) [, L] R n and where F : Y M n,n R), G : Y R n and B : Y Y R n are sufficiently smooth functions see property 6.16)). We assume that the system is strictly hyperbolic which means that for each Y Y, the matrix F Y) has distinct real eigenvalues. Furthermore, those eigenvalues are supposed to not vanish in Y. Stability condition for the H 2 -norm for systems with positive characteristic velocities In this subsection, for simplicity, we treat only the case where all eigenvalues of F Y) are strictly positive for all Y Y. Therefore, for all x [, L], with Y x) the steady state such that F Y x))y xx) + GY x)) =, the matrix F Y x)) can be diagonalized: Nx) M n,n R) such that Nx)F Y x)) = Λx)Nx), )} with Λx) diag {λ 1 F Y x)) ),..., λ n F Y x)) D n +, where λ i F ) is the i-th eigenvalue of F. We define the following change of coordinates: Zt, x) Nx) Yt, x) Y x)), Z = Z 1,..., Z n ) T. In the Z coordinates, the system 6.7), 6.8) is rewritten 6.9) 6.1) Z t + AZ, x)z x + BZ, x) =, B N) 1 Zt, ) + Y ), NL) 1 Zt, L) + Y L) ) =,

188 188 Chapter 6. Quasi-Linear Hyperbolic Systems where AZ, x) Nx)F Nx) 1 Z + Y x))nx) 1 with A, x) = Λx), [ BZ, x) Nx) F N 1 x)z + Y x))yxx) Nx) 1 N x)nx) 1 Z) ] + GNx) 1 Z + Y x)). Since, by definition of the steady state, [ ] B, x) = Nx) F Y x))yxx) + GY x)) =, it follows that there exists a matrix MZ, x) M n n R) such that 6.9) may be rewritten as 6.11) Z t + AZ, x)z x + MZ, x)z =, with M, x) B, x). Z As motivated in Chapter 1, it is assumed that the boundary condition 6.1) can be solved for Zt, ) and therefore written into the form 6.12) Zt, ) = H Zt, L) ). Our concern is to analyze the exponential stability of the steady state Zt, x) of the system 6.11) under the boundary condition 6.12) and under an initial condition 6.13) Z, x) = Z o x), x [, L]. which satisfies the compatibility conditions 6.14) Z o ) = H Z o L) ), 6.15) AZ o ), ) x Z o ) + MZ o ), )Z o ) = H Z o L) ) AZ o L), L) x Z o L) + MZ o L), L)Z o L) ), where H denotes the Jacobian matrix of the map H. The analysis is carried on under the property 6.16) A, M, H are of class C 2. The matrix K is defined as the linearization of the map H at the steady state: K H ). The well-posedness of the Cauchy problem and the existence of a unique classical solution results from the following theorem which is a direct extension of Theorem 4.9 see Appendix B for details).

189 6.2. Stability of general quasi-linear hyperbolic systems 189 Theorem 6.4. There exists δ > such that, for every Z o H 2, L); R n ) satisfying Z o H 2,L);R n ) δ and the compatibility conditions 6.14) and 6.15), the Cauchy problem 6.9), 6.12), 6.13) has a unique maximal classical solution with T, + ]. Moreover, if Z C [, T ), H 2, L); R n )) Zt, ) H2,L);R n ) δ, t [, T ), then T = +. The definition of the exponential stability is as follows. Definition 6.5. The steady state Zt, x) of the system 6.11), 6.12) is exponentially stable for the H 2 norm) if there exist δ >, ν > and C > such that, for every Z o H 2, L); R n ) satisfying Z o H 2,L);R n ) δ and the compatibility conditions 6.14), 6.15), the unique) solution Z of the Cauchy problem 6.11), 6.12), 6.13) is defined on [, + ) [, L] and satisfies Zt,.) H 2,L);R n ) Ce νt Z o H 2,L);R n ), t [, + ). We then have the following stability theorem. Theorem 6.6. The steady state Zt, x) of the system 6.11), 6.12) is exponentially stable for the H 2 -norm if there exists a map Q C 1 [, L]; D + n ) such that the following Matrix Inequalities hold: i) the matrix QL)ΛL) K T Q)Λ)K is positive semi-definite; ii) the matrix is positive definite x [, L]. Qx)Λx) ) x + Qx)M, x) + MT, x)qx) In order to define an appropriate Lyapunov function, we need the following lemma which is a straightforward generalization of Assumption Lemma 6.7. Let DZ, x) be the diagonal matrix whose diagonal entries are the eigenvalues λ i Z, x), i = 1,..., n, of the matrix AZ, x). There exist a positive real number η and a map E : B η [, L] M n,n R) of class C 2 such that 6.17) 6.18) EZ, x)az, x) = DZ, x)ez, x), Z B η, x [, L], E, x) = I n, x [, L], where I n is the identity matrix of M n,n R).

190 19 Chapter 6. Quasi-Linear Hyperbolic Systems In order to prove Theorem 6.6, we define the following candidate Lyapunov function: 6.19) V V 1 + V 2 + V 3, with 6.2) V 1 Z T E T Z, x)qx)ez, x)zdx, 6.21) V 2 Z T te T Z, x)qx)ez, x)z t dx, 6.22) V 3 Z T tte T Z, x)qx)ez, x)z tt dx. The proof of Theorem 6.6 will then be based on estimates of the time derivatives dv i /dt) i=1,2,3) along the system solutions. As usual see e.g. Comment 4.6), we assume that the solutions Z are of class C 3. Indeed, using a density argument similar to Comment 4.6, the estimates of dv i /dt given below remain valid, in the distribution sense with Z C [, T ], H 2, L); R n )) see the statement of Theorem 6.4). Estimate of dv 1 /dt The time derivative of V 1 along the solutions of 6.11), 6.12) is dv 1 dt = = = 2Z T E T Z, x)qx) [ EZ, x)z ] t dx [EZ, 2Z T E T ] ) Z, x)qx) x) Z + EZ, x)z t t dx [EZ, 2Z T E T ] Z, x)qx) x) Z t ) EZ, x)az, x)z x EZ, x)bz, x) dx. Using equality 6.17), we have dv 1 dt = [EZ, 2Z T E T ] Z, x)qx) x) Z t ) DZ, x)ez, x)z x EZ, x)bz, x) dx. Then, using integrations by parts, we get with 6.23) T 11 dv 1 dt = T 11 + T 12, [ Z T E T Z, x)qx)dz, x)ez, x)z ] L,

