Lyapunov-based Stability Analysis for a MIMO Counter-Propagating Raman Amplifier. Daniel Beauchamp

Transcription

1 Lyapunov-based Stability Analysis for a MIMO Counter-Propagating Raman Amplifier by Daniel Beauchamp A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical & Computer Engineering University of Toronto Copyright 214 by Daniel Beauchamp

2 Abstract Lyapunov-based Stability Analysis for a MIMO Counter-Propagating Raman Amplifier Daniel Beauchamp Master of Applied Science Graduate Department of Electrical & Computer Engineering University of Toronto 214 We consider the power stabilization problem for a Raman amplifier with an arbitrary number of pumps and signals. The system under consideration is counter-propagating, in the sense that the pump and signal powers propagate in opposite directions along the length of the amplifier. Such a system is modeled as a set of hyperbolic partial differential equations in space and time with Lotka-Volterra type nonlinearities. Lyapunov functionals are constructed, and boundary controllers are designed to guarantee exponential stability of the closed-loop system in the L 2 norm. Simulations are done in MATLAB to confirm the results. ii

3 Acknowledgements First and foremost, I gratefully acknowledge my supervisor, Professor Lacra Pavel. Her tremous knowledge and consistent guidance throughout the course of my research is what made this thesis possible. From her encouragement, I have gained invaluable experience by presenting my work to others in the field. Overall, she has immensely exceeded my expectations of a supervisor. I also gratefully acknowledge my family for their constant love and support. Finally, I thank the Department of Electrical & Computer Engineering at the University of Toronto for providing the knowledge and financial support which made this thesis possible. iii

4 Contents 1 Introduction Motivation Optical Communication Transient Control Problem Literature Review Contribution Organization Background Raman Scattering Raman Amplifier Signal & Pump Light Amplifier Configurations Mathematical Model Function Spaces C r Spaces of Continuous Functions L p Spaces Sobolev Spaces Derivatives Defined on Banach Spaces Problem Model Normalized Systems Fully-actuated Model Change of Variables Under-actuated Model iv

5 3.2 General Systems Feedback Controllers for Fully-Actuated Systems Proportional Controller Dynamic Controller Feedback Controllers for Under-Actuated Systems Proportional Controller Dynamic Controller Existence and Uniqueness of Classical Solutions Results for the Nonlinear System Results for the Linearized System Lyapunov Functional for Linearized Systems Lyapunov Functional Introduction Derivative of the Lyapunov Functional Lyapunov Functional Properties First Property Second Property for Fully-Actuated Systems Second Property for Under-Actuated Systems Lyapunov Functional for Nonlinear Systems Lyapunov Functional for Normalized Systems Derivative of the Lyapunov Functional Lyapunov Functional Properties First Property Second Property for Fully-Actuated Systems Extension to General Systems Stability Analysis Stability Analysis for Fully-Actuated Systems Proportional Controller Dynamic Controller Stability Results for Under-Actuated Systems Proportional Controller Dynamic Controller v

6 7 Simulations Fully-actuated Systems Steady-state Stabilization Output Tracking Under-actuated Systems Steady-state Stabilization Output Tracking Conclusion 11 Bibliography 13 9 Appix PDE Solver Sample Steady-state Generation Code Sample Output Tracking Code vi

7 List of Figures 1.1 Typical WDM System Light Scattering Co-Propagating Raman Amplifier [11] MIMO Counter-Propagating Raman Amplifier Normalized Raman gain spectrum for bulk silica with a pump wavelength around 15 nm [1] Closed-loop system Raman gain (solid trace) of a Raman amplifier pumped with six lasers with different wavelengths and input powers (vertical bars) Plot of g(µ) versus µ Signal steady-state profiles - 4x Pump steady-state profiles - 4x Signal evolution in time at z=25km - 4x4 - Proportional Controller Signal evolution in time at z=5km - 4x4 - Proportional Controller Signal evolution in time at z=1km - 4x4 - Proportional Controller Signal evolution in time at z=1km - 4x4 - Dynamic Controller Signal Evolution - 4x4 - Dynamic Controller Pump Evolution - 4x4 - Dynamic Controller σ i (t) for ɛ = σ i (t) for ɛ = σ i (t) for ɛ = Signal evolution in time at z=1km - 4x4 - Dynamic Controller, ɛ = Plots of σ i (t) for ɛ = vii

8 7.14 Signal evolution in time at z=1km - 4x4 - Dynamic Controller, ɛ = Signal 2 Evolution - 4x4 Dynamic Controller, ɛ = Pump 2 Evolution - 4x4 Dynamic Controller, ɛ = σ i (t) for ɛ = Signal steady-state profiles - 3x Pump steady-state profile - 3x Signal evolution in time at z=25km - 3x3- Proportional Controller Signal evolution in time at z=5km - 3x3 - Proportional Controller Signal evolution in time at z=1km - 3x3 - Proportional Controller Signal evolution in time at z=1km - 3x3 - Dynamic Controller σ 1 (t), n = σ 1 (t), n= σ 1 (t), n= σ 1 (t), n= viii

9 List of Notations α i Attenuation per unit length w i (z) Steady-state signal or pump power λ i Group velocity H k (Ω) Space of functions with first k spatial derivatives in L 2 (Ω) L 2 (Ω) Space of square integrable functions defined on Ω Ω σ i (t) Spatial domain of the Raman amplifier State variable for the dynamic controller τ i Group delay per unit length w(t) State vector representing the deviation from steady-state C (Ω) Space of continuous functions on Ω c ij Coupling coefficients E, F Lyapunov candidates for nonlinear systems k i, ˆk i, ɛ, r Boundary controller parameters p i (t, z) Signal or pump power V, W Lyapunov candidates for linearized systems y d,i Reference for the i th signal ix

10 Chapter 1 Introduction The goal of the thesis is to develop a new standardized method for stabilizing power transients of Raman amplifiers using a Lyapunov-based approach. We begin by discussing general fiber-optic communication systems, and the need for optical amplification. The motivation for the thesis is then discussed, and in particular the problem behind controlling the transient behavior of the Raman amplifier. Following this, previous research on this control problem is introduced. Lastly, we state the contribution and organization of the thesis. 1.1 Motivation Optical Communication A well known advantage of optical communication is the ability to transmit information at a relatively high bandwidth, as illustrated in Table 1.1 [9]. Table 1.1: Comparison of communication methods Communication medium Carrier frequency Bandwidth Copper cable 1 MHz 1 khz Coaxial cable 1 MHz 1 MHz Radio frequencies 5 khz - 1 MHz 1 MHz Microwave frequencies 2 GHz 2 GHz Optical fiber cable 1 THz - 1 THz 4 THz In addition, crosstalk between fiber cables is minimal due to opaque coatings and total internal reflection [9]. This also adds a layer of security to the information under transmission. The light weight, low cost, and small diameter of the fiber cables are other features that make optical communication 1

11 Chapter 1. Introduction 2 desirable. Current fiber-optic communication systems use several optical channels, each with its own center wavelength and defined bandwidth [21]. The channels can be combined, and then sent to propagate along the length of a fiber-optic cable without inter-channel interference. Such systems are commonly referred to as wavelength-division multiplexing (WDM) systems. A typical WDM system with three channels having center wavelengths λ 1, λ 2, λ 3 is illustrated in Figure 1.1. Figure 1.1: Typical WDM System The electronic signals are first are combined using a multiplexer. The TX block converts the combined electronic signals into optical signals. The optical signals are then combined using an optical multiplexer, and sent to propagate along the length of the amplifier. Unfortunately, such signals become attenuated in power as they propagate long distances. Thus, power amplification is required to avoid this problem. An optical demultiplexer is then used to split the optical combination into its respective channels. The RX block converts the optical signals back into electronic signals, which are then sent to a demultiplexer for the final output. Popular choices for the AMP block in Figure 1.1 are the erbium-doped fibre amplifier (EDFA) and the Raman amplifier. The Raman amplifier has several advantages. First, the gain spectrum is not dopant depent like its EDFA counterpart. Instead, it is depent on the wavelength of a controllable pump laser. In addition, the gain spectrum is nonresonant over a relatively large bandwidth [23] Transient Control Problem There are two critical quantities of interest in the operation of the Raman amplifier. The first is the power of the information signal under transmission, which we denote as signal light. The second

12 Chapter 1. Introduction 3 is the power of a controllable laser which we denote as pump light. Both quantities are a function of space and time. Energy is transferred from the pumps to the signals by a phenomenon known as stimulated Raman scattering [11]. Essentially, the signal light is amplified in the presence of the pump light. There have been many recent efforts to control transient cross-talk of signal power using PID feedback, feedforward control, or all-optical gain clamping [2], [32], [8], [14]. However, these designs rely on empirical selection of control parameters, which gives rise to an open research problem. That is, developing a systematic approach for regulating the transient signal and pump power of the Raman amplifier. 1.2 Literature Review Previous research involves Lyapunov-based techniques to systematically stabilize power transients for the Raman amplifier [24]. However, only the simplest 2x2 co-propagating system is considered. Such a system is modeled as a set of nonlinearly coupled, hyperbolic partial differential equations of space and time with Lotka-Volterra type nonlinearity. In this case, the signal and pump power propagates in the same direction along a normalized length. After a change of variables, a proportional controller is introduced in order to stabilize the system to a desired steady-state power profile. Existence and uniqueness of global in time classical solutions are investigated in Sobolev spaces. It is shown in [25] that for any initial condition in H 1, there exists a unique local in time solution for the closed-loop system. In addition, the author proves that if a uniform apriori bound can be derived on the H 1 norm of the local in time solution, then global in time existence and uniqueness is guaranteed for the closed-loop system. The authors in [24] exploit these results by choosing initial conditions inside invariant sets, which are bounded in the H 1 norm. Such sets were found by using an entropy-like Lyapunov function together with a comparison lemma for differential inequalities. The time derivative of the Lyapunov function is made strictly negative by an appropriate choice of controller gains. Subsequently, asymptotic (exponential) stability is shown in the C (L 2 ) norms. Regulation of signal power at the output of the Raman amplifier is another key contribution from [24]. In this case, the system is assumed to be in steady-state, and the output reference is changed. The goal of the controller is to track this change. The system was simulated in MATLAB using a hyperbolic PDE solver. The simulations agree with the analysis in the sense that the system stabilizes to a desired steady-state spatial profile. However, the proportional controller produced large offsets during the output regulation experiment. A dynamic controller with integral action was introduced to accommodate this, and performed with much higher tracking accuracy.

13 Chapter 1. Introduction 4 Note that the authors in [24] operate directly on the infinite-dimensional dynamical system. Other authors attempt to solve the control problem using finite-dimensional approximation techniques [7]. However, it is not always ideal to apply such approximations to an underlying infinite dimensional system. Hyperbolic systems, in particular, have properties such as finite speed of propagation that require higher order approximations [17]. 1.3 Contribution The contribution of this thesis generalizes the Lyapunov-based results from [24], which considered only the simplest 2x2 co-propagating model. In this thesis, we apply Lyapunov methods to solve the transient control problem for the general N xn counter-propagating Raman amplifier. Raman amplifiers employed for practical use are mostly counter-propagating. The reason for this is due to the quality of the amplified signal regarding RIN transfer noise from pump lasers, polarization depence of gain, and nonlinear effects in the amplifier [13]. Furthermore, a typical setup has a MIMO configuration [7]. That is, several pumps and signals. In some cases, the system may even be under-actuated, in the sense that there are less pumps (controllers) than signals. In addition, each pump and signal may have non-normalized attenuation and coupling coefficients, which is not considered in [24]. The main results are presented in the form of theorems, which are outlined in Chapter 6. Such theorems provide a systematic guideline for choosing controller parameters which guarantee stability of the nonlinear closed-loop systems herein. The generalized case introduces changes in the structure of the PDE model, as shown in Chapter 3. In turn, the analysis changes in a non-trivial manner. In fact, the SISO counter-propagating model alone poses challenges if one decides to work in H 1 spaces as in [24]. In this case, existence and uniqueness results are used in H 2 spaces after a linearization about the steady-state solution. A new Lyapunov functional is used to treat this linearized case. Furthermore, the sign changes inherent in the counterpropagating model prevent the cancellation of key terms in the Lyapunov analysis. To accommodate this, new constraints are introduced in order to guarantee stability of the closed-loop system. The MIMO case is treated by constructing new controllers. Such controllers are often described as boundary controllers, as they essentially become the boundary conditions of the Raman amplifier PDE model. In a physical sense, they are situated at either the input or output of the amplifier. The latter is the case for a counter-propagating configuration, the reason for which becomes clear in the Lyapunov analysis. These controllers are different deping on whether or not the system is fully-actuated. Lastly, the non-normalized case is treated by introducing a novel entropy-like function.

14 Chapter 1. Introduction Organization The remaining chapters of the thesis are organized as follows: ˆ Chapter 2: This chapter provides the reader with background knowledge on Raman scattering, different amplifier configurations, and mathematical preliminaries. ˆ Chapter 3: Several Raman amplifier models are introduced, as they will each be analyzed separately in subsequent chapters. Boundary controllers are introduced, and a change of variables is applied. Existence and uniqueness results are discussed for the closed-loop models. We conclude by discussing the control objective in a mathematical framework. ˆ Chapter 4: We carry out Lyapunov analysis for the linearized models introduced in the previous chapter. A Lyapunov candidate is introduced, and its properties are discussed. Key lemmas are derived for the main stability results, which are later presented in Chapter 6. ˆ Chapter 5: A Lyapunov candidate for the normalized nonlinear model is introduced, and its properties are discussed. We ext the lemmas from the previous chapter to the nonlinear system by following an assumption. Following this comes the introduction of a modified Lyapunov candidiate, which is used to treat more general systems. ˆ Chapter 6: In this chapter, stability results are derived using results from the previous chapters. In particular, it is shown how to choose controller conditions to guarantee exponential stability in the L 2 -norm. ˆ Chapter 7: We derive the main results of the thesis by applying lemmas constructed in previous chapters. The main results are coupled with MATLAB simulations through several different experiments. ˆ Chapter 8: The main results of the thesis are summarized, and its limitations are highlighted. We conclude the thesis with suggestions for future research to improve the results herein.

