Docteur De L'

Université De

Chiraz Hussein

Kassem Thèse Dirigée Par

Kaïs Ammari

Stéphane Gerbi

Ali Wehbe

M Ahmed

Bchatnia Professeur

M Manuel Gonzalez-Burgos

Professeur

M Kaïs Ammari

M Stéphane Gerbi

Amina Mme

Mortada

M Ali Wehbe

M Abbes

Benaissa Professeur

M Kim

Dang Phung

Ahmad Bchat- Nia

Manuel Gonzàlez-Burgos

Abbes Benaissa

Dan Kim

Amina Phung

THÈSE Pour obtenir le grade de

Keywords: C 0 -Semigroup, Exponential stability, Polynomial stability, Frequency domain method, Multiplier method, Observability inequality, Exact Controllability, HUM method, Bresse system, Finite difference discretization

Stability and controllability

Stabilité et contrôlabilité de quelques systèmes localement couplés

Résumé

Cette thèse est consacrée à l'étude de la stabilité et de la contrôlabilité de quelques systèmes localement couplés. D'abord, nous avons étudié la stabilisation d'un système de deux équations d'ondes couplées par les termes des vitesses avec un seul amortissement localisé et sous des conditions géométriques appropriées. Pour le cas où les ondes se propagent à la même vitesse, nous avons établi un taux de décroissance exponentielle de l'énergie. Cependant, dans le cas physique naturel où les ondes ne se propagent pas à la même vitesse, nous avons montré que notre système n'est pas uniformément stable et nous avons établi le taux de décroissance polynomial optimal de l'énergie.

Après, nous avons traité la contrôlabilité exacte d'un système des équations d'ondes localement couplées. L'outil principal est le résultat de A. Haraux dans [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] par lequel l'inégalité d'observabilité est équivalente à la stabilité exponentielle. Plus précisément, nous avons fourni une analyse complète de la stabilité exponentielle du système dans deux espaces d'Hilbert différents et sous des conditions géométriques convenables. Ensuite, en utilisant la méthode HUM, nous avons prouvé que le système est exactement contrôlable. Nous avons aussi effectué des études numériques pour valider nos résultats théoriques obtenus.

Finalement, nous avons analysé la stabilité d'un système de Bresse avec un amortissement local de type Kelvin-Voigt avec des conditions aux bords Dirichlet ou Dirichlet-Neumann-Neumann. Dans le cas de trois amortissements locaux, nous avons établi un taux de décroissance exponentielle ou polynomiale de l'énergie. Cependant, lorsque les ondes ne sont soumises qu'à un ou deux amortissements et que, dans les conditions aux bords sont de type Dirichlet-Neumann-Neumann, nous avons démontré que le système n'est pas uniformément stable. Dans le cas d'un seul amortissement local, nous avons établi un taux de décroissance polynomiale de l'énergie.

Dans cette thèse, la méthode de domaine fréquentielle et la technique des multiplicateurs ont été utilisées.

Mots-clés

Avant-propos

La théorie du contrôle et de la stabilisation d'un système physique gouverné par des équations mathématiques, en particulier par des EDP, peut être décrit comme étant le processus qui consiste à influer le comportement asymptotique du système pour atteindre un but désiré, principalement par l'utilisation d'un contrôle qui modifie son état final. Cette théorie est appliquée dans un large éventail de disciplines scientifiques et techniques comme la réduction du bruit, la vibration de structures, les vagues et les tremblements de terre sismiques, la régulation des systèmes biologiques comme le système cardiovasculaire humain, la conception des systèmes robotiques, le contrôle laser mécanique quantique, les systèmes moléculaires, etc.

Chapitre 1 Introduction

The theory of stability and controllability of mathematical systems involved from engineering and physicals problems (wave equation, beam equation, Schrödinger equation, plates equation, etc) has recently received the attention of many authors. This thesis treats the stability and exact controllability of some locally coupled systems with different types of internal damping. In practice, it is often not possible to control all the components of the state, either because cost's reasons or technological limitations. Mathematically, this means that some equations of the coupled system are not directly stabilized. Which creates mathematical difficulties, that requires to answer new questions, especially the transmission of the informations (the effect of the control) from the damped equation to the undamped one through the coupling.

For more details about the indirect stabilization or controllability studies of some coupled systems, we refer you to see [START_REF] Alabau-Boussouira | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF][START_REF] Alabau-Boussouira | Indirect internal stabilization of weakly coupled evolution equations[END_REF][START_REF] Alabau-Boussouira | A general formula for decay rates of nonlinear dissipative systems[END_REF][START_REF] Ammari | Stabilization of coupled systems[END_REF][START_REF] Bassam | Polynomial stability of the Timoshenko system by one boundary damping[END_REF][START_REF] Bassam | Stability results of some distributed systems involving Mindlin-Timoshenko plates in the plane[END_REF][START_REF] Wehbe | Exponential and polynomial stability of an elastic bresse system with two locally distributed feedback[END_REF][START_REF] Soufyane | Uniform stabilization for the Timoshenko beam by a locally distributed damping[END_REF][START_REF] Wehbe | Stabilization of the uniform Timoshenko beam by one locally distributed feedback[END_REF][START_REF] Najdi | Weakly locally thermal stabilization of bresse systems[END_REF][START_REF] Kapitonov | Uniform stabilization and exact controllability for a class of coupled hyperbolic systems[END_REF][START_REF] Loreti | Optimal energy decay rate for partially damped systems by spectral compensation[END_REF][START_REF] Ammar-Khodja | Stabilization of the nonuniform Timoshenko beam[END_REF].

Description of the Thesis :

This thesis is devoted to study the stabilization and exact controllability of some locally coupled systems and it contains three chapters.

First, in chapter 2, we study the stability of a system of two wave equations coupled by velocities with only one localized damping and under appropriate geometric conditions. First, we establish the strong stability without geometric conditions. We then study the energy decay rate of our system by distinguishing two cases. The first one is when the waves propagate at same speed. In this case, under appropriate geometric conditions named by Piecewise multiplier geometric condition (PMGC in short), we establish an exponential energy decay rate for usual initial data. Next, in the general case, when the waves are not assumed to propagate at the same speed, we prove the non uniform (exponential) stability and under the same geometric conditions, we establish a polynomial energy decay rate of type 1 t for smooth initial data. Finally, in one space dimension, using the real part of the asymptotic expansion of the eigenvalues of the system, we show that the obtained polynomial decay is optimal.

Next, chapter 3 is devoted to the study of the exact controllability of locally coupled wave

Principal methods used

As the analysis in this thesis is based on the semigroup theory, in this section, we exhibit and talk about many recent results on the strong, exponential and polynomial stability of a C 0 -semigroup that will be used to prove our main results in the next chapters. Next, we present some results about observability and exact controllability. Finally we recall some geometric conditions needed in our work. For more details we refer to [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF][START_REF] Benchimol | A note on weak stabilizability of contraction semigroups[END_REF][START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF][START_REF] Foguel | Powers of a contraction in Hilbert space[END_REF][START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF][START_REF] Nagy | Harmonic Analysis of Operators on Hilbert Space[END_REF][START_REF]I of standards and Digital library of mathematical functions[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF][START_REF] Russell | Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods[END_REF].

Semigroups, Existence and uniqueness of solution

Let (X, • X ) be a Banach space over C and H be a Hilbert space equipped with the inner product < •, • > H and the induced norm • H .

In this subsection, we start by the definition of semigroups since the vast majority of the evolution equations can be reduced to the form

U t (x, t) = AU (x, t), t > 0, U (0) = U 0 ∈ H, (1.1.1)
where A is the infinitesimal generator of a C 0 -semigroup (S(t)) t≥0 in a Hilbert space H.

Definition 1.1.1. A family (S(t)) t≥0 of bounded linear operators on X is a strongly continuous semigroup (in a short, a C 0 -semigroup) if -S(0) = I.

-S(t + s) = S(t)S(s) ∀s, t ≥ 0.

-lim t→0 S(t) -I X = 0.

Definition 1.1.2. The linear operator A defined by

Ax = lim t→0 S(t)x -x t , ∀x ∈ D(A),
where

D(A) = x ∈ X; lim t→0 S(t)x -x t exists
is the infinitesimal generator of the semigroup (S(t)) t≥0 .

Some properties of semigroup and its generator operator A are given in the following theorems :

Theorem 1.1.3. (Pazy [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]) Let (S(t)) t≥0 be a C 0 -semigroup on a Hilbert space H. Then there exist two constants ω ≥ 0 and M ≥ 1 such that S(t) L(H) ≤ M e ωt , ∀t ≥ 0.

If ω = 0, the semigroup (S(t)) t≥0 is called uniformly bounded. Moreover, if M = 1, then it is called a C 0 -semigroup of contractions.

Theorem 1.1.4. If A generates a C 0 -semigroup on H. Then -D(A) = H.

-A is closed.

Definition 1.1.5. An unbounded linear operator (A, D(A)) on X is said to be dissipative if (λI -A) x X ≥ λ x X ∀x ∈ D(A) and ∀λ > 0.

Proposition 1.1.6. Let (A, D(A)) be an unbounded linear operator on H, then

A is dissipative if and only if Ax, x ≤ 0, ∀x ∈ D(A).

Definition 1.1.7. An unbounded linear operator (A, D(A)) on X is said to be maximal dissipative (m-dissipative) if

• A is a dissipative operator.

• ∃ λ 0 such that R(λ 0 I -A) = X, i.e.∀ x ∈ X, ∃ u ∈ D(A) such that λ 0 u -Au = x.

For the existence of solution of problem (1.1.1), we typically use the following Hille-Yosida and Lumer-Phillips theorems from [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] :

Theorem 1.1.8. (Hille-Yosida) An unbounded linear operator (A, D(A)) on X generates a C 0 -semigroup of contractions (S(t)) t≥0 if and only if -A is closed and D(A) = X.

-The resolvent set ρ(A) contains (0, ∞) and ∀ λ > 0, (λI -A) -1 L(X) 1 λ .

Theorem 1.1.9. (Lumer-Phillips) Let (A, D(A)) be an unbounded linear operator on X with dense domain D(A) in X. A is the infinitesimal generator of a C 0 -semigroup of contractions (S(t)) t≥0 if and only if it is a m-dissipative operator.

Corollary 1.1.10. Let (A, D(A)) be an unbounded linear operator on H. A is the infinitesimal generator of a C 0 -semigroup of contractions (S(t)) t≥0 if and only if it is a m-dissipative operator.

Consequently, the existence of solution is justified by the following corollary which follows from Lumer-Phillips theorem. -For U 0 ∈ H, the problem (1.1.1) has a unique weak solution U (t) ∈ C([0, +∞), H).

Stability of semigroups

In order to show the strong stability of a C 0 -semigroup, we apply the next theorem due to Arendt and Batty in [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF].

Theorem 1.1.12. Assume that A is the infinitesimal generator of a strongly continuous semigroup of contractions (S(t)) t≥0 on X. If A has no pure imaginary eigenvalues and if σ(A) ∩ iR is countable, then (S(t)) t≥0 is strongly stable. Now, when the C 0 -semigroup is strongly stable, we look for a necessary and sufficient conditions for which a semigroup is exponentially stable. We recall here only the following frequency domain approach method obtained by Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Prüss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF].

Theorem 1.1.13. (Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Prüss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]) Let (S(t)) t≥0 be a C 0 -semigroup of contractions on H and A be its infinitesimal generator. Then, (S(t)) t≥0 is exponentially stable if and only if -iR ⊆ ρ(A), -lim sup β∈R,|β|→∞

(iβI -A) -1 L(H) < ∞.
Since some studied systems in this thesis do not achieve the exponential stability, therefore we look for a polynomial one. In general, polynomial stability results are obtained using different methods like : multipliers method, frequency domain approach, Riesz basis approach, Fourier analysis or a combination of them (see [START_REF] Komornik | Exact Controllability and Stabilization : The Multiplier Method[END_REF][START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF][START_REF] Littman | Stabilization of a hybrid system of elasticity by feedback boundary damping[END_REF]). In this thesis, we recall only the frequency domain approach obtained by Borichev-Tomilov in [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF]Theorem 2.4].

Theorem 1.1.14. (Borichev-Tomilov [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] ) Let (S(t)) t≥0 be a bounded C 0 -semigroup of contractions on H generated by A. If iR ⊂ ρ(A), then for a fixed l > 0, the following conditions are equivalent :

-lim sup β→+∞β∈R 1 |β| l (iβI -A) -1 L(H) < +∞.

-S(T )U 0 H ≤ c t l -1 U 0 D(A) , ∀t > 0, ∀U 0 ∈ D(A), for some c > 0.

Finally, in order to study the optimality of the obtained decay rate, we refer to a Theorem 3.4.1 in [START_REF] Najdi | Étude de la stabilisation exponentielle et polynomiale de certains systèmes d'équations couplées par des contrôles indirects bornés ou non bornés[END_REF].

Theorem 1.1.15. (Wehbe, Najdi 2016) Let A be the infinitesimal generator of a C 0semigroup of contractions (S(t)) t 0 . Let (λ k,n ) the eigenvalues of A and (e k,n ) eigenvectors. Assume that there exist µ k,n → +∞, α k > 0, β k > 0 such that (λ k,n ) ∼ -

β k µ α k k,n , | (λ k,n )| ∼ µ k,n
, iR ⊂ ρ(A) and for any u 0 ∈ D(A), there exists constant M > 0 such that S(t)u 0 H ≤ M t

1 l k u 0 D(A) , l k = max 1≤k≤k 0 (α k ), ∀t > 0. (1.1.2)
Then the decay rate (1.1.2) is optimal.

Observability and exact controllability

In this part, we present briefly the duality between the notion of observability and controllability, which lies at the basis of the Hilbert uniqueness method (HUM) of J .L. Lions [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF]. First, we consider the following system :

   U t (x, t) = AU (x, t) + Bv(x), on Ω × (0, +∞), U (x, 0) = U 0 (x), (1.1.3)
where Ω ⊂ R d (d ∈ N * ), U is a scalar or vector-valued function, A is a set of partial differential operators, linear or non linear (at least for the time being), v denotes the control and B maps the "space of controls" into the "state space". The partial differential equation (1.1.3) should include boundary conditions. We do not make them explicit here.

They are supposed to be contained in the abstract formulation (1.1.3).

The control v can be either applied inside the domain Ω (in this case v is said to be internal control), or on the boundary Γ of Ω or on part of it (in this case v is said to be a boundary control). If v is applied at points of Ω, v is said to be pointwise control.

It will be assumed that, given v (in a suitable space), problem (1.1.3) uniquely defines a solution. This solution is a function (scalar or vector-valued) of x ∈ Ω, t > 0 and of U 0 and v. Now, we can introduce the notion of controllability, either exact or approximate. Let T > 0 be given and let U T (the target function) be a given element of the state space. We want to "drive the System" from initial state U 0 at t = 0 to final state U T at t = T , that is, we want to find a suitable control v such that

U (x, T ) = U T (x), x ∈ Ω.
If this is possible for any target function U T in the state place, one can say that the System is controllable (or exactly controllable). For more details for controllability see [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF].

In [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], J. L. Lions introduce the Hilbert Uniqueness Method (HUM) to solve controllability problems for linear partial differential equations. This method is closely related to duality between controllability and observability.

We now present a result of A. Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] to get the observability inequality. If A is an unbounded, self adjoint, positive and coercive linear operator on a Hilbert space H and B a bounded linear operator on H such that B = B * ≥ 0, he established a logical equivalence between the exponential decay of solutions of the second order evolution equation U tt + AU + BU t = 0, uniformly on bounded subsets of D(A 1/2 ) × H and a "B 1/2controllability" property of the system governed by the undamped equation ϕ tt + Aϕ = 0 on some time interval (see Proposition 1 and 2 in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF]).

Remark 1.1.16. The result of A. Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] still valid for a first order evolution equation.

Geometric conditions

This part is devoted to recall some geometric conditions since we shall use them along our work. We begin by reviewing the Geometric Control Conditions GCC introduced by Rauch and Taylor in [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF] for manifolds without boundaries and by Bardos, Lebeau and Rauch in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] for domains with boundaries. Definition 1.1.17. We say that a subset ω of Ω satisfies the GCC if every ray of the geometrical optics starting at any point x ∈ Ω at t = 0 enters the region ω in finite time T.

Next, we recall the Piecewise Multipliers Geometric Condition introduced by K. Liu in [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF].

Definition 1.1.18. We say that ω satisfies the Piecewise Multipliers Geometric Condition (PMGC in short) if there exist Ω j ⊂ Ω having Lipschitz boundary Γ j = ∂Ω j and x j ∈ R N , j = 1, ..., J such that Ω j ∩ Ω i = ∅ for j = i and ω contains a neighborhood in Ω of the set ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j where γ j (x j ) = {x ∈ Γ j : (x -x j ) • ν j (x) > 0} and ν j is the outward unit normal vector to Γ j .

Remark 1.1. [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF]. The PMGC is the generalization of the Multipliers Geometric Condition (MGC in short) introduced by Lions in [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], saying that ω contains a neighborhood in Ω of the set {x ∈ Γ : (x -x 0 ) • ν(x) > 0}, for some x 0 ∈ R N , where ν is the outward unit normal vector to Γ = ∂Ω. However, the PMGC is much more restrictive than the GCC.

Chapter 2 : Local Indirect Stabilization of two coupled wave equations under geometric conditions

Let Ω be a non empty open bounded domain of R N with boundary Γ of class C 2 . In chapter 2, we consider the following coupled wave equation :

u tt -a∆u + c(x)u t + b(x)y t = 0 in Ω × R + , (1.1.4) y tt -∆y -b(x)u t = 0 in Ω × R + , (1.1.5) u = y = 0 on Γ × R + , (1.1.6)
with the following initial data :

u(x, 0) = u 0 , y(x, 0) = y 0 , u t (x, 0) = u 1 and y t (x, 0) = y , x ∈ Ω, where a > 0 constant and b(x) ∈ C 0 (Ω; R) is a non-zero function. The damping term c(x) ∈ C 0 (Ω, R + ) is only applied at the first equation and the second equation is indirectly damped through the coupling between the two equations. This type of indirect control was introduced by D.L. Russel [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF] and since this time, it attracted the attention of many authors.

Preceding results :

In [START_REF] Kapitonov | Uniform stabilization and exact controllability for a class of coupled hyperbolic systems[END_REF], B. Kapitonov studied the stability of system (1.1.4)- (1.1.6) in the case when the support of b coincide with the support of c. When the waves propagate at the same speed (i.e. a = 1), he established an exponential decay of the energy. While when the waves propagate at different speeds, no decay rate was discussed.

In [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF], F. Alabau-Boussouira et al. considered the energy decay of the following system :

u tt -a∆u + ρ(x, u t ) + b(x)y t = 0 in Ω × R * + , (1.1.7) y tt -∆y -b(x)u t = 0 in Ω × R * + , (1.1.8) u = y = 0 on Γ × R * + , (1.1.9)
where a > 0 constant, b ∈ C 0 (Ω, R) and ρ(x, u t ) is a non linear damping. Using an approach based on multiplier techniques, weighted nonlinear inequalities and the optimalweight convexity method (developed in [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF]), the authors established an explicit energy decay formula in terms of the behavior of the nonlinear feedback close to the origin in the case that the three following conditions are satisfied : the waves propagate at the same speed (a = 1) and the coupling coefficient b(x) is small positive (0 ≤ b(x) ≤ b 0 , b 0 ∈ (0, b ] where b is a constant depending on Ω and on the control region) and both the coupling and the damping regions satisfying an appropriated geometric conditions named Piecewise Multipliers Geometric Conditions (PMGC, in short). But the contrary case, when the waves are not assumed to propagate with equal speed ( a is not necessarily equal to 1) and/or b is not assumed to be small and positive has been left as open question even in the linear case i.e. ρ(x, u t ) = c(x)u t . This open question will be our target in chapter 2 in the linear case. The main novelty in this chapter is that the waves are not necessarily propagating at the same speed and the coupling coefficient is not assumed to be positive and small. First, we begin to study the existence, uniqueness and regularity of the solution of our system using the semigroup approach. Let (u, u t , y, y t ) be a regular solution of the system (1.1.4)- (1.1.6), its associated total energy is defined by

E(t) = 1 2 Ω |u t | 2 + a|∇u| 2 + |y t | 2 + |∇y| 2 dx.
(1.1.10)

A direct computation gives

d dt E(t) = - Ω c(x)|u t | 2 dx ≤ 0.
(1.1.11)

Consequently, system (1.1.4)-(1.1.6) is dissipative in the sense that its energy is nonincreasing with respect to the variable time t. Next, we define the energy space H = ( H 1 0 (Ω) × L 2 (Ω)) 2 equipped, for all U = ( u, v, y, z), U = ( u, v, y, z) ∈ H, by the scalar product :

(U, U ) H = a Ω (∇u • ∇ u)dx + Ω v vdx + Ω (∇y • ∇ y)dx + Ω z zdx.
Setting U = (u, u t , y, y t ), system (1.1.4)-(1.1.6) may be rewritten as : U t = AU, in (0, +∞), U (0) = (u 0 , u 1 , y 0 , y 1 ), where the unbounded operator A : D(A) ⊂ H → H is given by :

D(A) = (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) 2
(1.1.12) and AU = ( v, a∆u -bz -cv, z, ∆y + bv ), ∀ U = (u, v, y, z) ∈ D(A).

(1.1.13)

The operator A is m-dissipative in H and generates a C 0 -semigroup of contractions (e tA ) t≥0 . So, system (1.1.4)-(1.1.6) is well posed in H.

Then, we move to study the asymptotic behavior of E(t). For this aim, we assume that there exists a non empty open ω c + ⊂ Ω satisfying the following condition We first prove that our system is strongly stable without geometric condition. This is given by the following theorem : Now, we study the energy decay rate by using a frequency domain approach combined with piecewise multiplier technique in two cases : the first one when the waves are assumed to propagate at the same speed (i.e. a = 1) and the second case when a = 1.

{x ∈ Ω : c(x) > 0} ⊃ ω c + . (
The first main result is the following one : Then there exist positive constants M ≥ 1, θ > 0 such that for all initial data (u 0 , u 1 , y 0 , y 1 ) ∈ H the energy of the system (1.1.4)-(1.1.6) satisfies the following decay rate :

E(t)
≤ M e -θt E(0), ∀t > 0.

(1.1.15)

Remark 1.1.22. Note that in the previous theorem we have no restriction on the upper bound and the sign of the function b. This theorem is then a generalization in the linear case of the result of [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] where the coupling coefficient considered have to satisfy 0 ≤ b(x) ≤ b 0 , b 0 ∈ (0, b ] where b is a constant depending on Ω and on the control region. Nevertheless, the problem still be open in the nonlinear case.

The condition of equal speed propagation is a necessary and sufficient condition for the exponential stability of our system. In fact, in the case a = 1, we construct a sequence (U n ) of elements in D(A) and a real sequence (µ n ) such that U n = 1 and (iµ n I -A)U n H → 0. Hence, the resolvent of A is not uniformly bounded on the imaginary axis. Following a result of Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Prüss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF] we conclude that the semigroup (e tA ) t≥0 is not uniformly stable in H. So it is natural to look for a polynomial decay of the energy. Consequently, our second main result when the wave propagate at different speeds (a = 1) can be stated as follows : Then there exists a positive constant C > 0 such that for all initial data (u 0 , u 1 , y 0 , y 1 ) ∈ D(A) the energy of the system (1.1.4)-(1.1.6) satisfies the following polynomial decay rate : Finally, in one space dimension (i.e. N = 1), a = 1 and b is a constant, we prove that there exist n 0 ∈ N sufficiently large and a sequence λ n of simple eigenvalues of the operator A satisfying the following asymptotic behavior

E(t) ≤ C 1 t U (
λ n = inπ - ib 2 2(a -1)nπ - cb 2 2(a -1) 2 n 2 π 2 + O 1 n 3 , ∀ |n| ≥ n 0 .
(1.1.17)

It follows that the obtained polynomial decay rate is optimal (see Theorem 3.4.1 in [START_REF] Najdi | Étude de la stabilisation exponentielle et polynomiale de certains systèmes d'équations couplées par des contrôles indirects bornés ou non bornés[END_REF]).

Chapter 3 : Exact controllability and stabilization of locally coupled wave equations

The aim of this chapter is to investigate the exact controllability of the following system :

u tt -a∆u + b(x)y t = c(x)v(t) in Ω × R * + , (1.1.18) y tt -∆y -b(x)u t = 0 in Ω × R * + , (1.1.19) u = y = 0 on Γ × R * + , (1.1.20)
with the following initial data

u(x, 0) = u 0 , y(x, 0) = y 0 , u t (x, 0) = u 1 and y t (x, 0) = y 1 , x ∈ Ω, (1.1.21)
under appropriate geometric conditions. Here, a > 0 constant, b ∈ C 0 (Ω, R), c ∈ C 0 (Ω, R + ) and v is an appropriate control. The idea is to use a result of A. Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] for which the observability of the homogeneous system associated to (1.1.18)-(1.1.20) is equivalent to the exponential stability of system (1.1.4)-(1.1.6) in an appropriate Hilbert space. So, we provide a complete analysis for the exponential stability of system (1.1.4)-(1.1.6) in different Hilbert spaces.

Preceding results :

In chapter 2, we studied the stabilization of system (1.1.4)-(1.1.6) in two cases. In the first one, when the waves are assumed propagating at the same speed (i.e. a = 1), under the assumption that the coupling region and the damping region have a non empty intersection and satisfying the PMGC condition. In this case, we established an exponential decay rate for weak initial data. On the contrary (i.e. a = 1 ) we first proved the lack of the exponential stability of the system. However, under the same geometric condition, an optimal energy decay rate of type 1 t was established for smooth initial data. Our aim in this chapter is to prove the exponential stability of system (1.1.4)- (1.1.6) in two different Hilbert spaces by using geometric conditions more general than that used in chapter 2. And consequently, by using Proposition 2 of A. Haraux [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF], we obtain the observability of the homogeneous system associated to (1.1.18)- (1.1.20).

Principal results of the chapter.

First, we need to study the asymptotic behavior of E(t) associated to (1.1.4)-(1.1.6) and given by equation (1.1.10). For this aim, we suppose that there exists a non empty open ω c + ⊂ Ω satisfying the following condition lim t→+∞ e tA (u 0 , u 1 , y 0 , y 1 ) H = 0 ∀(u 0 , u 1 , y 0 , y 1 ) ∈ H.

{x ∈ Ω : c(x) > 0} ⊃ ω c + . ( LH1
Then, when the waves propagate at the same speed (i.e., a = 1), under the condition that the coupling region includes in the damping region and satisfying the called Geometric Control Condition (GCC in Short), we establish the exponential stability of system (1.1.4)- (1.1.6) given by the following theorem Theorem 1.1.24. (Exponential decay rate) Let a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω b ⊂ ω c + satisfies the geometric control conditions GCC and that b, c ∈ W 1,∞ (Ω). Then there exist positive constants M ≥ 1, θ > 0 such that for all initial data (u 0 , u 1 , y 0 , y 1 ) ∈ H the energy of the system (1.1.4)-(1.1.6) satisfies the following decay rate :

E(t) ≤ M e -θt E(0), ∀t > 0. (1.1.22)
Remark 1.1.25. The geometric situations covered by Theorem 1.1.24 are richer than that considered in Chapter 2 and [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF]. Indeed, in the previous references, the authors consider the PMGC geometric conditions that are more restrictive than GCC. On the other hand, unlike the results in [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF], we have no restriction in Theorem 1.1.24 on the upper bound and the sign of the coupling function coefficient b. This theorem is then a generalization in the linear case of the result of [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] where the coupling coefficient considered have to satisfy

0 ≤ b(x) ≤ b 0 , b 0 ∈ (0, b ]
where b is a constant depending on Ω and on the control region.

Consequently, using Proposition 2 of A. Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF], an observability inequality of the solution of the homogeneous system associated to (1.1.18)- (1.1.20) in the space (H 1 0 (Ω) × L 2 (Ω))

2
is established. This leads, by the HUM method introduced by Lions in [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], to the exact controllability of system (1.

1.18)-(1.1.20) in the space (H -1 (Ω) × L 2 (Ω)) 2 .
Furthermore, on the contrary when the waves propagate at different speeds, (i.e., a = 1), For this, we introduce the following weak energy space

D = H 1 0 (Ω) × L 2 (Ω) × L 2 (Ω) × H -1 (Ω), equipped with the scalar product (U, Ũ ) = Ω (a∇u.∇ũ + vṽ + y ỹ + (-∆) -1/2 z(-∆) -1/2 z)dx,
for all U = (u, v, y, z) ∈ D and Ũ = (ũ, ṽ, ỹ, z) ∈ D. Next, we define the unbounded linear operator

A d : D(A d ) ⊂ D → D by A d U = ( v, a∆u -bz -cv, z, ∆y + bv ), D(A d ) = (H 1 0 (Ω) ∩ H 2 (Ω)) × H 1 0 (Ω) × H 1 0 (Ω) × L 2 (Ω) , ∀ U = (u, v, y, z) ∈ D(A d ).
Its total mixed energy is defined by

E m (t) = 1 2 a ∇u 2 L 2 (Ω) + u t 2 L 2 (Ω) + y t 2 H -1 (Ω) + y 2 L 2 (Ω) .
Then, we move to study the asymptotic behavior of E m (t). For this aim, we need to assume that ω c + satisfies the geometric conditions PMGC, then there exist ε > 0, subsets Ω j ⊂ Ω, j = 1, ..., J, with Lipschitz boundary Γ j = ∂Ω j and points

x j ∈ R N such that Ω i ∩Ω j = ∅ if i = j and ω + c ⊃ N ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j ∩Ω with N (O) = {x ∈ R N : d(x, O) < ε} where O ⊂ R N , γ j (x j ) = {x ∈ Γ j : (x -x j ) • ν j (x) > 0}
where ν j is the outward unit normal vector to Γ j and that ω b satisfies the GCC condition and

ω b ⊂ Ω \ ∪ J j=1 Ω j . (LH3)
Now, our second main result when the waves propagate at different speed (i.e. a = 1) can be stated as follows :

Theorem 1.1.26. (Exponential decay rate) Let a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω c + satisfies the geometric conditions PMGC, ω b satisfies GCC condition and (LH3) and b, c ∈ L ∞ (Ω). Then there exist positive constants M ≥ 1, θ > 0 such that for all initial data (u 0 , u 1 , y 0 , y 1 ) ∈ D the energy of system (1.1.4)-(1.1.6) satisfies the following decay rate :

E m (t) ≤ M e -θt E m (0), ∀t > 0. (1.1.23)
Consequently, using Proposition 2 of A. Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF], an observability inequality of the solution of the homogeneous system associated to (1.1.18)-(1.1.20) is established. This leads, by the HUM method, to the exact controllability of system (1.

1.18)-(1.1.20) in the space L 2 (Ω) × H -1 (Ω) × H -1 (Ω) × (H 2 (Ω) ∩ H 1 0 (Ω))
, where the duality is according to L 2 (Ω). Finally, we perform numerical tests in the 1-D case to insure the theoretical results obtained here and in chapter 2. In fact, the numerical results show a better behavior that the one expected by the theoretical results.

Chapter 4 : Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface

This chapter is devoted to study the stability of an elastic Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface under fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. The system defined on (0, L)×(0, +∞) is governed by the following partial differential equations :

           ρ 1 ϕ tt -[k 1 (ϕ x + ψ + lw) + D 1 (ϕ xt + ψ t + lw t )] x -lk 3 (w x -lϕ) -lD 3 (w xt -lϕ t ) = 0, ρ 2 ψ tt -[k 2 ψ x + D 2 ψ xt ] x + k 1 (ϕ x + ψ + lw) + D 1 (ϕ xt + ψ t + lw t ) = 0, ρ 1 w tt -[k 3 (w x -lϕ) + D 3 (w xt -lϕ t )] x + lk 1 (ϕ x + ψ + lw) + lD 1 (ϕ xt + ψ t + lw t ) = 0, (1.1.24) The coefficients ρ 1 , ρ 2 , k 1 , k 2 ,

Preceding results :

Kelvin-Voigt material is a viscoelastic structure having properties of both elasticity and viscosity. The Kelvin-Voigt damping can be globally or locally distributed. But the case we are interested in is when it is localized on an arbitrary subinterval of the domain. The regularity and stability properties of a solution depend on the properties of the damping coefficients. Indeed, the system is more effectivelly controled by the local Kelvin-Voigt damping when the coefficient changes more smoothly near the interface. Recently, X. Tian and Q. Zhang in [START_REF] Tian | Stability of a timoshenko system with local kelvin-voigt damping[END_REF] considered the following Timoshenko system defined on (0, L) × (0, +∞) with fully Dirichlet boundary conditions :

   ρ 1 ϕ tt -[k 1 (ϕ x + ψ) x + D 1 (ϕ xt -ψ t )] x = 0, ρ 2 ψ tt -(k 2 ψ x + D 2 ψ xt ) x + k 1 (ϕ x + ψ) x + D 1 (ϕ xt -ψ t ) = 0.
(1.1.25) They studied this system with locally or globally distributed Kelvin-Voigt damping when coefficient functions D 1 , D 2 ∈ C ([0, L]). First, when the damping is globally distributed, they showed that the Timoshenko system (1.1.25) under fully Dirichlet boundary conditions is analytic stable. Next, when the damping are locally distributed near the boundary, they analyzed the exponential or polynomial stability according to the properties of coefficient functions D 1 , D 2 . Unlike the results of [START_REF] Tian | Stability of a timoshenko system with local kelvin-voigt damping[END_REF], in this chapter, we studied the Bresse system (1.1.24) subjected to either the fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions and in the case of Kelvin-Voigt dampings localized on any arbitrary subinterval of the domain.

Principal results of the chapter :

First, we begin to study the well-posedness of our system using the semigroup approach. Let (ϕ, ϕ t , ψ, ψ t , w, w t ) be a regular solution of system (1.1.24), its total energy is defined 17 Chapitre 1. Introduction by

E (t) = 1 2 L 0 ρ 1 |ϕ t | 2 + ρ 2 |ψ t | 2 + ρ 1 |w t | 2 + k 1 |ϕ x + ψ + lw| 2 dx + L 0 k 2 |ψ x | 2 + k 3 |w x -lϕ| 2 dx . (1.1.26)
Hence a straightforward computations gives

E (t) = - L 0 D 1 |ϕ xt + ψ t + lw t | 2 + D 2 |ψ xt | 2 + D 3 |w xt -lϕ t | 2 dx ≤ 0. (1.1.27)
Thus, the system (1.1.24) is dissipative in the sense that its energy is non-increasing with respect to the variable time t. Next, we define the following energy spaces :

H 1 = H 1 0 × L 2 3 and H 2 = H 1 0 × L 2 × H 1 * × L 2 * 2 ,
where

L 2 * = {f ∈ L 2 (0, L) : L 0 f (x)dx = 0} and H 1 * = {f ∈ H 1 (0, L) : L 0 f (x)dx = 0}.
We define the unbounded linear operators A j in H j , j = 1, 2 by

D (A 1 ) = U ∈ H 1 | v 2 , v 4 , v 6 ∈ H 1 0 (0, L) , [k 1 (v 1 x + v 3 + lv 5 ) + D 1 (v 2 x + v 4 + lv 6 )] x ∈ L 2 (0, L) , [k 2 v 3 x + D 2 v 4 x ] x ∈ L 2 (0, L), [k 3 (v 5 x -lv 1 ) + D 3 (v 6 x -lv 2 )] x ∈ L 2 (0, L) , k 1 (v 1 x + v 3 + lv 5 ) + D 1 (v 2 x + v 4 + lv 6 ) x + lk 3 (v 5 x -lv 1 ) + lD 3 (v 6 x -lv 2 ) v 4 ρ -1 2 (k 2 v 3 x + D 2 v 4 x ) x -k 1 v 1 x + v 3 + lv 5 -D 1 (v 2 x + v 4 + lv 6 ) v 6 ρ -1 1 k 3 (v 5 x -lv 1 ) + D 3 (v 6 x -lv 2 ) x -lk 1 v 1 x + v 3 + lv 5 -lD 1 (v 2 x + v 4 + lv 6 )              for all U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) T ∈ D (A j ).
Setting U = (ϕ, ϕ t , ψ, ψ t , w, w t ) T , system (1.1.24) may be rewritten as :

U t = A j U, j = 1, 2 U = (ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , w 0 , w 1 ) T .
Then, we prove that the operator A j is m-dissipative in the energy space H j . Therefore, thanks to Lumer-Phillips Theorem, we deduce that A j generates a C 0 -semigroup of contractions e tA j in H j , and then the problem is well posed in H j . Later, using a general criteria of Arendt and Batty [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF], we show that the C 0 -semigroup e tA j is strongly stable in the absence of the compactness of the resolvent of A j and in the presence of at least one local Kelvin-Voigt damping. Now, we need to study the energy decay rate by using a frequency domain approach combined with piecewise multiplier technique in several cases depending on the regularity of the coefficient functions D 1 , D 2 , D 3 , the localization of their supports and their numbers.

In the case when the three dampings are globally distributed, we prove an analytic stability. Then, in the case when the positive continuous functions D i , i = 1, 2, 3 satisfy the following condition :

∃ d 0 > 0 such that D i ≥ d 0 > 0 for every x ∈ (α, β) , 0 < α < β < L, (1.1.28)
we establish the uniform stability of the C 0 -semigroup e tA j given by the following theorem :

Theorem 1.1.27. (Exponential decay rate) Assume that (1.1.28) is satisfied and D 1 , D 2 and D 3 ∈ W 1,∞ (0, L). The C 0 -semigroup e tA j is exponentially stable, i.e., there exist constants M ≥ 1 and > 0 independent of U 0 such that

e tA j U 0 H j ≤ M e -t U 0 H j , t ≥ 0, j = 1, 2.
While in the case when the positive functions D i ∈ L ∞ (0, L), i = 1, 2, 3 satisfy the following condition :

ω = suppD 1 ∩ suppD 2 ∩ suppD 3 = (α, β) ⊂ (0, L) such that mes(ω) > 0, (1.1.29)
we establish a polynomial decay given by the following theorem : Assume also that D 1 , D 2 and D 3 ∈ L ∞ (0, L). Then, there exists a positive constant c > 0 such that for all U 0 ∈ D(A j ), j = 1, 2, the energy of the system satisfies the following decay rate : is not exponentially stable. Then, we prove that this system is also not exponentially stable under the following hypothesis

E(t) ≤ c t U 0 2 D(A j ) . ( 1
D 1 = D 3 = 0 and D 2 = 1 on (0, L). (1.1.32)
In fact, to prove the non uniform stability, we construct a sequence (V n ) of elements in D(A 2 ) and a real sequence (λ n ) such that V n → +∞ and (iλ n I -A 2 )U n H 2 is bounded as n → +∞. Hence, the resolvent of A 2 is not uniformly bounded on the imaginary axis.

Following Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Pruss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF] we conclude that the semigroup e tA 2 is not uniformly stable in H 2 .

Finally, in the case of one local damping, i.e. we assume that there exists d 2 > 0 such that

D 1 = D 3 = 0 in (0, L) and D 2 ≥ d 2 > 0 in (α, β) ⊂ (0, L), (1.1.33)
we prove the following theorem : Assume also that D 2 ∈ L ∞ (0, L). Then, there exists a positive constant c > 0 such that for all U 0 ∈ D(A j ), j = 1, 2 the energy of system (1.1.24) satisfies the following decay rate :

E(t) ≤ c √ t U 0 2 D(A j ) . (1.1.34) 1.2 Introduction in French 1.2.

Principales méthodes utilisées

Le fait que l'analyse de cette thèse est basée sur la théorie de semigroupe, dans cette partie, nous exposons et discutons de nombreux résultats récents sur la stabilité forte, exponentielle et polynomiale d'un C 0 -semigroupe qui servira à prouver nos résultats dans les chapitres suivants. Ensuite, nous présentons quelques résultats sur l'observabilité et la contrôlabilité exacte. Enfin, nous rappelons certaines conditions géométriques nécessaires à notre travail. Pour plus de détails nous nous référons à [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF][START_REF] Benchimol | A note on weak stabilizability of contraction semigroups[END_REF][START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF][START_REF] Foguel | Powers of a contraction in Hilbert space[END_REF][START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF][START_REF] Nagy | Harmonic Analysis of Operators on Hilbert Space[END_REF][START_REF]I of standards and Digital library of mathematical functions[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF][START_REF] Russell | Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods[END_REF].

Semigroupes, Existence et Unicité de la solution

Soit (X, • X ) un espace de Banach sur C et H un espace de Hilbert muni du produit scalaire < •, • > H et la norme induite • H . La majorité des équations d'évolution peuvent être réduites sous la forme

U t (x, t) = AU (x, t), t > 0, U (0) = U 0 ∈ H, (1.2
(λI -A) x X ≥ λ x X ∀x ∈ D(A) et ∀λ > 0.
Proposition 1.2.4. Soit (A, D(A)) un opérateur linéaire non borné sur H, alors

A est dissipatif si et seulement si Ax, x ≤ 0, ∀x ∈ D(A). Définition 1.2.5. Un opérateur linéaire non borné (A, D(A)) sur X est dit maximal dissipatif (m-dissipatif ) si • A est un opérateur dissipatif. • ∃ λ 0 tel que R(λ 0 I -A) = X, i.e.∀ x ∈ X, ∃ u ∈ D(A) tel que λ 0 u -Au = x.
Pour l'existence de la solution du problème (1.2.1), nous utilisons typiquement les théorèmes de Hille-Yosida et de Lumer-Phillips suivants de [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] :

Théorème 1.2.3. (Hille-Yosida) Un opérateur linéaire non borné (A, D(A)) sur X engendre un C 0 -semigroupe de contractions (S(t)) t≥0 si et seulement si

-A est fermé et D(A) = X. -L'ensemble résolvant ρ(A) contient (0, ∞) et ∀ λ > 0, (λI -A) -1 L(X) 1 λ .
U (t) = S(t)U 0 ∈ C([0, +∞), D(A)) ∩ C 1 ([0, +∞), H).
-Pour U 0 ∈ H, le problème (1.2.1) admet une solution unique faible

U (t) ∈ C([0, +∞), H).

Stabilité des semigroupes

Afin de démontrer la stabilité forte d'un C 0 -semigroupe, nous appliquons le théorème suivant dû à Arendt et Batty dans [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF].

(iβI -A) -1 L(H) < ∞.
Puisque certains systèmes étudiés dans cette thèse n'atteignent pas la stabilité exponentielle, nous cherchons donc un taux polynomial. En général, les résultats de la stabilité polynomiale sont obtenus à l'aide de différentes méthodes telles que : méthode des multiplicateurs, méthode fréquentielle, base de Riesz, analyse de Fourier ou une combinaison de ces méthodes (Voir [START_REF] Komornik | Exact Controllability and Stabilization : The Multiplier Method[END_REF][START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF][START_REF] Littman | Stabilization of a hybrid system of elasticity by feedback boundary damping[END_REF]). Dans cette thèse, nous utilisons la méthode fréquentielle obtenue par Borichev-Tomilov dans [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF]Theorem 2.4].