191 6.2. Stability of general quasi-linear hyperbolic systems ) T 12 From 6.23), we have Z T E T Z, x)q x)dz, x)ez, x)z Z T [E T Z, x)qx)dz, x)ez, x)] x Z [EZ, ) + 2Z T E T Z, x)qx) x) ]t Z EZ, x)bz, x) dx, 6.25) T 11 = Z T t, L)E T Zt, L), L)QL)DZt, L), L)EZt, L), L)Zt, L) + Z T t, )E T Zt, ), )Q)DZt, ), )EZt, ), )Zt, ). Let us introduce a notation in order to deal with estimates on higher order terms. We denote by OX; Y ), with X and Y, quantities for which there exist C > and ε >, independent of Z, Z t and Z tt, such that Y ε) OX; Y ) CX). Then from 6.18) and 6.25), using the boundary condition 6.12), we have [ ] 6.26) T 11 = Z T t, L) QL)ΛL) K T Q)Λ)K Zt, L) + O Zt, L) 3 ; Zt, L) ), and from 6.18), 6.24) we have 6.27) T 12 = Z T[ Qx)Λx) ) ] + x MT, x)qx) + Qx)M, x) Z dx + O Z 3 + Z Z t 2 ) )dx; Zt,.). Recall that for f C [, L]; R n ), we denote f = max{ fx) ; x [, L]}, see Section 4.1). Estimate of dv 2 /dt By time differentiation of the system equations 6.11), 6.12), Z t can be shown to satisfy the following hyperbolic dynamics: 6.28) [ ] A Z tt + AZ, x)z tx + diag Z Z, x)z t Z x + B Z Z, x)z t =, 6.29) Z t t, ) = H Zt, L))Z t t, L). In 6.28), the matrix A/ Z is defined as the matrix where the i, j entry is A ij / Z j, while the matrix diag[ A/ Z)Z t ] stands for the diagonal matrix whose diagonal entries are the components of the vector A/ Z)Z t. The time derivative of V 2 along the solutions of 6.11), 6.12), 6.28), 6.29) is dv 2 dt = 2Z T te T Z, x)qx) [ EZ, x)z t ]t dx

192 192 Chapter 6. Quasi-Linear Hyperbolic Systems = = Using equality 6.17), we have [EZ, ) 2Z T te T Z, x)qx) x) ]t Z t + EZ, x)z tt dx [EZ, 2Z T te T ] Z, x)qx) x) t Z t EZ, x)az, x)z tx EZ, x) [ ] A diag Z Z, x)z t Z x + B Z Z, x)z ) ) t dx. dv 2 dt = [EZ, 2Z T te T ] Z, x)qx) x) t Z t DZ, x)ez, x)z tx EZ, x) [ ] A diag Z Z, x)z t Z x + B Z Z, x)z ) ) t dx. Then, using integrations by parts, we get with 6.3) T 21 dv 2 dt = T 21 + T 22, [ ] L Z T te T Z, x)qx)dz, x)ez, x)z t, 6.31) T 22 Z T te T Z, x)q x)dz, x)ez, x)z t Z T t[e T Z, x)qx)dz, x)ez, x)] x Z t e µx [EZ, + 2Z T te T ] Z, x)qx) x) Z t t EZ, x) [ ] A diag Z Z, x)z t From 6.3), we have Z x + B Z Z, x)z ) ) t dx. 6.32) T 21 = Z T tt, L)E T Zt, L), L)QL)DZt, L), L)EZt, L), L)Z t t, L) + Z T tt, )E T Zt, ), )Q)DZt, ), )EZt, ), )Z t t, ). Then, using the boundary condition 6.29), we get [ ] 6.33) T 21 = Z T tt, L) QL)ΛL) K T Q)Λ)K Z t t, L) Moreover T 22 is written + O Z t t, L) 2 Zt, L) ; Zt, L) ).

193 6.2. Stability of general quasi-linear hyperbolic systems ) T 22 = Z T t [ Qx)Λx) ) ] + x MT, x)qx) + Qx)M, x) Z t dx + O Z t 2 ) Z t + Z )dx; Zt,.). Estimate of dv 3 /dt By time differentiation of the system equations 6.28), 6.29), Z tt can be shown to satisfy the following hyperbolic dynamics: [ ] [ ] A A 6.35) Z ttt + AZ, x)z ttx + 2diag Z Z, x)z t Z tx + diag Z Z, x)z t Z x t ] + B [ B Z Z, x)z tt + Z, x) Z 6.36) Z tt t, ) = H Zt, L))Z tt t, L) + H Zt, L))Z t t, L), Z t t, L)). Z t =, t The time derivative of V 3 along the C 3 -solutions of 6.11), 6.12), 6.28), 6.29), 6.35), 6.36) is dv 3 dt = = = Using equality 6.17), we have dv 3 dt = 2Z T tte T Z, x)qx) [ EZ, x)z tt ]t dx [EZ, ) 2Z T tte T Z, x)qx) x) ]t Z tt + EZ, x)z ttt dx 2Z T tte T Z, x)qx) [EZ, x) ] [ A EZ, x) 2diag Z Z, x)z t ] t Z tt EZ, x)az, x)z ttx [ ] A Z tx + diag Z Z, x)z t Z x t + B [ ] B ) ) Z Z, x)z tt + Z, x) Z t dx. Z t 2Z T tte T Z, x)qx) [EZ, x) ] [ A EZ, x) 2diag Z Z, x)z t Then, using integration by parts, we get ] t Z tt DZ, x)ez, x)z ttx [ ] A Z tx + diag Z Z, x)z t Z x t + B [ ] B ) ) Z Z, x)z tt + Z, x) Z t dx. Z t 6.37) dv 3 dt = T 31 + T 32,