15 Chapter 2 Background We begin by introducing the concept of Raman scattering. This is inted to give some information behind the physics of the Raman amplifier. Such information is used as intuition when investigating the system models presented in later chapters. Following this, we clarify terms that are used frequently throughout the thesis. Different Raman amplifier configurations are discussed, and their differences highlighted. The chapter concludes with a review of mathematical preliminaries, and in particular, the critical function spaces used throughout the thesis. 2.1 Raman Scattering As light propagates through a medium, it undergoes scattering, which may take place in an elastic or inelastic manner. The former is referred to as Rayleigh scattering, and the latter as Raman scattering. It is illustrated by Figure 2.1 that after Raman scattering, light may propagate with an altered frequency. At room temperature, the most dominant effect is a decrease in frequency, which is referred to as Stokes scattering. Such scattering takes place spontaneously as light propagates through a medium. That is, at random time intervals. We typically denote the original wave as the pump and the Stokes wave as the signal. If the intensity of the pump is increased, a phenomenon known as stimulated Raman scattering (SRS) occurs [21]. In this case, the rate of Stokes scattering is increased. Since the scattering rate is proportional to the amount of Stokes photons present, a positive feedback-like effect occurs. As a result, the intensity of the Stokes wave is increased and consequently its power as well. Specifically, we say that the signal is amplified in the presence of the pump. Throughout the SRS process, energy is essentially transferred from the pumps to the signals. 6

16 Chapter 2. Background 7 Figure 2.1: Light Scattering Another way to exploit the SRS effect is to inject Stokes photons, or signal light, (that has been previously generated by spontaneous Raman scattering) together with the original pump light. For reasons mentioned previously, this effectively increases the Raman scattering rate beyond that of spontaneous Raman scattering. The Raman amplifier exploits this method of SRS, as outlined in the subsequent section. Remark 1. Stimulated Raman scattering can be described as a nonlinear interaction involving the thirdorder nonlinear susceptibility χ (3) [16] 2.2 Raman Amplifier We begin the section by clarifying the basis under which the Raman amplifier operates. That is, the pump and signal light sources. Furthermore, we introduce the two most common types of Raman amplifier configurations. Specifically, we discuss the co-propagating and counter-propagating schemes. Lastly, we illustrate the amplification process in a mathematical framework using a simple, intuitive example Signal & Pump Light The terms signal and pump are used quite frequently throughout the thesis, so they deserve some clarification. Both are similar in the sense that they are light sources propagating at some particular frequency in an optical medium. Recall that the goal of optical communication is to transmit information

17 Chapter 2. Background 8 from one point to another. In the case of the Raman amplifier, such information is carried by the signal light. In 1962, it was observed that the power of the signal light can be amplified by propagating an intense optical field in the same medium [11]. Such a field is denoted as the pump light, and comes from a frequency adjustable laser source. This process is commonly known as the nonlinear stimulated Raman scattering (SRS) effect, as described in the previous section. In Section 2.2.3, the SRS process is described in a mathematical framework using coupled ordinary differential equations Amplifier Configurations In a Raman amplifier, the signal and pump light propagate in a particular optical medium. The SRS effect was observed in silica fibers in 1972; soon after, losses of such fibers were reduced to acceptable levels [11], [12]. Without loss of generality, the reader may assume the optical media in this thesis are silica fibers. The two most common Raman amplifier configurations are co-propagating, and counter-propagating. In the co-propagating case, the signals and pumps propagate in the same direction along the length of the fiber. For example, Figure 2.2 illustrates a co-propagating configuration with signal and pump frequencies ω s and ω p respectively. Figure 2.2: Co-Propagating Raman Amplifier [11] The purpose of the coupler is combine the pump and signal frequency sources for transmission along the length of the fiber. The filter passes the signal beam, but blocks the residual pump [11]. For a configuration as in Figure 2.2, the input of the amplifier is the where the coupler meets the fiber. Conversely, the output is where the fiber meets the filter. In the counter-propagating configuration, the opposite scenario occurs, where the signals and pumps propagate in opposite directions. In fact, Raman amplifiers employed for practical use are mostly counter-propagating. The reason for this is due to the quality of the amplified signal regarding RIN transfer noise from pump lasers, polarization depence of gain, and nonlinear effects in the amplifier [13]. In Figure 2.3 we show a typical MIMO counter-propagating Raman amplifier with n signals and m

18 Chapter 2. Background 9 pumps propagating along a fiber of length L. In the positive z direction, the signal power P sig (t, ) R n propagates with group velocities λ 1... λ n. In the opposite direction, the pump power P pump (t, ) R m propagates with group velocities λ n+1... λ N, where N := n m. Figure 2.3: MIMO Counter-Propagating Raman Amplifier Throughout the thesis, we denote input and output of the Raman amplifier as z = and z = L respectively. 2.3 Mathematical Model For simplicity, consider a SISO (one pump, one signal) counter-propagating Raman amplifier in steadystate. The power evolution of the pump P p and signal P s can be modeled by the following coupled ordinary differential equations [1] dp s dz = g RP p P s α s P s (2.1) dp p dz = ω p ω s g R P p P s α s P p (2.2) where g R (W 1 m 1 ) is the Raman gain coefficient of the fiber normalized with respect to the

19 Chapter 2. Background 1 effective fiber area A eff. The coefficients α s, α p represent the attenuation coefficients at the signal and pump wavelengths with angular frequencies ω s, ω p. The first term on the right hand side of 2.1 (2.2) represents the signal gain (pump loss) due to the SRS effect, and the second term represents the intrinsic signal (pump) loss [1]. Figure 2.3 illustrates the normalized peak gain coefficient versus the pump-signal frequency shift for bulk silica. The dotted curve suggests that the gain diminishes when the pump and signal are polarized in an orthogonal manner. However, when the pump and signal are co-polarized, the gain exists over a considerably large frequency range (up to 4 THz). Figure 2.4: Normalized Raman gain spectrum for bulk silica with a pump wavelength around 15 nm [1] By introducing more pumps, the overall spectral gain becomes a superposition of the individual pump spectra (see Figure 3.4). Such a superposition can shape the overall spectrum to have constant gain over a large bandwidth. In fact, the gain spectrum can be dynamically adjusted by tuning the power of each pump laser. This key feature of the Raman amplifier is not offered by its EDFA counterpart. 2.4 Function Spaces In this thesis, we operate directly on infinite-dimensional dynamical systems. It is thus natural to introduce some functional analysis preliminaries. For the Raman amplifier models, we are primarily concerned with PDE solutions in Banach spaces. That is, complete normed spaces. There are three Banach spaces that are of particular interest. The purpose of this section is to introduce these spaces, along with the norms that accompany them. We conclude the section with the notion of derivatives defined on Banach spaces. In this section, we denote Ω as a general open subset of R n with closure Ω and boundary Ω. However, in the Raman amplifier models we use a normalized spatial domain Ω = (, 1).

20 Chapter 2. Background C r Spaces of Continuous Functions We denote C (Ω) as the space of all continuous functions on Ω. Note that functions in C (Ω) need not be bounded, whereas if Ω is bounded those in C (Ω) are both bounded and uniformly continuous [27]. The space C (Ω) is equipped with the usual supremum norm u = sup u(z) (2.3) z Ω If Ω R, then we define the spaces C r (Ω) = { } u : dα u dz α C (Ω) α r (2.4) equipped with the norm u C k (Ω) = k d α u dz α (2.5) α= The space C r (Ω) consists of functions u whose derivatives up to and including order r are continuous. For the Raman amplifier, we are primarily concerned with functions in C 1 (Ω) at every point in time L p Spaces For the case p = 2, let L 2 (Ω; R n ) denote the Lebesgue space of R n -valued square-integrable functions on Ω, and use L 2 (Ω) := L 2 (Ω; R n ) as a compact notation for this space of vector-valued functions. L 2 (Ω) is a Hilbert space with the inner product denoted <, > and L 2 -norm denoted by L 2 u 2 L := u(z) 2 dz (2.6) 2 Ω where denotes the Euclidean norm in R n, and the integral (2.6) is taken in the Lebesgue sense. A useful interpretation of (2.6) is the notion of energy for the function u. In later chapters we show exponential convergence in the L 2 (Ω)-norm for various Raman amplifier models. For the case p =, let L (Ω) denote the Lebesgue space of R n -valued measurable functions on Ω

21 Chapter 2. Background 12 that are essentially bounded with the norm u L := ess sup u(z) < (2.7) z Ω where u = max i u i in R n. The L -norm arises in existence and uniqueness results, which are discussed in Chapter Sobolev Spaces Sobolev spaces are a class of function spaces that consist of functions with conditions on their derivatives. The general definition of the Sobolev space for functions defined over Ω is given as W k,p (Ω) := {u : D α u L p (Ω), α k} (2.8) where D denotes the weak differential operator. In cases where u is continuously differentiable, the weak derivative coincides with the classical derivative. For the Raman amplifier problem, we are primarily concerned with such cases. In addition, the case p = 2 arises most naturally in many applications [27]. For this reason, we introduce the space H k (Ω) := W k,2 (Ω) = { } u : dα u dz α L2 (Ω), α k (2.9) equipped with the norm u H k := k n= dn u dz n L 2 (2.1) The solutions to many PDEs naturally exist in Sobolev spaces, including the Raman amplifier models presented in this thesis. In Chapter 3, we devote a section to discuss existence and uniqueness of classical solutions in Sobolev spaces.

22 Chapter 2. Background Derivatives Defined on Banach Spaces The Lyapunov functions we construct in later chapters are defined on Banach spaces. It is therefore important to discuss what it means to differentiate such functions. We begin by introducing the definition of the Fréchet derivative from [3]. Definition 1. Let T be a transformation defined on an open domain D(T ) in a normed space (X, X ) with range R(T ) in a normed space (Y, Y ). If for a fixed x D(T ) there exists a bounded linear operator (dt ) x such that T (x + h) T (x ) (dt ) x h Y lim = (2.11) h X h X then T is said to be Fréchet differentiable at x and (dt ) x is said to be the Fréchet derivative of T at x. as If the Fréchet derivative exists at x, then it coincides with the Gâteaux derivative which we define Definition 2. The Gateaux derivative of T at x in the direction φ X is given by T (x + τφ) T (x ) dt (x ; φ) = lim = d τ τ dτ T (x + τφ) (2.12) τ= provided the limit exists.

23 Chapter 3 Problem Model Recall that previous research considered only the SISO co-propagating system with normalized coefficients. In order to fully generalize the analysis, we must consider the MIMO counter-propagating system with non-normalized coefficients. Such systems may even be under-actuated, in the sense that there are less pumps than signals. Every step beyond the simple SISO system poses its own unique challenges, each of which we model separately. The purpose of this chapter is to introduce these models, along with the control problem in a mathematical framework. The transient control objective discussed in Chapter 1 naturally becomes an initial boundary-value problem (IBVP) for a nonlinear hyperbolic PDE system. We begin by assuming the existence of a steadystate solution in some normed space (X, X ). Boundary feedback controllers are then proposed with the objective of guiding the system to steady-state as time progresses. The system is then written in terms of coordinates that are used for Lyapunov analysis in the remaining chapters. A section is devoted to characterizing the proposed controllers. The first is a static proportional controller, and the second a dynamic controller with integral action. The latter serves as the ideal controller for the output regulation problem, which is also discussed in further detail. The chapter concludes with an important discussion on existence and uniqueness of solutions in Sobolev spaces, and in particular the appropriate choice of X for the various PDE models herein. 3.1 Normalized Systems In the following section, we consider a Raman amplifier model in which the attenuation and coupling coefficients are normalized. The non-normalized case is discussed in the subsequent section. 14

24 Chapter 3. Problem Model Fully-actuated Model Consider a fully-actuated counter-propagating Raman amplifier with n signals, where N := 2n. We define the power vector for pumps and signals as T [ p(t, z) = p + (t, z) p (t, z)] (3.1) [ := p 1 (t, z)... p n (t, z) p n+1 (t, z)... p N (t, z) ] T where if i {1,..., n} then p i (t, z) denotes the i th signal power which is the i th entry to the vector p + (t, z) R n. If i {n + 1,..., N} then p i (t, z) denotes the (i n) th pump power which is the (i n) th entry to the vector p (t, z) R n. The signals propagate along a spatial domain Ω := (, 1) with characteristic speeds < λ 1 < < λ n. The pumps propagate in the opposite direction, with characteristic speeds < λ n+1 < < λ N. Without loss of generality we assume λ n+1 > λ n so that {λ 1,..., λ N } forms an increasing sequence. Such a configuration is outlined in Figure 2.3, and may be modeled by the following set of nonlinearly coupled hyperbolic PDEs of space and time p i (t, z) t p j (t, z) t = λ i p i (t, z) z = λ j p j (t, z) z p i (t, z) + p j (t, z) j=n+1 p i (t, z)p j (t, z), i {1,..., n} (3.2) p j (t, z)p i (t, z), j {n + 1,..., N} p i (, z) = p i (z), i {1,..., N} where t, z Ω. Remark 2. Note that the system is indeed counter-propagating, hence the opposite signs for the spatial differential operators. This is not the case in [24]. Remark 3. We emphasize that the state of the system belongs to a function space. That is, p(t, ) (X, X ) for some normed space (X, X ) of R N -valued functions. In the section on existence and uniqueness, we give the specifics of X and discuss the well-posedness of the closed-loop systems introduced in this chapter. Remark 4. Note that the system is coupled by nonlinear Lotka-Volterra terms. Such nonlinearities are commonly used to model predator-prey systems [26], but in our case they are used to model the SRS

25 Chapter 3. Problem Model 16 effect [2], [7]. Furthermore, we apply the reasonable non-depleted pump (NDP) approximation [1]. In doing so, we neglect the coupling between the pumps and signals themselves. Typically the signals at the input are held constant p i (t, ) = ū s,i. Each signal has a desired set point at the output y d,i. That is, we require p i (t, 1) y d,i as t i {1,..., n}. We propose a feedback controller based on the signals at the output by manipulating only the pumps at their input. This gives rise to the following boundary conditions that hold t, i {1,..., n} for system (3.2) p i (t, ) = ū s,i (3.3) p n+i (t, 1) = u c,i (t) := g i (p i (t, 1)) where u c,i (t) are the control actions to be designed based on the signal power at the output p i (t, 1). For constant control actions u c,i (t) = ū c,i, let T [ p(z) := p 1 (z)... p n (z) p n+1 (z)... p N (z)] (3.4) denote the steady-state solution that satisfies the set point [ T [ ] T p 1 (1)... p n (1)] = y d,1... y d,n (3.5) along with boundary conditions i {1,..., n} p i () = ū s,i (3.6) p n+i (1) = ū c,i Remark 5. In this case, the set point (3.5) corresponds to the desired steady-state solution (3.4). However, in practical situations the designer may wish to track a new reference y d,i p(1). In such situations, it is important to choose the type of controller carefully. This is discussed later in the chapter after characterizing the controllers. Boundary conditions (3.3) give rise to the following initial-boundary compatibility conditions i

26 Chapter 3. Problem Model 17 {1,..., n} for system (3.2) p i () = ū s,i (3.7) p n+i (1) = g i (p i (, 1)) as Using boundary conditions (3.3) and the initial condition in (3.2), we write the closed-loop system p i (t, z) t p j (t, z) t = λ i p i (t, z) z = λ j p j (t, z) z p i (t, z) + p j (t, z) j=n+1 p i (t, z)p j (t, z), i {1,..., n} (3.8) p j (t, z)p i (t, z), j {n + 1,..., N} p i (, z) = p i (z), i {1,..., N} p i (t, ) = ū s,i i {1,..., n} p n+i (t, 1) = g i (p i (t, 1)) i {1,..., n} and if we denote [ ] T y d := y d,1... y d,n R n (3.9) as the power vector of desired references, then we have the following block diagram in Figure 3.1 for the closed-loop system (3.8) Figure 3.1: Closed-loop system

