Boundary control synthesis for hyperbolic systems: A singular perturbation approach

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1 Boundar control snthesis for hperbolic sstems: A singular perturbation approach Ying Tang Christophe Prieur Antoine Girard To cite this version: Ying Tang Christophe Prieur Antoine Girard. Boundar control snthesis for hperbolic sstems: A singular perturbation approach. 53rd IEEE Conference on Decision and Control (CDC 24) Dec 24 Los Angeles (CA) United States. pp <.9/CDC >. <hal-4435> HAL Id: hal Submitted on 2 Apr 25 HAL is a multi-disciplinar open access archive for the deposit and dissemination of scientific research documents whether the are published or not. The documents ma come from teaching and research institutions in France or abroad or from public or private research centers. L archive ouverte pluridisciplinaire HAL est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche publiés ou non émanant des établissements d enseignement et de recherche français ou étrangers des laboratoires publics ou privés.

2 Boundar control snthesis for hperbolic sstems: a singular perturbation approach Ying TANG Christophe PRIEUR and Antoine GIRARD Abstract In this paper we consider the problem of boundar control of a class of linear hperbolic sstems of conservation laws based on the singular perturbation method. The full hperbolic sstem is written as two subsstems namel the reduced sstem representing the slow dnamics and the boundar-laer sstem standing for the fast dnamics. B choosing the boundar conditions for the reduced sstem as zero the slow dnamics is stabilized in finite time. The main result is illustrated with a design of boundar control for a linearized Saint-Venant Exner sstem. The stabilization of the full sstem is achieved with different boundar conditions for the fast dnamics. I. INTRODUCTION Man distributed phsical sstems are described b hperbolic PDEs. This class of sstems with infinite dimensional dnamics is relevant for a wide range of phsical sstems having an engineering interest for instance hdraulic networks for irrigation or navigation [] gas flow in pipelines [2] networks of electrical transmission [3] or road traffic networks [4]. The significant importance of these applications motivates man research works on optimal control and controllabilit of hperbolic sstems as considered in [5] [6] [7]. The singular perturbation techniques started at the beginning of the 2th centur. A great deal of the earl motivation in this area arose from the studies of phsical problems exhibiting both fast and slow dnamics for instance DCmotor model voltage regulator in [8]. In late 98s the research works in the singularl perturbed partial differential equations occurred. This kind of sstems is interesting for analsis because of its relevance to man important phenomenon in different domains as reported in the surve paper [9] where a comprehensive bibliograph is involved. The present paper focuses on the boundar control of linear hperbolic sstems. Our main contribution here is to achieve the boundar control snthesis using singular perturbation method. The full hperbolic sstem of conservation laws is decomposed into two subsstems the reduced sstem and the boundar-laer sstem. B selecting K r = as the boundar conditions matrix for the reduced sstem the slow dnamics converges to the origin in finite time. The boundar conditions matrix K for the full sstem can be chosen such Y. TANG and C. PRIEUR are with Department of Automatic Control Gipsa-lab rue des Mathematiques BP Saint Martin d Heres France ing.tang@gipsa-lab.fr christophe.prieur@gipsa-lab.fr. A. GIRARD is with Department of Automatic Control Gipsa-lab rue des Mathematiques BP Saint Martin d Heres France and Laboratoire Jean Kuntzmann Université de Grenoble B.P Grenoble France. antoine.girard@imag.fr. that the stabilit condition of the full dnamics is satisfied. Using Tikhonov theorem in [] the full sstem converges to a small neighborhood of the equilibrium in finite time. In this paper the main result is applied to the Saint- Venant Exner model for the regulation of the water level in a channel. This problem has attracted the attention of man researchers for a long time for instance in [] [2] where the Lapunov methods are used to stabilize such sstems. In [3] the robust boundar control is designed for