194 194 Chapter 6. Quasi-Linear Hyperbolic Systems with 6.38) T 31 [ ] L Z T tte T Z, x)qx)ez, x)az, x)z tt, 6.39) T 32 Z T tte T Z, x)q x)ez, x)az, x)z tt Z T tt[e T Z, x)qx)ez, x)az, x)] x Z tt e µx [EZ, + 2Z T tte T ] [ ] A Z, x)qx) x) t Z tt EZ, x) 2diag Z Z, x)z t From 6.38), we have Z tx [ ] A + diag Z Z, x)z t Z x + B [ ] B ) ) t Z Z, x)z tt + Z, x) Z t dx. Z t T 31 = Z T ttt, L)E T Zt, L), L)QL)EZt, L), L)AZt, L), L)Z tt t, L) + Z T ttt, )E T Zt, ), )Q)EZt, ), )AZt, ), )Z tt t, ). Then, using the boundary condition 6.36), T 31 is written [ ] 6.4) T 31 = Z T ttt, L) QL)ΛL) K T Q)Λ)K Z tt t, L) Moreover T 32 is written 6.41) T 32 = + O Z tt t, L) 2 Zt, L) + Z tt t, L) Z t t, L) 2 + Z t t, L) 4 ; Zt, L) ). Z T tt [ + O Qx)Λx) ) ] + x MT, x)qx) + Qx)M, x) Z tt dx Z tt 2 Z t + Z ) + Z tt Z t 2) dx; Zt,.) + Z t t,.) ). In the next lemma, we shall now use these estimates to show that the Lyapunov function exponentially decreases along the system trajectories. Lemma 6.8. There exist positive real constants α, β and δ such that, for every Z such that Z + Z t δ, we have 6.42) 6.43) 1 β Z 2 + Z t 2 + Z tt 2 )dx V β dv dt αv. Z 2 + Z t 2 + Z tt 2 )dx,

195 6.2. Stability of general quasi-linear hyperbolic systems 195 Proof. Inequalities 6.42) follow directly from the definition of V and straightforward estimations. Let us introduce the following compact matrix notations: 6.44) 6.45) K QL)ΛL) K T Q)Λ)K, [ Lx) Qx)Λx) ) ] x + MT, x)qx) + Qx)M, x). Then, it follows from 6.26), 6.27), 6.33), 6.34), 6.4), 6.41) that 6.46) dv dt = ZT t, L)K Zt, L) Z T tt, L)K Z t t, L) Z T ttt, L)K Z tt t, L) + O Zt, L) Zt, L) 2 + Z t t, L) 2 + Z tt t, L) 2 ) + Z tt t, L) Z t t, L) 2 + Z t t, L) 4 ; Zt, L) ) Z T Lx) Z + Z T tlx) Z t + Z T ttlx) Z tt ) dx + O Z 2 Z + Z 2 Z t + Z t 2 Z + Z t 2 Z t + Z tt 2 Z t ) + Z t 2 Z tt + Z tt 2 ) Z t dx; Zt,.) + Z t t,.). Then by assumption i) of Theorem 6.6 and from 6.44), there exists δ 1 > such that if Zt, L) + Z t t, L) < δ 1 then Z T t, L)K Zt, L) Z T tt, L)K Z t t, L) Z T ttt, L)K Z tt t, L) 6.47) +O Zt, L) Zt, L) 2 + Z t t, L) 2 + Z tt t, L) 2 ) + Z tt t, L) Z t t, L) 2 + Z t t, L) 4 ; Zt, L) ). Let us recall the following Sobolev inequality see e.g. Brezis 1983) ): for a function ϕ C 1 [, L]; R n ), there exixts C 1 > such that 6.48) ϕ C 1 ϕx) 2 + ϕ x) 2 )dx. Moreover, from 6.9) and 6.28), we know also that there exist δ 2 > and C 2 > such that, if Zt, x) + Z t t, x) < δ 2, then 6.49) 6.5) 6.51) Z t t, x) C 2 Zt, x) + Zx t, x) ), Z tt t, x) C 2 Zt, x) + Zx t, x) + Z xx t, x) ), Z x t, x) C 2 Zt, x) + Zt t, x) ),

196 196 Chapter 6. Quasi-Linear Hyperbolic Systems 6.52) Z xx t, x) C 2 Zt, x) + Zt t, x) + Z tt t, x) ). By using repeatedly inequalities 6.48) to 6.52), it follows that there exists δ 3 C 3 > such that, if Zt,.) + Z t t,.) < δ 3, then > and 6.53) O Z 2 Z + Z 2 Z t + Z t 2 Z + Z t 2 Z t + Z tt 2 Z t + Z t 2 Z tt + Z tt 2 ) Z t )dx; Zt,.) + Z t t,.) C 3 Zt,.) + Z t t,.) )V. Using assumption ii) of Theorem 6.6, there exists γ > such that Z T Lx) Z + Z T tlx) Z t + Z T ttlx) Z tt ) dx 2γVZt,.), Z t t,.), Z tt t,.)). It follows from 6.46) that, if δ < minδ 1, δ 2, δ 3 ) is taken sufficiently small, then α > can be selected such that dv dt = 2γ + C 3 Zt,.) + Z t t,.) )V αv, for every Z such that Z + Z t δ. This concludes the proof of Lemma 6.8. Proof of Theorem 6.6. The proof follows from Lemma 6.8 exactly as the proof of Theorem 4.11 follows from Lemma Remark 6.9. Semi-linear systems. Here above, we have analyzed the exponential stability of general quasi-linear hyperbolic systems of the form Z t + AZ, x)z x + BZ, x) =. In this equation, A depends on Z and it is necessary to address the exponential stability for the H 2 norm. It is however interesting to point out that there are many examples where the system is semi-linear, i.e. A is constant or depends only on x but not on Z, as for instance in the models of Raman amplifier, plug flow reactors or chemotaxis presented in Chapter 1. In that special case of a semi-linear system of the form Z t + Λx)Z x + BZ, x) =, it is possible to establish the exponential stability in H 1 norm under assumptions i) and ii) of Theorem 6.6. The details of the analysis when Λ is constant can be found in Bastin and Coron 216).