27 Chapter 3. Problem Model 18 where p + (t, z) and p (t, z) are defined as in (3.1) Change of Variables The closed-loop system (3.8) describes the absolute power of the pumps and signals. However, we are more interested in the deviation from a desired steady-state, or set point. We therefore define the following [ T w(t, z) := p(t, z) p(z)] R N (3.1) define where p(t, z), p(z) are defined as in (3.1) and (3.4) respectively. For simplicity of notation, we also [ T w(z) := w 1 (z)... w N (z)] = p(z) T R N (3.11) It is clear from definition (3.1) that (w(t, z) ) = (p(t, z) p(z)). The Lyapunov analysis in later chapters focuses on stabilizing the origin in w(t, z) coordinates. We therefore write the closed-loop system (3.8) as such. Assume the existence and uniqueness of a steady-state solution p(z) in some normed space (X, X ). Such a solution must also satisfy the following set of ordinary differential equations inherent in (3.8) λ i d w i dz = w i + λ j d w j (z) dz = w j z Ω with boundary conditions (3.6). j=n+1 w i w j, i {1,..., n} (3.12) w j w i, j {n + 1,..., N} Remark 6. Note that we have implicitly defined w i := w i (z) for simplicity of notation. Throughout the thesis, we generally refer to functions f := f(t, z) and f := f(z) unless stated otherwise. Using the fact that p(t, z) = w(t, z) + w(z) we write the first two lines of the closed-loop system (3.8) as follows

28 Chapter 3. Problem Model 19 (w i + w i ) t (w j + w j ) t = λ i (w i + w i ) z = λ j (w j + w j ) z (w i + w i ) + (w j + w j ) j=n+1 (w i + w i )(w j + w j ), i {1,..., n} (3.13) (w j + w j )(w i + w i ), j {n + 1,..., N} and after some manipulation of (3.13) by using the steady-state ODEs (3.12) we have w i t + λ w i i z + w j t λ j w j z + 1 ( 1 + j=n+1 w j w i ) w i w j + j=n+1 w i w j = and the overall system is written in vector form as j=n+1 w i w j i {1,..., n} (3.14) w j w i = w j w i j {n + 1,..., N} t w(t, z) + a w(t, z) + b(z)w(t, z) = F (w(t, z)) (3.15) z t, z Ω with initial condition [ T w(, z) := w (z) = p(, z) p(z)] R N (3.16) and boundary conditions i {1,..., n} w i (t, ) = (3.17) w n+i (t, 1) = g i (w i (t, 1)) g i ( w i (1)) where a := diag {λ 1,...λ n, λ n+1,..., λ N } (3.18)

29 Chapter 3. Problem Model 2 and b(z) R NxN is defined in terms of the following nxn block matrices b(z) := B 11(z) B 21 (z) B 12 (z) B 22 (z), (3.19) B 11 (z) = diag 1 N w j (z),..., 1 j=n+1 j=n+1 w j (z) (3.2) w 1 (z)... w 1 (z) B 12 (z) :=.. w n (z)... w n (z) (3.21) w n+1 (z)... w n+1 (z) B 21 (z) :=.. w N (z)... w N (z) (3.22) { } B 22 (z) = diag 1 + w k (z),..., 1 + w k (z) k=n+1 k=n+1 (3.23) with nonlinear reaction F (w(t, z)) R N defined as N k=n+1 F i (w(t, z)) := w kw i if i {1,..., n} n k=1 w kw i if i {n + 1,..., N} (3.24) Under-actuated Model In the case of system (3.8), the designer has access to as many pumps as there are signals. Consequently, we characterize the system as being fully-actuated. In some cases however, there may be less pumps than signals available. We refer to such systems as being under-actuated, since the pumps are viewed as controllers. Consider a system with m pumps to serve as controllers for n signals, where m < n. The

30 Chapter 3. Problem Model 21 model for the closed-loop system is identical to (3.15) - (3.17) with N := n + m. In Section 3.4, we propose feedback controllers g i (p i (t, 1)) specific to the under-actuated case. 3.2 General Systems Here we model the more general Raman amplifier, where the attenuation, coupling, and group delay coefficients may not be unity. Consider the more general N xn counter-propagating Raman amplifier with attenuation coefficients α i >, coupling coefficients c ij R, and group delay τ i > where (i, j) {1,..., N} 2. In addition, we neglect self-coupling so that c ii = i. Such a system is modeled as in [7] by the following PDEs p i (t, z) t p j (t, z) t = λ i p i (t, z) z = λ j p j (t, z) z α i τ i p i (t, z) + 1 τ i α j τ j p j (t, z) 1 τ j c ij p i (t, z)p j (t, z), i {1,..., n} (3.25) j=1 c ji p j (t, z)p i (t, z), j {n + 1,..., N} p i (, z) = p i (z), i {1,..., N} p i (t, ) = ū s,i i {1,..., n} p n+i (t, 1) = g i (p i (t, 1)) i {1,..., n} where z Ω, t. Following the same procedure as in Section with w(t, z) R N defined by (3.1), the system is written as follows t w(t, z) + a w(t, z) + b(z)w(t, z) = F (w(t, z)) (3.26) z t, z Ω with initial condition [ T w(, z) := w (z) = p(, z) p(z)] R N (3.27) and boundary conditions i {1,..., n}

31 Chapter 3. Problem Model 22 w i (t, ) = (3.28) w n+i (t, 1) = g i (w i (t, 1)) g( w i (1)) where a := diag {λ 1,... λ n, λ n+1,..., λ N } (3.29) and b(z) R NxN is defined as follows b(z) := [b ij (z)] (3.3) such that i {1,..., n}, j {1,..., N} α i τ i 1 N τ i k=1 b ij (z) = c ik w k if i = j cij τ i w i if i j (3.31) and i {n + 1,..., N}, j {1,..., N} α i τ i + 1 N τ i k=1 b ij (z) = c ik w k if i = j c ij τ i w i if i j (3.32) with nonlinear reaction F (w(t, z)) R N defined as 1 N τ i k=1 F i (w(t, z)) := c ikw i w k if i {1,..., n} 1 N τ i k=1 c ikw i w k if i {n + 1,..., N} (3.33) In Chapter 5, we carry out Lyapunov analysis on (3.26) - (3.28). The analysis changes considerably in comparison to the normalized system (3.15) - (3.17).

32 Chapter 3. Problem Model Feedback Controllers for Fully-Actuated Systems Up until now, the controllers g i (p i (t, 1)) in (3.8), (3.25) were assumed to be general. Here we propose controllers for fully-actuated systems, and in particular a proportional controller and dynamic controller with integral action. At this point, we emphasize that the controller parameters are determined analytically by Lyapunov analysis in later chapters. The goal is to derive sufficient conditions on such parameters in order to guarantee set point tracking p i (t, 1) y d,i and closed-loop stability in some justifiable norm Proportional Controller Consider a fully-actuated NxN counter-propagating Raman amplifier as modeled by (3.8) or (3.25). The following controller is proposed i {1,..., n} for closed-loop system (3.8) with block diagram outlined in Figure 3.1. p i (t, ) = ū s,i (3.34) p n+i (t, 1) = k i (p i (t, 1) y d,i ) + ū c,i where k i R. Note that we have (p i (t, 1) y d,i ) = (p i (t, 1) p(1)) as desired. Controller (3.34) is written i {1,..., n} in terms of w(t, z) for the closed-loop system (3.15)-(3.17) as w i (t, ) = (3.35) w n+i (t, 1) = k i w i (t, 1) Dynamic Controller Consider a fully-actuated NxN counter-propagating Raman amplifier as modeled by (3.8) or (3.25). The following controller is proposed i {1,..., n} for closed-loop system (3.8) with block diagram outlined in Figure 3.1.

33 Chapter 3. Problem Model 24 p n+i (t, 1) = k i p i (t, 1) + ˆk i σ i (t) (3.36) σ i (t) = ɛσ i (t) + p i (t, 1) y d,i p i (t, ) = ū s,i where k i, ˆk i R. Note that for sufficiently small ɛ > we have ( σ(t) ) = (p i (t, 1) y d,i ) as desired. Controller (3.36) is written i {1,..., n} in terms of w(t, z) coordinates for the closed-loop system (3.15)-(3.17) as w n+i (t, 1) = k i w i (t, 1) + ˆk i σ i (t) (3.37) σ i (t) = ɛσ i (t) + w i (t, 1) where σ i (t) stands for σ i (t) σ i and σ i := lim t σ i (t). 3.4 Feedback Controllers for Under-Actuated Systems The purpose of this section is to propose controllers for under-actuated systems. Proportional and integral controllers are still considered. However, having less pumps than signals changes the structure of the controllers in comparison to the fully-actuated case. It is obvious that having multiple pumps supports a wider bandwidth for signal amplification. For example, consider Figure 3.4 from [1]

34 Chapter 3. Problem Model 25 Figure 3.2: Raman gain (solid trace) of a Raman amplifier pumped with six lasers with different wavelengths and input powers (vertical bars) For the fully-actuated case, each pump uses feedback from only one signal. In the under-actuated case, at least one pump requires feedback from more than one signal. In this section, the intuition is to use such pumps to operate on a linear combination of signal power. Consequently, it is natural to expect a decrease in set-point tracking accuracy for the under-actuated system Proportional Controller Consider an under-actuated N xn counter-propagating Raman amplifier as modeled by (3.8) or (3.25). Assume the system has m pumps and n signals where m < n so that N := n + m. Denote the difference between the number of signals and pumps by γ := n m. Without loss of generality, assume that the first pump uses feedback from the first (γ + 1) signals, and the remaining (m 1) pumps each use feedback from one of the remaining (m 1) signals. The following controller is proposed for closed-loop system (3.8) with block diagram outlined in Figure 3.1. γ+1 p n+1 (t, 1) = k i (p i (t, 1) y d,i ) + ū c,1 (3.38) and for the remaining pumps i {2,..., n} p n+i (t, 1) = k i (p i (t, 1) y d,i ) + ū c,i (3.39)

35 Chapter 3. Problem Model 26 with constant signal input i {1,..., n} p i (t, ) = ū s,i (3.4) where k i R. Note that we have (p i (t, 1) y d,i ) = (p i (t, 1) p(1)) as desired. Controller (3.38)-(3.4) is written in w(t, z) coordinates for the closed-loop system (3.15)- (3.17) as γ+1 w n+1 (t, 1) = k i w i (t, 1) (3.41) and for the remaining pumps i {2,..., m} m w n+i (t, 1) = k i w i (t, 1) (3.42) i=γ+2 with constant signal input i {1,..., n} w i (t, ) = (3.43) Dynamic Controller Consider an under-actuated N xn counter-propagating Raman amplifier as modeled by (3.8) or (3.25). Assume the system has m pumps and n signals where m < n so that N := n + m. Without loss of generality, assume that the first pump uses feedback from the first (γ + 1) signals, and the remaining (m 1) pumps each use feedback from one of the remaining (m 1) signals. The following controller is proposed for closed-loop system (3.8) with block diagram outlined in Figure 3.1.

36 Chapter 3. Problem Model 27 γ+1 p n+1 (t, 1) = k i p i (t, 1) + ˆk 1 σ 1 (t) (3.44) γ+1 σ 1 (t) = ɛσ 1 (t) + (p i (t, 1) y d,i ) and for the remaining pumps i {2,..., m} we have p n+i (t, 1) = k i s γ+i (t, 1) + ˆk i σ i (t) (3.45) σ i (t) = ɛσ i (t) + s γ+i (t, 1) y d,γ+i with constant signal input i {1,..., n} p i (t, ) = ū s,i (3.46) where k i, ˆk i R. Note that for sufficiently small ɛ > we have ( σ(t) ) = (p i (t, 1) y d,i ) as desired. Controller (3.44)-(3.46) is written in w(t, z) coordinates for the closed-loop system (3.15)- (3.17) as γ+1 w n+1 (t, 1) = k i w i (t, 1) + ˆk 1 σ 1 (t) (3.47) γ+1 σ 1 (t) = ɛσ 1 (t) + w i (t, 1) and for the remaining pumps i {2,..., m} we have w n+i (t, 1) = k i w γ+i (t, 1) + ˆk i σ i (t) (3.48) σ i (t) = ɛσ i (t) + w γ+i (t, 1)

37 Chapter 3. Problem Model 28 where σ i (t) stands for σ i (t) σ. 3.5 Existence and Uniqueness of Classical Solutions This chapter presents various PDE models for the closed-loop counter-propagating Raman amplifier. It is thus important to discuss sufficient conditions under which these models are well-posed. As previously mentioned, this thesis is primarily concerned with classical solutions in Sobolev spaces. For the copropagating system considered in [24], existence and uniqueness of classical solutions in H 1 spaces follows from the results of [25]. Fortunately, these results can also be applied to counter-propagating systems under a key assumption. Such an assumption is clarified later in the section. If no assumptions are to be made, then we may linearize the system about the steady-state solution and use results from [4]. Hence, this section is split into two parts. The first part aims to treat the nonlinear system, and the second part considers the system linearized about the steady-state solution Results for the Nonlinear System The author in [25] considers the following nxn hyperbolic system with nonlinear source term w t (t, z) + a(z) w (t, z) + b(z)w(t, z) = f(w(t, z)) (3.49) z t, z Ω := (, 1), w(t, z) = [w i (t, z)] R n with boundary conditions t w(t, ) = G(w(t, 1)) (3.5) and initial condition z Ω w(, z) = w (z) (3.51) Ω and where it is assumed that the coefficients a(z) = [a ij (z)] and b(z) = [b ij (z)] are sufficiently smooth on

38 Chapter 3. Problem Model 29 a ij K a, b ij K b (3.52) for some K a, K b >. The co-propagating case where a(z) is a symmetric matrix with elements < λ 1 (z) < < λ n (z) is considered, however the counter-propagating case can be treated similarly. It is also assumed that G is a smooth map of class C 2 with G() = satisfying the Lipschitz property G(x) G(y) K g x y x, y (3.53) The nonlinear source terms f(x) = [f i (x)] R n are assumed to be of Lotka-Volterra type such that x R n, i {1,..., n} f i (x) = j x i m ij (z) x j, (3.54) where the coefficients m ij (z) are sufficiently smooth and m ij K m (3.55) The following space with embedded boundary condition is defined H 1 bc(ω) := { u H 1 (Ω) u() = G(u(1)) } (3.56) In [25], w is regarded as a function of t with values in a Banach space over z. That is, for each t, let t w(t) := w(t, ). As a result, the closed-loop system (3.49)-(3.51) is written as ẇ(t) + Lw(t) = F (w(t)), (3.57) w(t, ) = G(w(t, 1)), (3.58) w() = w (3.59)

