Claude Universite

Bernard -Lyon

M Le Professeur

Frédéric Fleury

Hamda Ben

Hadid M Le Professeur Didier

Revel M Le Professeur

Philippe Chevalier

M Fabrice

Vallée Mme

Dominique Marchand

Composantes Sante

G Rode

La Mme

C Professeure

Burillon

D Bourgeois

Vinciguerra

M X Perrot

A-M Professeure

Schott

Une autre contribution de la thèse consiste en la construction des programmes de Malab permettant d'effectuer des simulations numériques pour la validation des correcteurs conçus. Pour la résolution numérique des EDP hyperboliques nous avons discrétisé nos systèmes par le schéma numérique de Preissmann. Nous avons chaque fois un système d'équations algébriques non-linéaires à résoudre de façon récurrente. L'apport des simulations numériques nous permet de mieux comprendre la méthodologie applicative de la théorie du contrôle en dimension infinie.

Mots-Clés

Systèmes EDP hyperboliquessystèmes non-linéairescontrôle au bordfeedback dynamiquestabilitérégulation de mesure -PI contrôleurfonctionnelle de Lyapunovréseau en étoileréseau en cascadesimulation numériqueschéma Preissmann.

COMPOSANTES ET DEPARTEMENTS DE SCIENCES ET TECHNOLOGIE

Résumé

Dans cette étude on s'intéresse à la dynamique d'une classe de systèmes non-linéaires décrits par des équations aux dérivées partielles (EDP) du type hyperbolique. L'objectif de l'étude est de construire des lois de contrôle par feedback dynamique de la sortie afin de stabiliser le système autour d'un point d'équilibre d'une part, et, d'autre part, de réguler la sortie vers le point de consigne.

Nous considérons la classe des systèmes de dimension infinie gouvernés par des EDP quasi-linéaires à deux variables indépendantes (une variable temporelle et une variable spatiale). Le modèle mathématique des systèmes prend la forme suivante en termes de EDP : où désigne un intervalle connexe borné de R, l'état , et la matrice . Nous supposons que la matrice est diagonalisable ou plus simplement diagonale ayant toutes les valeurs propres réelles et non nulles. Pour le bien-posé du système dynamique non seulement l'état initial mais aussi certaines conditions frontières doivent être prescrites en cohérence avec les EDP. Nous supposons que l'observation et le contrôle sont ponctuels. Autrement dit l'action du contrôle intervient dans le système via les conditions frontières et l'observation est effectuée aux points de la frontière. La trajectoire du système évolue en fonction du temps dans un espace fonctionnel défini sur l'intervalle Ω.

Notre étude est motivée par l'observation que de nombreux processus physiques sont modélisés par ce type d'équations EDP. Nous citons, par exemple, des processus tels que flux trafique en transport, flux de gaz dans un réseau de pipeline, échangeurs thermiques en génie des procédés, équations de télégraphe dans des lignes de transmission, canaux d'irrigation en génie civil etc.

Nous commençons l'étude par une EDP non-linéaire scalaire. Dans ce cas-là nous proposons un correcteur intégral stabilisant qui assure la régulation de la sortie avec l'erreur statique nulle. Nous prouvons la stabilisation locale du système non-linéaire par le correcteur intégral en construisant une fonctionnelle de Lyapunov appropriée.

La conception des correcteurs proportionnels et intégraux (PI) que nous proposons est étendue dans un cadre de systèmes de deux EDP. Nous prouvons la stabilisation du système en boucle fermée à l'aide d'une nouvelle fonctionnelle de Lyapunov. La synthèse des correcteurs PI stabilisants se poursuit dans un cadre de réseaux formés d'un nombre fini de systèmes à deux EDP : réseau étoilé et réseau série en cascade. Les contrôles et les observations se trouvent localisés aux différents noeuds de connexion. Pour ces configurations nous présentons un ensemble de correcteurs PI stabilisants qui assurent la régulation vers le point de consigne. Les correcteurs PI que nous concevons sont validés par des simulations numériques à partir des modèles non-linéaires EDP.

La contribution de la thèse, par rapport à la littérature existante, consiste en l'élaboration de nouvelles fonctionnelles de Lyapunov pour une classe de systèmes stabilisés par correcteur PI. En effet une grande quantité de résultats ont été obtenus sur la stabilisation des systèmes hyperboliques par feedback statique de la sortie. Toutefois il existe encore peu de résultats sur la stabilisation de ces systèmes par feedback dynamique de la sortie. L'étude de la thèse est consacrée sur l'élaboration des fonctionnelles de Lyapunov permettant d'obtenir des correcteurs PI stabilisants. L'approche de Lyapunov direct que nous avons proposée a pour l'avantage de permettre d'étudier la robustesse des lois de feedback de la sortie PI vis-à-vis de la non-linéarité.

Notations

R

The set of real numbers

R

PDE hyperbolic systems of conservation laws

In nature, a large number of phenomena or physical processes are modelized by systems of conservation laws. For example they describe the state evolution of open channels in hydraulic engineering such as the Saint-Venant equations in [START_REF] Bounit | Regulation of an irrigation channel through the semigroup approach[END_REF][START_REF] Coron | A Lyapunov approach to control irrigation canals modeled by Saint Venant equations[END_REF][START_REF] Pham | Predictive control with guaranteed stability for hyperbolic systems of conservation laws[END_REF][START_REF] Trinh | Multivariable pi controller design for 2 x 2 systems governed by hyperbolic partial differential equations with lyapunov techniques[END_REF][START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF], traffic flow (see the Burger equation or Lighthill-Whitham-Richards model in [START_REF] Bayen | Adjoint-based control of a new Eulerian network model of air traffic flow[END_REF][START_REF] Goatin | Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity[END_REF][START_REF] Lighthill | On kinematic waves. ii. a theory of traffic flow on long crowded roads[END_REF]), gas flow in pipeline networks (see the Euler equations in [START_REF] Banda | Coupling conditions for gas networks governed by the isothermal Euler equations[END_REF][START_REF] Banda | Gas flow in pipeline networks[END_REF][START_REF] Euler | Principes gnraux du mouvement des fluides[END_REF][START_REF] Gugat | Gas flow in fan-shaped networks: classical solutions and feedback stabilization[END_REF]), heat exchanger processes in [START_REF] Friedly | Dynamic Behavior of Processes[END_REF][START_REF] Xu | A robust PI controller for infinite dimensional systems[END_REF][START_REF] Xu | Exponential stability of the heat exchanger equation[END_REF], and the telegrapher equation of current and voltage along electrical transmission lines in [START_REF] Harrington | Time-Harmonic Electromagnetic Fields[END_REF], to cite but a few. A very nice literature on the modelling applications by systems of conservation laws can be seen in [4, Chapter 1].

Systems of conservation laws are often considered in three main forms as follows:

• Derivative form:

∂ ∂t φ(x, t) + ∂ ∂x f (φ(x, t)) = 0
where space variable x ∈ W, time variable t ∈ [0, ∞), the conserved state φ(x, t) ∈ R n , and the flux function f : R n → R n .

• Hyperbolic form: in the case that function f from the derivative form above is differentiable, by noting A(φ) the n × n Jacobien matrix of function f at point φ, we get the following hyperbolic form

∂ ∂t φ(x, t) + A(φ) ∂ ∂x φ(x, t) = 0
Chapter 1. General introduction 2

• Integral form: by integrating the derivative form over a given space interval [x 1 , x 2 ] one obtains ∂ ∂t

x 2 x 1 φ(x, t)dx = f (φ(x 1 , t)) -f (φ(x 2 , t))
This equation implies that the quantity φ is conserved because the total amount of φ in any interval [x 1 , x 2 ] can change only due to the flow f at boundary points (see in [START_REF] Bressan | Hyperbolic systems of conservation laws: An illustrated tutorial[END_REF]).

In the thesis, we consider the systems of conservation laws in the hyperbolic form, i.e the dynamic systems governed by 1D homogeneous first-order hyperbolic partial differential equations (PDE) as follows:

∂ ∂t φ(x, t) + A(φ) ∂ ∂x φ(x, t) = 0 φ(0, x) = φ 0 (x) , (1.1)
where x ∈ [0, L] and t ∈ [0, ∞) are respectively space and time variables; φ : [0, L] × [0, ∞] → R n is the vector state, for each (x, t) → φ(x, t) is a mapping which is in X the state space of the associated Cauchy problem; and matrix A : X → R n×n such that A(φ) has n real eigenvalues λ i (φ) for i = 1, n. In the case that these eigenvalues are all distinct, then system (1.1) is strictly hyperbolic. We distinguish two cases: (i) if A(φ)

does not depend on φ, system (1.1) is called linear hyperbolic; (ii) if A(φ) depends on φ, we call (1.1) nonlinear system or quasi linear system.

We consider some examples of PDE hyperbolic equations of conservation laws in details:

• The first one is the Saint-Venant equations (see [START_REF] De Saint | Théorie du mouvement non permanent des eaux, avec application aux crues rivières et à l'introduction des marées dans leur lit[END_REF]) derived from the conservation of mass and conservation of linear momentum, which describe an open channel of neglected friction slope,

∂ ∂t H Q + ⎛ ⎜ ⎜ ⎝ 0 1 B Q 2 BH + gB 2Q BH ⎞ ⎟ ⎟ ⎠ ∂ ∂x H Q = 0 ,
where H(x, t) and Q(x, t) represent the water level and the flow discharge respectively; B and g denote the base width and the Newton gravitation constant. This model will be studied as the direct application with simulations in Chapter 3 for a single system and in Chapter 4 for networks.

• The second one is the Lighthill-Witham-Richards equation (see [START_REF] Lighthill | On kinematic waves. ii. a theory of traffic flow on long crowded roads[END_REF]) which modelizes traffic flow on long road

∂ ∂t ρ + f (ρ) ∂ ∂x ρ = 0 ,
Chapter 1. General introduction 3 with conserved quantity ρ(x, t) represents the vehicle density. This scalar system will be studied more in Chapter 2.

It should be remarked that in many cases, system (1.1) can be diagonalized by an appropriate change of coordinates (usually using Riemann invariants) to obtain a characteristic form. It means that we can find a change of coordinates ψ(φ) = ψ 1 (φ),

• • • , ψ n (φ) such that ∂ ∂t ψ(x, t) + D(ψ) ∂ ∂x ψ(x, t) = 0
where

D(ψ) = diag λ 1 (ψ), λ 2 (ψ), • • • , λ n (ψ) .
Note also that this change of coordinates always exits if system (1.1) is linear and all eigenvalues of matrix A are non-zero, or if system (1.1) is quasi-linear with distinct eigenvalues of A and n = 2, see in [4]. This characteristic form will be used in Chapter 3 and Chapter 4 for linear hyperbolic systems.

Boundary control problem

One of the popular topic in studying PDE hyperbolic systems is concerning boundary conditions. Firstly, the boundary conditions play an important role to guarantee the existence and unique solution for the PDE hyperbolic system (1.1) in the state space X. It implies that in our case, special conditions on the boundary interval x ∈ [0, L] need to be added. In addition, these boundary conditions are also useful to control PDE hyperbolic systems, for example to obtain the stability of un equilibrium for controlled closed-loop systems, or the output regulation with the presence of disturbances.

To be more specific, we consider system (1.1) with the following boundary condition F φ(0, t), φ(L, t), U(t) = 0 (1.2)

where F : R n × R n × R q → R n is C 1 function which is obtained from physical constraints on the system (see [4,[START_REF] Lamare | Control of Hyperbolic Systems by Lyapunov Analysis[END_REF]). The function U (t) ∈ R q denotes the feedback control action on the boundary. We distinguish here two kinds of control action U :

• Static state feedback control law: if the control action is a function of state on the boundary, i.e U (t) = U φ(0, t), φ(L, t) (see for instantce [START_REF] Coron | Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems[END_REF][START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF][START_REF] Greenberg | The effect of boundary damping for the quasilinear wave equation[END_REF][START_REF] Hale | Introduction to functional differential equations[END_REF][START_REF] Li | Global classical solutions for quasilinear hyperbolic systems[END_REF]). The kind of control is not robust to disturbances (for example constant disturbances).

• Dynamic state feedback control law: if some other dynamic terms are added in the control action, i.e U (t) = g φ(0, t), φ(L, t), z(t) where ż(t) = h φ(0, t), φ(L, t) , with h and g are functions in R q (see for example [3,[START_REF] Pham | Contribution to predictive control for systems of conservation laws[END_REF][START_REF] Perrollaz | Finite time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks[END_REF][START_REF] Trinh | Multivariable pi controller design for 2 x 2 systems governed by hyperbolic partial differential equations with lyapunov techniques[END_REF][START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF]). With this control type, the closed-loop system is more complicated because it becomes a couple system of PDE and ordinary differential equation (ODE). However, and as this is already the case for finite dimensional controled system, adding this new freedom degree in the control law allows to extend the class of system for which output feedback stabilization may be achieved. Moreover, it may increase the robustness of the closed loop system. This robustness aspect is the aspect in which we are interested in the following.

To resume, the general objective of boundary control problem is to find boundary conditions in the form (1.2) such that:

• the well-posed problem : System (1.1)-(1.2) has a unique solution in a normed vector state space X with the norm denoted by | • | X .

• The stability problem : The equilibrium point of system (1.1)-(1.2) is stable in one of three kinds as follows:

-Globally asymptotically stable in X if there exists a class KL function β such that for all initial condition ϕ in X for which there exists a (weak) solution denoted φ(x, t) |φ(x, t)| X β(|ϕ| X , t) , ∀t 0.

-Locally asymptotically stable in X if there exists a positive real number d such that former property is true only for |ϕ| X d -Locally and globally Exponentially stable if there exists k and λ such that the function β(s, t) = k exp(-λt)s.

On top of these two classical problems in boundary control problem, we can consider another one which is the problem of robustness with respect to external disturbances. This aspect will be addressed in the thesis and precisely defined in the following Section.

Let us now mention some important results in the literature concerning the boundary control problem.

Almost all results of boundary control for general quasi-linear systems are given by static control law. In this case, supposing that the eigenvalues λ i > 0 for i = 1, k and λ i < 0 for

i = k + 1, n, boundary condition (1.
2) with static state feedback control law is written under the following form (see in [START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF])

φ + (0, t) φ -(L, t) = G φ + (L, t) φ -(0, t) (1.3)
where the G : R n → R n satisfies that G(0) = 0, φ + ∈ R k and φ -(L, t) ∈ R n-k are two vector elements of φ. Many interesting results in the literature are based on finding sufficient conditions for G in (1.3) such that the equilibrium point φ ≡ 0 is exponentially stable in terms of some norms.

The first result in this topic is given by Li in [START_REF] Li | Global classical solutions for quasilinear hyperbolic systems[END_REF], and de Halleux et al. in [START_REF] De Halleux | Boundary control in networks of open channels[END_REF] by the method of characteristic. A sufficient condition for the exponential stability in norm C 1 (0, L) is introduced as follows: The equilibrium point φ ≡ 0 of quasi-linear system (1.1) and (1.3) is exponentially stable in the state space C 1 (0, L) if the following sufficient condition is satisfied:

ρ ∞ G (0) < 1 (1.4)
where Regarding the sufficient condition (1.4), in [START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF], the authors give another proof by using Lyapunov direct method to prove the exponential stability of φ ≡ 0 in also C 1 norm.

ρ ∞ : R n×n → R , ρ ∞ (P ) := Inf R ∞ ( P -1 ), ∈ D n,+ with R p (K) := Max ||σ||p=1 { ||Kσ|| p },
Another remarkable result is developed by Coron et al. in [START_REF] Coron | Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems[END_REF]. In this work, another sufficient boundary condition is proposed to have the stability of the equilibrium solution φ ≡ 0 in the norm H 2 (0, L): The equilibrium solution φ ≡ 0 of quasi-linear system (1.1) and (1.3) is exponentially stable in the state space H 2 (0, L) if we have the following sufficient condition:

ρ 2 G (0) < 1 (1.5)
where

ρ 2 : R n×n → R , ρ 2 (P ) := Inf R 2 ( P -1 ), ∈ D n,+
It is proved in [START_REF] Coron | Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems[END_REF] that the condition in (1.5) is weaker than the one in (1.4), i.e

ρ 2 G (0) ρ ∞ G (0) ∀n 1.
In the particular case that system (1.1) is linear where A is a nonzero distinct diagonal matrix and boundary condition (1.3) becomes

φ + (0, t) φ -(L, t) = K φ + (L, t) φ -(0, t)
with K ∈ R n×n , it should be mentioned the result of Silkowski (see in [START_REF] Silkowski | Star shaped region of stability in hereditary systems[END_REF]) and also of Hale and Verduyn Lunel in [START_REF] Hale | Introduction to functional differential equations[END_REF]), in which they gave a necessary and sufficient boundary condition for the exponential stability of the zero equilibrium point φ ≡ 0 in the norm L 2 (0, L): The equilibrium point φ ≡ 0 of linear system is exponentially stable in the L 2

Chapter 1. General introduction 6 norm if and only if

ρ 3 (K) := max S diag(e jθ 1 , • • • , e jθn )K ; θ i ∈ R ∀i = 1, n < 1 , (1.6)
where j ∈ C and j 2 = -1, and S(N ) is the spectral radius of matrix N ∈ C n×n .

Note that in general ρ 3 (K) ρ 2 (K) (see in [START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF]).

Concerning boundary control problem of hyperbolic systems with dynamic control laws, in the case of quasi-linear systems it is still an open question up to now to find appropriate conditions for exponential stability of the equilibrium points. There exist only a few results for some particular systems such as:

• nonlinear scalar systems (i.e n = 1) in [65, Chapter 5] by using Proportional-Integral (PI) control with damping terms, and in [START_REF] Pham | Receding horizon boundary control of nonlinear conservation laws with shock avoidance[END_REF] by applying a boundary integrator with predictive control.

• 2 × 2 quasi-linear systems (i.e n = 2) in [START_REF] Perrollaz | Finite time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks[END_REF] with a dynamic boundary control called finite-time stabilizer.

Most of the stabilization results with dynamic control laws in the literature are concerned with linear hyperbolic systems inspired from the studies of general infinite-dimensional linear systems with implementation of integral control action (see [START_REF] Pohjolainen | Robust multivariable PI-controllers for infinite dimensional systems[END_REF] and [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions[END_REF]). Most results have been developed in recent publications for linearized hyperbolic systems: the predictive control combined with integral action for a class of 2 × 2 linear hyperbolic systems and their cascaded network in [START_REF] Pham | Contribution to predictive control for systems of conservation laws[END_REF][START_REF] Pham | Predictive control with guaranteed stability for hyperbolic systems of conservation laws[END_REF]; multivariable PI control design for some classes of 2 × 2 systems by using Lyapunov techniques in [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF][START_REF] Trinh | Multivariable pi controller design for 2 x 2 systems governed by hyperbolic partial differential equations with lyapunov techniques[END_REF], and integral controller design by using Laplace transformation frequency method in [3,[START_REF] Coron | Feedback Stabilization for a scalar conservation law with PID boundary control[END_REF], and integral control laws by using operator and semi-group approach in [START_REF] Trinh | Pi regulation control of a fluid flow model governed by hyperbolic partial differential equations[END_REF][START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF]. The backstepping method has been exploited in [START_REF] Deutscher | Backstepping design of robust state feedback regulators for linear 2×2 hyperbolic systems[END_REF][START_REF] Lamare | Control of 2 × 2 linear hyperbolic systems: backstepping-based trajectory generation and PI-based tracking[END_REF] to elaborate output regulators in the same context. These approaches will be considered in details in Section 1.3.

Output regulation with PI controllers and motivation 1.2.1 Output regulation problem with PI control design

In the above Section, we have described the boundary control problem with main objective as stabilization of closed-loop systems with boundary control feedback. In this Section, we consider another problem which is the output regulation, i.e., the output measurement needs to track a reference defined a priori despite of the presence of constant disturbances. Therefore, the control input must be designed such that the controlled disturbed system is robust in the senses that the output measurement is regulated to the desired reference.

At first, we explain how the constant disturbances lead us to considered output regulation problem. As a matter of fact, the disturbances are inevitable in practical model, that may come from the error of modelling, linearisation, sensors,... as in Figure 1.1. These disturbances result in errors between the measured output noted by y(t) ∈ R q and the references y r ∈ R q : e(t) = y(t)y r , see in Figure 1.2. Our task is to find input control

law u(t) ∈ R q such that lim t→∞ y(t) = y r (i.e lim t→∞ ||e(t)|| R q = 0).
In industrial applications, the PI controller is considered as one of the most effective approaches to remove steady state errors and oscillations of the controlled systems, see in [START_REF] Astrom | Advanced PID Control[END_REF]. This motivates us to introduce the integral action control in the form of PI controllers to resolve the output regulation problem. A PI controller admits the general form as follows: The ideas of designing PI control for PDE hyperbolic systems have been introduced in the literature by the works of Pohjolainen, Xu and Jerbi [START_REF] Pohjolainen | Robust multivariable PI-controllers for infinite dimensional systems[END_REF][START_REF] Xu | A robust PI controller for infinite dimensional systems[END_REF] for some classes of linear hyperbolic systems. Later on, many results on the same topic were developed for nonlinear hyperbolic systems [START_REF] Tamasoiu | Stabilization and boundary control for balance and conservation laws[END_REF][START_REF] Trinh | Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations[END_REF] and for linear systems [3,[START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF][START_REF] Trinh | Multivariable pi controller design for 2 x 2 systems governed by hyperbolic partial differential equations with lyapunov techniques[END_REF][START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF] with different techniques that will be analysed in Section 1.3.

u(t) = K P y(t) -y r + K I t 0 y(τ ) -y r dτ + w c ,

Motivation

The motivation here is employing PI control as a dynamic feedback control law to resolve simultaneously two tasks: boundary feedback stabilization problem and output regulation problem, i.e., (i) the asymptotic stabilization of closed-loop system controlled by PI controllers, (ii) the output regulation to the desired references.