Théorème 1.2.7. (Borichev-Tomilov [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] ) Soit (S(t)) t≥0 un C 0 -semigroupe de contractions borné sur H engendré par A. Si iR ⊂ ρ(A), alors pour l > 0 fixé, les conditions suivantes sont équivalentes :

-lim sup β∈R,|β|→∞

1 |β| l (iβI -A) -1 L(H) < +∞. -S(T )U 0 H ≤ c t l -1 U 0 D(A) , ∀t > 0, ∀U 0 ∈ D(A), pour certains c > 0.
Enfin, pour étudier l'optimalité du taux de décroissance obtenu, nous nous référons au Théorème 3.4.1 de [START_REF] Najdi | Étude de la stabilisation exponentielle et polynomiale de certains systèmes d'équations couplées par des contrôles indirects bornés ou non bornés[END_REF].

k,n → +∞, α k > 0, β k > 0 tel que (λ k,n ) ∼ - β k µ α k k,n , | (λ k,n )| ∼ µ k,n , iR ⊂ ρ(A) et pour tout u 0 ∈ D(A), il existe une constant M > 0 tel que S(t)u 0 H ≤ M t 1 l k u 0 D(A) , l k = max 1≤k≤k 0 (α k ), ∀t > 0. (1.2.2)
Alors le taux de décroissance (1.2.2) est optimal.

Observabilité et contrôlabilité exacte

Dans cette partie, nous présentons brièvement la dualité entre les notions d'observabilité et de contrôlabilité, qui est à la base de la Méthode d'Unicité de Hilbert (HUM) Chapitre 1. Introduction établie par J. L. Lions dans [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF]. Tout d'abord, nous considérons le système suivant :

   U t (x, t) = AU (x, t) + Bv(x), sur Ω × (0, +∞), U (x, 0) = U 0 (x), (1.2.3) où Ω ⊂ R d (d ∈ N * ), U
de x ∈ Ω, t > 0 et de U 0 et v.
Maintenant, nous pouvons introduire la notion de contrôlabilité, exacte ou approximative. Soit T > 0 et U T (la cible) un élément donné de l'espace d'état. Nous voulons "conduire le système" de l'état initial U 0 à t = 0 à l'état final U T à t = T , c'est-à-dire que nous voulons trouver un contrôle approprié v tel que

U (x, T ) = U T (x), x ∈ Ω.
Si cela est possible pour n'importe quelle cible U T à la place de l'état, on peut dire que le système est contrôlable (ou exactement contrôlable). Pour plus de détails sur la contrôlabilité, voir [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF].

Nous présentons maintenant un résultat de A. Haraux dans [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] pour obtenir l'inégalité d'observabilité. Si A est un opérateur linéaire non borné, auto-adjoint, positif et coercive sur un espace de Hilbert H et B un opérateur linéaire sur H tel que B = B * ≥ 0, il a établi une équivalence logique entre la décroissance exponentielle des solutions de l'équation d'évolution du deuxième ordre U tt + AU + BU t = 0, uniformément sur des sous-ensembles de D(A 1/2 ) × H et une propriété "B 1/2 -contrôlabilité" du système non contrôllable ϕ tt + Aϕ = 0 sur un intervalle de temps (voir les propositions 1 et 2 dans [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF]).

Remarque 1. Le résultat de A. Haraux dans [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] reste valable pour une équation d'évolution du premier ordre.

Conditions géométriques

Cette section est consacrée à rappeler certaines conditions géométriques nécessaires tout au long de notre travail.

Nous commençons par les conditions du contrôle géométrique GCC introduites par Rauch et Taylor dans [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF] pour des variétés sans frontière et par Bardos, Lebeau et Rauch dans [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] pour les domaines avec frontière. Définition 1.2.8. On dit qu'un sous ensemble ω de Ω satisfait le GCC si chaque rayon de l'optique géométrique partant d'un point quelconque x ∈ Ω à t = 0 doit rencontrer la région ω en temps fini T .

Ensuite, nous rappelons la condition de multiplicateurs géométriques par morceaux introduite par K. Liu dans [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF]. Définition 1.2.9. On dit que ω satisfait la condition de multiplicateurs géométriques par morceaux (PMGC) s'ils existent Ω j ⊂ Ω ayant une frontière Lipschitzienne Γ j = ∂Ω j et

x j ∈ R N , j = 1, ..., J tel que Ω j ∩ Ω i = ∅ pour j = i et ω contient un voisinage dans Ω de l'ensemble ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j où γ j (x j ) = {x ∈ Γ j : (x -x j ) • ν j (x) > 0} et ν j est le vecteur normal unitaire extérieure à Γ j .
Remarque 2. La condition PMGC est la généralisation de la condition géométrique des multiplicateurs (MGC) introduite par Lions dans [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], en disant que ω contient un voisinage dans Ω de l'ensemble {x ∈ Γ : (x -x 0 ) • ν(x) > 0}, pour certains x 0 ∈ R N , où ν est le vecteur unitaire normal extérieur à Γ = ∂Ω. Cependant, la condition PMGC est beaucoup plus restrictive que celle de GCC.

Après cette introduction générale, nous allons maintenant détailler les résultats de cette thèse dans les trois sections suivantes.

Chapitre 2 : Stabilité indirecte locale de deux équations d'ondes couplées sous des conditions géométriques

Soit Ω ⊂ R N un ouvert borné non vide avec une frontière Γ de class C 2 . Dans chapitre 2, nous sommes intéréssés à étudier la stabilité du système suivant de deux équations d'ondes couplées :

u tt -a∆u + c(x)u t + b(x)y t = 0 in Ω × R + , (1.2.4) y tt -∆y -b(x)u t = 0 in Ω × R + , (1.2.5) u = y = 0 on Γ × R + , (1.2.6)
avec les données initiales suivantes : ). Il a établi un taux de décroissance exponentiel de l'énergie dans le cas où les trois conditions suivantes sont satisfaites : les ondes se propagent à la même vitesse (a = 1), le support de b coïncide avec celui de c et le domaine, où l'amortissement est effectif, satisfait certaines conditions géométriques. En revanche, dans le cas général, quand les vitesses sont différentes le taux de décroissance de l'énergie n'est pas étudié.

u(x, 0) = u 0 , y(x, 0) = y 0 , u t (x, 0) = u 1 and y t (x, 0) = y , x ∈ Ω, où a > 0 est une constante et b(x) ∈ C 0 (Ω; R)
Dans [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF], F. Alabau-Boussouira et al. ont étudié la stabilité du système suivant :

u tt -a∆u + ρ(x, u t ) + b(x)y t = 0 in Ω × R * + , (1.2.7) y tt -∆y -b(x)u t = 0 in Ω × R * + , (1.2.8) u = y = 0 on Γ × R * + , (1.2.9) où a > 0 constante, b(x) ∈ C 0 (Ω, R) et ρ(x,

Principaux résultats obtenus

La principale nouveauté dans ce chapitre est que les ondes ne se propagent pas nécessairement à la même vitesse et que le coefficient de couplage n'est supposé ni positif ni petit. D'abord, nous commençons à étudier l'existence, l'unicité et la régularité de la solution de notre système en utilisant la théorie de semigroupe. Soit U = (u, u t , y, y t ) une solution régulière du système (1.2.4)-(1.2.6), son énergie associée est définie par :

E(t) = 1 2 Ω |u t | 2 + a|∇u| 2 + |y t | 2 + |∇y| 2 dx. (1.2.10)
Par un calcul direct, on obtient :

d dt E(t) = - Ω c(x)|u t | 2 dx ≤ 0. (1.2.11)
Maintenant, nous définissons l'espace de l'énergie H = ( H 1 0 (Ω) × L 2 (Ω)) 2 muni, pour tout U = ( u, v, y, z), U = ( u, v, y, z) ∈ H du produit scalaire :

(U, U ) H = a Ω (∇u • ∇ u)dx + Ω v vdx + Ω (∇y • ∇ y)dx + Ω z zdx.
Ensuite, nous définissons l'opérateur linéaire non borné A : D(A) ⊂ H → H par : Ensuite, nous nous intéressons à l'étude du taux de décroissance de l'énergie en utilisant la méthode fréquentielle combinée avec la technique de multiplicateur et en distinguant deux cas : le premier quand les ondes sont supposées propagées à la même vitesse (i.e. a = 1) et le deuxième quand a = 1. Théorème 1.2.10. (Taux de décroissance exponentiel) Soit a = 1. Supposons que l'hypothèse de localisation (LH1) est satisfaite. Supposons de plus que l'ouvert non vide ω = ω c + ∩ ω b + (ou ω = ω c + ∩ ω b -) satisfait les conditions géométriques PMGC et que b, c ∈ W 1,∞ (Ω), alors il existe des constantes θ > 0 et M ≥ 1 telles que, pour toute donnée initiale U 0 = (u 0 , u 1 , y 0 , y 1 ) ∈ H, l'énergie E(t) du système (1.2.4)-(1.2.6), vérifie l'estimation suivante :

D(A) = (H 2 (Ω) ∩ H 1 0 (Ω) × H 1 0 (Ω)) 2 , AU = ( v,
b + ∪ ω b -⊂ Ω tel que {x ∈ Ω : b(x) > 0} ⊃ ω b + ou {x ∈ Ω : b(x) < 0} ⊃ ω b -. (
E(t) ≤ M e -θt E(0), ∀ t > 0. ( 1
U n H = 1 et (iµ n I -A)U n H → 0.
Ainsi, la résolvante de A n'est pas uniformément bornée sur l'axe imaginaire. D'après un résultat de Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] et Prüss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF], nous concluons que le semigroupe e tA n'est pas uniformément stable sur H. Il est donc naturel de penser à un taux de décroissance polynomial de l'énergie du système ; nous établissons, alors, l'estimation suivante : Théorème 1.2.11. ( Taux de décroissance polynomial) Soit a = 1. Supposons que l'hypothèse de localisation (LH1) est satisfaite. Supposons de plus que l'ouvert non vide ω =

ω c + ∩ ω b + (ou ω = ω c + ∩ ω b -) satisfait les conditions géométriques PMGC et que b, c ∈ W 1,∞ (Ω), alors il existe une constante C > 0 telle que, pour toute donnée initiale U 0 = (u 0 , u 1 .y 0 , y 1 ) ∈ D(A), l'énergie E(t) du système (1.2.4)-(1.2.6), vérifie l'estimation sui- vante : E(t) ≤ C t U 0 2 D(A) , ∀ t > 0. (1.2.14)
Notons que dans le cas a = 1 les ondes ne se propagent pas à la même vitesse. Ce théorème généralise, dans le cas linéaire, les résultats de [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] et [START_REF] Kapitonov | Uniform stabilization and exact controllability for a class of coupled hyperbolic systems[END_REF].

Finalement, dans le cas monodimensiel d'espace (i.e. N = 1) , a = 1 et b est une constante, nous montrons qu'il existe n 0 ∈ N suffisamment grand et une suite λ n de valeurs propres simples associées à l'opérateur A satisfaisant le développement asymptotique suivant :

λ n = inπ - ib 2 2(a -1)nπ - cb 2 2(a -1) 2 n 2 π 2 + O 1 n 3 , ∀ |n| ≥ n 0 . ( 1
u tt -a∆u + b(x)y t = c(x)v(t) in Ω × R * + , (1.2.16) y tt -∆y -b(x)u t = 0 in Ω × R * + , (1.2.17) u = y = 0 on Γ × R * + , (1.2.18)
avec les données initiales suivantes :

u(x, 0) = u 0 , y(x, 0) = y 0 , u t (x, 0) = u 1 and y t (x, 0) = y 1 , x ∈ Ω, (1.2.19) où a > 0 constante, b ∈ C 0 (Ω, R), c ∈ C 0 (Ω, R + ) et v
⊂ Ω tel que {x ∈ Ω : b(x) = 0} ⊃ ω b . (LH2) Si ω = ω c + ∩ ω b = ∅ et l'
D = H 1 0 (Ω) × L 2 (Ω) × L 2 (Ω) × H -1 (Ω), muni de produit scalaire (U, Ũ ) = Ω (a∇u.∇ũ + vṽ + y ỹ + (-∆) -1/2 z(-∆) -1/2 z)dx, pour tout U = (u, v, y, z) ∈ D et Ũ = (ũ, ṽ, ỹ, z) ∈ D.
Ensuite, nous définissons l'opérateur linéaire non borné

A d : D(A d ) ⊂ D → D par A d U = ( v, a∆u -bz -cv, z, ∆y + bv ), D(A d ) = (H 1 0 (Ω) ∩ H 2 (Ω)) × H 1 0 (Ω) × H 1 0 (Ω) × L 2 (Ω) , ∀ U = (u, v, y, z) ∈ D(A d ). Son énérgie mixte totale est définie par E m (t) = 1 2 a ∇u 2 L 2 (Ω) + u t 2 L 2 (Ω) + y t
ε > 0, sous ensemble Ω j ⊂ Ω, j = 1, ..., J, avec frontière Lipchtizienne, Γ j = ∂Ω j et points

x j ∈ R N tel que Ω i ∩ Ω j = ∅ si i = j et ω + c ⊃ N ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j ∩ Ω avec N (O) = {x ∈ R N : d(x, O) < ε} où O ⊂ R N , γ j (x j ) = {x ∈ Γ j : (x -x j ) • ν j (x) > 0} où ν j est le vecteur normal extérieur à Γ j et que ω b satisfait le GCC et que ω b ⊂ Ω \ ∪ J j=1 Ω j . ( LH3
(Ω) × H -1 (Ω) × H -1 (Ω) × (H 2 (Ω) ∩ H 1 0 (Ω)) , où la dualité est par rapport à L 2 (Ω).
Enfin, nous effectuons des tests numériques dans le cas 1-D pour valider notre résultats théoriques obtenus ici et au chapitre 2. En fait, les résultats numériques montrent un meilleur comportement que celui attendu par les résultats théoriques. [START_REF] Tian | Stability of a timoshenko system with local kelvin-voigt damping[END_REF] ont consideré le système de Timoshenko suivant défini sur (0, L) × (0, +∞) avec des conditions de Dirichlet aux bords :

           ρ 1 ϕ tt -[k 1 (ϕ x + ψ + lw) + D 1 (ϕ xt + ψ t + lw t )] x -lk 3 (w x -lϕ) -lD 3 (w xt -lϕ t ) = 0, ρ 2 ψ tt -[k 2 ψ x + D 2 ψ xt ] x + k 1 (ϕ x + ψ + lw) + D 1 (ϕ xt + ψ t + lw t ) = 0, ρ 1 w tt -[k 3 (w x -lϕ) + D 3 (w xt -lϕ t )] x + lk 1 (ϕ x + ψ + lw) + lD 1 (ϕ xt + ψ t + lw t ) = 0, ( 1
   ρ 1 ϕ tt -[k 1 (ϕ x + ψ) x + D 1 (ϕ xt -ψ t )] x = 0, ρ 2 ψ tt -(k 2 ψ x + D 2 ψ xt ) x + k 1 (ϕ x + ψ) x + D 1 (ϕ xt -ψ t ) = 0. ( 1

Principaux résultats obtenus :

D'abord, nous commençons à étudier l'existence, l'unicité et la régularité de la solution de notre système en utilisant la théorie de semigroupe. Soit (ϕ, ϕ t , ψ, ψ t , w, w t ) une solution régulière du système (1.2.22), son énergie associée est définie par :

E (t) = 1 2 L 0 ρ 1 |ϕ t | 2 + ρ 2 |ψ t | 2 + ρ 1 |w t | 2 + k 1 |ϕ x + ψ + lw| 2 dx + L 0 k 2 |ψ x | 2 + k 3 |w x -lϕ| 2 dx .
(1.2.24)

Par un calcul direct, on obtient

E (t) = - L 0 D 1 |ϕ xt + ψ t + lw t | 2 + D 2 |ψ xt | 2 + D 3 |w xt -lϕ t | 2 dx ≤ 0. ( 1
H 1 = H 1 0 × L 2 3 et H 2 = H 1 0 × L 2 × H 1 * × L 2 où L 2 * = {f ∈ L 2 (0, L) : L 0 f (x)dx = 0} et H 1 * = {f ∈ H 1 (0, L) : L 0 f (x)dx = 0}.
Nous définissons les opérateurs linéaires non bornés A j dans H j , j = 1, 2 par

D (A 1 ) = U ∈ H 1 | v 2 , v 4 , v 6 ∈ H 1 0 (0, L) , [k 1 (v 1 x + v 3 + lv 5 ) + D 1 (v 2 x + v 4 + lv 6 )] x ∈ L 2 (0, L) , [k 2 v 3 x + D 2 v 4 x ] x ∈ L 2 (0, L), [k 3 (v 5 x -lv 1 ) + D 3 (v 6 x -lv 2 )] x ∈ L 2 (0, L) , D (A 2 ) = U ∈ H 2 | v 2 ∈ H 1 0 (0, L) , v 4 , v 6 ∈ H 1 * (0, L) , v 3 x | (0,L) = v 5 x | (0,L) = 0, [k 1 (v 1 x + v 3 + lv 5 ) + D 1 (v 2 x + v 4 + lv 6 )] x ∈ L 2 (0, L) , [k 2 v 3 x + D 2 v 4 x ] x ∈ L 2 * (0, L), [k 3 (v 5 x -lv 1 ) + D 3 (v 6 x -lv 2 )] x ∈ L 2 * (0, L) , et A j             v 1 v 2 v 3 v 4 v 5 v 6             =              v 2 ρ -1 1 k 1 (v 1 x + v 3 + lv 5 ) + D 1 (v 2 x + v 4 + lv 6 ) x + lk 3 (v 5 x -lv 1 ) + lD 3 (v 6 x -lv 2 ) v 4 ρ -1 2 (k 2 v 3 x + D 2 v 4 x ) x -k 1 v 1 x + v 3 + lv 5 -D 1 (v 2 x + v 4 + lv 6 ) v 6 ρ -1 1 k 3 (v 5 x -lv 1 ) + D 3 (v 6 x -lv 2 ) x -lk 1 v 1 x + v 3 + lv 5 -lD 1 (v 2 x + v 4 + lv 6 )              pour tout U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 )
T ∈ D (A j ). Soit U = (ϕ, ϕ t , ψ, ψ t , w, w t ) T , système (1.2.22) peut se reécrire de la forme

U t = A j U, j = 1, 2 U = (ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , w 0 , w 1 ) T .
Ensuite, nous démontrons que l'opérateur A j est m-dissipatif dans l'espace de l'énergie H j . Par conséquent, grâce au théorème de Lumer-Phillips, nous déduisons que A j engendre un C 0 -semigroupe de contractions e tA j dans H j , et que le problème est bien posé dans H j . En plus, en utilisant une critère générale de Arendt et Batty [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF], nous montrons que le C 0 -semigroupe e tA j est fortement stable en l'absence de la compacité du résolvant de A j et la présence d'au moins un amortissement local de Kelvin-Voigt.

Maintenant, notre but est d'étudier la décroissance de l'énergie en utilisant la méthode fréquentielle combinée avec la technique des multiplicateurs dans plusieurs cas dépendant de la régularité des coefficients D 1 , D 2 et D 3 , de la localisation de leurs supports et de leurs nombres.

33 Chapitre 1. Introduction Dans le cas où les trois amortissements sont globalement distribués, nous démontrons une stabilité analytique de l'énergie. Ensuite, dans le cas où les fonctions D i , i = 1, 2, 3 sont continues et satisfaisantes la condition suivante :

∃ d 0 > 0 tel que D i ≥ d 0 > 0 pour tout x ∈ (α, β) , 0 < α < β < L, ( 1
e tA j U 0 H j ≤ M e -t U 0 H j , t ≥ 0, j = 1, 2.
Cependant dans le cas où D i ∈ L ∞ (0, L), i = 1, 2, 3 satisfait la condition suivante :

ω = suppD 1 ∩ suppD 2 ∩ suppD 3 = (α, β) ⊂ (0, L) tel que mes(ω) > 0, ( 1
D 1 = 0 et D 2 = D 3 = 1 dans (0, L), (1.2.29)
n'est pas uniformément stable. De même, nous démontrons aussi que le système n'est pas exponentiellement stable sous la condition suivante :

D 1 = D 3 = 0 et D 2 = 1 dans (0, L). (1.2.30)
En effet, pour démontrer la stabilité non uniforme, nous construisons une suite (V n ) des éléments dans

D(A 2 ) et une suite réelle (λ n ) tel que V n → +∞ et (iλ n I -A 2 )U n H 2 est borné quand n → +∞.
Par conséquent, la résolvante de A 2 n'est pas uniformément bornée sur l'axe imaginaire. D'aprés Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] et Pruss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF] nous concluons que la semigroupe e tA 2 n'est pas uniformément stable dans H 2 .

Finalement, dans le cas d'un seul amortissement local, i.e. supposons qu'il existe d 2 > 0 tel que

D 1 = D 3 = 0 dans (0, L) et D 2 ≥ d 2 > 0 dans (α, β) ⊂ (0, L), (1.2
E(t) ≤ c √ t U 0 2 D(A j ) . (1.2.32)

Introduction

Let Ω be a nonempty bounded open set of R N having a boundary Γ of class C 2 . In [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] , F. Alabau-Boussouira et al. considered the energy decay of a system of two wave equations coupled by velocities :

u tt -a∆u + b(x)y t + ρ(x, u t ) = 0 in Ω × R + , (2.1.1) y tt -∆y -b(x)u t = 0 in Ω × R + , (2.1.2) u = y = 0 on Γ × R + , (2.1.3)
with the following initial data :

u(x, 0) = u 0 , y(x, 0) = y 0 , u t (x, 0) = u 1 and y t (x, 0) = y , x ∈ Ω
where a > 0 constant and b ∈ C 0 (Ω; R) is a non-zero function. The damping term ρ is applied at the first equation and the second equation is indirectly damped through the coupling between the two equations. In [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF], using an approach based on multiplier techniques, weighted nonlinear inequalities and the optimal-weight convexity method (developed in [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF]), the authors established an explicit energy decay formula in terms of the behavior of the nonlinear feedback close to the origin. Their results are obtained in the case when the following three conditions are satisfied : the waves propagate at the same speed (a = 1), the coupling coefficient b(x) is small positive (0

≤ b(x) ≤ b 0 , b 0 ∈ (0, b ]
where b is a constant depending on Ω and on the control region) and both the coupling and the damping regions satisfying an appropriated geometric conditions named Piecewise Multipliers Geometric Conditions (introduced in [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF], used in [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF] and denoted by PMGC, in short). Then the stabilization of system (2.1.1)-(2.1.3) in the case where the waves are not assumed to be propagated with equal speeds (a is not necessarily equal to 1) and/or when the coupling coefficient b(x) is not assumed to be positive and small has been left as an open problem even when the damping term ρ is linear with respect to the second variable. In this chapter, we are interested to answer this open question and to provide a stability analysis for system (2.1.1)-(2.1.3) when the damping term ρ is linear with respect to the second variable i.e. ρ(x, u t ) = c(x)u t where c ∈ C 0 (Ω; R + ). So, we consider the stability of the following system :

u tt -a∆u + b(x)y t + c(x)u t = 0 in Ω × R + , (2.1.4) y tt -∆y -b(x)u t = 0 in Ω × R + , (2.1.5)
u = y = 0 on Γ × R + , (2.1.6)
with the following initial data :

u(x, 0) = u 0 , y(x, 0) = y 0 , u t (x, 0) = u 1 and y t (x, 0) = y , x ∈ Ω.
The notion of indirect damping mechanisms has been introduced by D.L. Russell in [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF], and since then, it attracted the attention of many authors. In particular, the stabilization of systems of two second order equations coupled through displacements when only one equation is effectively damped by internal or boundary feedback, has been initiated and studied in [START_REF] Alabau-Boussouira | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF][START_REF] Alabau-Boussouira | Indirect internal stabilization of weakly coupled evolution equations[END_REF][START_REF] Alabau-Boussouira | Indirect boundary stabilization of weakly coupled hyperbolic systems[END_REF], and further studied by many authors, for instance [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF][START_REF] Liu | Frequency domain approach for the polynomial stability of a system of partially damped wave equations[END_REF][START_REF] Ammar-Khodja | Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force[END_REF]. Recall that the exponential or polynomial energy decay rate occurs in many control problems, we quote [START_REF] Bassam | Stability results of some distributed systems involving Mindlin-Timoshenko plates in the plane[END_REF][START_REF] Soufyane | Uniform stabilization for the Timoshenko beam by a locally distributed damping[END_REF] for the Timoshenko system in bounded or unbounded domains. Here, we focus our attention only on the literature of the indirect internal stability of coupled wave equations. In [START_REF] Kapitonov | Uniform stabilization and exact controllability for a class of coupled hyperbolic systems[END_REF], B. Kapitonov studied the stabilization of a system of two coupled hyperbolic equations involving (2.1.4)-(2.1.6). He established an exponential energy decay rate for usual initial data in the case that the waves propagate at same speed, (a = 1) and the damping and coupling coefficients have the same support. For the other cases, when a = 1 and/or support of b does not coincide with that of c no energy decay rate has been discussed. In [START_REF] Alabau-Boussouira | Indirect internal stabilization of weakly coupled evolution equations[END_REF], F. Alabau et al. studied the indirect stabilization of a system of two evolution equations coupled through displacements where the damping is effective in the whole domain. Using the method of higher order energies initiated in [START_REF] Alabau-Boussouira | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF], they established polynomial energy decay depending on the smoothness of initial data. These results have been generalized by F. Alabau and M. Léautaud in [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF] to the case when the Chapitre 2. Local indirect stabilization of N-d system of two coupled wave equations under geometric conditions coupling and the damping coefficients are localized in Ω and both satisfying the PMGC conditions. In addition, without geometric conditions, using an interpolation inequality for elliptic system (see Proposition 5.1 in [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF]) together with the resolvent estimates of G. Lebeau in [START_REF] Lebeau | Équation des ondes amorties[END_REF], the authors proved that the energy decay of smooth initial data is at least logarithmic when the coupling and the damping regions intersect in a nonempty subdomain ω ⊂ Ω. However, when ω = ∅, the question of the stability or the null controllability of the system, is still an open problem. Indeed, F. Alabau and M. Léautaud in [START_REF] Alabau-Boussouira | Indirect controllability of locally coupled wave-type systems and applications[END_REF], solved partially this problem by proving that the system is null controllable provided that both the coupling and the damping regions satisfy the optimal geometric condition named Geometric Control Condition introduced by Bardos et al. in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. Finally, we refer to [START_REF] Alabau-Boussouira | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF][START_REF] Alabau-Boussouira | Indirect internal stabilization of weakly coupled evolution equations[END_REF][START_REF] Alabau-Boussouira | A general formula for decay rates of nonlinear dissipative systems[END_REF][START_REF] Ammari | Stabilization of coupled systems[END_REF][START_REF] Bassam | Polynomial stability of the Timoshenko system by one boundary damping[END_REF][START_REF] Wehbe | Exponential and polynomial stability of an elastic bresse system with two locally distributed feedback[END_REF][START_REF] Soufyane | Uniform stabilization for the Timoshenko beam by a locally distributed damping[END_REF][START_REF] Wehbe | Stabilization of the uniform Timoshenko beam by one locally distributed feedback[END_REF][START_REF] Najdi | Weakly locally thermal stabilization of bresse systems[END_REF][START_REF] Kapitonov | Uniform stabilization and exact controllability for a class of coupled hyperbolic systems[END_REF][START_REF] Loreti | Optimal energy decay rate for partially damped systems by spectral compensation[END_REF][START_REF] Ammar-Khodja | Stabilization of the nonuniform Timoshenko beam[END_REF] for the indirect stabilization and the indirect exact controllability of distributed systems with different kinds of damping.

In this chapter, we study the stability of system (2.1.4)-(2.1.6) when the coupling region and the damping region intersect in ω ⊂ Ω. First, we establish the strong stability without geometric conditions. We then study the energy decay rate of our system by distinguishing two cases. The first one is when the waves propagate at same speed, i.e. a = 1. In this case, under the hypothesis that ω satisfies the geometric conditions PMGC (see below), we establish an exponential energy decay rate for usual initial data. Next, in the general case, when a = 1, we prove the non uniform (exponential) stability and under the same geometric conditions, we establish a polynomial energy decay rate of type 1 t for smooth initial data. Finally, in one space dimension, using the real part of the asymptotic expansion of the eigenvalues of the system, we show that the obtained polynomial decay is optimal.

Well posedness and strong stability

In this Section, we will study the strong stability of system (2.1.4)-(2.1.6) without additional geometric conditions. First, we will study the existence, uniqueness, and regularity of the solution of our system.

Well posedness of the problem

Let U = (u, u t , y, y t ) be a regular solution of (2.1.4)-(2.1.6), its associated energy is defined by

E(t) = 1 2 Ω |u t | 2 + a|∇u| 2 + |y t | 2 + |∇y| 2 dx. (2.2.1)
So, a direct computation gives

d dt E(t) = - Ω c(x)|u t | 2 dx ≤ 0. (2.2.2)
Consequently, system (2.1.4)-(2.1.6) is dissipative in the sense that its energy is nonincreasing.

First, we define the energy space H = ( H 1 0 (Ω) ×L 2 (Ω)) 2 equipped, for all U = ( u, v, y, z), U = ( u, v, y, z) ∈ H, by the scalar product :

(U, U ) H = a Ω (∇u • ∇ u)dx + Ω v vdx + Ω (∇y • ∇ y)dx + Ω z zdx.
Next, we define the unbounded linear operator A : D(A) → H by :

D(A) = (H 2 (Ω) ∩ H 1 0 (Ω) × H 1 0 (Ω)) 2 , AU = ( v, a∆u -bz -cv, z, ∆y + bv ).
Note that, using the fact that c(x) ≥ 0, then A is m-dissipatif and generates a C 0 semigroup of contractions e tA on the energy space H. As the system (2.

1.4)-(2.1.6) is equivalent to U t = AU in H, t > 0, U (0) = U 0 (2.2.3)
with U = (u, u t , y, y t ), we deduce its well posedeness character. So, we have the following existence results :

Theorem 2.2.1. Let U 0 ∈ H then, problem (2.2.3) admits a unique weak solution U satisfies U (t) ∈ C 0 R + , H . Moreover, if U 0 ∈ D(A) then, problem (2.2.
3) admits a unique strong solution U satisfies

U (t) ∈ C 1 R + , H ∩ C 0 (R + , D(A)).
Secondly, we will study the strong stability of our system.

Strong stability

In this subsection we study the asymptotic behavior of E(t). For this aim, we assume that there exists a nonempty open ω c + ⊂ Ω satisfying the following condition :

{x ∈ Ω : c(x) > 0} ⊃ ω c + (LH1)
On the other side, as b(x) is a non-zero continuous function, then there exists a nonempty open

ω b + ∪ ω b -⊂ Ω such that {x ∈ Ω : b(x) > 0} ⊃ ω b + and {x ∈ Ω : b(x) < 0} ⊃ ω b - (LH2)
Our main result in this part is the following Theorem 2.2.2. (Strong Stability) Assume that a > 0, condition (LH1) holds and that

ω = ω c + ∩ ω b + = ∅ or ω c + ∩ ω b -= ∅.
Then the semi group of contractions e tA is strongly stable on the energy space H, i.e. for any U 0 ∈ H, we have

lim t→+∞ e tA U 0 H = 0. ( 2

.2.4) under geometric conditions

In [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF], the authors considered the stabilization of a system of two wave equations coupled in displacements with one localized internal damping. They showed that, under the assumption that the damping region and the coupling region have a non-empty intersection in Ω i.e.

ω = ω c + ∩ ω b + = ∅ (or ω c + ∩ ω b -= ∅)
, the energy of smooth solutions decays logarithmically to zero as t goes to infinity. This result still holds in the case where the two wave equations are coupled through the velocities. Indeed, following the method introduced by G. Lebeau and L. Robbiano in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], M. Léautaud in [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF] established an interpolation inequality for the associated elliptic system. This interpolation inequality implies the resolvent estimates of G. Lebeau in [START_REF] Lebeau | Équation des ondes amorties[END_REF] (see also [START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF]) that provide the logarithmic energy decay rate for smooth initial data. So, using the density of D(A) in H and the contraction property of the C 0 semigroup e tA , we deduce that the energy of system (2.1.4)-(2.1.6) decays asymptotically to zero as t goes to infinity for all usual initial data.

Then we are interested, in this chapter, to study the energy decay rate by distinguishing two cases.

Exponential stability, the case a = 1

This section is devoted to the study of the exponential stability of system (2.1.4)-(2.1.6) in case the waves propagate at same speed (in the case a = 1) and under appropriated geometric conditions. For that purpose, we will use a frequency domain approach combined with piecewise multiplier technique.

Before presenting our main result of this section, we recall the Piecewise Multipliers Geometric Condition introduced by K. Liu in [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF].

Definition 2.3.1. We say that ω satisfies the Piecewise Multipliers Geometric Condition (PMGC in short) if there exist Ω j ⊂ Ω having Lipschitz boundary Γ j = ∂Ω j and x j ∈ R N , j = 1, ..., J such that Ω j ∩ Ω i = ∅ for j = i and ω contains a neighborhood in Ω of the set

∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j where γ j (x j ) = {x ∈ Γ j : (x -x j ) • ν j (x) > 0} and ν j is the outward unit normal vector to Γ j . Remark 2.3.2.
The PMGC is the generalization of the Multipliers Geometric Condition (MGC in short) introduced by Lions in [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], saying that ω contains a neighborhood in Ω of the set {x ∈ Γ : (x -x 0 ) • ν(x) > 0}, for some x 0 ∈ R N , where ν is the outward unit normal vector to Γ = ∂Ω. Now, we are in position to present the main result of this section.

= ω c + ∩ ω b + (or ω = ω c + ∩ ω b -) satisfies the geometric conditions PMGC and that b, c ∈ W 1,∞ (Ω).
Then there exist positive constants M ≥ 1, θ > 0 such that for all initial data (u 0 , u 1 , y 0 , y 1 ) ∈ H the energy of the system (2.1.4)-(2.1.6) satisfies the following decay rate :

E(t) ≤ M e -θt E(0), ∀t > 0. (2.3.1)
Remark 2.3.4. Note that in Theorem 2.3.3 we have no restriction on the upper bound and the sign of the function b. This theorem is then a generalization in the linear case of the result of [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] where the coupling coefficient considered have to satisfy

0 ≤ b(x) ≤ b 0 , b 0 ∈ (0, b ]
where b is a constant depending on Ω and on the control region. Nevertheless, the problem still be open in the nonlinear case.

In order to prove the above theorem, we apply a result of F. L. Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and J. Pruss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF] : A C 0 -semigroup of contraction (e tA ) t 0 in a Hilbert space H is uniformly stable if and only if

iR ⊆ ρ(A) (H1)
and sup

β ∈R (iβI -A) -1 < +∞ (H2) hold.
Since the resolvent of A is compact and 0 ∈ ρ(A), then from Theorem 2.2.2, we deduce that condition (H1) is satisfied. We now prove that condition (H2) holds, using an argument of contradiction. For this aim, we suppose that there exist a real sequence β n with β n → +∞ and a sequence

U n = (u n , v n , y n , z n ) ∈ D(A) such that (u n , v n , y n , z n ) H = 1, (2.3.2)
and lim

n→∞ (iβ n I -A)U n H = 0. (2.3.3)
Now, detailing equation (2.3.3), we get

iβ n u n -v n = f 1 n → 0 in H 1 0 (Ω), (2.3.4) iβ n v n -∆u n + b(x)z n + c(x)v n = g 1 n → 0 in L 2 (Ω), (2.3.5) iβ n y n -z n = f 2 n → 0 in H 1 0 (Ω), (2.3.6) iβ n z n -∆y n -b(x)v n = g 2 n → 0 in L 2 (Ω). (2.3.7)
Eliminating v n and z n from the previous system, we obtain the following reduced system

β 2 n u n + ∆u n -iβ n b(x)y n -iβ n c(x)u n = -g 1 n -b(x)f 2 n -iβ n f 1 n -c(x)f 1 n , (2.3.8)
β 2 n y n + ∆y n + iβ n b(x)u n = -iβ n f 2 n + b(x)f 1 n -g 2 n .
(2.3.9)

On the other side, using equation ( 2

iβ n U n 2 -(AU n , U n ) = Ω c(x)|v n | 2 dx = o(1). (2.3.12)
Under condition (LH1), it follows that

ωc + |v n | 2 dx = o(1). (2.3.13)
So, using equations (2.3.12) and (2.3.4), we get

Ω c(x)|β n u n | 2 dx = o(1). (2.3.14)
Consequently, we have

ωc + |β n u n | 2 dx = o(1).
The proof is thus complete. Now, the subset ω satisfies the PMGC. Hence, denoting by Ω j and x j , j = 1, ..., J the sets and the points given by the PMGC, we have

ω ⊃ N ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j ∩ Ω. In this expression, N (O) = {x ∈ R N : d(x, O) < ε} with d(•, O) is the usual euclidean distance to the subset O of R N and γ j (x j ) = {x ∈ Γ j : (x -x j ) • ν j (x) > 0} where ν j is the outward unit normal vector to Γ j = ∂Ω j . Let the reals 0 < ε 1 < ε 2 < ε 3 < ε and define V i = N ε i ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j , i = 1, 2, 3. Since Ω \ V 3 ∩ V 2 = ∅, then we may define the function η ∈ C ∞ 0 (R N ) by η(x) = 0 if x ∈ Ω \ V 3 , 0 ≤ η(x) ≤ 1, η(x) = 1 if x ∈ V 2 . Lemma 2.3.6. The solution (u n , v n , y n , z n ) ∈ D(A) of system (2.3.4)-(2.3.7) satisfies the following estimation Ω η(x) |∇u n | 2 dx = o(1)
and

V 2 ∩Ω |∇u n | 2 dx = o(1). ( 2

.3.15)

Proof: Multiplying equation (2.3.8) by ηū n . Then, using Green's formula and the fact that u n = 0 on Γ, we obtain

Ω η(x)|β n u n | 2 dx - Ω η(x) | ∇u n | 2 dx - Ω u n (∇η • ∇u n )dx -i β n Ω b(x)ηy n u n d -iβ n Ω c(x)η(x)|u n | 2 dx = Ω (-g 1 n -b(x)f 2 n -iβ n f 1 n -c(x)f 1 n )ηu n dx. (2.3.16) As f 1 n and f 2 n converge to zero in H 1 0 (Ω), g 1 n converges to zero in L 2 (Ω), the sequences (β n u n ), (β n y n ), (∇u n ) are uniformly bounded in L 2 (Ω) and u n = o(1), we get Ω η(x)|β n u n | 2 dx - Ω η(x) | ∇u n | 2 dx = o(1).
(2.3.17)

By using the definition of η and equations (2.3.11) and (2.3.17), we deduce

Ω η(x) | ∇u n | 2 dx = o(1)
and

V 2 ∩Ω | ∇u n | 2 dx = o(1).
The proof is thus complete

Lemma 2.3.7. The solution (u n , v n , y n , z n ) ∈ D(A) of system (2.3.4)-(2.3.7) satisfies the following estimation Ω η(x)|∇y n | 2 dx = o(1)
and

V 2 ∩Ω |∇y n | 2 dx = o(1). ( 2

.3.18)

Proof: The proof contains three points.

(i) Notice that, from equation (2.3.9), 1 βn ∆ȳ n is uniformly bounded in L 2 (Ω). So, multiplying equation (2.3.8) by 1 βn η∆ȳ n . Using Green's formula and the fact that

u n = y n = f 1 n = 0 on Γ, we get - Ω β n η(x)(∇y n • ∇u n )dx - Ω β n u n (∇η • ∇y n )dx + Ω 1 β n η(x)∆y n ∆u n dx +i Ω b(x)η(x) | ∇y n | 2 dx + i Ω η(x)y n (∇b • ∇y n )dx + i Ω b(x)y n (∇η • ∇y n )dx +i Ω c(x)η(x)(∇u n • ∇y n )dx + i Ω c(x)u n (∇η • ∇y n )dx +i Ω η(x)u n (∇c • ∇y n )dx = + Ω (-g 1 n -b(x)f 2 n -c(x)f 1 n )( 1 β n η(x)∆y n )dx +i Ω η(x)(∇f 1 n • ∇y n )dx + i Ω f 1 n (∇η • ∇y n )dx. (2.3.19) First, since f 1 n , f 2 n converge to zero in H 1 0 (Ω), g 1 n converges to zero in L 2 (Ω) and (∇y n ), ( 1 βn ∆y n ) are uniformly bounded in L 2 (Ω), then we have Ω (-g 1 n -bf 2 n -cf 1 n )( 1 β n η∆y n )dx + i Ω η(∇f 1 n • ∇y n )dx + i Ω f 1 n (∇η • ∇y n )dx = o(1
- Ω β n u n (∇η • ∇y n )dx + i Ω η(x)y n (∇b • ∇y n )dx +i Ω b(x)y n (∇η • ∇y n )dx + i Ω c(x)η(x)(∇u n • ∇y n )dx +i Ω c(x)u n (∇η • ∇y n )dx + i Ω η(x)u n (∇c • ∇y n )dx = o(1
- Ω β n η(∇y n • ∇u n )dx + Ω 1 β n η∆y n ∆u n dx + i Ω bη | ∇y n | 2 dx = o(1). (2.3.22)
(ii) Multiplying (2.3.9) by the bounded sequence 1 βn η∆u n , integrating over Ω and use the fact that u n = y n = f 2 n = 0 on Γ , we get

- Ω β n η(x)(∇y n • ∇u n )dx -β n Ω y n (∇η • ∇u n )dx + Ω 1 β n η(x)∆y n ∆u n dx -i Ω η(x)u n (∇b • ∇u n )dx -i Ω η(x)b(x)|∇u n | 2 dx -i Ω b(x)u n (∇u n • ∇η)dx = i Ω η(x) ∇f 2 n • ∇u n + f 2 n (∇η • ∇u n ) dx + Ω b(x)f 1 n -g 2 n 1 β n η(x)∆u n dx. (2.3.23) First, since f 1 n , f 2 n converge to zero in H 1 0 (Ω), g 2 n converges to zero in L 2 (Ω) and ∇u n , 1 βn ∆u n are uniformly bounded in L 2 (Ω), then we have i Ω η(x) ∇f 2 n • ∇u n + f 2 n (∇η • ∇u n ) dx + Ω b(x)f 1 n -g 2 n 1 β n η(x)∆u n dx = o(1). (2.3.24)
Next, using equation (2.3.15) and the fact that

u n = o(1) and (β n y n ) is uniformly bounded in L 2 (Ω), we deduce that -β n Ω y n (∇η • ∇u n )dx -i Ω η(x)u n (∇b • ∇u n )dx -i Ω η(x)b(x)|∇u n | 2 dx -i Ω b(x)u n (∇u n • ∇η)dx = o(1
Ω b(x)η(x)|∇y n | 2 dx = o(1). (2.3.27)
Using the definition of function η and condition (LH2), we deduce that

Ω η(x)|∇y n | 2 dx = o(1)
and

V 2 ∩Ω |∇y n | 2 dx = o(1).
The proof is thus complete.

Lemma 2.3.8. The solution (u n , v n , y n , z n ) ∈ D(A) of system (2.3.4)-(2.3.7) satisfies the following estimation Ω η(x)|β n y n | 2 dx = o(1)
and

V 2 ∩Ω |β n y n | 2 dx = o(1). ( 2

.3.28)

Proof: Multiplying equation (2.3.9) by η ȳn and integrating over Ω. Then, using Green's formula and the fact that y n = o(1) and y n = 0 on Γ, we get

Ω η(x)|β n y n | 2 dx - Ω y n (∇y n • ∇η)dx - Ω η(x) | ∇y n | 2 dx + iβ n Ω b(x)η(x)y n u n dx = o(1).
(2.3.29)

Combining equations (2.3.11), (2.3.18) and (2.3.29) and using the fact that y n = o(1), we get

Ω η(x)|β n y n | 2 dx = o(1)
and

V 2 ∩Ω |β n y n | 2 dx = o(1).
The proof is thus complete.