39 Chapter 3. Problem Model 3 where the linear operator L is defined z Ω as (Lu)(z) := a(z) du(z) dz + b(z) u(z), L : D(L) L 2 (Ω) L 2 (Ω) (3.6) with domain D(L) = H 1 (Ω), and F is a nonlinear operator F : D(F ) L 2 (Ω) L 2 (Ω) defined i {1,..., n} by F (u) = [F i (u)], F i (u) := j u i m ij u j, (3.61) The notion of a classical solution is then defined as Definition 3. Let T >, and denote by C 1 ([, T ]; X ) the space of continuously differentiable functions on [, T ] with values in X. A function w : [, T ] X is a classical solution of (3.57)-(3.59) on [, T ] if w C 1 ([, T ]; X ), w(t) Hbc 1 (Ω) t [, T ]. The function w(t) is a classical solution on [, ) if w(t) is a classical solution on [, T ] for every T. The author shows that for every initial condition w Hbc 1 (Ω) there exists some T > such that (3.57)-(3.59) has a unique classical solution w(t) defined for every t [, T ] such that w C ([, T ]; H 1 bc(ω)) C 1 ([, T ]; L 2 (Ω)) Furthermore, it is also shown that (3.57)-(3.59) admits a unique global in time classical solution provided the following uniform apriori bound holds for t [, T ] w(t) K(T ) (3.62) where < T T for some K(T ) > indepent of T. The proof involves using energy-like estimates along with Gronwall s inequality to show that w(t) H 1 cannot blow up in finite time if (3.62) is satisfied. For the co-propagating case, the authors in [24] apply this result by choosing initial conditions inside subsets that are invariant and bounded in the C (Ω)-norm. Such subsets were shown to exist by investigating sublevel sets of an entropy-like Lyapunov function. As a result, asymptotic properties of the solution could be studied, and Lyapunov techniques were applied to guarantee stability in both the L 2 (Ω)

40 Chapter 3. Problem Model 31 and C (Ω) norms. In the co-propagating case, a scalar comparison lemma for differential inequalities was a key ingredient in showing (3.62). Unfortunately, this cannot be applied to the counter-propagating system due to the opposite spatial differential operator signs. Therefore, we use the following assumption to treat the nonlinear counter-propagating system Assumption 1. ( ɛ > )( δ > ) such that if w H 1 < δ then on the whole existence domain of the solution to Cauchy problem (3.15)-(3.17) we have w(t) ɛ, t This assumption is similar to what is used in the pioneer work [19] pg However, this author does not consider boundary conditions and derives results in a C 1 framework. Following Assumption 1 indirectly assumes global in time existence and uniqueness of classical solutions to the nonlinear closedloop system (3.15)-(3.17) by the results of [25]. As a result, asymptotic properties of the solution can be studied. This is of course desirable, since the goal is to have w(t) as t. In the next section, we investigate the linearization of system (3.15)-(3.17) about the steady-state solution Results for the Linearized System Some work has been done for hyperbolic systems of conservation laws in H 2 spaces (see [5], [29], [4]). Such systems are similar to (3.15)-(3.17) in the sense that there is a counter-propagating configuration. The following nxn hyperbolic Cauchy problem is considered in [4] u t + F (u)u x =, x [, 1], t [, ), (3.63) u +(t, ) = G +(t, 1) (3.64) u (t, 1) u (t, ) u(, x) = u (x) x [, 1]. (3.65) where u + (t, z) = [u i +(t, z)] R m, u (t, z) = [u i (t, z)] R n m. Similar to (3.5), G is assumed to be a map of class C 2 which satisfies G() =. In the context of a counter-propagating Raman amplifier, we regard F (u) as

41 Chapter 3. Problem Model 32 F (u) := diag {λ 1,..., λ m, λ m+1,..., λ n } (3.66) Under the assumption that appropriate initial-boundary compatibility conditions are satisfied, the authors propose the following Proposition 1. There exists a δ > such that for every u H 2 ((, 1), R n )) satisfying u H 2 ((,1),R n ) δ (3.67) the Cauchy problem (3.63)-(3.65) has a unique maximal classical solution with T [, + ]. Moreover, if u(t, ) H 2 ((,1),R n ) δ t [, T ), (3.68) then T = + The proof of this proposition is outlined in various works; see for example [22], [18]. The wellposedness of the Cauchy problem (3.63)-(3.65) has also been studied in a C 1 framework [19], [2]. However, in [22] and [18] the case of boundary conditions is not considered. Furthermore, [19] and [2] work in a C 1 framework, as opposed to the Sobolev space framework we are interested in. The authors in [4] explain how to adapt these proofs to treat the case with boundary conditions in a Sobolev space framework. Just as in [25], a fixed point method is used. The proof begins by considering the case where T (, min { λ 1 1,..., } λ 1 N ). For R > and for u H 2 ((, 1), R n ), the authors define the set C R (u ) as u L ((, T ), H 2 ((, 1), R n )) W 1, ((, T ), H 1 ((, 1), R n )) W 2, ((, T ), L 2 ((, 1), R n )) (3.69) such that

42 Chapter 3. Problem Model 33 u L ((,T ),H 2 ((,1),R n )) R, (3.7) u W 1, ((,T ),H 1 ((,1),R n )) R, u W 2, ((,T ),L 2 ((,1),R n )) R, u(, 1) H 2 ((, 1), R n ), u(, 1) H2 ((,1),R n ) R 2, u(, ) = u, u t (, ) = F (u )u x The mapping F : C R (u ) L ((, T ), H 2 ((, 1), R n ) is then defined by F(ũ) = u where u is the solution of the linear hyperbolic Cauchy problem u t + F (ũ) =, u(t, ) = G(ũ(t, 1)), t [, T ], (3.71) u(, x) = u (x), x [, 1]. Note that if u H 2 ((,1),R n is sufficiently small, then the set C R(u ) is a non-empty closed subset of L ((, T ), L 2 (, 1), R n ). By using standard energy estimates along with the finite speed of propagation in (3.71), the authors show that the map F has a fixed point u C R (u ). Therefore, there exists a solution u C R (u ) to the Cauchy problem (3.63)-(3.65). The general case of T (, + ) is treated by applying the above to [, T 1 ], [T 1, 2T 1 ], [2T 1, 3T 1 ],... with T 1 given in (, min { λ 1 1,..., } λ 1 n ). The above results can be exted to systems with nonzero right hand side h(u) with the map h : R n R n of class C 2 and vanishing at zero [4]. This is indeed the case for system (3.15)-(3.17) or (3.26)-(3.28) linearized about the steady-state solution p(z), which we write as follows t w(t, z) + a w(t, z) + b(z)w(t, z) = (3.72) z t, z Ω with initial condition

43 Chapter 3. Problem Model 34 [ T w(, z) := w (z) = p(, z) p(z)] R N (3.73) and boundary conditions i {1,..., n} w i (t, ) = (3.74) w n+i (t, 1) = g i (w i (t, 1)) g( w i (1)) where a, b(z) are defined as in (3.18) and (3.19) respectively. Note that upon linearization, the Lotka-Volterra terms in (3.15) vanish. Using the results from [4] we propose the following Proposition 2. If appropriate initial-boundary compatibility conditions are satisfied then there exists a sufficiently small δ > such that if w H 2 < δ then the Cauchy problem (3.72)-(3.74) admits a unique maximal classical solution w(t) C ([, T ), H 2 (Ω)) (3.75) for any given T (, ). In [4], Lyapunov analysis is done for homogeneous systems of conservation laws. Since our system is the inhomogeneous case, it is worth it to explicitly carry out the analysis, which is done in the next chapter.

44 Chapter 4 Lyapunov Functional for Linearized Systems In this chapter, we present key lemmas regarding a Lyapunov functional for the linearized closed-loop system (3.72)-(3.74). The chapter begins by introducing a quadratic-like functional V (w(t)), along with its basic properties. Following this, another functional V (w(t)) is introduced based on the Gâteaux derivative of V (w(t)). We then state Lemma 1, which involves bounding V (w(t)) by terms involving w(t) L 2. Lemma 2 then follows for fully-actuated systems, which presents an upper bound on V (w(t)) in terms of the boundary controller parameters. The chapter concludes with Lemma 3, which is analogous to Lemma 2, but specific to under-actuated systems. These lemmas serve as key ingredients for the stability analysis presented in Chapter Lyapunov Functional Introduction Consider the following quadratic-like function H : R R H(t) := 1 2 w 2 i (t, z)e µz dz wi 2 (t, z)e µz dz (4.1) i=n+1 and the Lyapunov candidate functional V : R n R 35

45 Chapter 4. Lyapunov Functional for Linearized Systems 36 V (w(t)) := 1 2 w 2 i e µz dz w i e µz dz (4.2) i=n+1 where we define w i := w i (t, z) for simplicity of notation. Note that V is positive definite in a neighborhood of the origin. That is, given some t > we have V (w(t)) = w(t) = and V (w(t)) > otherwise. Furthermore, V is well-defined and continuous for any given w(t) H 2 (Ω). In the next section, we relate the Gâteaux derivative of (4.2) to the time derivative of (4.1). Quadratic-like functionals similar to (4.2) are used in various works involving the stability of hyperbolic systems [5], [29], [6]. In fact, a common approach to constructing Lyapunov functionals for hyperbolic systems works as follows. A function φ(w(t)) is first defined based on intuition behind the structure of the system at hand. Such a function becomes part of two separate integrals over the spatial domain Ω. The first integral treats the part of the solution propagating in the positive direction, and the second treats the counter-propagating part. These integrals are then summed together to form the Lyapunov candidate functional V (w(t)) := φ i (w i )e µz dz + i=n+1 φ i (w i )e µz dz. (4.3) For the case of the linearized system, the function φ(w(t)) is quadratic-like. However, such a function becomes insufficient in the next chapter when dealing with the nonlinear system. In the next section, we discuss properties of Lyapunov functional (4.2). 4.2 Derivative of the Lyapunov Functional It is important to discuss the derivative of the functional V. For simplicity of notation, let u := w(t) and consider (4.2) as a function of u H 2 (Ω) V (u) := 1 2. u 2 i e µz dz u i e µz dz (4.4) i=n+1 In addition, we apply (2.12) in order to define the functional V (u) as the Gâteaux derivative of (4.4) at u in the direction of η := u/ t

46 Chapter 4. Lyapunov Functional for Linearized Systems 37 V (u) := dv (u; η) = d dτ V (u + τη) τ= (4.5) u 1 i u i = u i t e µz dz + u i t eµz dz. (4.6) i=n+1 Since u := w(t) we have V (w(t)) = w 1 i w i t e µz dz + i=n+1 w i w i t eµz dz. (4.7) Note that the functional V evaluated along solutions of the closed-loop system is equivalent to the time derivative of (4.1). That is, V (w(t)) = dh dt (t). 4.3 Lyapunov Functional Properties In this section, we present two key properties of the Lyapunov functional (4.2). Such properties are stated in the form of lemmas which are used for stability analysis in Chapter First Property The following lemma illustrates the first property of V (w(t)). Lemma 1. The following holds µ > 1 2 e µ w(t) 2 L 2 V (w(t)) 1 2 eµ w(t) 2 L 2 (4.8) where V (w(t)) is defined by (4.2). Proof. The proof is simple, and follows from the definitions of the L 2 (Ω)-norm and V (w(t)) and it is also clear that e µ w(t) 2 L 2 = N e µ w 2 i dz V (w(t))

47 Chapter 4. Lyapunov Functional for Linearized Systems 38 V (w(t)) and hence we arrive at (4.8). N e µ w 2 i dz = e µ w(t) 2 L 2 In Chapter 6, we investigate the stability of the origin in the L 2 (Ω)-norm, and (4.8) becomes crucial Second Property for Fully-Actuated Systems We now consider a fully-actuated N xn counter-propagating Raman amplifier. Consider a system with n signals and n pumps so that N := 2n. Assume that such a configuration is accurately modeled by (3.72)-(3.74) in a sufficiently small neighborhood of the steady-state solution p(z), and consider the following lemma Lemma 2. For sufficiently small w H 2, let w(t) denote the unique maximal classical solution of closed-loop system (3.72)-(3.74) with duration T >. Assume the steady-state solution satisfies j=n+1 w j < 1. If we have λ 1 n w ( ) 1 (1 θ) µ + e2µ µ for some µ > and θ (, 1), then the following holds t [, T ]. V (w(t)) µλ 1 θ V (w(t)) + ( λn+i 2 w2 n+i(t, 1)e µ λ ) i 2 w2 i (t, 1)e µ (4.9) Refer to Section (Proposition 2) for a discussion on the duration of the unique maximal classical solution T >. Remark 7. It is worth emphasizing the significance of Lemma 2. Notice how the upper bound (4.9) is written in terms of the boundary controller parameters. Hence, the idea is to eventually choose controller parameters which make the summation term vanish. What then remains is to apply Gronwall s inequality along with Lemma 1 in order to show exponential stability of the origin in the L 2 (Ω)-norm. In Chapter 6, this is done in detail. The proof for Lemma 2 is as follows.