The application of PI controllers in controlled models is illustrated in Figure 1.3. More precisely we consider the PDE hyperbolic system (1.1) with the boundary condition (1.2), and the output measurement taken on the boundary, i.e.,

y(t) := l φ(0, t), φ(L, t), w o where w o ∈ R q is the constant output disturbance, function l : R n × R n × R q → R q .
We write the control law by

u(t) = K P l (φ(0, t), φ(L, t), w o ) -y r + K I z(t) + w c , ż(t) = l φ(0, t), φ(L, t), w o -y r
The main objective here is to find the conditions for boundary function l and matrix gains K P , K I such that the equilibrium point of the coupled PDE/ODE closed-loop system is asymptotically stable, and y(t) → y r as t → ∞.

In the next Section we will give a general introduction about some popular methods to prove the stability of the the coupled PDE/ODE closed-loop systems.

Stability analysis methods in the literature

Lyapunov direct method

The Lyapunov direct approach called also the second method of Lyapunov in [START_REF] Lyapunov | The general problem of the stability of motion[END_REF] is widely used to analyze the stability of hyperbolic PDE systems in both linear cases and nonlinear ones. The advantage of this method is: (i) The stability proof is less complicate because of using more direct computations compared to other approaches;

and (ii) it gives a direct extension from linear systems to nonlinear systems (note that in PDE systems, the stability of linearized models does not imply the one of nonlinear systems with even a very small variation, see in [START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF]).

The pioneer result of using Lyapunov direct method for 1D hyperbolic PDE systems goes back to [START_REF] Coron | A Lyapunov approach to control irrigation canals modeled by Saint Venant equations[END_REF] where entropies are used as Lyapunov functionals. By this Lyapunov functional type, their time derivatives are only semi-definite negative. Therefore the precompactness of trajectories is required to apply the LaSalle invariant principle to get the asymptotic stability. However, this condition is difficult to obtain in nonlinear PDEs (see [START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF]). Following this, the strict Lyapunov functionals [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF][START_REF] Coron | Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems[END_REF][START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF][START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF][START_REF] Prieur | An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems[END_REF][START_REF] Trinh | Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations[END_REF] are proposed to avoid the drawback of the former Lyapunov functionals so that the time derivative along the trajectories is strictly definite negative. To be detailed, let denote extended state ϕ(., t) of extended closed-loop systems in state space X, which is equipped with a norm ||.|| X . One must construct a strict Lyapunov functional

V (ϕ(t)) ∈ C 1 (X, R)
such that with the boundary conditions the followings hold:

• There exists a constant M > 0 such that

1 M ||ϕ|| X V (ϕ) M ||ϕ|| X
• There exists some constant α > 0 such that the time derivative of V (ϕ(t)) along the trajectories of the system satisfies

dV (ϕ(t)) dt -αV (ϕ(t))
A large number of works using the strict Lyapunov functional technique can be found in the literature. Some Lyapunov functionals of nonlinear hyperbolic systems with static state feedback control laws, in which the Lyapunov inequalities are only derived with ||ϕ|| X small enough have been considered in [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF][START_REF] Coron | Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems[END_REF][START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF] equivalent to H 2 -norm or C 1norm. Other Lyapunov functionals (equivalent to H 2 -norm) have been proposed in [START_REF] Tamasoiu | Stabilization and boundary control for balance and conservation laws[END_REF][START_REF] Trinh | Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations[END_REF] for scalar nonlinear hyperbolic systems with integral control action. In the linear case, Lyapunov functionals (equivalent to L 2 norm) for a general class of linear Chapter 1. General introduction 10 symmetric hyperbolic systems with static boundary control have been constructed [START_REF] Xu | Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems[END_REF],

for the 2 × 2 linear hyperbolic systems with boundary PI control [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF][START_REF] Lamare | Control of 2 × 2 linear hyperbolic systems: backstepping-based trajectory generation and PI-based tracking[END_REF][START_REF] Trinh | Multivariable pi controller design for 2 x 2 systems governed by hyperbolic partial differential equations with lyapunov techniques[END_REF], and for some networks of linear hyperbolic systems in [START_REF] Bastin | Andréa Novel On Lyapunov stability of linearised Saint-Venant equations for a sloping channel[END_REF][START_REF] Trinh | Boundary pi controllers for a star-shaped network of 2 × 2 systems governed by hyperbolic partial differential equations[END_REF].

The Lyapunov direct method is the principal tool utilized along this thesis.

Frequency method with Laplace transform

Another method to be mentioned here is the frequency analysis method with Laplace transform. This method is only used for liner hyperbolic systems, and has the advantage that we can establish necessary and sufficient conditions for the stability analysis. The principal idea is that, by employing Laplace transform, one gets a characteristic equation (i.e, denominator of the transfer function). In the case of 1D linear hyperbolic system, the controlled system is exponentially stable if and only if all roots of the characteristic equation (called poles of the system) have negative real parts (see [START_REF] Coron | Feedback Stabilization for a scalar conservation law with PID boundary control[END_REF][START_REF] Lichtner | Spectral mapping theorem for linear hyperbolic systems[END_REF]). Some works with this approach can be found in [4, Chapter 2] for hyperbolic systems of two conservation laws with static boundary control, for the control of 2 × 2 linear hyperbolic systems with a PI control action in [3], and for linear scalar hyperbolic systems with integral control action in [START_REF] Coron | Feedback Stabilization for a scalar conservation law with PID boundary control[END_REF][START_REF] Trinh | Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations[END_REF].

As a matter of fact, the closed-loop systems in these works can be considered as the delay-differential equations of neutral type, and we have the characteristic equation in the form of a polynomial which contains some exponential terms. To guarantee all roots of a characteristic equation to be in the left half-plane, one can apply the direct computation as in [START_REF] Trinh | Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations[END_REF]Appendix], or some results presented in [START_REF] Hale | Introduction to functional differential equations[END_REF]Appendix] and [11, Chapter 13], or a variant of the Walton-Marshall procedure as in [3,[START_REF] Walton | Direct method for TDS stability analysis[END_REF].

The frequency method will be used in Section 2.3.1.1 Chapter 2 to study exponential stability of the linear scalar hyperbolic system with integral controller.

Operator and semi-group method

Concerning the stability analysis of linear hyperbolic systems, operator and semi-group approach is also a well-known approach. Indeed, this approach is developed for general linear infinite-dimensional systems, and therefore applied for linear hyperbolic systems to determine stabilities. In general, the closed-loop systems is written by the extended system φ = Aϕ where ϕ is extended state containing all state and dynamic terms of control action, and the linear operator A is the generator of a C 0 semi-group on the extended state space of the controlled system. Then, with the boundary control action and characteristics of the original system, one may prove that A generates an exponentially stable C 0 semigroup on the extended state space.

The approach has been made use of to construct robust proportional integral (PI) controllers in [START_REF] Pohjolainen | Robust multivariable PI-controllers for infinite dimensional systems[END_REF] for a class of infinite holomorphic semigroup systems, and then for strongly continuous semigroup systems in [START_REF] Pohjolainen | Robust controller for systems with exponentially stable strongly continuous semigroups[END_REF][START_REF] Xu | A robust PI controller for infinite dimensional systems[END_REF]. The method has been applied to a particular model of 2 × 2 hyperbolic systems governed by the Saint-Venant equations with constant control in [START_REF] Xu | Linearization method to stability analysis for nonlinear hyperbolic systems[END_REF], and with PI control in [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF].

Backstepping method

The last method mentioned to analyze the stability of hyperbolic systems with boundary control is so-called backstepping technique. A nice introduction about this approach can be found in [START_REF] Krstic | Boundary Control of PDEs: A Course on Backstepping Designs[END_REF] and [4,Chapter 7]. The key idea of the method is that, one needs to find a coordinate transformation along with an adaptive mapping for control action to transform the controlled system to a target system for which the stability condition is well-known a priori. From the stability criteria of the target system, by inverse transformation, one can find the solution of boundary control for the original system.

Many results in the literature have been developed based on this technique, such as in [START_REF] Vazquez | Backstepping boundary stabilization and state estimation of a 2 × 2 linear hyperbolic system[END_REF] for the static boundary control of a class of 2 × 2 linear hyperbolic systems, or in [START_REF] Deutscher | Backstepping design of robust state feedback regulators for linear 2×2 hyperbolic systems[END_REF][START_REF] Lamare | Adding an Integrator to Backstepping: Output Disturbances Rejection for Linear Hyperbolic Systems[END_REF][START_REF] Lamare | Control of 2 × 2 linear hyperbolic systems: backstepping-based trajectory generation and PI-based tracking[END_REF] for integral control actions of some class of 2 × 2 linear hyperbolic systems.

Contribution of the thesis

In this thesis, we consider the boundary control and output regulation problems with dynamic control laws for some classes of PDE hyperbolic systems as follows:

∂ ∂t φ(x, t) + A(φ) ∂ ∂x φ(x, t) = 0 φ(0, x) = φ 0 (x) , F φ(0, t), φ(L, t), U(t) = 0 y(t) = l φ(0, t), φ(L, t), w o
Firstly, in Chapter 2, we consider a nonlinear scalar hyperbolic system where A(.) > 0,

φ ∈ H 2 (0, L), F φ(0, t), φ(L, t), U(t) = φ(0, t) -U (t)
= (φ 1 , φ 2 ) ∈ (L 2 (0, L)) 2 , A(φ) = λ 1 0 0 -λ 2
with two real positive constants λ 1 and Finally, in Appendix, we develop a numerical Preissmann scheme that can be used for general PDE hyperbolic systems of conservation laws. Based on an Preissmann scheme, we discretize the PDE models to get an implicit discrete equation. Then we use Newton Raphson method to find numerical solution of the discrete equation. This approach is employed in details to carry out all the simulations in Chapters 2-4.

λ 2 , F φ(0, t), φ(L, t), U(t) = φ 1 (0, t) -R 0 φ 2 (0, t) -u 1 (t) φ 2 (L, t) -R L φ 1 (L,

Publications

During the thesis, the following publications have been realized:

Chapter 2

Scalar hyperbolic systems with boundary integral controllers

Introduction

The chapter deals with the control and regulation by integral controllers for the nonlinear systems governed by scalar quasi-linear hyperbolic partial differential equations. Both the control input and the measured output are located on the boundary. This type of systems appear in many industrial applications and in study of traffic flow. For instance, scalar quasi-linear hyperbolic equations include Burgers equations [START_REF] Burgers | A mathematical model illustrating the theory of turbulence[END_REF] which are employed in modeling turbulent fluid motion. Another example is given by the equation employed by Lighthill-Whitham in [START_REF] Lighthill | On kinematic waves. ii. a theory of traffic flow on long crowded roads[END_REF] to describe traffic flow on long crowded roads. Finally scalar conservation laws can also be regarded as a particular simpler case of quasi-linear hyperbolic systems under some regularity assumptions (see for instance [4,[START_REF] Bressan | Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem[END_REF]).

In this work, we consider a 1D scalar conservation law with strictly increasing flux. We are interested in the classical solutions of the system around an equilibrium state. The objective is to propose a boundary integral output feedback controller to asymptotically stabilize the nonlinear closed-loop system and regulate the disturbed output measurement to the desired reference. The local stability and the regulation effect for the nonlinear closed-loop system are proven in the H 2 topology by using Lyapunov techniques. To be more detailed, the closed-loop stabilization of the linearized model with the designed integral controller is proved first by using the method of spectral analysis and then by the Lyapunov direct method. Based on the elaborated Lyapunov function inspired from [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF], we prove local exponential stability of the nonlinear closed-loop system with the same controller. The output regulation to the set-point with zero static error by the integral controller is shown upon the nonlinear system. Then, numerical simulations by the Preissmann scheme are carried out to validate the robustness performance of the designed controller in nonlinear closed-loop PDE system to face to unknown constant disturbances.

The chapter is organized as follows.

Problem statement and existing results

Statement of the problem

In this chapter, we consider a 1D quasi-linear hyperbolic system of the form :

∂ψ ∂t (x, t) + F (ψ(x, t)) ∂ψ ∂x (x, t) = 0, x ∈ (0, L), t ∈ R + , (2.1)
where L is a positive constant, ψ : (0,

L) × R + → R is the state in C([0, ∞), H 2 (0, L)),
and

F : R → R is a C 2 function such that F (σ) > 0 ∀ σ ∈ R. The initial condition is given by ψ(•, 0) = ψ 0 ∈ H 2 (0, L).
Notice that H 2 (0, L) is the usual Sobolev space defined by

H 2 (0, L) = {f ∈ L 2 (0, L) | f , f ∈ L 2 (0, L)}
where L 2 (0, L) denotes the usual Hilbert space of square summable functions on the open set (0, L). The Sobolev space H 2 (0, L) is normed by

f 2 H 2 = L 0 (|f (x)| 2 + |f (x)| 2 + |f (x)| 2 )dx ∀ f ∈ H 2 (0, L).
We consider the control u on the boundary x = 0, i.e.,

ψ(0, t) = u(t) , t ∈ R + .
The output we wish to regulate is also located on the boundary and eventually corrupted by an additive unknown disturbance, i.e.

y(t) = ψ(L, t) + w o , t ∈ R + ,
where w o ∈ R is an unknown constant. Our control objective is to design a dynamic output feedback control law in order to achieve asymptotic stabilization of the closed loop system and to ensure that the output y(t) converges to a desired set-point y r ∈ R, as t → ∞.

In our study, the control action u(t) has the structure of an integral controller. We assume that an unknown constant disturbance may corrupt the control. Hence we write the control law as follows

u(t) = -k I ζ(t) + w c , ζ(t) = y(t) -y r
where w c ∈ R is an unknown constant and k I is a positive constant called tuning parameter.

To summarize, the closed-loop system with disturbances is governed by the following

PDE : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ψ ∂t (x, t) = -F (ψ(x, t)) ∂ψ ∂x (x, t) ζ(t) = ψ(L, t) -y r + w o ψ(0, t) = -k I ζ(t) + w c ψ(x, 0) = ψ 0 (x), ζ(0) = ζ 0 . (2.2)
We are studying a nonlinear infinite-dimensional system controlled by an integral controller faced with unknown constant disturbances on the control and the output. The

ψ 0 (0) = -k I ζ 0 + w c F (ψ 0 (0))ψ 0 (0) = k I (ψ 0 (L) -y r + w o ). (2.4)

Existing results

There are many results in literature concerning this topic. The first work to be mentioned is in [START_REF] Coron | Feedback Stabilization for a scalar conservation law with PID boundary control[END_REF], in which a PI controller with damping for linear scalar PDE hyperbolic systems is addressed with Lyapunov direct method and frequency method. By frequency analysis, the authors also prove that PID controllers in general with the derivative terms can not be applied because of the instability of closed-loop system. Moreover in [START_REF] Tamasoiu | Stabilization and boundary control for balance and conservation laws[END_REF]Chapter 5],

the same PI controller with damping is extended for nonlinear scalar PDE hyperbolic systems. However in both works, they loose the regulation effect because of the damping required in the PI controller.

It is also interesting to note that asymptotic stabilization of entropy solutions to scalar conservation laws has been recently studied by Perrollaz in [START_REF] Perrollaz | Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law[END_REF] and by Balandin et al.

in [START_REF] Blandin | Regularity and Lyapunov stabilization of weak solutions to scalar conservation laws[END_REF]. In particular they have considered the stabilization problem of weak entropy solutions by boundary control and internal control around a constant equilibrium state for a scalar 1D conservation law with strictly convex flux. In [START_REF] Perrollaz | Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law[END_REF] an internal state feedback control law has been designed to asymptotically stabilize the entropy solutions around a constant equilibrium in the topologies L 1 and L ∞ . A stabilizing nonlocal boundary control law (depending on time and the whole initial data) has been proposed in [START_REF] Blandin | Regularity and Lyapunov stabilization of weak solutions to scalar conservation laws[END_REF] to get asymptotic stability of the constant equilibrium in the L 2 topology.

Main result and its proof

To be simple the initial data with the compatibility condition satisfied up to the required order are called compatible initial data throughout the chapter. From now on, the state space X for (2.2) is the Hilbert space X = H 2 (0, L) × R equipped with the norm

(f, z) 2 X = f 2 H 2 + z 2 .
Note that due to the constants w 0 , w c and y r , (ψ, ζ) = (0, 0) is not a steady state of the closed-loop system. In fact the equilibrium denoted

(ψ ∞ , ζ ∞ ) is defined as follows ψ ∞ = y r -w o , ζ ∞ = k -1 I (w o + w c -y r ). (2.5) Let B X ((f, z), δ) denote the open ball in X centered at (f, z) with radius δ > 0, i.e., B X ((f, z), δ) = {(ψ, ζ) ∈ X | (ψ, ζ) -(f, z) X < δ} .
Then the main result of the chapter is stated as follows.

Theorem 2.1. There exist positive real constants k * I and δ such that, for each k I ∈ (0, k * I ), and for every (y r , w o , w c ) ∈ R 3 and every compatible

(ψ 0 , ζ 0 ) ∈ B X ((ψ ∞ , ζ ∞ ), δ
), the following assertions hold true :

1. The closed-loop system (2.2) has a unique solution (ψ, ζ) ∈ C([0, ∞), X);
2. The solution of the closed-loop system (2.2) converges exponentially to the equilibrium state (ψ ∞ , ζ ∞ ) in the state space X as t → ∞, and the disturbed output is regulated to the desired set-point y r , i.e.,

lim t→∞ |y(t) -y r | = 0 .
3. There exist real constants M > 0 and ω > 0 such that

(ψ(•, t) -ψ ∞ , ζ(t) -ζ ∞ ) X Me -ωt (ψ 0 -ψ ∞ , ζ 0 -ζ ∞ ) X ∀t 0.

Remark :

The equilibrium state is some constant state determined by y r , w o and w c .

Though it is unknown a priori, the state of the closed-loop system is bounded because of the asymptotic stability property of the equilibrium. Moreover the output is always regulated to the set-point independently of the unknown disturbances. It is the virtue of the integral controller that allows to suppress the static error and hence achieves output regulation. A more general situation is explained in the chapter [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF].

Remark :

The solution considered in Theorem 2.1 is a classical solution in the sense of Li and Yu [START_REF] Li | Boundary value problems for quasi linear hyperbolic equations[END_REF]. However the topology used here is the topology induced by the Hilbert H 2 norm instead of the C 1 norm. Moreover, we have only local exponential stability of the equilibrium of the closed-loop system. For initial compatible data outside some neighborhood of the equilibrium, the classical solution to the Cauchy problem (2.2) may not be extended on the whole positive time axis.

The proof of Theorem 2.1 is given in the next section. Our proof is based on the construction of an appropriate Lyapunov functional. The direct Lyapunov approach allows us to consider the tuning parameter k I relatively bigger than the classical method (see [START_REF] Davison | Multivariable tuning regulators: the feedforward and robust control of a general servomechanism problem[END_REF], [START_REF] Pohjolainen | Robust multivariable PI-controllers for infinite dimensional systems[END_REF] and [START_REF] Xu | A robust PI controller for infinite dimensional systems[END_REF]). The proposed upper bound on k I is computed directly from the given system. This may be the advantage of our approach with respect to that of the literature [START_REF] Pohjolainen | Robust multivariable PI-controllers for infinite dimensional systems[END_REF][START_REF] Xu | A robust PI controller for infinite dimensional systems[END_REF][START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF].

To prove Theorem 2.1 we consider the following transformation :

φ(x, t) = ψ(x, t) -ψ ∞ , ξ(t) = ζ(t) -ζ ∞ (2.6)
where ψ ∞ and ζ ∞ are defined in (2.5). Then we obtain a perturbation free nonlinear closed-loop system as follows :

φ t (x, t) = -F (φ(x, t))φ x (x, t) (2.7) ξ(t) = φ(L, t) (2.8) φ(0, t) = -k I ξ(t) (2.9) φ(x, 0) = φ 0 (x) = ψ 0 (x) -ψ ∞ (2.10) ξ(0) = ξ 0 = ζ 0 -ζ ∞ (2.11)
where φ t (x, t) denotes the time partial derivative of φ(x, t), and we have defined

F (φ) = F (φ + ψ ∞ ).
In the new coordinates, the output is written as

y(t) = φ(L, t) + y r .
Hence the output regulation to y r is achieved if

lim t→∞ |φ(L, t)| = 0.
To guarantee the output regulation of the disturbed nonlinear system (2.2), we design the integral controller so as to ensure local asymptotic stabilization to the origin of the equivalent system (2.7)-(2.11).

In the following, the integral stabilization problem of the equivalent system is considered first for the linearized case in Section 2.3.1 and then for the nonlinear case in Section 2.3.2. Finally the complete proof of Theorem 2.1 is presented in Section 2.3.3.