197 6.3. References and further reading 197 Stability condition for the H p -norm for any p N {, 1} We now consider the most general class of quasi-linear hyperbolic systems with m positive and n m negative characteristic velocities represented by the equations 6.54) 6.55) 6.56) Z t + AZ, x)z x + BZ, x) =, t [, + ), x [, L], Z + ) t, ) Z + ) t, L) Z = H t, L) Z, t [, + ) t, ) Z, x) = Z o x), x [, L], with A, x) Λ + x) Assumption 6.16 is now replaced by Λ x) 6.57) A, M, H are of class C p. ), B, x) =, x [, L]. Using the same approach as in Section 4.5 of Chapter 4, the conditions for the H 2 norm given in Theorem 6.6 of the previous section can be generalized to the stability for any H p norm if the definition of exponential stability involves an appropriate extension of the compatibility conditions of order p 1 see page 143). With the usual notation K H ), the stability theorem may then be stated as follows. Theorem 6.1. The steady state Zt, x) of the system 6.54), 6.55) is exponentially stable for the H p -norm if there exists a map Q diag { Q +, Q } with Q + C 1 [, L]; D m) + and Q C 1 [, L]; D n m) + such that the following Matrix Inequalities hold: i) the matrix Q + L)Λ + ) 6.58) Q )Λ is positive semi-definite; K T Q + )Λ + ) Q L)Λ K ii) the matrix is positive definite x [, L]. Q x)λ + Qx)M, x) + M T, x)qx) The proof of this theorem is omitted References and further reading The generalization of Theorem 6.1 to the case of a non-autonomous coefficient matrix AZ, x, t) explicitly depending on both x and t has been carried out by Diagne and Drici

198 198 Chapter 6. Quasi-Linear Hyperbolic Systems 212). Moreover the issue of the input-to-state stability is addressed by Prieur and Mazenc 212). An interesting application of boundary control of the Raman amplifier system 1.21) is presented by Pavel and Chang 212). In this paper, the authors use an entropy Lyapunov function in order to prove the global existence of a C solution and its exponential convergence to the desired steady state. Extremum seeking control of cascaded Raman amplifiers is also addressed by Dower et al. 28). This chapter has dealt with the stability of the smooth C p or H p solutions of systems of nonlinear balance laws. Some results for entropic solutions have also been recently obtained. The boundary feedback stabilization of entropic solutions of scalar nonlinear balance laws is addressed by Perollaz 213). Moreover, in Coron et al. 215), the condition ρ 1 K) < 1 is shown to be sufficient for the exponential stability of the steady state of systems of two conservation laws in the framework of BV entropic solutions when the two characteristic velocities are positive.

199 Chapter 7 Backstepping Control IN THIS CHAPTER, we address the problem of boundary stabilization of linear hyperbolic systems of balance laws by full state feedback and by dynamic output feedback in observer-controller form. We consider only the case of systems of two balance laws as in Section 5.3. The control design problem is solved by using a backstepping method where the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The backstepping method for hyperbolic PDEs was initially introduced by Krstic and Smyshlyaev 28a), Krstic and Smyshlyaev 28b) and Smyshlyaev et al. 21). The present chapter is essentially based on Vazquez et al. 211) and Coron et al. 213) Motivation and problem statement We have seen in Section 5.3 that there is always a proper coordinate transformation such that any system of two linear balance laws can be written in the general form 7.1) t S 1 + λ 1 x) x S 1 + ax)s 2 =, t S 2 λ 2 x) x S 2 + bx)s 1 =, t [, + ), x [, L], where λ 1, λ 2 are in C 1 [, L]; R + ) and a, b are in C 1 [, L]; R). considered under the boundary conditions This system is here 7.2) S 1 t, ) = ut), S 2 t, L) = γs 1 t, L), t [, + ), where γ is a real constant. This is a control system where ut) R is the command signal. From Section 5.6, we know that, if this system is open-loop unstable, there is a limitation to the stabilization with a static boundary output feedback i.e a feedback of the state values at the boundaries only): there is a maximal length L above which the stabilization by a static boundary output feedback is impossible. In this chapter, we show how this limitation can be bypassed by using a dynamic boundary output feedback in so-called observer-controller form. In a first step, we design a fullstate feedback control law. Then we design a boundary feedback state observer. Finally the stabilizing output feedback controller is built by combining both designs. We define a linear reference model of the following form: 7.3) t S 1 + λ 1 x) x S 1 =, t S 2 λ 2 x) x S 2 =, t [, + ), x [, L], 199

200 2 Chapter 7. Backstepping Control with boundary conditions 7.4) S 1t, ) = ks 2t, ), S 2t, L) = γs 1t, L), t [, + ), where k is a tuning parameter selected such that kγ < 1. According to Theorem 2.4, this reference model is exponentially stable for the L 2 norm. The design method is then to seek a feedback control law which transforms the closed loop system into this reference model which, for this reason, is also called target system in the literature) Full-state feedback We introduce the vector and matrix notations ) ) S1 S, S S 1, S 2 S 2 ) λ1 x) Λx), Mx) λ 2 x) With these notations, the system 7.1) is written 7.5) S t + Λx)S x + Mx)S =. ) ax). bx) The state transformation called backstepping transformation) is then defined as follows: 7.6) S t, x) St, x) x P x, ξ)st, ξ)dξ, where the map P : X M 2,2, with X {x, ξ); x ξ L}. The map St,.) S t,.) is a Volterra transformation of the second kind. The transformation is linear, continuous and bijective from L 2, L); R 2 ) into L 2, L); R 2 ), see Volterra 1896) and Evans 191). Using 7.5) and 7.6), we have 7.7) S t t, x) = S t t, x) + = S t t, x) = S t t, x) + x x Moreover, using again 7.6), we have 7.8) S x t, x) = S xt, x) + x P x, ξ)s t t, ξ)dξ ) P x, ξ) Λξ)S x t, ξ) + Mξ)St, ξ) dξ ) P ξ x, ξ)λξ) + P x, ξ)λ x ξ) P x, ξ)mξ) St, ξ)dξ P x, L)ΛL)St, L) + P x, x)λx)st, x). x P x x, ξ)st, ξ)dξ P x, x)st, x).