48 Chapter 4. Lyapunov Functional for Linearized Systems 39 Proof. Consider the time derivative of (4.1), or equivalently the functional V (w(t)) V (w(t)) = dh dt (t) = w 1 i w i t e µz dz + i=n+1 w i w i t eµz dz (4.1) From (3.72) we have w i t w i t = λ i w i z + b ii(z)w i + = λ i w i z b ii(z)w i j=n+1 w i w j i {1,..., n} (4.11) w i w j i {n + 1,..., N}, j=1 from which, we write (4.1) as V (w(t)) = dh dt (t) = + i=n+1 ( ) w i w i λ i z b ii(z)w i e µz dz + ( ) w i w i λ i z b ii(z)w i e µz dz j=n+1 i=n+1 j=1 w i w i w j e µz dz (4.12) w i w i w j e µz dz We define the following speed weighted quadratic functions q + := q(w(t, z)), q := q (w(t, z)) q + (w(t, z)) := λ i 2 w2 i, q (w(t, z)) := i=n+1 λ i 2 w2 i (4.13) and compute their spatial derivatives dq + dz = n λ i w i w i z, dq dz = N i=n+1 λ i w i w i z, (4.14) from which, we write (4.12) as follows

49 Chapter 4. Lyapunov Functional for Linearized Systems 4 V (w(t)) = dq + dz e µz dq + dz eµz b ii (z)wi 2 e µz dz + i=n+1 b ii (z)w 2 i e µz dz j=n+1 i=n+1 j=1 w i w i w j e µz dz (4.15) w i w i w j e µz dz. Since by assumption in the lemma statement we have N j=n+1 w j < 1, this implies b ii (z) > i (refer to (3.19)). V (w(t)) Therefore, we may drop the square terms and write the following upper bound on V (w(t)) dq + dz e µz dz + dq + dz eµz dz j=n+1 i=n+1 j=1 w i w i w j e µz dz w i w i w j e µz dz. Integrating by parts the terms involving q + and q we have V (w(t)) q + e µz 1 µ q + e µz dz + + q e µz 1 µ q e µz dz w i w i w j e µz dz (4.16) j=n+1 w i w i w j e µz dz. i=n+1 j=1 The terms that appear after the integration by parts are bounded as follows. First, since λ 1 = min i λ i then clearly µ q + e µz dz µ q e µz dz µλ 1 V (w(t)). (4.17) Furthermore, since q + (w(t, )) = from boundary conditions (3.74) and q (w(t, )) > we have

50 Chapter 4. Lyapunov Functional for Linearized Systems 41 q + e µz 1 + q e µz 1 q +(w(t, 1))e µ + q (w(t, 1))e µ (4.18) ( λn+i = 2 w2 n+i(t, 1)e µ λ ) i 2 w2 i (t, 1)e µ. Using (4.17) and (4.18), we write the following upper bound on V (w(t)) ( λn+i V (w(t)) µλ 1 V (w(t)) + 2 w2 n+i(t, 1)e µ λ ) i 2 w2 i (t, 1)e µ + w i w i w j e µz dz + w i w i w j eµz dz. j=n+1 i=n+1 j=1 (4.19) where the product terms in (4.16) are bounded by their absolute values. Estimates are derived on the product terms as follows i=n+1 j=1 w i w i w j w w 2 n w 2 i=n+1 j=1 i=n+1 j=1 w i w j (4.2) ( w 2 i + wj 2 ) (4.21) wi 2. (4.22) In (4.2) we used the fact that w i (z) w i, z Ω. In addition, Young s inequality ([3] pg. 43) is applied in (4.21). Using the same methodology, we also have j=n+1 w i w i w j n w 2 wi 2. (4.23) so that (4.19) becomes

51 Chapter 4. Lyapunov Functional for Linearized Systems 42 V (w(t)) µλ 1 V (w(t)) + + n w 2 w 2 i ( λn+i 2 w2 n+i(t, 1)e µ λ ) i 2 w2 i (t, 1)e µ ( e µz + e µz) dz. (4.24) Using the fact that max z Ω (e µz + e µz ) = e µ + e µ along with the definition of the L 2 (Ω)-norm (2.6) we have V (w(t)) µλ 1 V (w(t)) + + n w 2 ( e µz + e µz) w(t) 2 L 2 ( λn+i 2 w2 n+i(t, 1)e µ λ ) i 2 w2 i (t, 1)e µ (4.25) We equivalently write (4.25) as follows θ (, 1) V (w(t)) µ(1 θ)λ 1 V (w(t)) µθλ 1 V (w(t)) (4.26) ( λn+i + 2 w2 n+i(t, 1)e µ λ ) i 2 w2 i (t, 1)e µ + n w 2 ( e µ + e µ) w(t) 2 L 2 Remark 8. Splitting the Lyapunov functional into two parts in (4.26) is analogous to [15] pg.177 in order to derive conditions in which the positive terms on the right hand side vanish. Note that the following holds by assumption in the lemma statement λ 1 n w ( ) 1 (1 θ) µ + e2µ µ = n w g(µ) (4.27) (1 θ) where g(µ) := 1 µ + e2µ µ. Multiplying both sides of (4.27) by 1 2 µ(1 θ)e µ w(t) 2 L 2 we have e µ 2 w(t) 2 L 2µλ 1(1 θ) n w 2 (e µ + e µ ) w(t) 2 L 2 (4.28) and using the left hand side of Lemma 1 (4.8) we have

52 Chapter 4. Lyapunov Functional for Linearized Systems 43 µλ 1 (1 θ)v (w(t)) n w 2 ( e µ + e µ) w(t) 2 L2. (4.29) Hence, the last term in (4.26) is dominated and we have V (w(t)) µθλ 1 V (w(t)) (4.3) ( λn+i + 2 w2 n+i(t, 1)e µ λ ) i 2 w2 i (t, 1)e µ which completes the proof of the lemma. Remark 9. Note that (4.27) imposes a lower bound on λ 1, the slowest signal. However, from a practical perspective this condition should always be satisfied for an appropriate range of µ. Indeed, consider the following plot of g(µ) versus µ Figure 4.1: Plot of g(µ) versus µ One may verify that min g(µ) occurs at µ min Since the λ i are generally of µ (, ) much higher order, and w on much lower order, we see that (4.27) does not impose a strict practical constraint for an appropriate value of θ. For example, the author of [7] considers signal power on the order of 1dBm, and characteristic speeds on the order of 1 8.

53 Chapter 4. Lyapunov Functional for Linearized Systems Second Property for Under-Actuated Systems Consider an under-actuated N xn counter-propagating Raman amplifier. Assume the system has n signals and m pumps where m < n so that N := n + m. In addition, assume that such a configuration is accurately modeled by (3.72)-(3.74) in a sufficiently small neighborhood of the steady-state solution p(z). Denote the difference between the number of pumps and signals by γ := n m. Without loss of generality, assume that the first pump uses feedback from the first (γ + 1) signals, and the remaining (m 1) pumps each use feedback from one of the remaining (m 1) signals. We state the following lemma without proof Lemma 3. For sufficiently small w H 2, let w(t) denote the unique maximal classical solution of closed-loop system (3.72)-(3.74) with duration T >. Assume the steady-state solution satisfies j=n+1 w j < 1. If we have λ 1 n w ( ) 1 (1 θ) µ + e2µ µ for some µ > and θ (, 1), then the following holds t [, T ] V (w(t)) µλ 1 θ V (w(t)) + + ( λ n+1 m i=2 γ+1 λ i 2 w2 i (t, 1)e µ 2 w2 n+1(t, 1)e µ ( λn+i 2 w2 n+i(t, 1)e µ λ γ+i 2 w2 γ+i(t, 1)e µ ) ) (4.31) T >. Refer to (Proposition 2) for a discussion on the duration of the unique maximal classical solution Note that Lemma 3 is analogous to Lemma 2, but modified to treat the under-actuated case. The proof follows the same methodology as the proof of Lemma 2. Note how in (4.31), the first pump is grouped with the (γ + 1) signals it uses feedback from. In later chapters, the under-actuated controller w n+1 attempts to dominate this term. The remaining (m 1) pumps are coupled with the remaining (m 1) signals in a fully-actuated manner.

54 Chapter 5 Lyapunov Functional for Nonlinear Systems The goal of this chapter is to present lemmas analogous to those in the previous chapter, but for nonlinear systems. Such lemmas serve as key ingredients for the stability analysis presented in Chapter 6. We present two different entropy-like Lyapunov functionals; one for normalized systems (3.15)-(3.17) and the other for general systems (3.26)-(3.28). We first introduce the functional E(w(t)) for normalized fully-actuated systems, along with its basic properties. Following this, we introduce the functional Ė(w(t)), which is based on the Gâteaux derivative of E(w(t)). Two key properties of the functional (similar to those in the previous chapter) are discussed. Lastly, we naturally ext such properties to general systems using a modified Lyapunov functional. Throughout the chapter, Assumption 1 is strictly followed to ensure all PDE models are well-posed (refer to Chapter 3 on Existence and Uniqueness of Classical Solutions). Remark 1. Since several lemmas are introduced in this chapter, it is worth explaining their logical flow. Thus far, we have introduced lemmas 1-3. Lemmas 4 and 5 are analogous to lemmas 1 and 2 respectively, but for the normalized nonlinear system. Lemma 6 is specific to the general nonlinear system. Lemmas 7 and 8 are also analogous to lemmas 1 and 2 respectively, but for the general nonlinear system. 5.1 Lyapunov Functional for Normalized Systems Consider an NxN counter-propagating Raman amplifier as modeled by (3.15)-(3.17). Assume the system has n pumps and n signals so that N := 2n. First, we recall an entropy-like function used in [24] for the 45

55 Chapter 5. Lyapunov Functional for Nonlinear Systems 46 co-propagating system. For any given real number ū > let v : I = ( ū, ) R + be given by ( v(u) := u ū ln 1 + ū ). (5.1) u Such a function is positive definite on a neighborhood of u =. That is, v() = and v(u) > on I\ {}. In addition, if we have ā, b > such that 1/ b < ū < 1/ā, then the following holds on the domain R := {u 1/ b ū u 1/ā ū} 1 2ā u 2 v(u) 1 2 b u 2 (5.2) The proof of this property is outlined in [24]. Using this entropy function, consider the function I : R + R + I(t) := v i (w i (t, z))e µz dz + i=n+1 v i (w i (t, z))e +µz dz (5.3) from which we introduce the following Lyapunov candidate functional E(w(t)) := v i (w i (t, z))e µz dz + i=n+1 v i (w i (t, z))e +µz dz (5.4) where ( v i (w i (t, z)) := w i (t, z) w i (z) ln 1 + w ) i(t, z) w i (z) (5.5) with domain of definition w i (t, z) > min z Ω w i (z). In the next section, we relate (5.3) to the Gâteaux derivative of E(w(t)) in a particular direction. Unlike the quadratic-like Lyapunov functional considered in (4.2), the entropy-like functional (5.4) matches the structure of the Lotka-Volterra nonlinearity in system (3.15)-(3.17).

56 Chapter 5. Lyapunov Functional for Nonlinear Systems Derivative of the Lyapunov Functional For simplicity of notation, let u := w(t) and consider the functional (5.4) written in terms of u E(u) = v i (u i )e µz dz + i=n+1 v i (u i )e +µz dz. (5.6) We now apply (2.12) in order to define the functional Ė(u), the Gâteaux derivative of E at u in the direction of η := ( u)/( t) Ė(u) := de(u; η) = d dτ E(u + τη) τ= (5.7) dv i u 1 i dv i u i = du i t e µz dz + du i t e+µz dz. (5.8) i=n+1 Since u = w(t) we have Ė(w(t)) = dv i w 1 i dw i t e µz dz + i=n+1 dv i dw i w i t e+µz dz. (5.9) Note that the functional Ė evaluated along solutions of the closed-loop system is equivalent to the time derivative of (5.3). That is, Ė(w(t)) = di dt (t). 5.3 Lyapunov Functional Properties In this section, we present two key properties for the Lyapunov functional (5.4). Such properties are illustrated by lemmas that are used for stability analysis in the next chapter First Property The following lemma illustrates the first property of E(w(t)). Lemma 4. Let w m := min i min z Ω w i (z) i {1,..., N}. Consider any ā, b > such that (1/ b) < w m, (1/ā) > w. For any µ > let α := āe µ and β := be µ. Then for any w(t) H 1 (Ω) such that w(t) d, where d := min {(1/ā) w, w m (1/ b)}, the following holds

57 Chapter 5. Lyapunov Functional for Nonlinear Systems α w(t) 2 L 2 E(w(t)) 1 2 β w(t) 2 L 2 (5.1) Proof. Consider the left-hand side of (5.1) 1 2 α w(t) 2 L = 1 2 2āe µ 1 2ā wi 2 dz + 1 2āe µ wi 2 e µz dz + 1 2ā i=n+1 i=n+1 w 2 i dz (5.11) w 2 i e µz dz (5.12) Since w(t) < d, by (5.2) we have α w(t) 2 L 2 v i (w i )e µz dz + i=n+1 v i (w i )e µz dz = E(w(t)). (5.13) The proof for the right-hand side of (5.1) follows analogously Second Property for Fully-Actuated Systems We now consider a fully-actuated N xn counter-propagating Raman amplifier. Assume the system has n signals and n pumps so that N := 2n. Following Assumption 1, we assume w H 1 is sufficiently small so that w(t) < d where d is defined in the Lemma 4 statement. We can then prove the following lemma. Lemma 5. For sufficiently small w H 1, let w(t) denote the unique global in time classical solution of closed-loop system (3.15)-(3.17). Assume the steady-state solution satisfies j=n+1 w j < 1. If we have λ 1 ( ) n 1 (1 θ) µ + e2µ µ for some µ > and θ (, 1), then t >

58 Chapter 5. Lyapunov Functional for Nonlinear Systems 49 Ė(w(t)) µλ 1 θ E(w(t)) + ( bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ (5.14) Proof. Consider the time derivative of (5.3), or equivalently the functional Ė(w(t)) Ė(w(t)) = di dt (t) = dv i w 1 i dw i t e µz dz + i=n+1 dv i dw i w i t eµz dz. (5.15) From (3.15) we have w i t w i t = λ i w i z b i, (z)w(t, z) + = λ i w i z b i, (z)w(t, z) j=n+1 j=n+1 w i w j i {1,..., n} (5.16) w i w j i {n + 1,..., N} where b i, (z) is an N-dimensional row vector which denotes the i th row of b(z). Using the above, we write (5.15) as Ė(w(t)) = + i=n+1 dv i dw i dv i dw i w i λ i z b i, (z)w(t, z) + w i w j e µz dz (5.17) j=n+1 w i λ i z b i, w(t, z) w i w j e µz dz j=1 Similar to the linearized case, we define the following speed weighted functions q + := q(w(t, z)) and q := q (w(t, z)) q + (w(t, z)) := q (w(t, z)) := λ i v i (w i (t, z)) (5.18) i=n+1 λ i v i (w i (t, z)) and compute their spatial derivatives as

59 Chapter 5. Lyapunov Functional for Nonlinear Systems 5 dq + dz = dq dz = n N i=n+1 λ i dv i dw i w i z + n λ i dv i dw i w i z + dv i d w i λ i d w i dz N i=n+1 λ i dv i d w i d w i dz. (5.19) Using (5.19) we write (5.17) in terms of q + and q as follows Ė(w(t)) = dq + dz e µz dz (5.2) dv i d w i λ i d w i dz + dv i b i, (z)w(t, z) + w i w j e µz dz dw i j=n+1 dq dz eµz dz (5.21) dv i d w i λ i d w i dz + dv i b i, (z)w(t, z) w i w j e µz dz. dw i i=n+1 j=1 In order to write the above in a more compact form, we define the following functions dv i d w n i T + (t, z) := λ i d w i dz + dv i b i, (z)w(t, z) + w i w j (5.22) dw i j=n+1 dv i d w N i T (t, z) := λ i d w i dz + dv i b i, (z)w(t, z) w i w j (5.23) dw i i=n+1 i=n+1 j=1 so that (5.2) becomes Ė(w(t)) = dq + dz e µz dz + T + (t, z)e µz dz + dq dz eµz dz + T (t, z)e µz dz. (5.24) and after integrating by parts the first and third terms we have

60 Chapter 5. Lyapunov Functional for Nonlinear Systems 51 Ė(w(t)) = q + e µz 1 µ q + e µz dz + q e µz 1 µ q e µz dz (5.25) + T + (t, z)e µz dz + T (t, z)e µz dz. The terms that appear after the integration by parts are bounded as follows. First, since λ 1 = min i λ i then clearly µ q + e µz dz µ q e µz dz µλ 1 E(w(t)). (5.26) Furthermore, since q + (w(t, )) = from boundary conditions (3.17) and q (w(t, )) < we have q + e µz 1 + q e µz 1 q +(w(t, 1))e µ + q (w(t, 1))e µ (5.27) ( λn+i = 2 v n+i(w n+i (t, 1))e µ λ ) i 2 v i(w i (t, 1))e µ. and since w(t) d, we may use property (5.2) so that ( λn+i 2 v n+i(w n+i (t, 1))e µ λ ) i 2 v i(w i (t, 1))e µ ( bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ. (5.28) Therefore, using upper bounds (5.26) and (5.28) in (5.25) we have Ė(w(t)) µλ 1 E(w(t)) (5.29) ( + bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ + T + (t, z)e µz dz + T (t, z)e µz dz. It remains to treat the integral terms. We begin by simplifying the first terms of (5.22). For clarity, we denote such terms by