Linear hyperbolic system

The purpose of this Section is to study stability property of the origin for the nonlinear hyperbolic system with an integral controller on the boundary as described in (2.7)-(2.11). To begin with, we consider the particular case where the system is linear, i.e., F does not depend on φ. This is the case if for instance the considered system is obtained by the tangent linearization of the nonlinear system around the equilibrium state. In Chapter 2. Scalar hyperbolic systems with boundary integral controllers 20 this subsection, we consider the following linear system :

φ t = -rφ x , r > 0 (2.12) ξ = φ(L, t), φ(0, t) = -k I ξ(t) (2.13) φ(x, 0) = φ 0 (x), ξ(0) = ξ 0 . (2.14)
To the system (2.12)-(2.14) is associated the state space Z which is the Hilbert space

Z = L 2 (0, L) × R equipped with the scalar product (φ a , ξ a ), (φ b , ξ b ) Z = L 0 φ a (x)φ b (x)dx + ξ a ξ b , ∀ ((φ a , ξ a ), (φ b , ξ b )) ∈ Z 2 .
and we denote by • Z its associated norm.

Frequency analysis method

The first stability result is obtained by employing the Laplace transform approach.

Proposition 2.2. The closed-loop linear system (2.12)

-(2.14) is exponentially stable in Z w.r.t. L 2 norm if and only if k I ∈ 0, rπ 2L .
The proof of this result can also be found in [START_REF] Bellman | Differential-difference equations[END_REF]p.444,Chapter 13] or in [START_REF] Hale | Introduction to functional differential equations[END_REF]Appendix Theorem A.5]. Our another proof is given in the following Proof : A necessary and sufficient condition for exponential stability of the system (2.12)-(2.14) is that all the poles of the transfer function have negative real part (see [START_REF] Xu | Exponential stability of the heat exchanger equation[END_REF], [START_REF] Bellman | Differential-difference equations[END_REF]Chapter 13], or [START_REF] Hale | Introduction to functional differential equations[END_REF]Appendix Theorem 3.5]). To formulate the transfer function, we set v(t) as the new control input with y(t) as the output :

φ(0, t) = -k I ξ(t) + v(t), y(t) = φ(L, t). (2.15)
By taking the Laplace transform in (2.12)-(2.14), we obtain:

s φ + r φx = 0 (2.16
)

s ξ = φ(L, s), (2.17) φ(0, s) = -k I ξ(s) + v(s) , ŷ(s) = φ(L, s) (2.18)
From (2.16) we have the solution φ(x, s) = φ(0, s)e -sr -1 x . Combining it with (2.17) and

(2.18), we obtain:

ŷ(s) = φ(L, s) = φ(0, s)e -sLr -1 = e -sLr -1 (-k I ξ(s) + v(s)) = e -sLr -1 -k I ŷ(s) s + v(s) Hence, 1 + k I s e -sLr -1 ŷ(s) = e -sLr -1 v(s)
Therefore we get the transfer function as follows:

G(s) = ŷ(s) v(s) = s k I + se sLr -1
The poles of transfer function are solutions of the following equation :

k I + se sLr -1 = 0 (2.19)
We set

μ = sLr -1 and α = k I Lr -1 . ( 2.20)
Note that α > 0. The characteristic equation now becomes

α + μe μ = 0 (2.21)
The proposition is proved if we show that the equation (2.21) has all the solutions μ in the left-half complex plane e(μ) < 0 if and only if α ∈ (0, π 2 ).

Let set μ = σ + iη, where σ, η ∈ R. Then (2.21) is rewritten as follows :

(σ + iη)e σ+iη + α = 0 By separating the real part and the imaginary part, we obtain:

-e σ (σcos(η) -ηsin(η)) = α, (2.22) ηcos(η) + σsin(η) = 0. (2.23)
We consider the following two cases.

• If sin(η) = 0, by (2.23), η cos(η) = 0 implies η = 0. From (2.22), we have α = -σe σ . The last equation has no solution σ 0 whatever is α > 0. Hence each solution σ is negative if and only if α ∈ (0, π/2).

• If sin(η) = 0, from (2.23), σ = - ηcos(η) sin(η) (2.

24)

Thus α = H(η) where

H(η) = η sin(η) exp -ηcos(η) sin(η)
Because H(η) is a pair function, we only need to consider the case where η > 0.

Thus α > 0 if and only if sin(η) > 0. As η > 0 and sin(η) > 0, we set η = γ + 2kπ, where γ ∈ (0, π) and k ∈ N.

Now considering the function H(η), we have :

∂H(η) ∂η = e -ηcos(η) sin(η) sin 2 (η) -ηsin(2η) + η 2 sin 3 (η)
One can easily check that sin 2 (η)ηsin(2η) + η 2 0 for all η > 0. Therefore, ∂H(η) ∂η 0. Hence, on each interval (2kπ, 2kπ+π), the function H(η) is continuous and monotonic increasing. Moreover we have lim

η→(2k+1)π - H(η) = +∞, lim η→2kπ+ π 2 H(η) = 2kπ + π 2 .
In addition, lim π). Therefore σ < 0 if and only if the equation H(η)α = 0 has all its solutions in ∪ ∞ k=0 (2kπ, 2kπ + π 2 ). Since α needs to be in one interval including 0, we have α ∈ (0, π 2 ).

if γ ∈ (0, π 2 ). Obviously σ 0 if γ ∈ [ π 2 ,
From the two cases, the proposition is proved.

Remark : By frequency-domain analysis it is possible to establish some necessary and sufficient conditions on the parameter k I for asymptotic stability of the equilibrium to the linear closed-loop system (2.12)-(2.14). However the approach is no longer applicable when dealing with a general nonlinear system. This is the reason why we introduce a Lyapunov functional for the linear system which allows us to tackle the nonlinear hyperbolic system in the following section.

Lyapunov direct method

In this Section, the following Lyapunov functional candidate V : Z → R is considered:

V (φ, ξ) = L 0 φ 2 (x)e -μx + q 1 ξφ(x)e -μx 2 dx + q 2 ξ 2 (2.25)
where μ > 0 and

q i > 0 ∀ i = 1, 2. Consider the function Π : [0, 2] → R such that Π(z) = z(2 -z)e -z/2 . We have Π(2 - √ 2) 0.3395 that is the maximum value of Π(z) in [0, 2].
Given T > 0 and a function φ

: (0, L) × (0, T ) → R, we use the notation φ(t) := φ(•, t)
when there is no ambiguity. Assume that the initial condition is smooth enough so that the solution of (2.12)-(2.14) is continuously differentiable with respect to time t and space x. Then, by differentiating V (φ(t), ξ(t)) with time along the solution and by using integration by parts we get

V (φ(t), ξ(t)) = -re -μL φ 2 (L, t) -k I r(q 1 -k I )ξ 2 (t) -μr L 0 e -μx φ 2 (x, t)dx + 2q 2 -q 1 re -μL 2 ξ(t)φ(L, t) - μq 1 r 2 ξ(t) L 0 e -μx 2 φ(x, t)dx + q 1 φ(L, t) L 0 e -μx 2 φ(x, t)dx. (2.26) Lemma 2.3. Let k * I = r 2L Π(2 - √ 2). Take k I ∈ (0, k * I ) and μ ∈ (0, (2 - √ 2)/L] such that r 2L Π(μL) > k I .
Let q 1 = 2k I and let q 2 = rk I e -μL/2 . Then there exist positive constants M 1 and α > 0 such that

M -1 ||(φ, ξ)|| 2 Z V (φ, ξ) M ||(φ, ξ)|| 2 Z ∀ (φ, ξ) ∈ Z, (2.27)
and for every smooth compatible

(φ 0 , ξ 0 ) ∈ Z V (φ(t), ξ(t)) -αV (φ(t), ξ(t)) - re -μL 2 φ 2 (L, t).
(2.28)

Proof : Rewrite V (φ, ξ) as follows V (φ, ξ) = L 0 φ(x)e -μx/2 ξ √ L P φ(x)e -μx/2 ξ √ L dx where P = 1 √ Lq 1 2 √ Lq 1 2 q 2 .
We claim that the matrix P is positive definite. Indeed, we have

det(P ) = Lk I rΠ(μL) 2L -k I + r 2L e -μL/2 2 -μL(2 -μL) .
Since r 2L Π(μL) > k I and μL < 2, it is easy to see that det(P ) rk I 2 e -μL 2 . Hence there is some real constant M 1 such that the inequality (2.27) holds.

By substituting the given q 1 and q 2 into (2.26) we have the following

V (φ(t), ξ(t)) = -re -μL φ 2 (L, t) -μr L 0 e -μx φ 2 (x, t)dx -k 2 I rξ 2 (t) -μrk I ξ(t) L 0 e -μx 2 φ(x, t)dx + 2k I φ(L, t) L 0 e -μx 2 φ(x, t)dx.
(2.29)

By using the Young and Cauchy-Schwarz inequalities we get

2k I φ(L, t) L 0 e -μx 2 φ(x, t)dx re -μL 2 φ 2 (L, t) + 2Lk 2 I e μL r L 0 e -μx φ 2 (x, t)dx (2.30)
and

μrk I ξ(t) L 0 e -μx 2 φ(x, t)dx rk 2 I 2 ξ 2 (t) + rμ 2 L 2 L 0 e -μx φ 2 (x,
V (φ(t), ξ(t)) - re -μL 2 φ 2 (L, t)- k 2 I r 2 ξ 2 (t)- r 2L Π(μL) + k I r 2L Π(μL) -k I J φ,ξ (2.32)
where

J φ,ξ = 2L r e μL L 0 e -μx φ 2 (x, t)dx.
By the choice of μ, we have r 2L Π(μL)k I > 0. It follows from (2.32) that there exists a positive real number M 1 such that Theorem 2.4. Assume that k I = 0. Then there exist no μ > 0 and q 2 > 0 such that the following function (derived from function V without the coupled term, i.e q 1 = 0)

V (φ(t), ξ(t)) -M 1 ξ 2 (t) + L 0 e -μx φ 2 (x, t)dx -re -μL 2 φ 2 (L, t). ( 2
U (φ, ξ) = L 0 φ 2 (x, t)e -μx dx + q 2 ξ 2 (t)
is a strict Lyapunov function for the closed-loop system (2.12)- (2.14).

Therefore, by adding the new coupled term L 0 q 1 ξ(t)φ(x, t)e -μx 2 dx allows us to construct strict Lypunov functionals for the closed-loop system (2.12)-(2.14).

Nonlinear system

In this section, we consider the problem for the nonlinear system (2.7)-(2.11) with F (0) = r > 0. By the designed integral controller the nonlinear closed-loop system (2.7)- (2.11) is written as follows

φ t + F (φ)φ x = 0 ξ = φ(L, t) φ(0, t) = -k I ξ φ(x, 0) = φ 0 (x), ξ(0) = ξ 0 .
(2.34)

Let us set :

s(x, t) = φ x (x, t) , p(x, t) = φ xx (x, t).
By successive derivatives and compatibility conditions we find that the dynamics of s(x, t) and p(x, t) are governed by the following PDE, respectively,

s t + F (φ)s x = -F (φ) s 2 F (φ(0, t))s(0, t) = k I φ(L, t) s(x, 0) = φ 0 (x) (2.35)
and

p t + F (φ)p x = -3F (φ) s p -F (φ) s 3 F 2 (φ(0, t))p(0, t) = k I F (φ(L, t))s(L, t) -2k I F (φ(0, t))φ(L, t)s(0, t) p(x, 0) = φ 0 (x).
(2.36)

Now we use the idea presented in [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] to extend the Lyapunov functional from the linear system (in L 2 norm) to the nonlinear system (in H 2 norm). Therefore local asymptotic stability of the equilibrium state and the set-point output regulation will be proved for the nonlinear closed-loop system (2.34). To do that, we consider the Lyapunov functional candidate S :

X → R + such that S(φ, ξ) = V (φ, ξ) + q 3 V 1 (φ x ) + q 4 V 1 (φ xx ) (2.37)
where V (φ, ξ) is defined in (2.25) with q 1 and q 2 given in Lemma 2.3 and

V 1 (φ x ) = L 0 e -μx φ 2 x (x)dx (2.38)
with the real positive constants q 3 and q 4 to be determined later.

For the moment we assume that all the required regularity is satisfied and carry out formal computations.

Lemma 2.5. The time derivative of V (φ(t), ξ(t)) along each regular trajectory of the nonlinear system (2.34) is written as follows

V (φ(t), ξ(t)) = -re -μL φ 2 (L, t) -k 2 I rξ 2 (t) -μr L 0 e -μx φ 2 (x, t)dx -μrk I ξ(t) L 0 e -μx 2 φ(x, t)dx + 2k I φ(L, t) L 0 e -μx 2 φ(x, t)dx -φ 3 (L, t)F 1 (φ(L, t))e -μL + φ 3 (0, t)F 1 (φ(0, t)) + L 0 e -μx F (0) + φ(x, t)F 2 (φ(x, t)) φ x (x, t)φ 2 (x, t)dx -μ L 0 e -μx F 1 (φ(x, t))φ 3 (x, t)dx -2k I ξ(t) L 0 e -μx/2 F 1 (φ(x, t))φ(x, t)φ x (x, t)dx (2.39)
where

F (z) = F (0) + F 1 (z)z F (z) = F (0) + F 2 (z)z (2.40) with F 1 (z) = 1 0 F (λz)dλ and F 2 (z) = 1 0 F (λz)dλ.

Proof :

By differentiating V (φ(t), ξ(t)) along each regular trajectory of (2.34) the following identity holds true

V (φ(t), ξ(t)) = - L 0 2e -μx φ(x, t)F (φ(x, t))φ x (x, t)dx -q 1 L 0 e -μx/2 F (φ(x, t))φ x (x, t)ξ(t)dx + q 1 L 0 e -μx/2 φ x (x, t)dxφ(L, t) + 2q 2 ξ(t)φ(L, t).
By integration by parts and by using the boundary condition (2.34) and the parameters q 1 and q 2 given in Lemma 2.3 as well as the relations (2.40) we prove the required identity (2.39).

Similarly we may prove the following lemmas.

Lemma 2.6. With the same notations as in Lemma 2.5, the time derivative of

V 1 (φ x (t))
along every regular trajectory of the nonlinear system (2.34)- (2.35) is written as follows

V1 (φ x (t)) = -re -μL s 2 (L, t) + r -1 k 2 I φ 2 (L, t) -rμ L 0 e -μx s 2 (x, t)dx -k 2 I F 3 (φ(0, t))φ(0, t)φ 2 (L, t) -e -μL F 1 (φ(L, t))φ(L, t)s 2 (L, t) - L 0 [(F (φ(x, t))) x + μF 1 (φ(x, t))φ(x, t)] e -μx s 2 (x, t)dx (2.41)
where

F 3 (z) = 1 0 F (λz) F 2 (λz) dλ.
Lemma 2.7. With the same notations as in Lemma 2.5, the time derivative of

V 1 (φ xx (t))
along each regular trajectory of the nonlinear system (2.34)-(2.36) is written as follows

V1 (φ xx (t)) = -e -μL F (φ(L, t))p 2 (L, t)+ k 2 I F 2 (φ(L, t)) F 3 (φ(0, t)) s 2 (L, t)-μ L 0 e -μx F (φ(x, t))p 2 (x, t)dx + 4k 2 I (F (φ(0, t)) 2 F 3 (φ(0, t)) φ 2 (L, t)s 2 (0, t) - 4k 3 I F (φ(L, t))F (φ(0, t)) F 4 (φ(0, t)) s(L, t)φ 2 (L, t) -5 L 0 e -μx F (φ(x, t))s(x, t)p 2 (x, t)dx -2 L 0 e -μx F (φ(x, t))s 3 (x, t)p(x, t)dx. (2.42) Let T > 0. For each function (φ, ξ) ∈ C([0, T ]; C 1 [0, L] × R) we define (φ, φ x , ξ) T,∞ = sup t∈[0,T ] |ξ(t)| + sup x ∈ [0, L] t ∈ [0, T ] |φ(x, t)| + sup x ∈ [0, L] t ∈ [0, T ] |φ x (x, t)|.
By combining results of Lemmas 2.3-2.7 the following theorem is obtained. Theorem 2.8. Let the parameters k I , μ, q 1 and q 2 be determined as in Lemma 2.3. Then there are positive real constants q 3 , q 4 , δ and β such that, for each function [START_REF] Harrington | Time-Harmonic Electromagnetic Fields[END_REF]) and the condition (φ, φ x , ξ) T,∞ < δ, the following differential inequality holds true

(φ, ξ) ∈ C([0, T ]; C 3 [0, L] × R) ∩ C 1 ([0, T ]; C 2 [0, L] × R) satisfying the PDE (2.34)-(2.
Ṡ(φ(t), ξ(t)) -βS(φ(t), ξ(t)) ∀ t ∈ [0, T ].
(2.43)

Moreover there exists a positive constant K 1 such that

K -1 (φ, ξ) 2 X S(φ, ξ) K (φ, ξ) 2 X ∀ (φ, ξ) ∈ X. (2.

44)

Proof : Without loss of generality we assume that δ 1. For the sake of simplicity we write C T = (φ, φ x , ξ) T,∞ . By Lemma 2.5, Lemma 2.3 and the Cauchy-Schwarz inequality there exists a positive constant K 1 > 0 such that V (φ(t), ξ(t)) -αV (φ(t), ξ(t)) -(r/2)e -μL φ 2 (L, t)

+ K 1 C T L 0 e -μx φ 2 (x, t)dx + ξ 2 (t) + φ 2 (L, t) . (2.45)
Similarly, by Lemma 2.6 there exists a positive constant

K 2 > 0 such that V1 (φ x (t)) -(re -μL -K 2 C T )s 2 (L, t) + r -1 k 2 I φ 2 (L, t) -rμ L 0 e -μx s 2 (x, t)dx + K 2 C T φ 2 (L, t) + L 0 e -μx s 2 (x, t)dx . (2.46)
Similarly, by Lemma 2.7 there exists a positive constant

K 3 > 0 such that V1 (φ xx (t)) -(re -μL -K 3 C T )p 2 (L, t) + (r -2 k 2 I + K 3 C T )s 2 (L, t) + K 3 C T φ 2 (L, t) + K 3 C T L 0 e -μx s 2 (x, t)dx -(rμ -K 3 C T ) L 0 e -μx p 2 (x, t)dx. (2.47)
Chapter 2. Scalar hyperbolic systems with boundary integral controllers 29

As C T can be made as small as we like with δ, adding the inequalities (2.45)-(2.47) and taking δ, q 3 and q 4 sufficiently small lead us directly to the following differential relation

Ṡ(φ(t), ξ(t)) - α 2 V (φ(t), ξ(t))- q 3 rμ 2 L 0 e -μx s 2 (x, t)dx- q 4 rμ 2 L 0 e -μx p 2 (x, t)dx.
(2.48)

Therefore the theorem is proved by using (2.27), (2.37) and (2.48).

Proof of Theorem 2.1

With Theorem 2.8, we are now ready to prove the main result of the chapter.

Proof of Theorem 2.1 :

We first prove the local existence of a unique solution to the closed-loop system (2.7)-(2.11) for each compatible initial state (φ

0 , ξ 0 ) in H 2 (0, L) × R.
The closed-loop control system (2.7)-(2.11) is governed by the following PDE coupled with an ODE through the boundary as follows:

⎧ ⎨ ⎩ φ t = -F (φ)φ x , ξ = φ(L, t) φ(0, t) = -k I ξ, (φ(x, 0), ξ(0)) = (φ 0 (x), ξ 0 ). (2.49) Recall that X = H 2 (0, L) × R is equipped with the norm (f, z) 2 X = f 2 H 2 + z 2 .
Assume that the initial condition (φ 0 , ξ 0 ) is in B X (0, δ), δ > 0 and satisfies the C 0 and C 1 compatibility conditions as in (2.34) and (2.35).

By using the Theorem 1.2 and the Propositions 1.3-1.5 in [66, pp.362-365], or [39, Theorem II] we deduce the existence of a unique solution to (2.49) for some δ > 0 and

T > 0 : (φ, ξ) ∈ C([0, T ]; H 2 (0, L) × R) ∩ C 1 ([0, T ]; H 1 (0, L) × R).
The reader is referred to [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] and [4, Appendix B] for a rigorous proof to the initial boundary case. Now we prove local exponential stability of the null state to (2.49). Notice that each compatible w 0 = (φ 0 , ξ 0 ) ∈ X admits a sequence of w 0,n = (φ 0,n , ξ 0,n ) ∈ H 4 (0, L) × R satisfying the C k compatibility condition, k = 0, 1, 2, 3, such that lim n→∞ w 0,nw 0 X = 0 (cf. [15, p.130]). Hence it is sufficient for us to prove the exponential stability for w 0 ∈ H 4 (0, L) × R. As the solution depends continuously on the initial condition (see [39, Theorem III]), then the exponential decay of solution from compatible w 0 ∈ X is proved by taking the limit. Indeed, take a compatible w 0 ∈ (H 4 (0, L) × R) ∩ B X (0, δ). As stated above the system (2.49) has a unique solution w(t) in H 4 (0, L) × R (cf. [START_REF] Taylor | Partial Differential Equation III Nonlinear Equation[END_REF]) such that

w ∈ C([0, T ]; H 4 (0, L) × R) ∩ C 1 ([0, T ]; H 3 (0, L) × R)
where w(t) = (φ(t), ξ(t)). By the continuous embedding (cf. [15, p.167

]) H n (0, L) → C n-1 [0, L] ∀ n 1 integer, we have the solution w ∈ C([0, T ]; C 3 [0, L] × R) ∩ C 1 ([0, T ]; C 2 [0, L] × R).
Let w 0 X < δ 1 for some δ 1 > 0. We choose δ 1 > 0 sufficiently small such that w 0 X < Kδ 1 implies (φ, φ x , ξ) T,∞ < δ with smaller T if necessary (cf. [

w(t) X Me -ωt w 0 X ∀ w 0 ∈ B X (0, δ 1 ).
The regulation effect is automatically guaranteed, since

w ∈ C([0, ∞); H 2 (0, L) × R).
Hence the proof of Theorem 2.1 is complete. 2

Numerical simulations 2.4.1 Discretization with Preissmann scheme and numerical solutions

To make numerical simulations, we firstly use the Preissmann scheme (see in Appendix)

to discretize the nonlinear system (2.1) looped by the integral controller designed from Theorem 2.1.