Saint-Venant equations with small perturbations. The paper is outlined as follows. Section II recalls the class of singularl perturbed sstems of conservation laws and the Tikhonov theorem for linear hperbolic sstems. In Section III the boundar controller established in [] is snthesized b the singular perturbation method. More precisel when the boundar condition of the reduced sstem is chosen as zero then the slow dnamics of the singularl perturbed sstem converges to the equilibrium in finite time. In Section IV the main result is illustrated b an application that is the design of boundar controls for the Saint-Venant Exner equations. Finall concluding remarks end the paper. Due to space limitation some proofs have been omitted. Notation. Given a matrix G G and G T represent the inverse and the transpose matrix of G respectivel. For a smmetric matrix S the minimum eigenvalue of the matrix S is denoted b λ min (S). For a positive integer n I n is the identit matrix in R n n. denotes the usual Euclidean norm in R n and is associated with the matrix norm. L 2 denotes the associated norm in ( ) L 2 ( ) space defined b f L 2 = f 2 2 dx for all functions f L 2 ( ). Similarl the associated norm in H 2 ( ) space is denoted b H 2 defined for all functions ( ) h H 2 ( ) b h H 2 = h 2 + h x 2 + h xx 2 2 dx. Following [4] we introduce the notation for all matrices K R (n+m) (n+m) ρ (K) = inf{ K D (n+m)+ } where D (n+m)+ denotes the set of diagonal positive matrix in R (n+m) (n+m). II. LINEAR SINGULARLY PERTURBED SYSTEM OF CONSERVATION LAWS We consider the following singularl perturbed sstem of conservation laws for a small positive perturbation parameter ε t (x t) + Λ x (x t) = () εz t (x t) + Λ 2 z x (x t) =

3 where x [ ] t [ + ) : [ ] [ + ) R n z : [ ] [ + ) R m Λ and Λ 2 are diagonal positive matrices in R n n and R m m respectivel. The boundar conditions for sstem () are written as follows (( ) ( ) t) ( t) = K t [ + ) (2) z( t) z( t) ( ) K K where K = 2 is a constant matrix in (n+m) K 2 K 22 (n + m) with K in R n n K 2 in R n m K 2 in R m n K 22 in R m m. Given two functions : [ ] R n and z : [ ] R m the initial conditions are: ( (x ) z(x ) ) = ( ) (x) z x [ ]. (3) (x) ( Remark ) : According to Proposition 2. in [4] ( for all z L 2 ( ) there exists a unique solution z) C ([ + ) L 2 ( )) for the Cauch ( ) problem ()-(3). B Proposition 2. in [5] for ever z H 2 ( ) satisfing the following compatibilit conditions: ( ) ( ) () z = K () () z (4) () ( ) ( ) Λ x() Λ ε Λ 2 zx() = K x() ε Λ 2 zx() (5) the Cauch ( problem ()-(3) has a unique maximal classical solution C z) ([ + ) H 2 ( )). Let us compute the reduced and boundar-laer subsstems for ()-(2) adapting the approach of [8] to the infinite dimensional case. B setting ε = in sstem () the reduced sstem is computed from: t (x t) + Λ x (x t) = z x (x t) =. (6a) (6b) Substituting (6b) into the boundar conditions (2) and assuming (I m K 22 ) invertible ields: ( t) = (K + K 2 (I m K 22 ) K 2 )( t) z(. t) = (I m K 22 ) K 2 ( t). The reduced sstem in R n is defined as ȳ t (x t) + Λ ȳ x (x t) = x [ ] t [ + ) (7) with the boundar condition ȳ( t) = K r ȳ( t) t [ + ) (8) where K r = K +K 2 (I m K 22 ) K 2 whereas the initial condition is given as ȳ(x ) = (x) x [ ]. (9) To define the boundar-laer sstem let us first perform a change of variable z = z (I m K 22 ) K 2 ( t) () this shifts the equilibrium of z to the origin. The boundar-laer sstem in R m is defined as z τ (x τ) + Λ 2 z x (x τ) = x [ ] τ [ + ) () with the boundar condition: z( τ) = K 22 z( τ) τ [ + ) (2) where τ = t ε is a stretching time scale. In τ time scale ( t) in () is handled as a fixed parameter with respect to time. The initial condition of the boundar-laer sstem is z(x ) = z (x) (I m K 22 ) K 2 () x [ ]. (3) Assuming ρ (K) < which implies in particular I m K 22 invertible we next state Tikhonov theorem for linear singularl perturbed sstem of conservation laws: Theorem : [] Consider the linear singularl perturbed sstem of conservation laws ()-(2). Assume that the boundar conditions matrix K satisfies ρ (K) < then for all initial conditions H 2 ( ) satisfing the compatibilit conditions for the reduced sstem () = K r () Λ x() = K r Λ x() and z L 2 ( ) there exist positive values ε C C and ω such that for all < ε < ε and for all t (. t) ȳ(. t) 2 L 2 Cεe ωt (4) z(. t) (I m K 22 ) K 2 ȳ( t) 2 L 2dt C