201 7.2. Full-state feedback 21 Hence, from 7.5) and 7.7), 7.8), we get with We denote S t t, x) + Λx)S xt, x) = E 1 x)st, x) + E 2 t, x) E 1 x) Mx) + P x, x)λx) Λx)P x, x), E 2 t, x) P x, L)ΛL)St, L), x E 3 x, ξ)st, ξ)dξ, E 3 x, ξ) P ξ x, ξ)λξ) + Λx)P x x, ξ) + P x, ξ)λ x ξ) P x, ξ)mξ). ) p p P 1. p 1 p 11 Let us now show how the map P can be selected such that E 1 =, E 2 = and E 3 =. We see that E 1 = is equivalent to 7.9) p 1 x, x) = ax) λ 1 x) + λ 2 x), p bx) 1x, x) = λ 1 x) + λ 2 x). Using the boundary condition S 2 t, L) = γs 1 t, L), we have E 2 t, x) = if 7.1) p x, L)λ 1 L) γp 1 x, L)λ 2 L) =, p 1 x, L)λ 1 L) γp 11 x, L)λ 2 L) =. Then, in order to have E 3 t, x) =, the matrix function P x, ξ) is defined as the solution, in the domain X {x, ξ); x ξ L}, of the matrix hyperbolic partial differential equation 7.11) P ξ x, ξ)λξ) + Λx)P x x, ξ) + P x, ξ)λ x ξ) P x, ξ)mξ) = under the boundary conditions 7.9) and 7.1). The well-posedness of the system 7.9), 7.1) and 7.11) is established in Vazquez et al. 211) for γ. In the special case where γ =, the reference model is modified as follows: t S 1 + λ 1 x) x S 1 gx)s 2L, t) =, t S 2 λ 2 x) x S 2 =, where the function gx) has to be selected adequately, see Vazquez et al. 211). From now on, we assume that γ. Let us now look at the boundary conditions for S. From the definition 7.6) of the state transformation and using the boundary condition 7.2), we have S 1t, ) = ut) S 2t, L) = γs 1t, L), ) p, ξ)s 1 t, ξ) + p 1, ξ)s 2 t, ξ) dξ,

202 22 Chapter 7. Backstepping Control S 2t, ) = S 2 t, ) ) p 1, ξ)s 1 t, ξ) + p 11, ξ)s 2 t, ξ) dξ. In order to realize the dissipative boundary condition 7.4), the feedback control law is defined as 7.12) ut) = ks 2 t, ) + p, ξ) kp 1, ξ) ) S 1 t, ξ) + p 1, ξ) kp 11, ξ) ) ) S 2 t, ξ) dξ, where k is a tuning parameter selected such that kγ < 1. Hence this control law exponentially stabilizes the control system 7.1), 7.2). In the special case of a dead-beat control i.e. k = ), the steady state is reached in finite time t F = 1 λ 1 ξ) + 1 ) dξ. λ 2 ξ) Obviously, from 7.12), the practical implementation of this feedback control law needs the on-line knowledge of the full state St, x) on [, L]. In the next section, we shall see how this knowledge can be provided by a state-observer that uses a boundary on-line measurement of S 2 t, ) only Observer design and output feedback The objective is now to design an observer for the on-line estimation of St, x). Assuming that the output S 2 t, ) is measured on-line, the observer is a copy of the system 7.1) with additional so-called output injection terms: ) 7.13) Ŝ t t, x) + Λx)Ŝxt, x) + Mx)Ŝ + υ1 x) S2 t, ) υ 2 x) Ŝ2t, ) ) = with the boundary conditions 7.14) Ŝ 1 t, ) = ut), Ŝ 2 t, L) = γŝ1t, L). In these equations, the estimates are denoted by a hat accent while υ 1 x) and υ 2 x) are the output injection gains. We define the estimation errors S 1 S 1 Ŝ1, S2 S 2 Ŝ2. Then the so-called error system is obtained by subtraction of the observer equations 7.13) from the system equations 7.5): 7.15) St t, x) + Λx) S x t, x) + Mx) S Nx) St, ) =, with boundary conditions 7.16) S1 t, ) =, S2 t, L) = γ S 1 t, L).

203 7.3. Observer design and output feedback 23 The matrix Nx) is defined as 7.17) Nx) ) υ1 x). υ 2 x) In order to find the output injection gains υ i x), we use a backstepping transformation of the following form: x 7.18) St, x) S t, x) + P x, ξ)s t, ξ)dξ. where S t, x) is the state of the following reference model: t S 1 + λ 1x) x S 1 =, t S 2 λ 2x) x S 2 =, S 1 t, ) =, S 2 t, L) = γs 1t, L). Using 7.15) and 7.18), we have 7.19) x S t t, x) = S tt, x) + P x, ξ)s tt, ξ)dξ x = S tt, x) P x, ξ)λξ)s xt, ξ)dξ. Moreover, using again 7.18), we have x 7.2) Sx t, x) = S xt, x) + P x x, ξ)s t, ξ)dξ + P x, x)s t, x). Hence, from 7.15), 7.19) and 7.2), we get with x At, x) + x P x, ξ)at, ξ)dξ = E 1 x)s t, x) + E 2 t, x) E 3 x, ξ)s t, ξ)dξ, At, x) S tt, x) + Λx)S xt, x) E 1 x) P x, x)λx) Λx) P x, x) Mx), E 2 t, x) Nx) P x, )Λ))S t, ), E 3 x, ξ) P ξx, ξ)λξ) + P x, ξ)λ x ξ) + Mx) P x, ξ) + Λx) P x x, ξ). Denoting ) p p P 1, p 1 p 11