Let recall some principal parameters used in Preissmann scheme: N is the number of discretized space intervals; Δt and Δx are time discretization step and space discretization step, respectively; and θ ∈ [0.5, 1] is the weight parameter. In this Section, the nonlinear system (2.1) is considered with the following flux function

F (ψ) = ψ 2 + 3.
The following nonlinear equation is obtained with the Preissmann scheme

G X(k + 1) = 0 (2.50)
where

F : R N → R N defined by G = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ g 1 g 2 ... g N -1 g N ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, and

X(k + 1) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ψ k+1 1 ψ k+1 2 ... ψ k+1 N ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
.

with the N -1 sub-functions g i (i = 1, N -1) received from the discretization of N -1 PDE equations in (2.1)

g i = ψ k+1 i+1 + ψ k+1 i -ψ k i+1 -ψ k i + Δt Δx M i θ(ψ k+1 i+1 -ψ k+1 i ) + (1 -θ)(ψ k i+1 -ψ k i ) , ( 2.51)
where

M i = 6 + θ (ψ k+1 i+1 ) 2 + (ψ k+1 i ) 2 + (1 -θ) (ψ k i+1 ) 2 + (ψ k i ) 2 ,
and a boundary conditions

g N = ψ k+1 1 -u.
Note that the values at instant k of ψ i are supposed to be computed previously and considered known values.

The next step is to resolve the implicit nonlinear equation (3.25) by Newton-Raphson method, see the detail of this method in Appendix. To apply this iterative approach, it is obligatory to compute the Jacobien matrix of function G as following:

J = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∂g 1 ∂ψ k+1 1 ∂g 1 ∂ψ k+1 2 0 0 ... 0 0 0 0 ∂g 2 ∂ψ k+1 2 ∂g 2 ∂ψ k+1 3 0 ... 0 0 0 ... 0 0 0 0 ... 0 ∂g N -1 ∂ψ k+1 N -1 ∂g N -1 ∂ψ k+1 N ∂g N ∂ψ k+1 1 0 0 0 ... 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ where ∂g N ∂ψ k+1 1 = 1; and ∂g i ∂ψ k+1 i , ∂g i ∂ψ k+1 i are computed from the formula (2.51) (∀ i = 1, N -1).
Finally, we receive a numerical solution at instant k + 1 of ψ 1 , ψ n ,• • • , ψ n . With a sufficient time horizon, we get a numerical solution for the nonlinear system (2.1).

Conclusions

We have considered the design of stabilizing integral controllers for the nonlinear systems described by scalar hyperbolic PDE. First we have proposed an interval of integral gain for stabilization and then proved exponential stability of the linearized system controlled by the designed controller. Moreover, for the linearized system we have been able to establish a necessary and sufficient condition for the integral gain to get exponential stability of the controlled system in the L 2 norm. Then we have proved local exponential stability of the nonlinear controlled system by the same integral controller The results of this chapter can also be found in the publication [START_REF] Trinh | Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations[END_REF].

Chapter 3

Multiple input -multiple output 2 × 2 systems with PI control

Introduction

This chapter is concerned with the boundary control and output regulation of a class of linear multiple input -multiple output (MIMO) systems described by two hyperbolic partial differential equations with multivariable PI controllers. Both the input control and the output measurement are situated on the boundary. First, the system is transformed into the characteristic form of Riemann invariants, and the PI controller design is proposed for the Riemann invariant system. Then, a Lyapunov functional is constructed to prove stabilization and regulation of the closed-loop system. Furthermore, we apply the designed PI controller for a nonlinear Saint-Venant model and carry out numerical simulations to evaluate the performances of the designed PI controller.

We consider the following linear system governed by two hyperbolic partial differential equations:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ ∂t z 1 (x, t) z 2 (x, t) + G ∂ ∂x z 1 (x, t) z 2 (x, t) = 0, h 1 (z 1 (0, t), z 2 (0, t)) = U 0 (t), h 2 (z 1 (L, t), z 2 (L, t)) = U L (t), y(t) = (z 1 (0, t), z 2 (L, t)) , (3.1)
with the initial condition: open channel in the field of hydraulics (see in [START_REF] Bounit | Regulation of an irrigation channel through the semigroup approach[END_REF], [START_REF] Coron | A Lyapunov approach to control irrigation canals modeled by Saint Venant equations[END_REF], [START_REF] Xu | Linearization method to stability analysis for nonlinear hyperbolic systems[END_REF] and [START_REF] Trinh | Pi regulation control of a fluid flow model governed by hyperbolic partial differential equations[END_REF]), or the model of counter flow heat exchanger process in [START_REF] Xu | A robust PI controller for infinite dimensional systems[END_REF], [START_REF] Xu | Symmetric hyperbolic systems and applications to exponential stability of heat exchanges and irrigation canals[END_REF].

z 1 (x, 0) = z 0 1 (x), z 2 (x, 0) = z 0 2 (x),
Our control problem is to find two dynamic feedback control inputs on the boundary U 0 (t) and U L (t) which depend only on the two available measured outputs y(t) = (z 1 (0, t), z 2 (L, t)) such that the closed-loop system is stable and the output y(t) is regulated to the desired set-point y r = (y r1 , y r2 ).

The studied problem and existing results are described in the Section 3.2. In Section

Problem statement and review of existing results

Statement of the problem

Following the usual approach when dealing with hyperbolic systems, we first perform a change of coordinates (for example Riemann coordinates) around the desired set-point y r , and we rewrite the system in characteristic form

∂ ∂t φ 1 φ 2 + λ 1 0 0 -λ 2 ∂ ∂x φ 1 φ 2 = 0, (3.2)
where φ i : [0, L] × R + → R, and λ i > 0 for i = 1, 2. In addition, combining with the first order approximation, the boundary conditions and the measured outputs now have the following forms:

⎧ ⎨ ⎩ φ 1 (0, t) = R 0 φ 2 (0, t) + u 1 (t), φ 2 (L, t) = R L φ 1 (L, t) + u 2 (t), (3.3
)

y(t) -y r = aφ 1 (0, t) + bφ 2 (0, t) cφ 1 (L, t) + dφ 2 (L, t) , ( 3.4)
where R 0 , R L , a, b, c and d are real constants obtained from the linearization around the set-point. The new control input u(t) = (u 1 (t), u 2 (t)) ∈ R 2 is obtained by a linear transformation from (U 0 , U L ) (see in Section 4.5.2.3 as an example). In the chapter, we consider the case in which abcd < 0. Notice that the system (3.2), (3.3), (3.4) is the general form of the linearized systems used in [START_REF] Xu | Linearization method to stability analysis for nonlinear hyperbolic systems[END_REF], [START_REF] Trinh | Pi regulation control of a fluid flow model governed by hyperbolic partial differential equations[END_REF], [3], [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF] and [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF] (when the viscous friction slope is null).

In this work, the control action u(t) = (u 1 (t), u 2 (t)) designed by the PI controller on the boundary has the following form:

u(t) = K p (y(t) -y r ) + K I t 0 (y(τ ) -y r )dτ, (3.5)
where K p ∈ R 2×2 and K I ∈ R 2×2 are matrix gains which are the parameters of PI controller to be designed. They have to be selected to guarantee the asymptotic stability and regulation of the closed-loop system (3.2)-(3.5).

Existing results

In the simple case where u 1 (t) = u 2 (t) = 0, the system is said to be in open-loop. It is

shown in [START_REF] Xu | Linearization method to stability analysis for nonlinear hyperbolic systems[END_REF] by operator and semi-group method that the origin of open-loop system is exponentially stable in C 1 norm if and only if the following boundary condition is

satisfied |R 0 R L | < 1.
In other work [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF], they consider u 1 (t) and u 2 (t) as static feedback control laws, and after the linearization of boundary control in (3.3), they obtain the following boundary

condition φ 1 (0, t) = R * 0 φ 2 (0, t) and φ 2 (L, t) = R * L φ 1 (L, t)
where R * 0 and R * L are two real constants. With a strict Lypunov functional, they prove that the origin of nonlinear closed-loop network is exponentially stable in

H 2 norm if |R * 0 R * L | < 1.
Furthermore, linear hyperbolic systems with boundary PI control only from one side (in other word, single input single output hyperbolic system) are considered in [3,[START_REF] Lamare | Control of 2 × 2 linear hyperbolic systems: backstepping-based trajectory generation and PI-based tracking[END_REF].

In [3], by frequency method with Laplace transform, the authors give a necessary and sufficient condition of PI controllers at boundary x = 0 to obtain exponential stability of the closed-loop system. In [START_REF] Lamare | Control of 2 × 2 linear hyperbolic systems: backstepping-based trajectory generation and PI-based tracking[END_REF], by using Lyapunov direct method, a scalar PI controller is constructed at boundary x = L from scalar output measurement.

It should be mentioned that a multivariable PI control design has been proposed for linear 2 × 2 hyperbolic systems by using operator and semi-group techniques [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF]. Moreover, with Lyapunov techniques, Dos Santos et al. [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF] have addressed a multivariable PI controller design for a special class of 2 × 2 linear hyperbolic system. Chapter 3. Multiple input -multiple output 2 × 2 systems with PI control 38

Main result and its proof with Lyapunov technique

Main result

To describe the closed-loop system, let us introduce new state variable ξ(t) = (ξ 1 (t), ξ 2 (t))

where ξ(t) = y(t)y r . The closed-loop system (3.2) -(3.5) can be rewritten as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ ∂t φ 1 φ 2 = -λ 1 0 0 λ 2 ∂ ∂x φ 1 φ 2 , ∂ ∂t ξ 1 ξ 2 = aφ 1 (0, t) + bφ 2 (0, t) cφ 1 (L, t) + dφ 2 (L, t) , φ 1 (0, t) φ 2 (L, t) = R 0 φ 2 (0, t) R L φ 1 (L, t) + K p (y(t) -y r ) + K I ξ 1 ξ 2 , y(t) -y r = (aφ 1 (0, t) + bφ 2 (0, t), cφ 1 (L, t) + dφ 2 (L, t)) .
(3.6)

Let X = (L 2 (0, L)) 2 × R 2 be the state space of the closed-loop system (3.6) equipped with the following norm:

||v|| 2 X = ||v 1 || 2 L 2 (0,L) + ||v 2 || 2 L 2 (0,L) + |v 3 | 2 + |v 4 | 2 where v = (v 1 , v 2 , v 3 , v 4 ) ∈ X.
We consider now the closed-loop system (3.6), under the initial condition (φ 0 1 (x), φ 0 2 (x), ξ 0 1 , ξ 0 2 ). For each initial condition (φ 0 1 (x), φ 0 2 (x), ξ 0 1 , ξ 0 2 ) ∈ X satisfying the C 0 and C1 compatibility conditions, the closed-loop system (3.6) has a unique solution 1 (φ 1 , φ 2 , ξ 1 , ξ 2 ) ∈ X (see [START_REF] Coron | Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems[END_REF] and [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF] for a proof with dynamic boundary conditions). In the chapter, we look for asymptotic stability of the closed-loop system in X with the output regulation guaranteed.

The system (3.6) is called exponentially stable in X if there exits positive constants M > 0 and α > 0 such that for each initial condition (φ 0 1 (x), φ 0 2 (x), ξ 0 1 , ξ 0 2 ) ∈ X, the solution of (3.6) satisfies the following inequalities:

||(φ 1 (•, t), φ 2 (•, t), ξ 1 (t), ξ 2 (t))|| 2 X Me -αt ||(φ 0 1 , φ 0 2 , ξ 0 1 , ξ 0 2 )|| 2 X , ∀t ∈ R + .
Remark : It is interesting to note that in our work, the PI controller ensures stabilization of the origin in X without requiring a priori the classical constraint

|R 0 R L | < 1.
The main result of the chapter is stated as follows:

Theorem 3.1. Let a, b, c and d be real constants such that their product abcd < 0.

Then there exists μ * > 0 such that, for each μ ∈ (0, μ * ), and each PI controller with proportional gain K p and integral gain K I as follows:

K p = ⎛ ⎜ ⎝ -R 0 b 0 0 -R L c ⎞ ⎟ ⎠ , K I = -μ ⎛ ⎜ ⎝ ( aR 0 b + 1)a 0 0 ( dR L c + 1)d ⎞ ⎟ ⎠ , (3.7)
we have:

• the closed-loop system (3.6) with the PI controller design in (3.7) is exponentially stable toward the origin in X.

• for each initial condition

(φ 0 1 (x), φ 0 2 (x), ξ 0 1 , ξ 0 2 ) ∈ (H 1 (0, L)) 2 × R 2

Remark :

Since the weak solution depends continuously on the initial data, the exponential decay of the weak solution can be proved by a Lyapunov functional approach together with the classical density argument.

Lyapunov functional for the closed-loop system

By applying the PI controller from (3.7), the closed-loop system (3.6) is governed by the following partial differential equations : where

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ ∂t φ 1 φ 2 = -λ 1 0 0 λ 2 ∂ ∂x φ 1 φ 2 , ∂ ∂t ξ 1 ξ 2 = aφ 1 (0, t) + bφ 2 (0, t) cφ 1 (L, t) + dφ 2 (L, t) , φ 1 (0, t) = -k 1 ξ 1 , φ 2 (L, t) = -k 2 ξ 2 , y(t) -y r = (aφ 1 (0, t) + bφ 2 (0, t), cφ 1 (L, t) + dφ 2 (L, t)) ,
k 1 = aμ, k 2 = dμ, (3.9) with μ > 0; φ 1 , φ 2 ∈ L 2 (0, L) and ξ 1 , ξ 2 ∈ R.
The candidate Lyapunov function V : X → R is defined as follows:

V (φ 1 , φ 2 , ξ 1 , ξ 2 ) = L 0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ φ 1 e -μx 2 φ 2 e μx 2 ξ 1 ξ 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ T P ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ φ 1 e -μx 2 φ 2 e μx 2 ξ 1 ξ 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ dx , ( 3.10)
where

P = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 0 q 4 0 q 1 q 5 0 0 q 5 q 2 0 q 4 0 0 q 3 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ( 3.11)
and q i ∈ R with i = 1, • • • , 5 are parameters that have to be designed.

In the following, by properly selecting the parameters μ and the (q i )'s, it is shown that the functional V is positive definite and that its time derivative along the solution of the system (3.8) is negative. The proof of Theorem 3.1 is established thereafter.

In order to show that the functional V is positive definite, note that the following lemma may be easily obtained.

Lemma 3.2. The matrix P is symmetric positive definite if and only if the following three conditions are satisfied : q 1 > 0, q 1 q 2 > q 2 5 , q 3 > q 2 4 .

(3.12)

Proof : Applying the Sylvester criterion, the matrix P defined in (3.11) is symmetric positive definite if the following three determinants are positive:

P 1 = det 1 0 0 q 1 , P 2 = det ⎛ ⎜ ⎜ ⎝ 1 0 0 0 q 1 q 5 0 q 5 q 2 ⎞ ⎟ ⎟ ⎠ , P 3 = det(P ) .
Direct computations give us P 1 = q 1 , P 2 = q 1 q 2q 2 5 , P 3 = (q 3q 2 4 )P 2 .

This leads to the proof of Lemma 3.2.

In order to simplify the reading, all along the chapter, the notation

V (t) := V (φ 1 (., t), φ 2 (., t), ξ 1 (t), ξ 2 (t))
is employed. The following result can be obtained.

Lemma 3.3. Assume abcd < 0. Let k 1 , k 2 and q i , for i = 1, . . . , 5 be defined as follows:

k 1 = aμ , k 2 = dμ, q 1 = - bdλ 1 acλ 2 e -μL , q 2 = λ 1 μ L , q 3 = - bdλ 1 acL μ , q 4 = - bd a μe μL 2 , q 5 = λ 1 b λ 2 μ.
(3.13)

Then there exists μ * > 0 such that for every μ ∈ (0, μ * ), the following holds:

1. there exists K > 0 such that ∀ (φ 1 , φ 2 , ξ 1 , ξ 2 ) in X:

1 K V (φ 1 , φ 2 , ξ 1 , ξ 2 ) ||(φ 1 , φ 2 , ξ 1 , ξ 2 )|| 2 X KV (φ 1 , φ 2 , ξ 1 , ξ 2 ) . (3.14)
2. there exists α > 0 such that along the solution of (3.8), for all t at which the solution is well defined

V (t) -αV (t) -M φ 2 1 (L, t) + φ 2 2 (0, t) . ( 3

.15)

Proof : To begin with, we prove the first assertion. Since abcd < 0, it yields that ρ = -bd ac > 0. Employing (3.13), one can find that:

q 1 = ρ λ 1 λ 2 e -μL , q 1 q 2 -q 2 5 = μ(ρ λ 2 1 Lλ 2 e -μL -μ λ 2 1 b 2 λ 2 2 ) , q 3 -q 2 4 = μ( ρλ 1 L -μρ 2 c 2 e μL ) .
It is clearly that when μ is small enough, q 1 > 0, q 1 q 2q 2 5 > 0 and q 3q 2 4 > 0. Therefore, using Lemma 3.2, it yields that the matrix P is symmetric positive definite.

Hence, there exits σ 1 , σ 2 > 0 such that

σ 1 L 0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ φ 1 e -μx 2 φ 2 e μx 2 ξ 1 ξ 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ T ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ φ 1 e -μx 2 φ 2 e μx 2 ξ 1 ξ 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ dx V (φ 1 , φ 2 , ξ 1 , ξ 2 ) σ 2 L 0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ φ 1 e -μx 2 φ 2 e μx 2 ξ 1 ξ 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ T ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ φ 1 e -μx 2 φ 2 e μx 2 ξ 1 ξ 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ dx.
As a result, there exists K > 0 such that (3.14) holds. Now we prove the second part. Substituting q i and k 1 , k 2 from (3.13), the time derivative of V along the solution of the system (3.8) has the following form:

V = - L 0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ φ 1 (x, t)e -μx 2 φ 2 (x, t)e μx 2 ξ 1 (t) ξ 2 (t) φ 1 (L, t) φ 2 (0, t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ T Q ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ φ 1 (x, t)e -μx 2 φ 2 (x, t)e μx 2 ξ 1 (t) ξ 2 (t) φ 1 (L, t) φ 2 (0, t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ dx where Q = 1 L ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ μλ 1 L 0 0 T φ 1 ,ξ 2 T φ 1 ,φ 1 L 0 0 μq 1 λ 2 L T φ 2 ,ξ 1 0 0 T φ 2 ,φ 2 0 0 T φ 2 ,ξ 1 T ξ 1 T ξ 1 ,ξ 2 0 T φ 2 0,ξ 1 T φ 1 ,ξ 2 0 T ξ 1 ,ξ 2 T ξ 2 T φ 1 L,ξ 2 0 T φ 1 ,φ 1 L 0 T φ 1 L,ξ 1 T φ 1 L,ξ 2 λ 1 e -μL 0 0 T φ 2 ,φ 2 0 T φ 2 0,ξ 1 T φ 2 0,ξ 2 0 λ 2 q 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ with T ξ 1 = 2aq 2 Lk 1 -k 2 1 λ 1 , T ξ 2 = 2dq 3 Lk 2 -k 2 2 λ 2 q 1 e μL , T φ 1 ,ξ 2 = dk 2 q 4 L + q 4 λ 1 2 μL, T φ 1 ,φ 1 L = -cq 4 L, T φ 2 ,ξ 1 = ak 1 q 5 L + q 5 λ 2 2 μL, T φ 2 ,φ 2 0 = -bq 5 L, T ξ 1 ,ξ 2 = λ 2 q 5 k 2 e μL 2 + λ 1 k 1 q 4 , T φ 2 0,ξ 1 = -q 2 bL + λ 2 q 5 , T φ 1 L,ξ 2 = -q 3 cL + λ 1 q 4 e -μL 2 .
Therefore, V can be rewritten as follows

V = - L 0 ⎛ ⎜ ⎜ ⎝ φ 1 (x, t)e -μx 2 ξ 2 (t) φ 1 (L, t) ⎞ ⎟ ⎟ ⎠ T Q 1 ⎛ ⎜ ⎜ ⎝ φ 1 (x, t)e -μx 2 ξ 2 (t) φ 1 (L, t) ⎞ ⎟ ⎟ ⎠ dx - L 0 ⎛ ⎜ ⎜ ⎝ φ 2 (x, t)e μx 2 ξ 1 (t) φ 2 (0, t) ⎞ ⎟ ⎟ ⎠ T Q 2 ⎛ ⎜ ⎜ ⎝ φ 2 (x, t)e μx 2 ξ 1 (t) φ 2 (0, t) ⎞ ⎟ ⎟ ⎠ dx,
where

Q 1 = 1 L ⎛ ⎜ ⎜ ⎝ μλ 1 L d k 2 q 4 L + q 4 λ 1 2 μL -cq 4 L dk 2 q 4 L + q 4 λ 1 2 μL ρλ 1 e μL d 2 μ 2 0 -cq 4 L 0 λ 1 e -μL ⎞ ⎟ ⎟ ⎠ , Q 2 = 1 L ⎛ ⎜ ⎜ ⎝ μq 1 λ 2 L ak 1 q 5 L + q 5 λ 2 2 μL -bq 5 L ak 1 q 5 L + q 5 λ 2 2 μL λ 1 a 2 μ 2 0 -bq 5 L 0 λ 2 q 1 ⎞ ⎟ ⎟ ⎠ .
In the following it is proven that by picking μ small enough, the two matrices Q 1 and Q 2 are symmetric positive definite.