ε. (5) Proof: The full proof can be seen in [] and it is based on the analsis of a Lapunov function for the sstem which contains the error of slow dnamics between the full sstem and the reduced sstem and the error of the fast dnamics between the full sstem and its equilibrium point. III. BOUNDARY CONTROL SYNTHESIS BASED ON THE SINGULAR PERTURBATION METHOD The boundar control snthesis method used in this paper relies on the singular perturbation technique. In the previous section we have recalled the linear singularl perturbed sstem ()-(2). Obviousl the ideal choice of the boundar conditions for the full sstem is K =. Such boundar conditions make the solutions converge to the equilibrium in finite time. However in the actual phsical problems the boundar conditions are not alwas free to be chosen for instance see the example in Section IV where the structure of boundar conditions matrix is prescribed b phsical constraint. In this section we consider a singular perturbation approach for the boundar condition snthesis. More precisel we first choose the boundar conditions matrix K r for the reduced sstem (7)-(8) as it makes the slow dnamics converge to the equilibrium in finite time and the fast dnamics are not modified. For example in Section IV the boundar conditions matrix K r = for the reduced sstem is achieved b a suitable choice of the control actions. Then the boundar conditions matrix K for the full sstem ()-(2) can be chosen

4 based on the boundar conditions for the slow dnamics such that the stabilit condition ρ (K) < for the full sstem is satisfied. Before introducing the stabilit result let us first give the following definition: Definition : The reduced sstem (7)-(8) is convergent in finite time if there exists positive value T such that for ever initial condition ȳ H 2 ( ) satisfing the compatibilit conditions () = K r () Λ x() = K r Λ x() the solution to the sstem (7)-(8) equals zero for all t T : ȳ(. t) = t [T + ). Proposition : If the boundar conditions matrix K r = then the reduced sstem (7)-(8) is convergent in finite time T where T is given b T = λ min (Λ ). (6) Corollar : If the boundar conditions matrix K r for the reduced sstem (7)-(8) is and if the boundar conditions matrix K for the linear singularl perturbed sstem of conservation laws ()-(2) satisfies ρ (K) < then for ever initial condition H 2 ( ) satisfing the compatibilit conditions () = and x() = and for all z L 2 ( ) there exist positive values ε C C ω and T = λ such that for all < ε < min(λ ) ε and for all t T (. t) 2 L 2 Cεe ωt (7) T z(. t) 2 L 2dt C ε. (8) Proof: The proof of this corollar is based on Theorem and Proposition. Using Proposition it follows that ȳ(x t) converges to the origin within time T and using (4) in Theorem we get that (7) holds. Moreover it is deduced from (5) + T T and thus T z(. t) (I m K 22 ) K 2 ȳ( t) 2 L 2dt z(. t) (I m K 22 ) K 2 ȳ( t) 2 L 2dt C ε z(. t) (I K 22 ) K 2 ȳ( t) 2 L 2dt C ε. (9) Similarl using Proposition in (9) we get that (8) holds. This concludes the proof of Corollar. This new method for boundar control snthesis is effective. Instead of choosing K = for the full sstem the boundar conditions matrix K r for the reduced sstem is selected as. The slow dnamics converges to the equilibrium in finite time. The boundar conditions for the full sstem can be chosen based on that for the reduced sstem such that the stabilit condition ρ (K) < is satisfied. Then the full sstem converges to a small neighborhood of the origin in finite time. IV. DESIGN OF BOUNDARY CONTROL FOR THE SAINT-VENANT EXNER MODEL In this section we appl the main result of the previous section to the Saint-Venant Exner equation which is an example of a singularl perturbed sstem of conservation laws. We consider a prismatic open channel with a rectangular cross-section and a unit width where all the friction losses are neglected. The effect of the sediment on the flow is handled in this model. The dnamics of the sstem are described b the Saint-Venant equation in [6] and Exner equation in [7] [8] [9]: H t + V H x + HV x = (2a) V t + V V x + gh x + gb x = x [ ] t (2b) B t + av 2 V x = (2c) where the state variables are the water level H(x t) the water velocit V (x t) the bathmetr B(x t) which is the sediment laer above the channel bottom. The gravit constant is g and the constant parameter which represents the porosit