Now, since (Ω j \ V 2 ) ∩ V 1 = ∅, then we may define the function ψ j ∈ C ∞ 0 (R N ) by : ψ j (x) = 0 if x ∈ V 1 , 0 ψ j 1, ψ j (x) = 1 if x ∈ Ω j \ V 2 . For m j (x) = (x -x j ), we define h j (x) = ψ j (x)m j (x). Lemma 2.3.9. The solution (u n , v n , y n , z n ) ∈ D(A) of system (2.3.4)-(2.3.7) satisfies the following estimation N Ω\(V 2 ∩Ω) |β n u n | 2 dx + (2 -N ) Ω\(V 2 ∩Ω) |∇u n | 2 dx +2Re i J j=1 Ω j \(V 2 ∩Ω j ) β n b(x)y n (m j • ∇u n )dx o (1).
(2.3.30) under geometric conditions Proof: Multiplying equation (2.3.8) by 2(h j • ∇u n ) and integrating over Ω j , we obtain

2β 2 n Ω j u n (h j • ∇u n )dx + 2 Ω j ∆u n (h j • ∇u n )dx -2i Ω j β n b(x)y n (h j • ∇u n )dx = 2 Ω j (-g 1 n -b(x)f 2 n -c(x)f 1 n )(h j • ∇u n )dx -2i Ω j β n f 1 n (h j • ∇u n )dx.
(2.3.31) (i) Estimation of the second member of (2.3.31). First, using Green's formula, the fact that u n = 0 on (Γ j \ γ j ) ∩ Γ and that h j = 0 on γ j , we get -2i

Ω j β n f 1 n (h j • ∇u n )dx = 2i Ω j β n u n (h j • ∇f 1 n )dx + 2i Ω j β n u n f 1 n (divh j ) dx.(2.3.32)
It follows, from the convergence of f 1 n to zero in H 1 0 (Ω) and the uniformly boundedness in

L 2 (Ω) of β n u n , that -2i Ω j β n f 1 n (h j • ∇u n )dx = o(1). (2.3.33) Next, as f 1 n , f 2 n converge to zero in H 1 0 (Ω), g 1 n converges to zero in L 2 (Ω) and (∇u n ) is uniformly bounded in L 2 (Ω), we deduce that 2 Ω j (-g 1 n -b(x)f 2 n -c(x)f 1 n )(h j • ∇u n )dx = o(1). (2.3.34)
Finally, we deduce that the second member of equation (2.3.31) is o(1).

(ii) Estimation of the first integral of equation (2.3.31). Using Green's formula, we get Re 2

Ω j β 2 n u n (h j • ∇u n )dx = - Ω j (divh j )|β n u n | 2 dx + Γ j (h j • ν j )|β n u n | 2 dΓ j . (2.3.35)
Since Ψ j = 0 on γ j and u n = 0 on (Γ j \ γ j ) ∩ Γ, then we have Re 2

Ω j β 2 n u n (h j • ∇u n )dx = - Ω j (divh j )|β n u n | 2 dx. ( 2
Ω j ∆u n (h j • ∇u n ) = -2Re N i,k=1 Ω j ∂ i h k j ∂ i u n ∂ k ūn dx + Ω j (divh j )|∇u n | 2 dx (2.3.37) - Γ j (h j • ν j )|∇u n | 2 dΓ j + 2Re Γ j (∂ ν j u n )(h j • ∇u n )dΓ j .
According to the choice of Ψ j , only the boundary terms over (Γ j \ γ j ) ∩ Γ are non vanishing in (2.3.37). But on this part of the boundary u n = 0, and consequently

∇u n = (∂ ν u n ) • ν = (∂ ν j u n )ν j . Then, we have - Γ j (h j • ν j )|∇u n | 2 dΓ j + 2Re Γ j (∂ ν j u n )(h j • ∇u n )dΓ j = (Γ j \γ j )∩Γ (ψ j m j • ν j )|∂ ν j u n | 2 dΓ j 0. ( 2
Ω j ∆u n (h j • ∇u n ) ≤ -2Re N i,k=1 Ω j ∂ i h k j ∂ i u n ∂ k ūn dx + Ω j (divh j )|∇u n | 2 dx.
(

ψ j = 0 on V 1 , we get Ω j \(V 1 ∩Ω j ) div(ψ j m j )(|β n u n | 2 -|∇u n | 2 )dx +2Re Ω j \(V 1 ∩Ω j ) N i,k=1 ∂ i (ψ j m k j )∂ i u n ∂ k ūn dx +2iRe Ω j \(V 1 ∩Ω j ) β n b(x)y n (ψ j m j • ∇u n )dx o(1). (2.3.40)
Thus, summing over j and using the fact that ψ j = 1 on Ω j \ V 2 , we get

N Ω\(V 2 ∩Ω) |β n u n | 2 dx + (2 -N ) Ω\(V 2 ∩Ω) |∇u n | 2 dx +2Re i J j=1 Ω j \(V 2 ∩Ω j ) β n b(x)y n (m j • ∇u n )dx - J j=1 V 2 ∩Ω j div(ψ j m j )(|β n u n | 2 -|∇u n | 2 )dx + 2 N i,k=1 ∂ i (ψ j m k j )∂ i u n ∂ k u n dx -2i J j=1 V 2 ∩Ω j β n by n (ψ j m j • ∇u n )dx + o(1
- J j=1 V 2 ∩Ω j div(ψ j m j )(|β n u n | 2 -|∇u n | 2 )dx + 2 N i,k=1 ∂ i (ψ j m k j )∂ i u n ∂ k u n dx -2i J j=1 V 2 ∩Ω j β n b(x)y n (ψ j m j • ∇u n )dx = o(1
N Ω\V 2 ∩Ω |β n y n | 2 dx + (2 -N ) Ω\V 2 ∩Ω |∇y n | 2 dx +2Re i J j=1 Ω j \V 2 ∩Ω j β n b(x)y n (m j • ∇u n )dx o (1).
(2.3.43)

Proof: Multiplying equation (2.3.9) by 2( h j • ∇y n ), we obtain

2β 2 n Ω j y n (h j • ∇y n )dx + 2 Ω j ∆y n (h j • ∇y n )dx + 2i Ω j β n b(x)u n (h j • ∇y n )dx = -2i Ω j β n f 2 n (h j • ∇y n )dx + 2 Ω j (b(x)f 1 n -g 2 n )(h j • ∇y n )dx.
(2.3.44) (i) Estimation of the seconde member in (2.3.44). Using Green's formula and the fact that y n = 0 on (Γ j \ γ j ) ∩ Γ and h j = 0 on γ j ∩ Γ, we obtain -2i

Ω j β n f 2 n ( h j • ∇y n )dx = 2i Ω j β n y n (h j • ∇f 2 n )dx + 2i Ω j β n y n f 2 n (divh j ) dx.
It follows, since f 2 n converges to zero in H 1 0 (Ω) and β n y n is uniformly bounded in L 2 (Ω), that -2i

Ω j β n f 2 n ( h j • ∇y n )dx = o(1). (2.3.45)
As f 1 n converges to zero in H 1 0 (Ω), g 2 n converges to zero in L 2 (Ω) and (∇y n ) is uniformly bounded in L 2 (Ω), we deduce that 2

Ω j (b(x)f 1 n -g 2 n )(h j • ∇y n )dx = o(1
2β 2 n Ω j y n (h j • ∇y n )dx + 2 Ω j ∆y n (h j • ∇y n )dx + 2i Ω j β n b(x)u n (h j • ∇y n )dx = o(1).
(2.3.47) (ii) Estimation of first member of (2.3.44). Using Green's formula in (2.3.47) and following the same technique used in Lemma (2.3.9), we get

N Ω\V 2 ∩Ω |β n y n | 2 dx + (2 -N ) Ω\V 2 ∩Ω |∇y n | 2 dx -2Re i J j=1 Ω j \V 2 ∩Ω j β n b(x)u n (m j • ∇y n )dx o (1).
(2.3.48)

(iii) Estimation of the third integral of (2.3.48). Integrating by parts and using the fact that

u n = y n = 0 on ∂(Ω j \ V 2 ∩ Ω j ) ⊂ Γ, we obtain 2i J j=1 Ω j \V 2 ∩Ω j β n b(x)u n (m j • ∇y n )dx = -2i J j=1 Ω j \V 2 ∩Ω j β n by n (m j • ∇u n )dx -2i J j=1 Ω j \V 2 ∩Ω j β n u n y n (m j • ∇b)dx -2i J j=1 Ω j \V 2 ∩Ω j b(x)div(m j )β n u n y n dx.
(2.3.49)

Using the fact that β n u n is uniformly bounded in L 2 (Ω) and y n = o(1) in the right hand side of (2.3.49), we deduce 2i

J j=1 Ω j \V 2 ∩Ω j β n b(x)u n (m j • ∇y n )dx = -2i J j=1 Ω j \V 2 ∩Ω j β n b(x)y n (m j • ∇u n )dx + o(1
N Ω\V 2 ∩Ω |β n y n | 2 dx + (2 -N ) Ω\V 2 ∩Ω |∇y n | 2 dx +2Re i J j=1 Ω j \V 2 ∩Ω j β n b(x)y n (m j • ∇u n )dx o(1).
The proof is thus complete.

N Ω\(V 2 ∩Ω) |β n u n | 2 + |β n y n | 2 dx + (2 -N ) Ω\(V 2 ∩Ω) |∇u n | 2 + |∇y n | 2 dx ≤ o(1).
(2.3.52) Now, multiplying (2.3.8) by (1 -N )u n , integrating on Ω, using Green's formula, the fact that (

β n u n ) is uniformly bounded in L 2 (Ω), u n = o(1) and y n = o(1), we obtain (1 -N ) Ω |β n u n | 2 dx -(1 -N ) Ω |∇u n | 2 dx = o(1). ( 2
|β n u n | 2 dx -(1 -N ) Ω\(V 2 ∩Ω) |∇u n | 2 dx = o(1
|β n y n | 2 dx -(1 -N ) Ω\(V 2 ∩Ω) |∇y n | 2 dx = o(1
Ω\(V 2 ∩Ω) |∇u n | 2 + |β n u n | 2 + |∇y n | 2 + |β n y n | 2 dx = o(1).
The proof is thus complete.

Non uniform stability in the case a = 1

The aim of this section is to show that system (2.1.4)-(2.1.6) is not uniformly (i.e. not exponentially) stable when the waves propagate with different speeds (i.e. a = 1), since it is already the case for c and b are constants in the whole domain, as shown below. Our result is the following

Theorem 2.4.1. Assume that c = c 0 > 0, b = b 0 = 0 in Ω and that a = 1.
Then the energy of system (2.1.4)-(2.1.6) does not decrease exponentially to zero as t goes to infinity.

Proof: We will show that the resolvent of the operator A is not uniformly bounded on the imaginary axis i.e. condition (H2) does not hold. So, to prove the non uniform stability, it suffices to construct a sequence β n ∈ R and a sequence

U n = (u n , v n , y n , z n ) ∈ D(A) such that β n → +∞, (2.4.1) U n H → +∞ (2.4.2) and (iβ n I -A)U n H ≤ C < +∞. (2.4.3)
For this aim, let µ 2 n > 0 be an eigenvalue of the Laplacian with Dirichlet boundary condition and ϕ n its associated eigenfunction :

-∆ϕ n = µ 2 n ϕ n , in Ω, ϕ n = 0, on Γ. (2.4.4) Set β n = µ n and U n = (u n , v n , y n , z n ) = (b n ϕ n , iβ n b n ϕ n , a n ϕ n , iβ n a n ϕ n ) where a n = (a -1) b 2 + ic b 2 µ n + i bµ n and b n = 1 iµ n b . (2.4.5)
This implies that

U n = (u n , v n , y n , z n ) ∈ D(A) and v n -iµ n u n = 0, (2.4.6) a∆u n -bz n -cv n -iµ n v n = ϕ n , (2.4.7) z n -iµ n y n = 0, (2.4.8) ∆y n + bv n -iµ n z n = ϕ n .
(2.4.9)

It follows that U n is a solution of the equation

AU n -iµ n U n = V n , (2.4.10)
where

V n = (0, ϕ n , 0, ϕ n ) ∈ H.
Finally, from (2.4.10), we deduce that

iµ n U n -AU n 2 H = (0, ϕ n , 0, ϕ n ) 2 H = 2. (2.4.11)
On the other side, we have

U n 2 H = 2(|a n | 2 + |b n | 2 )µ 2 n ∼ (a -1) 2 b 4 µ 2 n → +∞.
(2.4.12)

Consequently, the sequences

β n = µ n and U n = (b n ϕ n , iβ n b n ϕ n , a n ϕ n , iβ n a n ϕ n ) satisfy the conditions (2.4.1)-(2.4.
3). So, using Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Pruss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF], system (2.1.4)-(2.1.6) is not uniformly stable in the energy space H. The proof is thus complete.

Polynomial stability in the case a = 1

The condition of equal speed is then a necessary and sufficient condition for the exponential stability of our system. Therefore, we look for a polynomial energy decay rate. Our second main result when the waves propagate at different speeds (a = 1) can be stated as follows.

Theorem 2.5.1. (Polynomial decay rate) Let a = 1. Assume that all assumptions of Theorem 2.3.3 are satisfied. Then there exists a positive constant C > 0 independent of U 0 such that for all initial data U 0 = (u 0 , u 1 , y 0 , y 1 ) ∈ D(A) the energy of the system (2.1.4)-(2.1.6) satisfies the following decay rate : and sup

E(t) ≤ C 1 t U (0) 2 D(A) , ∀t > 0. ( 2
|β|≥1 1 β 2 (iβI -A) -1 < +∞ (H3)
hold. As condition (H1) was already checked in Theorem 2.2.2, we now prove that condition (H3) holds, using an argument of contradiction. For this aim, we suppose that there exist a real sequence (β n ), with β n -→ +∞ and a sequence

U n = (u n , v n , y n , z n ) ∈ D(A) such that U n H = 1 (2.5.2)
and lim

n→∞ β 2 n (iβ n I -A)U n H = 0. (2.5.3)
Next, by detailing equation (2.5.3), we obtain

iβ 3 n u n -β 2 n v n = f 1 n → 0 in H 1 0 (Ω), (2.5.4) iβ 3 n v n -aβ 2 n ∆u n + b(x)β 2 n z n + c(x)β 2 n v n = g 1 n → 0 in L 2 (Ω), (2.5.5) iβ 3 n y n -β 2 n z n = f 2 n → 0 in H 1 0 (Ω), (2.5.6) iβ 3 n z n -β 2 n ∆y n -b(x)β 2 n v n = g 2 n → 0 in L 2 (Ω). (2.5.7)
Eliminating v n and z n from (2.5.4)-(2.5.7), we get

β 2 n u n + a∆u n -iβ n b(x)y n -iβ n c(x)u n = -g 1 n -bf 2 n -iβ n f 1 n -cf 1 n β 2 n , (2.5.8)
β 2 n y n + ∆y n + iβ n b(x)u n = -iβ n f 2 n + bf 1 n -g 2 n β 2 n .
(2.5.9)

In addition, from equation (2.5.2), (2.5.4) and (2.5.6), we deduce that

u n L 2 (Ω) = O(1) β n and y n L 2 (Ω) = O(1) β n . ( 2
U n 2 -β 2 n (AU n , U n ) = β 2 n Ω c(x)|v n | 2 dx = o(1).
(2.5.12)

Under condition (LH1), it follows

ωc + |v n | 2 dx = o(1) β 2 n .
(2.5.13) So, using equations (2.5.4) and (2.5.12), we get

Ω c(x)|u n | 2 dx = o(1) β 4 n . ( 2
Ω η|β 2 n u n | 2 dx -aβ 2 n Ω η | ∇u n | 2 dx -β 2 n Ω au n (∇η • ∇u n )dx -iβ 3 n Ω ηby n u n dx = o(1).
(2.5.16) Using (2.5.11) and the fact that the sequences (β n y n ) and (∇u n ) are uniformly bounded in L 2 (Ω), we deduce from (2.5.16) that

Ω η | β n ∇u n | 2 dx = o(1)
and

V 2 ∩Ω | β n ∇u n | 2 dx = o(1).
The proof is thus complete.

Lemma 2.5.4. The solution (u n , v n , y n , z n ) ∈ D(A) of system (2.5.4)-(2.5.7) satisfies the following estimation

Ω η|β n y n | 2 dx = o(1)
and

V 2 ∩Ω |β n y n | 2 dx = o(1).
(2.5.17)

Proof: Multiplying equation (2.5.8) by β n η ȳn . Using Green's formula and the fact that y n = 0 on Γ, we obtain Using the definition of function η and condition (LH2), we deduce

Ω β 3 n ηy n u n dx -a Ω β n η(∇u n • ∇y n )dx -a Ω β n y n (∇u n • ∇η)dx -i Ω b(x)η|β n y n | 2 dx -iβ 2 n Ω c(x)ηu n y n dx = o(1
Ω η|β n y n | 2 dx = o(1)
and

V 2 ∩Ω |β n y n | 2 dx = o(1).
The proof is thus complete.

Lemma 2.5.5. The solution (u n , v n , y n , z n ) ∈ D(A) of system (2.5.4)-(2.5.7) satisfies the following estimation

Ω η | ∇y n | 2 dx = o(1)
and

V 2 ∩Ω | ∇y n | 2 dx = o(1).
(2.5.20)

Proof: Multiplying equation (2.5.9) by ηy n . Then, using Green's formula, the condition y n = 0 on Γ, we get

Ω η|β n y n | 2 dx - Ω η|∇y n | 2 dx - Ω (∇η • ∇y n )y n dx + iβ n Ω bu n ηy n = o(1) (2.5.21)
Using (2.5.10) and the fact that (β n y n ) and (∇y n ) are bounded in L 2 (Ω) in (2.5.21), we get

Ω η|β n y n | 2 dx - Ω η | ∇y n | 2 dx = o(1). (2.5.22)
Finally, from (2.5.17), we deduce

Ω η|∇y n | 2 dx = o(1)
and

V 2 ∩Ω | ∇y n | 2 dx = o(1).
The proof is thus complete. Proof of Theorem 2.5. Remark 2.5.6. Note that our results in Theorem 2.5.1 might be more general because the waves are not assumed to be propagated with the same speed. So, this theorem generalize the results of [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] and [START_REF] Kapitonov | Uniform stabilization and exact controllability for a class of coupled hyperbolic systems[END_REF].

Optimality of the polynomial energy decay rate

We study here the optimality of the polynomial decay rate obtained for the N -dimensional coupled wave system in Theorem 2.5.1. For this aim, we will study the asymptotic behavior of the eigenvalues of the operator A in the one dimensional case for b and c are constants. Indeed, we consider the 1-dimensional version of system (2.1.4)-(2.1.6) :

u tt -au xx + by t + cu t = 0, in (0, 1) × (0, +∞), (2.6.1)

y tt -y xx -bu t = 0, in (0, 1) × (0, +∞), (2.6.2) u(0, t) = u(1, t) = y(0, t) = y(1, t) = 0, in (0, +∞) (2.6.3)
with the following initial data :

u(x, 0) = u 0 , y(x, 0) = y 0 , u t (x, 0) = u 1 and y t (x, 0) = y 1 , x ∈ (0, 1)
where 1 = a > 0, c > 0 and b ∈ R . From subsection 2.1, we express system (2.6.1)-(2.6.3) as an evolution equation of type (2.2.3) with H = (H 1 0 (0, 1) × L 2 (0, 1)) 2 and A :

(H 2 (0, 1) ∩ H 1 0 (0, 1) × H 1 0 (0, 1)) 2 → H, defined by A(u, v, y, z) := (v, au xx -bz -cv, z, y xx + bv) .
The aim of this section is the following result :

Theorem 2.6.1. Assume that N = 1, a = 1, b = b 0 = 0 and c = c 0 > 0.
The energy decay rate (2.5.1) is optimal in the sense that for any ε > 0, we can not expect the decay rate 1 t 1+ε for all initial data U 0 ∈ D(A).

For the proof of the Theorem 2.6.1, we first study the asymptotic behavior of the eigenvalues of the operator A. Since A is dissipative, we fix α 0 > 0 small enough and we study the asymptotic behavior of the eigenvalues λ of A in the strip

S = {λ ∈ C : -α 0 ≤ Re(λ) ≤ 0} .
So, let λ ∈ C be an eigenvalue of A with associated eigenvector U = (u, v, y, z) ∈ D(A).

Then AU = λU and equivalently

           v = λu, au xx -bz -cv = λv, z = λy, y xx + bv = λz, u(0) = u(1) = y(0) = y(1) = 0.
(2.6.4)

Eliminating v and z from (2.6.4), we get From the second equation of (2.6.5), we have

       au xx -λ(λ + c)u -bλy = 0, y xx -λ 2 y + bλu = 0, u(0) = u(1) = 0, y(0) = y(1) = 0.
u = 1 bλ [λ 2 y -y xx ].
(2.6.6) Substituting (2.6.6) in the first equation of (2.6.5), we get

   ay xxxx -[λ 2 (a + 1) + cλ]y xx + λ 2 (λ 2 + cλ + b 2 )y = 0, y(0) = y(1) = 0, y xx (0) = y xx (1) = 0.
(2.6.7)

The characteristic equation associated with system (2.6.7) is given by

Q(r) := ar 4 -[λ 2 (a + 1) + cλ]r 2 + λ 2 (λ 2 + cλ + b 2 ) = 0.
In order to proceed, we set the following notation. Here and below, in the case where z is a non zero non-real number, we define (and denote) by √ z the square root of z ; i.e., the unique complex number with positive real part whose square is equal to z. The general solution of the first equation of (2.6.7) is given by

y ( x) = 4 i=1 c i e r i (λ)x , where r 1 (λ) = 1 √ 2a λ λ(a + 1) + c + (a -1) 2 λ 2 -2c(a -1)λ -4ab 2 + c 2 , r 2 (λ) = -r 1 (λ),
(2.6.8)

r 3 (λ) = 1 √ 2a λ λ(a + 1) + c -(a -1) 2 λ 2 -2c(a -1)λ -4ab 2 + c 2 , r 4 (λ) = -r 3 (λ).
(2.6.9) For simplicity, here and below, we denote r i (λ) by r i . Hence, the general solution is given by

y(x) = A 1 sinh(r 1 x) + A 2 cosh(r 1 x) + A 3 sinh(r 3 x) + A 4 cosh(r 3 x).
Using the boundary condition y(0) = y xx (0) = 0, we get

A 2 + A 4 = 0, A 2 r 2 1 + A 4 r 2 3 = 0,
(2.6.10)

which implies A 2 = A 4 = 0, since r 2 3 -r 2 1 = 0. Therefore y(x) = A 1 sinh(r 1 x) + A 3 sinh(r 3 x).
The boundary conditions y(1) = y xx (1) = 0 may be written as the following system

M (λ)c(λ) = sinh(r 1 ) sinh(r 3 ) r 2
1 sinh(r 1 ) r 2 3 sinh(r 3 )

A 1 A 3 = 0. (2.6.11)
So, set F (λ) = det(M (λ)). We have the following results : of simple roots of F (that are also simple eigenvalues of A) satisfying the following asymptotic behavior

λ (0) n = inπ - ib 2 2(a -1)nπ - cb 2 2(a -1) 2 n 2 π 2 + O 1 n 3 , ∀ |n| ≥ n 0
(2.6.12) and λ

(1)

n = in π √ a - c 2 + O 1 n , ∀ |n | ≥ n 0 .
(2.6.13)

Proof: It is easy to see that system (2.6.11) has a non-trivial solution (A 1 , A 3 ) = (0, 0) if and only if λ is solution of the following equation

F (λ) = (r 2 3 -r 2 1 ) sinh(r 1 ) sinh(r 3 ) = 0.
Since r 2 3 -r 2 1 = 0 then sinh(r 1 ) = 0 or sinh(r 3 ) = 0. Thus

r 1 = inπ or r 3 = in π, n, n ∈ Z.
It follows, from the asymptotic expansion in (2.6.8) and (2.6.9), that

λ - b 2 2(a -1)λ - cb 2 2(a -1) 2 λ 2 + O( 1 λ 3 ) = inπ, or λ √ a + c 2 √ a + O( 1 λ ) = in π.
Hence, since n ∼ n ∼ λ, we obtain two branches λ (0)

n and λ

n of eigenvalues of the operator A which satisfy the asymptotic behavior (2.6.12) and (2.6.13). The proof is thus complete.

n , with n ≥ n 0 , be the sequence of eigenvalues of the operator A described in Proposition 2.6.2 and let U n ∈ D(A) be the associated normalized eigenfunction. Moreover, we introduce the following sequence

β n = (λ (0) n ), ∀n ≥ n 0 .
Next, using (2.6.12) we have

(iβ n I -A)U n = cb 2 2(a -1) 2 n 2 π 2 + O 1 n 3 U n , ∀ n ≥ n 0 , 57
Chapitre 2. Local indirect stabilization of N-d system of two coupled wave equations under geometric conditions and therefore

β 2-2 l n (iβ n I -A)U n H ∼ cb 2 2(a -1) 2 π 2 × 1 n 2 1+ , ∀ n ≥ n 0 .
Thus, we deduce that lim

n→+∞ β 2-2 l n (iβ n I -A)U n H = 0.
Thanks to Theorem 2.4 of Borichev-Tomilov in [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], we deduce that for U 0 ∈ D(A), e tA U 0 H decays slower that 1

t 1 2-2
l as the time t → +∞. The proof is thus complete. Remark 2.6.4. We can use Theorem 3.4.1 in [START_REF] Najdi | Étude de la stabilisation exponentielle et polynomiale de certains systèmes d'équations couplées par des contrôles indirects bornés ou non bornés[END_REF] and equation (2.6.12), to deduce the optimality of the polynomial energy decay rate (2.5.1) in the case N = 1.

Chapitre 3 Exact controllability and stabilization of locally coupled wave equations

Abstract : In this chapter, we study the exact controllability and stabilization of a system of two wave equations coupled by velocities with an internal locally control acted at only one equation. We distinguish two cases. In the first one, when the waves propagate at the same speed : using frequency domain approach combined with multiplier technique, we prove that the system is exponentially stable when the coupling region satisfies the geometric control condition GCC. Following a result of Haraux ([31]), we establish the main indirect observability inequality. This results leads, by the HUM method, to prove that the total system is exactly controllable by means of locally distributed control. On the contrary case, when the waves propagate at different speed, we establish an exponential decay rate in the weak energy space. Consequently, the system is exactly controllable using a result of [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF]. Finally, numerically, we provide results that insure the theoretical results of [START_REF] Kassem | Stabilisation locale indirecte d'un système n-d de deux équations d'ondes couplées sous conditions géométriques[END_REF].

Introduction

Motivation and aims.

Let Ω be an non empty connected open subset of R N having a boundary Γ of class C 2 . In [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF], F. Alabau et al. considered the energy decay of a system of two wave equations coupled by velocities

u tt -a∆u + ρ(x, u t ) + b(x)y t = 0 in Ω × R * + , (3.1.1) y tt -∆y -b(x)u t = 0 in Ω × R * + , (3.1.2) u = y = 0 on Γ × R * + , (3.1.3)
where a > 0 constant, b ∈ C 0 (Ω, R) and ρ(x, u t ) is a non linear damping. Using an approach based on multiplier techniques, weighted nonlinear inequalities and the optimalweight convexity method (developed in [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF]), the authors established an explicit energy decay formula in terms of the behavior of the nonlinear feedback close to the origin. Their results are obtained in the case when the following three conditions are satisfied : the waves propagate at the same speed (a = 1), the coupling coefficient b(x) is small positive (0

≤ b(x) ≤ b 0 , b 0 ∈ (0, b ]
where b is a constant depending on Ω and on the control region) and both the coupling and the damping regions satisfying an appropriated geometric conditions named Piecewise Multipliers Geometric Conditions (introduced in [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF] and denoted by PMGC in short). In their work, the case where the waves are not assumed to be propagated with equal speeds (a is not necessarily equal to 1) and/or the coupling coefficient b(x) is not assumed to be positive and small has been left as an open problem even when the damping term ρ is linear with respect to the second variable. Recently, C. Kassem et al. in [START_REF] Kassem | Stabilisation locale indirecte d'un système n-d de deux équations d'ondes couplées sous conditions géométriques[END_REF], answered this important open question by studying the stabilization of the following linear system :

u tt -a∆u + c(x)u t + b(x)y t = 0 in Ω × R * + , (3.1.4) y tt -∆y -b(x)u t = 0 in Ω × R * + , (3.1.5) u = y = 0 on Γ × R * + , (3.1.6)
in the case where the waves propagate with equal or different speeds and the coupling coefficient is not assumed to be positive and small. Indeed, they distinguished two cases. The first one is when the waves propagate at the same speed (i.e. a = 1), but unlike the works of [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF], the coupling coefficient function b is not necessarily assumed to be positive and small. In this case, under the condition that the coupling region and the damping region have non empty intersection satisfying the PMGC conditions, they established an exponential energy decay rate for weak initial data. On the contrary (i.e. a = 1 ) they first proved the lack of the exponential stability of the system. However, under the same geometric condition, an optimal energy decay rate of type 1 t was established for smooth initial data. The aim of this chapter is to investigate the exact controllability of the following system :

u tt -a∆u + b(x)y t = c(x)v(t) in Ω × R * + , (3.1.7) y tt -∆y -b(x)u t = 0 in Ω × R * + , (3.1.8) u = y = 0 on Γ × R * + , (3.1.9)
with the following initial data

u(x, 0) = u 0 , y(x, 0) = y 0 , u t (x, 0) = u 1 and y t (x, 0) = y 1 , x ∈ Ω, (3.1.10)
under appropriate geometric conditions. Here,

a > 0 constant, b ∈ C 0 (Ω, R), c ∈ C 0 (Ω, R + )
and v is an appropriate control. The idea is to use a result of A. Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] for which the observability of the homogeneous system associated to

(Ω) × L 2 (Ω)) 2 is established.
This leads, by the HUM method introduced by J. L. Lions in [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], to the exact controllability of system (3.1.7)-(3.1.9) in the space (H -1 (Ω) × L 2 (Ω)) 2 . Noting that, the geometric situations covered here are richer than those considered in [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] and [START_REF] Kassem | Stabilisation locale indirecte d'un système n-d de deux équations d'ondes couplées sous conditions géométriques[END_REF]. Furthermore, on the contrary when the waves propagate at different speeds, (i.e., a = 1), we establish the exponential stability of system (3.1.4)-(3.1.6) in the space H 1 0 (Ω)×L 2 (Ω)×L 2 (Ω)×H -1 (Ω) provided that the damping region satisfies the PMGC condition while the coupling region includes in the damping region and satisfying the GCC conditions. Consequently, an observability inequality of the solution of the homogeneous system associated to (3.1.7)-(3.1.9) is established. This leads, by the HUM method, to the exact controllability of system (3.1.7)-(3.1.9) in the space

L 2 (Ω) × H -1 (Ω) × H -1 (Ω) × (H 2 (Ω) ∩ H 1 0 (Ω))
, where the duality is according to L 2 (Ω). Finally, we perform numerical tests in the 1-D case to insure the theoretical results obtained here and in [START_REF] Kassem | Stabilisation locale indirecte d'un système n-d de deux équations d'ondes couplées sous conditions géométriques[END_REF]. In fact, the numerical results show a better behavior that the one expected by the theoretical results.

Literature

Since the work of J. L. Lions in [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], the observability and controllability of coupled wave equations have been studied by an intensive number of publications. In [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], J. L. Lions studied the complete and partial observability and controllability of coupled systems of either hyperbolic-hyperbolic type or hyperbolic-parabolic type. These results assume that the coupling parameter is sufficiently small. In [START_REF] Alabau-Boussouira | Observabilité frontière indirecte de systèmes faiblement couplés[END_REF] and [START_REF] Alabau-Boussouira | A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems[END_REF], F. Alabau studied the indirect boundary observability of an abstract system of two weakly coupled second order evolution equations where the coupling coefficient is strictly positive in the whole domain. In particular, using a piecewise multiplier method, she proved that, for a sufficiently large time T , the observation of the trace of the normal derivative of the first component of the solution on a part of the boundary allows us to get back a weakened energy of the initial data. Consequently, using Hilbert Uniqueness Method, she proved that the system is exactly controllable for small coupling parameter by means of one boundary control. Noting that, the situation where the waves propagate with different speeds is not covered. Later, the indirect boundary controllability of a system of two weakly coupled one-dimensional wave equations has been studied by Z. Liu and B. Rao in [START_REF] Liu | A spectral approach to the indirect boundary control of a system of weakly coupled wave equations[END_REF]. Using the non harmonic analysis, they established several weak observability inequalities which depend on the ratio of the wave propagation speeds and proved the indirect exact controllability. The null controllability of the reaction diffusion system has been studied by F. Ammar-Khodja et al. in [START_REF] Ammar-Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF], by deriving an observability estimate for the linearized problem. The exact controllability of a system of weakly coupled wave equations with an internal locally control acted on only one equation has been studied by A. Wehbe and W. Youssef in [START_REF] Wehbe | Observabilité et contrôlabilité exacte indirecte interne par un contrôle localement distribué de systèmes d'équations couplées[END_REF] and [START_REF] Wehbe | Indirect locally internal observability and controllability of weakly coupled wave equations[END_REF]. They showed that, for sufficiently large time, the observation of the velocity of the first component of the solution on a neighborhood of a part of the boundary allows us to get back a weakened energy of initial data of the second component, this if the coupling parameter is sufficiently small, but non-vanishing and by the HUM method, they proved that the total system is exactly controllable. F. Alabau and M. Léautaud in [START_REF] Alabau-Boussouira | Indirect controllability of locally coupled wave-type systems and applications[END_REF], considered a symmetric systems of two wave-type equations only one of them being controlled. The two equations are coupled by zero order terms, localized in part of the domain. They obtained an internal and a boundary controllability result in any space dimension, provided that 3.2. Well posedeness and strong stability 62 both the coupling and the control regions satisfy the Geometric Control Condition.

Description of the chapter

This chapter is organized as follows : In section (3.2), first, we show that the system (3.1.4)-(3.1.6) can be reformulated into a first order evolution equation and we deduce the well posedness property of the problem by the semi group approach. Second, by using Theorem 2.2 of [START_REF] Kassem | Stabilisation locale indirecte d'un système n-d de deux équations d'ondes couplées sous conditions géométriques[END_REF], we show that our problem is strongly stable without geometric conditions. In section 3.3, we show the exponential decay rate of system (3.1.4)-(3.1.6) when the coupling region b is a subset of the damping region c and satisfying the geometric control condition GCC. After that, we show that our system is exactly controllable by using Proposition 2 of A. Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF]. In section 3.4, we show the exponential decay rate of system (3.1.4)-(3.1.6) in the weak energy space provided that the damping region satisfies the PMGC condition while the coupling region is a subset of the damping region and satisfying the GCC condition. Section 5 is devoted to the numerical approximation of the problem by a finite difference discretization and to the validation of the theoretical results stated in the previous sections.

Well posedeness and strong stability

Let us define the energy space

H = H 1 0 (Ω) × L 2 (Ω) 2
equipped with the following inner product and norm, respectively

(U, U ) H = a Ω (∇u • ∇ u)dx + Ω v v dx + Ω (∇y • ∇ y)dx + Ω z zdx, U H = (U, U ) H ,
for all U = (u, v, y, z), U = ( u, v, y, z) ∈ H.

Let (u, u t , y, y t ) be a regular solution of the system (3.1.4)-(3.1.6). Its associated energy is defined by

E(t) = 1 2 Ω |u t | 2 + a|∇u| 2 + |y t | 2 + |∇y| 2 dx.
A straight forward computations gives

E (t) = - Ω c(x)|u t | 2 dx ≤ 0.
Consequently, system (3.1.4)-(3.1.6) is dissipative in the sense that its energy is nonincreasing with respect to t. Setting U = (u, u t , y, y t ), system (3.1.4)-(3.1.6) may be recast as :

U t = AU, in (0, +∞), U (0) = (u 0 , u 1 , y 0 , y 1 ),
where the unbounded operator A : D(A) ⊂ H → H is given by :

D(A) = (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) 2 (3.2.1)
and

AU = ( v, a∆u -bz -cv, z, ∆y + bv ), ∀ U = (u, v, y, z) ∈ D(A). (3.2.2)
Noting that due to the fact that c(x) ≥ 0, the operator A is m-dissipative in H and generates a C 0 -semigroup of contractions (e tA ) t≥0 . So, system (3.1.4)-(3.1.6) is well posed in H.

We need now to study the asymptotic behavior of E(t). For this aim, we suppose that there exists a non empty open ω c + ⊂ Ω satisfying the following condition ) is strongly stable using Theorem 2.2 in [START_REF] Kassem | Stabilisation locale indirecte d'un système n-d de deux équations d'ondes couplées sous conditions géométriques[END_REF], i.e.

{x ∈ Ω : c(x) > 0} ⊃ ω c + . ( LH1
lim t→+∞ e tA (u 0 , u 1 , y 0 , y 1 ) H = 0 ∀(u 0 , u 1 , y 0 , y 1 ) ∈ H.

Exponential stability and exact controllability in

the case a = 1

Exponential stability

This subsection is devoted to study the exponential stability of system (3.1.4)-(3.1.6) in the case when the waves propagate at the same speed, i.e., a = 1 under an appropriate geometric conditions. Before we state our results, we recall the Geometric Control Conditions GCC introduced by Rauch and Taylor in [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF] for manifolds without boundaries and by Bardos, Lebeau and Rauch in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] for domains with boundaries. Definition 3.3.1. We say that a subset ω of Ω satisfies the GCC if every ray of the geometrical optics starting at any point x ∈ Ω at t = 0 enters the region ω in finite time T.

We recall also the Piecewise Multipliers Geometric Condition introduced by K. Liu in [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF]. Definition 3.3.2. We say that ω satisfies the Piecewise Multipliers Geometric Condition (PMGC in short) if there exist Ω j ⊂ Ω having Lipschitz boundary Γ j = ∂Ω j and x j ∈ R N , j = 1, ..., J such that Ω j ∩ Ω i = ∅ for j = i and ω contains a neighborhood in Ω of the set ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j where γ j (x j ) = {x ∈ Γ j : (x -x j ) • ν j (x) > 0} and ν j is the outward unit normal vector to Γ j . Remark 3.3.3. The PMGC is the generalization of the Multipliers Geometric Condition (MGC in short) introduced by Lions in [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], saying that ω contains a neighborhood in Ω of the set {x ∈ Γ : (x -x 0 ) • ν(x) > 0}, for some x 0 ∈ R N , where ν is the outward unit normal vector to Γ = ∂Ω. Now, we are in position to state our first main result by the following theorem : Theorem 3.3.4. (Exponential decay rate) Let a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω b ⊂ ω c + satisfies the geometric control conditions GCC and that b, c ∈ W 1,∞ (Ω). Then there exist positive constants M ≥ 1, θ > 0 such that for all initial data (u 0 , u 1 , y 0 , y 1 ) ∈ H the energy of the system (3.1.4)-(3.1.6) satisfies the following decay rate :

E(t) ≤ M e -θt E(0), ∀t > 0. (3.3.1)
Remark 3.3.5. The geometric situations covered by Theorem 3.3.4 are richer than that considered in [START_REF] Kassem | Stabilisation locale indirecte d'un système n-d de deux équations d'ondes couplées sous conditions géométriques[END_REF] and [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF]. Indeed, in the previous references, the authors consider the PMGC geometric conditions that are more restrictive than GCC. On the other hand, unlike the results in [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF], we have no restriction in Theorem 3.3.4 on the upper bound and the sign of the coupling function coefficient b. This theorem is then a generalization in the linear case of the result of [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] where the coupling coefficient considered have to satisfy

0 ≤ b(x) ≤ b 0 , b 0 ∈ (0, b ]
where b is a constant depending on Ω and on the control region.

In order to prove Theorem 3.3.4, we apply a result of Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Prüss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF].

(iβI -A) -1 L(H) < ∞ (H2) hold.
Since the resolvent of A is compact and 0 ∈ ρ(A), then from the fact that our system is strongly stable, we deduce that condition (H1) is satisfied. We now prove that condition (H2) holds, using an argument of contradiction. For this aim, we suppose that there exist a real sequence β n with β n → +∞ and a sequence

U n = (u n , v n , y n , z n ) ∈ D(A) such that U n H = 1, (3.3.2)
and lim

n→∞ (iβ n I -A)U n H = 0. (3.3.3)
Next, detailing equation (3.3.3), we get

iβ n u n -v n = f 1 n → 0 in H 1 0 (Ω), (3.3.4) iβ n v n -∆u n + bz n + cv n = g 1 n → 0 in L 2 (Ω), (3.3.5) iβ n y n -z n = f 2 n → 0 in H 1 0 (Ω), (3.3.6) iβ n z n -∆y n -bv n = g 2 n → 0 in L 2 (Ω). (3.3.7)
Eliminating v n and z n from the previous system, we obtain the following system

β 2 n u n + ∆u n -iβ n by n -iβ n cu n = -g 1 n -bf 2 n -iβ n f 1 n -cf 1 n , (3.3.8)
β 2 n y n + ∆y n + iβ n bu n = -iβ n f 2 n + bf 1 n -g 2 n . ( 3
iβ n U n 2 H -(AU n , U n ) H = Ω c(x)|v n | 2 dx = o(1). (3.3.12)
Under condition (LH1), it follows that

ωc + |v n | 2 dx = o(1). (3.3.13)
Then, using equations (3.3.12) and (3.3.4), we get

Ω c|β n u n | 2 dx = o(1). (3.3.14)
Consequently, we have

ωc + |β n u n | 2 dx = o(1).
The proof is thus complete. Proof: Multiplying equation (3.3.8) by cu n , integrating by parts and using the fact that u n = 0 on Γ, we get

Ω c|β n u n | 2 dx - Ω c|∇u n | 2 dx - Ω (∇c • ∇u n )u n dx -i Ω β n by n cu n dx -i Ω β n cu n u n dx = Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n )cu n dx. (3.3.16)
Using the fact that f 1 n , f 2 n converge to zero in H 1 0 (Ω), g 1 n converges to zero in L 2 (Ω) and β n u n is uniformly bounded in L 2 (Ω), we obtain

Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n )cu n dx = o(1). (3.3.17)
Using the fact that ∇u n , β n y n , β n u n are uniformly bounded in L 2 (Ω) and u n = o(1), we get The proof is thus complete. Proof: The proof contains three points. i) First, multiplying equation (3.3.8) by 1 βn ∆y n , then using Green's formula and the fact that u n = f 1 n = 0 on Γ, we obtain

- Ω (∇c • ∇u n )u n dx -i Ω β n by n cu n dx -i Ω β n cu n u n dx = o(1). ( 3
Ω c|β n u n | 2 dx - Ω c|∇u n | 2 dx = o(1). (3.3.19)
- Ω β n (∇u n • ∇y n )dx + 1 β n Ω ∆u n ∆y n dx + i Ω (∇b • ∇y n )y n dx + i Ω b|∇y n | 2 dx + i Ω (∇c • ∇y n )u n dx + i Ω c(∇u n • ∇y n )dx (3.3.21) = Ω (-g 1 n -bf 2 n -cf 1 n ) 1 β n ∆y n dx + i Ω (∇f 1 n • ∇y n )dx.
As f 1 n , f 2 n converge to zero in H 1 0 (Ω), g 1 n converges to zero in L 2 (Ω) and the fact that 1 βn ∆y n , ∇y n are uniformly bounded in L 2 (Ω), we have

Ω (-g 1 n -bf 2 n -cf 1 n ) 1 β n ∆y n dx + i Ω (∇f 1 n • ∇y n )dx = o(1). (3.3.22)
Using the fact that ∇y n is uniformly bounded in L 2 (Ω),

u n L 2 (Ω) = o(1), y n L 2 (Ω) = o(1)
and using the estimation (3.3.15), we get

i Ω (∇b • ∇y n )y n dx + i Ω (∇c • ∇y n )u n dx + i Ω c(∇u n • ∇y n )dx = o(1). ( 3
- Ω β n (∇u n • ∇y n )dx + 1 β n Ω ∆u n ∆y n dx + i Ω b|∇y n | 2 dx = o(1). (3.3.24)
ii) Similarly, multiplying equation (3.3.9) by 1 βn ∆u n , then using Green's formula and the fact that y n = f 2 n = 0 on Γ, we obtain

- Ω β n (∇y n • ∇u n )dx + 1 β n Ω ∆y n ∆u n dx -i Ω (∇b • ∇u n )u n dx -i Ω b|∇u n | 2 dx = Ω (bf 1 n -g 2 n ) 1 β n ∆u n dx + i Ω (∇f 2 n • ∇u n )dx. (3.3.25)
Using the fact that f 1 n , f 2 n converge to zero in H 1 0 (Ω), g 2 n converges to zero in L 2 (Ω) and the fact that 1 βn ∆u n , ∇u n are uniformly bounded in L 2 (Ω), we get

Ω (bf 1 n -g 2 n ) 1 β n ∆u n dx + i Ω (∇f 2 n • ∇u n )dx = o(1). (3.3.26)
Also, using the fact that ∇u n is uniformly bounded in L 2 (Ω), u n L 2 (Ω) = o(1), we have The proof is thus complete. Proof: Multiplying equation (3.3.9) by by n . Then using Green's formula and the fact that y n = 0 on Γ, we obtain

-i Ω (∇b • ∇u n )u n dx = o(1). ( 3
- Ω β n (∇y n • ∇u n )dx + 1 β n Ω ∆y n ∆u n dx -i Ω b|∇u n | 2 dx = o(1). ( 3
Ω b|β n y n | 2 dx - Ω b|∇y n | 2 dx - Ω (∇b • ∇y n )y n dx + i Ω b 2 β n u n y n dx = Ω (-iβ n f 2 n + bf 1 n -g 2 n )by n dx. (3.3.31) As f 1 n , f 2 n converge to zero in H 1 0 (Ω), g 2 n converges to zero in L 2 (Ω) and β n y n is uniformly bounded in L 2 (Ω), we get Ω (-iβ n f 2 n + bf 1 n -g 2 n )by n dx = o(1). (3.3.32)
Using the fact that β n u n and ∇y n are uniformly bounded in L 2 (Ω) and This yields

y n L 2 (Ω) = o(1), we get Ω (∇b • ∇y n )y n + i Ω b 2 β n u n y n dx = o(1). ( 3
ω b |β n y n | 2 dx = o(1).
The proof is thus complete.