Regarding matrix Q 1 , from the Sylvester criterion, Q 1 is symmetric positive definite if and only if

D 1 = det μλ 1 L d k 2 q 4 L + q 4 λ 1 2 μL dk 2 q 4 L + q 4 λ 1 2 μL ρλ 1 e μL d 2 μ 2 > 0, D 2 = det(Q 1 ) > 0.
From (3.13) we have that

D 1 = μ 3 Ld 2 e μL ρλ 2 1 -μ b 2 L 4a 2 (2d 2 + λ 1 ) 2 ,
and

D 2 = λ 1 e -μL D 1 -c 2 q 2 4 L 2 λ 1 ρe μL d 2 μ 2 , = μ 3 Ld 2 λ 1 ρλ 2 1 -μ b 2 L 4a 2 (2d 2 + λ 1 ) 2 -μLe 2μL b 2 d 2 c 2 a 2 .
Taking μ small enough, it yields that the two terms D 1 , D 2 are both positive. Consequently, the matrix Q 1 is symmetric positive definite.

Similarly, it can be shown that the matrix Q 2 is symmetric positive definite when μ is small enough.

Chapter 3. Multiple input -multiple output 2 × 2 systems with PI control 44 Therefore, if μ is small enough, there exists a positive real number M > 0 such that for all t at which the solution is well defined we have:

V (t) -M ||(φ 1 , φ 2 , ξ 1 , ξ 2 )|| 2 X + φ 2 1 (L, t) + φ 2 2 (0, t) .
With (3.14) the former inequality implies that we can find α > 0 such that (3.15) holds.

This completes the proof of Lemma 3.3.

Remark :

Our result is an extension of the one in [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] in which a static feedback is considered and a Lyapunov function is constructed. In our work, since we have added some new dynamics ξ 1 (t) and ξ 2 (t) the Lyapunov function has to be modified to take into account these new states. This is also the case in [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF] in which a similar topic is studied. Note however that the Lyapunov function that we have introduced in (3.10) is different than the one of [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF] since it contains coupling terms between the state variables (φ 1 , φ 2 ) and the feedback variables (ξ 1 , ξ 2 ). This allows to consider a larger class of system since the unique condition on the physical parameters that has to be considered is that abcd < 0. This is an improvement compared to the result in [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF] in which the constraints on the parameter are :

a > 0 , b > 0 , c > 0 , d = -c , b a < 1, a + b = 1.
Note that another PI control for the case of a = b = c = 0 and d = 1 is solved with Lyapunov functional in [START_REF] Lamare | Control of 2 × 2 linear hyperbolic systems: backstepping-based trajectory generation and PI-based tracking[END_REF].

Proof of the main result

In this section, by using the results of Lemma 3.3, we state the proof of Theorem 1.

Stability analysis

Firstly, we prove that the closed-loop system (3.8) is exponentially stable toward the origin in X. From Lemma 3.3, there exists α > 0 such that:

V (t) V (0)e -αt .
Employing (3.14), it implies that there exists S > 0 such that for all initial conditions (φ 0 1 (x), φ 0 2 (x), ξ 0 1 , ξ 0 2 ) ∈ X and satisfying the C 0 and C 1 compatibility conditions, the solution are defined for all positive time and satisfies:

||(φ 1 , φ 2 , ξ 1 , ξ 2 )|| 2 X Se -αt ||(φ 0 1 , φ 0 2 , ξ 0 1 , ξ 0 2 )|| 2 X . (3.16)
As a result, the origin of the closed-loop system (3.6) with PI controller design in (3.7) is exponentially stable in X.

Proof of output regulation

It remains to show that the regulation is also obtained. First of all, if the initial con-

dition (φ 0 1 (x), φ 0 2 (x), ξ 0 1 , ξ 0 2 ) is in (H 1 (0, L)) 2 × R 2
and satisfies C 0 and C 1 compatibility conditions, then the solution of (3.8

) (φ 1 (•, t), φ 2 (•, t), ξ 1 (t), ξ 2 (t)) is in (H 1 (0, L)) 2 × R 2 for all t.
Now, by adding some high-order terms to the Lyapunov functional in (3.10), we prove the following lemma

Lemma 3.4. If the initial condition (φ 0 1 (x), φ 0 2 (x), ξ 0 1 , ξ 0 2 ) is in (H 1 (0, L)) 2 × R 2
and satisfies C 0 and C 1 compatibility conditions, then the following holds true:

lim t→∞ ||φ 1 (x, t)|| H 1 (0,L) + ||φ 2 (x, t)|| H 1 (0,L) = 0 .
(3.17)

Proof : Let us denote φ 1x = ∂ x φ 1 and φ 2x = ∂ x φ 2 . The dynamics of φ 1x (x, t) and φ 2x (x, t) are given by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t φ 1x φ 2x + λ 1 0 0 -λ 2 ∂ x φ 1x φ 2x = 0 , φ 1x (0, t) = k 1 λ 1 (bφ 2 (0, t) -ak 1 ξ 1 (t)) φ 2x (L, t) = - k 2 λ 2 (cφ 1 (L, t) -dξ 2 (t)) (3.18)
with the initial conditions

(φ 1x (x, 0), φ 2x (x, 0)) ∈ (L 2 (0, L)) 2 .
Now, considering the functional of φ 1x and

φ 2x V 1 (φ 1x , φ 2x ) = q L 0 φ 2 1x e -μx + φ 2 2x e μx dx
where q is a positive number.

Analysis the time derivative of V 1 along the solution of (3.18), one gets that

V1 (t) = -μq L 0 λ 1 φ 2 1x e -μx + λ 2 φ 2 2x e μx dx -φ 2 1x (L, t)e -μL qλ 1 -φ 2 2x (0, t)λ 2 q + k 2 1 q λ 1 (bφ 2 (0, t) -ak 1 ξ 1 (t)) 2 + k 2 2 q λ 2 (cφ 1 (L, t) -dξ 2 (t)) 2 . (3.19)
With (3.19), one can find N > 0, K > 0 such that

V1 (t) -qN ||φ 1x || L 2 (0,L) + ||(φ 2x || L 2 (0,L) + φ 2 1x (L, t) + φ 2 2x (0, t) + qK φ 2 1 (L, t) + φ 2 2 (0, t) + ξ 2 1 (t) + ξ 2 2 (t) (3.20)
With (3.15), there exists q > 0 and α > 0 such that

V (t) + V1 (t) -αV -qN ||φ 1x || L 2 (0,L) + ||(φ 2x || L 2 (0,L)
It implies that there exists β > 0 such that

∂ t (V + V 1 ) -β(V + V 1 )
With the similar analysis in the proof of stability, it is easily to prove that the origin of (3.18) is also exponentially stable in L 2 (0, L). This leads to the proof of (3.17).

Finally, we have the following lemma which is a special case of the Sobolev embedding theorem.

Application and numerical simulations 3.4.1 Application to the Saint Venant nonlinear model

In this Section, we apply our PI control design to the Saint Venant model of a reach canal for shallow water (see in Figure 3.1) studied in [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF][START_REF] Pham | Predictive control with guaranteed stability for hyperbolic systems of conservation laws[END_REF][START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF] with the friction slope neglected:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂H(x, t) ∂t + ∂ ∂x ( Q(x, t) B ) = 0 , ∂Q(x, t) ∂t + ∂ ∂x ( Q 2 (x, t) BH(x, t) ) + gBH(x, t) ∂H(x, t) ∂x = 0 y(t) = (H(0, t), Q(L, t)) (3.21)
equipped with two boundary conditions:

Q(0, t) = U 0 (t) , H(L, t) = U L (t)
where H(x, t) and Q(x, t) represent the water level and the flow discharge respectively; B and g denote the base width and the Newton gravitation constant; y(t) is the measured output on the boundary; U 0 and U L are control actions for the system.

Let define y r = (H r , Q r ) be the desired set-point satisfying the sub-critical condition

gB 2 H 3 r -Q 2 r > 0, and denoting h = H -H r and q = Q -Q r ,
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ ∂t h q + ⎛ ⎜ ⎜ ⎝ 0 1 B -Q 2 r BH 2 r + gBH r 2Q r BH r ⎞ ⎟ ⎟ ⎠ ∂ ∂x h q = 0 U 0 (t) = Q r + q(0, t) U L (t) = H r + h(L, t) y(t) = (h(0, t), q(L, t)) + y r , (3.22)
In addition, the characteristic form in (3.2) is obtained by using the following change of coordinates:

h = φ 1 + φ 2 , q = (B gH r + Q r H r )φ 1 -(B gH r - Q r H r )φ 2
The two boundary conditions and the output y(t) are rewritten in the following form:

φ 1 (0, t) = R 0 φ 2 (0, t) + u 1 (t) φ 2 (L, t) = R L φ 1 (L, t) + u 2 (t) y(t) -y r = aφ 1 (0, t) + bφ 2 (0, t) cφ 1 (L, t) + dφ 2 (L, t)
where

R 0 = BH r √ gH r -Q r BH r √ gH r + Q r , R L = -1, a = b = 1, c = B gH r + Q r H r , d = -B gH r + Q r H r ,
and the new control u(t) = (u 1 (t), u 2 (t)) defined by

u 1 (t) = U 0 (t) -Q r B √ gH r + Q r H r , u 2 (t) = U L (t) -H r (3.23)
Finally, applying Theorem 3.1, we design the PI controller as follows:

u(t) = u 1 (t) u 2 (t) = K p (y(t) -y r ) + K I t 0 (y(τ ) -y r )dτ, (3.24)
where the matrices K p and K I are given by Chapter 3. Multiple input -multiple output 2 × 2 systems with PI control 49

K p = ⎛ ⎜ ⎜ ⎝ Q r -BH r √ gH r Q r + BH r √ gH r 0 0 H r BH r √ gH r + Q r ⎞ ⎟ ⎟ ⎠ , K I = μ ⎛ ⎜ ⎜ ⎝ -2BH r √ gH r Q r + BH r √ gH r 0 0 2Q r (Q r -BH r √ gH r ) H r Q r + BH 2 r √ gH r ⎞ ⎟ ⎟ ⎠ .
When the tuning parameter μ is chosen small enough, the PI controller stabilizes the linearized model (4.56) and regulates the output to the desired set-point.

Numerical simulations and results

Discretization by Preissmann scheme and numerical solution

In this Section, applying Preissmann scheme (see in Appendix), we give the discretized nonlinear equations of the model (3.21) by

F X(k + 1) = 0 (3.25)
where

F : R 2N → R 2N defined by F = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ f 1 g 1 ... f N -1 g N -1 f N g N ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, and

X(k + 1) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Z k+1 1 Q k+1 1 ... Z k+1 N Q k+1 N ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .
with the 2N -2 sub-functions f i and g i (i = 1, N -1) received from the discretization of 2N -2 PDE equations in (3.21) .26) and two ODE boundary conditions

f i = Z k+1 i+1 + Z k+1 i -Z k i+1 -Z k i + 2 Δt Δx θ(Q k+1 i+1 -Q k+1 i ) + (1 -θ)(Q k i+1 -Q k i , g i = Q k+1 i+1 + Q k+1 i -Q k i+1 -Q k i + 2 Δt Δx [θM i + (1 -θ)H i ] . ( 3
f N = Z k+1 N -U L , g N = Q k+1 1 -U 0 ,
Chapter 3. Multiple input -multiple output 2 × 2 systems with PI control

50
where

M i = ( Q 2 BZ ) k+1 i+1 -( Q 2 BZ ) k+1 i + gB(θ Z k+1 i+1 + Z k+1 i 2 + (1 -θ) Z k i+1 + Z k i 2 )(Z k+1 i+1 -Z k+1 i
),

H i = ( Q 2 BZ ) k i+1 -( Q 2 BZ ) k i + gB(θ Z k+1 i+1 + Z k+1 i 2 + (1 -θ) Z k i+1 + Z k i 2 )(Z k i+1 -Z k i ).
Note also that the value at instant k of Z and Q are supposed to be computed previously and considered known values.

To continue, the implicit nonlinear equation (3.25) is resolved by Newton Raphson method, see the details in Appendix. To employ this iterative method, we need to compute the Jacobien matrix of function F as follows:

J = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∂f 1 ∂Z k+1 1 ∂f 1 ∂Q k+1 1 ∂f 1 ∂Z k+1 2 ∂f 1 ∂Q k+1 2 ... 0 0 0 0 ∂g 1 ∂Z k+1 1 ∂g 1 ∂Q k+1 1 ∂g 1 ∂Z k+1 2 ∂g 1 ∂Q k+1 2 ... 0 0 0 0 ... 0 0 0 0 ... ∂g N -1 ∂Z k+1 N -1 ∂g k+1 N -1 ∂Q k+1 N -1 ∂g N -1 ∂Z k+1 N ∂g N -1 ∂Q N 0 0 0 0 ... ∂f N -1 ∂Z k+1 N -1 ∂f N -1 ∂Q k+1 N -1 ∂f N -1 ∂Z k+1 N ∂f N -1 ∂Q k+1 N 0 0 0 0 ... 0 0 ∂f N ∂Z k+1 N ∂f N ∂Q k+1 N ∂g N ∂Z k+1 1 ∂g N ∂Q k+1 1 0 0 ... 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
where

∂f i ∂Z k+1 i = 1 , ∂f i ∂Z k+1 i+1 = 1 , ∂f i ∂Q k+1 i = -2 Δt Δx θ , ∂f i ∂H k+1 i+1 = 2 Δt Δx θ , ∂g i ∂Z k+1 i = 2 Δt Δx (θ ∂M i ∂Z k+1 i + (1 -θ) ∂H i ∂Z k+1 i ) , ∂g i ∂Z k+1 i+1 = 2 Δt Δx (θ ∂M i ∂Z k+1 i+1 + (1 -θ) ∂H i ∂Z k+1 i+1
),

∂g i ∂Q k+1 i = 1 + 2 Δt Δx (θ ∂M i ∂Q k+1 i + (1 -θ) ∂H i ∂Q k+1 i ) , ∂g i ∂Q k+1 i+1 = 1 + 2 Δt Δx (θ ∂M i ∂Q k+1 i+1 + (1 -θ) ∂H i ∂Q k+1 i+1 ) , ∂f N ∂Z k+1 N = 1 , ∂f N ∂Q k+1 N = 0 ∂g N ∂Z k+1 1 = 0 , ∂g N ∂Q k+1 1 = 1 . Chapter 3. Multiple input -multiple output 2 × 2 systems with PI control 51 With ∂M i ∂Z k+1 i = (Q k+1 i ) 2 B(Z k+1 i ) 2 -gB θZ k+1 i + (1 -θ) Z k i+1 + Z k i 2 , ∂M i ∂Q k+1 i = - 2Q k+1 i BZ k+1 i , ∂M i ∂Z k+1 i+1 = - (Q k+1 i+1 ) 2 B(Z k+1 i+1 ) 2 + gB(θZ k+1 i+1 + (1 -θ) Z k i+1 + Z k i 2 ) , ∂M i ∂Q k+1 i+1 = 2Q k+1 i+1 BZ k+1 i+1 , ∂H i ∂Z k+1 i = gB θ 2 (Z k+1 i+1 -Z k+1 i ) , ∂H i ∂Q k+1 i = 0 , ∂H i ∂Z k+1 i+1 = gB θ 2 (Z k+1 i+1 -Z k+1 i ) , ∂H i ∂Q k+1 i+1 = 0 .

Simulation results

In this section, we perform some numerical simulations for the model (3.21). The data used are the following consider the following: L = 50m, H r = 10m, Q r = 5m 3 /s, B = 1m. Note that the subcritical condition is satisfied. The parameters for Preismann schema to make the simulations are weighting parameter θ = 0.6, the ratio between space and time discretization steps Δt/Δx = 0.5, and the number of the discretized space point N = 100.

The PI controller designed in (3.24) with tunning with μ = 0.001. These disturbances lead to the error between the output y(t) and the desired set-point

y r = (H r , Q r )
in the open-loop system at right simulations. However, by using PI controller in closed-loop system, this error is totally cancelled as seeing in simulations on the left.

In addition, the evolutions of two boundary control inputs U 0 (t) = Q(0, t) and U L (t) = H(L, t) are depicted in figure 3.4. Another remark is that, the bigger the integral gain μ is, the more the regulation is rapid, but the more the profile is oscillatory; and a big value of μ (for instance μ = 0.005) results to the instability of the closed-loop system.

Conclusions

In this chapter, we have considered a class of linear 2 × 2 multiple input multiple output systems governed by hyperbolic partial differential equations with the problem of designing dynamic feedback under the form of multivariable PI controller on the boundary.

The Lyapunov functional in the general quadratic form of state variables and feedback variables is used to prove the stability of the closed-loop system in L 2 norm and the regulation of the output to the desired set-point. The idea of constructing the Lyapunov function can be used for the higher order systems, not only 2 × 2 systems. Furthermore, an explicit PI controller given in the Theorem 3.1 is easily applied for many linearized model of hyperbolic systems, such as the Saint-Venant linerized model in the Section 4.5.2.3. In the future, the work is to extend for the nonlinear models with the nonlinear boundary conditions; and the optimization in the performance of the closed-loop system by controller will be considered as well.

The problem studied in this chapter was almost published in [START_REF] Trinh | Multivariable pi controller design for 2 x 2 systems governed by hyperbolic partial differential equations with lyapunov techniques[END_REF].

Chapter 4

Networks of hyperbolic systems:

Stability and Output regulation 4.1 Introduction

In this chapter, we consider some networks of n systems described by the following two hyperbolic partial differential equations (PDE)

⎧ ⎪ ⎨ ⎪ ⎩ ∂ t ψ i1 (x, t) + λ i1 ∂ x ψ i1 (x, t) = 0 ∂ t ψ i2 (x, t) -λ i2 ∂ x ψ i2 (x, t) = 0 , x ∈ [0, L], t ∈ [0, ∞), i = 1, n (4.1)
under the initial conditions:

ψ i1 (x, 0) = ψ 0 i1 (x) , ψ i2 (x, 0) = ψ 0 i2 (x).
where ψ i1 , ψ i2 : [0, L]×[0, ∞] → R are two states and λ i1 , λ i2 are two positive constants.

Here ∂ t , ∂ x denote the time and space partial derivative respectively.

Each system in (4.1) is a 2× 2 linear hyperbolic PDE model. This type of dynamics may be used to describe various physical phenomena. Indeed, a large number of models can be transformed into the form (4.1) by some change of coordinates as shown for example in [3,4,[START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF][START_REF] Pham | Predictive control with guaranteed stability for hyperbolic systems of conservation laws[END_REF][START_REF] Perrollaz | Finite time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks[END_REF].

Our goal in the chapter is to study two kinds of network: star-shaped network and cascaded one; and to design the proportional-integral (PI) dynamic control law in order to guarantee the stability of closed-loop networks and regulate output measurements to desired set-points.

The chapter is organized as follows. Section 4.2 is to review existing results in the literatures and present our contribution in general. In Section 4.3 and Section 4.4, we give the main results and their proofs based on Lyapunov direct method for the starshaped network and cascaded one, respectively. Then we explain in details in Section 4.5.2.3 how to apply our general PI control designs for practical networks of n fluid flow models. Moreover, numerical simulations are carried out to validate the theoretical results. Finally, conclusions are addressed in Section 4.6.

Existing results and motivation

The problem of controlling hyperbolic networks with both inputs and outputs on the junctions becomes an important subject with a lot of studies in the literature. Most of results have established the boundary conditions at each junction to stabilize all subsystems around the equilibrium state. Let us review significant results in the literature in this domain:

• In [START_REF] De Halleux | Boundary control in networks of open channels[END_REF], a simple network of water flow in open channels with three reaches in cascade is studied. The static control law is given by using the Li Ta-Tsien theorem Chapter 5] to prove the stability of the closed-loop system in

(Theorem 1.3 in [46,
C 1 norm.
In addtion, by employing the same theorem of Li Ta-Tsien, a static stabilization feedback control law is done for fan-shaped networks of gas flow pipe in [START_REF] Gugat | Gas flow in fan-shaped networks: classical solutions and feedback stabilization[END_REF].

• Based on Lyapunov technique, in [5], [START_REF] Bastin | Andréa Novel On Lyapunov stability of linearised Saint-Venant equations for a sloping channel[END_REF] they have demonstrated the stability in L 2 norm of the open-water channel cascaded network described by Saint-Venant equations with the static boundary control design.

• The finite time stabilization method is considered in [START_REF] Perrollaz | Finite time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks[END_REF] for tree-shaped networks of 2 × 2 hyperbolic systems of conservation laws.

• The PI boundary control was applied for the networks of density-flow systems governed by linear hyperbolic PDE equations with the frequent approach in [3].

• In [START_REF] Pham | Contribution to predictive control for systems of conservation laws[END_REF], the receding horizon optimal control approach which is analysed for a network of 2 × 2 hyperbolic systems in cascade.

In the chapter, control inputs U (t) and measured outputs Y (t) are on the boundary for the star-shaped network, and on junctions for cascaded network. Our purpose is to design dynamic feedback control laws U (t) with structure of PI controllers such that the networks of closed-loop systems are exponentially stable, and output measurements In the following, we give the results and their proofs in details, for the star-shaped network in Section 4.3, and for cascaded network in Section 4.4.