and viscosit effects on the sediment dnamics is denoted b a. The space variable is x [ ] and the time variable is t. A. Sstem linearization Let us consider a constant in space stead-state H V B. More precisel (2c) gives Vx = and we get successivel Hx = and Bx = from (2a) and (2b). Let us define the deviations of the state H V and B with respect to the stead-state for all x [ ] and t h = H H v = V V b = B B. The linearization of sstem (2) around the stead-state ields h t + V h x + H v x = v t + gh x + gb x + V v x = (2) b t + av 2 v x =. B. Dnamics in Riemann coordinates Let us perform a change of variable for the linearized sstem (2). More precisel following [8] [9] the characteristic coordinates are defined for each k = 2 3 b W k = ((V λ i)(v λ j)+gh ) h+h λ k v+gh b (λ k λ i)(λ k λ j) k i j { 2 3}. (22) Using the new variables W k sstem (2) can be rewritten as W t + ΛW x = (23) where W = (W W 2 W 3 ) T and Λ = diag(λ λ 2 λ 3 ) for all x [ ] t [ + ). According to [8] [9] the three eigenvalues of Λ are such that λ < < λ 2 < λ 3. (24)

5 In [8] [9] λ and λ 3 represent the velocities of the water flow and λ 2 represents the velocit of the sediment motion. The sediment motion is much slower than the water flow then we get that λ 2 << λ and λ 2 << λ 3. In this case b performing the change of spatial variable W ( x t) = W (x t) we ma assume without loss of generalit that λ > thus Λ is diagonal positive. Let us define a small positive value ε = λ2 λ 3 and a new time scale t = λ 2 t the sstem (23) is rewritten as the following singularl perturbed sstem for all x [ ] and for all t εw t + λ λ 3 W x = W + W 2 t 2x = (25) εw + W 3 t 3x =. C. Boundar conditions We assume that the channel is equipped with hdraulic control devices such as pumps valves spillwas gates etc. The water levels at upstream and downstream of the channel are assumed to be measured. The control action is provided b the control devices. In the present paper we introduce the following three boundar conditions (these are the same boundar conditions as in [2]): ) The first boundar condition describes the value of the channel inflow rate which is denoted b c ( t). Here we consider c ( t) as a control input (see [2]): H( t)v ( t) = c ( t). 2) The second boundar condition is given b gate operation at outflow of the reach. A gate model can be expressed as follows (see [2]): H( t)v ( t) = α [H( t) c ( t)] 3 where α is a positive constant coefficient. The control input is denoted b c ( t). 3) The third boundar condition is a phsical constraint on the bathmetr (see [9]): B( t) = B where B is a constant value. After the linearization of these boundar conditions we derive the following boundar conditions for sstem (2): H v( t) + V h( t) = c ( t) c (26) H v( t) + V h( t) = 3α(h( t) + c c ( t)) H c 2 (27) b( t) = (28) where c and c are constant control actions at the steadstate ( H V B ) T. The boundar conditions for sstem (25) are given as follows: W ( t) = k 2 W 2 ( t) + k 3 W 3 ( t) (29) W 2 ( t) = k 2 W ( t) (3) W 3 ( t) = ξ(k 2 )W ( t) (3) with ξ(k 2 ) = [(λ V ) 2 gh ] + k 2 [(λ 2 V ) 2 gh ] (λ 3 V ) 2 gh. (32) Proposition 2: The boundar conditions (26)-(28) for the sstem (2) are equivalent to the boundar conditions (29)- (3) for the sstem (25) with the following boundar control inputs for all t [ c ( t) = c + h( t) V + gh (k 2 φ φ 2 ) φ 2 λ 2 k 2 φ λ + k 2φ (V λ 2 )(V λ 3 ) φ 2 (V λ )(V ] λ 3 ) φ 2 λ 2 k 2 φ λ (33) [ c ( t) = c 2V + h( t) 3α H c 2H ( gh 3α ( φ + k 2 φ 2 + k 3 φ 3 ) H c H (λ φ k 2 λ 2 φ 2 k 3 λ 3 φ 3 ) φ (V λ 2 )(V λ 3 ) k 2 φ 2 (V λ )(V λ 3 ) H (λ φ k 2 λ 2 φ 2 k 3 λ 3 φ 3 ) + k 3φ 3 (V λ )(V )] λ 2 ) H (λ φ k 2 λ 2 φ 2 k 3 λ 3 φ 3 ) [ ] b( t) 2H 3α g( φ + k 2 φ 2 + k 3 φ 3 ) H c λ φ k 2 λ 2 φ 2 k 3 λ 3 φ 3 (34) where H c and φ k = (λ k λ i)(λ k λ j) for (i j k) in { 2 3} 3. Adopting the definitions of the reduced sstem and the boundar-laer sstem in Section II the two subsstems are computed as follows. The reduced sstem is with the boundar condition W 2 t + W 2x = (35) W 2 ( t) = K r W2 ( t) (36) where K r = k2k2 k. 3ξ(k 2) Let us perform the following change of variables: W = W k 2 k W 3ξ(k 2) 2( t) W 3 = W 3 k2ξ(k2) k W 3ξ(k 2) 2( t). The boundar-laer sstem is ( ) ( W λ ) ( ) + λ 3 W W 3 W 3 τ x (37) = (38) with the boundar conditions ( ) ( ) W ( τ) W = K ( τ) W 3 ( τ) 22 (39) W 3 ( τ) ( ) k3 where K 22 = τ = t ξ(k 2 ) ε.