Lemma 3.3.10. Let f n be a bounded sequence in L 2 (Ω). Then the solution φ n ∈ H 1 0 (Ω) ∩ H 2 (Ω) of the following system

β 2 n φ n + ∆φ n -ibβ n φ n = f n in Ω, φ n = 0 on Γ, (3.3.34)
verifies the following estimation

Ω (|β n φ n | 2 + |∇φ n | 2 )dx ≤ C Ω |f n | 2 dx, (3.3.35)
where C is a constant independent of n.

Proof: Consider the following wave equation

φ tt -∆φ + bφ t = 0 in Ω, φ = 0 on Γ. (3.3.36)
System (3.3.36) is well posed in the space H = H 1 0 (Ω) × L 2 (Ω) and since ω b verifies GCC condition then it is exponentially stable (see [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]). Therefore, following Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Pruss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF], we deduce that the resolvent of its corresponding operator

A aux : D(A aux ) -→ H 1 0 (Ω) × L 2 (Ω) defined by D(A aux ) = (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) and A aux (φ, φ) = ( φ, ∆φ -b φ
) is uniformly bounded on the imaginary axis.

On the other hand, system (3.3.34) can be rewritten in the form :

iβ n φ n -φn = 0, iβ n φn -∆φ n + b φn = -f n . (3.3.37)
So,

iβ n -A aux φ n φn = 0 -f n . (3.3.38)
Equivalently,

φ n φn = iβ n -A aux -1 0 -f n . (3.3.39)
This yields

(φ n , φn ) 2 H ≤ (iβ n -A aux ) -1 2 L(H) (0, -f n ) 2 H ≤ C Ω |f n | 2 dx, (3.3.40)
where C is a constant independent of n. Consequently, we deduce

Ω (|β n φ n | 2 + |∇φ n | 2 )dx ≤ C Ω |f n | 2 dx.
The proof is thus complete. n φ n where φ n is a solution of (3.3.34). Then using Green's formula and the fact that u n = φ n = 0 on Γ, we obtain

Ω β 2 n u n (β 2 n φ n + ∆φ n )dx -i Ω bβ n y n β 2 n φ n dx -i Ω cβ n u n β 2 n φ n dx = Ω (-g 1 n -bf 2 n -cf 1 n )β 2 n φ n dx -i Ω β n f 1 n β 2 n φ n dx. (3.3.42)
Substituting the first equation of system (3.3.34) into the first term of (3.3.42), we get

Ω |β n u n | 2 dx -i Ω β 2 n φ n bβ n u n dx -i Ω bβ n y n β 2 n φ n dx -i Ω cβ n u n β 2 n φ n dx = Ω (-g 1 n -bf 2 n -cf 1 n )β 2 n φ n dx -i Ω β n f 1 n β 2 n φ n dx. (3.3.43) As f 1 n , f 2 n converge to zero in H 1 0 (Ω) and β 2 n φ n is uniformly bounded in L 2 (Ω) due to (3.3.35), we get Ω (-g 1 n -bf 2 n -cf 1 n )β 2 n φ n dx = o(1). (3.3.44)
From the first equation of (3.3.34), we have

β 2 n φ n = u n -∆φ n -ibβ n φ n . Consequently, we have -i Ω β n f 1 n β 2 n ϕ n dx = -i Ω β n f 1 n (u n -∆φ n -ibβ n φ n )dx = -i Ω β n f 1 n u n dx -i Ω β n (∇φ n • ∇f 1 n )dx - Ω bf 1 n β 2 n φ n , (3.3.45)
which yields

-i Ω β n f 1 n β 2 n ϕ n dx = o(1), (3.3
Ω |β n u n | 2 dx -i Ω β 2 n φ n bβ n u n dx -i Ω bβ n y n β 2 n φ n dx -i Ω cβ n u n β 2 n φ n dx = o(1). (3.3.47)
Finally, using estimations (3.3.11), (3.3.30) and the fact that β 2 n φ n is uniformly bounded in L 2 (Ω) into the previous equation, we obtain

Ω |β n u n | 2 dx = o(1). (3.3.48)
The proof is thus complete. Proof: Taking f n = y n in Lemma 3.3.10. Multiplying equation (3.3.9) by β 2 n φ n where φ n is a solution of (3.3.34). Then using Green's formula and the fact that y n = φ n = 0 on Γ, we obtain

Ω β 2 n y n (β 2 n φ n + ∆φ n )dx + i Ω bβ n u n β 2 n φ n dx = -i Ω β n f 2 n β 2 n φ n + Ω (bf 1 n -g 2 n )β 2 n φ n dx. (3.3.50)
Then, substituting the first equation of problem (3.3.34) into the first term of (3.3.50), we get

Ω |β n y n | 2 dx -i Ω bβ 2 n φ n β n y n dx + i Ω bβ n u n β 2 n φ n dx = -i Ω β n f 2 n β 2 n φ n + Ω (bf 1 n -g 2 n )β 2 n φ n dx. (3.3.51) Since β 2 n φ n is uniformly bounded in L 2 (Ω), f 1 n converges to zero in H 1 0 (Ω) and g 2 n converges to zero in L 2 (Ω), we have Ω (-bf 1 n -g 2 n )β 2 n φ n dx = o(1). (3.3.52)
Moreover, using the first equation of problem (3.3.34) and integrating by parts yields

-i Ω β n f 2 n β 2 n φ n dx = -i Ω (y n -∆φ n -ibβ n φ n )β n f 2 n dx = -i Ω f 2 n β n y n dx -i Ω β n (∇φ n .∇f 2 n )dx - Ω bf 2 n β 2 n φ n dx. (3.3.53)
Using the fact that β n y n , β 2 n φ n and β n ∇φ n are uniformly bounded in L 2 (Ω) and f 2 n converges to zero in H 1 0 (Ω) in (3.3.53), we get

-i Ω β n f 2 n β 2 n φ n dx = o(1). ( 3
Ω |β n y n | 2 dx -i Ω bβ 2 n φ n β n y n dx + i Ω bβ n u n β 2 n φ n dx = o(1). (3.3.55)
Finally, using (3.3.30), (3.3.41) and the fact that β 2 n φ n is uniformly bounded in L 2 (Ω), we deduce

Ω |β n y n | 2 dx = o(1).
The proof is thus complete. (3.3.56)

Proof: Multiplying equation (3.3.8) by u n , applying Green's formula and using the fact that u n = 0 on Γ, we get Similarly, multiplying equation (3.3.9) by y n and applying Green's formula and using the fact that y n = 0 on Γ, we get

Ω |β n u n | 2 dx - Ω |∇u n | 2 dx -i Ω β n by n u n dx -i Ω β n cu n u n dx = o(1). ( 3
Ω |β n y n | 2 dx - Ω |∇y n | 2 dx + i Ω β n bu n y n dx = o(1). ( 3

Observability and exact controllability

First, we consider the following homogeneous system associated to (3.1.7)-(3.1.9) for a = 1 by :

ψ tt -∆ψ + b(x)ϕ t = 0 in Ω × R + , (3.3.61) ϕ tt -∆ϕ -b(x)ψ t = 0 in Ω × R + , (3.3.62) ψ = ϕ = 0 on Γ × R + , (3.3.63) ψ(•, 0) = ψ 0 , ψ t (•, 0) = ψ 1 , ϕ(•, 0) = ϕ 0 , ϕ t (•, 0) = ϕ 1 in Ω. (3.3.64)
Let Φ = (ψ, ψ t , ϕ, ϕ t ) be a regular solution of system (3.3.61)-(3.3.63), its associated total energy is given by : It is also well posed and admits a unique solution in the energy space H. Now, we establish the direct and indirect inequality given by the following theorem :

E(t) = 1 2 Ω |ψ t | 2 + |∇ψ| 2 + |ϕ t | 2 + |∇ϕ| 2 dx. ( 3
Theorem 3.3.14. Let a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω b ⊂ ω c + satisfying the geometric control condition GCC and that b, c ∈ W 1,∞ (Ω).

Then there exists a time T 0 such that for all T > T 0 , there exist two constants M 1 > 0, M 2 > 0 such that the solution of system (3.3.61)-(3.3.63) satisfies the following observability inequalities :

M 1 Φ 0 2 H ≤ T 0 Ω c(x) |ψ t | 2 dxdt ≤ M 2 Φ 0 2 H , (3.3.67)
for all Φ 0 = (ψ 0 , ψ 1 , ϕ 0 , ϕ 1 ) ∈ H.

Proof:

The direct inequality follows from the definition of the total energy for all T > 0.

While the proof of the inverse inequality is a direct consequence of Proposition 2 of Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] for which the exponentially stability (3.3.1) implies the existence of a time T 0 > 0 such that for all T > T 0 there exist two constants M 1 > 0 and M 2 > 0 such that (3.3.67) holds. The proof is thus complete. Now, we are ready to study the exact controllability of a system (3.1.7)-(3.1.9) by using the HUM method. First, thanks to the direct inequality, the solution of the system (3.1.7)-(3.1.9) can be obtained as usual by the method of transposition (see [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF]). Let v 0 ∈ L 2 (0, T ; L 2 (w c + )), we choose the control

v(t) = - d dt v 0 (t) ∈ [H 1 (0, T ; L 2 (ω c + ))] , (3.3.68)
where the derivative d dt is not taken within the meaning of the distributions but within the meaning of the duality between H 1 (0, T ; L 2 (ω c + )) and its dual [H 1 (0, T ; L 2 (ω c + ))] , i.e.,

- T 0 d dt v 1 (t)µ(t)dt = T 0 v 1 (t) d dt µ(t)dt, ∀µ ∈ H 1 (0, T ; L 2 (ω c + ).
Then we have the following result :

Theorem 3.3.15. Let T > 0 and a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω b ⊂ ωc + satisfying the geometric control condition GCC and that b, c ∈ W 1,∞ (Ω). Given

U 0 = (u 0 , u 1 , y 0 , y 1 ) ∈ (L 2 (Ω) × H -1 (Ω)) 2 , v = - d dt v 0 ∈ [H 1 (0, T ; L 2 (ω c + ))] ,
the controlled system (3.1.7)-(3.1.9) has a unique weak solution

U = (u, u t , y, y t ) ∈ C 0 ([0, T ], (L 2 (Ω) × H -1 (Ω)) 2 ).
Proof: Let (ψ, ψ t , ϕ, ϕ t ) be the solution of (3.3.61)-(3.3.63) associated to Φ 0 = (ψ 0 , ψ 1 , ϕ 0 , ϕ 1 ).

Multiplying the first equation of (3.1.7)-(3.1.9) by ψ and the second by ϕ and integrating by parts, we obtain

                     Ω y t (T )ϕ(T )dx + Ω u t (T )ψ(T )dx - Ω y(T )ϕ t (T )dx - Ω u(T )ψ t (T )dx - Ω bu(T )ϕ(T )dx + Ω by(T )ψ(T )dx = Ω y t (0)ϕ(0)dx + Ω u t (0)ψ(0)dx - Ω ϕ t (0)y(0)dx - Ω ψ t (0)u(0)dx - Ω bu(0)ϕ(0)dx + Ω by(0)ψ(0)dx + T 0 Ω c(x)v(t)ψdxdt.
Noting that H = (H -1 (Ω) × L 2 (Ω)) 2 . Then we have

   (u t (T, x), -u(T, x), y t (T, x), -y(T, x)), Φ(T ) H ×H = (u 1 , -u 0 , y 1 , -y 0 ), Φ 0 H ×H + T 0 Ω cv(t)ψdxdt = L(Φ 0 ). (3.3.69)
Using the direct observability inequality (3.3.67), we deduce that

L L(H,R) ≤ v 0 L 2 (0,T ;L 2 (ωc + )) + U 0 H . (3.3.70)
Using Riesz representation theorem, there exists an element Z(x, t) ∈ H solution of

L(Φ 0 ) = Z, Φ 0 H ×H , ∀Φ 0 ∈ H. (3.3.71)
Then, define the weak solution U (x, t) of system (3.1.7)-(3.1.9) by S A (t) U (x, t) = Z(x, t).

The proof is thus complete. Next, we consider the indirect locally internal exact controllability problem : For given T > 0 (sufficiently large) and initial data U 0 , does there exists a suitable control v that brings back the solution to equilibrium at time T , that is such the solution of (3.1.7)-(3.1.9) satisfies u(T ) = u t (T ) = y(T ) = y t (T ) = 0. Indeed, applying the HUM method, we obtain the following result. Theorem 3.3.16. Let a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω b ⊂ ω c + satisfying the geometric control condition GCC and that b, c ∈ W 1,∞ (Ω). For every T > M 1 , where M 1 is given in Theorem 3.3.14 and for every

U 0 ∈ (L 2 (Ω) × H -1 (Ω)) 2 , there exists a control v(t) ∈ [H 1 (0, T ; L 2 (ω c + ))] ,
such that the solution of the controlled system (3.1.7)-(3.1.9) satisfies u(T ) = u t (T ) = y(T ) = y t (T ) = 0.

Proof: We will apply the HUM method. Thanks to the indirect observability inequalities (3.3.67), we consider the seminorme defined by

Φ 0 2 H = T 0 ω b |ψ t | 2 dxdt,
where Φ = (ψ, ψ t , ϕ, ϕ t ) designate the solution of the homogeneous problem (3.3.61)- (3.3.63). Take the control v = d dt ψ t . Now, we solve the following retrograde problem :

   ζ tt -∆ζ + bχ t = c d dt ψ t in (0, T ) × Ω, χ tt -∆χ -bζ t = 0 in (0, T ) × Ω, χ(T ) = χ t (T ) = ζ(T ) = ζ t (T ) = 0.
(3.3.72) By Theorem 3.3.15, the system (3.3.72) admits a solution

Ψ(x, t) = (ζ, ζ t , χ, χ t ) ∈ C 0 ([0, T ], H ).
We define the linear operator Λ by :

Λ : H = (H 1 0 (Ω) × L 2 (Ω)) 2 → (H -1 (Ω) × L 2 (Ω)) 2 , where ΛΦ 0 = (ζ t (0), -ζ(0), χ t (0), -χ(0)) ∀ Φ 0 ∈ (H 1 0 (Ω) × L 2 (Ω)
). In addition, we define the following linear form ΛΦ 0 , Φ0 = T 0 ωc

ψ t ψt dxdt = (Φ 0 , Φ0 ) H , ∀ Φ0 ∈ H, (3.3.73)
where (., .) H is the scalar product associated to the norm . H .

Using Cauchy-Schwarz in (3.3.73) , we deduce that

| ΛΦ 0 , Φ0 H×H | Φ 0 H Φ0 H , ∀ Φ 0 , Φ0 ∈ H. (3.3.74)
In particular, we have

| ΛΦ 0 , Φ 0 H×H | = Φ 0 2 H ∀ Φ 0 ∈ H.
Then the inverse inequality in Theorem 3.3.14 implies that the operator Λ is coercive and continuous over H. Thanks to Lax-Milgram theorem, we have Λ is an isomorphism from H into H . In particular, for every

U 0 ∈ (L 2 (Ω) × H -1 (Ω)) 2 , there exists a solution Φ 0 ∈ H, such that Λ(Φ 0 ) = -U 0 = (ζ t (0), -ζ(0), χ t (0), -χ(0)).
It follows from the uniqueness of the solution of problem (3.3.72) that

U = Ψ.
Consequently, we have

u(T ) = u t (T ) = y(T ) = y t (T ) = 0.
The proof is thus complete.

Exponential stability and exact controllability in

the case a = 1

Exponential stability in the weak energy space

The aim of this subsection is to show the exponential stability of system (3.1.4)-(3.1.6) in a weak energy space in the case when the waves are not assumed to propagate with same speed, i.e., a = 1. For this, we define the weak energy space

D = H 1 0 (Ω) × L 2 (Ω) × L 2 (Ω) × H -1 (Ω)
equipped with the scalar product (U, Ũ ) = Ω (a∇u.∇ũ + vṽ + y ỹ + (-∆) -1/2 z(-∆) -1/2 z)dx, for all U = (u, v, y, z) ∈ D and Ũ = (ũ, ṽ, ỹ, z) ∈ D.

Next, we define the unbounded linear operator

A d : D(A d ) ⊂ D → D by A d U = ( v, a∆u -bz -cv, z, ∆y + bv ), D(A d ) = (H 1 0 (Ω) ∩ H 2 (Ω)) × H 1 0 (Ω) × H 1 0 (Ω) × L 2 (Ω) , ∀ U = (u, v, y, z) ∈ D(A d ).
We define the partial energy associated to a solution U = (u, u t , y, y t ) of (3.1.4)-(3.1.6) by

e 1 (t) = 1 2 a ∇u 2 L 2 (Ω) + u t 2 L 2 (Ω) .
We define also the weakened partial energy by

ẽ2 (t) = 1 2 y t 2 H -1 (Ω) + y 2 L 2 (Ω)
and the total mixed energy by

E m (t) = e 1 (t) + ẽ2 (t).
In order to study the exponential decay rate, we need to assume that ω c + satisfies the geometric conditions PMGC, then there exist ε > 0, subsets Ω j ⊂ Ω, j = 1, ..., J, with Lipschitz boundary Γ j = ∂Ω j and points

x j ∈ R N such that Ω i ∩ Ω j = ∅ if i = j and ω + c ⊃ N ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j ∩ Ω with N (O) = {x ∈ R N : d(x, O) < ε} where O ⊂ R N , γ j (x j ) = {x ∈ Γ j : (x -x j ) • ν j (x) > 0}
where ν j is the outward unit normal vector to Γ j and that ω b satisfies the GCC condition and

ω b ⊂ Ω \ ∪ J j=1 Ω j . (LH3)
Now, we are ready to establish the following main theorem of this section :

Theorem 3.4.1. (Exponential decay rate) Let a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω c + satisfies the geometric conditions PMGC, ω b satisfies GCC condition and (LH3) and b, c ∈ L ∞ (Ω). Then there exist positive constants M ≥ 1, θ > 0 such that for all initial data (u 0 , u 1 , y 0 , y 1 ) ∈ D the energy of system (3.1.4)-(3.1.6) satisfies the following decay rate :

E m (t) ≤ M e -θt E m (0), ∀t > 0. (3.4.1)
In order to prove the above theorem, we apply a result of Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Prüss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]

(iβI -A d ) -1 L(D) < ∞ (H2)
hold.

Condition (H1) was already proved. We now prove that condition (H2) holds, using an argument of contradiction. For this aim, we suppose that there exist a real sequence β n with β n → +∞ and a sequence

U n = (u n , v n , y n , z n ) ∈ D(A d ) such that U n D = 1, (3.4.2)
iβ n u n -v n = f 1 n → 0 in H 1 0 (Ω), (3.4.4) iβ n v n -a∆u n + bz n + cv n = g 1 n → 0 in L 2 (Ω), (3.4.5) iβ n y n -z n = f 2 n → 0 in L 2 (Ω), (3.4.6) iβ n z n -∆y n -bv n = g 2 n → 0 in H -1 (Ω). (3.4.7)
Eliminating v n and z n from the previous system, we obtain the following system Under condition (LH1), it follows

β 2 n u n + a∆u n -iβ n by n -iβ n cu n = -g 1 n -bf 2 n -iβ n f 1 n -cf 1 n in L 2 (Ω), (3.4.8)
β 2 n y n + ∆y n + iβ n bu n = -iβ n f 2 n + bf 1 n -g 2 n in H -1 (Ω). ( 3
ωc + |β n u n | 2 dx = o(1).
The proof is thus complete. Now as ω c + satisfies the PMGC condition, let the reals 0 < ε 1 < ε 2 < ε and define Proof: First, multiplying equation (3.4.8) by ηū n . Then, using Green's formula and the fact that u n = 0 on Γ, we obtain

Q i = N ε i ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j , i = 1, 2. Since R N \ ω c + ∩ Q 2 = ∅, we can construct a function η ∈ C ∞ 0 (Ω) defined by η(x) = 0 if x ∈ Ω \ ω c + , 0 ≤ η(x) ≤ 1, η(x) = 1 if x ∈ Q 2 .
Ω η|β n u n | 2 dx -a Ω η | ∇u n | 2 dx -a Ω u n (∇η • ∇u n )dx -i β n Ω bηy n u n dx -iβ n Ω cη|u n | 2 dx = Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n )ηu n dx. (3.4.14) As f 1 n converges to zero in H 1 0 (Ω), f 2 n , g 1 n converge to zero in L 2 (Ω) and β n u n is uniformly bounded in L 2 (Ω), we get Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n )ηu n dx = o(1). (3.4.15)
Using the fact that ∇u n , y n are uniformly bounded in L 2 (Ω), u n L 2 (Ω) = o(1) and estimation (3.4.10), we will have

Ω η|β n u n | 2 dx-a Ω u n (∇η•∇u n )dx-i β n Ω bηy n u n dx-iβ n Ω cη|u n | 2 dx = o(
Q 2 ∩Ω | ∇u n | 2 dx = o(1).
The proof is thus complete. (3.4.17)

Proof: The proof contains two steps.

Step 1. (Boundedness of 1 βn ∇y n ). Multiplying equation (3.4.9) by 1

β 2 n y n , we obtain Ω |y n | 2 dx+ < ∆y n , 1 β 2 n y n > H -1 (Ω)×H 1 0 (Ω) = -i Ω 1 β n f 2 n y n dx + Ω bf 1 n 1 β 2 n y n dx (3.4.18) -< g 2 n , 1 β 2 n y n > H -1 (Ω)×H 1 0 (Ω) .
Since f 1 n converges to zero in H 1 0 (Ω), f 2 n converges to zero in L 2 (Ω) and y n is uniformly bounded in L 2 (Ω), we get

-i Ω 1 β n f 2 n y n dx + Ω bf 1 n 1 β 2 n y n dx = o(1). ( 3
dx = Ω |y n | 2 dx+ < g 2 n , 1 β 2 n y n > H -1 (Ω)×H 1 0 (Ω) +o(1).
Using Cauchy-Schwartz and Young inequalities in the previous equation, we obtain that 1 2

∇y n β n 2 L 2 (Ω) ≤ y n 2 L 2 (Ω) + 1 2 g 2 n 2 H -1 (Ω) + o(1).
It follows, from the uniform boundedness of y n in L 2 (Ω) and g 2 n in H -1 (Ω), that

∇y n β n 2 = O(1). (3.4.20)
Step 2. (Main asymptotic estimation). Multiplying equation (3.4.8) by η 1 βn y n . Later, using Green's formula and the fact that y n = 0 on Γ, we get Using the fact that f 1 n converges to zero in H 1 0 (Ω), f 2 n , g 1 n converge to zero in L 2 (Ω) and y n is uniformly bounded in L 2 (Ω), we will have The proof is thus complete. Proof: Noting that ω b satisfies the GCC condition, so we can taking f n = y n in Lemma 3.3.10. Multiplying equation (3.4.9) by φ n . Then, we have

Ω ηβ n u n y n dx -a Ω 1 β n η(∇u n • ∇y n )dx -a Ω 1 β n (∇η • ∇u n )y n dx -i Ω bη|y n | 2 dx -i Ω cu n ηy n dx = Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n ) η β n y n dx. ( 3
Ω (-g 1 n -bf 2 n -iβ n f 1 n -cf 1 n ) η β n y n dx = o(1). ( 3
Ω β 2 n φ n y n dx-< ∆y n , φ n > H -1 (Ω)×H 1 0 (Ω) +i Ω β n bu n φ n dx = -i Ω β n f 2 n φ n dx + Ω bf 1 n φ n dx-< g 2 n , φ n > H -1 (Ω)×H 1 0 (Ω) . (3.4.26)
Using the fact that φ n ∈ H 2 (Ω) ∩ H 1 0 (Ω) and y n ∈ H 1 0 (Ω), then we have The proof is thus complete. Proof: Multiplying equation (3.4.9) by (-∆) -1 y n , then integrating by parts and using the fact that y n = 0 on Γ, we get

-< ∆y n , φ n > H -1 (Ω)×H 1 0 (Ω) = Ω y n ∆φ n dx. ( 3
Ω |y n | 2 dx = i Ω bβ n φ n y n dx -i Ω β n bu n φ n d -i Ω β n f 2 n φ n dx + Ω bf 1 n φ n dx-< g 2 n , φ n > H -1 (Ω)×H 1 0 (Ω) . ( 3
Ω |β n (-∆) -1/2 y n | 2 dx = Ω |y n | 2 dx -i Ω β n bu n (-∆) -1 y n dx -i Ω β n (-∆) -1/2 f 2 n (-∆) -1/2 y n dx + Ω bf 1 n (-∆) -1 y n dx -< g 2 n , (-∆) -1 y n > H -1 (Ω)×H 1 0 (Ω) (3.4.31)
Using Cauchy-Schwartz and Poincaré inequalities, we get

| Ω β n (-∆) -1/2 f 2 n (-∆) -1/2 y n dx| ≤ (-∆) -1/2 f 2 n L 2 (Ω) β n (-∆) -1/2 y n L 2 (Ω) (3.4.32) ≤ c 0 f 2 n L 2 (Ω) β n y n H -1 (Ω) .
It follows, from the convergence to zero of f 2 n in L 2 (Ω) and the boundedness of

β n y n in H -1 (Ω), that Ω β n (-∆) -1/2 f 2 n (-∆) -1/2 y n dx = o(1). (3.4.33)
Similarly, we have

|< g 2 n , (-∆) -1 y n > H -1 (Ω)×H 1 0 (Ω) | = Ω (-∆) -1/2 g 2 n (-∆) -1/2 y n dx (3.4.34) ≤ (-∆) -1/2 g 2 n L 2 (Ω) (-∆) -1/2 y n L 2 (Ω) ≤ g 2 n H -1 (Ω) y n H -1 (Ω)
. It follows, from the convergence of g 2 n and y n to zero in

H -1 (Ω), that < g 2 n , (-∆) -1 y n > H -1 (Ω)×H 1 0 (Ω) = o(1). (3.4.35)
Noting that (-∆) -1 is compact operator from L 2 to L 2 , then (-∆)

Ω |β n (-∆) -1/2 y n | 2 dx = o(1).
The proof is thus complete.

Ω\(Q 2 ∩Ω) |∇u n | 2 + |β n u n | 2 )dx = o(1). (3.4.36)
Proof:

Since (Ω j \ Q 2 ) ∩ Q 1 = ∅, we define the function ψ j ∈ C ∞ 0 (R N ) by : ψ j (x) = 0 if x ∈ Q 1 , 0 ψ j 1, ψ j (x) = 1 if x ∈ Ω j \ Q 2 .
For m j (x) = (x -x j ), we define h j (x) = ψ j (x)m j (x).

Multiplying equation (3.4.8) by 2(h j •∇u n ) and integrating over Ω j , using the dissipation (3.4.10) and the fact that ∇u n is uniformly bounded in L 2 (Ω), we obtain

2β 2 n Ω j u n (h j • ∇u n )dx + 2a Ω j ∆u n (h j • ∇u n )dx -2i Ω j β n by n (h j • ∇u n )dx = 2 Ω j (-g 1 n -bf 2 n -cf 1 n )(h j • ∇u n )dx -2i Ω j β n f 1 n (h j • ∇u n )dx. (3.4.37)
i) Estimation of the second member of (3.4.37). First, using Green's formula and the fact that u n = 0 on (Γ j \ γ j ) ∩ Γ and h j = 0 on γ j , we get -2i

Ω j β n f 1 n (h j • ∇u n )dx = 2i Ω j β n u n (h j • ∇f 1 n )dx + 2i Ω j β n u n f 1 n (divh j )dx.(3.4.38)
So, from the fact that f 1 n converges to zero in H 1 0 (Ω) and β n u n is uniformly bounded in L 2 (Ω), we obtain

-2i Ω j β n f 1 n (h j • ∇u n )dx = o(1). (3.4.39)
Next, as f 1 n converges to zero in H 1 0 (Ω), f 2 n , g 1 n converge to zero in L 2 (Ω) and the sequence (∇u n ) is uniformly bounded in L 2 (Ω), we deduce 2

Ω j (-g 1 n -bf 2 n -cf 1 n )(h j .∇u n )dx = o(1). (3.4.40)
Finally, we deduce that the second member of (3.4.37) is o(1).

ii) Estimation of the first integral of equation (3.4.37). Using Green's formula, we get Re 2

Ω j β 2 n u n (h j • ∇u n )dx = - Ω j (divh j )|β n u n | 2 dx + Γ j (h j • ν j )|β n u n | 2 dΓ j . (3.4.41)
Since Ψ j = 0 on γ j and u n = 0 on (Γ j \ γ j ) ∩ Γ, then we have Re{2

Ω j β 2 n u n (h j • ∇u n )dx} = - Ω j (divh j )|β n u n | 2 dx. (3.4.42)
iii) Estimation of the second integral of equation (3.4.37). Using Green's formula, we get Re 2a

Ω j ∆u n (h j • ∇u n ) = -2aRe N i,k=1 Ω j ∂ i h k j ∂ i u n ∂ k u n dx + (3.4.43) a Ω j (divh j )|∇u n | 2 dx -a Γ j (h j • ν j )|∇u n | 2 dΓ j + 2aRe Γ j ∂ ν j u n (h j • ∇u n )dΓ j .
According to the choice of ψ j , only the boundary terms over (Γ j \ γ j ) ∩ Γ are non vanishing in (3.4.43). But on this part of the boundary u n = 0, and consequently

∇u n = (∂ ν u n ) • ν = (∂ ν j u n )ν j . Then, we have -a Γ j (h j • ν j )|∇u n | 2 dΓ j + 2aRe Γ j (∂ ν j u n )(h j • ∇u n )dΓ j = (3.4.44) a (Γ j \γ j )∩Γ (ψ j m j • ν j )|∂ ν j u n | 2 dΓ j 0.
Inserting (3.4.44) into (3.4.43), we get Re 2a

Ω j ∆u n (h j • ∇u n ) ≤ -2aRe N i,k=1 Ω j ∂ i h k j ∂ i u n ∂ k u n dx (3.4.45) + a Ω j (divh j )|∇u n | 2 dx.
iv) The main estimation. Inserting equations (3.4.39), (3.4.40), (3.4.42) and (3.4.45) into (3.4.37) and using the fact that ψ j = 0 on Q 1 , we get

Ω j \(Q 1 ∩Ω j ) div(ψ j m j )(|β n u n | 2 -a|∇u n | 2 )dx + 2a N i,k=1 ∂ i (ψ j m k j )∂ i u n ∂ k u n dx +2i Ω j \(Q 1 ∩Ω j )
β n by n (ψ j m j • ∇u n )dx o(1).

Thus, summing over j and using the fact that ψ j = 1 on Ω j \ Q 2 , we get

N Ω\(Q 2 ∩Ω) |β n u n | 2 dx + (2 -N )a Ω\(Q 2 ∩Ω) |∇u n | 2 dx + 2Re i J j=1 Ω j \(Q 1 ∩Ω j ) β n by n (ψ j m j • ∇u n )dx (3.4.46) - J j=1 Q 2 ∩Ω j div(ψ j m j )(|β n u n | 2 -a|∇u n | 2 )dx + 2a N i,k=1 ∂ i (ψ j m k j )∂ i u n ∂ k u n dx + o(1).
Using (3.4.10) and (3.4.13), we deduce

- J j=1 Q 2 ∩Ω j div(ψ j m j )(|β n u n | 2 -a|∇u n | 2 )dx + 2a N i,k=1 ∂ i (ψ j m k j )∂ i u n ∂ k u n dx = o(1
|β n u n | 2 dx + (2 -N )a Ω\(Q 2 ∩Ω) |∇u n | 2 dx+ (3.4.48) 2Re i J j=1 Ω j \(Q 1 ∩Ω j ) β n by n (ψ j m j .∇u n )dx o(1).
Under condition (LH3) and the definition of ψ j , we will have

2Re i J j=1 Ω j \(Q 1 ∩Ω j )
β n by n (ψ j m j .∇u n )dx = 0.

Inserting the previous estimation into (3.4.48), we get

N Ω\(Q 2 ∩Ω) |β n u n | 2 dx + (2 -N )a Ω\(Q 2 ∩Ω) |∇u n | 2 dx ≤ o(1). ( 3
Ω\(Q 2 ∩Ω) |β n u n | 2 dx -(1 -N )a Ω\(Q 2 ∩Ω) |∇u n | 2 dx = o(1). ( 3

Observability and exact controllability

First, we consider the following homogeneous system associated to (3.1.4)-(3.1.6) for a = 1 by :

ψ tt -a∆ψ + b(x)ϕ t = 0 in Ω × R + , (3.4.52) ϕ tt -∆ϕ -b(x)ψ t = 0 in Ω × R + , (3.4.53) ψ = ϕ = 0 on Γ × R + , (3.4.54) ψ(•, 0) = ψ 0 , ψ t (•, 0) = ψ 1 , ϕ(•, 0) = ϕ 0 , ϕ t (•, 0) = ϕ 1 in Ω. (3.4.55)
Let Φ = (ψ, ψ t , ϕ, ϕ t ) be a regular solution of system (3.3.61)-(3.3.63), its associated total energy is given by : It is also well posed and admits a unique solution in the energy space D. Now, we establish the direct and indirect inequality given by the following theorem :

E m (t) = 1 2 a ∇ψ 2 L 2 (Ω) + ψ t 2 L 2 (Ω) + ϕ t 2 H -1 (Ω) + ϕ 2 L 2 (Ω) . ( 3
Theorem 3.4.8. Let 0 < a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that ω c + satisfies the PMGC, ω b satisfies GCC condition and (LH3) and b, c ∈ L ∞ (Ω).

Then there exists a time T 0 such that for all T > T 0 , there exist two constants C 1 > 0, C 2 > 0 such that the solution of system (3.4.52)-(3.4.54) satisfies the following observability inequalities :

C 1 Φ 0 2 D ≤ T 0 Ω c(x)|ψ t | 2 dxdt ≤ C 2 Φ 0 2 D dx, (3.4.58)
for all Φ 0 = (ψ 0 , ψ 1 , φ 0 , φ 1 ) ∈ D.

Proof: The direct inequality follows from the definition of the total energy for all T > 0.

While the proof of the inverse inequality is a direct consequence of Proposition 2 of Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] for which the exponentially stability (3.4.1) implies the existence of a time T 0 > 0 such that for all T > T 0 there exist two constants C 1 > 0 and C 2 > 0 such that (3.4.58) holds.

It is well known that observality of the homogeneous system (3.4.52)-(3.4.54) implies the exact controllability of the associated system to (3.1.7)-(3.1.9).

Numerical approximation : Validation of the theoretical results

This section is devoted to the numerical approximation of the problem by a finite difference discretization and to the validation of the theoretical results stated in the previous sections. We will firstly construct in detail a discretization in the 1D case and we will define its corresponding discrete energy. Numerical experiments are performed to validate the theoretical results. In fact, the numerical results in 1D show an exponential stabilization in any case when a = 1 and a polynomial stabilization in any case in the case a = 1. They are better than expected.

Finite difference scheme in one dimensional space

We firstly introduce the finite difference scheme we will work on then we will construct the corresponding energy and finally we will perform numerical experiments. Let us firstly recall the problem we are considered.

Consider Ω = [0, 1]. We are interested to study the controllability of the following coupled wave equations by velocities :

   u tt -au xx + b(x)y t + c(x)u t = 0 x ∈ (0, 1), t > 0 y tt -y xx -b(x)u t = 0 x ∈ (0, 1), t > 0 u(0, t) = u(1, t) = y(0, t) = y(1, t) = 0 t > 0, (3.5.1)
with the following initial data u(x, 0) = u 0 (x), and y(x, 0) = y 0 (x) x ∈ (0, 1) (3.5.2)

Practical implementation and CFL condition

Let us denote λ = ∆t 2 ∆x 2 . We easily remark that the discrete scheme (3.5.4) is composed of N linear systems of two equations which can be written under the form :

for j = 1, . . . , N , M j •   u n+1 j y n+1 j   =   A j B j   (3.5.10)
where

M j =      1 + c j ∆t 2 b j ∆t 2 -b j ∆t 2 1      A j = 2(1 -aλ)u n j + ( c j 2 ∆t -1)u n-1 j + aλ(u n j+1 + u n j-1 ) + b j 2 ∆ty n-1 j and B j = 2(1 -λ)y n j + λ(y n j+1 + y n j-1 ) -y n-1 j - b j 2 ∆tu n-1 j .
Thanks to the hypothesis ∀x ∈ (0, 1) , c(x) ≥ 0, for j = 1, . . . , N the determinant of M j given by

|M j | = 1 + c j ∆t 2 + b j ∆t 2 2 ,
is a strictly positive quantity. Consequently, system (3.5.10) admits a unique solution given by : for j = 1, . . . , N , u n+1 j = (1-aλ)α j u n j +λβ j (u n j+1 +u n j-1 )+γ j u n-1 j -(1-λ) j y n j -λξ j (y n j+1 +y n j-1 )+κ j y n-1 j (3.5.11)

y n+1 j = (1-λ) α j y n j +λ β j (y n j+1 +y n j-1 )+ γ j y n-1 j +(1-aλ) j u n j +λ ξ j (u n j+1 +u n j-1 )+ κ j u n-1 j (3.5.12)
where we have set :

α j = 2 1 + c j 2 ∆t + b j ∆t 2 2 , β j = a 1 + c j 2 ∆t + b j ∆t 2 2 , γ j = c j 2 ∆t + b j 2 ∆t 2 -1 1 + c j 2 ∆t + b j ∆t 2 2 , j = b j ∆t 1 + c j 2 ∆t + b j ∆t 2 2 , ξ j = b j ∆t 2 1 + c j 2 ∆t + b j ∆t 2 2 , κ j = b j ∆t 1 + c j 2 ∆t + b j ∆t 2 2 α j = 2 - (b j ∆t) 2 2 1 + c j 2 ∆t + b j ∆t 2 2 , β j = 1 - (b j ∆t) 2 4 1 + c j 2 ∆t + b j ∆t 2 2 , γ j = (b j ∆t) 2 2 1 + c j 2 ∆t + b j ∆t 2 2 -1 , j = b j ∆t 1 + c j 2 ∆t + b j ∆t 2 2 , ξ j = ab j ∆t 2 1 + c j 2 ∆t + b j ∆t 2 2 , κ j =      c j 2 ∆t + b j ∆t 2 2 -1 1 + c j 2 ∆t + b j ∆t 2 2 -1      b j ∆t 2 .
The implementation of the numerical discretization of the problem (3.5.1) consists finally of equations (3.5.5), (3.5.6), (3.5.11), (3.5.12) where (u -1 , y -1 ) used for n = 0, are defined by (3.5.8), (3.5.9). By a standard Von Neumann stability analysis (that is a discrete Fourier analysis, see for instance [START_REF] Ames | Numerical methods for partial differential equations[END_REF]), the numerical scheme is stable if and only if, the following Courant-Friedrichs-Lewy, CFL, condition holds :

∆t 2 ≤ ∆x 2 and a ∆t 2 ≤ ∆x 2 which is equivalent to ∆t ≤ min 1, 1 √ a ∆x . (3.5.13)
The number min 1, 1 √ a is called the CFL number and is denoted in the following by CF L.

Discrete energy : definition and dissipation.

The aim of this section is to design a discrete energy that might be preserved in the case c = 0 and to obtain the dissipation of the discrete energy in the case c > 0. To this end, let us define :

-the discrete kinetic energy for u as :

E n k,u = 1 2 N j=1 u n+1 j -u n j ∆t 2
-the discrete potential energy for u as : The total discrete energy is then defined as

E n p,u = a 2 N j=0 u n j+1 -u n j ∆x u n+1 j+1 -u n+1
E n = E n k,u + E n p,u + E n k,y + E n p,u . (3.5.14)
Let us prove now that this definition of the energy fulfill the two properties stated above. For this sake, we multiply the first equation of (3.5.4) by (u n+1 j -u n-1 j ) and we sum over j = 1, . . . , N . We obtain :

N j=1 u n+1 j -2u n j + u n-1 j ∆t 2 (u n+1 j -u n-1 j ) -a N j=1 u n j+1 -2u n j + u n j-1 ∆x 2 (u n+1 j -u n-1 j ) + N j=1 b j y n+1 j -y n-1 j 2∆t (u n+1 j -u n-1 j ) + N j=1 c j (u n+1 j -u n-1 j ) 2 2∆t = 0. (3.5.15)
Estimation of the first term of (3.5.15) We firstly have :

N j=1 u n+1 j -2u n j + u n-1 j ∆t 2 (u n+1 j -u n-1 j ) = N j=1 u n+1 j -u n j -(u n j -u n-1 j ) ∆t 2 (u n+1 j -u n j + u n j -u n-1 j ) = N j=1 u n+1 j -u n j ∆t 2 - N j=1 u n+1 j -u n-1 j ∆t 2 = 2(E n k,u -E n-1 k,u ). (3.5.16)
Estimation of the second term of (3.5.15). Using the same trick we have : 2 .