61 Chapter 5. Lyapunov Functional for Nonlinear Systems 52 T +,1 (t, z) := T,1 (t, z) := dv i d w i λ i d w i dz i=n+1 dv i d w i λ i d w i dz (5.3) First, note that from the steady-state ODEs (3.12) we have λ i d w i dz = b i,i(z) w i i {1,..., n} (5.31) λ i d w i dz = b i,i(z) w i i {n + 1,..., N} and from (5.5) we compute (dv i /d w i ) as dv i = w ( ) i wi ln + 1. (5.32) d w i w i + w i w i Using (5.31) and (5.32), we write (5.3) after some manipulation as follows T +,1 (t, z) = T,1 (t, z) = [ wi 2 b i,i (z) wi 2 + w i [ wi 2 b i,i (z) i=n+1 w 2 i + w i ] v i (w i (t, z)) ] v i (w i (t, z)) (5.33) (5.34) We now focus on the second terms in (5.22). For clarity, we denote such terms by T +,2 (t, z) := T,2 (t, z) := i=n+1 dv i dw i dv i dw i b i, (z)w(t, z) + b i, (z)w(t, z) j=n+1 w i w j (5.35) w i w j. j=1 From (5.5) we compute (dv i /dw i ) as

62 Chapter 5. Lyapunov Functional for Nonlinear Systems 53 dv i = w i (5.36) dw i w i + w i which we then substitute into (5.35) which yields after some manipulation T +,2 (t, z) := wi 2 b i,i (z) + w i + w i wi 2 T,2 (t, z) := b i,i (z) w i + w i i=n+1 w i w j (5.37) j=n+1 w i w j. i=n+1 j=1 Using (5.37) along with (5.33) we have T + (t, z) = b i,i (z)v i (w i (t, z)) + w i w j (5.38) j=n+1 T (t, z) = b i,i (z)v i (w i (t, z)) w i w j j=n+1 and substituting (5.38) into (5.29) yields Ė(w(t)) µλ 1 E(w(t)) (5.39) ( + bλ n+i w 2 n+i(t, 2 1)e µ āλ ) i 2 w2 i (t, 1)e µ + b i,i (z)v i (w i (t, z)) + w i w j e µz dz + b i,i (z)v i (w i (t, z)) j=n+1 j=n+1 w i w j e µz dz. In the lemma statement we have N j=n+1 w j < 1 which implies b i,i (z) >. Therefore, we may drop the terms in the upper bound (5.39) to yield

63 Chapter 5. Lyapunov Functional for Nonlinear Systems 54 Ė(w(t)) µλ 1 E(w(t)) (5.4) ( + bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ + + j=n+1 j=n+1 w i w j e µz dz w i w j e µz dz. where the product terms are bounded by their absolute values. Estimates are derived on the product terms using Young s inequality as follows w i w j 1 2 i=n+1 j=1 = n 2 ( w 2 i + wj 2 ) j=n+1 wi 2 (5.41) so that (5.4) becomes Ė(w(t)) µλ 1 E(w(t)) (5.42) ( + bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ + n 2 w 2 i ( e µz + e µz) dz and using the fact that max z Ω (e µz + e µz ) = e µ + e µ we have Ė(w(t)) µλ 1 E(w(t)) (5.43) ( + bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ + ( e µ + e µ) n 2 w(t) 2 L 2. We equivalently write (5.43) as follows θ (, 1)

64 Chapter 5. Lyapunov Functional for Nonlinear Systems 55 Ė(w(t)) µλ 1 θ E(w(t)) µλ 1 (1 θ) E(w(t)) (5.44) ( + bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ + ( e µ + e µ) n 2 w(t) 2 L 2. Note that the following holds by assumption in the lemma statement λ 1 ( ) n 1 (1 θ) µ + e2µ µ = n w g(µ) (5.45) (1 θ) where g(µ) := 1 µ + e2µ µ. Multiplying both sides of (5.45) by 1 2 µ(1 θ)e µ w(t) 2 L 2 we have µλ 1 (1 θ)e µ w(t) L 2 n 2 2 (e µ + e µ ) w(t) 2 L (5.46) 2 and using the left hand side of Lemma 4 (5.1) we have µλ 1 (1 θ)e(w(t)) n 2 ( e µ + e µ) w(t) 2 L2. (5.47) Hence, the positive term in (5.44) is dominated and we have Ė(w(t)) µλ 1 θe(w(t)) + ( bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ (5.48) which completes the proof of the lemma. Remark 11. Note that this property is immediately exted to under-actuated systems just as in Chapter 4, Lemma 3. Remark 12. Note the difference in the bound on λ 1 in the Lemma 2 and Lemma 5 statements. In Lemma 2, the factor w arises when deriving estimates on the product terms in (4.2). This is not the case for the derivation of such terms in Lemma 5.

65 Chapter 5. Lyapunov Functional for Nonlinear Systems Extension to General Systems In this section, we use a modified functional to treat the more general system (3.26)-(3.28). Throughout the section, we denote α i := αi τ i and c ij := cij τ i i, j in order to simplify the notation. We begin by introducing the following modified entropy function v : ( ū, ) R + for some c > v(u) := u c ū (1 c ln + ū ). (5.49) u Such a function was used to treat the general SISO co-propagating system in [31]. Consider the following lemma Lemma 6. For any ā, b > such that (1/c b) < ū < (1/cā) the following holds on the domain R := { u R 1 c b ū u 1 } cā ū 1 2āu2 v(u) 1 2 bu2 (5.5) The proof follows the same methodology as in [24] (Lemma 1, Part 2). However, we present it here since the entropy function is now modified. Proof. Consider f(u) := v(u) 1 2āu2. The left hand side of the inequality follows by showing that f(u) is positive definite in a neighborhood of u =. The function f(u) is continuous and twice differentiable on u ( ū, ) with derivatives f (u) = u c(u + ū) āu, f ū (u) = ā. (5.51) c(u + ū) 2 Note that f (u) = when u = or u = r(ā) where r(ā) = (1/cā) ū. For any ā such that ū < (1/cā), it follows that r(ā) > which implies f () > and f (r(ā)) <. Therefore, f has a local minimum at u = and local maximum at u = r(ā). Hence, f(u) u ( ū, r(ā)). The right-hand side of the inequality follows similarly. Now consider the following Lyapunov candidate functional, inspired by the function (5.49)

66 Chapter 5. Lyapunov Functional for Nonlinear Systems 57 E(w(t)) := v ij (w i (t, z))e µz dz + j=1 j=1 i=n+1 v ij (w i (t, z))e +µz dz (5.52) where v ij (w i (t, z)) := w i c ij w ( i c ij ln 1 + w ) i, (i, j) {1,..., N} 2, i j. (5.53) w i v ij (w i (t, z)) =, i = j. and c ij := cij τ i i, j. Note that (5.53) is well-defined t, z Ω such that w i (t, z) > min z Ω w i. The first property for E(w(t)) (5.52) is stated as follows. Lemma 7. Let w m := min i min z Ω w i (z) i {1,..., N}. Consider any ā, b > such that (i, j) {1,..., N} 2, i j (1/ c ij b) < w m, (1/ c ij ā) > w. For any µ > let α := Nāe µ and β := N be µ. Then for any w(t) H 1 (Ω) such that w(t) d, where d := min { } 1 max (i,j) {1,...,N} 2 c ij ā w 1, w m min (i,j) {1,...,N} 2 c ij b the following holds 1 2 α w(t) 2 L 2 E(w(t)) 1 2 β w(t) 2 L 2 (5.54) Remark 13. Note that Lemma 7 is analogous to Lemma 4. However, there are some subtle differences. First, a factor of N arises in α and β due to the summation over the j indices in (5.52). In addition, the definition of d is changed so that (5.5) can be used. The proof of Lemma 7 is very similar to that of Lemma 4, and we omit it for this reason. Without loss of generality, consider a fully-actuated N xn counter-propagating Raman amplifier. Assume the system has n signals and n pumps so that N := 2n. Furthermore, assume w H 1 is sufficiently small so that w(t) < d where d is defined as in the Lemma 7 statement. The second property for E(w(t)) (5.52) is as follows. Lemma 8. For sufficiently small w H 1, let w(t) denote the unique global in time classical solution of

67 Chapter 5. Lyapunov Functional for Nonlinear Systems 58 closed-loop system (3.26)-(3.28). Assume the steady-state solution satisfies j=n+1 c ij w j < α i i. If we have λ 1 2N ( ) 1 ā(1 θ) µ + e2µ µ for some µ > and θ (, 1) then t > Ė(w(t)) µλ 1 θ E(w(t)) + N ( bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ (5.55) Remark 14. See Remark 12 for an analogous discussion on the different λ 1 bounds in Lemma 8 and Lemma 5. Proof. Following the same methodology as in the proof of Lemma 5, we arrive at the following upper bound Ė(w(t)) µλ 1 E(w(t)) (5.56) ( + N bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ + T + (t, z)e µz dz + T (t, z)e µz dz. where T + (t, z) := T (t, z) := j=1 j=1 λ i dv ij d w i d w i dz + n i=n+1 λ i dv ij d w i d w i dz + dv ij dw i N i=n+1 b i, w(t, z) + dv ij dw i j=n+1 b i, w(t, z) c ij w i w j (5.57) c ij w i w j. j=1 We begin by manipulating the first terms of (5.57). For simplicity, we define these terms as follows

68 Chapter 5. Lyapunov Functional for Nonlinear Systems 59 T +,1 (t, z) := T,1 (t, z) := j=1 j=1 i=n+1 dv ij d w i λ i d w i dz (5.58) λ i dv ij d w i d w i dz. (5.59) From the steady-state ODEs inherent in (3.25), we have λ i d w i dz = α i w i + λ i d w i dz = α i w i j=n+1 c ij w i w j i {1,..., n} (5.6) c ij w i w j i {n + 1,..., N} (5.61) j=1 which is equivalently written using the definition of b(z) (3.3) as λ i d w i dz = b i,i(z) w i i {1,..., n} (5.62) λ i d w i dz = b i,i(z) w i i {n + 1,..., N}. From the definition of v ij (w i (t, z)) (5.53) we compute (dv ij /d w i ) (i, j) {1,..., N} 2 as dv ij w i = d w i c ij (w i + w i ) 1 ( c ij ln 1 + w ) i. (5.63) w i Using (5.63) and (5.62) we write (5.58) after some manipulation as follows T +,1 (t, z) = T,1 (t, z) = j=1 j=1 i=n+1 ( b i,i (z) ( b i,i (z) w 2 i c ij (w i + w i ) v ij(w i ) ) w 2 i c ij (w i + w i ) v ij(w i ) ). (5.64) We now focus on the remaining terms of (5.57). For simplicity, we denote these terms as follows

69 Chapter 5. Lyapunov Functional for Nonlinear Systems 6 dv ij T +,2 (t, z) = b i, w(t, z) + c ij w i w j (5.65) dw j=1 i j=1 dv ij T,2 (t, z) = b i, w(t, z) + c ij w i w j. dw i j=1 i=n+1 j=1 From the definition of v ij (w i (t, z)) (5.53) we compute (dv ij /dw i ) (i, j) {1,..., N} 2 as dv ij w i = dw i c ij (w i + w i ) (5.66) Substituting (5.66) along with the definition of b(z) (3.3) into (5.65) yields T +,2 (t, z) = T,2 (t, z) = j=1 j=1 j=n+1 w i c ij (w i + w i ) w i c ij (w i + w i ) b i,i w i + c ij w i w j + c ij w i w j (5.67) j=1 j=1 j=1 b i,i w i c ij w i w j c ij w i w j j=1 and after some manipulation of (5.67) we have T +,2 (t, z) = j=1 T,2 (t, z) = w b 2 N i i,i c ij (w i + w i ) + j=1 w b 2 N i i,i c j=1 i=n+1 ij (w i + w i ) j=1 sgn ( c ij )w i w j (5.68) sgn ( c ij )w i w j. Using (5.68) and (5.64), the overall terms T + (t, z), T (t, z) become T + (t, z) = T +,1 (t, z) + T +,2 (t, z) (5.69) = b i,i (z)v ij (w i ) + N sgn ( c ij )w i w j j=1 j=1 and

70 Chapter 5. Lyapunov Functional for Nonlinear Systems 61 T (t, z) = T,1 (t, z) + T,2 (t, z) (5.7) = b i,i (z)v ij (w i ) N sgn ( c ij )w i w j. j=1 i=n+1 i=n+1 j=1 Using (5.69) and (5.7), we write (5.56) as Ė(w(t)) µλ 1 E(w(t)) (5.71) ( + N bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ + N j=1 w i w j e µz dz + N i=n+1 j=1 w i w j e µz dz. We now derive estimates on the product terms using Young s inequality as follows j=1 w i w j 1 2 j=1 ( w 2 i + w 2 j ) = = N Nw i 2 + j=1 Nw i 2 + w 2 i j=1 w 2 j w 2 j so that (5.71) becomes Ė(w(t)) µλ 1 E(w(t)) (5.72) ( + N bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ + N 2 w 2 i ( e µz + e µz) dz and using the fact that max z Ω (e µz + e µz ) = (e µ + e µ ) we have

71 Chapter 5. Lyapunov Functional for Nonlinear Systems 62 Ė(w(t)) µλ 1 E(w(t)) (5.73) ( + N bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ + N 2 (e µ + e µ ) w(t) 2 L 2. We equivalently write (5.73) θ (, 1) as Ė(w(t)) µθλ 1 E(w(t)) µ(1 θ)λ 1 E(w(t)) (5.74) ( + N bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ + N 2 (e µ + e µ ) w(t) 2 L 2. Recall by assumption in the lemma statement we have λ 1 2N( 1 µ + e 2µ µ ) ā(1 θ) (5.75) and multiplying both sides (5.75) by N 2 µ(1 θ)āe µ w(t) 2 L 2 yields N 2 µλ 1āe µ w(t) 2 L > N 2 (e µ + e µ ) w(t) 2 2 L2. (5.76) Using the left-hand side of Lemma 7 (5.54) we have 1 2 µλ 1(1 θ)e(w(t)) > N 2 2 (e µ + e µ ) w(t) 2 L (5.77) 2 which is used to dominate the positive term in the upper bound (5.74) to yield Ė(w(t)) µθλ 1 E(w(t)) + N ( bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ (5.78)