6 D. Boundar control snthesis The boundar conditions matrix K r for the reduced sstem (35)-(36) need to be chosen as. Assuming that k 3 ξ(k 2 ) K r = holds as soon as k 2 = or k 2 =. Proposition 3: Consider the boundar conditions matrix k 2 k 3 K = (4) ξ() with k 2 k 3 in R assume that k 3 ξ() and that there exist positive values d 2 d 3 such that d 3 ξ() d 2 k 2 d 3 k 3 k 2 k 3 > (4) d 2 ξ() d 3 is satisfied. Consider the boundar conditions matrix k 3 K 2 = k 2 (42) ξ(k 2 ) with k 2 k 3 in R assume that k 3 ξ(k 2 ) and that there exist positive values d 2 d 3 such that k 2 d 2 ξ(k 2 )d 3 d 2 d 3 k 3 k 3 > (43) k 2 d 2 d 2 ξ(k 2 )d 3 d 3 is satisfied. Then Corollar can be applied to sstem (25) with either boundar conditions matrix K or K 2 defining the boundar conditions (29)-(3). To solve (4) we can compute ξ() from (32) then (4) is a linear matrix inequalit (LMI) which can be solved. Similarl b choosing ξ(k 2 ) = in (43) k 2 is computed from (32) then LMI (43) can be solved. E. Numerical simulation Using the numerical values in [22] the equilibrium is chosen as H =.365 V = 4.65 B =. We take the gravit constant g = 9.8. The eigenvalues of matrix Λ are also given in [22] as λ = λ 2 = λ 3 = 3. Using Yalmip toolbox [23] on Matlab to solve LMI (4) and (43). The obtained boundar conditions matrix K is K = (44) 4 and K 2 is K 2 =.95. (45) To numericall compute the solutions of sstem (25) with the boundar conditions matrix (44) or (45) we discretize them b using a two-step variant of the Lax-Wendroff method (see [24] and [25]). Precisel the space domain [] is divided into intervals of identical length the final time is chosen as 2. We take a time-step that satisfies the CFL condition and select the initial conditions as follows for all x [ ] W (x) = + cos(4πx) W 2 (x) = + cos(2πx) W 3 (x) = cos(4πx). Fig. shows the dnamics W 2 for the reduced sstem (35). It converges to the origin within time T 3s for the boundar condition matrix K r =. The finite time of convergence obtained in Proposition is T = λ 2 which is close to the numericall computed finite time T 3s. The slow dnamics W 2 for sstem (25) with the boundar conditions matrix K given b (44) in Fig. 2 is roughl the same graph as W 2 in Fig.. Fig. 3-4 show the time evolutions of the fast dnamics for sstem (25) with the boundar conditions matrix K. It is observed that the solutions converge to as time increases and that the depend on the evolution of the slow dnamics. Fig. : Time evolution of W 2 for sstem (35) with K r = Fig. 2: Time evolution of the slow dnamics W 2 for sstem (25) with K

7 Fig. 3: Time evolution of the fast dnamics W for sstem (25) with K Fig. 4: Time evolution of the fast dnamics W 3 for sstem (25) with K Similar results are obtained for sstem (25) with the boundar conditions matrix K 2 b numerical simulations. V. CONCLUSIONS In this paper the boundar control snthesis of a class of linear hperbolic sstems has been studied based on the singular perturbation method. The slow dnamics is stabilized within time T b choosing the boundar conditions for the reduced sstem as zero. The main result is applied to a boundar control design for a hperbolic sstem represented b the Saint-Venant Exner equation. The simulation example shows the effectiveness of the contribution of this work. This work could be applied to different kinds of phsical sstems governed b singularl perturbed sstems of conservation laws such as the flow control in [26]. These applications will be considered in the future works. REFERENCES [] G. Bastin J.