Star-shaped network 4.3.1 Problem statement

In this section, we consider a star-shaped network of n PDE systems in (4.1). The connection between systems is depicted in Fig. 4.1, with n -1 "inlet" systems (1th to (n -1)th) and an "outlet" system (nth channel). To describe the star-shaped network, we suppose the following boundary conditions:

ψ j1 (0, t) = R j1 ψ j2 (0, t) + u j (t), ψ j2 (L, t) = R j2 ψ j1 (L, t), (4.3)
• For the outlet system:

ψ n1 (0, t) = R n1 ψ n2 (0, t) + n-1 j=1 α j ψ j1 (L, t), ψ n2 (L, t) = R n2 ψ n1 (L, t) (4.4)
where α j , R j1 , R j2 , R n1 and R n2 are real constants depending on physical models; u j (t)

is the dynamic feedback control law located at x = 0, which we want to design by PI controller for jth channel.

Moreover, n -1 measured outputs are considered for each inlet system:

y j (t) -y jr = a j ψ j1 (0, t) + b i ψ j2 (0, t), (4.5)
where a j , b j are real constants, and y jr is desired reference for channel j.

To summarize, the dynamic feedback PI control laws u j (t) are as follows

u j (t) = K jP (y j (t) -y jr ) + K jI L 0 (y j (s) -y jr )ds (4.6)
such that the network of closed-loop systems is stabilized toward the origin and the outputs y j (t) of inlet channels are regulated to the reference y jr (∀j = 1, n -1).

Remark : It should be mentioned here that the stability of the origin of closed-loop network implies the output regulation. This is because if we can prove that lim t→∞ ψ i1 (., t) = 0 and lim t→∞ ψ i2 (., t) = 0 for all x ∈ [0, L], by employing the output form in (4.29), one deduces that lim t→∞ y j (t) = y jr for all j = 1, n -1.

Main result

Denoting the new state variables ξ j (t) where ∂ t ξ j = y j (t)-y jr , the network of closed-loop systems (4.1), (4.3), (4.4) and (4.6) is governed by :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t ψ i1 (x, t) = -λ i1 ∂ x ψ i1 , ∂ t ψ i2 (x, t) = λ i2 ∂ x ψ i2 , ∂ t ξ j = a j ψ j1 (0, t) + b j ψ j2 (0, t) ψ i2 (L, t) = R i2 ψ i1 (L, t), ψ j1 (0, t) = R j1 ψ j2 (0, t) + K jP (y j (t) -y jr ) + K jI ξ j (t), ψ n1 (0, t) = R n1 ψ n2 (0, t) + n-1 j=1 α j ψ j1 (L, t). y j (t) -y jr = a j ψ j1 (0, t) + b j ψ j2 (0, t). ( 4.7)
The closed-loop system (4.7) is completed by the following initial conditions

ψ 0 11 (x), ψ 0 12 (x), • • • , ψ 0 n1 (x), ψ 0 n2 (x), ξ 0 1 , • • • , ξ 0 n-1 ∈ (L 2 (0, L)) 2n × R n-1 ( 4.8)
Let E = (L 2 (0, L)) 2n × R n-1 be the state space of the closed-loop system (4.7) equipped with the following norm :

||Y || 2 E = 2n i=1 ||Y i (., t)|| 2 L 2 (0,L) + 3n-1 j=2n+1 |Y j (t)| 2 ,
where

Y = (Y 1 , Y 2 , • • • , Y 3n-1 ) ∈ E.
The main result of this Section is given in the following theorem Theorem 4.1. There exists

μ * > 0 such that for each μ ∈ (0, μ * ), each R n2 ∈ -λ n1 λ n2 , λ n1 λ n2
and for each PI controller with the following proportional gain K jP and the integral gain

K jI K jP = -R j1 b j , K jI = -μ (a j + b j R j2 e μL )(b j + a j R j1 ) b j (4.9)
then the two following properties hold true :

• The network of closed-loop systems (4.7) with the initial condition in (4.8) is exponentially stable toward the origin in E.

S = n-1 j=1 V j + V n (4.10)
where

V j = L 0 F T j P j F dx with F j = ⎛ ⎜ ⎜ ⎝ ψ j1 e -μx 2 ψ j2 e μx 2 ξ j ⎞ ⎟ ⎟ ⎠ , P j = ⎛ ⎜ ⎜ ⎝ 1 0 q j3 0 q j1 q j4 q j3 q j4 q j2 ⎞ ⎟ ⎟
⎠ , and

V n = q n L 0 qψ 2 n1 e -μx + ψ 2 n2 e μx dx
Note that each sub-functional V j is independent on others. Therefore we consider at first each sub-functional V j and design the parameter q j1 , q j2 , q j3 , q j4 such that each V j is a Lyapunov functional. Then, by taking appropriate values of q n and q, we prove that S is also a Lyapunov functional.

Lyapunov candidate functional for each inlet system

To simplify the writing, in this Section we omit the symbol j, and all notations used imply for jth system. At first, applying the PI controller design in (4.9), each closed-loop of each inlet system can be described as follows Let denote the set X = (L 2 (0, L)) 2 × R associated with the following norm

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t ψ 1 ψ 2 = -λ 1 0 0 λ 2 ∂ x ψ 1 ψ 2 ∂ t ξ = aψ 1 (0, t) + bψ 2 (0, t) ψ 1 (0, t) = -k I ξ(t) ψ 2 (L, t) = R 2 ψ 1 (L, t) y(t) -y r = aψ 1 (0, t) + bψ 2 (0, t).
||v|| 2 X = ||v 1 || 2 L 2 (0,L) + ||v 2 || 2 L 2 (0,L) + |v 3 | 2 where v = (v 1 , v 2 , v 3 ) ∈ X.
The following lemma gives the design of each Lyapunov sub-functional.

Lemma 4.2. Let k I be defined in (4.12) and q 1 , q 2 , q 3 , q 4 be defined by

q 1 = λ 1 e -2μL 2λ 2 R 2 2 , q 2 = μλ 1 , q 3 = μbR 2 e μL , q 4 = μ bλ 1 λ 2 . (4.13)
Then there exists γ > 0 and μ * > 0 such that for every μ ∈ (0, μ * ), we have :

1. There exists M > 0 such that ∀ (ψ 1 , ψ 2 , ξ) in X : 1 M V (ψ 1 , ψ 2 , ξ) ||(ψ 1 , ψ 2 , ξ)|| 2 X MV (ψ 1 , ψ 2 , ξ). ( 4

.14)

2. There exists β > 0 such that along the solution of each closed-loop inlet system, for all t at which the solution is well defined

dV dt -βV -ψ 2 1 (L, t) λ 1 e -μL 2 . ( 4

.15)

Proof : Proof of the first part

To begin with, we prove that matrix P is definite positive. Applying the Sylvester criterion, matrix P is positive definite if and only if :

q 1 > 0, q 2 > q 2 3 + q 2 4 q 1 .
Employing (4.13), one can find that :

q 1 > 0, q 2 > 0 , q 2 -q 2 3 - q 2 4 q 1 = μ λ 1 -μb 2 R 2 2 e 2μL -μb 2 R 2 2 e 2μL 2λ 1 λ 2 .
It is clearly that if μ is small enough, q 2q 2 3 -q 2 4 q 1 > 0. It therefore yields that the matrix P is symmetric positive definite. Hence, there exits σ 1 , σ 2 > 0 such that

σ 1 L 0 ⎛ ⎜ ⎜ ⎝ ψ 1 e -μx 2 ψ 2 e μx 2 ξ ⎞ ⎟ ⎟ ⎠ T ⎛ ⎜ ⎜ ⎝ ψ 1 e -μx 2 ψ 2 e μx 2 ξ ⎞ ⎟ ⎟ ⎠ dx V (ψ 1 , ψ 2 , ξ) σ 2 L 0 ⎛ ⎜ ⎜ ⎝ ψ 1 e -μx 2 ψ 2 e μx 2 ξ ⎞ ⎟ ⎟ ⎠ T ⎛ ⎜ ⎜ ⎝ ψ 1 e -μx 2 ψ 2 e μx 2 ξ ⎞ ⎟ ⎟ ⎠ dx.
As a result, there exists M > 0 such that (4.14) holds.

Proof of the second part

The time derivative of V along the solution of each inlet closed-loop system has the following form:

V = - L 0 Λ T QΛ -L(t) , where Λ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ψ 1 e -μx 2 ψ 2 e μx 2 ξ ψ 2 (0, t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, and

L(t) = ψ 2 1 (L, t) λ 1 e -μL 2 + ψ 2 2 (0, t) λ 1 e -2μL 4R 2 2 + ξ 2 (t) λ 1 k 2 i 2 - γk 2 i λ 1 (bψ 2 (0, t) -ak i ξ(t)) 2 , Q = 1 L ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ μλ 1 L 0 T ψ 1 ,ξ T ψ 1 ,ψ 20 0 μq 1 λ 2 L T ψ 2 ,ξ T ψ 2 ,ψ 20 T ψ 1 ,ξ T ψ 2 ,ξ T ξ 0 T ψ 1 ,ψ 20 T ψ 2 ,ψ 20 0 λ 2 q 1 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ with T ξ = μ 2 λ 1 c 2 2 , T ψ 1 ,ψ 2 0 = -μLb 2 R 2 e μL , T ψ 2 ,ψ 2 0 = -μL b 2 λ 1 λ 2 , T ψ 1 ,ξ = μ 2 LbR 2 e μL 2ac + λ 1 2 , T ψ 2 ,ξ = μ 2 Lbλ 1 2ac + λ 2 2λ 2 , c = a + bR 2 e μL .
At first, one can easily see that if taking γ small enough, we have L(t) ψ 2 1 (L, t) λ 1 e -μL 2 .

In the following, we prove that by picking μ small enough, matrix Q is symmetric positive

D 1 = det ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ μλ 1 L 0 T ψ 1 ,ξ 0 μq 1 λ 2 L T ψ 2 ,ξ T ψ 1 ,ξ T ψ 2 ,ξ T ξ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ > 0, D 2 = det ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ μλ 1 L 0 T ψ 1 ,ξ T ψ 1 ,ψ 20 0 μq 1 λ 2 L T ψ 2 ,ξ T ψ 2 ,ψ 20 T ψ 1 ,ξ T ψ 2 ,ξ T ξ 0 T ψ 1 ,ψ 20 T ψ 2 ,ψ 20 0 λ 2 q 1 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ > 0.
By the direct computing, we have that

D 1 = μ 4 f 1 (μ) , D 2 = μ 4 f 2 (μ) with lim μ→0 f 1 (μ) = 1 2 L 2 λ 2 1 λ 2 c 2 q 1 > 0 and lim μ→0 f 2 (μ) = 1 4 L 2 λ 2 1 λ 2 2 c 2 q 2 1 > 0.
Taking μ small enough, it yields that the two terms D 1 , D 2 are both positive. Consequently, the matrix Q is symmetric positive definite.

Therefore, with the adaptive choice of γ and μ, there exists a positive real number K > 0 such that for all t at which the solution is well defined we have:

V (t) -K||(ψ 1 , ψ 2 , ξ)|| 2 X -ψ 2 1 (L, t) λ 1 e -μL 2 .
With (3.14) the former inequality implies that we can find β > 0 such that (4.15) holds.

This completes the proof of Lemma 4.2.

Remark : Lemma 4.2 shows a Lypunov functional constructed for a single system in the case of single input-single output. A more complex situation of multi inputs-multi outputs is considered in chapter 3. In [3], by frequency method, another result of PI control for single system is presented. Another work in the same context with Lyapunov direct method is given in [4]. However the Lyapunov functional constructed in this work has not coupled term (the term q 3 , q 4 in our Lyapunov functional), therefore the result obtained is restrained for only the case of parameter |R 2 | < 1.

Proof of the theorem

Proof of the stability

Employing the Lemma 4.2, we can design the parameters q j1 , • • • , q j4 for every j = 1, n -1 such that

• There exists M j > 0 such that

1 M j V j ||(ψ j1 , ψ j2 , ξ j )|| 2 X M j V j . ( 4

.16)

• There exists β j > 0 such that

Vj (t) -β j V j (t) -ψ 2 j1 (L, t)
λ j1 e -μL 2 .

(4.17)

Now analysis the time derivative of V n along the solution of the closed-loop network,

Vn (t) = -μq n L 0 λ n1 qψ 2 n1 e -μx + λ n2 ψ 2 n2 e μx dx -ψ 2 n1 (L, t)e -μL q n (λ n1 q -λ n2 R 2 n2 ) -ψ 2 n2 (0, t)λ 2n q n + λ n1 q n q ⎛ ⎝ R n1 ψ n2 (0, t) + n-1 j=1 α j ψ j1 (L, t) ⎞ ⎠ 2 . (4.18) Since |R n2 | < λ n1 λ n2 , one can choose q such that R 2 n2 λ n2 λ n1 < q < λ n1 λ n2
, and take q n > 0 small enough, we have

- n-1 j=1 ψ 2 j1 (L, t) λ j1 e -μL 2 -ψ 2 n2 (0, t)λ n2 q n +λ n1 q n q ⎛ ⎝ R n1 ψ n2 (0, t) + n-1 j=1 α j ψ j1 (L, t) ⎞ ⎠ 2 0 , -ψ 2 n1 (L, t)e -μL (λ n1 q -λ n2 R 2 n2 )q n 0 . ( 4

Proof of the output regulation

We use the same technique with the output regulation proof in Section 3.3.3.2 from Chap 3. We prove that with the initial condition (ψ j1 (x, 0),

ψ j2 (x, 0)) ∈ (H 1 (0, L)) 2 , then ∀x ∈ [0, L] lim t→∞ ||ψ j1 (x, t)|| H 1 (0,L) = 0 , lim t→∞ ||ψ j2 (x, t)|| H 1 (0,L) = 0. (4.22)
Let us put ψ j1x = ∂ x ψ j1 and ψ j2x = ∂ x ψ j2 . The dynamics of ψ j1x (x, t) and ψ j2x (x, t) are given by

∂ t ψ j1x ψ j2x + λ j1 0 0 -λ j2 ∂ x ψ j1x ψ j2x = 0 , (ψ j1x (x, 0), ψ j2x (x, 0)) ∈ (L 2 (0, L)) 2 , ψ j1x (0, t) = k I λ j1 (a j ψ j1 (0, t) + bψ j2 (0, t)) ψ j2x (L, t) = - R j2 λ j1 λ j2 ψ j1x (L, t) (4.23)
Now, considering the following functional of ψ j1x and ψ j2x

V j1 (ψ j1x , ψ j2x ) = γ L 0 q 5 ψ 2 j1x e -μx + ψ 2 j2x e μx dx
where γ and q 5 are positive numbers.

Analysis the time derivative of V j1 along the solution of closed-loop system j, we have

dV j1 dt = -μγ L 0 q 5 λ j1 ψ 2 j1x e -μx + λ j2 ψ 2 j2x e μx dx -ψ 2 j1x (L, t)γ e -μL q 5 λ j1 - R 2 j2 λ 2 j1 λ 2 j2 -ψ 2 j2x (0, t)γλ j2 + k 2 I γq 5 λ j1 (-k I a j ξ j (t) + b j ψ j2 (0, t)) 2 . (4.24)
From (4.24), if choosing q 5 > R 2 j2 λ j1 λ 2 j2 e μL one can find N j > 0, K j > 0 such that

dV j1 dt -γN j ||ψ j1x || L 2 (0,L) + ||(ψ j2x || L 2 (0,L) + ψ 2 j1x (L, t) + ψ 2 j2x (0, t) + γK j ψ 2 j2 (0, t) + ξ 2 j (t) (4.25)
With (4.17), there exists ζ > 0 such that

d(V j + V j1 ) dt -ζ(V + V j1 )
With the similar analysis in the proof of stability, it is easily to prove that the origin of each closed-loop inlet system is also exponentially stable in H 1 (0, L). Employing the Sobolev embedding theorem in Lemma 3.5, this leads to the proof of (4.22).

As a result, lim

t→∞ |y j (t) -y jr | = 0. ∀i = 1, n -1.
This completes the proof of Theorem 4.1.

Cascaded network

Problem statement

In this Section we are concerned with a class of cascaded networks of n PDE hyperbolic systems (4.1) (see Figure 4.2). In addition, we suppose that the boundary conditions define a cascaded network. More precisely, the following boundary conditions are defined at the junctions • For the first junction,

ψ 11 (0, t) = R 11 ψ 12 (0, t) (4.26)
• For n-1 intermediate junctions,

ψ i2 (L, t) =R i2 ψ i1 (L, t) + u i (t) ψ (i+1)1 (0, t) = R (i+1)1 ψ (i+1)2 (0, t) + α i ψ i1 (L, t) + δ i ψ i2 (L, t) , i = 1, n -1 (4.27)
• For the last junction, The measured outputs that need to be regulated are located at the junctions and disturbed by some unknown output perturbations, i.e

ψ n2 (L, t) = R n2 ψ n1 (L, t) + u n (t) (4.28)
y i (t) = a i ψ i1 (L, t) + b i ψ i2 (L,

Main result

Denoting the new state variables z i (t) where ∂ t z i = y i (t)y ir , the network of closedloop systems (4.1)-(4.30) becomes a PDE-ODE system and is governed by (i = 1, n and j = 2, n):

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t ψ i1 (x, t) = -λ i1 ∂ x ψ i1 , ∂ t ψ i2 (x, t) = λ i2 ∂ x ψ i2 , ∂ t z i = y i (t) -y ir , ψ 11 (0, t) = R 11 ψ 12 (0, t), ψ i2 (L, t) = R i2 ψ i1 (L, t) + K iP (y i (t) -y ir ) + K iI z i (t) + w ic , ψ j1 (0, t) = R j1 ψ j2 (0, t) + α j ψ (j-1)1 (L, t) + δ j ψ (j-1)2 (L, t), y i (t) =a i ψ i1 (L, t) + b i ψ i2 (L, t) + w io .
(4.31)

The closed-loop system (4.31) is completed by the following initial conditions

ψ 0 11 (x), ψ 0 12 (x), z 0 1 , • • • , ψ 0 n1 (x), ψ 0 n2 (x), z 0 n ∈ (L 2 (0, L)) 2 × R n , ∀x ∈ [0, L]
which satisfy the C 0 and C 1 compatibility conditions in (4.31).

Let E = (L 2 (0, L)) 2 × R n be the state space of the closed-loop system (4.31) equipped with the following norm:

||Y || 2 E = n i=1 ||Y 3i-2 || 2 L 2 (0,L) + ||Y 3i-1 || 2 L 2 (0,L) + Y 2 3i where Y = (Y 1 , Y 2 , • • • , Y 3n ) ∈ E.
For each initial condition ψ 0 11 (x), ψ 0 12 (x), z 0 1 , • • • , ψ 0 n1 (x), ψ 0 n2 (x), z 0 n in E satisfying C 0 and C 1 compatibility conditions, then there exits a unique smooth solution of (4.31) in E for all t (see in [START_REF] Coron | Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems[END_REF]). We study therefore the exponential stability of the closed-loop system (4.31) in E and the output regulation to the desired references.

Let us define iteratively equilibrium points ψ i1∞ , ψ i2∞ and z i∞ as follows

ψ i2∞ = y ir -w io + a i D i b i + a i R i1 , ψ i1∞ = R i1 ψ i2∞ -D i , z i∞ = ψ i2∞ -R i2 ψ i1∞ -w ic K iI . (4.32)
where

D 1 = 0 , D j = α j ψ (j-1)1∞ + δ j ψ (j-1)2∞ , ∀ j = 2, n.
We suppose the following hypothesises for the output measures:

H1: a i = 0 ∀i = 1, n.

H2:

a i + b i R i2 = 0 ∀i = 1, n.
i + a i R i1 = 0 ∀i = 1, n.
Then the main result of our paper is given in the following theorem.

Theorem 4.3. Assume that the three hypotheses H1, H2 and H3 are satisfied. Then, there exists μ * > 0 such that for each initial condition in E satisfying the C 0 and C 1 compatibility conditions, each μ ∈ (0, μ * ) and each PI controller with the following proportional gain K iP and the integral gain K iI :

K iP = -R i2 a i , K iI = -μ (b i + a i R i1 e μL )(a i + b i R i2 ) a i , ∀i = 1, n (4.33)
the following two properties hold true: • For smooth initial condition

• The equilibrium state (ψ 11∞ , ψ 12∞ , z 1∞ , • • • , ψ n1∞ ,
ψ 0 11 (x), ψ 0 12 (x), z 0 1 , • • • , ψ 0 n1 (x), ψ 0 n2 (x), z 0 n ∈ (H 1 (0, L)) 2 × R

Remark :

We need to impose Hypothesis H1 for our PI controller to exist. On the contrary, if H2 is not satisfied, then our PI controller can still be implemented.

However the integral term K i disappears and consequently the PI control law becomes only proportional control law. In that case, even so stability of the closed loop system may still be obtained, the integral effect of our PI controllers that leads to the output regulation is lost. Up to now, removing these conditions is an open question.

In addition, hypothesis H3 is a necessary condition for the existence of an equilibrium point of the closed-loop system (4.31) for all values of disturbances w io and w ic . Without this assumption, the procedure introduced in (4.32) is no longer valid. Hence, H3 is a necessary condition for the output regulation by integral action.