204 24 Chapter 7. Backstepping Control let us now show how the map P and the output gains υ i x) can be selected such that E 1 =, E 2 = and E 3 =. We see that E 1 = is equivalent to 7.21) p 1 x, x) = ax) λ 1 x) + λ 2 x), p bx) 1x, x) = λ 1 x) + λ 2 x). Using the boundary condition S 1 t, ) =, we see that E 2 t, x) = if ) Nx) P x, )Λ)) = for any S S 2 t, ) 2 t, ), and therefore if the output injection gains are chosen such that 7.22) υ 1 x) = λ 2 ) p 1 x, ), υ 2 x) = λ 2 ) p 11 x, ). Moreover, in order to satisfy the boundary condition S 2 t, L) = γs 1 t, L), we impose 7.23) p 1 L, ξ) = γ p L, ξ), p 11 L, ξ) = γ p 1 L, ξ). Then, in order to have E 3 t, x) =, the matrix function P x, ξ) is defined as the solution, in the domain X {x, ξ); x ξ L}, of the matrix hyperbolic partial differential equation 7.24) Pξ x, ξ)λξ) + P x, ξ)λ x ξ) Mx) P x, ξ) + Λx) P x x, ξ) = under the boundary conditions 7.21) and 7.23). The well-posedness of the system 7.21), 7.23), 7.24) is established in Vazquez et al. 211). The exponential stability of the error system 7.15) follows. This implies that the state estimate Ŝt, x) exponentially converges to the real state St, x), and even in finite time in the special case where γ =. A stabilizing output feedback controller is then obtained by combining the full state feedback controller 7.12) and the observer 7.13) as 7.25) ut) = ks 2 t, ) + and we have the following stability theorem. p, ξ) kp 1, ξ) ) Ŝ 1 t, ξ) + p 1, ξ) kp 11, ξ) ) ) Ŝ 2 t, ξ) dξ, Theorem 7.1. Consider the system 7.1) with boundary condition 7.2), control law 7.25) and initial condition S o L 2, L); R 2 ). Then, for any k such that kγ < 1, there exist ν > and C > such that St,.) L2,L);R 2 ) C e νt S o L2,L);R 2 ), t [, + ). Furthermore, the equilibrium S = is reached in finite time when γ =.

205 7.4. Backstepping control of systems of two balance laws Backstepping control of systems of two balance laws In this section, we briefly examine how the backstepping control approach can be extended to the nonlinear case, with a local stability property. We consider a general system of two balance laws in quasilinear form: 7.26) Z t + AZ, x)z x + BZ, x) =, where Z : [, L] [, + ) R 2 and A : R 2 [, L] M 2,2 R), B : R 2 [, L] R 2, AZ, x) and BZ, x) are twice continuously differentiable w.r.t. Z and x, A, x) = diag{λ 1 x), λ 2 x)} with λ i x) > and B, x) = x [, L]. Remark that these assumptions imply that Z may be a steady state of the system. With the notation Z Z 1, Z 2 ) T, the system 7.26) is considered under boundary conditions of the following form: 7.27) Z 1 t, ) = ut), Z 2 t, L) = GZ 1 t, L)), where the map G is assumed to be twice differentiable with G) =. The system 7.26), 7.27) is an open-loop control system when ut) R is an exogenous command signal. In order to design a backstepping observer-controller, we first rewrite the quasilinear system in a form which is, up to the nonlinear terms, identical to 7.1), 7.2). For that, we introduce the notation B, x) Z and we use the transformation 5.27): [ ] γ1 x) δ 1 x) = Γx) γ 2 x) δ 2 x) 7.28) St, x) = Φx)Zt, x) with Φx) ϕ1 x) ϕ 2 x) where the functions ϕ 1 and ϕ 2 are given by 5.26). Then, it can be shown for the details see Coron et al. 213)), that the quasilinear system 7.26) is written as follows in the S coordinates: 7.29) S t + Λx)S x + Mx)S + fs, x) = ) where and ) ) λ1 x) ax) Λx), Mx), λ 2 x) bx) ax) ϕ 1x) ϕ 2 x) δ 1x), bx) ϕ 2x) ϕ 1 x) γ 2x),

206 26 Chapter 7. Backstepping Control [ fs, x) Φx) AΦ 1 x)s, x) Λx) ] x Φ 1 x)s ) In the S coordinates, the boundary conditions 7.27) are then written + BΦ 1 x)s, x) Γx)Φ 1 x)s. 7.3) S 1 t, ) = ut), S 2 t, L) = γs 1 t, L) + gs 1 t, L)), with γ ϕ 2L) ϕ 1 L) G ) and gs 1 ) ϕ 2 L)G ϕ 1 L) ) γs 1. It is clear that, in these equations, the nonlinear terms satisfy f, x) =, g) = and g ) =. Then we see that the linear parts of 7.29) and 7.3) are indeed identical to the linear system equations 7.1) and 7.2) in Section 7.1. Therefore, it is quite natural to apply the linear observer-controller designed in the previous sections to the quasi-linear system case and to get a local stability property in H 2 norm in the same vein as what has been done in Section 6.2. The details of the analysis can be found in Coron et al. 213). S References and further reading Backstepping is a technique originally developed from 199 for designing stabilizing controls for nonlinear dynamical systems, see e.g. Coron, 27, Section 12.5) and the tutorial textbook by Krstic et al. 1995). The first extensions to PDEs were published by Coron and d Andréa-Novel 1998) for the beam equation and by Liu and Krstic 2) for discretized PDEs. Later on, Krstic and his collaborators introduced a modification of the method by means of an integral Volterra transformation of the second kind. This invertible transformation maps the original PDE control system into a reference system which is exponentially stable and the exponential decay can be made arbitrarily fast by choosing suitably the reference system). In this framework, the first backstepping designs were proposed for the heat equation in Liu 23) and Smyshlyaev and Krstic 24). The applications to wave equations appeared later in Krstic et al. 28), Smyshlyaev and Krstic 29), Smyshlyaev et al. 21). An excellent introduction to the backstepping method for PDEs is the book Boundary control of PDEs: A course on backstepping designs by Krstic and Smyshlyaev 28b). In this chapter we have only considered systems of two hyperbolic balance laws. More general systems have also been considered in the literature: linear systems with only one negative characteristic velocity in Di Meglio et al. 213), systems of three linear balance laws in Hu and Di Meglio 215), systems of n linear balance laws in Hu et al. 215a) and systems of n nonlinear balance laws in Hu et al. 215b). Note that in all these papers the reference models target systems) need to be more complicated than in this chapter. Let us finally mention the following interesting contributions on the backstepping approach for hyperbolic systems. An extension of backstepping to Navier-Stokes two-dimensional time-varying problems can be found in Vazquez et al. 28)