-M. Coron and B. d Andréa Novel. Using hperbolic sstems of balance laws for modeling control and stabilit analsis of phsical networks. In Lecture notes for the Pre-Congress Workshop on Complex Embedded and Networked Control Sstems Seoul Korea 28. 7th IFAC Word Congress. [2] M. Dick M. Gugat and G. Leugering. Classical solutions and feedback stabilization for the gas flow in a sequence of pipes. Networks and Heterogeneous Media 5(4): [3] M. Gugat. Boundar feedback stabilization of the telegraph equation: Deca rates for vanishing damping term. Sstems and Control Letters 66: [4] B. Haut and G. Bastin. A second order model of road junctions in fluid models of traffic networks. Networks and Heterogeneous Media 2(2): [5] M. Gugat and G. Leugering. Global boundar controllabilit of the Saint-Venant sstem for sloped canals with friction. Annales de l Institut Henri Poincare (C) Non Linear Analsis 26(): [6] M. Gugat M. Hert A. Klar and G. Leugering. Optimal control for traffic flow networks. Journal of Optimization Theor and Applications 26(3): [7] R.M. Colombo G. Guerra M. Hert and V. Schleper. Optimal control in networks of pipes and canals. SIAM Journal on Control and Optimization 48: [8] P. Kokotović H.K. Khalil and J. O Reill. Singular pertrubation methods in control: analsis and design. Academic Press 986. [9] M.K. Kadalbajoo and K.C. Patidar. Singularl perturbed problems in partial differential equations: a surve. Applied Mathematics and Computation 34: [] Y. Tang C. Prieur and A. Girard. Tikhonov theorem for linear hperbolic sstems. Submitted for publication archives-ouvertes.fr/hal [] M. Krstic and A. Smshlaev. Backstepping boundar control for first-order hperbolic PDEs and application to sstems with actuator and sensor delas. Sstems and Control Letters 57(75-758) 28. [2] C. Prieur and J. de Halleux. Stabilization of a -D tank containing a fluid modeled b the shallow water equations. Sstems and Control Letters 52(3-4): [3] C. Prieur J. Winkin and G. Bastin. Robust boundar control of sstems of conservation laws. Math. Control Signals Sstem 2(2): [4] J-M. Coron G. Bastin and B. d Andréa-Novel. Dissipative boundar conditions for one-dimensional nonlinear hperbolic sstems. SIAM Journal on Control and Optimization 47(3): [5] J-M. Coron. Control and nonlinearit volume 36 of Mathematical Surves and Monographs. Americam Mathematical Societ 27. [6] W.H. Graf. Fluvial hdraulics. Wile 998. [7] W.H. Graf. Hdraulics of sediment transport. Water Resources Publications Highlands Ranch Colorado USA 984. [8] J. Hudson and P.K. Sweb. Formulations for numericall approximating hperbolic sstems governing sediment transport. Journal of Scientific Computing 9: [9] A. Diagne G. Bastin and J-M. Coron. Lapunov exponential stabilit of -D liear hperbolic sstems of balance laws. Automatica 48: [2] C. Prieur and F. Mazenc. ISS-Lapunov functions for time-varing hperbolic sstems of balance laws. Math. Control Signals Sstem 24(): [2] G. Bastin J.-M. Coron and B. d Andréa Novel. On Lapunov stabilit of linearised Saint-Venant equations for a sloping channel. Networks and Heterogeneous Media 4(2): [22] V. Dos Santos and C. Prieur. Boundar control of open channels with numerical and experimental validations. IEEE Trans. Control Sst. Tech. 6(6): [23] J. Löfberg. Yalmip: A toolbox for modeling and optimization in MATLAB. In In Proc. of the CACSD Conference Taipei Taiwan 24. [24] L.F. Shampine. Solving hperbolic PDEs in Matlab. Appl. Numer. Anal. Comput. Math. 2: [25] L.F. Shampine. Two-step Lax-Friedrichs method. Applied Mathematics Letters 8: [26] F. Castillo E. Witrant and L. Dugard. Contrôle de Température dans un Flux de Poiseuille. IEEE Conférence Internationale Francophone d Automatique Grenoble France 22.