72 Chapter 5. Lyapunov Functional for Nonlinear Systems 63 which completes the proof of the lemma.

73 Chapter 6 Stability Analysis In this chapter, we derive controller conditions which guarantee stability of the origin for the various PDE models herein. The infinite-dimensional nature of these models suggests this must be done in some particular norm. Based on the lemmas constructed in the previous chapters, the L 2 (Ω)-norm seems most natural. The systems in this thesis may be fully-actuated or under-actuated. In either case, a proportional or dynamic controller may be implemented. Therefore, this chapter is organized into four major parts. We first begin by analyzing the stability of fully-actuated systems using the proportional controller. Following this, we carry out analysis using the dynamic controller. The stability analysis with the proportional controller operates on the linearized system, and the dynamic controller on the nonlinear one. The reason being that the proportional controller is generally applied in cases where the deviation from the origin is relatively small. In cases where large offsets are applied, the dynamic controller is considered. The remainder of the chapter treats the under-actuated case in a similar manner. 6.1 Stability Analysis for Fully-Actuated Systems In this section, we present stability analysis using both proportional and dynamic controllers for fullyactuated systems. Specifically, we derive controller conditions which guarantee exponential stability of the origin in the L 2 (Ω)-norm Proportional Controller Without loss of generality, consider a normalized, fully-actuated N xn counter-propagating Raman amplifier. Assume that such a configuration is accurately modeled by (3.72)-(3.74) in a sufficiently small neighborhood of the steady-state solution p(z). We can then prove the following theorem. 64

74 Chapter 6. Stability Analysis 65 Theorem 1. Assume w H 2 is chosen sufficiently small so that system (3.72)-(3.74) using controller (3.35) admits a unique maximal classical solution with duration T >. If the conditions of Lemma 2 are satisfied, and k i are chosen such that k 2 i < λ i λ n+i e 2µ i {1,..., n} (6.1) then following holds t [, T ] w(t) L 2 ce σt w L 2 (6.2) with c = e 2µ, σ = θµλ 1. Proof. Since the conditions of Lemma 2 are satisfied, we have V (w(t)) µλ 1 θ V (w(t)) + ( λn+i 2 w2 n+i(t, 1)e µ λ ) i 2 w2 i (t, 1)e µ t [, T ] (6.3) with V (w(t)) defined as in (4.2). From the definition of controller (3.35), we have w n+i (t, 1) = k i w i (t, 1) i {1,..., n} so that V (w(t)) µλ 1 θ V (w(t)) + ( λn+i 2 k2 i wi 2 (t, 1)e µ λ ) i 2 w2 i (t, 1)e µ t [, T ] (6.4) and with k i chosen as in the lemma statement, the upper bound on V (w(t)) becomes V (w(t)) µλ 1 θ V (w(t)) t [, T ]. (6.5) Applying Gronwall s inequality [27] to (6.5) yields V (w(t)) V (w )e µθλ1t t [, T ] (6.6)

75 Chapter 6. Stability Analysis 66 and using the left-hand side of Lemma 1 t [, T ] and the right-hand side for t = we have e µ 2 w(t) L 2 V (w(t)) V (w )e µθλ1t eµ e µθλ1t 2 w L 2 t [, T ]. (6.7) Thus, from (6.7) we have the following w(t) L 2 ce σt w L 2 t [, T ] (6.8) with c = e 2µ and σ = θµλ 1, which completes the proof of the theorem. Remark 15. By Theorem 1, we conclude that the closed-loop system (3.72)-(3.74) is exponentially stable in the L 2 -norm t [, T ] for any given T (, ) provided w H 2 is sufficiently small. In other words, the pumps and signals approach the desired steady-state solution p(z) in the L 2 sense. Also note the trade off between θ and the minimum speed requirement for λ 1 in Lemma 2. A larger value of θ results in faster decay to the steady-state solution, but it also strengthens the constraint on λ Dynamic Controller Consider a general fully-actuated N xn counter-propagating Raman amplifier, as modeled by (3.26)- (3.28). Assume the system has n signals, so that N := 2n. We can then prove the following theorem. Theorem 2. Consider the closed-loop system (3.26)-(3.28) using controller (3.37), and assume the conditions in the Lemma 8 statement are satisfied. Choose ā, b > as in the Lemma 7 statement, and r > in the range < r < 2λ 1 āe µ. If k i, ˆk i are chosen such that k 2 i < āλ i bλ n+i e 2µ i {1,..., n} (6.9) K d,i < ˆk i < K + d,i i {1,..., n} (6.1) where K d,i, K+ d,i denote the roots of the following quadratic in ˆk i

76 Chapter 6. Stability Analysis 67 ˆk 2 i ā bλ i λ n+i 2r be µ λ n+i k iˆki r 2 + 2rɛ(āe µ λ i be µ λ n+i k 2 i ) = (6.11) then following holds t w(t) L 2 ce σt w L 2 (6.12) for some c, σ >. Proof. We begin by introducing a Lyapunov candidate based on the augmented state (w(t), σ(t)) F (w(t), σ(t)) := E(w(t)) + Nr 2 σ 2 (t) (6.13) where r > and E(w(t)) is defined as in (5.52). Using the same methodology as in the Lemma 8 proof, we have F (w(t)) µλ 1 θ E(w(t)) + N ( bλ n+i w 2 2 n+i(t, 1)e µ āλ ) i 2 w2 i (t, 1)e µ and using the definition of controller (3.37), + Nr σ(t) σ(t). (6.14) F (w(t), σ(t)) µθλ 1 F (w(t), σ(t)) N 2 ] [w i σ i M i w i (6.15) σ i where M i := λ iāe µ λ n+i k 2i beµ r k iˆki λ n+i beµ r k iˆki λ n+i beµ 2rɛ λ n+i beµˆk i 2 i {1,..., n}. (6.16) In order to simplify the upper bound on F, it is desirable to have the M i be positive definite. For this to be the case, we require the M i to have positive diagonal entries, and det(m i ) >. Choosing controller

77 Chapter 6. Stability Analysis 68 conditions accordingly, we have ki 2 < and āλ i bλ n+i e 2µ i {1,..., n} (6.17) ˆk 2 i < 2rɛ bλ n+i e µ i {1,..., n} (6.18) which makes the diagonal entries of the matrices M i (6.16) positive. We now compute the determinant of the matrices M i (6.16) as det(m i ) = ( λ i āe µ λ n+i k 2i beµ) ) 2 (2rɛ λ n+i beµˆk2 i ( r k iˆki λ n+i beµ) (6.19) = ˆk 2 i ā bλ i λ n+i 2r be µ λ n+i k iˆki r 2 + 2rɛ(āe µ λ i be µ λ n+i k 2 i ). Consider (6.19) as a quadratic function in ˆk i. For such a function to have real and distinct roots, we require dis(det(m i )) = (4r bλ n+i e µ ) [r bλ n+i e µ k 2 i āλ i e µ r + 2e µ āɛλ i (āe µ λ i be µ k 2 i ) ] (6.2) = (4 be µ λ n+i ) (āe µ λ i be µ λ n+i k 2 i ) ( 2ɛāe µ λ i r ) > if i {1,..., n} where dis( ) returns the discriminant of a quadratic function. Note that (6.2) holds < r < 2ɛāe µ λ i i {1,..., n} (6.21) Given k i, r, ɛ, denote the roots of the function (6.19) as ˆK + i, ˆK i. If ˆk i is chosen such that ˆK i < ˆk i < ˆK + i (6.22)

78 Chapter 6. Stability Analysis 69 then the matrix M i (6.16) is positive definite. It remains to show that (6.22) implies (6.18). We begin by computing an upper bound on the roots of (6.19), starting with ˆK + i ˆK + i = rk i āe µ dis(det(m i)) λ i 2 bλ i λ n+i ā (6.23) < rk i āe µ λ i < reµ āλ i āλ i bλ e 2µ n+i r = 2 e 2µ āλ i ā 2 λ 2 i + ie 2µ bλn 2rɛ = beµ λ n+i where we used conditions (6.17) and (6.21) in the second and fourth lines of (6.23) respectively. Following the same methodology, a lower bound on ˆK i is computed as ˆK i > 2rɛ λ n+i (6.24) and hence (6.22) implies (6.18). Thus, if k i is chosen as in (6.17) and ˆk i as in (6.22), then M i (6.16) is positive definite and (6.15) becomes F (w(t), σ(t)) µθλ 1 F (w(t), σ(t)). (6.25) Note that Lemma 7 (5.54) can be naturally exted for F (w(t), σ(t)). That is, α, β > such that 1 2 α w(t) 2 L F (w(t)) β w(t) 2 L2. (6.26) After an application of Gronwall s inequality along with (6.26), we conclude that w(t) L 2 β α e µλ1θ t w L 2 (6.27)

79 Chapter 6. Stability Analysis 7 which proves Theorem 2 (6.12) for c = β α and σ = µλ 1θ. 6.2 Stability Results for Under-Actuated Systems In this section, we present stability analysis using both proportional and dynamic controllers for underactuated systems. Specifically, we derive controller conditions which guarantee exponential stability of the origin in the L 2 (Ω)-norm. We consider NxN systems with n signals and m pumps so that N := n+m. Denote the difference between the number of pumps and signals by γ := n m. In the results that follow, we assume the system is under-actuated in the sense that the first pump uses feedback from the first (γ + 1) signals, and the remaining (m 1) pumps each use feedback from one of the remaining (m 1) signals. Such an assumption is without loss of generality Proportional Controller Consider an under-actuated counter-propagating Raman amplifier, as modeled by (3.72)-(3.74) in a sufficiently small neighborhood of the steady-state solution w(z). We implement controller (3.41)-(3.43), so that such a system is under a closed-loop configuration. Consider the following theorem Theorem 3. Assume w H 2 is chosen sufficiently small so that system (3.72)-(3.74) using controller (3.41)-(3.43) admits a unique maximal classical solution with duration T >. If the conditions of Lemma 3 are satisfied, and k i are chosen such that k i < λ i λ n+1 γ+1 j=1 k j 1 e 2µ i {1,..., γ + 1} (6.28) and k 2 i < λ γ+i λ m+i e 2µ i {γ + 2,..., n} (6.29) then following holds t [, T ] w(t) L 2 ce σt w L 2

80 Chapter 6. Stability Analysis 71 with c = e 2µ, σ = θµλ 1. Proof. Since the conditions of Lemma 3 are satisfied, the following holds t [, T ] V (w(t)) µλ 1 θ V (w(t)) + + [ λ n+1 i=2 γ+1 λ i 2 w2 i (t, 1)e µ 2 w2 n+1(t, 1)e µ m [ λn+i 2 w2 n+i(t, 1)e µ λ ] γ+i 2 w2 γ+i(t, 1)e µ. ] (6.3) Using the definition of controller (3.41)-(3.43) and the multinomial theorem, we have w 2 n+1(t, 1) = ( γ+1 2 γ+1 γ+1 k i w i (t, 1)) = ki 2 wi 2 (t, 1) + 2 k i k j w i (t, 1)w j (t, 1). (6.31) j>i Using Young s inequality on (6.31) yields the following upper bound γ+1 γ+1 wn+1(t, 2 1) ki 2 wi 2 ( (t, 1) + k i k j w 2 i (t, 1) + wj 2 (t, 1) ) (6.32) j>i γ+1 γ+1 = k i k j wi 2 (t, 1) j=1 Using (6.32) along with the definition of controller (3.41)-(3.43), we write (6.3) as V (w(t)) µλ 1 θ V (w(t)) + 1 γ+1 γ m i=2 j=1 γ+1 λ n+1 e µ k i k j wi 2 (t, 1) [ λn+i 2 k2 γ+iw 2 γ+i(t, 1)e µ λ γ+i 2 w2 γ+i(t, 1)e µ λ i wi 2 (t, 1)e µ (6.33) ] and after factoring wi 2(t, 1) and w2 γ+i (t, 1), V (w(t)) µλ 1 θ V (w(t)) γ+1 γ+1 wi 2 (t, 1) λ n+1 e µ k i k j λ i e µ (6.34) j=1 m wγ+i(t, 2 1) ( λ n+i kγ+ie 2 µ λ γ+i e µ). i=2

81 Chapter 6. Stability Analysis 72 If we choose k i such that k i < λ i λ n+1 γ+1 j=1 k j 1 e 2µ i {1,..., γ + 1} (6.35) and k 2 i < λ γ+i λ m+i e 2µ i {γ + 2,..., n} (6.36) then the upper bound on V (6.34) becomes V (w(t)) µλ 1 θ V (w(t)). (6.37) The remainder of the proof follows analogously to Theorem Dynamic Controller Consider an under-actuated counter-propagating Raman amplifier, as modeled by (3.26)-(3.28). We implement controller (3.47)-(3.48), so that such a system is under a closed-loop configuration. Consider the following theorem. Theorem 4. Consider the closed-loop system (3.26)-(3.28) using controller (3.47)-(3.48). Assume the conditions in the Lemma 8 statement are satisfied. Choose ā, b > as in the Lemma 7 statement. If k i, ˆk i are chosen such that k i < ki 2 < āλ i bλ e 2µ n+1 āλ i bλ n+i γ+1 j=1 k j 1 i {1,..., γ + 1} (6.38) e 2µ i {γ + 2,..., n} ˆK i < ˆk 1 < ˆK + i i {1,..., γ + 1} ˆK i < ˆk i < ˆK + i i {γ + 2,..., n}

82 Chapter 6. Stability Analysis 73 where K i, K+ i denote the roots of the following quadratics γ+1 φ i (ˆk 1 ) := ˆk 1 2 ā bλ i λ n+1 b 2 λ 2 n+1e 2µ k i j i k j ˆk 1 (2r be µ λ n+1 k i ) r 2 + ( 2rɛ āe µ λ i be µ ) γ+1 λ n+1 k i j=1 k j γ + 1 i {1,..., γ + 1}, and φ i (ˆk i ) := ˆk 2 i ā bλ i λ n+i 2r be µ λ n+i k iˆki r 2 + 2rɛ(āe µ λ i be µ λ n+i k 2 i ) i {γ + 2,..., n}, with r chosen such that < r < 2ɛ(āe µ λ 1 be µ λ n+1 k i j i k j) γ + 1 i {1,..., γ + 1} then following holds t w(t) L 2 ce σt w L 2 for some c, σ >. Proof. We begin by introducing a Lyapunov candidate based on the augmented state (w(t), σ(t)) F (w(t), σ(t)) := E(w(t)) + Nr 2 σi 2 (t) (6.39) where r > and E(w(t)) is defined as in (5.52). Using the same methodology as in the Lemma 8 proof, we have [ ] F (w(t)) µλ 1 θ F (w(t)) + N γ+1 2 beµ λ n+1 wn+1(t, 2 1) λ i āe µ wi 2 (t, 1) + rσ 1 σ 1 m + N [ beµ λ n+i wn+i(t, 2 1) āλ i wi 2 (t, 1)e µ ] + rσ i σ i i=2 (6.4) Remark 16. Notice how we group the first pump with the (γ + 1) signals it uses feedback from. The

83 Chapter 6. Stability Analysis 74 remaining pumps are grouped with their respective signals in a fully-actuated manner. The reason for this is due to first pump being different than the others, and hence gives rise to a term that is analyzed separately. Using the definition of controller (3.47)-(3.48), we write (6.4) as follows F (w(t)) µλ 1 θ F (w(t)) (6.41) ( + N γ+1 2 γ+1 be µ λ n+1 k i w i (t, 1) + 2 ˆk 1 σ 1 (t)) λ i āe µ wi 2 (t, 1) + rσ 1 σ 1 N 2 m ] [w i σ i i=2 M i w i (6.42) σ i where M i := λ iāe µ λ n+i k 2i beµ r k iˆki λ n+i beµ r k iˆki λ n+i beµ 2rɛ λ n+i beµˆk i 2 i {γ + 2,..., n}. (6.43) If controller conditions are chosen such that k 2 i < āλ i bλ n+i e 2µ i {γ + 2,..., n} (6.44) ˆK i < ˆk i < ˆK + i i {2,..., m} (6.45) where i {2,..., m} ˆK i, ˆK + i denote the roots of det(m i ) (which are real and distinct if r is chosen as in (6.21), then the third line of (6.41) vanishes so that F (w(t)) µλ 1 θ F (w(t)) (6.46) ( + N γ+1 2 γ+1 be µ λ n+1 k i w i (t, 1) + 2 ˆk 1 σ 1 (t)) λ i āe µ wi 2 (t, 1) + rσ 1 σ 1.