-a N j=1 u n j+1 -2u n j + u n j-1 ∆x 2 (u n+1 j -u n-1 j ) = -a N j=1 u n j+1 -u n j -(u n j -u n j-1 ) ∆x 2 (u n+1 j -u n-1 j ) = -a N j=1 (u n j+1 -u n j )(u n+1 j -u n-1 j ) ∆x 2 + a N +1 j=1 (u n j -u n j-1 )(u n+1 j -u n-1 j ) ∆x
91 Chapitre 3. Exact controllability and stabilization of locally coupled wave equations So, by translation of index in the second term in the previous sum, we will have :

-a N j=1 u n j+1 -2u n j + u n j-1 ∆x 2 (u n+1 j -u n-1 j ) = -a N j=0 (u n j+1 -u n j )(u n+1 j -u n-1 j ) ∆x 2 + a N j=0 (u n j+1 -u n j )(u n+1 j+1 -u n-1 j+1 ) ∆x 2 = a N j=0 (u n+1 j+1 -u n+1 j )(u n j+1 -u n j ) ∆x 2 -a N j=0 (u n-1 j+1 -u n-1 j )(u n j+1 -u n j ) ∆x 2 = 2(E n p,u -E n-1 p,u ).
(3.5.17) Substituting (3.5. 16) and (3.5.17) into (3.5.15), we get

2 E n k,u + E n p,u -E n-1 k,u -E n-1 p,u +2∆t N j=1 c j u n+1 j -u n-1 j 2∆t 2 + N j=1 b j y n+1 j -y n-1 j 2∆t (u n+1 j -u n-1 j ) = 0.
(3.5.18) Similarly, by multiplying the second equation of (3.5.4) by (y n+1 j -y n-1 j ), and using the same algebraic tricks, we will get :

2 E n k,y + E n p,y -E n-1 k,y -E n-1 p,y - N j=1 b j u n+1 j -u n-1 j 2∆t (y n+1 j -y n-1 j ) = 0. (3.5.19)
Using the definition of the total discrete energy, (3.5.14), and the two equations (3.5.18), (3.5.19) leads to :

E n -E n-1 + ∆t N j=1 c j u n+1 j -u n-1 j 2∆t 2 = 0. (3.5.20)
Consequently, the total discrete energy of system (3.5.4) is decreasing along time.

Numerical experiments : validation of the theoretical results

In every experiments, we have chosen :

u 0 (x) = x(x -1) , u 1 (x) = x(x -1) , y 0 (x) = -x(x -1) , y 1 (x) = -x(x -1).
The mesh size is chosen as N = 100 so that ∆x = 0.01 and the time step is chosen as ∆t ∆x = CF L.

In order to validate the different theoretical results, we have chosen different functions b and c synthesized in the list below :

-No coupling : b 1 (x) = 0 or no dissipation c 1 (x) = 0, -Full coupling b 2 (x) = 1 1 (0,1) (x) or full dissipation c 2 (x) = 1 1 (0,1) (x), -Partial coupling b 3 (x) = 1 [0.1,0.2]∪[0.8,0.9] (x) or partial dissipation c 3 (x) = 1 1 [0.1,0.2]∪[0.8,0.9] (x), -Partial coupling b 4 (x) = 1 1 [0.1,0.2] (x) or partial dissipation c 4 (x) = 1 1 [0.1,0.2] (x), -Partial coupling b 5 (x) = 1 1 [0.4,0.6] (x) or partial dissipation c 5 (x) = 1 1 [0.4,0.6] (x).
Combining the different choices of the coupling and damping functions in order to have or not ω b ∩ ω c + = ∅ will permit us to validate the theoretical results.

Let us notice that in the special case of the dimension 1, the geometric control conditions GCC holds as soon as ω c + = ∅.

Same propagation speed : a = 1

For every numerical simulation, the final time T is chosen as T = 500.

No damping : conservation of the total energy Firstly, let us verify that when no damping are present, the discrete energy is conserved. We present in figure 3.2 the numerical experiment when c = c 1 = 0 and b = b 3 = 1 1 [0.1,0.2]∪[0.8,0.9] (x). Indeed, the total energy is conserved along time.

Remark 3.5.1. This numerical test where no damping is applied shows that without a damping term, the total energy is completely conserved. This fact suggests that the numerical scheme does not produce numerical dissipation. So the numerical behavior observed thereafter is only due to the considered model.

ω b ∩ ω c + = ∅. Exponential stability.
Let us now verify the theoretical results when we suppose that ω b ∩ ω c + = ∅. For this sake, we present in figure 3.3, the total energy and the quantity -ln E(t) /t versus time t for large time, where we have chosen b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 3 (x) = 1 1 [0.1,0.2]∪[0.8,0.9] (x). This choice verifies the assumption that ω b ∩ ω c + = ∅ and in figure 3.3, it is shown that the energy is decreasing and an exponential decay is observed since it seems that -ln E(t) /t tends to a constant as t → +∞. The final time profile confirms that u and y are small and the final profiles of u and y are smooth as expected (high frequency oscillations are exponentially dissipated).

ω b ∩ ω c + = ∅.
Unpredicted behavior. At the numerical level, we are interested in the long time behavior of the solution (u, y) when we suppose that ω b ∩ ω c + = ∅. For this sake, we present in figure 3.4, the total energy and the quantity -ln E(t) /t versus time t for large time, where we have chosen b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 5 (x) = 1 1 [0.4,0.6] (x). This choice verifies the assumption that ω b ∩ ω c + = ∅. In figure 3.4, it is shown that the energy is decreasing and an exponential decay is observed since it seems that -ln E(t) /t tends to a constant as t → +∞. The final time profile confirms that u and y are small and again the couple of solution (u, y) is smooth . This result is surprising since it is not predicted by the theoretical results.

So we decided to confirm this strange behavior by choosing b = b 5 (x) = 1 1 [0.4,0.6] (x) and c = c 4 (x) = 1 1 [0.1,0.2] (x). This choice verifies also the assumption that ω b ∩ ω c + = ∅. In figure 3.5, it is shown that the energy is decreasing and an exponential decay is observed since it seems that -ln E(t) /t tends to a constant as t → +∞. The final time profile confirms that u and y are small and again the couple of solution (u, y) is smooth. Remark 3.5.2. Let us notice that when the propagation speeds are the same for u and y, the final profiles of the solution u , y presented in figure 3.3(c), figure 3.4(c) and in figure 3.5(c) have the same form as the initial one, that is no spurious oscillations due to high frequency are present.

Different propagation speed : a = 2

We investigate now the long time behavior of (u, y) when the propagation speeds are different. We firstly investigate the case when the propagation speed for u is greater than the one of y namely a > 1. We have chosen a = 2.

ω b ∩ ω c + = ∅.
Polynomial stability Let us now verify the theoretical results when we suppose that ω b ∩ ω c + = ∅. For this sake, we present in figure 3.6, the total energy where we have chosen

b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 3 (x) = 1 1 [0.1,0.2]∪[0.8,0.9] (x).
When taking as final time T = 500, it seems that the energy does not tend to zero as shown in figure 3.6(a). This is the reason why we have chosen for the case when a = 1 take as final time T = 500 000 and figure 3.6(b) shows that the energy suddenly goes to zero. To explore the speed of convergence to zero, we have plotted in figure 3.7 -ln E(t) /t , t•E(t) and finally -ln E(t) / ln(t) versus t. Figure 3.7(a) shows clearly that -ln E(t) /t tends to zero and it permits to conclude that E(t) tends to zero slower than an exponential. Figure 3.7(b) permits to conclude that E(t) tends to zero faster than 1/t. Finally figure 3.7(c) shows that E(t) tends to zero as 1/t α with α 1.4. The final time profile presented in figure 3.7(d) confirms that u and y are small but it shows also that high frequencies for the unknown y are not completely controlled. ω b ∩ ω c + = ∅. Unpredicted behavior At the numerical level, we are interested in the long time behavior of the solution (u, y) when we suppose that ω b ∩ ω c + = ∅. For this sake, we present in figure 3.8, the total energy where we have chosen

b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 5 (x) = 1 1 [0.4,0.6] (x).
Again, when taking as final time T = 500, it seems that the energy does not tend to zero as shown in figure 3.8(a). Taking as final time T = 500 000 , figure 3.8(b) shows that the energy suddenly goes to zero. To explore the speed of convergence to zero, we have plotted in figure 3.9 -ln E(t) /t , t•E(t) and finally -ln E(t) / ln(t) versus t. Figure 3.9(a) shows clearly that -ln E(t) /t tends to zero and it permits to conclude that E(t) tends to zero slower than an exponential but figure 3.9(b) shows that E(t) tends to zero slower than 1/t. This fact is confirmed by figure 3.9(c) which shows that E(t) tends to zero as 1/t α with α 0.9. Eventually, taking a larger time could conclude that the convergence is like 1/t. Again, the final time profile presented in figure 3.9(d) confirms that u and y are small but it shows also that high frequencies for the unknown y are not completely controlled. As for the case when the two propagation speed were identical this results is surprising since it is not predicted by the theoretical results. So we decided to confirm this strange behavior by choosing b = b 5 (x) = 1 1 [0.4,0.6] (x) and c = c 4 (x) = 1 1 [0.1,0.2] (x). Again, when taking as final time T = 500, it seems that the energy does not tends to zero as shown in figure 3.10(a). Taking as final time T = 500 000, figure 3.8(b) shows that the energy suddenly goes to zero. To explore the speed of convergence to zero, we have plotted in figure 3.11 -ln E(t) /t , t•E(t) and finally -ln E(t) / ln(t) versus t. Figure 3.11(a) shows clearly that -ln E(t) /t tends to zero and it permits to conclude that E(t) tends to zero slower than an exponential and figure 3.11(b) permits to conclude that the convergence is faster than 1/t. Finally figure 3.11(c) shows that E(t) tends to zero as 1/t α with α 1.19.

Remark 3.5.3. The final time profile presented in figure 3.7(d) , figure 3.9(d) and figure 3.11(d) confirms that u and y are small but it shows also that high frequencies for the unknown y are not completely controlled.

Different propagation speed : a = 0.5 When a = 1, in order to see if the same behavior occurs no matter if a is greater or less than 1, we investigate now the long time behavior of (u, y) when the propagation speeds is less than the one of y namely a > 1. We have chosen a = 0.5.

ω b ∩ ω c + = ∅.
Polynomial stability Let us now verify the theoretical results when we suppose that ω b ∩ ω c + = ∅. For this sake, we present in figure 3.12(a), the total energy where we have chosen b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 3 (x) = 1 1 [0.1,0.2]∪[0.8,0.9] (x). When taking as final time T = 500, it seems that the energy does not tend to zero as shown in figure 3.12(a). Taking as final time T = 500 000, figure 3.12(b) shows that the energy suddenly goes to zero. To explore the speed of convergence to zero, we have plotted in figure 3.13 -ln E(t) /t , t•E(t) and finally -ln E(t) / ln(t) versus t. Figure 3.13(a) shows clearly that -ln E(t) /t tends to zero slower than an exponential. Figure 3.13(b) permits to conclude that E(t) tends to zero faster than 1/t. Finally figure 3.13(c) shows that E(t) tends to zero as 1/t α with α 1.5. The final time profile confirms that u and y are small but it shows also that high frequencies for the unknown y are not completely controlled. Again, when taking as final time T = 500, it seems that the energy does not tend to zero as shown in figure 3.14(a). Taking as final time T = 500 000, figure 3.14(b) shows that the energy suddenly goes to zero. To explore the speed of convergence to zero, we have plotted in figure 3.15 -ln E(t) /t , t•E(t) and finally -ln E(t) / ln(t) versus t. Figure 3. 15(a) shows clearly that -ln E(t) /t tends to zero slower than an exponential. But figure 3.15(b) shows that E(t) tends to zero faster than 1/t. Finally figure 3.15(c) shows that E(t) tends to zero as 1/t α with α 1.25. Again, the final time profile presented in figure 3.15(d) confirms that u and y are small but it shows also that high frequencies for the unknown y are not completely controlled. This result is surprising since it is not predicted by the theoretical results. So we decided to confirm this strange behavior by choosing b = b 5 (x) = 1 1 [0.4,0.6] (x) and c = c 4 (x) = 1 1 [0.1,0.2] (x). Again, when taking as final time T = 500, it seems that the energy does not tend to zero as shown in figure 3.16(a). Taking as final time T = 500 000, figure 3. 16(b) shows that the energy suddenly goes to zero. To explore the speed of convergence to zero, we have plotted in figure 3.17 -ln E(t) /t , t•E(t) and finally -ln E(t) / ln(t) versus t. Figure 3.17(a) shows clearly that -ln E(t) /t tends to zero and it permits to conclude that E(t) tends to zero slower than an exponential but figure 3.17(b) shows that E(t) tends to zero faster than 1/t. Finally figure 3.17(c) shows that E(t) tends to zero as 1/t α with α 1.15. Again, the final time profile presented in figure 3.17(d) confirms that u and y are small but it shows also that high frequencies for the unknown y are not completely controlled. Remark 3.5.4. The final time profile presented in figure 3.13(d) , figure 3.15(d) and figure 3.17(d) confirms that u and y are small but it shows also that high frequencies for the unknown y are not completely controlled. Remark 3.5.5. When the propagation speeds are not equal, the solution (u, y) has the same behavior no matter if a > 1 or a < 1. The polynomial convergence is numerically better than 1/t but it will be probably be 1/t for greater time. For reason of computation time, we did not perform very long simulation to confirm.

General conclusion

In this work, we have obtain theoretical results for waves equations coupled by an order one term and a dissipation term of order one. We have completely investigate the two cases when the propagation speed are the same and when they are different. In the case when they are the same, we have obtain theoretical results no matter the size of the damping and the coupling term. This result is up to our knowledge new. Moreover when the propagation speeds are different, we have investigate the convergence behavior and proved a polynomial decay. Numerical simulations in the 1D case confirm the theoretical results and are even better since when the intersection between the support of the coupling and damping are empty, the same long time behavior is observed. This fact will be investigated in the 2D dimension. or with Dirichlet-Neumann-Neumann boundary conditions

ϕ (0, •) = ϕ (L, •) = ψ x (0, •) = ψ x (L, •) = w x (0, •) = w x (L, •) = 0 in R + , (4.1.3)
in addition to the following initial conditions

ϕ (•, 0) = ϕ 0 (•) , ψ (•, 0) = ψ 0 (•), w (•, 0) = w 0 (•) , ϕ t (•, 0) = ϕ 1 (•) , ψ t (•, 0) = ψ 1 (•) , w t (•, 0) = w 1 (•) , in (0, L). (4.1.4)
The functions ϕ, ψ, and w model the vertical, shear angle, and longitudinal displacements of the filament. The coefficients ρ 1 , ρ 2 , k 1 , k 2 , k 3 , l are positive constants. D 1 , D 2 and D 3 are positive functions over (0, L).

The Bresse system is usually considered in studying elastic structures of the arcs type (see [START_REF] Lagnese | Modeling, analysis and control of dynamic elastic multi-link structures[END_REF]). It can be expressed by the equations of motion

ρ 1 ϕ tt = Q x + lN ρ 2 ψ tt = M x -Q ρ 1 w tt = N x -lQ where N = k 3 (w x -lϕ) + D 3 (w xt -lϕ t ) Q = k 1 (ϕ x + ψ + lw) + D 1 (ϕ xt + ψ t + lw t ) M = k 2 ψ x + D 2 ψ xt
are the stress strain relations for elastic behavior. Here

ρ 1 = ρA, ρ 2 = ρI, k 1 = k GA, k 3 = EA, k 2 = EI, l = R -1
where ρ is the density of the material, E is the modulus of elasticity, G is the shear modulus, k is the shear factor, A is the cross-sectional area, I is the second moment of area of the cross-section, and R is the radius of curvature. ϕ, ψ, and w are the vertical, shear angle, and longitudinal displacements. We note that when R → ∞, then l → 0 and the Bresse model reduces to well-known Timoshenko beam equations.

The stability of the elastic Bresse system with different types of dissipative has been intensively studied. We start by recall some results. In [START_REF] Guesmia | Bresse system with infinite memories[END_REF], Guesmia et al. considered Bresse system with infinite memories acting in the three equations of the system. They established asymptotic stability results under some conditions on the relaxation functions regardless the speeds of propagation. Wehbe and Youssef in [START_REF] Wehbe | Exponential and polynomial stability of an elastic bresse system with two locally distributed feedback[END_REF] considered an elastic Bresse system subject to two locally internal dissipation laws. They proved that the system is exponentially stable if and only if the waves propagate at the same speed. Otherwise, a polynomial decay holds. Alabau et al in [START_REF] Boussouira | Stability to weak dissipative bresse system[END_REF] considered the same system with one globally distributed dissipation law. The authors proved the existence of polynomial decays with rates that depend on some particular relation between the coefficients. Moreover, for the thermoelastic Bresse system, we quote [START_REF] Fatori | Rates of decay to weak thermoelastic bresse system[END_REF], [START_REF] Liu | Energy decay rate of the thermoelastic bresse system[END_REF] and [START_REF] Najdi | Weakly locally thermal stabilization of bresse systems[END_REF]. In [START_REF] Liu | Energy decay rate of the thermoelastic bresse system[END_REF], Liu and Rao considered the Bresse system with two thermal dissipation laws. The authors proved an exponential decay rate when the wave speed of the vertical displacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement. Otherwise, they showed polynomial decays depending on the boundary conditions. These results are improved by Fatori and Rivera in [START_REF] Fatori | Rates of decay to weak thermoelastic bresse system[END_REF] where they considered the case of one thermal dissipation law 115 Chapitre 4. Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface globally distributed on the displacement equation. Wehbe and Najdi in [START_REF] Najdi | Weakly locally thermal stabilization of bresse systems[END_REF] extended and improved the results of [START_REF] Fatori | Rates of decay to weak thermoelastic bresse system[END_REF], when the thermal dissipation is locally distributed.

The purpose of this chapter is to study the Bresse system in the presence of local Kelvin-Voigt damping with non-smooth coefficient at interface and under fully Dirichlet boundary conditions or Dirichlet-Neumann-Neumann boundary conditions. First, we study the strong stability of the Bresse system under the condition of the existence of at least one local damping. Note that we give the proof only in the case of one local damping applied at the shear angle displacement under, either the boundary condition (4.1.2) or (4.1.3) since the other cases are similar to prove it. Next, we consider the case when the Kelvin-Voigt damping are globally distributed. Here, we show that the Bresse system (4.1.1) is analytic stable. Later, in the presence of three local Kelvin-Voigt dampings, we analyse the exponential and polynomial stability according to the properties of coefficient functions D 1 , D 2 and D 3 . Indeed, if D 1 , D 2 , D 3 ∈ W 1,∞ (0, L) and their supports coincide, using the frequency domain approach combined with multiplier technique, we prove that the Bresse system (4.1.1) is exponentially stable (see Theorem 4.5.1). Otherwise, if D 1 , D 2 , D 3 ∈ L ∞ (0, L) and their supports have a non-empty intersection, we establish a polynomial stability of type 1 t (see Theorem 4.6.1). Moreover, in the absence of at least one damping, we prove the lack of uniform stability for the system (4.1.1) subjected to (4.1.3). Finally, in the presence of only one local damping D 2 acting on the shear angle displacement, we establish a polynomial decay rate of type 1 √ t (see Theorem 4.8.1). In these cases, we conjecture the optimality of the obtained decay rate.

Furthermore, in addition to the previously cited papers, we rapidly recall some previous studies done on the Timoshenko system. The stability of the Timoshenko system with different kinds of dampings has been studied in [START_REF] Fernández Sare And R Racke | On the stability of damped timoshenko systems : Cattaneo versus fourier law[END_REF], [START_REF] Fatori | The timoshenko system with history and cattaneo law[END_REF], [START_REF] Santos | The stability number of the timoshenko system with second sound[END_REF] and [START_REF] Tian | Stability of a timoshenko system with local kelvin-voigt damping[END_REF]. Hago and al. in [START_REF] Fernández Sare And R Racke | On the stability of damped timoshenko systems : Cattaneo versus fourier law[END_REF] proved that the Timoshenko system with history type damping is not exponentialy stable under Cattaneo's law, while under Fourier's law, an exponential decay can be obtained if the speeds are equal. Moreover, if the speeds are different, no decay rate has been discussed. This result has been improved by Fatori and al in [START_REF] Fatori | The timoshenko system with history and cattaneo law[END_REF], where an exponential decay can be attained with Cattaneo's law if and only if a few condition on the wave speed of propagation is verified. In [START_REF] Tian | Stability of a timoshenko system with local kelvin-voigt damping[END_REF], Tian and Zhang studied the Timoshenko system with Kelvin-Voigt damping. They proved that the energy decays exponentially or polynomially and the decay rate depends on material of coefficient function. Last but not least, we exhibit some studies done on systems with Kelvin-Voigt damping. In [START_REF] Hassine | Stability of elastic transmission systems with a local kelvin-voigt damping[END_REF], Hassine considered the longitudinal and transversal vibrations of the transmission Euler-Bernoulli beam with Kelvin-Voigt damping distributed locally on any subinterval of the region occupied by the beam and only in one side of the transmission point. He proved that the semigroup associated with the equation for the transversal motion of the beam is exponentially stable, although the semigroup associated with the equation for the longitudinal motion of the beam is polynomially stable. Hassine in [START_REF] Hassine | Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping[END_REF] discussed the asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. He proved that sufficiently smooth solutions decay logarithmically at infinity even the feedback affects a small open subset of the interior. Also, in [START_REF] Hassine | Energy decay estimates of elastic transmission wave/beam systems with a local kelvin-voigt damping[END_REF], Hassine considered a beam and a wave equations coupled by an elastic beam through transmission condition. The damping which is locally distributed acts only at one equation. First, he considered the case where the dissipation acts through the beam equation, he showed a precise polynomial energy decay rate. Second, in the case where damping acts through the wave equation, he provided a precise polynomial energy decay rate. In both cases, he proved the lack of exponential stability. In [START_REF] Hassine | Logarithmic stabilization of the euler-bernoulli transmission plate equation with locally distributed kelvin-voigt damping[END_REF], Hassine studied the asymptotic behavior of the energy decay of a transmission plate equation with locally distributed Kelvin-Voigt damping. More precisely, he proved that the energy decay at least logarithmically over the time. Recently, Ammari et al in [START_REF] Ammari | Stabilization for the wave equation with singular Kelvin-Voigt damping[END_REF] considered the wave equation with Kelvin-Voigt damping in a bounded domain. In their work, they proposed to deal with the geometrical condition by considering a singular Kelvin-Voigt damping which is localized far away from the boundary. In this particular case, they showed that the energy of the wave equation decreases logarithmically to zero as time goes to infinity. This chapter is organized as follows : In Section 4.2, we prove the well-posedness of system (4.1.1) with either the boundary conditions (4.1.2) or (4.1.3). Next, in Section 4.3, we prove the strong stability of the system in the lack of the compactness of the resolvent of the generator. In Section 4.4, we prove the analytic stability when the three Kelvin-Voigt dampings are globally distributed. Later, Sections 4.5 and 4.6 are devoted to analyze the stability of the system provided the existence of three local dampings by distinguishing two cases : in the first one, when the coefficient functions D 1 , D 2 , and D 3 are smooth, in Section 4.5, we prove the exponential stability of the system. In the second one, when the coefficient functions D 1 , D 2 , and D 3 are non smooth, in Section 4.6, we prove the polynomial stability of type 1 t . Last but not least, in Section 4.7, under boundary conditions (4.1.3), we prove the lack of uniform stability of the system in the absence of at least one damping. Finally, in section 4.8, we prove the polynomial energy decay rate of type 1 √ t

for the system in the case of one local non-smooth damping D 2 acting on the shear angle displacement.

Well-posedness of the problem

In this part, using a semigroup approach, we establish well-posedness result for the system (4.1.1). Its energy is given by

E (t) = 1 2 L 0 ρ 1 |ϕ t | 2 + ρ 2 |ψ t | 2 + ρ 1 |w t | 2 + k 1 |ϕ x + ψ + lw| 2 dx + L 0 k 2 |ψ x | 2 + k 3 |w x -lϕ| 2 dx , (4.2.1)
and it is dissipated according to the following law

E (t) = - L 0 D 1 |ϕ xt + ψ t + lw t | 2 + D 2 |ψ xt | 2 + D 3 |w xt -lϕ t | 2 dx ≤ 0. (4.2.2)
Now, we define the following energy spaces :

H 1 = H 1 0 × L 2 3 and H 2 = H 1 0 × L 2 × H 1 * × L 2 * 2 ,
117 Chapitre 4. Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface where

L 2 * = {f ∈ L 2 (0, L) : L 0 f (x)dx = 0} and H 1 * = {f ∈ H 1 (0, L) : L 0 f (x)dx = 0}.
Both spaces H 1 and H 2 are equipped with the inner product which induces the energy norm

U 2 H j = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) 2 H j , j = 1, 2, = ρ 1 v 2 2 + ρ 2 v 4 2 + ρ 1 v 6 2 + k 1 v 1 x + v 3 + lv 5 2 +k 2 v 3 x 2 + k 3 v 5 x -lv 1 2 . (4.2.3)
Here and after • denotes the norm of L 2 (0, L) .

Remark 4.2.1. In the case of boundary condition (4.1.2), it is easy to see that expression (4.2.3) defines a norm on the energy space H 1 . But in the case of boundary condition (4.1.3) the expression (4.2.3) define a norm on the energy space H 2 if L = nπ l for all positive integer n. Then, here and after, we assume that there exists no n ∈ N such that L = nπ l when j = 2. Next, we define the linear operator A j in H j by

D (A 1 ) = U ∈ H 1 | v 2 , v 4 , v 6 ∈ H 1 0 (0, L) , [k 1 (v 1 x + v 3 + lv 5 ) + D 1 (v 2 x + v 4 + lv 6 )] x ∈ L 2 (0, L) , [k 2 v 3 x + D 2 v 4 x ] x ∈ L 2 (0, L), [k 3 (v 5 x -lv 1 ) + D 3 (v 6 x -lv 2 )] x ∈ L 2 (0, L) , D (A 2 ) = U ∈ H 2 | v 2 ∈ H 1 0 (0, L) , v 4 , v 6 ∈ H 1 * (0, L) , v 3 x | (0,L) = v 5 x | (0,L) = 0, [k 1 (v 1 x + v 3 + lv 5 ) + D 1 (v 2 x + v 4 + lv 6 )] x ∈ L 2 (0, L) , [k 2 v 3 x + D 2 v 4 x ] x ∈ L 2 * (0, L), [k 3 (v 5 x -lv 1 ) + D 3 (v 6 x -lv 2 )] x ∈ L 2 * (0, L)
and

A j             v 1 v 2 v 3 v 4 v 5 v 6             =              v 2 ρ -1 1 k 1 (v 1 x + v 3 + lv 5 ) + D 1 (v 2 x + v 4 + lv 6 ) x + lk 3 (v 5 x -lv 1 ) + lD 3 (v 6 x -lv 2 ) v 4 ρ -1 2 (k 2 v 3 x + D 2 v 4 x ) x -k 1 v 1 x + v 3 + lv 5 -D 1 (v 2 x + v 4 + lv 6 ) v 6 ρ -1 1 k 3 (v 5 x -lv 1 ) + D 3 (v 6 x -lv 2 ) x -lk 1 v 1 x + v 3 + lv 5 -lD 1 (v 2 x + v 4 + lv 6 )              for all U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) T ∈ D (A j ). If U = (ϕ, ϕ t , ψ, ψ t , w, w t )
T is the state of (4.1.1), then the Bresse beam system is transformed into a first order evolution equation on the Hilbert space H j :

U t (x, t) = A j U (x, t), U (x, 0) = U 0 (x), (4.2.4)
where U 0 (x) = (ϕ 0 (x) , ϕ 1 (x) , ψ 0 (x), ψ 1 (x) , w 0 (x) , w 1 (x)) T .

Remark 4.2.2. It is easy to see that there exists a positive constant c 0 such that

k 1 ϕ x + ψ + lw 2 + k 2 ψ x 2 + k 3 w x -lϕ 2 ≤ c 0 ϕ x 2 + ψ x 2 + w x 2 . (4.2.5)
On the other hand, we can show by a contradiction argument the existence of a positive constant c 1 such that, for any (ϕ, ψ, w) ∈ (H 1 0 (0, L)) 3 for j = 1 and for any (ϕ, ψ, w) ∈

H 1 0 (0, L) × (H 1 * (0, L)) 2 for j = 2, c 1 ϕ x 2 + ψ x 2 + w x 2 ≤ k 1 ϕ x + ψ + lw 2 + k 2 ψ x 2 + k 3 w x -lϕ 2 . (4.2.6)
Therefore the norm on the energy space H j given in (4.2.3) is equivalent to the usual norm on H j .

Proposition 4.2.3. Assume that coefficients functions D 1 , D 2 and D 3 are non negative. Then, the operator A j is m-dissipative in the energy space H j , for j = 1, 2.

Proof: For all U ∈ D (A j ), by a straight forward calculation, we have

Re (A j U, U ) H j = - L 0 D 1 v 2 x + v 4 + lv 6 2 + D 2 v 4 x 2 + D 3 v 6 x -lv 2 2 dx. (4.2.7)
As D 1 ≥ 0, D 2 ≥ 0 and D 3 ≥ 0 , we get that A j is dissipative. Now, let

F = f 1 , f 2 , f 3 , f 4 , f 5 , f 6 T ∈ H j ,
we prove the existence of

U = v 1 , v 2 , v 3 , v 4 , v 5 , v 6 T ∈ D (A j )
unique solution of the equation

-A j U = F.
Equivalently, we have the following system

-v 2 = f 1 , (4.2.8)
-k 1 (v 1 x + v 3 + lv 5 ) + D 1 (v 2 x + v 4 + lv 6 ) x -lk 3 (v 5 x -lv 1 ) -lD 3 (v 6 x -lv 2 ) = ρ 1 f 2 , (4.2.9)
-v 4 = f 3 , (4.2.10)
-(k 2 v 3 x + D 2 v 4 x ) x + k 1 v 1 x + v 3 + lv 5 + D 1 (v 2 x + v 4 + lv 6 ) = ρ 2 f 4 , (4.2
-v 6 = f 5 , (4.2.12)
-k 3 (v 5 x -lv 1 ) + D 3 (v 6 x -lv 2 ) x + lk 1 v 1 x + v 3 + lv 5 + lD 1 (v 2 x + v 4 + lv 6 ) = ρ 1 f 6 . (4.2.13)
Let (ϕ 1 , ϕ 3 , ϕ 5 ) ∈ (H 1 0 (0, L)) 3 for j = 1 and (ϕ 1 , ϕ 3 , ϕ 5 ) ∈ (H 1 0 (0, L) × (H 1 * (0, L)) 2 ) for j = 2 be a test function. Multiplying (4.2.9), (4.2.11) and (4.2.13) by ϕ 1 , ϕ 3 and ϕ 5 respectively. Consequently, (4.2.8) -(4.2.13) can be written after integrating by parts in the following form

     k 1 v 1 x + v 3 + lv 5 ϕ 1 x -lk 3 v 5 x -lv 1 ϕ 1 = h 1 , k 2 v 3 x ϕ 3 x + k 1 v 1 x + v 3 + lv 5 ϕ 3 = h 3 , k 3 v 5 x -lv 1 ϕ 5 x + lk 1 v 1 x + v 3 + lv 5 ϕ 5 = h 5 , (4.2.14)
where [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]), we deduce that (4.2.14) admits a unique solution in (H 1 0 (0, L)) 3 for j = 1 and in (H 1 0 (0, L) × (H 1 * (0, L)) 2 ) for j = 2. Thus, -A j U = F admits a unique solution U ∈ D (A j ) and consequently 0 ∈ ρ(A j ). Then, A j is closed and consequently ρ (A j ) is open set of C (see Theorem 6.7 in [START_REF] Kato | Perturbation theory for linear operators[END_REF] ). Hence, we easily get R(λI -A j ) = H j for sufficiently small λ > 0. This, together with the dissipativeness of A j , imply that D (A j ) is dense in H j and that A j is m-dissipative in H j (see Theorems 4.5, 4.6 in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). Thus, the proof is complete.

h 1 = ρ 1 f 2 ϕ 1 + D 1 f 1 x + f 3 + lf 5 ϕ 1 x -lD 3 (f 5 x -lf 1 )ϕ 1 , h 3 = ρ 2 f 4 ϕ 3 + D 2 f 3 x ϕ 3 x + D 1 f 1 x + f 3 + lf 5 ϕ 3 , and h 5 = ρ 1 f 6 ϕ 5 + D 3 f 5 -lf 1 ϕ 5 x + lD 1 f 1 x + f 3 + lf 5 ϕ 5 . Using Lax-Milgram Theorem (see
Thanks to Lumer-Phillips Theorem (see [START_REF] Liu | Semigroups associated with dissipative systems / zhuangyi liu, songmu zheng[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]), we deduce that A j generates a C 0semigroup of contraction e tA j in H j and therefore problem is well-posed. Then, we have the following result. Theorem 4.2.4. For any U 0 ∈ H j , problem (4.2.4) admits a unique weak solution

U ∈ C (R + ; H j ) . Moreover, if U 0 ∈ D (A j ) , then U ∈ C (R + ; D (A j )) ∩ C 1 (R + ; H j ) .

Strong stability of the system

In this part, we use a general criteria of Arendt-Batty in [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF] to show the strong stability of the C 0 -semigroup e tA j associated to the Bresse system (4.1.1) in the absence of the compactness of the resolvent of A j . Our main result is the following Theorem. Theorem 4.3.1. Assume hat there exists at least one localized damping D i for i = 1, 2 or 3 in (0, L). Then the C 0 -semigroup e tA j is strongly stable in H j , j = 1, 2, i.e., for all U 0 ∈ H j , the solution of (4.2.4) satisfies lim t→+∞ e tA j U 0 H j = 0.

For the proof of Theorem 4.3.1, we need the following two lemmas. Proof: We will prove Lemma 4.3.2 in the case D 1 = D 3 = 0 on (0, L) and D 2 ≥ a 0 > 0 on (α, β) ⊂ (0, L) and the other cases are similar to prove. First, from Proposition 4.2.3, we claim that 0 ∈ ρ (A j ) . We still have to show the result for λ ∈ R * . Suppose that there exist a real number λ = 0 and 0

= U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) T ∈ D (A j ) such that A j U = iλU. (4.3.2)
Our goal is to find a contradiction by proving that U = 0. Taking the real part of the inner product in H j of A j U and U , we get

Re (A j U, U ) H j = - L 0 D 2 v 4 x 2 dx = 0. (4.3.3)
Since by assumption D 2 ≥ a 0 > 0 on (α, β), it follows from equality (4.3.3) that

v 4 x = 0 in (α, β). (4.3.4)
Detailing (4.3.2) we get

v 2 = iλv 1 , (4.3.5) k 1 v 1 x + v 3 + lv 5 x + lk 3 v 5 x -lv 1 = iρ 1 λv 2 , (4.3.6)
v 4 = iλv 3 , (4.3.7)
k 2 v 3 x + D 2 v 4 x x -k 1 v 1 x + v 3 + lv 5 = iρ 2 λv 4 , (4.3.8)
v 6 = iλv 5 , (4.3.9)
k 3 v 5 x -lv 1 x -lk 1 v 1 x + v 3 + lv 5 = iρ 1 λv 6 . (4.3.10)
Next, inserting (4.3.4) in (4.3.7) and using the fact that λ = 0, we get

v 3 x = 0 in (α, β). ( 4
     ρ 1 λ 2 v 1 + k 1 (v 1 x + v 3 + lv 5 ) x + lk 3 (v 5 x -lv 1 ) = 0, ρ 2 λ 2 v 3 + (k 2 v 3 x + iD 2 λv 3 x ) x -k 1 (v 1 x + v 3 + lv 5 ) = 0, ρ 1 λ 2 v 5 + k 3 (v 5 x -lv 1 ) x -lk 1 (v 1 x + v 3 + lv 5 ) = 0.
v i = v i x ,
where it is easy to see that v i ∈ H 1 (0, L). It follows from equations (4.3.4) and (4.3.11) that

v 3 = v 4 = 0 in (α, β) (4.3.13)
and consequently system (4.3.12) will be after differentiating it with respect to x

ρ 1 λ 2 v 1 + k 1 v 1 x + l v 5 x + lk 3 v 5 x -l v 1 = 0 in (α, β), (4.3.14)
v 1 x + l v 5 = 0 in (α, β), (4.3.15)
ρ 1 λ 2 v 5 + k 3 v 5 x -l v 1 x -lk 1 v 1 x + l v 5 = 0 in (α, β). ( 4
ρ 1 λ 2 v 1 + lk 3 v 5 x -l v 1 = 0 in (α, β), (4.3.17)
v 1 x + l v 5 = 0 in (α, β), (4.3.18)
ρ 1 λ 2 v 5 + k 3 v 5 x -l v 1 x = 0 in (α, β). (4.3.19)
Differentiating equation (4.3.17) with respect to x, a straightforward computation with equation (4.3.19) yields

ρ 1 λ 2 v 1 x -l v 5 = 0 in (α, β). Equivalently v 1 x -l v 5 = 0 in (α, β). (4.3.20)
Hence, from equations (4.3.18) and (4.3.20), we get

v 5 = 0 and v 1 x = 0 in (α, β). (4.3.21)
The fact that v 5 = 0, from (4.3.17), we get

ρ 1 λ 2 -l 2 k 3 v 1 = 0. (4.3.22)
To finish our proof, we are against two cases to discuss :

Case 1 : λ = l k 3 ρ 1 . Using equation (4.3.22) , we deduce that v 1 = 0 in (α, β). Setting V = ( v 1 , v 1 x , v 3 , v 3 x , v 5 , v 5 x )
T . By continuity of v i on (0, L), we deduce that V (α) = 0. Then system (4.3.12) could be given as

V x = BV, in (0, α) V (α) = 0, (4.3.23)
where

B =              0 1 0 0 0 0 -λ 2 ρ 1 + l 2 k 3 k 1 0 0 -1 -l(k 1 + k 3 ) k 1 0 0 0 0 1 0 0 0 k 1 k 2 + iλD 2 k 1 -λ 2 ρ 2 k 2 + iλD 2 0 lk 1 k 2 + iλD 2 0 0 0 0 0 0 1 0 l(k 3 + k 1 ) k 3 lk 1 k 3 0 l 2 k 1 -λ 2 ρ 1 k 3 0              . (4.3.24)
Using ordinary differential equation theory, we deduce that system (4.3.23) has the unique trivial solution V = 0 in (0, α). Same argument as above leads us to prove that V = 0 on (β, L). Consequently, we obtain v 1 = v 3 = v 5 = 0 on (0, L). It follows that v 2 = v 4 = v 6 = 0 on (0, L), thus U = 0. This gives that U = C, where C is a constant. Finally, since U (0) = 0, we deduce that U = 0.

Case 2 : λ = l k 3 ρ 1 . The fact that v 1

x = 0 on (α, β), we get v 1 = c on (α, β), where c is a constant. By continuity of v 1 on (0, L), we deduce that v 1 (α) = c. We know also that v 3 = v 5 = 0 on (α, β) from (4.3.13) and (4.3.21). Hence, setting V (α) = (c, 0, 0, 0, 0, 0) T = V 0 , we can rewrite system (4.3.12) on (0, α) under the form

V x = BV, V (α) = V 0 , where B =                0 1 0 0 0 0 0 0 0 -1 -l(k 1 + k 3 ) k 1 0 0 0 0 1 0 0 0 k 1 k 2 + il k 3 ρ 1 D 2 k 1 -λ 2 ρ 2 k 2 + il k 3 ρ 1 λD 2 0 lk 1 k 2 + il k 3 ρ 1 D 2 0 0 0 0 0 0 1 0 l(k 3 + k 1 ) k 3 lk 1 k 3 0 l 2 (k 1 -k 3 ) k 3 0                . Introducing V = ( v 1 x , v 3 , v 3 x , v 5 , v 5 x ) T and B =              0 0 -1 -l(k 1 + k 3 ) k 1 0 0 0 1 0 0 k 1 k 2 + il k 3 ρ 1 D 2 k 1 -λ 2 ρ 2 k 2 + il k 3 ρ 1 λD 2 0 lk 1 k 2 + il k 3 ρ 1 D 2 0 0 0 0 0 1 l(k 3 + k 1 ) k 3 lk 1 k 3 0 l 2 (k 1 -k 3 ) k 3 0              . 123
Chapitre 4. Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface

Then system (4.3.12) could be given as

V x = B V , in (0, α), V (α) = 0. (4.3.25)
Using ordinary differential equation theory, we deduce that system (4.3.25) has the unique trivial solution V = 0 in (0, α). This implies that on (0, α), we have v 3 = v 5 = 0. Consequently, v 3 = c 3 and v 5 = c 5 where c 3 and c 5 are constants. But using the fact that v 3 (0) = v 5 (0) = 0, we deduce that v 3 = v 5 = 0 on (0, α). Substituting v 3 and v 5 by their values in the second equation of system (4.3.12), we get that v 1 x = 0. This yields v 1 = c 1 , where c 1 is a constant. But as v 1 (0) = 0, we get v 1 = 0 on (0, α). Thus U = 0 on (0, α). Same argument as above leads us to prove that U = 0 on (β, L) and therefore U = 0 on (0, L). Thus the proof is complete. Proof: We will prove Lemma 4.3.3 in the case D 1 = D 3 = 0 on (0, L) and D 2 ≥ a 0 > 0 on (α, β) ⊂ (0, L) and the other cases are similar to prove. Since 0 ∈ ρ (A j ), we still need to show the result for λ ∈ R * . For any

F = f 1 , f 2 , f 3 , f 4 , f 5 , f 6 T ∈ H j , λ ∈ R * ,
we prove the existence of

U = v 1 , v 2 , v 3 , v 4 , v 5 , v 6 T ∈ D (A j ) solution of the following equation ( iλI -A j ) U = F.
Equivalently, we have the following system

iλv 1 -v 2 = f 1 , (4.3.26) ρ 1 iλv 2 -k 1 v 1 x + v 3 + lv 5 x -lk 3 v 5 x -lv 1 = ρ 1 f 2 , (4.3.27)
iλv 3 -v 4 = f 3 , (4.3.28) ρ 2 iλv 4 -(k 2 v 3 x + D 2 v 4 x ) x + k 1 v 1 x + v 3 + lv 5 = ρ 2 f 4 , (4.3.29)
iλv 5 -v 6 = f 5 , (4.3.30) ρ 1 iλv 6 -k 3 v 5 x -lv 1 x + lk 1 v 1 x + v 3 + lv 5 = ρ 1 f 6 . (4.3.31)
From (4.3.26),(4.3.28) and (4.3.30), we have

v 2 = iλv 1 -f 1 , v 3 = iλv 3 -f 3 , v 6 = iλv 5 -f 5 . ( 4
       -λ 2 v 1 -k 1 ρ -1 1 v 1 x + v 3 + lv 5 x -lk 3 ρ -1 1 v 5 x -lv 1 = h 1 , -λ 2 v 3 -ρ -1 2 (k 2 + iλD 2 ) v 3 xx + k 1 ρ -1 2 v 1 x + v 3 + lv 5 = h 3 , -λ 2 v 5 -k 3 ρ -1 1 v 5 x -lv 1 x + lk 1 ρ -1 1 v 1 x + v 3 + lv 5 = h 5 , (4.3.33)
where

h 1 = f 2 + iλf 1 , h 3 = f 4 + iλf 3 -ρ -1 2 D 2 f 3 xx , h 5 = f 6 + iλf 5 . For all v = (v 1 , v 3 , v 5 ) T ∈ (H 1 0 (0, L)) 3 for j = 1 and v = (v 1 , v 3 , v 5 )
T ∈ H 1 0 (0, L)×H 1 * (0, L) 2 for j = 2, we define the linear operator L by

Lv =     -k 1 ρ -1 1 v 1 x + v 3 + lv 5 x -lk 3 ρ -1 1 v 5 x -lv 1 -ρ -1 2 (k 2 + iλD 2 )v 3 xx + k 1 ρ -1 2 v 1 x + v 3 + lv 5 -k 3 ρ -1 1 v 5 x -lv 1 x + lk 1 ρ -1 1 v 1 x + v 3 + lv 5     .
Using Lax-Milgram theorem, it is easy to show that L is an isomorphism from (H 1 0 (0, L)

) 3 onto (H -1 (0, L)) 3 . Let v = (v 1 , v 3 , v 5 )
T and h = (h 1 , h 3 , h 5 ) T , then we transform system (4.3.33) into the following form

v -λ 2 L -1 v = L -1 h. (4.3.34)
Using the compactness embeddings from L 2 (0, L) into H -1 (0, L) and from H 1 0 (0, L) into L 2 (0, L), we deduce that the operator L -1 is compact from (L 2 (0, L))

3 into (L 2 (0, L)) 3 . Consequently, by Fredholm alternative, proving the existence of v solution of (4.3.34) reduces to proving Ker (I -λ 2 L -1 ) = 0. Indeed, if ṽ = (ṽ 1 , ṽ3 , ṽ5 )

T ∈ Ker(I -λ 2 L -1 ), then we have λ 2 ṽ -Lṽ = 0. It follows that

       -ρ 1 λ 2 ṽ1 -k 1 ṽ1 x + ṽ3 + lṽ 5 x -lk 3 ṽ5 x -lṽ 1 = 0, -ρ 2 λ 2 ṽ3 -(k 2 + iλD 2 ) ṽ3 xx + k 1 ṽ1 x + ṽ3 + lṽ 5 = 0, -ρ 1 λ 2 ṽ5 -k 3 ṽ5
x -lṽ 1 x + lk 1 ṽ1

x + ṽ3 + lṽ 5 = 0. Now, it is easy to see that if (ṽ 1 , ṽ3 , ṽ5 ) is a solution of system (4.3.35), then the vector Ṽ defined by Ṽ = ṽ1 , iλṽ 1 , ṽ3 , iλṽ 3 , ṽ5 , iλṽ 5 T belongs to D(A j ) and we have iλ Ṽ -A j Ṽ = 0.