207 7.5. References and further reading 27 In some cases it is useful to use more general integral transformations than integral Volterra transformations of the second kind: see Smyshlyaev et al. 29) for an Euler-Bernoulli beam equation and Coron and Lü 214) for a Korteweg-de Vries equation. In the special case of systems of two linear balance laws, it is possible to find the exact analytical solution to a Goursat PDE system governing the kernels of a backsteppingbased boundary control law that stabilizes the system, see Vazquez and Krstic 213). The backstepping method can also be extended to design adaptive output feedback controllers for hyperbolic systems with the goal of disturbance rejection, see Aamo 213), or to account for unknown parameters, see Bernard and Krstic 214). In Lamare et al. 215a), the backstepping method is used for trajectory generation and the design of PI controllers for 2 2 linear hyperbolic systems with non-uniform coefficients.

208 28 Chapter 7. Backstepping Control

209 Chapter 8 Case Study: Control of Navigable Rivers HE OBJECTIVE of this chapter is to emphasize the main technological features that may T occur in real live applications of boundary feedback control of hyperbolic systems of balance laws. The issue is presented through the specific case study of the control of navigable rivers with a particular focus on the Meuse river in Wallonia south of Belgium) Geographic and technical data According to Wikipedia the Meuse see Fig.8.1) is a major European river, rising in France and flowing through Belgium and the Netherlands before draining into the North Sea. It has a total length of 925 km. In this chapter we shall mainly report on the control implementation in a stretch of the Meuse river which is called Haute Meuse and is located between the city of Givet at the Belgian-French border) and the city of Namur in Belgium). A map of the basin of the Meuse river is shown in Fig.8.2. As in most natural navigable rivers, the riverbed has been transformed such that the water flows through successive pools separated by weirs. The locations and the names of the Haute Meuse weirs can also be seen in Fig.8.2 while the profile of the river is shown in Fig.8.3. Fig.8.1: The Meuse river in the city of Liege Belgium). From

210 21 Chapter 8. Case Study : Control of Navigable Rivers La Plante Tailfer Rivière Hun Houx Dinant Anseremme Hastière Waulsort Chooz Fig.8.2: Meuse river basin and zoom on the Haute-Meuse stretch with the weir names and locations The figure is adapted from en.wikipedia.org/wiki/meuseriver)).

211 8.1. Geographic and technical data Chooz Hastière Waulsort 211 Anseremme Dinant 29m Houx Hun Rivière Tailfer La Plante 44.9 km Fig.8.3: Profile of the Haute Meuse stretch Each weir is provided with three or four parallel automated gates that are used to regulate the water-level in the pools. This is illustrated in Fig.8.4 with the last weir of the Haute Meuse La Plante). The gates can be operated in both overflow and underflow modes as sketched in Fig.8.5 and illustrated in Fig.8.6. The overflow mode is used in case of low flow rates less than 3 m3 /sec) while the underflow mode is used in the range 3-8 m3 /sec. Fig.8.4: La Plante weir with four parallel gates. Top view on the left from Google maps) and panoramic view from the Namur fortress on the right Fig.8.5: Each gate can be operated in both overflow and underflow mode.

212 212 Chapter 8. Case Study : Control of Navigable Rivers a) Overflow mode b) Underflow mode Fig.8.6: Automated control gates in the Meuse river Belgium).

213 8.2. Modelling and simulation 213 The river bathymetry has been recorded by using a swath sonar system see Fig.8.7 and Dal Cin et al. 25) for more details). A typical example of a cross-section of the riverbed obtained with swath bathymetry are shown in Fig.8.8. Fig.8.7: Bathymetric recording with the swath sonar system from Dal Cin et al. 25)) m 13 Fig.8.8: Example of a Meuse cross section obtained with swath bathymetry Modelling and simulation The dynamics of each pool are represented by the following general Saint-Venant equations see Equations 1.29) in Section 1.4): t A + x Q =, t Q + x Q 2 A + ga [ H A xa S b ) + S f ] =, where At, x) is the cross-sectional area of the water in the channel, Qt, x) is the flow rate or discharge), HA) is the water depth, S f is the friction term, S b x) is the bottom

214 214 Chapter 8. Case Study : Control of Navigable Rivers slope and g is the constant gravity acceleration. The friction term S f is represented by the Manning 1891) formula ) QP A)) 2/3 2 S f A 5/3 ν where ν is the so-called Strickler constant coefficient and P A) is the perimeter of the cross-sectional area. For simulation purpose, the Saint-Venant equations are integrated numerically using a standard Preissman scheme see e.g. Litrico and Fromion, 29, Section 2.2.2) ) with a spatial step size x = 1 m, a time step t = 1 s and a Strickler coefficient ν = 33 m 1/3 /s. For the simulations, the boundary conditions are given by the hydraulic model of the gates see Equations 1.25) and 1.26) in Section 1.4) written here in a form that covers both underfow and overflow modes in a single expression: Q g t, L) = w 2g k G1 max{, Ht, L) Uog } ) 3 + k G2 U ug H }{{}}{{} overflow underflow where see Fig.8.9) Ht, L) is the regulated water-level and H is the waterfall, w is the width of the gate, U og is the elevation of the top the gate and U ug is the depth of the underflow aperture of the gate, k G1 =.39 and k G2 =.7 are the adimensional discharge coefficients, Q g t, L) is the flow rate of one gate which has to be multiplied by the number of gates to get the total discharge at the boundary i.e. at the corresponding weir). Ht, L) U og H U ug Fig.8.9: The hydraulic gates determine the boundary conditions. In the next sections, we shall see that the simulation model is a fundamental tool for the practical set-up of the control system. In particular, it will be used to rationalize the selection of the set-points, of the control tuning parameters and of the time step for the digital implementation of the control law.