Proof of the main result

To prove Theorem 4.3, the following coordinate transformation is considered

φ i1 (x, t) = ψ i1 (x, t) -ψ i1∞ , φ i2 (x, t) = ψ i2 (x, t) -ψ i2∞ , ξ i (t) = z i (t) -z i∞ (4.34)
where ψ i1∞ , ψ i2∞ and z i∞ are defined in (4.32). With the new coordinates defined in (4.34) and applying the PI controller design (4.33) in Theorem 4.3, then we obtains the following network of closed-loop systems without perturbations

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t φ i1 (x, t) = -λ i1 ∂ x φ i1 , ∂ t φ i2 (x, t) = λ i2 ∂ x φ i2 , ∂ t ξ i = a i φ i1 (L, t) + b i φ i2 (L, t) φ i2 (L, t) = -k i ξ i (t), φ i1 (0, t) = R i1 φ i2 (0, t) + α i φ (i-1)1 (L, t) + β i ξ i-1 (t). y i (t) -y ir = a i φ i1 (L, t) + b i φ i2 (L, t). (4.35)
completed by the initial conditions in E:

φ i1 (x, 0) = φ 0 i1 (x) , φ i2 (x, 0) = φ 0 i2 (x) , ξ i (0) = ξ 0 i ,
where

k i = μ(b i + a i R i1 e μL ) (4.36)
and α 1 = 0, β 1 = 0, β j = -k j-1 δ j for j = 2, n.

In the new coordinates, y i (t)y ir = a i φ i1 (L, t)

+ b i φ i2 (L, t), the output regulation is obtained if lim t→∞ |a i φ i1 (L, t) + b i φ i2 (L, t)| = 0 , ∀i = 1, n.
Hence, to prove the asymptotic stability and the output regulation for the disturbed network systems in (4.31) with the PI control design in (4.30), we must naturally prove the stability of equivalent system (4.35) to the origin.

In the following, the Lyapunov candidate function is given in Section 4.4.3.1 and then the proof of the Theorem 4.3 by using direct Lyapunov method is presented in Section 4.4.3.2.

Lyapunov candidate functional

In the paper, we construct the following Lyapunov candidate functional:

V(φ 11 , φ 12 , ξ 1 , • • • , φ n1 , φ n2 , ξ n ) = n i=1 p i V i (4.37)
where V i is defined by

V i (φ i1 , φ i2 , ξ i ) = L 0 F T i P i F i dx with F i = ⎛ ⎜ ⎜ ⎝ φ i1 e -μx 2 φ i2 e μx 2 ξ i ⎞ ⎟ ⎟ ⎠ and P i = ⎛ ⎜ ⎜ ⎝ 1 0 q i3 0 q i1 q i4 q i3 q i4 q i2 ⎞ ⎟ ⎟ ⎠ .
Here p i and q i1 , • • • , q i4 are positive real number that be designed later on.

To begin with, consider the set X = (L 2 (0, L)) 2 × R. The following lemma for the constructing of each sub-function V i can be obtained.

Lemma 4.4. Let k i be defined in (4.36) and q i1 , q i2 , q i3 , q i4 be defined as follows:

q i1 > 3λ i1 R 2 i1 λ i2 , q i2 = μe μL λ i2 q i1 , q i3 = μe 3μL 2 a i λ i2 q i1 λ i1 , q i4 = μe 3μL 2 a i R i1 q i1 . (4.38)
Then there exists μ * > 0 such that for every μ ∈ (0, μ * ), we have 1. There exists M i > 0 such that ∀ (φ i1 , φ i2 , ξ i ) in X:

1 M i V i (φ i1 , φ i2 , ξ i ) ||φ i1 (., t)|| 2 L 2 (0,L) + ||φ i2 (., t)|| 2 L 2 (0,L) + ξ 2 i (t) M i V i (φ i1 , φ i2 , ξ i ) (4.

39)

2. There exists γ i > 0 such that along the solution of (4.35), for all t such that the solution is well defined

Vi (t) -γ i V i (t)- 1 4 ξ 2 i (t) k 2 i λ i2 q i1 e μL -φ 2 i1 (L, t) λ i1 e -μL 2 + φ 2 (i-1)1 (L, t)A i + ξ 2 i-1 (t)B i , ( 4

.40)

where

A i = λ i1 α 2 i 3 + 4λ 2 i1 q 2 i3 e -μL k 2 i λ i2 q i1 , B i = λ i1 β 2 i 3 + 4λ 2 i1 q 2 i3 e -μL k 2 i λ i2 q i1
. Proof : Proof of the first property 4.39

To begin with, we prove that the matrix P i is positive definite. Making use of the Sylvester criterion, matrix P i is positive definite if and only if:

q i1 > 0, q i2 > q 2 i3 + q 2 i4 q i1 .
Employing (4.38), one can find that:

q i1 > 0 , q i2 -q 2 i3 - q 2 i4 q i1 = μe μL q i1 λ i2 -μe 2μL a 2 i λ 2 i2 q i1 λ 2 i1 -μe 2μL a 2 i R 2 i1 .
It is clearly that if μ is small enough, q i2q 2 i3 -q 2 i4 q i1 > 0. It therefore yields that the matrix P i is symmetric positive definite. Hence, there exits σ i1 , σ i2 > 0 such that

σ i1 L 0 ⎛ ⎜ ⎜ ⎝ φ i1 e -μx 2 φ i2 e μx 2 ξ i ⎞ ⎟ ⎟ ⎠ T ⎛ ⎜ ⎜ ⎝ φ i1 e -μx 2 φ i2 e μx 2 ξ i ⎞ ⎟ ⎟ ⎠ dx V i (φ i1 , φ i2 , ξ i ) σ i2 L 0 ⎛ ⎜ ⎜ ⎝ φ i1 e -μx 2 φ i2 e μx 2 ξ i ⎞ ⎟ ⎟ ⎠ T ⎛ ⎜ ⎜ ⎝ φ i1 e -μx 2 φ i2 e μx 2 ξ i ⎞ ⎟ ⎟ ⎠ dx.
As a result, there exists M i > 0 such that (4.39) holds.

Proof of the second property 4.40

The time derivative of V along the solution of the system (4.35) has the following form:

Vi (t) = - L 0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ φ i1 (x, t)e -μx 2 φ i2 (x, t)e μx 2 ξ i (t) φ i1 (L, t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ T Q i ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ φ i1 (x, t)e -μx 2 φ i2 (x, t)e μx 2 ξ i (t) φ i1 (L, t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ dx -F (t) (4.41)
where

F (t) = φ 2 i1 (L, t) λ i1 e -μL 2 + φ 2 i2 (0, t)λ i2 q i1 + 3 4 ξ 2 i (t)k 2 i λ i2 q i1 e μL -λ i1 R i1 φ i2 (0, t) + α i φ (i-1)1 (L, t) + β i ξ i-1 (t) 2 -2ξ i (t)q i3 λ i1 α i φ (i-1)1 (L, t) + β i ξ i-1 (t) , (4.42)
and,

Q i = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ μλ i1 0 T φ i1 ,ξ i T φ i1 ,φ i1L 0 μq i1 λ i2 T φ i2 ,ξ i T φ i2 ,φ i1L T φ i1 ,ξ i T φ i2 ,ξ i T ξ i 0 T φ i1 ,φ i1L T φ i2 ,φ i1L 0 λ i1 e -μL 2L ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
T ξ i = μ 2 c 2 λ i2 q i1 e μL 4L , T φ i1 ,φ i1L = -μe 3μL 2 a 2 i λ i2 q i1 λ i1 , T φ i2 ,φ i1L = -μe 3μL 2 a 2 i R i1 q i1 , T φ i1 ,ξ i = μ 2 e 3μL 2 a i λ i2 q i1 (2b i c + λ i1 ) 2λ i1 , T φ i2 ,ξ i = μ 2 e 3μL 2 a i R i1 q i1 2b i c + λ i2 2λ i2 , and c = b i + a i R i1 e μL .
At first, we consider the boundary terms F (t) in (4.42). Applying the Cauchy-Schwarz inequality, one can find that

R i1 φ i2 (0, t) + α i φ (i-1)1 (L, t) + β i ξ i-1 (t) 2 3 R 2 i1 φ 2 i2 (0, t) + α 2 i φ 2 (i-1)1 (L, t) + β 2 i ξ 2 i-1 (t) (4.43) 2ξ i (t)q i3 λ i1 α i φ (i-1)1 (L, t) + β i ξ i-1 (t) 1 2 ξ 2 i (t) k 2 i λ i2 q i1 e μL + 4λ 2 i1 q 2 i3 e -μL k 2 i λ i2 q i1 α 2 i φ 2 (i-1)1 (L, t) + β 2 i ξ 2 i-1 (t) (4.44)
From (4.42), (4.43), (4.44) and with the choice of q i1 in (4.38) it can be deduced that

F (t) φ 2 i1 (L, t) λ i1 e -μL 2 + 1 4 ξ 2 i (t) k 2 i λ i2 q i1 e μL -φ 2 (i-1)1 (L, t)A i -ξ 2 i-1 (t)B i , ( 4.45)
where

A i = λ i1 α 2 i 3 + 4λ 2 i1 q 2 i3 e -μL k 2 i λ i2 q i1 , B i = λ i1 β 2 i 3 + 4λ 2 i1 q 2 i3 e -μL k 2 i λ i2 q i1
In the following, we prove that by picking μ small enough, matrix Q is symmetric positive definite. From the Sylvester criterion, Q is symmetric positive definite if and only if

D 1 = det ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ μλ i1 0 T φ i1 ,ξ i 0 μq i1 λ i2 T φ i2 ,ξ i T φ i1 ,ξ i T φ i2 ,ξ i T ξ i ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ > 0, D 2 = det(Q i ) > 0.
By the direct computing, we have that

D 1 = μ 4 f 1 (μ) , D 2 = μ 4 f 2 (μ) , with lim μ→0 f 1 (μ) = 1 L λ i1 λ 2 i2 q 2 i1 c 2 e μL > 0 and lim μ→0 f 2 (μ) = 1 2L λ 2 i1 λ 2 i2 q 2 i1 c 2 > 0.
Taking μ small enough, it yields that the two terms D 1 , D 2 are both positive. Consequently, the matrix Q i is symmetric positive definite.

Therefore, with the adaptive choice of μ, there exists a positive real number K i > 0 such that for all t at which the solution is well defined we have:

Vi (t) -K i ||(φ i1 , φ i2 , ξ i )|| 2 X - 1 4 ξ 2 i (t) k 2 i λ i2 q i1 e μL -φ 2 i1 (L, t) λ i1 e -μL 2 + φ 2 (i-1)1 (L, t)A i + ξ 2 i-1 (t)B i .
With (4.39) the former inequality implies that we can find γ i > 0 such that (4.40) holds.

This completes the proof of Lemma 4.4. Now, by employing Lemma (4.4), the following lemma for the design of Lypunov candidate function V is given, Lemma 4.5. Let k i and q i1 , q i2 , • • • , q i4 be defined in Lemma 4.4 and p i be defined as follows

p 1 > 0 , p i+1 = p i . ( 4

.46)

Then there exists > 0 and μ * > 0 such that for every μ ∈ (0, μ * ), we have :

1. There exists M > 0 such that ∀ (φ 11 , φ 12 , ξ 1 , • • • , φ n1 , φ n2 , ξ n ) in E: 1 M V(φ 11 , φ 12 , ξ 1 , • • • , φ n1 , φ n2 , ξ n ) ||(φ 11 , φ 12 , ξ 1 , • • • , φ n1 , φ n2 , ξ n )|| 2 E M V(φ 11 , φ 12 , ξ 1 , • • • , φ n1 , φ n2 , ξ n ) . (4.

47)

2. There exists γ > 0 such that along the solution of (4.31), for all t at which the solution is well defined

V(t) -γV(t) -δ n i=1 ξ 2 i (t) + φ 2 i1 (L, t) , ( 4

.48)

we have used the notation

V(t) = V(φ 11 (•, t), φ 12 (•, t), ξ 1 (t), • • • , φ n1 (•, t), φ n2 (•, t), ξ n (t)).
Proof : To begin with, applying the property (4.39) of Lemma 4.4, with μ small enough, one easily finds that

n i=1 p i M i V i ||(φ 11 , φ 12 , ξ 1 , • • • , φ n1 , φ n2 , ξ n )|| 2 E n i=1 p i M i V i . ( 4
V(t) - n i=1 p i γ i V i (t) - n i=1 ξ 2 i (t) p i k 2 i λ i2 q i1 e μL 4 -p i+1 B i+1 - n i=1 φ 2 i1 (L, t) p i λ i1 e -μL 2 -p i+1 A i+1 . (4.50)
Since p i+1 = p i , by taking small enough, it is clearly to see that

p i k 2 i λ i2 q i1 e μL 4 -p i+1 B i+1 > 0 p i λ i1 e -μL 2 -p i+1 A i+1 > 0 (4.51)
From (4.50) and (4.51), there exits γ > 0 and δ > 0 such that

V(t) -γ n i=1 p i V i (t) -δ n i=1 ξ 2 i (t) + φ 2 i1 (L, t)
This inequality implies that (4.48) holds.

Remark : In our Lyapunov functional (4.37), compared to the one in [START_REF] Bastin | Andréa Novel On Lyapunov stability of linearised Saint-Venant equations for a sloping channel[END_REF], dynamic feedbacks are added with n new states ξ i (t). Some Lyapunov functional for dynamic feedback states have been studied for single hyperbolic PDE systems in literature [START_REF] Coron | Feedback Stabilization for a scalar conservation law with PID boundary control[END_REF][START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF], but it should be pointed out that our Lyapunov function is different since it contains coupling terms of the states φ i and feedback states ξ i . These coupling terms allows to avoid the damping term in PI controller as in [START_REF] Coron | Feedback Stabilization for a scalar conservation law with PID boundary control[END_REF]. It allows also to consider a larger classe of hyperbolic PDE systems than the one considered in [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF].

Note that in [START_REF] Trinh | Multivariable pi controller design for 2 x 2 systems governed by hyperbolic partial differential equations with lyapunov techniques[END_REF], a Lyapunov functional with the dynamic feedback terms and the coupled terms is considered only for the single PDE hyperbolic system, but cannot be extended for a cascaded network.

Proof of the theorem 4.3

In this section, by using the result of Lemma 4.5, we present the proof of Theorem 4.3.

First of all, we prove the exponential stability of the closed-loop system (4.31) to the equilibrium state (ψ 11∞ , ψ 12∞ , z 1∞ , • • • , ψ n1∞ , ψ n2∞ , z n∞ ). From Lemma 4.5, there exists γ > 0 such that

V(t) V(0)e -γt . (4.52)
With (4.47), it implies that there exists S > 0 such that for all initial conditions

φ 0 11 (x), φ 0 12 (x), ξ 0 1 , • • • , φ 0 n1 (x), φ 0 n2 (x), ξ 0 n ∈ E, ∀x ∈ [0, L]
and satisfying C 0 , C 1 compatibility conditions, the solution of (4.35) is defined for all positive time and satisfies that for all t 0

||(φ 11 (•, t), φ 12 (•, t), ξ 1 (t), • • • , φ n1 (•, t), φ n2 (•, t), ξ n (t))|| 2 E Se -γt ||(φ 0 11 , φ 0 12 , ξ 0 1 , • • • , φ 0 n1 , φ 0 n2 , ξ 0 n )|| 2 E . ( 4
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t v i1 (x, t) = -λ i1 ∂ x v i1 , ∂ t v i2 (x, t) = λ i2 ∂ x v i2 , ∂ t s i = a i v i1 (L, t) + b i v i2 (L, t) v i2 (L, t) = -k i s i (t), v i1 (0, t) = R i1 v i2 (0, t) + α i v (i-1)1 (L, t) + β i s i-1 (t). ( 4

.54)

From the hypothesis in theorem, we deduce that (4.54) has also smooth initial condition

(v 0 i1 (x), v 0 i2 (x), s 0 i ) in E (∀i = 1, n)
and satisfying the C 0 and C 1 compatibility conditions. Employing similar analysis, one finds that the origin of (4.54) is also exponentially stable in E. This allows us to prove that:

lim t→∞ ||φ i1 (•, t)|| H 1 (0,L) = 0 , lim t→∞ ||φ i2 (•, t)|| H 1 (0,L) = 0.
By the Sobolev embedding theorem, one gives that

lim t→∞ φ i1 (x, t) = 0, lim t→∞ φ i2 (x, t) = 0 ∀x ∈ [0, L].
Therefore, the output regulation is obtained, i.e lim

t→∞ |y i (t) -y ir | = 0 , ∀i = 1, n.
This completes the proof of the Theorem 4.3.

Numerical simulations

In this section, we consider networks with the structure of n horizontal channels which are described by Saint Venant equations with the neglected friction slope, studied for example in [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF][START_REF] Pham | Predictive control with guaranteed stability for hyperbolic systems of conservation laws[END_REF][START_REF] Trinh | Multivariable pi controller design for 2 x 2 systems governed by hyperbolic partial differential equations with lyapunov techniques[END_REF] for a single channel and [START_REF] Pham | Contribution to predictive control for systems of conservation laws[END_REF][START_REF] Perrollaz | Finite time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks[END_REF] for networks of n channels.

Without loss of generality, the lengths of channels are assumed identical and equal L.

The dynamics of each channel is governed by

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ B i ∂ t H i (x, t) + ∂ x Q i (x, t) = 0 ∂ t Q i (x, t) + ∂ x Q 2 i (x, t) B i H i (x, t) + 1 2 gB i H 2 i (x, t) = 0 , x ∈ [0, L], t ∈ [0, ∞), i = 1, n (4.55)
where H i (x, t) denote water level, Q i (x, t) water discharge at the position x and time t respectively, B i the channel width and g the gravitational constant for the channel i.

(H * i ) 3 -(Q * i ) 2
> 0, and is more discussed in the following for each kind of network). By the notation

h i = H i -H * i and q i = Q i -Q * i , the linearized network is described as follows ∂ ∂t h i q i + ⎛ ⎜ ⎜ ⎝ 0 1 B i -(Q * i ) 2 B i (H * i ) 2 + gB i H * i 2Q * i B i H * i ⎞ ⎟ ⎟ ⎠ ∂ ∂x h i q i = 0 (4.56)
Then by using the following change of coordinates

h i = ψ i1 + ψ i2 , q i = (B i gH * i + Q * i H * i )ψ i1 -(B i gH * i - Q * i H * i )ψ i2 (4.57)
One obtains the following form that is similar to the one in (4.1):

⎧ ⎪ ⎨ ⎪ ⎩ ∂ t ψ i1 (x, t) + λ i1 ∂ x ψ i1 (x, t) = 0 ∂ t ψ i2 (x, t) -λ i2 ∂ x ψ i2 (x, t) = 0
* i + Q * i B i H * i > 0 , λ i2 = gH * i - Q * i B i H * i > 0.
In the following, we present in details two kinds of network: star-shaped model and cascaded one.

Simulations for star-shaped network

Modeling of star network

In this section, we aim to realise simulations for a star-shaped network of n channels (n 3) whose dynamics are considered in (4.55). The connection between channels can be seen in Fig. 4.1, with n -1 inlet channels (j = 1, n -1) and an outlet channel (nth channel).

The following notations are used for the next : i = 1, n and j = 1, n -1.

The n -1 online measured outputs at each time t are

y j (t) = H j (0, t). ( 4.59)
We assume that the network is controlled by 2n-1 input controls, in which each channel is controlled by the discharge at x = L, i.e

Q i (L, t) = Q ic (t), (4.60)
and n -1 inlet channels is actively controlled at x = 0 by

Q j (0, t) = U j (t), (4.61)
where inputs U j are at x = 0, and Q jc (t) are inputs at x = L.

Note however that at the junction, the constraint of flow-rate conservation is

Q n (0, t) = n-1 j=1 Q j (L, t). (4.62)
The equation (4.62) implies that, the boundary x = 0 of the outlet channel (nth channel) cannot be controlled, but its dynamic can be deduced from the conservation of the flow.

PI controllers for the model

At first, we consider the equilibrium state of the network model. The last boundary condition comes from the control of the inflow discharge by an appropriate constant value Q 0 , i.e

Q 1 (0, t) = Q 0 . (4.70)
The n online disturbed outputs y i (t) are water levels at x = L, i.e y i (t) = H i (L, t) + w io (4.71) where w io is unknown output disturbance. Then, we consider the linearized model analysed in (4.56), and rewrite all boundary conditions in (4.68)-(4.71) in the new coordinates ψ i2 (L, t) = R i2 ψ i1 (L, t) + u i (t) , ψ i1 (0, t) = R i1 ψ i2 (0, t) + α i ψ (i-1)1 (L, t) + δ i ψ (i-1)2 (L, t) , y i (t) =ψ i1 (L, t) + ψ i2 (L, t) + H * i + w io , (4.72) where Remark : To control this network of n cascaded models, we only need n control inputs represented by n underflow gates at junctions. Moreover, to implement controllers, only water levels at the end of each channel are required for the output measurements. This is an advantage because in practical engineering, water level is much simpler to measure than water discharge.

R i1 = λ i2 λ i1 , R i2 = 2Q 0 (B i gH * i + Q 0 H * i ) -α i 2Q 0 (B i gH * i - Q 0 H * i ) + α i , ∀i = 1, n α 1 = δ 1 = 0 , α k = λ (k-1)1 λ k1 , δ k = - λ (k-

Numerical simulations

In this Section, in order to validate theoretical results we make numerical simulations for a cascade network of three channels with the following data: Finally, note that with the big value of tuning parameter μ (bigger than 0.002) in simulations, the controlled network becomes unstable.