Therefore, by Lemma 4.3.2, we get Ṽ = 0 and so

Ker(I -λ 2 L -1 ) = {0}.
Thanks to Fredholm alternative, the equation (4.3.34) admits a unique solution

v = (v 1 , v 3 , v 5 ) ∈ (H 1 0 (0, L))
3 . Thus, using (4.3.32) and a classical regularity arguments, we conclude that ( iλI -A j ) U = F admits a unique solution U ∈ D (A j ). Thus, the proof is complete. Proof of Theorem 4.3.1. Following a general criteria of Arendt-Batty in [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF], the C 0 -semigroup e tA j of contractions is strongly stable if A j has no pure imaginary eigenvalues and σ(A j ) ∩ iR is countable. By Lemma 4.3.2, the operator A j has no pure imaginary eigenvalues and by Lemma 4.3.3, R( iλ -A j ) = H j for all λ ∈ R. Therefore the closed graph theorem of Banach implies that σ(A j ) ∩ iR = ∅. Thus, the proof is complete. In this part, we prove the analytic stability of the Bresse system (4.1.1) provided that there exists a positive constant a 0 such that D 1 , D 2 and D 3 ≥ a 0 > 0 for every x ∈ (0.L).

(4.4.1)

Before we state our main result, we recall the following standard result see [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF], [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF] for part i), [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF] for ii) and [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] for iii).

U n = v 1 n , v 2 n , v 3 n , v 4 n , v 5 n , v 6 n T ∈ D (A j ) with U n H j = 1 (4.4.2) such that λ -1 n ( iλ n U n -A j U n ) = f 1 n , f 2 n , f 3 n , f 4 n , f 5 n , f 6 n T → 0 in H j , j = 1, 2 (4.4.3)
In the following we will check the condition (H2) by finding a contradiction with (4.4.2) such as U n H j = o(1). For clarity, we divide the proof into several lemmas. From now on, for simplicity, we drop the index n. First, remark that using (4.4.2) and (4. Proof: Taking the inner product of (4.4.3) with U in H j , then using the fact that U is uniformly bounded in H j , we get

λ -1 L 0 D 1 v 2 x 2 + D 2 v 4 x 2 + D 3 v 6 x 2 dx = λ -1 Re (A j U, U ) H j = -λ -1 Re ( iλU -A j U, U ) H j = o (1) .
Thanks to condition (4.4.1), we obtain the desired asymptotic equation (4.4.11). Thus, the proof is complete.

v 1 x = λ -1/2 v 2 x iλ 1/2 -if 1 x , v 3 x = λ -1/2 v 4 x iλ 1/2 -if 3 x and v 5 x = λ -1/2 v 6 x iλ 1/2 -if 5 x . (4.4.13)
Using the asymptotic estimation (4.4.11) and the fact that f 1 , f 3 and f 5 converge to zero in H 1 0 (0, L) in (4.4.13), we obtain the desired estimation (4.4.12). Proof: i) Multiplying (4.4.5) by -iv 2 in L 2 (0, L) and after integrating over x, we get

ρ 1 L 0 v 2 dx -iλ -1 k 1 L 0 v 1 x + v 3 + lv 5 v 2 x dx -iλ -1 L 0 D 1 v 2 x + v 4 + lv 6 v 2 x dx +iλ -1 lk 3 L 0 v 5 x -lv 1 v 2 dx + ilλ -1 L 0 D 3 v 6 x -lv 2 v 2 dx = -i L 0 ρ 1 f 2 v 2 dx.
Using (4.4.10), the first asymptotic estimation of (4.4.11), (4.4.12), the fact that v 2 , v 4 and v 6 are uniformly bounded in L 2 (0, L) and f 2 converges to zero in L 2 (0, L) in the above equation, we obtain that v 2 = o(1).

ii) Similarly, multiplying (4.4.7) by -iv 4 in L 2 (0, L), we get

ρ 2 L 0 v 4 dx -iλ -1 L 0 k 2 v 3 x + D 2 v 4 x v 4 x dx -iλ -1 k 1 L 0 v 1 x + v 3 + lv 5 v 4 dx -iλ -1 L 0 D 1 v 2 x + v 4 + lv 6 v 4 dx = -iρ 2 L 0 f 4 v 4 dx.
Using (4.4.10), (4.4.11), (4.4.12), the fact that v 4 and v 6 are uniformly bounded in L 2 (0, L) and f 4 converges to zero in L 2 (0, L) in the preceding equation, we deduce that v 4 = o(1).

iii) Finally, multiplying (4.4.9) by -iv 6 in L 2 (0, L), we get

ρ 1 L 0 v 6 dx -iλ -1 k 3 L 0 v 5 x -lv 1 v 6 x dx -iλ -1 L 0 D 3 v 6 x -lv 2 v 6 x dx -iλ -1 lk 1 L 0 v 1 x + v 3 + lv 5 v 6 dx -iλ -1 l L 0 D 1 v 2 x + v 4 + lv 6 v 6 dx = -iρ 1 L 0 f 6 v 6 dx.
Using (4.4.10), (4.4.11), (4.4.12), the fact that v 2 , v 4 and v 6 are uniformly bounded in L 2 (0, L) and f 6 converges to zero in L 2 (0, L) in the previous equation, we deduce that v 6 = o(1). Thus the proof is complete.

Proof of Theorem 4.4.2. Using Lemma 4.4.4 and Lemma 4.4.5, we obtain U H j = o (1) which contradicts (4.4.2). Therefore, (H2) holds and consequently we deduce the analytic stability of the system (4.1.1) in the case of three global dampings.

Exponential stability in the case of three local smooth dampings

In this section, we will establish the uniform stability of the C 0 -semigroup e tA j in the case when the positive continuous functions D i , i = 1, 2, 3 satisfy the following condition :

∃ d 0 > 0 such that D i ≥ d 0 > 0 for every x ∈ (α, β) , 0 < α < β < L.
(4.5.1)

Our main result in this part is the following stability estimate.

Theorem 4.5.1. Assume that (4.5.1) is satisfied and D 1 , D 2 and D 3 ∈ W 1,∞ (0, L). The C 0 -semigroup e tA j is exponentially stable, i.e., there exist constants M ≥ 1 and > 0 independent of U 0 such that

e tA j U 0 H j ≤ M e -t U 0 H j , t ≥ 0, j = 1, 2.
According to [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF], we have to check if the following conditions hold,

iR ⊆ ρ (A j ) (H1) and lim sup |λ|→+∞λ∈R (iλI -A j ) -1 L(H j ) = O (1) . (H2)
Condition iR ⊆ ρ (A j ) is already proved in Lemma 4.3.2 and Lemma 4.3.3. We will establish (H2) by contradiction. Suppose that there exist a sequence of real numbers (λ n ) n , with |λ n | → +∞ and a sequence of vectors

U n = v 1 n , v 2 n , v 3 n , v 4 n , v 5 n , v 6 n T ∈ D (A j ) with U n H j = 1 (4.5.2)
such that

iλ n U n -A j U n = f 1 n , f 2 n , f 3 n , f 4 n , f 5 n , f 6 n T → 0 in H j , j = 1, 2 (4.5.3)
detailed as

iλ n v 1 n -v 2 n = f 1 n , (4.5.4
)

iρ 1 λ n v 2 n -k 1 v 1 n x + v 3 n + lv 5 n + D 1 k 1 v 2 n x + v 4 n + lv 6 n x -lk 3 v 5 n x -lv 1 n -lD 3 v 6 n x -lv 2 n = ρ 1 f 2 n , (4.5.5) iλ n v 3 n -v 4 n = f 3 n , (4.5.6) iρ 2 λ n v 4 n -k 2 v 3 n x + D 2 k 2 v 4 n x x + k 1 v 1 n x + v 3 n + lv 5 n + D 1 v 2 n x + v 4 n + lv
iλ n v 5 n -v 6 n = f 5 n , (4.5.8
)

iρ 1 λ n v 6 n -k 3 v 5 n x -lv 1 n + D 3 k 3 v 6 n x -lv 2 n x + lk 1 v 1 n x + v 3 n + lv 5 n + lD 1 v 2 n x + v 4 n + lv 6 n = ρ 1 f 6 n . (4.5.9)
In the following we will check the condition (H2) by finding a contradiction with (4.5.2) such as U n H j = o(1). For clarity, we divide the proof into several lemmas. From now on, for simplicity, we drop the index n.

Lemma 4.5.2. Assume that condition (4.5.1) is verified. Then, we have

D 1/2 1 v 2 x + v 4 + lv 6 = o (1) , D 1/2 2 v 4 x = o (1) , D 1/2 3 v 6
x -lv 2 = o (1) (4.5.10) and

v 2 x + v 4 + lv 6 = o (1) , v 4 x = o (1) , v 6
x -lv 2 = o (1) in (α, β) . (4.5.11)

Proof: Taking the inner product of (4.5.3) with U in H j , we get

Re ( iλU -A j U, U ) H j = -Re (A j U, U ) H j (4.5.12) = L 0 D 1 |v 2 x + v 4 + lv 6 | 2 + D 2 |v 4 x | 2 + D 3 |v 6 x -lv 2 | 2 dx = o (1) .
Thanks to (4.5.1), we obtain the desired asymptotic equation (4.5.10) and (4.5.11). Thus the proof is complete.

Lemma 4.5.3. Assume that condition (4.5.1) is verified. Then, we have

v 1 x + v 3 + lv 5 = o (1) λ , v 3 x = o (1) λ , v 5 x -lv 1 = o (1) λ in (α, β) . (4.5.13)
Proof: First, using equations (4.5.4), (4.5.6) and (4.5.8), we obtain

λ v 1 x + v 3 + lv 5 = -i(v 2 x + f 1 x + v 4 + f 3 + lv 6 + lf 5 ). (4.5.14)
Consequently, The proof is thus complete.

β α λ 2 |v 1 x + v 3 + lv 5 | 2 dx ≤ 2 β α |v 2 x + v 4 + lv 6 | 2 dx + 2 β α |f 1 x + f 3 + lf 5 | 2 dx.
Here and after designate a fixed positive real number such that 0 < α+ < β -< L. We define the cut-off function Proof: First, multiplying equation (4.5.4) by iληv 1 in L 2 (0, L) and integrating by parts, we get

η ∈ C ∞ c (R) by η = 1 on [α + , β -] , 0 ≤ η ≤ 1, η = 0 on (0, L) \ (α, β) .
- L 0 η λv 1 2 dx -i L 0 ληv 2 v 1 dx = i L 0 λf 1 ηv 1 dx. (4.5.19)
As λv 1 is uniformly bounded in L 2 (0, L) and f 1 converges to zero in H 1 0 (0, L), we get that the term on the right hand side of (4.5.19) converges to zero and consequently

- L 0 η λv 1 2 dx -i L 0 ληv 2 v 1 dx = o(1). (4.5.20)
Moreover, multiplying (4.5.5) by ρ -1 1 ηv 1 in L 2 (0, L) and then integrating by parts, we obtain

i L 0 ληv 2 v 1 dx + k 1 ρ -1 1 L 0 v 1 x + v 3 + lv 5 + D 1 k 1 v 2 x + v 4 + lv 6 ηv 1 x dx -lk 3 ρ -1 1 L 0 v 5 x -lv 1 ηv 1 dx -lρ -1 1 L 0 D 3 v 6 x -lv 2 ηv 1 dx = L 0 f 2 ηv 1 dx. ( 4
+ L 0 ik 1 v 1 x + v 3 + lv 5 D 2 λv 4 dx + L 0 iD 1 v 2 x + v 4 + lv 6 D 2 λv 4 dx - L 0 iρ 2 f 4 D 2 λv 4 dx .
(i) Estimation of the second term of (4.5.24). Using (4.5.6), we have

k 2 L 0 (iλv 3 ) x D 2 v 4 x dx = k 2 L 0 v 4 x + f 3 x D 2 v 4 + D 2 v 4 x dx. (4.5.25)
Using the fact that v 4 is uniformly bounded in L 2 (0, L), f 3 converges to zero in H 1 0 (0, L) and D 2 v 4

x = o(1) due to (4.5.10) in the above equation, we deduce that

k 2 L 0 (iλv 3 ) x D 2 v 4 x dx = o(1). (4.5.26)
(ii) Estimation of the third term of (4.5.24). We have

Re L 0 D 2 v 4 x iλ D 2 v 4 x dx = Re L 0 D 2 v 4 x iλ D 2 v 4 + D 2 v 4 x dx = Re L 0 iλD 2 v 4 x v 4 D 2 dx + Re L 0 |D 2 | 2 iλ v 4 x 2 dx = Re L 0 iλD 2 v 4 x v 4 D 2 dx .
Let 1 be a positive constant. Using Young's inequality in the above equation and then using the second estimation of (4.5.10), we get (iii) Estimation of the fourth term of (4.5.24). Let 2 > 0. Using Young's inequality and the fact that v 1 x + v 3 + lv 5 is uniformly bounded in L 2 (0, L) due to (4.5.2) in the fourth term of (4.5.24), we obtain

Re L 0 D 2 v 4 x iλ D 2 v 4 x dx ≤ 1 L 0 D 2 |λv 4 | 2 dx + 1 1 L 0 D 2 |v 4 x | 2 |D 2 | 2 dx (4.
Re L 0 ik 1 v 1 x + v 3 + lv 5 D 2 λv 4 dx ≤ 2 L 0 D 2 λv 4 2 dx + 1 2 L 0 k 2 1 D 2 v 1 x + v 3 + lv 5 2 dx ≤ 2 L 0 D 2 λv 4 2 dx + O(1). (4.5.29)
(iv) Estimation of the fifth term of (4.5.24). Let 3 > 0. Using Young's inequality and the first estimation of (4.5.10) in the fifth term of (4.5.24), we obtain Proof: Multiplying (4.5.5) by iD 1 λv 2 and integrating by parts, we get

Re L 0 iD 1 v 2 x + v 4 + lv 6 D 2 λv 4 dx ≤ 3 L 0 D 2 λv 4 2 dx + 1 3 l 0 |D 1 | 2 v 2 x + v 4 + lv 6 2 D 2 dx ≤ 3 L 0 D 2 λv 4 2 dx + o(1). ( 4
ρ 1 L 0 D 1 |λv 2 | 2 dx = Re k 1 L 0 ((iλv 1 ) x + iλv 3 + liλv 5 )(D 1 v 2 ) x dx + L 0 D 1 (iλv 2 ) x (D 1 v 2 ) x dx + L 0 D 1 (v 4 + lv 6 )iλ(D 1 v 2 ) x dx - L 0 ilk 3 (v 5
x -lv 1 )D 1 λv 2 dx (4.5.34)

- L 0 ilD 3 (v 6 x -lv 2 )D 1 λv 2 - L 0 ρ 1 f 2 iλD 1 v 2 dx .
+ iλv 3 + liλv 5 )(D 1 v 2 ) x dx = Re k 1 L 0 (v 2 x + v 4 + lv 6 + f 1 x + f 3 + lf 5 )(D 1 v 2 + D 1 v 2 x )dx .
Consequently, using the fact that v 2 is uniformly bounded in L 2 (0, L), f 1 , f 3 , f 5 converge to zero in H 1 0 (0, L) and equation (4.5.10), we obtain

Re

k 1 L 0 ((iλv 1 ) x + iλv 3 + liλv 5 )(D 1 v 2 ) x dx = o(1). (4.5.35)
(ii) Estimation of the third term of (4.5.34). We have

Re L 0 D 1 v 2 x iλ(D 1 v 2 ) x dx = Re L 0 D 1 v 2 x iλ(D 1 v 2 + D 1 v 2 x )dx = Re L 0 D 1 v 2 x iλv 2 D 1 dx + Re L 0 iλ |D 1 | 2 |v 2 x | 2 dx = Re L 0 D 1 v 2 x iλv 2 D 1 dx .
Let 1 > 0. Using Young's inequality in the above equation and then using the first estimation of (4.5.10), we get

Re L 0 D 1 v 2 x iλ D 1 v 2 x dx ≤ 1 L 0 D 1 |λv 2 | 2 dx + 1 1 L 0 D 1 |v 2 x | 2 |D 1 | 2 dx ≤ 1 L 0 D 1 |λv 2 | 2 dx + o(1). (4.5.36)
(iii) Estimation of the fourth term of (4.5.34). We have

Re L 0 D 1 (v 4 + lv 6 )iλ(D 1 v 2 ) x dx =Re L 0 D 1 (v 4 + lv 6 )iλ(D 1 v 2 + D 1 v 2 x )dx =Re L 0 D 1 (v 4 + lv 6 )D 1 iλv 2 dx (4.5.37) + Re L 0 |D 1 | 2 (v 4 + lv 6 )iλv 2 x dx .
Now, we need to estimate each term of (4.5.37) as follows : 1) Let 2 > 0 and using Young's inequality and the fact that v 4 and v 6 are uniformly bounded in L 2 (0, L), we get

Re L 0 D 1 (v 4 + lv 6 )D 1 iλv 2 dx ≤ 2 L 0 D 1 |λv 2 | 2 dx + 1 2 L 0 D 1 |D 1 | 2 |v 4 + lv 6 | 2 dx ≤ 2 L 0 D 1 |λv 2 | 2 dx + O(1). (4.5.38) 2) We have Re L 0 |D 1 | 2 (v 4 + lv 6 )iλv 2 x dx = Re L 0 D 1 iλv 4 D 1 v 2 x dx (4.5.39) + Re L 0 |D 1 | 2 iλlv 6 v 2 x dx .
Hence, since suppD 1 = suppD 2 by using Lemma 4.5.5 and estimation (4.5.10), we get

Re L 0 D 1 iλv 4 D 1 v 2 x dx = o(1). (4.5.40)
On the other hand, after integrating by parts, then using Young's inequality, the fact that v 6 is uniformly bounded in L 2 (0, L) and the estimation (4.5.10), we get for 3 > 0, 4 > 0 (iv) Estimation of the fifth term of (4.5.34). For 5 > 0, by using Young's inequality and the fact that v 5

Re L 0 |D 1 | 2 iλlv 6 v 2 x dx = -Re L 0 2D 1 D 1 lv 6 iλv 2 dx -Re L 0 D 1 lv 6 x D 1 iλv 2 dx ≤ 3 L 0 D 1 |λv 2 | 2 dx + 1 3 L 0 4D 1 |D 1 | 2 l 2 |v 6 | 2 dx + 4 L 0 D 1 |λv 2 | 2 dx + 1 4 L 0 |D 1 | 3 l 2 |v 6 x | 2 dx (4.5.41) ≤( 3 + 4 ) L 0 D 1 |λv 2 | 2 dx + O(1).
x -lv 1 is uniformly bounded in L 2 (0, L), we get

Re L 0 ilk 3 (v 5 x -lv 1 )D 1 λv 2 dx ≤ 5 L 0 D 1 |λv 2 | 2 dx + 1 5 L 0 l 2 k 2 3 D 1 |v 5 x -lv 1 | 2 dx ≤ 5 L 0 D 1 |λv 2 | 2 dx + O(1). (4.5.43)
(v) Estimation of the sixth term of (4.5.34). For 6 > 0, by using Young's inequality and the third estimation of (4.5.10), we obtain (vi) Estimation of the last term of (4.5.34). For 7 > 0, by using Young's inequality and the fact that f 2 converges to zero in L 2 (0, L), we get Proof: First, multiplying (4.5.9) by iD 3 λv 6 and integrating by parts, we get (i) Estimation of the second term of (4.5.48). Using (4.5.4) and (4.5.8), we get

Re L 0 ilD 3 (v 6 x -lv 2 )D 1 λv 2 dx ≤ 6 L 0 D 1 |λv 2 | 2 dx + 1 6 L 0 l 2 D 1 |D 3 | 2 |v 6 x -lv 2 | 2 dx ≤ 6 L 0 D 1 |λv 2 | 2 dx + o(1
Re L 0 ρ 1 f 2 iλD 1 v 2 dx ≤ 7 L 0 D 1 λv 2 2 dx + 1 7 L 0 D 1 ρ 2 1 |f 2 | 2 dx ≤ 7 L 0 D 1 λv 2 2 dx + o(1). ( 4
ρ 1 L 0 D 3 |λv 6 | 2 dx = Re k 3 L 0 iλv 5 x -liλv 1 (D 3 v 6 ) x dx + L 0 D 3 v 6 x iλ(D 3 v 6 ) x dx - L 0 lv 2 D 3 iλ(D 3 v 6 ) x dx + L 0 ilk 1 v 1 x + v 3 +
Re k 3 L 0 iλv 5 x -liλv 1 (D 3 v 6 ) x dx = Re k 3 L 0 (v 6 x + f 5 x -lv 2 -lf 1 )(D 3 v 6 + D 3 v 6 x )dx , (4.5.49)
consequently, by using the fact that v 6 is uniformly bounded in L 2 (0, L), f 1 , f 5 converge to zero in H 1 0 (0, L) and the third estimation of (4.5.10), we get

Re k 3 L 0 iλv 5 x -liλv 1 (D 3 v 6 ) x dx = o(1). (4.5.50)
(ii) Estimation of the third term of (4.5.48). We have

Re L 0 D 3 v 6 x iλ(D 3 v 6 ) x dx = Re L 0 D 3 v 6 x iλ(D 3 v 6 + D 3 v 6 x )dx = Re L 0 D 3 v 6 x iλv 6 D 3 dx + Re L 0 iλ|D 3 | 2 |v 6 x | 2 dx = Re L 0 D 3 v 6 x iλv 6 D 3 dx .
Consequently, using Young's inequality in the above equation and then using the third estimation of (4.5.10), we get for 1 > 0 (iv) Estimation of the fifth term of (4.5.48). Let 2 > 0. Using Young's inequality and then the fact that v 1 x + v 3 + lv 5 is uniformly bounded in L 2 (0, L). we get (v) Estimation of the sixth term of (4.5.48). Let 3 > 0. Using Young's inequality and then the first estimation of (4.5.10), we obtain Now, let h ∈ H 1 0 (0, L). Lemma 4.5.8. Assume that condition (4.5.1) is verified. Then, we have

Re L 0 D 3 v 6 x iλ D 3 v 6 x dx ≤ 1 L 0 D 3 |λv 6 | 2 dx + 1 1 L 0 D 3 |v 6 x | 2 |D 3 | 2 dx ≤ 1 L 0 D 3 |λv 6 | 2 dx + o(1). ( 4
Re L 0 ilk 1 v 1 x + v 3 + lv 5 D 3 λv 6 dx ≤ 2 L 0 D 3 |λv 6 | 2 dx + 1 2 L 0 l 2 k 2 1 D 3 |v 1 x + v 3 + lv 5 | 2 dx
Re L 0 ilD 1 v 2 x + v 4 + lv 6 D 3 λv 6 dx ≤ 3 L 0 D 3 |λv 6 | 2 dx + 1 3 L 0 l 2 D 3 |D 1 | 2 |v 2 x + v 4 + lv 6 | 2 dx ≤ 3 L 0 D 3 |λv 6 | 2 dx + o(1). ( 4
1 2 ρ 1 L 0 h |v 2 | 2 dx + k 1 2 L 0 h |v 1 x + D 1 k 1 v 2 x + v 4 + lv 6 | 2 dx + Re -k 1 L 0 hv 3 x v 1 x dx -lk 1 L 0 v 5 x hv 1 x dx -lk 3 L 0 v 5 x hv 1 x dx = o(1). (4.5.58) Proof: First, let M = v 1 x + D 1 k 1 (v 2 x + v 4 + lv 6
). Multiplying (4.5.5) by hM then integrating by parts and using (4.5.10), the fact that v 1

x is uniformly bounded in L 2 (0, L) and f 2 converges to zero in L 2 (0, L), we get

Re L 0 iλρ 1 v 2 hM dx -k 1 L 0 v 3 x hM dx -lk 1 L 0 v 5 x hM dx + k 1 2 L 0 h |M | 2 dx -lk 3 L 0 v 5 x -lv 1 hM dx = o(1). (4.5.59)
(i) Estimation of the first term of (4.5.59). First, we have

Re L 0 iλρ 1 v 2 hM dx = -Re L 0 ρ 1 v 2 h(iλv 1 ) x dx (4.5.60) + Re L 0 iλv 2 ρ 1 h D 1 k 1 (v 2 x + v 4 + lv 6 )dx .
Next, we need to estimate each term of (4.5.60). For this, by using equation (4.5.4) we get

-Re L 0 ρ 1 v 2 h(iλv 1 ) x dx = -Re L 0 ρ 1 v 2 h(v 2 x + f 1 x )dx = 1 2 ρ 1 L 0 h |v 2 | 2 dx -Re L 0 ρ 1 v 2 hf 1 x dx . (4.5.61)
In addition, using the fact that v 2 is uniformly bounded in L 2 (0, L) and f 1 converges to zero in H 1 0 (0, L), we obtain (ii) Estimation of the second and third terms of (4.5.59). By using the fact that v 3

Re L 0 ρ 1 v 2 hf 1 x dx = o(1). ( 4
x is uniformly bounded in L 2 (0, L) and the first estimation of (4.5.10), we get 1). (4.5.66) Also, by using the fact that v 5

Re k 1 L 0 v 3 x hM dx = Re k 1 L 0 v 3 x hv 1 x dx + L 0 v 3 x D 1 (v 2 x + v 4 + lv 6 )dx = Re k 1 L 0 v 3 x hv 1 x dx + o(
x is uniformly bounded in L 2 (0, L) and the first estimation of (4.5.10), we get Re lk 1

L 0 v 5 x hM dx = Re l L 0 v 5 x hv 1 x dx + lk 1 L 0 v 5 x D 1 (v 2 x + v 4 + lv 6 )dx = Re lk 1 L 0 v 5
x hv 1 x dx + o(1). (4.5.67)

(iii) Estimation of the fifth term of (4.5.59). Using the fact that v 1 x and v 5

x are uniformly bounded in L 2 (0, L), v 1 = O( 1 λ ) and the first estimation of (4.5.10), we get Lemma 4.5.9. Assume that condition (4.5.1) is verified. Then, we have

Re -lk 3 L 0 v 5 x -lv 1 hM dx = Re -lk 3 L 0 v 5 x -lv 1 h v 1 x + D 1 k 1 (v 2 x + v 4 + lv 6 ) = Re -lk 3 L 0 v 5 x hv 1 x + o(1). ( 4
1 2 ρ 2 L 0 h |v 4 | 2 dx + k 2 2 L 0 h |v 3 x + D 2 k 2 v 4 x | 2 dx + Re k 1 L 0 v 1 x hv 3 x dx = o(1
: Let N = v 3 x + D 2 k 2 v 4
x . Multiplying (4.5.7) by hN then integrating by parts and using (4.5.10), the fact that v 3

x is uniformly bounded in L 2 (0, L) and f 4 converges to zero in L 2 (0, L), we get

Re L 0 iλρ 2 v 4 hN dx + k 2 2 L 0 h |N | 2 dx + k 1 L 0 v 1 x + v 3 -lv 5 hN dx = o(1). ( 4
+ Re L 0 iλv 4 ρ 2 h D 2 k 2 v 4 x dx .
Next, we need to estimate each term of (4.5.71). For this, by using equation (4.5.6) and then integrating by parts, we get Lemma 4.5.10. Assume that condition (4.5.1) is verified. Then, we have

-Re L 0 ρ 1 v 4 h(iλv 3 ) x dx = -Re L 0 ρ 1 v 4 h(v 4 x + f 3 x )dx = 1 2 ρ 2 L 0 h |v 4 | 2 dx -Re L 0 ρ 2 v 4 hf 3 x dx . ( 4
-Re L 0 ρ 2 v 4 h(iλv 3 ) x dx = 1 2 ρ 2 L 0 h |v 4 | 2 dx + o(1). ( 4
v 1 x + v 3 -lv 5 hN dx = Re k 1 L 0 v 1 x + v 3 -lv 5 h v 3 x + D 2 k 2 v 4 x dx = Re k 1 L 0 v 1 x hv 3 x dx + o(1). ( 4
1 2 ρ 1 L 0 h |v 6 | 2 dx + k 3 2 L 0 h |v 5 x + D 3 k 3 v 6 x -lv 2 | 2 dx + Re lk 1 L 0 v 1 x hv 5 x dx + lk 3 L 0 v 1 x hv 5 x dx = o(1). (4.5.78) Proof: Let T = v 5 x + D 3 k 3 (v 6
x -lv 2 ). Multiplying (4.5.9) by hT then integrating by parts and using (4.5.10), the fact that v 5

x is uniformly bounded in L 2 (0, L) and f 6 converges to zero in L 2 (0, L), we get

Re L 0 iλρ 1 v 6 hT dx + k 3 2 L 0 h |T | 2 dx + lk 3 L 0 v 1 x hT dx + lk 1 L 0 v 1 x + v 3 + lv 5 hT dx = o(1). ( 4
+ Re L 0 iλv 6 ρ 1 h D 3 k 3 (v 6
x -lv 1 )dx .

Next, we need to estimate each term of (4.5.80). For this, by using equation (4.5.8) we get (ii) Estimation of the third and fourth terms of (4.5.79). By using the fact that v 1

-Re L 0 ρ 1 v 6 h(iλv 5 ) x dx = -Re L 0 ρ 1 v 6 h(v 6 x + f 5 x )dx = 1 2 ρ 1 L 0 h |v 6 | 2 dx -Re L 0 ρ 1 v 6 hf 5 x dx . ( 4
x is uniformly bounded in L 2 (0, L) and the third estimation of (4.5.10), we get

Re lk

3 L 0 v 1 x hT dx = Re lk 3 L 0 v 1 x hv 5 x dx + l L 0 v 1 x hD 3 (v 6 x -lv 2 )dx = Re lk 3 L 0 v 1 x hv 5 x dx + o(1). ( 4
v 1 x + v 3 + lv 5 hT dx = Re lk 1 L 0 v 1 x + v 3 + lv 5 h v 5 x + D 3 k 3 (v 6 x -lv 2 ) = Re lk 1 L 0 v 1 x hv 5 x + o(1). ( 4
|v 2 | 2 dx + L β- |v 2 | 2 dx + k 1 2 α+ 0 |v 1 x | 2 dx + L β- |v 1 x | 2 dx 1 2 ρ 2 α+ 0 |v 4 | 2 dx + L β- |v 4 | 2 dx + k 2 2 α+ 0 |v 3 x | 2 dx + L β- |v 3 x | 2 dx (4.5.88) 1 2 ρ 1 α+ 0 |v 6 | 2 dx + L β- |v 6 | 2 dx + k 3 2 α+ 0 |v 5 x | 2 dx + L β- |v 5 x | 2 dx = o(1).
Proof: First, combining Lemma 4.5.8, Lemma 4.5.9 and Lemma 4.5.10, we get

1 2 ρ 1 L 0 h |v 2 | 2 dx + k 1 2 L 0 h |v 1 x + D 1 k 1 v 2 x + v 4 + lv 6 | 2 dx 1 2 ρ 2 L 0 h |v 4 | 2 dx + k 2 2 L 0 h |v 3 x + D 2 k 2 v 4 x | 2 dx (4.5.89) 1 2 ρ 1 L 0 h |v 6 | 2 dx + k 3 2 L 0 h |v 5 x + D 3 k 3 v 6 x -lv 2 | 2 dx = o(1).
Next, we define the cut-off function η ∈ C ∞ 0 (0, L) by η = 1 on (0, α + ), 0 ≤ η ≤ 1, η = 0 on (α + 2 , L)

and the cut-off function η ∈ C ∞ 0 (0, L) by η = 0 on (0, β -2 ), 0 ≤ η ≤ 1, η = 1 on (β -, L).
Taking h = xη + (x -L)η in (4.5.89). Then using estimations (4.5.10), (4.5.13), and (4.5.18), we get the desired estimation (4.5.88). Thus the proof is complete.

Proof of Theorem 4.5.1 By using (4.5.13), (4.5.18) and (4.5.88), we get U H j = o(1) on (0, L) which is a contradiction with (4.5.2). Therefore (H2) holds and so, by [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF], we deduce the exponential stability of the system (4.1.1) in the case of three local smooth dampings. The proof is thus complete.

Polynomial stability in the case of three local non smooth dampings

In this section, we will establish the polynomial stability of the C 0 -semigroup e tA j in the case when the positive functions D i ∈ L ∞ (0, L), i = 1, 2, 3 satisfy the following condition :

ω = suppD 1 ∩ suppD 2 ∩ suppD 3 = (α, β) ⊂ (0, L) such that mes(ω) > 0.
(4.6.1)

Our main result in this section can be given by the following theorem :

Theorem 4.6.1. Assume that condition (4.6.1) is satisfied. Assume also that D 1 , D 2 and D 3 ∈ L ∞ (0, L). Then, there exists a positive constant c > 0 such that for all U 0 ∈ D(A j ), for j = 1, 2, the energy of the system satisfies the following decay rate : Proof: Taking the inner product of (4.6.4) with U in H j , we get

E(t) ≤ c t U 0 2 D(A j ) . ( 4
Re iλ 3 U 2 -λ 2 (A j U, U ) H j = -λ 2 Re (A j U, U ) H j = λ 2 L 0 D 1 |v 2 x + v 4 + lv 6 | 2 + D 2 |v 4 x | 2 + D 3 |v 6 x -lv 2 | 2 dx = o (1) .
Thanks to (4.6.1), we obtain the desired asymptotic equation (4.6.12) and (4.6.13). Thus the proof is complete.

Lemma 4.6.3. Assume that condition (4.5.1) is verified. Then, we have

v 1 x + v 3 + lv 5 = o (1) λ 2 , v 3 x = o (1) λ 2 , v 5 x -lv 1 = o (1) λ 2 in (α, β) . (4.6.14)
Proof: First, using equations (4.6.5), (4.6.7) and (4.6.9), we obtain

λ v 1 x + v 3 + lv 5 = -i(v 2 x + f 1 x λ 2 + v 4 + f 3 λ 2 + lv 6 + l f 5 λ 2 ). (4.6.15)
Consequently,

β α λ 2 |v 1 x + v 3 + lv 5 | 2 dx ≤ 2 β α |v 2 x + v 4 + lv 6 | 2 dx + 2 β α |f 1 x + f 3 + lf 5 | 2 λ 4 dx. (4.6.16)
Using the first estimation of (4.6.13) and the fact that f 1 , f 3 , f 5 converge to zero in H 1 0 (0, L) in (4.6.16), we deduce

β α λ 2 |v 1 x + v 3 + lv 5 | 2 dx = o(1) λ 2 . ( 4
- L 0 η λv 1 2 dx -i L 0 ληv 2 v 1 dx = i L 0 f 1 λ 2 ηλv 1 dx. (4.6.20)
As λv 1 is uniformly bounded in L 2 (0, L) and f 1 converges to zero in H 1 0 (0, L), we get that the term on the right hand side of (4.6.20) converges to zero and consequently

- L 0 η λv 1 2 dx -i L 0 ληv 2 v 1 dx = o(1) λ 2 . (4.6.21)
Moreover, multiplying (4.6.6) by ρ -1 1 ηv 1 in L 2 (0, L), then integrating by parts we obtain

i L 0 ληv 2 v 1 dx + ρ -1 1 L 0 k 1 v 1 x + v 3 + lv 5 + D 1 v 2 x + v 4 + lv 6 ηv 1 x dx -lk 3 ρ -1 1 L 0 v 5 x -lv 1 ηv 1 dx -lρ -1 1 L 0 D 3 v 6 x -lv 2 ηv 1 dx = L 0 f 2 λ 2 ηv 1 dx. (4.6.22)
Using (4.6.12), (4.6.14), the fact that f 2 converges to zero in L 2 (0, L) and λv 1 , v 1 x are uniformly bounded in L 2 (0, L) in (4.6.22), we get In a same way, we show

i L 0 ληv 2 v 1 dx = o (1) λ .
β- α+ λv 3 2 dx = o (1) λ , β- α+ λv 5 2 dx = o (1) λ .
Lemma 4.6.5. Let a 1 (x), a 2 (x) and a verifies the following estimation

3 (x) ∈ L ∞ (0, L). Then the solution (u, y, z) ∈ (H 1 0 (0, L) ∩ H 2 (0, L)) 3 for j = 1 and (u, y, z) ∈ (H 1 0 (0, L) ∩ H 2 (0, L))×(H 1 * (0, L) ∩ H 2 (0, L)) 2 for j = 2 of the following system            ρ 1 λ 2 u + k 1 (u x + y + lz) x + lk 3 (z x -lu) -a 1 (x)iλu = v 1 , ρ 2 λ 2 y + k 2 y xx -k 1 (u x + y + lz) -a 2 (x)iλy = v 3 , ρ 1 λ 2 z + k 3 (z x -lu) x -lk 1 (u x + y + lz) -a 3 (x)iλz = v 5 , u (0) = u (L) = y (0) = y (L) = z (0) = z (L) = 0, for j = 1, u (0) = u (L) = y x (0) = y x (L) = z x (0) = z x (L) = 0, for j = 2,
L 0 ρ 1 |λu| 2 + ρ 2 |λy| 2 + ρ 1 |λz| 2 + k 2 |y x | 2 + k 1 |u x + y + lz| 2 + k 3 |z x -lu| 2 dx < C L 0 |v 1 | 2 + |v 3 | 2 + |v 5 | 2 dx, (4.6.25)
where C is a constant independent of n.

Proof: Consider the system

           ρ 1 u tt -k 1 (u x + y + lz) x -lk 3 (z x -lu) + a 1 (x)u t = 0, ρ 2 y tt -k 2 y xx + k 1 (u x + y + lz) + a 2 (x)y t = 0, ρ 1 z tt -k 3 (z x -lu) x + lk 1 (u x + y + lz) + a 3 (x)z t = 0. ( 4
= (H 1 0 (0, L) × L 2 (0, L))
3 and it is exponentially stable. Therefore, following Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Pruss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF], we deduce that the resolvent of our operator

A aux : D(A aux ) ⊂ H 1 → H 1 defined by A aux            u ũ y ỹ z z            =             ũ ρ -1 1 [k 1 (u x + y + lz) x + lk 3 (z x -lu) -a 1 (x)ũ] ỹ ρ -1 2 [k 2 y xx -k 1 (u x + y + lz) -a 2 (x)ỹ] z ρ -1 1 [k 3 (z x -lu) x -lk 1 (u x + z + lz) -a 3 (x)z]            
is uniformly bounded on the imaginary axis. On the other hand, system (4.6.24) can be rewritten in the following form :

             iλu -ũ = 0, iρ 1 λũ -k 1 (u x + y + lz) x -lk 3 (z x -lu) + a 1 (x)ũ = -v 1 , iλy -ỹ = 0, iρ 2 λỹ -k 2 y xx + k 1 (u x + y + lz) + a 2 (x)ỹ = -v 3 , iλz -z = 0, iρ 1 λz -k 3 (z x -lu) x + lk 1 (u x + z + lz) + a 3 (x)z = -v 5 . (4.6.27)      0 -1 ρ 1 v 1 0 -1 ρ 2 v 3 0 -1 ρ 1 v 1            . This yields (u, ũ, y, ỹ, z, z) 2 H 1 ≤ (iλ -A aux ) -1 L(H 1 ) (0, -1 ρ 1 v 1 , 0, -1 ρ 2 v 3 , 0, -1 ρ 1 v 5 ) H 1 ≤ C L 0 |v 1 | 2 + |v 3 | 2 + |v 5 | 2 dx, (4.6.28)
where C is a constant independent of n. Consequently, (4.6.25) holds. The proof is thus complete. Proof: For clarity of the proof, we divide the proof into several steps.