215 8.3. Control implementation Control implementation In this section, we address some practical issues that have to be considered for the control implementation. Local or nonlocal control? Two basic configurations to achieve set-point level regulation with PI boundary controllers are shown in Fig.8.1. a) Local Level measurement PI Controller Control action x Control action PI Controller Level measurement b) Nonlocal x Fig.8.1: PI control configurations for a pool of an open channel Local control, which is the usual structure in navigable rivers, means that the flow over a gate is used to regulate the water-level immediately upstream the same gate. In contrast, nonlocal control means that the flow over the upstream gate of a pool is used to regulate the water-level at the downstream gate of the pool. With nonlocal control, the effect of the control action is delayed which limits the achievable performance. However, in arid regions, this structure of control is generally used in irrigation networks to enhance water savings. Steady state and set-point selection For a constant flow rate Q, the profile of the steady state cross section A x) of the water flow is the solution of the differential equation [ Q 2 x A A 3 g H ] A xa S b ) + S f =. In order to set up the control system, the first task is to use the simulation model to compute the steady state profile in all pools of the river. This is illustrated in Fig.8.11 for the pool between Houx and Hun weirs. It can be seen that for low flow rates less than

216 216 Chapter 8. Case Study : Control of Navigable Rivers 89 Altitude meters) Houx steady-state level bottom slope =.4 5 km Level set-point at Hun for 16 m 3 /sec for 93 m 3 /sec Hun Fig.8.11: Steady state levels for low and high flow rates in the pool between Houx and Hun weirs. 2 m 3 /sec) the steady state profile is almost horizontal, while for high flow rates above 8 m 3 /sec) it is quasi-parallel to the bottom of the river albeit with a slight curvature). At each weir, the level set point must therefore be carefully selected in order to meet three conflicting objectives i) to guarantee a sufficient draft for the boats, that is a sufficient depth of water for a safe navigation see Fig.8.12); ii) to guarantee a sufficient air draft allowing the boats to pass safely under the bridges and other obstacles such as power lines; iii) to avoid water overflow on the banks of the river in the upper parts of the pools in case of high flow rates. For these reasons, as shown in Fig.8.11, the level set point is generally lower when the the flow rate is higher. Therefore, a schedule for the level set point is provided at each weir as a piecewise constant decreasing function of the flow rate. This also explains the counterintuitive observation that, in a regulated navigable river, the water-level is often lower when the flow is faster! Choice of the time step for digital control As already said above, in the Meuse river, the level set-point regulation is achieved with PI controllers. In practice, each weir is provided with a local Programmable Logic Controller PLC) wherein a numerical incremental PI control algorithm is implemented. The starting point is the following continuous time expression for PI control see Equation 2.25) in

217 8.3. Control implementation 217 air draft draft Fig.8.12: The level set point is selected to guarantee a safe navigation. Section 2.2): t Ut) U R + k P et) + k I eτ)dτ. In this equation, Ut) denotes the actual gate position and U R is a constant reference value. Moreover, et) H Ht, L) is the regulation error with H the level set point and Ht, L) the on-line measurement of the actual water-level immediately upstream of the gate see Fig.8.1a)). k P and k I are the tuning parameters. For the computer implementation, the continuous variables are approximated by their sampled counterparts, see e.g. Åström and Murray, 29, Section 1.5): Ut k ) U R + k P et k ) + k I It k ), where t k denotes the sampling instant. The sampling period is denoted t t k t k 1. In this expression, the integral term It k ) is given by approximating the integral with a sum: It k ) = It k 1 ) + t k I et k ). The incremental form is obtained by considering the time difference of the controller output as follows: Ut k ) = Ut k t) + k P etk ) et k t) ) + k I et k ). Regarding the choice of the time step value, it is well known that, for the efficiency and the robustness of the control, it is important to use the largest time step which is compatible with the system dynamics. A standard rule of thumb is to select t between one tenth and one fifth of the step response time of the system. Experimental step responses recorded at the weir of Dinant are shown in Fig.8.13 where it can be seen that the response time is around 2 minutes. However, the response time may be different from one place to another and, in addition, it is highly dependent on the flow rate value and on the number of gates that are in operation in a weir. This issue was investigated by intensive simulations for the various pools. For instance, the dependence of the step response time on the flow rate is shown in Fig.8.14 for the weir of Tailfer in the case where four gates are used, and also for the case

218 218 Chapter 8. Case Study : Control of Navigable Rivers t = 2 min Why? Fig.8.13: Water-level step responses recorded at Dinant on 9 and 19 october 211 for a flow rate of about 2 m 3 /sec. unit step response time usrt) computed with the Saint Venant model min 2 15 Hun Weir 3 gates) Tailfer Weir 4 gates) m 3 /sec flow rate Fig.8.14: Unit step response time versus flow rate computed by simulations. where only three gates are in operation at the weir of Hun. It can be seen that, depending on the number of gates and the flow rate, the step response time ranges from 12 to 2 minutes. Finally, it was decided to use a time step t = 2 minutes. This means that the gates move, during a short time, once every 2 minutes at most. This value has also the advantage not to exceed the limits of the patience of the human operators who are supervising the weir operation Control tuning and performance The role of PI control is to achieve set-point regulation, off-set errors cancellation and attenuation of load disturbances in a robust way see e.g. Åström and Murray, 29, Chapter 1) for a detailed motivation of PI controllers). In the Meuse river, the load disturbances mainly come from the river tributaries, the operation of electrical power plants along the river or, sometimes, from inadvertent actions of the human operators at the weirs.