84 Chapter 6. Stability Analysis 75 Using the fact that ( γ+1 ) 2 k γ+1 γ+1 iw i j=1 k ik j wi 2 from (6.32) and σ 1 = ɛσ 1 + γ+1 w i from the controller definition (3.41)-(3.43), we have the following upper bound F (w(t)) µλ 1 θ F (w(t)) (6.47) + N γ+1 [ ] M w 2 i σ 1 i w i σ 1 where M i := λ iāe µ γ+1 ki beµ λ n+1 j=1 k j r be µ λ n+1ˆk1 k i i {1,..., γ + 1}. (6.48) r be µ 2rɛ λ n+1ˆk1 k i γ+1 beµ λ n+1ˆk2 1 In order to simplify the upper bound on F, it is desirable to have the M i be positive definite. For this to be the case, we require the M i to have positive diagonal entries, and det(m i ) >. Choosing controller conditions accordingly, we have k i < āλ i bλ e 2µ n+1 γ+1 j=1 k j 1 i {1,..., γ + 1} (6.49) and ˆk 1 2 < 2rɛ b(γ +. (6.5) 1)e µ For simplicity of notation, define ã i := āλ i e µ, b := bλ n+1 e µ, u := γ+1 j=1 k j, C := j i k ik j. We then compute the determinant of the matrices M i (6.48) as det( M i ) = ˆk 2 1 ( ã b b ) 2 C ˆk 2rɛ (ã buk ) i 1 (2r bk i ) + r 2 + (6.51) γ + 1 which we regard as a quadratic in ˆk 1 with discriminant

85 Chapter 6. Stability Analysis 76 dis(det( M i )) = 4 br 2 ( bki u ã i ) + ( 8 brɛ ã bk ) ( ) i u ã i C b. (6.52) γ + 1 If we define G i : ã i bk i u, then G i > by (6.49), and we write (6.52) equivalently as follows dis(det( M i )) = 4rG i b [ r + 2ɛ(ã i bc) γ + 1 ] i {1,..., γ + 1}. (6.53) Note that the quadratic (6.51) has real and distinct roots if (6.53) is positive. That is, if we choose r such that < r < 2ɛ(ã 1 bc). (6.54) γ + 1 which also satisfies condition (6.21). Given k i, r, ɛ, we denote the roots of the quadratic (6.51) as ˆK + i, ˆK i. If ˆk 1 is chosen such that ˆK i < ˆk 1 < ˆK + i (6.55) i {1,..., γ + 1} then the matrix M i (6.51) is positive definite. It remains to show that (6.55) implies (6.5). We begin by computing an upper bound on the roots of (6.51), starting with ˆK + i ˆK + i = r bk i ã i b b2 C + dis(det( M i ) 2(ã b bc) rk i ã + rk i ã + 8rG i bɛ(ãi bc)/(γ + 1) 2rɛ b(γ + 1) 2 b(ã i bc) (6.56) Using (6.49), we have k i > ã i / b so that (6.56) becomes

86 Chapter 6. Stability Analysis 77 ˆK + i 2rɛ 2rɛ +. (6.57) b b(γ + 1) Furthermore, one can verify that 2rɛ 2rɛ + < b b(γ + 1) 2rɛ b(γ + 1) (6.58) γ > 3/4. Since γ >, we conclude ˆK + 2rɛ i < (6.59) b(γ + 1). Following the same methodology, a lower bound on ˆK i is computed as ˆK + i > 2rɛ b(γ + 1) (6.6) and hence (6.55) implies (6.5). Thus, if k i is chosen as in (6.49) and ˆk i as in (6.55), then M i (6.48) is positive definite and (6.47) becomes F (w(t), σ(t)) µθλ 1 F (w(t), σ(t)). (6.61) At this step, the remainder of the proof is analogous to that of Theorem 3. Remark 17. At this point, we may compare the various theorems in this chapter. Note that Theorem 1 is analogous to Theorem 3. However, the latter treats the under-actuated case. Therefore, we naturally expect some changes in the range of controller parameters which guarantee stability. Such changes arise in Theorem 3 due to the separate controller terms in upper bound (6.34). Specifically, condition (6.28) is stronger for the first (γ + 1) gains in Theorem 3 as opposed to the more relaxed (6.1) in Theorem 1. In addition, Theorem 2 is analogous to Theorem 4. The reason for having stronger conditions in

87 Chapter 6. Stability Analysis 78 Theorem 4 follows similarly. It is worth mentioning that in Theorem 4, the polynomial φ i (ˆk 1 ) must be positive i {1,..., γ + 1}. Equivalently, the parameter ˆk 1 must be chosen from the intersection of the sets ˆK i < ˆk 1 < ˆK + i i {1,..., γ + 1}.

88 Chapter 7 Simulations In this chapter, we simulate several Raman amplifier configurations using a hyperbolic PDE solver in MATLAB. Such a solver was used for the SISO co-propagating case [24] [31], and originates from [28]. Two types of experiments are conducted. The first involves stabilizing stabilizing the system to a desired steady-state power distribution. In this case, the linear model is sufficient for approximating the dynamics of the system. This is due to the fact that the designer has knowledge of the target distribution, and can choose initial conditions accordingly. The second experiment involves set-point tracking for signal power at the output. In this case, the nonlinear model is used, as changing the set-point may result in the system stabilizing to an unknown steady-state. Simulations are done for both fully-actuated and under-actuated systems. In both experiments, the goal is to choose controller parameters based off the stability analysis in Chapter Fully-actuated Systems In this section, we simulate the fully-actuated counter-propagating Raman amplifier. We begin by considering the steady-state stabilization problem for a 4x4 system using the proportional controller. In addition, we investigate the set-point tracking problem using the dynamic controller. In both cases, controller parameters are selected based off the analysis from Chapter Steady-state Stabilization Consider the following 4x4 counter-propagating Raman amplifier over a 1km fibre span 79

89 Chapter 7. Simulations 8 p i (t, z) t p i (t, z) t = λ i p i (t, z) z = λ i p i (t, z) z p i (z) := p(, z). α i τ i p i (t, z) + 1 τ i α i τ i p i (t, z) 1 τ i 4 c ij p i (t, z)p j (t, z) i = 1, 2 (7.1) j=1 4 c ij p i (t, z)p j (t, z) i = 2, 4 j=1 z (, ), with the following specifications (SI units in square parentheses) λ 1 = , λ 2 = , λ 3 = , λ 4 = [m/s] (7.2) τ 1 = , τ 2 = , τ 3 = , τ 4 = [s/m] α 1 = , α 2 = , α 3 = , α 4 = i [m 1 ] and coupling matrix C := [(W m) 1 ] (7.3) An open-loop configuration was run for 2µs to generate the following steady-state profiles Figure 7.1: Signal steady-state profiles - 4x4

90 Chapter 7. Simulations 81 Figure 7.2: Pump steady-state profiles - 4x4 After this, the following boundary controller was considered p 1 (t, ) =.6 (7.4) p 2 (t, ) =.4 p 3 (t, 1) = k 1 (p 1 (t, 1) p 1 (1)) + p 3 (1) p 4 (t, 1) = k 2 (p 2 (t, 1) p 2 (1)) + p 4 (1). in order to regulate the steady-state from a 5% offset at t =. Since the increase is relatively small, we assume that the linearized model accurately captures the dynamics of the system. Therefore, Theorem 1 may be used to choose controller parameters as in (6.1). The parameters k 1 = k 2 =.85 were chosen using µ =.1, and the following plots were generated at various points along the length of the amplifier Figure 7.3: Signal evolution in time at z=25km - 4x4 - Proportional Controller

91 Chapter 7. Simulations 82 Figure 7.4: Signal evolution in time at z=5km - 4x4 - Proportional Controller Figure 7.5: Signal evolution in time at z=1km - 4x4 - Proportional Controller From figures , we observe that the signals approach the desired steady-state. Note that Theorem 1 relies on the conditions in Lemma 2, which are indeed satisfied. To show this, we first adapt these conditions to account for non-normalized coefficients as follows λ 1 2N w c θ) τ(1 ( ) 1 µ e2µ µ (7.5) where c := τ := max c ij (7.6) (i,j) {1,...,N} 2 min i {1,...,N} τ i Note that we have c = , τ = , and w =.9. Using θ =.5 and µ =.1, condition (7.5) becomes λ which is indeed satisfied for λ 1 = The second condition is adapted by having

92 Chapter 7. Simulations 83 c w k (z) k=1 < α i i. (7.7) Based on the steady-state plots in figures and amplifier specifications (7.2), this condition is also satisfied Output Tracking It is often desirable to regulate the signal power at the output. In this case, the proportional controller is not sufficient. For simplicity, consider a SISO Raman amplifier in the steady-state ( p 1 (z), p 2 (z)). Without loss of generality, assume the following proportional controller guides the system to such a state p 2 (t, 1) = k 1 (p 1 (t, 1) p 1 (1)) + p 2 (1). (7.8) Now consider what happens if p 1 (1) is changed to some y d p 1 (1). In this case, the system may settle to an entirely new steady-state ( p 1(z), p 2(z)). Therefore, if p 2(1) p 2 (1), then tracking error is present. Consequently we consider the dynamic controller for the output tracking problem. Consider again the 4x4 Raman amplifier (7.1) with specifications (7.2) and coupling matrix (7.3). We begin by generating the steady-state in figures We then implement boundary controller (3.36) in an attempt to track a 15 % increase for both signals at the output of the amplifier (z = 1km). In order to choose controller gains, Theorem 2 is applied as follows. We begin by choosing ā, b as in the Lemma 7 statement. Using the fact that w m =.26, w =.9, max (i,j) {1,...,4} 2 c ij = 35, min (i,j) {1,...,4} 2 c ij = 15, we choose ā =.3 and b = Note that if θ =.5, the lower bound on λ 1 in the Lemma 8 statement is indeed satisfied for this choice of ā. Furthermore, recall that (7.7) holds. Thus, all of the conditions in the Lemma 8 statement are satisfied. Controller gains k 1 = k 2 =.2, ˆk 1 = ˆk 2 =.27 were chosen using ɛ = 5, µ =.1 based on Theorem 2 from the following script L 1 = 2.5e8 L 3 = 2.53e8 L 2 = 2.52e8

93 Chapter 7. Simulations 84 L 4 = 2.54e8 mu =.1 eps = 5 k1 =.95*sqrt((a*L 1)/(b*L 3))*exp( mu) k2 =.95*sqrt((a*L 2)/(b*L 4))*exp( mu) r1=.95*2*eps*(l 1*a*exp( mu)) c1 = a*b*l 1*L 3 c2 = 2*r1*b*exp(mu)*L 3*k1 c3 = r1ˆ2 + 2*r1*eps*(a*exp( mu)*l 1 b*exp(mu)*l 3*k1ˆ2) p1 = [c1 c2 c3] roots(p1) r2=.95*2*eps*(l 2*a*exp( mu)) d1 = a*b*l 2*L 4 d2 = 2*r2*b*exp(mu)*L 4*k2 d3 = r2ˆ2 + 2*r2*eps*(a*exp( mu)*l 2 b*exp(mu)*l 4*k2ˆ2) p2 = [d1 d2 d3] roots(p2) The following plots were generated at the output of the amplifier after running the simulation for 6µs Figure 7.6: Signal evolution in time at z=1km - 4x4 - Dynamic Controller In addition, the following surface plots were generated

94 Chapter 7. Simulations 85 Figure 7.7: Signal Evolution - 4x4 - Dynamic Controller

95 Chapter 7. Simulations 86 Figure 7.8: Pump Evolution - 4x4 - Dynamic Controller It is evident from the plots above that the dynamic controller performs with minimal error. Another way to investigate tracking performance is to observe the plots of σ i (t). Consider the definition of σ i (t) from (3.36). For tracking to be achieved, σ(t) approaches zero if and only if σ i (t) approaches a constant. This is indeed the case for our simulation, as outlined by the plot below

96 Chapter 7. Simulations 87 Figure 7.9: σ i (t) for ɛ = 5 The effect of ɛ on tracking performance is another matter to investigate. From the quadratic in Theorem 2, it is immediately obvious that changing ɛ will change the range of ˆk i. We begin by considering a decrease in ɛ. Consider the following plots of σ i (t) for ɛ = 1, ˆk 1 = ˆk 2 =.5. Figure 7.1: σ i (t) for ɛ = 1 From Figure 7.1, it is apparent that the settling time has increased. We continue the investigation with an increase in ɛ. Consider the following plots for ɛ = 1, ˆk 1 = ˆk 2 =.55

97 Chapter 7. Simulations 88 Figure 7.11: σ i (t) for ɛ = 1 It is apparent that the setting time has decreased with a larger value of ɛ. Since there is now overshoot present in Figure 7.11, we observe the following output tracking plots Figure 7.12: Signal evolution in time at z=1km - 4x4 - Dynamic Controller, ɛ = 1 Compare figures 7.6, 7.12, and notice that the former has less overshoot but a longer settling time. We now further increase ɛ. Consider the following plots with ɛ = 2, ˆk 1 = ˆk 2 = 1.1

98 Chapter 7. Simulations 89 Figure 7.13: Plots of σ i (t) for ɛ = 2 and output tracking plots Figure 7.14: Signal evolution in time at z=1km - 4x4 - Dynamic Controller, ɛ = 2 In comparison to Figure 7.12, the system is faster to enter a relatively small neighborhood of the steady-state, but has more overshoot. In addition, consider the following surface plots for ɛ = 2

99 Chapter 7. Simulations 9 Figure 7.15: Signal 2 Evolution - 4x4 Dynamic Controller, ɛ = 2 Figure 7.16: Pump 2 Evolution - 4x4 Dynamic Controller, ɛ = 2 and note the oscillatory behavior in comparison to figures Lastly, observe the following plot for the case where ɛ = 1, ˆk 1 = ˆk 2 = 5.5.