Step 1. First, multiplying (4.6.5) by iρ 1 λu, where u is a solution of system (4.6.24), we get

- L 0 ρ 1 λ 2 uv 1 dx -i L 0 ρ 1 λuv 2 dx = ρ 1 L 0 if 1 λ udx. ( 4
L 0 ρ 1 λuv 2 dx - L 0 k 1 u xx v 1 dx - L 0 lk 3 (-lu)v 1 dx + L 0 k 1 u x v 3 dx + L 0 lk 1 u x v 5 dx + L 0 lk 3 u x v 5 dx + L 0 D 1 (v 2 x + v 4 + lv 6 )u x dx - L 0 lD 3 (v 6 x -lv 2 )udx = ρ 1 L 0 f 2 λ 2 udx. ( 4
L 0 ρ 1 λ 2 u + k 1 u xx + lk 3 (-lu) v 1 dx - L 0 k 1 u x v 3 dx - L 0 lk 1 u x v 5 dx - L 0 lk 3 u x v 5 dx - L 0 D 1 (v 2 x + v 4 + lv 6 )u x dx + L 0 lD 3 (v 6 x -lv 2 )udx = -ρ 1 L 0 if 1 λ + f 2 λ 2 udx. ( 4
ρ 2 λ 2 y + k 2 y xx -k 1 y v 3 dx + L 0 k 1 y x v 1 dx - L 0 lk 1 yv 5 dx - L 0 D 2 v 4 x y x dx - L 0 D 1 (v 2 x + v 4 + lv 6 )ydx = -ρ 2 L 0 if 3 λ + f 4 λ 2 ydx. ( 4
.6.33)

Step 3. As in step1 and step2, by multiplying (4.6.9) by iρ 1 λz and (4.6.10) by z, where z is a solution of system (4.6.24), we get

L 0 ρ 1 λ 2 z + k 3 z xx -lk 1 (lz) v 5 dx + L 0 lk 3 z x v 1 dx + L 0 lk 1 z x v 1 dx - L 0 lk 1 zv 3 dx - L 0 D 3 v 6 x -lv 2 z x dx -l L 0 D 1 (v 2 x + v 4 + lv 6 )zdx = -ρ 1 L 0 if 5 λ + f 6 λ 2 zdx. ( 4
.6.34)

Step 4. First, combining (4.6.32), (4.6.33) and (4.6.34), we obtain

L 0 ρ 1 λ 2 u + k 1 (u x + y + lz) x + lk 3 (z x -lu) v 1 dx + L 0 ρ 2 λ 2 y + k 2 y xx -k 1 (u x + y + lz) v 3 dx + L 0 ρ 1 λ 2 z + k 3 (z x -lu) x -lk 1 (u x + y + lz) v 5 dx - L 0 D 1 (v 2 x + v 4 + lv 6 )u x dx + L 0 lD 3 (v 6 x -lv 2 )udx - L 0 D 2 v 4 x y x dx - L 0 D 1 (v 2 x + v 4 + lv 6 )ydx - L 0 D 3 v 6 x -lv 2 z x dx -l L 0 D 1 (v 2 x + v 4 + lv 6 )zdx = -ρ 1 L 0 if 1 λ + f 2 λ 2 udx -ρ 2 L 0 if 3 λ + f 4 λ 2 ydx -ρ 1 L 0 if 5 λ + f 6 λ 2 zdx. ( 4
L 0 |v 1 | 2 dx -i L 0 λua 1 (x)v 1 dx + L 0 |v 3 | 2 dx -i L 0 λya 2 (x)v 3 dx + L 0 |v 5 | 2 dx -i L 0 λza 3 (x)v 5 dx - L 0 D 1 (v 2 x + v 4 + lv 6 )u x dx + L 0 lD 3 (v 6 x -lv 2 )udx - L 0 D 2 v 4 x y x dx - L 0 D 1 (v 2 x + v 4 + lv 6 )ydx - L 0 D 3 v 6 x -lv 2 z x dx -l L 0 D 1 (v 2 x + v 4 + lv 6 )zdx = -ρ 1 L 0 if 1 λ + f 2 λ 2 udx -ρ 2 L 0 if 3 λ + f 4 λ 2 ydx -ρ 1 L 0 if 5 λ + f 6 λ 2 zdx. ( 4
L 0 |λv 1 | 2 dx -i L 0 λ 2 ua 1 (x)λv 1 dx + L 0 |λv 3 | 2 dx -i L 0 λ 2 ya 2 (x)λv 3 d + L 0 |λv 5 | 2 dx-i L 0 λ 2 za 3 (x)λv 5 dx- L 0 λD 1 (v 2 x +v 4 +lv 6 )λu x dx+ L 0 lD 3 (v 6 x -lv 2 )λ 2 udx - L 0 λD 2 v 4 x λy x dx - L 0 D 1 (v 2 x + v 4 + lv 6 )λ 2 ydx - L 0 λD 3 v 6 x -lv 2 λz x dx -l L 0 D 1 (v 2 x + v 4 + lv 6 )λ 2 zdx = -ρ 1 L 0 if 1 λ + f 2 udx -ρ 2 L 0 if 3 λ + f 4 ydx -ρ 1 L 0 if 5 λ + f 6 zdx. (4.6.37)
Next, we need to estimate each term of (4.6.37). Assuming now that for i = 1, 2, 3, there exists a 0 > 0 such that a

i (x) ≥ a 0 > 0 in (α + , β -) . ( 4
L 0 λ 2 ua 1 (x)λv 1 dx + i L 0 λ 2 ya 2 (x)λv 3 dx + i L 0 λ 2 za 3 (x)λv 5 dx = o(1) λ . ( 4
.6.39)

In addition, using (4.6.12) and the fact that λu x , λy x and λz x are uniformly bounded in L 2 (0, L) due to (4.6.25). we get

L 0 λD 1 (v 2 x + v 4 + lv 6 )λu x dx + L 0 λD 2 v 4 x λy x dx - L 0 λD 3 v 6 x -lv 2 λz x dx = o(1). ( 4
.6.40) Also, by using (4.6.12) and the fact that λ 2 u, λ 2 y and λ 2 z are uniformly bounded in L 2 (0, L) due to (4.6.25), we obtain

L 0 lD 3 (v 6 x -lv 2 )λ 2 udx + L 0 D 1 (v 2 x + v 4 + lv 6 )λ 2 ydx + l L 0 D 1 (v 2 x + v 4 + lv 6 )λ 2 zdx = o(1) λ . ( 4
.6.41)

Moreover, we have

-ρ 1 L 0 if 1 λ + f 2 udx -ρ 2 L 0 if 3 λ + f 4 ydx -ρ 1 L 0 if 5 λ + f 6 zdx = o(1), (4.6.42)
since f 1 , f 3 , f 5 converges to zero in H 1 0 (0, L), f 2 , f 4 , f 6 converges to zero in L 2 (0, L), and λ 2 u, λ 2 y, λ 2 z are uniformly bounded in L 2 (0, L). Finally, inserting (4.6.39) -(4.6.42) into (4.6.37), we get the desired estimation (4.6.29). Thus the proof is complete. Proof: First, multiplying (4.6.6) by v 1 and then integrating by parts, we get

i L 0 ρ 1 λv 2 v 1 dx + k 1 L 0 |v 1 x | 2 dx + k 1 L 0 v 3 + lv 5 v 1 x dx + L 0 D 1 v 2 x + v 4 + lv 6 v 1 x dx -lk 3 L 0 v 5 x -lv 1 v 1 dx -l L 0 D 3 v 6 x -lv 2 v 1 dx = ρ 1 L 0 f 2 λ 2 v 1 dx. (4.6.44)
Then, using (4.6.11), (4.6.12) and the fact that v 1 x , (v 5

x -lv 1 ) are uniformly bounded in L 2 (0, L) due to (4.6.3), we obtain

k 1 L 0 v 3 + lv 5 v 1 x dx + L 0 D 1 v 2 x + v 4 + lv 6 v 1 x dx -lk 3 L 0 v 5 x -lv 1 v 1 dx -l L 0 D 3 v 6 x -lv 2 v 1 dx = o(1). (4.6.45)
As f 2 converges to zero in L 2 (0, L) and λv 1 is uniformly bounded in L 2 (0, L), we have Similarly, one can prove that

ρ 1 L 0 f 2 λ 2 v 1 dx = o(1). ( 4
L 0 |v 3 x | 2 dx = o(1), L 0 |v 5 x | 2 dx = o(1).
Thus, the proof is complete.

Proof of Theorem 4.6.1 Using Lemma 4.6.6 and Lemma 4.6.7, we get that U H j = o(1). Therefore, we get a contradiction with (4.6.3) and consequently (H3) holds. Thus the proof is complete.

Lack of exponential stability

In this section, we consider the system (4.1.1) subject to the initial data (4.1.4) and to the boundary conditions (4.1.3). We prove first that system (4.1.1), subject to (4.1.3), under the following hypothesis

D 1 = 0 and D 2 = D 3 = 1 on (0, L) (4.7.1)
is not exponentially stable. Then, we prove that this system is also not exponentially stable under the following hypothesis

D 1 = D 3 = 0 and D 2 = 1 on (0, L). ( 4
(λ n ) ⊂ R with lim n→+∞ |λ n | = +∞, (V n ) ⊂ D(A 2 ), such that (iλ n I -A 2 )V n is bounded in H 2 and lim n→+∞ V n = +∞. Let F = (0, 0, 0, f 4 , 0, 0) ∈ H 2 with f 4 (x) = cos( nπx L ), λ = nπ √ ρ 2 k 2 Lρ 2
, n ∈ N.

We solve the following equations :

iλv 1 -v 2 = 0, (4.7.3) iλρ 1 v 2 -k 1 v 1 xx + v 3 x + lv 5 x -lk 3 v 5 x -lv 1 -l v 6
x -lv 2 = 0, (4.7.4)

iλv 3 -v 4 = 0, (4.7.5) iλρ 2 v 4 -k 2 v 3 xx + k 1 v 1 x + v 3 + lv 5 = ρ 2 f 4 , (4.7.6)
iλv 5 -v 6 = 0, (4.7.7)

iλρ 1 v 6 -k 3 v 5 xx -lv 1 x + lv 2 x + lk 1 v 1 x + v 3 + lv 5 = 0. ( 4
ρ 1 v 1 + k 1 v 1 xx + v 3 x + lv 5 x + l (k 3 + iλ) v 5
x -lv 1 = 0, (4.7.9) .7.11) This can be solved by the ansatz

λ 2 ρ 2 v 3 + k 2 v 3 xx -k 1 v 1 x + v 3 + lv 5 = -ρ 2 f 4 , (4.7.10) λ 2 ρ 1 v 5 + k 3 v 5 xx -lv 1 x -iλlv 1 x -lk 1 v 1 x + v 3 + lv 5 = 0. ( 4
v 1 = A sin( nπx L ), v 3 = B cos( nπx L ), v 5 = C cos( nπx L ) ( 4
L 2 k 1 -λ 2 ρ 1 + (k 3 + iλ) l 2 A+k 1 nπ L B +(k 1 + k 3 + iλ) l nπ L C = 0, (4.7.13) k 1 nπ L A + k 1 B + lk 1 C = ρ 2 , (4.7.14) (k 1 + k 3 + iλ) l nπ L A + lk 1 B + k 3 nπ L 2 -λ 2 ρ 1 + l 2 k 1 C = 0. (4.7.15) Equivalently,   nπ L 2 k 1 -λ 2 ρ 1 + (k 3 + iλ) l 2 k 1 nπ L (k 1 + k 3 + iλ) l nπ L k 1 nπ L k 1 lk 1 (k 1 + k 3 + iλ) l nπ L lk 1 k 3 nπ L 2 -λ 2 ρ 1 + l 2 k 1     A B C   =   0 ρ 2 0   .
This implies that

A = (k 2 ρ 1 -ρ 2 k 3 ) ρ 2 2 L π (k 2 ρ 2 1 -k 3 ρ 1 ρ 2 + ρ 2 l 2 ) k 2 n + O(n -2 ), (4.7.16) B = ρ 2 (k 1 k 3 ρ 2 2 + ((-k 1 -k 3 ) ρ 1 + l 2 ) k 2 ρ 2 + k 2 2 ρ 2 1 ) k 1 ((-k 3 ρ 1 + l 2 ) ρ 2 + k 2 ρ 2 1 ) k 2 + O(n -1
), (4.7.17)

C = ilρ 2 2 L √ ρ 2 k 2 π ((-k 3 ρ 1 + l 2 ) ρ 2 + k 2 ρ 2 1 ) k 2 n + O(n -2 ). (4.7.18) Now, let V n = (v 1 , iλv 1 , v 3 , iλv 3 , v 5 , iλv 5
), where v 1 , v

(iλI -A 2 )V n 2 H 2 = (0, 0, 0, ρ 2 f 4 -iλD 2 v 3 xx , 0, iλD 3 v 5 xx ) 2 H 2 ≤ c. Consequently, (iλI -A 2 )V n 2
|λ n | = +∞, (V n ) ⊂ D(A 2 ), such that (iλ n I -A)V n is bounded in H 2 and lim n→+∞ V n = +∞. We distinguish several cases : case 1. If k 1 ρ 1 = k 2 ρ 2 .
Let F = (0, 0, 0, f 4 , 0, 0) ∈ H 2 with

f 4 (x) = cos( nπx L ), , λ = nπ √ ρ 2 k 2 ρ 2 L
, n ∈ N.

We solve the following equations :

iλv 1 -v 2 = 0, (4.7.19
) (k

iλρ 1 v 2 -k 1 v 1 xx + v 3 x + lv 5 x -lk 3 v 5 x -lv 1 = 0, (4.7
λ 2 ρ 2 v 3 + k 2 v 3 xx -k 1 v 1 x + v 3 + lv 5 = -ρ 2 f 4 , (4.
1 + k 3 ) l nπ L k 1 nπ L k 1 lk 1 (k 1 + k 3 ) l nπ L lk 1 l 2 k 1 + k 3 nπ L 2 -λ 2 ρ 1     A B C   =   0 ρ 2 0   .
This implies that

A = ρ 2 2 L πρ 1 k 2 n
+ O(n -3 ), (4.7.32)

B = -k 1 ρ 2 2 + k 2 ρ 1 ρ 2 ρ 1 k 1 k 2 + O(n -2
), (4.7.33)

C = l ρ 2 2 (k 2 ρ 1 + k 3 ρ 2 ) L 2 π 2 ρ 1 (k 2 ρ 1 -k 3 ρ 2 ) k 2 n 2 + O(n -4 ).
(4.7.34)

Now, let V n = (v 1 , iλv 1 , v 3 , iλv 3 , v 5 , iλv 5 ), where v 1 , v Consequently, we deduce that (iλI -A 2 )V n 2 H 2 is bounded as n tends to +∞. Thus the proof is complete. case 2. If k 1 = k 3 . Let F = (0, 0, 0, 0, 0, f 6 ) ∈ H 2 with

f 6 (x) = cos( nπx L ), , λ = nπ √ ρ 1 k 3 ρ 1 L
, n ∈ N.

We solve the following equations : , n ∈ N.

iλv 1 -v 2 = 0, ( 4
k 1 nπ L A + k 1 + k 2 nπ L 2 -λ 2 ρ 2 B + lk 1 C = 0, (4.7.46)
We solve the following equations : This can be solved by the ansatz

iλv 1 -v 2 =
v 1 = A sin( nπx L ), v 3 = B cos( nπx L ), v 5 = C cos( nπx L ), (4.7.60)
where A, B and C depend on λ to be determined. Noting that nπ [START_REF] Wehbe | Exponential and polynomial stability of an elastic bresse system with two locally distributed feedback[END_REF] and [START_REF] Abdallah | Stability results of a distributed problem involving bresse system with history and/or cattaneo law under fully dirichlet or mixed boundary conditions[END_REF]) , where the energy decays exponentially provided that the wave speeds are equal, in the case of Kelvin-Voigt damping the energy does not decay uniformly even if the waves speed are equal. The main result of this section is given by the following theorem : Theorem 4.8.1. Assume that condition (4.8.1) is satisfied. Assume also that D 2 ∈ L ∞ (0, L). Then, there exists a positive constant c > 0 such that for all U 0 ∈ D(A j ), j = 1, 2, the energy of system (4.1.1) satisfies the following decay rate :

Equivalently,           l 2 k 3 k 3 nπ L 2k 3 lnπ L k 3 nπ L k 3 lk 3 2k 3 lnπ L lk 3 l 2 k 3                 A B C       =       0 ρ 2 0       . ( 4
E(t) ≤ c √ t U 0 2 D(A j ) . (4.8.2)
Referring to [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] In the following we will check the condition (H4) by finding a contradiction with (4.8.3) such as U n H j = o(1). For clarity, we divide the proof into several lemmas. From now on, for simplicity, we drop the index n. Thanks to (4.8.1), we obtain the desired asymptotic equation (4.8.12). Thus the proof is complete. x | 2 λ 8 dx. Using (4.8.12) and the fact that f 3 converges to zero in H 1 0 (0, L) in the above equation, we get the desired estimation (4.8.13). Thus the proof is complete.

Let be a positive constant such that 0 < α + < β -. We define the cut-off function η by η(x) = 1 in (α + , β -), 0 ≤ η(x) ≤ 1, η(x) = 0 in (0, L) \ (α, β). Consequently for 1 = 1 2 , we get the desired estimation (4.8.14). Thus the proof is complete. Then, using (4.8.12) and (4.8.13), v 3 = O( 1 λ ), the fact that (v 1 x + v 3 + lv 5 ) is uniformly bounded in L 2 (0, L) and the fact that f 3 , f 4 converge to zero in H 1 0 (0, L), L 2 (0, L) respectively, we deduce that The proof is thus complete. Finally, using (4.8.11), (4.8.27), the fact that (v 1 x + v 3 + lv 5 ), (v 5

x -lv 1 ) are uniformly bounded in L 2 (0, L) and f 1 , f 2 converge respectively to zero in H 1 0 (0, L), L 2 (0, L) in the right hand side of the above equation, we deduce that Step 2. Our aim here is to prove • Using the fact that f 4 converges to zero in L 2 (0, L) and v 1

x is uniformly bounded in L 2 (0, L), we get On the other hand, our target now is to prove (4.8.40). For this, multiplying (4. x + v 3 + lv 5 ), (v 5

x -lv 1 ) are uniformly bounded in L 2 (0, L) and the fact that f 1 , f 2 converge to zero respectively in H 1 0 (0, L), L 2 (0, L) in the right hand side of the above equation, we deduce that Step 3. Our target is to prove Using (4.8.11), (4.8.40), (4.8.54), the fact that (v 1 x + v 3 + lv 5 ), (v 5

η v 1 xx √ λ = O(1
x -lv 1 ) are uniformly bounded in L 2 (0, L) and the fact that f 1 , f 2 converge to zero respectively in H 1 0 (0, L), L 2 (0, L) in the right hand side of the above equation, we deduce that The proof is thus complete. x + v 3 + lv 5 ), (v 5

x -lv 1 ) are uniformly bounded in L 2 (0, L), f 6 , f 5 converge to zero respectively in H 1 0 (0, L), L 2 (0, L) in the right hand side of the above equation, we deduce Finally, using the definition of η, we get the desired estimation (4.8.67). The proof is thus complete.

Remark 4.8.9. It is easy to see that the results of Lemmas 4.5.8, 4.5.9, 4.5.10 still hold here, and consequently one may got the estimation (4.5.88) of Lemma 4.5.11.

Proof of Theorem 4.8.1 As we mention in Remark 4.8.9, the estimation (4.5.88) is also true here. It follows from estimations (4.8.13), (4.8.20), (4.8.25), (4.8.64), (4.8.67) and (4.5.88) that U n H j = o(1) which is a contradiction with (4.8.3) . Consequently, condition (H4) holds and the energy of smooth solutions of system (4.1.1) decays polynomially as t goes to infinity.

C 0

 0 -semigroupe, Stabilité exponentielle, Stabilité polynomiale, Méthode fréquentielle, Méthode des multiplicateurs, Inégalité d'observabilité, Contrôlabilité exacte, Méthode HUM, Système de Bresse, Différences finies.iv

1 Introduction 1 . 1

 111 Introduction in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Principal used methods . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Local Indirect Stabilization of two coupled wave equations under geometric conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Exact controllability and stabilization of locally coupled wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface . . . . . . . . . . . . . . . . . . 1.2 Introduction in French . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Méthodes principales utilisées . . . . . . . . . . . . . . . . . . . . . 1.2.2 Stabilité indirecte locale de deux équations d'ondes couplées sous des conditions géométriques . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Contrôllabilité exacte et stabilité des équations d'ondes localement couplées . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Chapitre 1 . Introduction Corollary 1 . 1 . 11 .

 711111 Let (A, D(A)) be an unbounded linear operator on H. Assume that A is the infinitesimal generator of a C 0 -semigroup of contractions (S(t)) t 0 .-For U 0 ∈ D(A), the problem (1.1.1) has a unique strong solutionU (t) = S(t)U 0 ∈ C([0, +∞), D(A)) ∩ C 1 ([0, +∞), H).

11 Chapitre 1 .

 111 IntroductionPrincipal results of the chapter.

  LH1)On the other hand, as b(x) is not identically null and continuous, then there exists a non empty open setsω b + ∪ ω b -⊂ Ω such that {x ∈ Ω : b(x) > 0} ⊃ ω b + and {x ∈ Ω : b(x) < 0} ⊃ ω b -.(LH2)

Theorem 1 . 1 . 20 .

 1120 (Strong Stability) Assume that a > 0, condition (LH1) holds and thatω = ω c + ∩ ω b + = ∅ (or ω c + ∩ ω b -= ∅).Then the semigroup of contractions (e tA ) t≥0 is strongly stable on the energy space H, i.e. for any U 0 ∈ H, we have lim t→+∞ e tA U 0 H = 0.(1.1.14)

Theorem 1 . 1 . 21 .

 1121 (Exponential decay rate) Let a = 1. Assume that condition (LH1) holds. Assume also that the nonempty open set ω = ω c + ∩ ω b + (or ω = ω c + ∩ ω b -) satisfies the geometric conditions PMGC and that b, c ∈ W 1,∞ (Ω).

Theorem 1 . 1 . 23 .

 1123 (Polynomial decay rate) Let a = 1. Assume that condition (LH1) holds. Assume also that the nonempty open set ω = ω c + ∩ ω b + (or ω = ω c + ∩ ω b -) satisfies the geometric conditions PMGC and that b, c ∈ W 1,∞ (Ω).

)

  On the other hand, as b(x) is not identically null and continuous, then there exists a non empty open ω b ⊂ Ω such that {x ∈ Ω : b(x) = 0} ⊃ ω b . (LH2) If ω = ω c + ∩ ω b = ∅ and condition (LH1) holds, then system (1.1.4)-(1.1.6) is strongly stable using Theorem 1.1.20, i.e.

  k 3 and l are positive constants. D 1 , D 2 and D 3 are positive functions over (0, L).

  prove that the system (1.1.24) with Dirichlet-Neumann-Neumann boundary condition and under the following D 1 = 0 and D 2 = D 3 = 1 on (0, L), (1.1.31)

1. 2 . 4

 24 Chapitre 4 : Stabilité d'un système de Bresse avec amortissement local Kelvin-Voigt et coefficient non régulière à l'interface Ce chapitre est consacré à l'étude de la stabilité d'un système de Bresse élastique avec amortissement local de type Kelvin-Voigt et de coefficient non-régulier à l'interface dans des conditions de Dirichlet ou Dirichlet-Neumann-Neumann aux bords. Le système défini sur (0, L) × (0, +∞) est donné par :

Theorem 2 . 3 . 3 .

 233 (Exponential decay rate) Let a = 1. Assume that condition (LH1) holds. Assume also that the nonempty open set ω

55 Chapitre 2 .

 552 Local indirect stabilization of N-d system of two coupled wave equations under geometric conditions

Proposition 2 . 6 . 2 .

 262 Assume that N = 1, a = 1, b = b 0 = 0 and c = c 0 > 0. Then, there exist n 0 ∈ N sufficiently large and two sequences λ

Remark 2 . 6 . 3 .

 263 The operator A has two branches of eigenvalues. The energy corresponding to the first branch λ (0) n decays polynomially while the energy corresponding to the second branch of eigenvalues λ (1) n decays exponentially. Proof of Theorem 2.6.1 Let > 0 and set l = 1 + . First, let λ

  (3.1.7)-(3.1.9) is equivalent to the exponential stability of system (3.1.4)-(3.1.6) in an appropriate Hilbert space. So, we provide a complete analysis for the exponential stability of system (3.1.4)-(3.1.6) in different Hilbert spaces. First, when the waves propagate at the same speed (i.e., a = 1), under the condition that the coupling region includes in the damping region and satisfying the called Geometric Control Condition (GCC in Short), we establish the exponential stability of system (3.1.4)-(3.1.6). Consequently, an observability inequality of the solution of the homogeneous system associated to (3.1.7)-(3.1.9) in the space (H 1 0

)

  On the other hand, as b(x) is not identically null and continuous, then there exists a non empty open ω b ⊂ Ω such that {x ∈ Ω : b(x) = 0} ⊃ ω b . (LH2) If ω = ω c + ∩ ω b = ∅ and condition (LH1) holds, then system (3.1.4)-(3.1.6

Lemma 3 . 3 . 7 .

 337 The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.3.4)-(3.3.7) satisfies the following estimation Ω c|∇u n | 2 dx = o(1) and ωc + |∇u n | 2 dx = o(1). (3.3.15)

Finally, using estimation ( 3 . 3 .

 33 11) in (3.3.19), we deduce ωc + |∇u n | 2 dx = o(1).

Lemma 3 . 3 . 8 .

 338 The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.3.4)-(3.3.7) satisfies the following estimation ω b |∇y n | 2 dx = o(1).(3.3.20)

  .3.28) iii) Finally, by combining (3.3.24) and(3.3.28) and taking the imaginary part, we obtain Ω b|∇y n | 2 dx = Ω b|∇u n | 2 dx + o(1). (3.3.29) Since ω b ⊂ ω c + , it follows from (3.3.15) and (3.3.29) that ω b |∇y n | 2 dx = o(1).

Lemma 3 . 3 . 9 .

 339 The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.3.4)-(3.3.7) satisfies the following estimation ω b |β n y n | 2 dx = o(1).(3.3.30)

.3. 33 )

 33 Inserting (3.3.32), (3.3.33) into (3.3.31), we obtain Ω b|β n y n | 2 dx -Ω b|∇y n | 2 dx = o(1). Using the estimation (3.3.20) in the previous equation, we get Ω b|β n y n | 2 dx = o(1).

Lemma 3 . 3 . 11 .

 3311 The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.3.4)-(3.3.7) satisfies the following estimation Ω |β n u n | 2 dx = o(1). (3.3.41) Proof: Taking f n = u n in Lemma 3.3.10 and multiplying equation (3.3.8) by β 2

Lemma 3 . 3 . 12 .

 3312 The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.3.4)-(3.3.7) satisfies the following estimation Ω |β n y n | 2 dx = o(1).(3.3.49)

Lemma 3 . 3 . 13 .

 3313 The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.3.4)-(3.3.7) satisfies the following estimation Ω |∇u n | 2 dx = o(1) and Ω |∇y n | 2 dx = o(1).

.3. 57 )

 57 Using the fact that β n u n , β n y n are uniformly bounded in L 2 (Ω), u n = o(1) and the estimation(3.3.41) in (3.3.57), we obtain Ω |∇u n | 2 dx = o(1). (3.3.58)

.3. 59 )

 59 Using the fact that β n u n is uniformly bounded in L 2 (Ω), y n = o(1) and (3.3.49) in (3.3.59), we obtain Ω |∇y n | 2 dx = o(1). (3.3.60) The proof is thus complete. Proof of Theorem 3.3.4 It follows from (3.3.41), (3.3.49) and (3.3.56) that U n H = o(1) which is a contradiction with (3.3.2). Consequently, condition (H2) holds and the energy of system (3.1.4)-(3.1.6) decays exponentially to zero. The proof is thus complete.

and lim n→∞ (

 n→∞ iβ n I -A d )U n D = 0. (3.4.3) Next, detailing equation (3.4.3), we get

Lemma 3 . 4 . 3 .

 343 The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (3.4.4)-(3.4.7) satisfies the following estimation Ω η | ∇u n | 2 dx = o(1) and Q 2 ∩Ω | ∇u n | 2 dx = o(1).(3.4.13)

  1). (3.4.16) Finally, inserting (3.4.15) and (3.4.16) into (3.4.14), we deduce Ω η | ∇u n | 2 dx = o(1) and

Lemma 3 . 4 . 4 .

 344 The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (3.4.4)-(3.4.7) satisfies the following estimation ω b |y n | 2 dx = o(1).

.4. 24 )

 24 Finally, inserting (3.4.22)-(3.4.24) into (3.4.21), we get Ω bη|y n | 2 dx = o(1). It follows, from condition (LH3), that ω b |y n | 2 dx = o(1).

Lemma 3 . 4 . 5 .

 345 The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (3.4.4)-(3.4.7) satisfies the following estimation Ω |y n | 2 dx = o(1). (3.4.25)

Lemma 3 . 4 . 6 .

 346 The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (3.4.4)-(3.4.7) satisfies the following estimation Ω |β n (-∆) -1/2 y n | 2 dx = o(1).(3.4.30)

Lemma 3 . 4 . 7 .

 347 The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (3.4.4)-(3.4.7) satisfies the following estimation

  .4.49) Multiplying(3.4.8) by (1 -N )u n . Then integrating on Ω, using Green's formula, the fact that y n and β n u n are bounded in L 2 (Ω) and the estimation (3.4.10), we obtain(1 -N ) Ω |β n u n | 2 dx -(1 -N )a Ω |∇u n | 2 dx = o(1). (3.4.50) Using (3.4.10) and (3.4.13) in (3.4.50), we deduce (1 -N )

2 -

 2 the discrete potential energy for u as : E n y,

  ω b ∩ ω c + = ∅ : Unpredicted behavior Again, the numerical level, we are interested in the long time behavior of the solution (u, y) when we suppose that ω b ∩ ω c + = ∅. For this sake, we present in figure 3.14(a), the total energy where we have chosen b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 5 (x) = 1 1 [0.4,0.6] (x).

Figure 3 . 1 -

 31 Figure 3.1 -Initial profiles

Figure 3 . 3 -

 33 Figure 3.3 -Long time behavior when ω b ∩ ω c + = ∅. b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 3 (x) = 1 1 [0.1,0.2]∪[0.8,0.9] (x).

  (a) Energy. (b) Exponential decay. (c) Final time profile.

Figure 3 . 4 -

 34 Figure 3.4 -Long time behavior when ω b ∩ ω c + = ∅. b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 5 (x) = 1 1 [0.4,0.6] (x).

  (a) Energy. (b) Exponential decay. (c) Final time profile.

Figure 3 . 5 -

 35 Figure 3.5 -Long time behavior when ω b ∩ ω c + = ∅. b = b 5 (x) = 1 1 [0.4,0.6] (x) and c = c 4 (x) = 1 1 [0.1,0.2] (x).

  (a) Final time T = 500. (b) Final time T = 500 000.

Figure 3 . 6 -

 36 Figure 3.6 -Energy when ω b ∩ ω c + = ∅. b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 3 (x) = 1 1 [0.1,0.2]∪[0.8,0.9] (x).

  (a) Exponential decay ? (b) Polynomial decay in 1/t ? (c) Which exponent if polynomial decay ? (d) Final time profile.

Figure 3 . 7 -

 37 Figure 3.7 -Long time behavior when ω b ∩ ω c + = ∅. b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 3 (x) = 1 1 [0.1,0.2]∪[0.8,0.9] (x).

  (a) Final time T = 500. (b) Final time T = 500 000.

Figure 3 . 8 -

 38 Figure 3.8 -Energy when ω b ∩ ω c + = ∅. b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 5 (x) = 1 1 [0.4,0.6] (x).

  (a) Exponential decay ? (b) Polynomial decay in 1/t ? (c) Which exponent if polynomial decay ? (d) Final time profile.

Figure 3 . 9 -

 39 Figure 3.9 -Long time behavior when ω b ∩ ω c + = ∅. b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 5 (x) = 1 1 [0.4,0.6] (x).

  (a) Exponential decay ? (b) Polynomial decay in 1/t ? (c) Which exponent if polynomial decay ? (d) Final time profile.

Figure 3 . 11 -

 311 Figure 3.11 -Long time behavior when ω b ∩ ω c + = ∅. b = b 5 (x) = 1 1 [0.4,0.6] (x) and c = c 4 (x) = 1 1 [0.1,0.2] (x).

  (a) Final time T = 500. (b) Final time T = 500 000.

Figure 3 . 12 -

 312 Figure 3.12 -Energy when ω b ∩ ω c + = ∅. b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 3 (x) = 1 1 [0.1,0.2]∪[0.8,0.9] (x).

  (a) Exponential decay ? (b) Polynomial decay in 1/t ? (c) Which exponent if polynomial decay ? (d) Final time profile.

Figure 3 . 13 -

 313 Figure 3.13 -Long time behavior when ω b ∩ ω c + = ∅. b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 3 (x) = 1 1 [0.1,0.2]∪[0.8,0.9] (x).

  (a) Final time T = 500. (b) Final time T = 500 000.

Figure 3 . 14 -

 314 Figure 3.14 -Energy when ω b ∩ ω c + = ∅. b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 5 (x) = 1 1 [0.4,0.6] (x).

  (a) Exponential decay ? (b) Polynomial decay in 1/t ? (c) Which exponent if polynomial decay ? (d) Final time profile.

Figure 3 . 15 -

 315 Figure 3.15 -Long time behavior when ω b ∩ ω c + = ∅. b = b 4 (x) = 1 1 [0.1,0.2] (x) and c = c 5 (x) = 1 1 [0.4,0.6] (x).

Figure 3 . 16 -

 316 Figure 3.16 -Energy when ω b ∩ ω c + = ∅. b = b 5 (x) = 1 1 [0.4,0.6] (x) and c = c 5 (x) = 1 1 [0.1,0.2] (x).

Figure 3 . 17 -

 317 Figure 3.17 -Long time behavior when ω b ∩ ω c + = ∅. b = b 5 (x) = 1 1 [0.4,0.6] (x) and c = c 4 (x) = 1 1 [0.1,0.2] (x).

Lemma 4 . 3 . 2 .

 432 Under the same condition of Theorem 4.3.1, we have ker ( iλI -A j ) = {0}, j = 1, 2, for all λ ∈ R. (4.3.1)

( 4 . 3 . 12 ) 121 Chapitre 4 .

 43121214 Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface Now, we introduce the functions v i , for i = 1, .., 6 by

Lemma 4 . 3 . 3 .

 433 Under the same condition of Theorem 4.3.1, ( iλI -A j ), j = 1, 2 is surjective for all λ ∈ R.

( 4 .

 4 3.35)

125 Chapitre 4 .

 4 Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface 4.4 Analytic stability in the case of three global dampings

4 . 4 )Lemma 4 . 4 . 3 .

 44443 ( respectively (4.4.6) and (4.4.8)), we deduce that v 1 = o(1), v 3 = o(1), v 5 = o(1). (4.4.10) Under condition (4.4.1), we have λ -1/2 v 2 x = o(1), λ -1/2 v 4 x = o(1) and λ -1/2 v 6 x = o(1). (4.4.11)

Lemma 4 . 4 . 4 .Chapitre 4 .

 4444 Under condition (4.4.1), we havev 1 x = o(1), v 3 x = o(1)and v 5 x = o(1). (4.4.12) 127 Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface Proof: Differentiating (4.4.4), (4.4.6) and (4.4.8) with respect to the variable x, we get

Lemma 4 . 4 . 5 .

 445 Under condition (4.4.1), we have v 2 = o(1), v 4 = o(1) and v 6 = o(1). (4.4.14)

( 4 . 5 . 15 )λ 2 λ 2

 451522 Using the first estimation of (4.5.11) and the fact that f 1 , f 3 , f 5 converge to zero in H 1 0 (0, L) in (4.5.15), we deduceβ α λ 2 |v 1 x + v 3 + lv 5 | 2 dx = o(1). (4.5.16) In a similar way, one can prove β α |v 3 x | 2 dx = o(1) and β α |v 5 x -lv 1 | 2 dx = o(1). (4.5.17)

Lemma 4 . 5 . 4 .

 454 Assume that condition (4.5.1) is verified. Then, we have β-α+ λv 1 2 dx = o (1) ,

0 ληv 2 v 1 dx = o( 1 4 . 4 . 5 . 5 . 2 L 0 D 2

 014455202 .5.21) Using (4.5.10),(4.5.13), the fact that f 2 converges to zero in L 2 (0, L) and λv 1 is uniformly bounded in L 2 (0, L) in (4.5.21), we get i L Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface Lemma Assume that condition (4.5.1) is verified. Then, we have D 2 λv 4 = O(1). (4.5.23) Proof: First, multiplying (4.5.7) by iD 2 λv 4 and integrating by parts, we get ρ λv 4 2 dx = Re k 2

5 . 27 ) ≤ 1 L 0 D 2

 527102 |λv 4 | 2 dx + o(1). (4.5.28)

0 iρ 2 f 4 D 2 λv 4 dx ≤ 4 L 0 D 2 λv 4 2 dx + 1 4 l 0 D 2 ρ 2 2 |f 4 | 2 dx ≤ 4 L 0 D 2 0 D 2

 04404040202 .5.30) (V) Estimation of the last term of (4.5.24). Let 4 > 0. Using Young's inequality and the fact that f 4 converges to zero in L 2 (0, L), we get Re L λv 4 2 dx + o(1). (4.5.31) Main estimation. Finally, inserting (4.5.26), (4.5.28), (4.5.29), (4.5.30) and (4.5.31) into (4.5.24), we get (ρ 2 -1 -2 -3 -4 ) L λv 4 2 dx ≤ O(1). (4.5.32) Taking 1 = 2 = 3 = 4 = ρ 2 8 in the above equation, we get the desired estimation (4.5.23). The proof is thus complete. Lemma 4.5.6. Assume that condition (4.5.1) is verified. Then, we have D 1 λv 2 = O(1). (4.5.33)

Finally, inserting ( 4 . 5 . 0 D 1 (v 4 + 0 D 1

 4501401 [START_REF] Kassem | Stabilisation locale indirecte d'un système n-d de deux équations d'ondes couplées sous conditions géométriques[END_REF], (4.5.40), (4.5.41) into (4.5.37), we obtain ReL lv 6 )iλ(D 1 v 2 ) x dx ≤ ( 2 + 3 + 4 ) L |λv 2 | 2 dx + O(1).(4.5.42)

.5. 45 ) 0 D 1

 4501 Main estimation. Inserting (4.5.35), (4.5.36), (4.5.42), (4.5.43), (4.5.44) and (4.5.45) into (4.5.34), we obtain(ρ 1 -1 -2 -3 -4 -5 -6 -7 ) L |λv 6 | 2 dx ≤ O(1).(4.5.46)Taking 1 = 2 = 3 = 4 = 5 = 6 = 7 = ρ 1 14in the above equation, we get the desired estimation (4.5.33). The proof is thus complete. Lemma 4.5.7. Assume that condition (4.5.1) is verified. Then, we have D 3 λv 6 = O(1). (4.5.47)

lv 5 D 0 ilD 1 v 2 x + v 4 + lv 6 D 3 λv 6 dx - L 0 iρ 1 f 6 D 3

 50246063 3 λv 6 dx (4.5.48) + L λv 6 dx .

0 lv 2 D 0 liλv 2 D 3 (D 3 v 6 + 0 lv 2 D

 02023602 .5.51) (iii) Estimation of the fourth term of (4.5.48). We have ReL 3 iλ(D 3 v 6 ) x dx = Re L D 3 v 6 x )dx ,(4.5.52)since suppD 3 = suppD 1 , using Lemma 4.5.6, the fact that v 6 is uniformly bounded in L 2 (0, L) and the third estimation of (4.5.10), we deduce that Re L 3 iλ(D 3 v 6 ) x dx = O(1). (4.5.53)

≤ 2 L 0 D 3

 203 |λv 6 | 2 dx + O(1). (4.5.54)

0 iρ 1 f 6 D 3 λv 6 dx ≤ 4 L 0 D 3 λv 6 2 dx + 1 4 l 0 D 3 ρ 2 2 |f 6 | 2 dx ≤ 4 L 0 D 3 0 D 3 Chapitre 4 .

 064040403034 .5.55) (vi) Estimation of the last term of (4.5.48). Let 4 > 0. Using Young's inequality and then the fact that f 6 converges to zero in L 2 (0, L), we obtain Re L λv 6 2 dx + o(1). (4.5.56) Main estimation. Inserting (4.5.50), (4.5.51), (4.5.53), (4.5.54), (4.5.55) and (4.5.56) into (4.5.48), we get (ρ 1 -1 -2 -3 -4 ) L |λv 6 | 2 dx ≤ O(1). (4.5.57) 137 Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface Taking 1 = 2 = 3 = 4 = ρ 1 8 in the above equation, we get the desired estimation (4.5.47). The proof is thus complete.

ρ 1 ρ 1 L 0 h 0 iλv 2 ρ 1 h D 1 k 1 (v 2 x + v 4 + 0 iλρ 1 v 2 hM dx = 1 2 ρ 1 L 0 h |v 2 |

 11001240102 v 2 h(iλv 1 ) x dx = 1 2 |v 2 | 2 dx + o(1). (4.5.63) Moreover, using Lemma 4.5.6 and the first estimation of (4.5.10), we deduce that Re L lv 6 )dx = o(1). (4.5.64) Finally, inserting (4.5.63) and (4.5.64) into (4.5.60), we get Re L 2 dx + o(1). (4.5.65)

  .5.68) (iv) Main estimation. Inserting (4.5.65), (4.5.66), (4.5.67) and (4.5.68) into (4.5.59), we obtain the desired estimation (4.5.58). Thus the proof is complete.

0 iλρ 2 v 4 0 ρ 2 v

 0402 .5.70) (i) Estimation of the first term of (4.5.70). First, we have Re L hN dx = -Re L 4 h(iλv 3 ) x dx (4.5.71)

0 ρ 2 v 4

 024 .5.72) In addition, using the fact that v 4 is uniformly bounded in L 2 (0, L) and f 3 converges to zero in H 1 0 (0, L), we obtain Re L hf 3 x dx = o(1). (4.5.73) Inserting (4.5.73) into (4.5.72), we get

0 iλρ 2 v 4 hN dx = 1 2 ρ 2 L 0 h

 020 .5.74) Moreover, using Lemma 4.5.5 and the second estimation of (4.5.10), we deduce that Re (4.5.74) and (4.5.75) into (4.5.71), we get Re L |v 4 | 2 dx + o(1). (4.5.76) (ii) Estimation of the third term of (4.5.70). Using the fact that v 1 x , v 3 x are uniformly bounded in L 2 (0, L), v 3 = O( 1 λ ), v 5 = O( 1 λ ) and the second estimation of (4.5.10), we get Re k 1 L 0

  .5.77) (iii) Main estimation. Inserting (4.5.76) and (4.5.77) into (4.5.70) we get the desired estimation (4.5.69). Thus the proof is complete.

0 iλρ 1 v 6 0 ρ 1 v

 0601 .5.79) (i) Estimation of the first term of (4.5.79). First, we have Re L hT dx = -Re L 6 h(iλv 5 ) x dx (4.5.80)

  .5.86) Using the fact that v 1 x , v 5 x are uniformly bounded in L 2 (0, L), v 3 = O 1 λ ,v 5 = O 1 λ and the third estimation of (4.5.10), we get Re lk 1 L 0

Lemma 4 . 6 . 6 . 0 |λv 1 | 0 |λv 3 | 0 |λv 5 |

 466010305 Assume that condition (4.6.1) is satisfied. Then we haveL 2 dx = o(1), L 2 dx = o(1), L 2 dx = o(1).(4.6.29)

Lemma 4 . 6 . 7 .

 467 Assume that condition (4.6.1) is satisfied. Then we have

7 . 26 ) λ 2 ρ 1 v 5 + k 3 v 5 xx -lv 1 x -lk 1 v 1 x + v 3 + lv 5 2 nπ L 2 - 2 k 1 -+ k 1 B 2 -λ 2 ρ 1 C2 k 1 -λ 2 ρ 1 + l 2 k 3 k 1

 72655352221121111 B and C depend on λ to be determined. Noting that k λ 2 ρ 2 = 0 and inserting (4.7.28) in (4.7.25)-(4.7.27) we obtain thatnπ L λ 2 ρ 1 + l 2 k 3 A + k 1 nπ L B + (k 1 + k 3 ) l nπ L C = 0, (4.7.29) + lk 1 C = ρ 2 , (4.7.30) (k 1 + k 3 ) l nπ L A + lk 1 B + l 2 k 1 + k 3 nπ L nπ L

(k 1 + k 3 ) l nπ L A + lk 1 B + l 2 k 1 C = ρ 1 .2 k 1 -λ 2 ρ 1 + l 2 k 3 k 1 2 H 2

 111111122 Lρ 1 (k 1 + k 3 ) πk 3 (3k 1 + k 3 )ln + O(n -3 ), (4.7.48) B = -2k 1 ρ 2 1 L 2 l(3k 1 + k 3 )(k 2 ρ 1 -k 3 ρ 2 )π 2 n 2 + O(n -4 ), (4.7.49) C = -(k 1 -k 3 )ρ 1 (3k 1 + k 3 )k 3 l 2 + O(n -2 ). (4.7.50) Now, let V n = (v 1 , iλv 1 , v 3 , iλv 3 , v 5 , iλv5 ), where v 1 , v 3 and v 5 are given by (4.7.44) and (4.7.48)-(4.7.50). It is easy to check V n → +∞ and (iλI -A 2 )V n is bounded as n tends to +∞. case 3. If k 1= k 3 and k 1 ρ 1 = k 2 ρ 2 . Let F = (0, 0, 0, f 4 , 0, 0) ∈ H 2 with f 4 (x) = cos( nπx L ), , λ = nπ √ ρ 2 k 2 ρ 2 L

L 2 k 3 -λ 2 ρ 1 = 0 and k 2 nπ L 2 -+ k 3 B

 3223 λ 2 ρ 2 = 0. Inserting (4.7.60) in (4.7.57)-(4.7.59) we obtain that l 2 k 3 A + k + lk 3 C = ρ 2 , (4.7.62) 2k 3 lnπ L A + lk 3 B + l 2 k 3 C = 0. (4.7.63)

4. 8

 8 Polynomial stability in the case of one local dampingThis section is devoted to show the polynomial stability of system (4.1.1) under the following hypothesisD 1 = D 3 = 0 in (0, L) and ∃ d 2 > 0 such that D 2 ≥ d 2 > 0 in (α, β) ⊂ (0, L).(4.8.1)

Lemma 4 . 8 . 2 . 4 L 0 D 2

 482402 Assume that condition (4.8.1) is verified. Then, we have D Taking the inner product of (4.8.4) with U in H j , we getRe iλ 5 U 2 -λ 4 (A j U, U ) H j = -λ 4 Re (A j U, U ) H j = λ |v 4x | 2 dx = o (1) .

Lemma 4 . 8 . 3 . 3 x = v 4 x + f 3 x λ 4 ,Chapitre 4 .

 4833344 Assume that condition (4.8.1) is verified. Then, we have Page 158 of 174 159 Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface and consequently

Lemma 4 . 8 . 4 . 0 η|λv 4 | 2 L 0 k 2 2 L 0 D 2 2 L 0 k 1 2 L 0 D 2 v 4 x iλ(ηv 4 x + η v 4 2 L 0 k 1 1 L 0 k 2 1 1 L 0 η|λv 4 | 2 dx ≤ 1 L 0 η|λv 4 |η|λv 4 |

 48404202202201204201111041044 Assume that condition (4.8.1) is verified. Then, we have √ ηλv 4 = O(1). (4.8.14) Proof: First, multiplying (4.8.8) by iρ -1 2 ηλv 4 and after integrating by parts, we getL 2 dx = Re ρ -1 iλv 3 x (ηv 4 x + η v 4 )dx + ρ -1 v 4 x iλ(ηv 4 x + η v 4 )dx + ρ -1 (v 1 x + v 3 + lv 5 )ηiλv 4 dx -i L 0 f 4 λ 3 ηv 4 dx . (4.8.15)Now, using (4.8.12), (4.8.13) and the fact that v 4 is uniformly bounded in L 2 (0, L), we get Re ρ (4.8.12) and the fact that v 4 is uniformly bounded in L 2 (0, L), we obtain Re ρ -1 Young's inequality and the fact that v 1 x + v 3 + lv 5 is uniformly bounded in L 2 (0, L), we get for 1 > 0 Re ρ -1 (v 1x + v 3 + lv 5 )ηiλv 4 dx ≤1 η|v 1 x + v 3 + lv 5 | 2 dx + 2 dx + O(1). (4.8.18)Also, using the fact that f 4 converges to zero in L 2 (0, L) and v 4 is uniformly bounded in L 2 (0, L), 2 dx ≤ O(1).

Lemma 4 . 8 . 5 . 8 ) 0 ηλv 4 v 3 dx + ρ -1 2 L 0 k 2 v 3 x + D 2 v 4 x η v 3 + ηv 3 x dx + ρ -1 2 L 0 k 1 0 η|λv 3 | 2 dx = ρ -1 2 L 0 k 2 v 3 x + D 2 v 4 x η v 3 + ηv 3 x dx + ρ -1 2 L 0 k 1 v 1 x + v 3 + lv 5 ηv 3

 485802034201032203420133 Assume that condition (4.8.1) is verified. Then, we have β-α+ |λv 3 | 2 dx = o (1) . (4.8.20) Proof: First, multiplying (4.8.7) by iηλv 3 , we get by ρ -1 2 ηv 3 , we obtain after integrating by parts i L v 1 x + v 3 + lv 5 ηv 3 dx = L 0 f 4 λ 4 ηv 3 dx. (4.8.22) Now, combining (4.8.21) and (4.8.22), we get L

3 x + D 2 v 4 x η v 3 + ηv 3 x dx + ρ -1 2 L 0 k 1 v 1 x + v 3 +λ 4 0 η|λv 3 |

 342013403 lv 5 ηv 3 dx ηv 3 dx = o(1). (4.8.24) Finally, inserting (4.8.24) into (4.8.23) and using the definition of η, we deduce L 2 dx = o(1) and β-α+ |λv 3 | 2 dx = o(1).

Lemma 4 . 8 . 6 .

 486 Assume that condition (4.8.1) is verified. Then, we have

L 0 η|λv 1 | 0 η|λv 1 | 2 dx -i L 0 ηλv 1 v 2 dx = i L 0 f 1 λ 3 0 ηλv 1 v 2 dx + ρ -1 1 L 0 k 1 η|λv 1 | 1 L 0 k 1 1 L 0 lk 3 λ 3

 010103010111011033 2 dx = o(1). (4.8.35) Multiplying (4.8.5) by iηλv 1 , we get -L ηv 1 dx. (4.8.36) Then, multiplying (4.8.6) by ρ -1 1 ηv 1 and integrating by parts, we get iL (v 1 x + v 3 + lv 5 )(η v 1 + ηv 1 x )dx -2 dx =ρ -1 (v 1 x + v 3 + lv 5 )(η v 1 + ηv 1 x )dx -ρ -1 (v5x -lv 1 )ηv 1 -ηv 1 dx. (4.8.38)

L 0 η|λv 1 |

 01 2 dx = o(1).

40 )L 0 λρ 2 v 4 η λv 1 dx -i L 0 λρ 2 v 4 x ηλv 1 dx + L 0 (k 2 λv 3 x + D 2 λv 4 x

 400004 To prove (4.8.39), multiplying (4.8.28) by λ, we get -i

  (

0 ηλ|λv 1 | 2 dx =ρ -1 1 L 0 k 1 1 L 0 lk 3 1 -

 011011031 8.38) by λ, we getL (v 1 x + v 3 + lv 5 )(η λv 1 + ηλv 1 x )dx -ρ -1 (v 5x -lv 1 )ηλv 11), (4.8.35), (4.8.39), the fact that (v 1

L 0 ηλ|λv 1 |

 01 2 dx = o(1).

L 0 ηλ 1+1/2 |λv 1 | 2 dx =ρ -1 1 L 0 k 1 -ρ -1 1 L 0 lk 3 λ 3 / 2

 0110110332 (v 1 x + v 3 + lv 5 )(η λ 1+1/2 v 1 + ηλ 1+1/2 v 1x )dx (4.8.63) (v 5 x -lv 1 )ηλ 1+1/2 v 1 -ηv 1 dx.

L 0 ηλ

 0 1+1/2 |λv 1 | 2 dx = o(1).

Step 5 . 2 . 0 η|λv 1 | 2

 52012 Using (4.8.27), (4.8.39), (4.8.54) and the definition of η, we deduce Using (4.8.35), (4.8.40), (4.8.55) and the definition of η, we deduce L

Lemma 4 . 8 . 7 . 5 x | 2 = -ρ 1 L 0 ηλ 2 v 1 v 5 x dx + k 1 L 0 v 1 x η v 5 x dx + k 1 Lλ v 5 xxLemma 4 . 8 . 8 . 0 η|λv 5 | 0 ηλv 5 v 6 dx = i L 0 f 5 λ 3 0 ηλv 5 v 6 dx + ρ -1 1 L 0 k 3 (v 5 x -lv 1 )(η v 5 + ηv 5 x )dx + ρ -1 1 L 0 lk 1 η|λv 5 | 2 dx =ρ -1 1 L 0 k 3 1 L 0 lk 1 (v 1 x + v 3 + lv 5 )ηv 1 - 4 .

 487521011548805030103110151031011314 Assume that condition (4.8.1) is verified. Then, we have Proof: First, substituting (4.8.5) into (4.8.6), we get-λ 2 ρ 1 v 1 -k 1 (v 1 x + v 3 + lv 5 ) x -lk 3 (v 5x -lv 1 ) = ρ 1 8.65) by ηv 5 x and integrating over (0, L), we get (lk 1 + lk 3 ) (4.8.13), (4.8.25), the fact that v 5x is uniformly bounded in L 2 (0, L) and 1 is uniformly bounded in L 2 (0, L) due to (4.8.10) in the right hand side of the previous equation, we get the desired estimation (4.8.64). The proof is thus complete. Assume that condition (4.8.1) is verified. Then, we have β-α+ |λv 5 | 2 dx = o (1) . (4.8.67)Proof: Multiplying (4.8.9) by iηλv 5 , we get-L 2 dx -i L ηv 5 dx.(4.8.68)Then, multiplying (4.8.10) by ηρ -1 1 v 5 and integrating by parts, we get iL (v 1 x + v 3 + lv 5 )ηv 5 dx = (v 5 x -lv 1 )(η v 5 + ηv 5 x )dx + ρ -1 Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface Using (4.8.11), (4.8.64), the fact that (v 1

L 0 η|λv 5 |

 05 2 dx = o(1).

  Maintenant, quand le C 0 -semigroupe est fortement stable, nous recherchons des conditions nécessaires et suffisantes pour lesquelles il est exponentiellement stable. Nous rappelons ici uniquement la méthode fréquentielle suivante obtenue par Huang[START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] et Prüss[START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF].

	Théorème 1.2.6. (Huang [36] and Prüss [60]) Soit (S(t)) t≥0 un C 0 -semigroupe de contrac-
	tions sur H et A son générateur infinitésimal. Alors, (S(t)) t≥0 est exponentiellement stable
	si et seulement si
	-iR ⊆ ρ(A),
	-lim sup
	β∈R,|β|→∞

Théorème 1.2.5. Supposons que A est le générateur infinitésimal d'un C 0 -semigroupe de contractions (S(t)) t≥0 fortement continu sur X. Si A n'a pas de valeurs propres imaginaires pures et si σ(A) ∩ iR est dénombrable, alors (S(t)) t≥0 est fortement stable.

  Théorème 1.2.8. (Wehbe, Najdi 2016) Soit A le générateur infinitésimal d'un C 0 -semigroupe de contractions (S(t)) t 0 . Soient (λ k,n ) les valeurs propres de A et (e k,n ) les vecteurs propres accossiés. Supposons qu'il existe µ

  est un scalaire ou une fonction à valeur vectorielle, A est un ensemble d'équations aux dérivées partielles, linéaires ou non linéaires (au moins pour le moment), v dénote le contrôle et B est la fonction de "l'espace des contrôles" dans "l'espace d'état". Le système (1.2.3) doit inclure les conditions aux bords. Nous ne les faisons pas explicitement ici. Elles sont supposées être contenues dans la formulation abstraite (1.2.3).Le contrôle v peut être appliqué à l'intérieur du domaine Ω (dans ce cas, on dit que v est un contrôle interne), ou sur la frontière Γ de Ω ou sur une partie de la frontière (dans ce

cas, v est considéré comme un contrôle frontière). Si v est appliqué en un point de Ω, v est dit contrôle ponctuel.

On supposera que, pour v (dans un espace approprié), le problème (1.2.3) définit de manière unique une solution. Cette solution est une fonction (valeur scalaire ou vectorielle)

  a∆u -bz -cv, z, ∆y + bv ).

	Puisque c(x) ≥ 0, l'opérateur A est m-dissipatif et engendre un C 0 semigroupe de contrac-
	tions e tA sur l'espace d'énergie H. Comme le système (1.2.4)-(1.2.6) est équivalent	à
	U t = AU dans H, t > 0, U (0) = U 0	(1.2.12)
	avec U = (u, u t , y, y t ), nous déduisons son caractère bien posé.
	Comme souligné précédemment nous allons étudier la stabilité et le taux de décroissance
	de l'énergie de notre système. Pour cela, nous supposons qu'il existe un ouvert non vide
	ω c + ⊂ Ω tel que :
	{x ∈ Ω : c(x) > 0} ⊃ ω c + .	(LH1)
	D'autre part, comme b(x) est non identiquement nulle et continue, alors il existe un ouvert
	non vide ω

  Notons que dans le théorème précédent nous n'avons pas de restriction sur la borne supérieure et le signe de la fonction b. Ce théorème est une généralisation dans le cas linéaire du résultat de[START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] où le coefficient de couplage considéré doit satisfaire 0 ≤ b(x) ≤ b 0 , b 0 ∈ (0, b ] où b est une constante qui ne dépend que de Ω et de la zone du contrôle. Néanmoins, le problème reste ouvert dans le cas non linéaire.

	Chapitre 1. Introduction
	Remarque 3. La condition d'égalité des vitesses de propagation est une condition nécessaire et suf-
	fisante pour la stabilité exponentielle de notre système. En effet, dans le cas a = 1,
	nous construisons une suite (U n ) d'éléments de D(A) et une suite réelle (µ n ) telles que
	.2.13)

  (Taux de décroissance exponentiel) Soit a = 1. supposons que les conditions (LH1) et (LH2) sont satisfaites. Supposons aussi que ω b ⊂ ω c + satisfait la condition du contrôle géométrique GCC et que b, c ∈ W 1,∞ (Ω). Alors, il existe des constantes positives M ≥ 1, θ > 0 tel que pour toute donnée initiale (u 0 , u 1 , y 0 , y 1 ) ∈ H l'énergie du Ce théorème est alors une généralisation dans le cas linéaire du résultat de[START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] où le coefficient de couplage considéré doit satisfaire 0 ≤ b(x) ≤ b 0 , b 0 ∈ (0, b ] où b est une constante dépendante de Ω et de la région du contrôle.

	29		Chapitre 1. Introduction
	Théorème 1.2.12. système (1.2.4)-(1.2.6) vérifie l'estimation suivante :
		E(t) ≤ M e -θt E(0),	∀t > 0.	(1.2.20)
	Remarque 4. Les situations géométriques gouvernées par le Théorème 1.2.12 sont plus
	riches que celles considérées dans le Chapitre 2 et [12]. En effet, dans les références pré-
	cédentes, les auteurs considèrent les conditions géométriques de PMGC qui sont plus res-
	trictives que GCC. D'autre part, contrairement aux résultats de [12], nous n'avons aucune
	restriction dans le Théorème 1.2.12 sur la borne supérieure et le signe du coefficient de
	la fonction de couplage b. En utilisant la Proposition 2 de A. Haraux dans [31], une inégalité d'observabilité de la
	solution du système homogène associé à (1.2.16)-(1.2.18) dans l'espace (H 1 0 (Ω) × L 2 (Ω)) 2
	est établie. Cela mène, par la méthode HUM introduite par J. L. Lions dans [46], à la
	contrôlabilité exacte du système (1.2.4)-(1.2.6) dans l'espace (H -1 (Ω) × L 2 (Ω)) 2 .
	De plus, lorsque les ondes se propagent avec des vitesses différentes, (i.e., a = 1), nous
	établissons un taux de décroissance exponentielle de l'énergie du système (1.2.4)-(1.2.6)
	dans l'espace d'énergie faible. Pour cela, on introduit l'espace d'énergie faible suivant
		hypothèse de localisation (LH1) est satisfaite, alors le système
	(1.2.4)-(1.2.6) est fortement stable en utilisant le Théorème 1.2.5, i.e.
	lim t→+∞	e tA (u 0 , u 1 , y 0 , y 1 ) H = 0 ∀(u 0 , u 1 , y 0 , y 1 ) ∈ H.
	Ensuite, quand les ondes se propagent à la même vitesse, sous la condition que la région de
	couplage est inclue dans la région d'amortissement (i.e ω b ⊂ ω c + ) et satisfaisant la condition
	du contrôle géométrique GCC, nous établissons un taux de décroissance exponentielle du
	système (1.2.4)-(1.2.6) :

  .2.22) Les coefficients ρ 1 , ρ 2 , k 1 , k 2 , k 3 et l sont des constantes positives. D 1 , D 2 et D 3 sont des fonctions positives sur (0, L).

	Chapitre 1. Introduction
	Résultat antérieur :
	Le matériel Kelvin-Voigt est une structure viscoélastique ayant des propriétés d'élas-
	ticité et de viscosité. L'amortissement Kelvin-Voigt peut être distribué globalement ou
	localement. Mais le cas qui nous intéresse est quand l'amortissement est localisé sur un in-
	tervalle quelconque du domaine. Les propriétés de régularité et de stabilité d'une solution
	dépendent des propriétés des coefficients d'amortissement. En effet, le système est plus
	efficacement contrôlé par l'amortissement local Kelvin-Voigt lorsque le coefficient est plus
	régulier à l'interface.
	Récemment, X. Tian et Q. Zhang dans

  .2.23) Ils ont étudié ce système avec un amortissement Kelvin-Voigt distribué localement ou globalement et les coefficients D 1 , D 2 ∈ C ([0, L]). D'abord, lorsque les amortissements sont globalement distribués, ils ont montré que le système (1.2.23) avec des conditions aux bords entièrement Dirichlet est analytique. Ensuite, quand les amortissements sont localisés au voisinage du bord, ils ont analysé la stabilité exponentielle et polynomiale dépendant des propriétés des coefficients D 1 , D 2 . Contrairement aux résultats de [66], dans ce chapitre, nous étudions le système de Bresse (1.2.22) avec des conditions aux bords de type Dirichlet ou de Dirichlet-Neumann-Neumann et dans le cas où les amortissements sont localisés sur un intervalle quelconque du domaine.

  , D 2 et D 3 ∈ W 1,∞ (0, L). Le C 0 -semigroupe e tA j est exponentiellement stable, i.e., il existe des constantes M ≥ 1 et > 0 indépendant de U 0 tel que

	.2.26)
	nous établissons un taux de décroissance exponentiel de l'énergie donnée par le théorème
	suivant :
	Théorème 1.2.14. (Taux de décroissance exponentiel) Supposons que (1.2.26) est satis-
	fait et D 1

  ).

	(2.3.21)
	Finally, inserting (2.3.20) and (2.3.21) into (2.3.19), we get

  ).

	Finally, inserting (2.3.24) and (2.3.25) into (2.3.23), we get
	-
	(2.3.25)

Ω β n η(x)(∇y n • ∇u n )dx + Ω 1 β n η(x)∆y n ∆u n dx = o(1

). (2.3.26) (iii) Summing the imaginary parts of equations (2.3.22) and (2.3.26), we obtain

  ).

	(2.3.41)
	Using equations (2.3.11), (2.3.15) and (2.3.28), we deduce

  ). Lemma 2.3.10. The solution (u n , v n , y n , z n ) ∈ D(A) of system (2.3.4)-(2.3.7) satisfies the following estimation

	Chapitre 2. Local indirect stabilization of N-d system of two coupled wave equations
	47	under geometric conditions
	Finally, inserting (2.3.42) into (2.3.41), we obtain the desired equation (2.3.30) and the
	proof is thus complete.
		(2.3.42)

  ). The main estimation. Combining equations (2.3.48) and (2.3.50), we obtain

	(2.3.50)
	(iii)

  According to Theorem 2.4 of Borichev-Tomilov in[START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], a C 0 -semigroup of contractions e tA on a Hilbert space H verifies (2.5.1) if the following conditions

	Chapitre 2. Local indirect stabilization of N-d system of two coupled wave equations
	51	under geometric conditions
	iR ⊆ ρ(A)	(H1)
		.5.1)

  The solution (u n , v n , y n , z n ) ∈ D(A) of system (2.5.4)-(2.5.7) satisfies the following estimation

				.5.14)
	This yields
	ωc +	|u n | 2 dx =	o(1) n β 4	.
	The proof is thus complete.
	Lemma 2.5.3.

Ω

η|β n ∇u n | 2 dx = o(1) and

V 2 ∩Ω |β n ∇u n | 2 dx = o(1).

(2.5.15)

Proof: First, multiplying equation (2.5.8) by β 2 n η ūn . Later, using Green's formula, (2.5.10), (2.5.11) and the fact that the sequences f 1 n , f 2 n , g 1 n converge to zero, respectively, in H 1 0 (Ω), H 1 0 (Ω) and L 2 (Ω), we get

  ). .5.10),(2.5.11),(2.5.15) and the fact that the sequences (β n y n ) and (∇y n ) are uniformly bounded in L 2 (Ω) in (2.5.18), we deduce

	Chapitre 2. Local indirect stabilization of N-d system of two coupled wave equations
	53	under geometric conditions
	Using (2
		(2.5.18)

Ω b(x)η|β n y n | 2 dx = o(1).

(2.5.

[START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF]

1

  As we mention in Remark 2.3.12, using Lemmas 2.5.2, 2.5.3, 2.5.4 and 2.5.5, we deduce that estimation (2.3.51) is also true in the case a = 1.

	It follows,
	from the estimations (2.5.11), (2.5.15), (2.5.17), (2.5.20) and (2.3.51) that U n H = o(1)
	which is a contradiction with (2.5.2). Consequently, condition (H3) holds and the energy of
	smooth solution of system (2.1.4)-(2.1.6) decays polynomially to zero as t goes to infinity.

  .3.9) On the other side, we notice that v n and z n are uniformly bounded in L 2 (Ω).

									It follows,
	from equations (3.3.4) and (3.3.6), that
	Ω	|y n | 2 dx =	O(1) β 2 n	and	Ω	|u n | 2 dx =	O(1) n β 2	.	(3.3.10)
	For clarity, we divide the proof into several Lemmas.

Lemma 3.3.6. The solution (u n , v n , y n , z n ) ∈ D(A) of system (3.3.4)-(3.3.7) satisfies the following estimation Ω c|β n u n | 2 dx = o(1) and ωc + |β n u n | 2 dx = o(1). (3.3.11) Proof: First, since U n is uniformly bounded in H, then from (3.3.3), we get Re

  .4.9) From (3.4.2), we have ∇u n , v n and y n are uniformly bounded in L 2 (Ω) and z n is uniformly bounded in H -1 (Ω). Lemma 3.4.2. The solution (u n , v n , y n , z n ) ∈ D(A d ) of system (3.4.4)-(3.4.7) satisfies the following estimation

	|β n u n | 2 dx = o(1).	(3.4.10)
	Using now (3.4.2) and (3.4.4), we deduce that β n u n is uniformly bounded in L 2 (Ω). In addition, using (3.4.2) and (3.4.6), we deduce that β n y n is uniformly bounded in H -1 (Ω). More precisely, u n L 2 (Ω) = O(1) Proof: First, since U n is uniformly bounded in D and using (3.4.3), we get Re iβ n U n 2 -(A d U n , U n ) = Ω c(x)|v n | 2 dx = o(1). (3.4.11) Next, using equations (3.4.11) and (3.4.4), we get Ω β ωc + c|β n u n | 2 dx = o(1). (3.4.12)

n = o(1) and y n H -1 (Ω) = O(1) β n = o(1)

.

Ω c|β n u n | 2 dx = o(1) and

  Theorem 4.4.1. Let A : D(A) ⊂ H → H generates a C 0 -semigroup of contractions e tA on H. Assume that iλ ∈ ρ(A), for all λ ∈ R. Then, the C 0 -semigroup e tA is : i) Exponentially stable if and only if Assume that (4.4.1) is satisfied and D 1 , D 2 and D 3 ∈ L ∞ (0, L). Then, the C 0 -semigroup e tA j is analytically stable for j = 1, 2.

	lim sup	(iλI -A) -1	L(H) < +∞.
	|λ|→+∞λ∈R
	ii) Polynomially stable of order 1 l (l > 0) if and only if
	lim sup	|λ| -l (iλI -A) -1	L(H) < +∞.
	|λ|→+∞λ∈R
	iii) Analytic stable if and only if
	lim sup	|λ| (iλI -A) -1	L(H) < +∞.
	|λ|→+∞λ∈R
	Now, we are in a position to establish the main result of this part by the following stability
	estimate.
	Theorem 4.4.2. To prove Theorem 4.4.2, we have to check if the following conditions
			iR ⊆ ρ (A j )	(H1)
	and
	lim sup λ→+∞λ∈R	|λ| (iλI -A j ) -1	L(H j ) = O (1) ,	(H2)
	hold.
	Condition (H1) is already proved in Lemma 4.3.2 and Lemma 4.3.3. To prove condition
	(H2), we use a contradiction argument. For this aim, suppose that there exist a sequence
	of real numbers (λ n ) n , with |λ n | → +∞ and a sequence of vectors

  First, multiplying equation (4.6.5) by iληv 1 in L 2 (0, L) and integrating by parts, we get

		Chapitre 4. Stability of a Bresse system with local Kelvin-Voigt damping and
	145											non-smooth coefficient at interface
	Proof:
														.6.17)
	In a similar way, one can prove
		β α	λ 2 |v 3 x | 2 dx =	o(1) λ 2 and	β α	λ 2 |v 5 x -lv 1 | 2 dx =	o(1) λ 2 .	(4.6.18)
	The proof is thus complete.
	Lemma 4.6.4. Assume that condition (4.6.1) is verified. Then, we have
	β-α+	λv 1 2 dx =	o (1) λ	,	β-α+	λv 3 2 dx =	o (1) λ	,	β-α+	λv 5 2 dx =	o (1) λ	.	(4.6.19)

  .6.32)Step 2. Similarly to Step 1, multiplying (4.6.7) by iρ 2 λy and (4.6.8) by y, where y is a solution of system (4.6.24), we get

L 0

  For the proof of Theorem 4.7.1, it suffices to show the existence of sequences

	.7.2)
	Theorem 4.7.1. Under hypothesis (4.7.1), system (4.1.1), subject to the boundary condi-
	tions (4.1.3), is not exponentially stable in the energy space H 2 .

Proof:

  H 2 is bounded as n tense to +∞. Thus the proof is complete. For the proof of Theorem 4.7.2, we will follow the same argument used in Theorem 4.7.1. It suffices to show that the existence of sequences (λ n ) ⊂ R with lim

	Theorem 4.7.2. Under hypothesis (4.7.2) , system (4.1.1), subject to the boundary condi-
	tions (4.1.3), is not exponentially stable in the energy space H 2 .
	n→+∞

Proof:

  .20) iλv3 -v 4 = 0, (4.7.21)iλρ 2 v 4 -k 2 v 3 xx + k 1 v 1 x + v 3 + lv 5 = ρ 2 f 4 ,(4.7.22)iλv5 -v 6 = 0, (4.7.23)iλρ 1 v 6 -k 3 v 5 xx -lv 1 x + lk 1 v 1 x + v 3 + lv 5 = 0. (4.7.24) Eliminating v 2 , v 4 and v 6 in (4.7.20), (4.7.22) and (4.7.24) by (4.7.19), (4.7.21) and (4.7.23) we get λ 2 ρ 1 v 1 + k 1 v 1 xx + v 3 x + lv 5 x + lk 3 v 5 x -lv 1 = 0, (4.7.25)

  3 and v 5 are given by (4.7.28) and (4.7.32)-(4.7.34). It is easy to check thatV n H 2 ≥ √ ρ 2 λv 3 ∼ |Bλ| ∼ |n| → +∞ as n → +∞.On the other hand, using (4.7.19)-(4.7.24), we deduce that(iλI -A 2 )V n 2 H 2 = (0, 0, 0, -iλD 2 v 3 xx + ρ 2 f 4 , 0, 0) 2 H 2 ≤ c.

  .7.35) iλρ 1 v 2 -k 1 v 1 xx + v 3 x + lv 5x -lk 3 v 5x -lv 1 = 0, (4.7.36)iλv 3 -v 4 = 0, (4.7.37) iλρ 2 v 4 -k 2 v 3 xx + k 1 v 1 x + v 3 + lv 5 = 0, (4.7.38)iλv5 -v 6 = 0, (4.7.39)iλρ 1 v 6 -k 3 v 5 xx -lv 1 x + lk 1 v 1 x + v 3 + lv 5 = ρ 1 f 6 . (4.7.40) 155 Chapitre 4. Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface Eliminating v 2 , v 4 and v 6 in (4.7.36), (4.7.38) and (4.7.40) by (4.7.35), (4.7.37) and (4.7.39) we getλ 2 ρ 1 v 1 + k 1 v 1 xx + v 3 x + lv 5 x + lk 3 v 5x -lv 1 = 0, (4.7.41)λ 2 ρ 2 v 3 + k 2 v 3 xx -k 1 v 1 x + v 3 + lv 5 = 0,(4.7.42)λ 2 ρ 1 v 5 + k 3 v 5 xx -lv 1 x -lk 1 v 1 x + v 3 + lv 5 = -ρ 1 f 6 .and C depend on λ to be determined. Noting that k 3 nπ L 2 -λ 2 ρ 1 = 0 and inserting (4.7.44) in (4.7.41)-(4.7.43) we obtain that nπ L 2 k 1 -λ 2 ρ 1 + l 2 k 3 A + k 1

									(4.7.43)
	This can be solved by the ansatz
	v 1 = A sin(	nπx L	), v 3 = B cos(	nπx L	), v 5 = C cos(	nπx L	)	(4.7.44)
	where A, B nπ L	B + (k 1 + k 3 ) l	nπ L	C = 0,	(4.7.45)

  0, (4.7.51)iλρ 1 v 2 -k 3 v 1 xx + v 3 x + lv 5 x -lk 3 v 5 x -lv 1 = 0, (4.7.52) iλv 3 -v 4 = 0, (4.7.53) iλρ 2 v 4 -k 2 v 3 xx + k 3 v 1 x + v 3 + lv 5 = ρ 2 f 4 ,(4.7.54)iλv5 -v 6 = 0, (4.7.55)iλρ 1 v 6 -k 3 v 5 xx -lv 1 x + lk 3 v 1 x + v 3 + lv 5 = 0. (4.7.56) Eliminating v 2 , v 4 and v 6 in (4.7.52), (4.7.54) and (4.7.56) by (4.7.51), (4.7.53) and (4.7.55) we getλ 2 ρ 1 v 1 + k 3 v 1 xx + v 3 x + lv 5 x + lk 3 v 5 x -lv 1 = 0,(4.7.57)λ 2 ρ 2 v 3 + k 2 v 3 xx -k 3 v 1 x + v 3 + lv 5 = -ρ 2 f 4 , (4.7.58) λ 2 ρ 1 v 5 + k 3 v 5 xx -lv 1x -lk 3 v 1 x + v 3 + lv 5 = 0. (4.7.59)

  .7.64) Now, let V n = (v 1 , iλv 1 , v 3 , iλv 3 , v5 , iλv 5 ), where v 1 , v 3 and v 5 are given by (4.7.60) and (4.7.65). It is easy to check V n → +∞ and (iλI -A 2 )V n 157 Chapitre 4. Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface Remark 4.7.3. Unlike the results in the static case (see

	This implies that
	A =	-ρ 2 L nπk 3	, B =	4ρ 2 k 3	-	ρ 2 L 2 l 2 k 3 π 2 n 2 , C =	-2ρ 2 lk 3	+	L 2 lρ 2 k 3 π 2 n 2 .	(4.7.65)

  , (4.8.2) is verified if the following conditions Condition iR ⊆ ρ (A j ) is already proved in Lemma 4.3.2 and Lemma 4.3.3. We will establish (H4) by contradiction. Suppose that there exist a sequence of real numbers (λ n ) n , with |λ n | → +∞ and a sequence of vectors

			1 λ n	), v 5 n = O(	1 λ n	).	(4.8.11)
		iR ⊆ ρ (A j )		(H1)
	and
	lim sup |λ|→+∞λ∈R	1 λ 4 (iλI -A j ) -1	L(H j ) = O (1)	(H4)
	hold.
	U n = v 1 n , v 2 n , v 3 n , v 4 n , v 5 n , v 6 n	T ∈ D (A j ) with U n H j = 1	(4.8.3)
	such that
	λ 4					, 2	(4.8.4)

n ( iλ n U n -A j U n ) = f 1 n , f 2 n , f 3 n , f 4 n , f 5 n , f 6 n T → 0 in H j , j = 1

  Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface Proof: For the clarity of the proof, we divide the proof into several steps.Step 1. Our first aim here is to prove Now, we need to estimate each term of (4.8.28) :• Using (4.8.11) and (4.8.14), we getλρ 2 v 4 η v 1 dx = o(1).(4.8.29)• Using (4.8.12) and the fact that λv 1 is uniformly bounded in L 2 (0, L), we obtain xx is uniformly bounded in L 2 (0, L). This fact combined with (4.8.12) and (4.8.13) yields• Using (4.8.12), (4.8.13) and the fact that v 1x is uniformly bounded in L 2 (0, L), we get• Using (4.8.11) and the fact that v 1x is uniformly bounded in L 2 (0, L), we obtain• Using the fact that f 4 converges to zero in L 2 (0, L) and v 1x is uniformly bounded in L 2 (0, L), we get

	161			Chapitre 4. L
																η|v 1 x | 2 dx = o(1).	(4.8.27)
													0
	For this, multiplying (4.8.8) by ηv 1 x and integrating by parts, we get
			L			L									L	L
	-i			λρ 2 v 4 η v 1 dx -i	λρ 2 v 4 x ηv 1 dx +	(k 2 v 3 x + D 2 v 4 x )(ηv 1 xx )dx +	(k 2 v 3 x + D 2 v 4 x )(η v 1 x )dx
			0				0									0	0
	+	0	L	ηk 1 |v 1 x | 2 dx +	0	L	ηk 1 v 3 v 1 x dx +	0	L	lk 1 ηv 5 v 1 x dx =	0	L	ρ 2	f 4 λ 4 ηv 1 x dx.	(4.8.28)
															L
										-i
													0
										-i	0	L	λρ 2 v 4 x ηv 1 dx =	o(1) λ 2 .	(4.8.30)
	• From (4.8.6), we remark that	1 λ	v 1
							0	L	(k 2 λv 3 x + D 2 λv 4 x )(η	v 1 xx λ	)dx =	o(1) λ	.	(4.8.31)
								0	L	(k 2 v 3 x + D 2 v 4 x )(η v 1 x )dx =	o(1) λ 2 .	(4.8.32)
							L								L
								ηk 1 v 3 v 1 x dx +	lk 1 ηv 5 v 1 x dx = o(1).	(4.8.33)
							0									0
												0	L	ρ 2	f 4 λ 4 ηv 1 x dx =	o(1) λ 4 .	(4.8.34)
	Finally, inserting equations (4.8.29)-(4.8.34) into (4.8.28) we get the desired estimate
	(4.8.27).
	Next, our second aim is to prove
																β-α+	|λv 1 | 2 dx =	o(1) λ 2 .	(4.8.26)

2 ,

  Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface • Using (4.8.12) and the fact that λv 1 is uniformly bounded in L 2 (0, L), we obtain• Using (4.8.12), (4.8.13) and the fact that v 1x is uniformly bounded in L 2 (0, L), we get• Using (4.8.27) and the fact that λv 3 and λv 5 are uniformly bounded in L 2 (0, L), we obtain

	163	Chapitre 4. -i	0	L	λρ 2 v 4 x ηλv 1 dx =	o(1) λ	.	(4.8.43)
	• Using the fact (4.8.12) and (4.8.13) yields 1 v 1 xx is uniformly bounded in L 2 (0, L) due to (4.8.6) combined with λ
		0	L	(k 2 λ 2 v 3 x + D 2 λ 2 v 4 x )(η	v 1 xx λ	)dx = o(1).	(4.8.44)
			0	L	(k 2 λv 3 x + D 2 λv 4 x )(η v 1 x )dx =	o(1) λ	.	(4.8.45)
		L							L
			ηk 1 λv 3 v 1 x dx +	lk 1 ηλv 5 v 1 x dx = o(1).	(4.8.46)
		0							0
									4.8.41)
	Now, we need to estimate each term of (4.8.41) as follows :
	• Using (4.8.14) and (4.8.35), we get
						-i			(4.8.42)

L 0 λρ 2 v 4 η λv 1 dx = o(1).

  Finally, using (4.8.52), the fact that v 3x , v 5 x , (v 5x -lv 1 ) are uniformly bounded in L 2 (0, L) and f 2 converges to zero in L 2 (0, L) in the previous equation, we get the desired estimation (4.8.49).Step 4. Our aim is to prove • Using (4.8.12) and λv 1 is uniformly bounded in L 2 (0, L), we obtain Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface• Using (4.8.12), (4.8.13) and (4.8.49), we obtainL 0 (k 2 λ 1+1/2 v 3 x + D 2 λ 1+1/2 v 4 x )(ηv 1 xx )dx =• Using (4.8.39) and the fact that λv 3 and λv 5 are uniformly bounded in L 2 (0, L), we obtainL 0 ηk 1 λv 3 λ 1/2 v 1 x dx + L 0 lk 1 ηλv 5 λ 1/2 v 1 x dx = o(1). (4.8.61) • Using (4.8.39) and the fact that f 4 converges to zero in H 1 0 (0, L), we get Finally, inserting equations (4.8.57)-(4.8.62) into (4.8.56), we deduce that On the other side, our aim now is to prove (4.8.55). For this aim, multiplying (4.8.38) by λ 1+1/2 , we get

	So, multiplying (4.8.5) by η 165 Chapitre 4. L √ λ, we get η √ λv 2 = iη √ λλv 1 -η 0 (k 2 λ 2 v 3 √ x + D 2 λ 2 v 4 λ f 1 λ 4 . x ) η	v 1 xx √ λ	(4.8.50) = o(1). (4.8.59)
	Then, integrating (4.8.50) over (0, L), we get • Using (4.8.12), (4.8.13) and (4.8.39), we get
						0	L	η 2 λ|v 2 | 2 dx ≤ 2 L 0 (k 2 λv 3 x + D 2 λv 4 L 0 η 2 λ|λv 1 | 2 dx + 2 x )(η λ 1/2 v 1 x )dx = 0	L o(1) η 2 |f 1 | 2 λ 7 dx. λ .	(4.8.51) (4.8.60)
	Using (4.8.40) and the fact that f 1 converges to zero in H 1 0 (0, L) in the previous equation,
	we deduce							η	√	λv 2 = o(1).	(4.8.52)
	Next, multiplying (4.8.6) by	η √ λ	, we get
			k 1 η	v 1 xx √ λ	= iρ 1 η	√	λv 2 -k 1	η √ λ L (v 3 x + lv 5 x ) -lk 3 0 ρ 1 f 4 λ 3 ηλ 1/2 v 1 x dx = η √ λ	(v 5 x -lv 1 ) -ρ 1 η o(1) λ 3 .	f 2 λ 4 √	λ	.	(4.8.53) (4.8.62)
												0 L L 0 η|v 1 η|v 1 x | 2 dx = x | 2 dx =	λ 1+1/2 o(1) o(1) λ 1+1/2 .	(4.8.54)
	and											0	L	η|λv 1 | 2 dx =	o(1) λ 1+1/2 .	(4.8.55)
	To prove (4.8.54), multiplying (4.8.28) by λ 1+1/2 , we get
												L
	-i											(k 2 λ 1+1/2 v 3 x + D 2 λ 1+1/2 v 4 x )(ηv 1 xx )dx
												0
			L									L	L
	+			(k 2 λv 3 x + D 2 λv 4 x )(η λ 1/2 v 1 x )dx +	ηk 1 λ 1+1/2 |v 1 x | 2 dx +	ηk 1 λv 3 λ 1/2 v 1 x dx
		0										0	0
	+	0	L	lk 1 ηλv 5 λ 1/2 v 1 x dx =	0	L	ρ 2	f 4 λ 3 ηλ 1/2 v 1 x dx.	(4.8.56)
	Next, we need to estimate each term of (4.8.56) as follows
	• Using (4.8.14) and (4.8.40), we get
												L
											-i	λρ 2 v 4 η λ 1+1/2 v 1 dx = o(1).	(4.8.57)
												0
											-i	0	L	). x ηλ 1/2 v 1 dx = λ 2 ρ 2 v 4	o(1) λ 1/2 .	(4.8.49) (4.8.58)

L 0 λρ 2 v 4 η λ 1+1/2 v 1 dx -i L 0 λ 2 ρ 2 v 4

x ηλ 1/2 v 1 dx +

D (A 2 ) = U ∈ H 2 | v 2 ∈ H 1 0 (0, L) , v 4 , v 6 ∈ H 1 * (0, L) , v 3 x | (0,L) = v 5 x | (0,L) = 0, [k 1 (v 1 x + v 3 + lv 5 ) + D 1 (v 2 x + v 4 + lv 6 )] x ∈ L 2 (0, L) , [k 2 v 3 x + D 2 v 4 x ] x ∈ L 2 * (0, L), [k 3 (v 5 x -lv 1 ) + D 3 (v 6 x -lv 2 )] x ∈ L 2

(a) Final time T = 500 (b) Final time T = 500 000.

Remerciements

and u t (x, 0) = u 1 (x) and y t (x, 0) = y 1 (x), x ∈ (0, 1) (3.5.3) where a > 0 constant, b ∈ C 0 ([0, 1], R) and c ∈ C 0 ([0, 1], R + ). We will study the two cases a = 1 and a = 1.

Construction of the numerical scheme

Let N be a non negative integer. Consider the subdivision of [0, 1] given by 0 = x 0 < x 1 < ... < x N < x N +1 = 1, i.e. x j = j∆x , j = 0, . . . , N + 1 .

Set t n+1 -t n = ∆t for all n ∈ N. For j = 0, . . . , N +1, we denote b j = b(x j ), c j = c(x j ). The explicit finite-difference discretization of system (3.5.1) is thus, for n ∈ N and j = 1, . . . , N :

) According to the initial conditions given by equations (3.5.2), we have firstly : for j = 1, . . . , N , u 0 j = u 0 (x j ) (3.5.5)

We can use the second initial conditions (3.5.3) to find the values of u and y at time t 1 = ∆t, by employing a "ghost" time-boundary (i.e. t -1 = -∆t) and the second-order central difference formula for j = 1, . . . , N :

Thus we have for j = 1, . . . , N :

We use the same discrete form of the initial conditions for y, for j = 1, . . . , N :

(3.5.9)

Setting n = 0, in the numerical scheme (3.5.4), the two previous equalities permits to compute u 1 j , y 1 j j=0,N . Finally, the solution (u, y) can be computed at any time t n .

Introduction

In this chapter, we study the stability of an elastic Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface. This system defined on (0, L) × (0, +∞) takes the following form

detailed as

Chapitre 4. Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coefficient at interface and lim sup

Condition iR ⊆ ρ (A j ) is already proved in Lemma 4.3.2 and Lemma 4.3.3. We will establish (H3) by contradiction. Suppose that there exist a sequence of real numbers (λ n ) n , with |λ n | → +∞ and a sequence of vectors

detailed as

)

)

)

n , (4.6.8)

)

.6.10) From (4.6.3), (4.6.5), (4.6.7) and (4.6.9), we deduce that

.6.11)

In the following we will check the condition (H3) by finding a contradiction with (4.6.3) such as U n H j = o(1). For clarity, we divide the proof into several lemmas. From now on, for simplicity, we drop the index n.

detailed as

)