•

Conclusions

In this chapter, we have study two kinds of PDE hyperbolic network: star-shaped one and cascaded one, accompanied by boundary control and output regulation with PI control. The PI controllers are designed at free extremities of each inlet channel for star-shaped network; and at junctions of each subsystem for cascaded network. The exponential stability for the closed-loop network and the output regulation are proven by using Lyapunov direct method. The PI control design can be applied for many practical networks with the same steps as presented in Section 4.5.2.3. Although the stability analysis in the chapter is considered almost for linear system networks in L 2 norm, it is possible to extend our Lyapunov functional for the nonlinear case with the idea in [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]. In the future, the objective is to study the PI control design for other network types of PDE hyperbolic systems, such as tree-shaped network; and consider the performance of closed-loop systems controlled by PI controllers.

The results of this chapter are published in [START_REF] Trinh | Boundary pi controllers for a star-shaped network of 2 × 2 systems governed by hyperbolic partial differential equations[END_REF] and [START_REF] Trinh | Stability and output regulation for a cascaded network of 2 × 2 hyperbolic systems[END_REF] In Chapter 4, we are concerned with the star-shaped network and the cascaded one of multi-systems described by 2 × 2 linear hyperbolic PDEs. The issues of stabilization and output regulation are studied for both two kinds of network with systematic PI control designs by using Lyapunov direct method. Then elaborate applications of our Chapter 5. Conclusions and Perspectives 90 control design for Saint-Venant channel models are presented with numerical simulations to evaluate the performance of the control methodology.

Perspectives

There are still many directions to extend and deepen the results obtained in this thesis concerning the control and output regulation of hyperbolic systems.

The first perspective to be mentioned is how to extend the control design in the thesis to the case of inhomogeneous hyperbolic systems, i.e the such systems

∂ ∂t φ(x, t) + A(φ) ∂ ∂x φ(x, t) = B(φ) ,
This is an interesting question because the equilibrium point in this case is no longer constant value but dependent on x, and stability analysis for nonlinear systems is more challenging, in particularly with Lyapunov direct method.

The second perspective worthy studying in the future is how to find a strict Lyapunov function for general hyperbolic systems with dynamic control laws. In fact, in the case of static control law, Coron et al. have proposed a general Lyapunov functional in their works [START_REF] Coron | Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems[END_REF][START_REF] Coron | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF]. And in the thesis, this issue is considered for some special classes of hyperbolic systems as nonlinear scalar systems, MIMO linear 2 × 2 systems, some networks of 2n states with PI control, based on strict Lyapunov functions that contain the coupled terms between states and feedback variables. However, constructing a such Lyapunov functional for general closed-loop systems coupled by PDE/ODE is until an open question.

Another direction that we can deepen from the work in the thesis is to improve the PI control design for hyperbolic systems. The first improvement that could be considered is concerning the optimal value of integral gain of PI controllers. For example one would like to find an exact interval of integral gain μ to guarantee the stability of closed-loop systems, or an optimal value in the sense of minimizing some cost function defined a priori. The second improvement may be how to extend the PI control design in the thesis for the case of inconstant disturbances.

The last perspective is that how to adapt our control design method in the thesis to apply for other types of infinite dimensional systems such as parabolic systems, elliptic systems. This issue is interesting because it allows to enlarge the domain of applications in the practical models.

Faculté

  Techniques des Activités Physiques et Sportives Observatoire des Sciences de l'Univers de Lyon Polytech Lyon Ecole Supérieure de Chimie Physique Electronique Institut Universitaire de Technologie de Lyon 1 Ecole Supérieure du Professorat et de l'Education Institut de Science Financière et d'Assurances Directeur : M. F. DE MARCHI Directeur : M. le Professeur F. THEVENARD Directeur : Mme C. FELIX Directeur : M. Hassan HAMMOURI Directeur : M. le Professeur S. AKKOUCHE Directeur : M. le Professeur G. TOMANOV Directeur : M. le Professeur H. BEN HADID Directeur : M. le Professeur J-C PLENET Directeur : M. Y.VANPOULLE Directeur : M. B. GUIDERDONI Directeur : M. le Professeur E.PERRIN Directeur : M. G. PIGNAULT Directeur : M. le Professeur C. VITON Directeur : M. le Professeur A. MOUGNIOTTE Directeur : M. N. LEBOISNE

  m×n The set of real matrices of dimension m × n C m Class of functions whose derivatives of order of to m are continuous L 2 (a, b) Class of functions that are square integrable on [a, b] H m (a, b) Space of absolutely continuous functions whose derivatives of order of to m are in L 2 (a, b)

σ p i 1 p

 1 and D n,+ is space of all diagonal matrix with positive diagonal terms; here ||σ|| p := n j=1 and 1 p ∞.

Figure 1 . 1 :Figure 1 . 2 :

 1112 Figure 1.1: Example of disturbances in practical model

Figure 1 . 3 :

 13 Figure 1.3: Schema of controlled systems with PI controllers

  purpose of the chapter is to find sufficient conditions on the control parameter k I > 0 such that the three objectives are realized : (a) the closed-loop system (2.2) is well posed; (b) asymptotic stability of the closed-loop system is guaranteed; and (c) the regulation property holds lim t→∞ |y(t)y r | = 0. (2.3) As only the classical solutions are considered, in the following we restrict ourselves to study the solutions from the initial data (ψ 0 , ζ 0 ) in H 2 (0, L) × R which satisfy the C 0 and C 1 compatibility conditions :

  ) = 0 ∀ k ∈ N and k > 0, and lim η→0 H(η) = e -1 . By (2.24) and η = γ + 2kπ,we have σ < 0 if and only

Figure 2 . 1 :Figure 2 . 2 :

 2122 Figure 2.1: Evolution of ψ(x, t)

Figure 2 . 3 :Figure 2 . 4 :Figure 2 . 1

 232421 Figure 2.3: Evolution of error |y(t)y r |

in the H 2

 2 norm. Both of two main proofs have used Lyapunov techniques with the Lyapunov functions in the quadratic form. The regulation of the output to the set-point is automatically guaranteed from the local exponential stability of the closed-loop system in H 2 norm. Numerical simulations for the nonlinear closed-loop system have been carried out to validate the performance of the controlled system. In the future, the work is to extend the design of stabilizing integral controllers for networks of scalar systems governed by nonlinear hyperbolic PDE.

  where x ∈ [0, L] and t ∈ R + are respectively space and time variable. The two state variables are z i : [0, L] × R + → R for i = 1, 2. The matrix G ∈ R 2×2 has two real eigenvalues with opposite signs. Two control inputs on the boundary are U 0 (t) and U L (t), with h 1 : R × R → R and h 2 : R × R → R. The output measurement y(t) ∈ R 2 is determined by two state space variables, the first one evaluated at the boundary x = 0 and the other one at x = L. System (3.1) is a MIMO system and represents for dynamics of many linearized models in industrial engineering. For instance, it can be used as the linearized model of the

4. 4 . 3 ,

 43 we present the main result of the chapter and its proof based on Lyapunov techniques. Section 4.5.2.3 is devoted to numerical simulations in which we apply the PI controller design for a nonlinear fluid flow model. The conclusions are addressed in Section 3.5.

  satisfying the C 0 and C 1 compatibility conditions, the output y(t) is regulated to the desired set-point y r , i.e, lim t→∞ ||y(t)y r || 2 R 2 = 0.

(3. 8 )Chapter 3 .

 83 Multiple input -multiple output 2 × 2 systems with PI control 40

Lemma 3 . 5 .

 35 Let f : [0, L] → R be a smooth function in H 1 (0, L), then there exists a positive real number C such that||f || L ∞ (0,L) C||f || H 1 (0,L) .Employing Lemma 3.4 and Lemma 3.5 for φ 1 (•, t) and φ 2 (•, t), one can find that for∀x ∈ [0, L] lim t→∞ φ 1 (x, t) = 0, lim t→∞ φ 2 (x, t) = 0 , As a result, lim t→∞ ||y(t)y r || 2R 2 = 0. This completes the proof of Theorem 3.1.

Figure 3 . 1 :

 31 Figure 3.1: Saint-Venant model

Figure 3 .2 and figure 3 . 3

 333 show the evolutions of two states: water level H(x, t) and water discharge Q(x,t) respectively. With the simulations, one can conclude about the stability obtained for the closed-loop system with PI controller.

Figure 3 . 5 and

 35 Figure 3.5 and Figure 3.6 present the performance of the PI controller against the unknown constant disturbances. We consider the system with the constant output disturbance w o and the control disturbance w c , that means y(t) = (H(0, t), Q(L, t)) + w o and (U 0 , U L ) = (H(L, t), Q(0, t)) + w c , where w o = (0.1, 0.05) and w c = (0.02, 0.05).

Figure 3 . 2 :Figure 3 . 3 :Figure 3 . 4 :Figure 3 . 5 :

 32333435 Figure 3.2: Evolution of H(x,t)

Figure 3 . 6 :

 36 Figure 3.6: Output Q(L, t) with output disturbance w o and control disturbance w c of closed-loop system (left) and open-loop system (right).

Chapter 4 .

 4 Networks of hyperbolic systems: Stability and Output regulation 57

Figure 4 . 1 :

 41 Figure 4.1: Star-shaped networks of n channels

Chapter 4 . 58 •

 458 Networks of hyperbolic systems: Stability and Output regulation For n -1 inlet systems: (j = 1, n -1)

Chapter 4 . 60 •4. 3 . 3 1

 460331 Networks of hyperbolic systems: Stability and Output regulation If the initial conditions of each inlet systems (ψ j1 (x, 0), ψ j2 (x, 0)) ∈ (H 1 (0, L))2 , then the measured outputs y j (t) are regulated to the desired set-points y jr , i.e lim t→∞ |y j (t)y jr | = 0 Proof of Theorem 4.To prove Theorem 4.1, we construct the following Lyapunov candidate functional for the closed-loop network (4.7):

(4. 11 )Chapter 4 .

 114 Networks of hyperbolic systems: Stability and Output regulation 61 where k I = μ(a + bR 2 e μL ) (4.12)

Chapter 4 .

 4 Networks of hyperbolic systems: Stability and Output regulation 63 definite. From the Sylvester criterion, Q is symmetric positive definite if and only if

Figure 4 . 2 :

 42 Figure 4.2: Cascaded network of n systems

Chapter 4 .

 4 Networks of hyperbolic systems: Stability and Output regulation 69 H3: b

n

  satisfying the C 0 and C 1 compatibility conditions, the n measured outputs y i (t) are regulated to the desired set-points y ir , i.e lim t→∞ |y i (t)y ir | = 0 , ∀i = 1, n.

Chapter 4 .

 4 Networks of hyperbolic systems: Stability and Output regulation 73 with

. 49 )

 49 This leads to the existence of M > 0 such that (4.47) holds. To prove (4.48), we analyze the time derivative of V along the solution of the network of closed-loop systems in (4.35), V(t) = n i=1 p i Vi (t) To simplify the writing in the following, we denote p n+1 , A n+1 and B n+1 such that p n+1 = A n+1 = B n+1 = 0. Now, by employing the property (4.40) of Lemma 4.4, one finds that

(4. 58 )Chapter 4 .

 584 Networks of hyperbolic systems: Stability and Output regulation 78 where λ i1 = gH

Figure 4 . 3 :Figure 4 . 4 :

 4344 Figure 4.3: H 1 (x, t) Star-shaped network

Figure 4 . 5 :Figure 4 . 6 :

 4546 Figure 4.5: H 2 (x, t) Star-shaped network

Figure 4 . 7 :Figure 4 . 8 :Figure 4 . 9 :

 474849 Figure 4.7: H 3 (x, t) Star-shaped network

Figure 4 . 10 :

 410 Figure 4.10: Output measurements y i (t) in Star-shaped network

Figure 4 . 11 :

 411 Figure 4.11: Cascaded network of n fluid flow channels

•

  Length of each channel is L = 100 m, with the same width B = 4 m. Level set-points H * 1 = 10 m, H * 2 = 8 m, H * 3 = 6.5 m, and constant discharge Q 0 = 7 m 3 /s. • Output disturbances w 1o = 0.1, w 2o = 0.2, w 2o = 0.15; and control disturbances w 1c = 0.02, w 2c = 0.03, w 2c = 0.01. Obviously, subcritical conditions are satisfied with above data. Tuning parameter μ is chosen by 0.001. The simulations are based on Preissmann schema with the weighting

Figure 4 .

 4 Figure 4.12: H 1 (x, t)

Figure 4 .

 4 Figure 4.13: Q 1 (x, t)

Figure 4 . 12 -

 412 Figure 4.12-Figure 4.17 verify the stability of linearized closed-loop network and show the evolutions states of the water level and water discharge for each channel controlled by three PI controllers at junctions.In Figure4.18, we see that three output measurements y i (t) = H i (L, t) are regulated to the desired references H * i in spite of output disturbances w io and control disturbances w ic (i = 1, 3).

Figure 4 .

 4 Figure 4.14: H 2 (x, t)

Figure 4 .

 4 Figure 4.15: Q 2 (x, t)

Figure 4 .

 4 Figure 4.16: H 3 (x, t)

Figure 4 .Figure 4 . 18 :

 4418 Figure 4.17: Q 3 (x, t)

  However the Lyapunov functional allows us to deal with nonlinear systems as it will be shown in the next Section.

	Remark : In our Lyapunov functional V (φ, ξ) in 2.25, the new added couple term
	between state φ and dynamic feedback ξ has a key role. To explain its role, it should be
	mentioned the following result whose proof can be found in [65, Theorem 4.7, Chapter
	4] or [21, Theorem 4.1]:
	.33)
	The required inequality (2.28) is true by (2.33) and (2.27).

Remark : It can be noticed that the set of parameter k I which makes the Lyapunov functional decreasing along solutions is smaller than the set of parameter obtained from Proposition 2.2. Hence, in the linear context our Lyapunov approach is conservative.

  53, Theorem 2.2 , p.46]). Notice that K and δ are defined in our Theorem 2.8. Then direct application of Theorem 2.8 allows us to get w(T ) X < Kδ 1 . Since the system is autonomous, the same argument can be used on the time interval [T, 2T ]. By successive iterations we obtain the differential inequality (2.43) satisfied for all t 0. By (2.43)-(2.44) we find positive constants M and ω such that

  the linearized model of Chapter 3. Multiple input -multiple output 2 × 2 systems with PI control 48 (3.21) around the desired set-point is given as follows:

  || (ψ 11 , ψ 12 , • • • , ψ n1 , ψ n2 , ξ 1 , • • • , ξ n-1 ) || E A S

	Moreover, one can find A > 0 such that
	1 A	S (4.21)
	From (4.20) and (4.21), it leads to the exponential stability toward the origin of network
	closed-loop system (4.7) in E.
					.19)
	From (4.17), (4.18) and (4.51), there exits δ > 0 such that
		n-1	⎛ n-1		⎞
		Vj + Vn -δ	⎝	V j + V n	⎠
		j=1	j=1
	This implies that
		Ṡ(t) -δS(t)		(4.20)

  t) + w io(4.29) where a i , b i are unknown constants dependant on chosen outputs and w io ∈ R are unknown constant perturbations.As mentioned in Section 4.2 above, we design n PI controllers located at the junctions with real gain parameters K iP , K iI , and corrupted by some unknown constant control

	disturbances w ic , i.e
		t
		u i (t) = K iP (y i (t) -y ir ) + K iI	(y i (s) -y ir )ds + w ic	(4.30)
		0
	such that the network of closed-loop systems (4.1)-(4.30) is exponentially stable and
	lim t→∞	y i (t) = y ir .

  ψ n2∞ , z n∞ ) defined in (4.32) of network of closed-loop systems (4.31) is exponentially stable in the state space E.

  j can be chosen as an equilibrium state with the appropriate controls U j and Q ic .The objective is to stabilize the linearised model of the network (4.56) and (4.59)-(4.62) to the set-points H * i , V * i and regulate the output measurements of each inlet channel to the desired references y jr = H * j . With the change of coordinates in (4.57), we return to the familiar form in (4.58).Inspired by results in Theorem 4.1, we then rewrite all outputs and boundary conditions in the new coordinates as follow:n -1 measured outputs y j (t) -H * j = φ j1 (0, t) + φ j2 (0, t) (4.63)and 2n -1 boundary conditionsφ j1 (0, t) = R j1 φ j2 (0, t) + u j (t), φ i2 (L, t) = R i2 φ i1 (L, t),Here, to implement control inputs U j , we only need n -1 online measurements y j which are water levels H j (0, t) of inlet channels. It is practical because measure flow discharge in reality is difficult. Now employing the Theorem 4.1, n -1 dynamic feedback PI controllers u j (t) are de-

		n-1 j=1	Q (4.64)
	where R j1 =	λ j2 λ j1
						n-1
						α j φ j1 (L, t) ,	(4.66)
						j=1
	where
			R n1 =	λ n2 λ n1	, α j =	B j (λ j1 + λ j2 ) B n λ jn	.
	signed as follows
		u

Each 2n constant values

H * i , Q * i satisfying the subcritical conditions gB 2 i (H * i ) 3 -(Q * i ) 2 > 0 and the conservation law condition Q * n = * , R i2

are arbitrary constants depending on Q ic , and u j (t) are new dynamic feedback control law deduced from the control inputs U j (t) by

u j (t) = U j (t) -Q * j B j λ j1 (4.65)

The last condition comes from the conservation constraint at the junction (4.62)

φ n1 (0, t) = R n1 φ n2 (0, t) + Remark : j (t) = K jP (y j (t) -H * j ) + K jI L 0

(y j (s) -H * j )ds (4.67)

  Inspired by the result in Theorem 4.3, we design n feedback control laws u i (t) at junctions by the form of PI controllers with unknown control disturbance w ic as followsu i (t) = K iP (y i (t) -H * i ) + K iI t 0 (y i (s) -H * i )ds + w icwhere K iP and K iI are computed byHere, the tuning parameter μ is chosen small enough. Then n PI controllers in (4.74) stabilize the linearized model (4.56) of channel network, and n measured outputs are regulated to the desired references.

	1)2 λ k1 i -2Q 0 (B i gH * Q 0 H * i α i ) + α i u i (t) = (U i (t) -U i ) and n new control inputs with U i = H * i -Q 2 0 -2Q 0 (B i gH * i + Q 0 H * i α K iP = 2Q 0 (B i gH * i -Q 0 ) + α i H * i K iI = -μ (1 + e μL gH * Q 0 i + B i H * i gH * i -Q 0 B i H * i 4Q 0 (B i gH * , ∀k = 2, n ) + α i i ) 2Q 0 (B i gH * i -Q 0 ) + α i H * i	(4.73) , ∀i = 1, n (4.74)

i is the constant equilibrium opening of the ith gate.

Without the compatibility condition, it is possible to show following[START_REF] Coron | Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems[END_REF] and[START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF] that weak solutions exist.

where

with μ is tuning parameter chosen small enough.

Numerical simulations

In this Section, we give numerical simulations for a star-shaped network of three channels with the following data: The evolutions of input with PI controllers are seen in Figure 4.9.

In Figure 4.10, we see that two output measurements y j (t) = H j (0, t) are regulated to the desired references H * j in spite of output disturbances w jo .

Simulation for cascaded network

Modelling of network

In this section, we make simulations for a cascaded network of n horizontal channels in The network is controlled by n underflow gates at the end of each channel, i.e

where α i are positive constant coefficient of gates and U i (t) represent the openings of the gates considering as control inputs.

Appendix A

Numerical simulations of PDE hyperbolic systems with

Preisssmann schema A.1 Introduction

In this Section, a methodology of numerical simulations for models described by hyperbolic partial differential equations (PDE) is presented. The main steps of the simulations are summarized as follows:

• Studying an appropriate scheme for discretizing hyperbolic PDE models. In the thesis, we choose the Preissmann scheme which has been considered in the literature, see in [START_REF] Georges | Automatique pour la gestion des ressources en eau[END_REF], [2], [START_REF] Lyn | Stability of a general Preissmann scheme[END_REF] and [START_REF] Trinh | Pi regulation control of a fluid flow model governed by hyperbolic partial differential equations[END_REF].

• Based on the approximate formulas of Preissmann schema, we compute the discretized expressions of functions and theirs derivatives.

• Formulating the problem under the form of a nonlinear algebraic equation F (X) = 0, where X is the unknown state variable.

• Applying the Newton-Raphson iterative method to find a numerical solution X of above equation. schema 92

A.3 Discretizing hyperbolic PDE models

We consider the following 1D hyperbolic PDE system:

Note that N denotes the number of spatial elements in [0, L]. We discretize the PDE equation (A.4) and boundary equation (A.5) with Preissmann scheme (A.1)-(A.3), one gets that

where control input u k+1 is taken online.

With equations (A.6) and (A.7), we obtain n × N nonlinear equations with n × N unknown variables, which are written in the form as follows

where X = (Y k+1 i ) ∈ R n×N ∀i = 1, N includes all unknown variables at instant k + 1, and an nonlinear function F : R n×N → R n×N .

In the following, we apply the Newton-Raphson method to compute a numerical solution X for the implicit nonlinear equation (A.8).

A.4 Newton-Raphson method

The Newton-Raphson method is a very popular iterative approach used to resolve an implicit nonlinear equation by giving an approximate solution with chosen tolerance.

To construct a solution for nonlinear equation (A.8), we follow the following steps

• Take an initial value X 0 ∈ R n×N for X and a small tolerance ε.

• Make successive iterations by

Appendix A. Numerical simulations of PDE hyperbolic systems with Preisssmann schema 94

where p = 0, 1, 2, ... denotes the iteration index and J(X p ) is the Jacobian matrix of function F computed at X p , i.e , J(X p ) = ∂F ∂X (X p ).

• For each iteration, compute the error vector X p+1 (k) -X p (k) and verify that whether. The algorithm stops when

Then, we set the approximate value for X: