Prof Serge Nicaise

Prof Wehbe Ali

Keywords: C 0 -semigroup, frequency domain approach, coupled wave equations, Bresse system, Kirchhoff plate equation, Kelvin-Voigt damping, past history damping, dynamical control, time delay, strong stability, exponential stability, polynomial stability iii

This dissertation is the result of the efforts and

First, we investigate the stabilization of locally coupled wave equations with non-smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Using a general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. However, by combining the frequency domain approach with the multiplier method, we prove a polynomial energy decay rate.

Second, we investigate the stabilization of locally coupled wave equations with local viscoelastic damping of past history type acting only on one equation via non-smooth coefficients. We prove the strong stability of our system. Next, we establish the exponential stability of the solution if the two waves have the same speed of propagation. In the case of different propagation speeds, we prove that the energy of our system decays polynomially. Moreover, we show the lack of exponential stability if the speeds of wave propagation are different with a global damping and a global coupling.

Third, we investigate the stabilization of a linear Bresse system with one discontinuous local internal viscoelastic damping of Kelvin-Voigt type acting on the axial force, under fully Dirichlet boundary conditions. We prove the strong and polynomial stabilities of our system. Finally, we consider two models of the Kirchhoff plate equation, the first one with delay terms on the dynamical boundary controls, and the second one where delay terms on the boundary control are added. For the first system, we prove its well-posedness, strong stability, non-exponential stability, and polynomial stability under a multiplier geometric control condition. For the second one, we prove its well-posedness, strong stability, and exponential stability under the same multiplier geometric control condition. Finally, we give some instability examples of the second system for some choices of delays.

Symbols and Notations

R

The set of real numbers. R +

The set of non negative real numbers. R *

The set of non zero real numbers.

N

The set of natural numbers. N *

The set of non zero natural numbers.

Z

The set of integer numbers.

C

The set of complex numbers. i

The imaginary unit. The real part. The imaginary part.

L p

The Lebesgue space.

H m

The sobolev space. C 0

The space of continuous function. C 1 The space of continuously differentiable functions. C 2 The space of twice continuously differentiable functions.

D(Ω)

The set of smooth functions in Ω D (Ω)

The space of distribution in Ω.

|•|

The modulus.

|•| X

The semi-norm in X.

• X

The norm in X. (•, •) X

The inner product in X. max

The maximum. min

The minimum. sup

The supreme. inf

The infimum.

f y = ∂ y f
The partial derivative of f with respect of y. f yy = ∂ yy f The second partial derivative of f with respect of y. g (s)

The derivative of g with respect to s.

A B

Means that there exists a constant C > 0 independent of A, B and a natural parameter n such that A ≤ CB.

viii

Introduction

In this thesis, we study the indirect stability of some coupled systems with different kinds of local discontinuous dampings. We also study the stability and the instability results of the Kirchhoff plate equation with delay terms on the boundary or dynamical boundary controls. This thesis is divided into five chapters.

In the first chapter, we recall some well-known results about semigroups, including some theorems about strong, exponential, polynomial, and analytic stability of a C 0 -semigroup. We also recall the definition of the multiplier geometric control condition denoted by MGC.

A wave is created when a vibrating source disturbs the medium. In order to restrain those vibrations, several dampings can be added such as frictional (viscous), Kelvin-Voigt, time delay, past history (infinite memory) dampings. However, time delays appear in several applications such as in physics, chemistry, biology, thermal phenomena not only depend on the present state but also on some past occurrences (see [START_REF] Ernst | Delay-induced multistable synchronization of biological oscillators[END_REF] and [START_REF] Kolmanovskii | Introduction to the theory and applications of functional-differential equations[END_REF]) . In the last years, the control of partial differential equations with time delays have become popular among scientists, since in many cases time delays induce some instabilities see [START_REF] Datko | Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks[END_REF][START_REF] Datko | Two examples of ill-posedness with respect to time delays revisited[END_REF][START_REF] Datko | An example of the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Dreher | Ill-posed problems in thermomechanics[END_REF].

The notion of indirect damping mechanisms has been introduced by Russell in [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF] and since this time, it retains the attention of many authors. In particular, the fact that only one equation of the coupled system is damped refers to the so-called class of "indirect" stabilization problems initiated and studied in [START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF][START_REF] Alabau | 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, see for instance [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wavetype 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] Zhang | Polynomial decay and control of a 1 -d hyperbolic-parabolic coupled system[END_REF] and the rich references therein. The study of such systems is also motivated by several physical considerations like Timoshenko and Bresse systems (see for instance [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], [START_REF] Akil | Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary[END_REF], [START_REF] Mercier | Decay rate of the Timoshenko system with one boundary damping[END_REF] and [START_REF] Najdi | Weakly locally thermal stabilization of Bresse systems[END_REF]). The Bresse system is a model for arched beams (see Fig. 1 for an illustration), see [START_REF] Lagnese | Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures[END_REF]Chap. 6]. 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, ( Bresse System)
where N = k 3 (w x -lϕ) is the axial force, Q = k 1 (ϕ x + ψ + lw) is the shear force, and M = k 2 ψ x is the bending moment. The functions ϕ, ψ, and w are respectively the vertical, shear angle, and longitudinal displacements. Here

ρ 1 ϕ tt -k 1 (ϕ x + ψ) x = 0, ρ 2 ψ tt -k 2 ψ xx + k 1 (ϕ x + ψ) = 0.
In chapters two, three, and four, we focus on strongly coupled systems with different kinds of indirect local dampings and non-smooth coefficients at the interface.

In chapter two, we investigate the stability of local coupled wave equations with singular localized viscoelastic damping of Kelvin-Voigt type and localized time delay. More precisely, we consider the following system:

                         u tt -[au x + b(x)(κ 1 u tx + κ 2 u tx (x, t -τ ))]
x + c(x)y t = 0, (x, t) ∈ (0, L) × (0, ∞), y tt -y xx -c(x)u t = 0, (x, t) ∈ (0, L) × (0, ∞), u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0, (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)), x ∈ (0, L), (y(x, 0), y t (x, 0)) = (y 0 (x), y 1 (x)), x ∈ (0, L), u t (x, t) = f 0 (x, t), (x, t) ∈ (0, L) × (-τ, 0),

where L, τ, a and κ 1 are positive real numbers, κ 2 is a non-zero real number and (u 0 , u 1 , y 0 , y 1 , f 0 ) belongs to a suitable space. We suppose that there exist 0 < α < β < γ < L INTRODUCTION and a non-zero constant c 0 , such that b(x) = 1, x ∈ (0, β), 0, x ∈ (β, L), and c(x) = c 0 , x ∈ (α, γ), 0, x ∈ (0, α) ∪ (γ, L).

In fact, there are few results concerning the stability of coupled wave equations with local Kelvin-Voigt damping and without time delay, especially in the absence of smoothness of the damping and coupling coefficients (see Subsection 2.1.2). This motivates our interest to study the stabilization of system (Sys1) in this chapter. As in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF], we introduce the following auxiliary change of variable η(x, ρ, t) := u t (x, t -ρτ ), x ∈ (0, β), ρ ∈ (0, 1), t > 0.

Then, system (Sys1) becomes

                               u tt -(S b (u, u t , η))
x + c(x)y t = 0, (x, t) ∈ (0, L) × (0, ∞), y tt -y xx -c(x)u t = 0, (x, t) ∈ (0, L) × (0, ∞), τ η t (x, ρ, t) + η ρ (x, ρ, t) = 0, (x, ρ, t) ∈ (0, β) × (0, 1) × (0, ∞), u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0, η(0, ρ, t) = 0, (ρ, t) ∈ (0, 1) × (0, ∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)), x ∈ (0, L), (y(x, 0), y t (x, 0)) = (y 0 (x), y 1 (x)), x ∈ (0, L), η(x, ρ, 0) = f 0 (x, -ρτ ), (x, ρ) ∈ (0, β) × (0,

where S b (u, u t , η) := au x + b(x)(κ 1 u tx + κ 2 u tx (x, t -τ )). Moreover, from the definition of b(x), we have S b (u, u t , η) = S 1 (u, u t , η) := au x + κ 1 u tx + κ 2 η x (•, 1, t), in (0, β), au x , in (β, L).

The energy of system (Sys2) is given by

E 1 (t) = E 1,u (t) + E 1,y (t) + E 1,η (t),
where

         E 1,u (t) = 1 2 L 0 |u t | 2 + a|u x | 2 dx, E 1,y (t) = 1 2 L 0 |y t | 2 + |y x | 2 dx and E 1,η (t) = τ |κ 2 | 2 INTRODUCTION
Then, system (Sys2) is dissipative in the sense that its energy is non-increasing with respect to time (i.e. E 1 (t) ≤ 0). Now, we write system (Sys2) as the following first order evolution equation

U t = A 1 U, U (0) = U 0 ,
where U 0 = (u 0 , u 1 , y 0 , y 1 , f 0 (•, -ρτ )) ∈ H 1 . The Hilbert space H 1 is defined by

H 1 := H 1 0 (0, L) × L 2 (0, L) 2 × W,
where W := L 2 ((0, 1); H 1 L (0, β)) and H 1 L (0, β) := η ∈ H 1 (0, β) | η(0) = 0 . The space W is a Hilbert space of H 1 L (0, β)-valued functions on (0, 1), equipped with the following inner product (η 1 , η 2 ) W :=

β 0 1 0 η 1 x η 2 x dρdx, ∀ η 1 , η 2 ∈ W.
The Hilbert space H 1 is equipped with the following inner product

U, U 1 H 1 = L 0 au x u 1 x + vv 1 + y x y 1 x + zz 1 dx + τ |κ 2 | β 0 1 0 η x η 1 x dρdx,
where U = (u, v, y, z, η) , U 1 = (u 1 , v 1 , y 1 , z 1 , η 1 ) ∈ H 1 . The linear unbounded operator A 1 : D(A 1 ) ⊂ H 1 -→ H 1 is defined by:

D(A 1 ) =    U = (u, v, y, z, η) ∈ H 1 | y ∈ H 2 (0, L) ∩ H 1 0 (0, L), v, z ∈ H 1 0 (0, L) (S b (u, v, η)) x ∈ L 2 (0, L), η ρ ∈ W, η(•, 0) = v(•) in (0, β)    and A 1       u v y z η       =       v (S b (u, v, η)) x -c(•)z z y xx + c(•)v -τ -1 η ρ      
, ∀U = (u, v, y, z, η) ∈ D(A 1 ).

In chapter three, we investigate the indirect stability of coupled elastic wave equations with localized past history damping. More precisely, we consider the following system:

                         u tt -au x -b(x)
∞ 0 g(s)u x (x, t -s)ds

x + c(x)y t = 0, (x, t) ∈ (0, L) × (0, ∞), y tt -y xx -c(x)u t = 0, (x, t) ∈ (0, L) × (0, ∞), u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0, (u(x, -s), u t (x, 0)) = (u 0 (x, s), u 1 (x)), (x, s) ∈ (0, L) × (0, ∞), (y(x, 0), y t (x, 0)) = (y 0 (x), y 1 (x)), x ∈ (0, L),

where L and a are positive real numbers. We suppose that there exist a non-zero constant c 0 and positive constants α, β, γ, and b 0 such that 0 < α < β < γ < L, and define b(x) = b 0 , x ∈ (0, β),

INTRODUCTION

The general integral term represents a history term with the relaxation function g that is supposed to satisfy the following hypotheses: According to the best of our knowledge, it seems that no result in the literature exists concerning the case of coupled wave equations with localized past history damping, especially in the absence of smoothness of the damping and coupling coefficients. This motivates our interest to study the stabilization of system (Sys3) in this chapter. As in [START_REF] Dafermos | Asymptotic stability in viscoelasticity[END_REF], we introduce the following auxiliary change of variable ω(x, s, t) := u(x, t) -u(x, t -s), (x, s, t) ∈ (0, β) × (0, ∞) × (0, ∞).

         g ∈ L 1 ([0, ∞)) ∩ C 1 ([0, ∞))
Then, system (Sys3) becomes

                                       u tt -S b(•) (u, ω)
x + c(•)y t = 0, (x, t) ∈ (0, L) × (0, ∞), y tt -y xx -c(•)u t = 0, (x, t) ∈ (0, L) × (0, ∞), ω t (x, s, t) + ω s (x, s, t) -u t = 0, (x, s, t) ∈ (0, β) × (0, ∞) × (0, ∞), u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0, ω(x, 0, t) = 0, (x, t) ∈ (0, β) × (0, ∞), ω(0, s, t) = 0, (s, t) ∈ (0, ∞) × (0, ∞), (u(x, -s), u t (x, 0)) = (u 0 (x, s), u 1 (x)), (x, s) ∈ (0, L) × (0, ∞), (y(x, 0), y t (x, 0)) = (y 0 (x), y 1 (x)), x ∈ (0, L), ω 0 (x, s) := ω(x, s, 0) = u 0 (x, 0) -u 0 (x, s), (x, s) ∈ (0, β) × (0, ∞),

where

S b(•) (u, ω) :=    S b 0 (u, ω) := b 0 u x + b 0 ∞ 0 g(s)ω x (x, s, t)ds, in (0, β), au x , in (β, L).
The energy of system (Sys4) is given by

E 2 (t) = E 2,u (t) + E 2,y (t) + E 2,ω (t),
where

         E 2,u (t) = 1 2 L 0 |u t | 2 + b(•)|u x | 2 dx, E 2,y (t) = 1 2

INTRODUCTION

According to Lemma 3.2.1, we have

d dt E 2 (t) = b 0 2 β 0 ∞ 0 g (s)|ω x (•, s, t)| 2 dsdx ≤ 0.
Then, system (Sys4) is dissipative in the sense that its energy is non-increasing with respect to time. Now, we write system (Sys4) as the following first order evolution equation

U t = A 2 U, U (0) = U 0 ,
where U 0 = (u 0 (•, 0), u 1 , y 0 , y 1 , ω 0 (•, s)) ∈ H 2 . The Hilbert space H 2 is defined by

H := H 1 0 (0, L) × L 2 (0, L) 2 × W g ,
where

W g := L 2 g ((0, ∞); H 1 L (0, β)) and H 1 L (0, β) := ω ∈ H 1 (0, β) | ω(0) = 0 .
The space W g is a Hilbert space of H 1 L (0, β)-valued functions on (0, ∞), equipped with the following inner product

(ω 1 , ω 2 ) Wg := β 0 ∞ 0 g(s)ω 1 x ω 2 x dsdx, ∀ ω 1 , ω 2 ∈ W g .
The Hilbert space H 2 is equipped with the following inner product

U, U 1 H 2 = L 0 b(•)u x u 1 x + vv 1 + y x y 1 x + zz 1 dx + b 0 β 0 ∞ 0 g(s)ω x ω 1 x dsdx,
where U = (u, v, y, z, ω) ∈ H 2 and U 1 = (u 1 , v 1 , y 1 , z 1 , ω 1 ) ∈ H 2 . Now, we define the linear unbounded operator A 2 : D(A 2 ) ⊂ H 2 -→ H 2 by:

D(A 2 ) =      U = (u, v, y, z, ω) ∈ H 2 | y ∈ H 2 (0, L) ∩ H 1 0 (0, L), v, z ∈ H 1 0 (0, L) S b(•) (u, ω) x ∈ L 2 (0, L), ω s ∈ W g , ω(•, 0) = 0 in (0, β)      and A 2       u v y z ω       =        v S b(•) (u, ω) x -c(•)z z y xx + c(•)v -ω s + v        , ∀U = (u, v, y, z, ω) ∈ D(A 2 ).
In chapter four, we investigate the stability of a Bresse system with only one discontinuous local internal Kelvin-Voigt damping on the axial force. More precisely, we consider the following INTRODUCTION system:

                                     ρ 1 ϕ tt -k 1 (ϕ x + ψ + lw) x -lk 3 (w x -lϕ)
-ld(x)(w tx -lϕ t ) = 0, (x, t) ∈ (0, L) × (0, ∞), ρ 2 ψ tt -k 2 ψ xx + k 1 (ϕ x + ψ + lw) = 0, (x, t) ∈ (0, L) × (0, ∞),

ρ 1 w tt -[k 3 (w x -lϕ) + d(x)(w tx -lϕ t )] x
+ lk 1 (ϕ x + ψ + lw) = 0, (x, t) ∈ (0, L) × (0, ∞), ϕ(x, t) = ψ(x, t) = w(x, t) = 0, (x, t) ∈ {0, L} × (0, ∞), ϕ(x, 0) = ϕ 0 (x), ϕ t (x, 0) = ϕ 1 (x), x ∈ (0, L), ψ(x, 0) = ψ 0 (x), ψ t (x, 0) = ψ 1 (x),

x ∈ (0, L), w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x),

x ∈ (0, L),

where ρ 1 , ρ 2 , k 1 , k 2 , k 3 , l and L are positive real numbers. We suppose that there exist 0 < α < β < L and a positive constant d 0 , such that

d(x) = d 0 if x ∈ (α, β), 0 if x ∈ (0, α) ∪ (β, L).
According to the best of our knowledge, it seems that no result in the literature exists concerning the case of Bresse system with only one discontinuous local internal Kelvin-Voigt damping on the axial force, especially under fully Dirichlet boundary conditions and without any condition on the curvature l. This motivates our interest to study the stabilization of system (Sys5) in this chapter. The energy of system (Sys5) is given by

E 3 (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 + k 2 |ψ x | 2 + k 3 |w x -lϕ| 2 dx.
A direct computation gives

E 3 (t) = - L 0 d(x)|w tx -lϕ t | 2 dx = -d 0 β α |w tx -lϕ t | 2 dx ≤ 0.
Thus, system (Sys5) is dissipative in the sense that its energy is non-increasing with respect to time. Now, we write system (Sys5) as the following first order evolution equation

U t = A 3 U, U (0) = U 0 ,
where U 0 = (ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , w 0 , w 1 ) ∈ H 3 . The Hilbert space H 3 is given by

H 3 := H 1 0 (0, L) × L 2 (0, L) 3 .
The Hilbert space H 3 is equipped with the following inner product and norm

(U, U ) H 3 = L 0 k 1 (v 1 x + v 3 + lv 5 )( v 1 x + v 3 + l v 5 ) + ρ 1 v 2 v 2 + k 2 v 3 x v 3 x + ρ 2 v 4 v 4 + k 3 (v 5
x -lv 1 )( v 5

x -l v 1 )dx + ρ 1 v 6 v 6 dx,

INTRODUCTION

and 6 ) ∈ H 3 and U = ( v 1 , v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ H 3 . Now, we define the linear unbounded operator A 3 : D(A 3 ) ⊂ H 3 -→ H 3 by:

U 2 H 3 = L 0 k 1 |v 1 x + v 3 + lv 5 | 2 + ρ 1 |v 2 | 2 + k 2 |v 3 x | 2 + ρ 2 |v 4 | 2 + k 3 |v 5 x -lv 1 | 2 + ρ 1 |v 6 | 2 dx, where U = (v 1 , v 2 , v 3 , v 4 , v 5 , v
D(A 3 ) =    U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ H 3 | v 1 , v 3 ∈ H 2 (0, L) ∩ H 1 0 (0, L) v 2 , v 4 , v 6 ∈ H 1 0 (0, L), k 3 v 5 x + d(x)(v 6 x -lv 2 ) x ∈ L 2 (0, L)    and A 3         v 1 v 2 v 3 v 4 v 5 v 6         =               v 2 k 1 ρ 1 (v 1 x + v 3 + lv 5 ) x + lk 3 ρ 1 (v 5 x -lv 1 ) + ld(x) ρ 1 (v 6 x -lv 2 ) v 4 k 2 ρ 2 v 3 xx - k 1 ρ 2 (v 1 x + v 3 + lv 5 ) v 6 1 ρ 1 k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) x - lk 1 ρ 1 (v 1 x + v 3 + lv 5 )               , for all U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A 3 ).
In Sections 2.2, 3.2, and 4.2, we prove that the operators A j are m-dissipative in H j , j ∈ {1, 2, 3}. Thus, according to Lumer-Phillips theorem (see Theorem 1.2.8), we deduce that the operators A j generate a C 0 -semigroup of contractions e tA j in H j which give the well-posedness of systems (Sys2), (Sys4), and (Sys5).

In Sections 2.3, 3.3, and 4.3, we use a general criteria of Arendt-Batty (see Theorem 1.3.3) to show the strong stability of the C 0 -semigroups e tA j associated with systems (Sys2), (Sys4), and (Sys5) in the absence of the compactness of the resolvents of A j . The tools used in these proofs are: In Section 2.3, by using a contradiction argument (see Remark 1.3.5) with the help of some multiplier techniques, we prove that iR ⊂ ρ(A 1 ), ρ(A 1 ) being the resolvent set of A 1 .

In Sections 3.3 and 4.3, by using Holmgren uniqueness theorem (see [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF]) and Fredholm alternative (see Theorem 1.1.4), for all λ ∈ R, we prove that ker(iλI -A j ) = {0}, j ∈ {2, 3}.

-R(iλI -A j ) = H j , j ∈ {2, 3}.

In Sections 2.4, 3.4, amd 4.4, by combining a frequency domain approach with a multiplier method (see Theorems 1.3.6 and 1.3.7), we prove that the energies of systems (Sys2), (Sys4), and (Sys5) decay exponentially or polynomially with the rates summarized in the following table :  INTRODUCTION System Energy decay rate (Sys2)

t -1 (Sys4) Exponential if a = 1 t -1 if a = 1 (Sys5) t -1 if k 1 ρ 1 = k 2 ρ 2 t -1 2 if k 1 ρ 1 = k 2 ρ 2
In other words, for all U 0 ∈ D(A j ), j ∈ {1, 2, 3}, there exists a constant C > 0 independent of U 0 such that for all t > 0, we have

E 1 (t) ≤ C t U 0 2 D(A 1 ) , E 2 (t) ≤ C t U 0 2 D(A 2 ) if a = 1, E 3 (t) ≤        C t U 0 2 D(A 3 ) if k 1 ρ 1 = k 2 ρ 2 , C √ t U 0 2 D(A 3 ) if k 1 ρ 1 = k 2 ρ 2 ,
and for all U 0 ∈ H 2 , there exist constants M ≥ 1 and > 0 independent of U 0 such that for all t > 0 we have

e tA 2 U 0 H 2 ≤ M e -t U 0 H 2 if a = 1.
In Section 3.5, we use Theorem 1.3.6 to prove the lack of exponential stability of system (Sys3) when b(x) = c(x) = 1 in case of different speeds of propagation, i.e., when a = 1.

In the last chapter, we study the boundary stabilization of the Kirchhoff plate equation with time delay. Let Ω ⊂ R 2 be a bounded open set with boundary Γ of class C 4 consisting of a clamped part Γ 0 = ∅ and a rimmed part Γ 1 = ∅ such that Γ 0 ∩ Γ 1 = ∅. In the first part of this chapter, we study the stability of the Kirchhoff plate equation with delay terms on the dynamical boundary controls, namely we consider

                                         u tt (x, t) + ∆ 2 u(x, t) = 0 in Ω × (0, ∞), u(x, t) = ∂ ν u(x, t) = 0 on Γ 0 × (0, ∞), B 1 u(x, t) + η(x, t) = 0 on Γ 1 × (0, ∞), B 2 u(x, t) -ξ(x, t) = 0 on Γ 1 × (0, ∞), η t (x, t) -∂ ν u t (x, t) + β 1 η(x, t) + β 2 η(x, t -τ 1 ) = 0 on Γ 1 × (0, ∞), ξ t (x, t) -u t (x, t) + γ 1 ξ(x, t) + γ 2 ξ(x, t -τ 2 ) = 0 on Γ 1 × (0, ∞), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) in Ω, η(x, 0) = η 0 (x), ξ(x, 0) = ξ 0 (x) on Γ 1 , η(x, t) = f 0 (x, t) on Γ 1 × (-τ 1 , 0), ξ(x, t) = g 0 (x, t) on Γ 1 × (-τ 2 , 0). (Sys6)
In the second part of this chapter, we study the stability of the Kirchhoff plate equation with

INTRODUCTION

delay terms on the boundary controls, by considering:

                         u tt (x, t) + ∆ 2 u(x, t) = 0 in Ω × (0, ∞), u(x, t) = ∂ ν u(x, t) = 0 on Γ 0 × (0, ∞), B 1 u(x, t) = -β 1 ∂ ν u t (x, t) -β 2 ∂ ν u t (x, t -τ 1 ) on Γ 1 × (0, ∞), B 2 u(x, t) = γ 1 u t (x, t) + γ 2 u t (x, t -τ 2 ) on Γ 1 × (0, ∞), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) in Ω, u t (x, t) = f 0 (x, t) on Γ 1 × (-τ 1 , 0), ∂ ν u t (x, t) = g 0 (x, t) on Γ 1 × (-τ 2 , 0). ( Sys7
)
Here and below, β 1 , γ 1 , τ 1 and τ 2 are positive real numbers, β 2 and γ 2 are non-zero real numbers, ν = (ν 1 , ν 2 ) is the unit outward normal vector along Γ, and τ = (-ν 2 , ν 1 ) is the unit tangent vector along Γ. The constant 0 < µ < 1 2 is the Poisson coefficient and the boundary operators B 1 and B 2 are defined by

B 1 f = ∆f + (1 -µ)C 1 f and B 2 f = ∂ ν ∆f + (1 -µ)∂ τ C 2 f,
where

C 1 f = 2ν 1 ν 2 f x 1 x 2 -ν 2 1 f x 2 x 2 -ν 2 2 f x 1 x 1 and C 2 f = (ν 2 1 -ν 2 2 )f x 1 x 2 -ν 1 ν 2 (f x 1 x 1 -f x 2 x 2 )
. In Section 5.2, we study the first system (Sys6). For this aim, as in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF], we introduce the following auxiliary variables

z 1 (x, ρ, t) := η(x, t -ρτ 1 ), x ∈ Γ 1 , ρ ∈ (0, 1), t > 0, z 2 (x, ρ, t) := ξ(x, t -ρτ 2 ), x ∈ Γ 1 , ρ ∈ (0, 1), t > 0. Then, system (Sys6) becomes                                                      u tt + ∆ 2 u = 0 in Ω × (0, ∞), u = ∂ ν u = 0 on Γ 0 × (0, ∞), B 1 u + η = 0 on Γ 1 × (0, ∞), B 2 u -ξ = 0 on Γ 1 × (0, ∞), η t -∂ ν u t + β 1 η + β 2 z 1 (•, 1, t) = 0 on Γ 1 × (0, ∞), ξ t -u t + γ 1 ξ + γ 2 z 2 (•, 1, t) = 0 on Γ 1 × (0, ∞), τ 1 z 1 t (•, ρ, t) + z 1 ρ (•, ρ, t) = 0 on Γ 1 × (0, 1) × (0, ∞), τ 2 z 2 t (•, ρ, t) + z 2 ρ (•, ρ, t) = 0 on Γ 1 × (0, 1) × (0, ∞), u(•, 0) = u 0 (•), u t (•, 0) = u 1 (•) in Ω, η(•, 0) = η 0 (•), ξ(•, 0) = ξ 0 (•) on Γ 1 , z 1 (•, ρ, 0) = f 0 (•, -ρτ 1 ) on Γ 1 × (0, 1), z 2 (•, ρ, 0) = g 0 (•, -ρτ 2 ) on Γ 1 × (0, 1). (Sys8)
The energy of system (Sys8) is given by

E(t) = 1 2 a(u, u) + Ω |u t | 2 dx + Γ 1 |η| 2 dΓ + Γ 1 |ξ| 2 dΓ + τ 1 |β 2 | Γ 1 1 0 z 1 (•, ρ, t) 2 dρdΓ + τ 2 |γ 2 |

INTRODUCTION

where the sesquilinear form a :

H 2 (Ω) × H 2 (Ω) -→ C is defined by a(f, g) = Ω f x 1 x 1 g x 1 x 1 + f x 2 x 2 g x 2 x 2 + µ f x 1 x 1 g x 2 x 2 + f x 2 x 2 g x 1 x 1 + 2(1 -µ)f x 1 x 2 g x 1 x 2 dx.
According to Lemma 5.2.1, we have

d dt E(t) ≤ -(β 1 -|β 2 |) Γ 1 |η| 2 dΓ -(γ 1 -|γ 2 |) Γ 1 |ξ| 2 dΓ ≤ 0.
In the sequel, we make the following assumptions

β 1 , γ 1 > 0, β 2 , γ 2 ∈ R * , |β 2 | < β 1 and |γ 2 | < γ 1 .
Then, system (Sys8) is dissipative in the sense that its energy is non-increasing with respect to time (i.e. E (t) ≤ 0).

In Subsection 5.2.1, we write system (Sys8) as the following first order evolution equation

U t = AU, U (0) = U 0 ,
where

U 0 = (u 0 , u 1 , η 0 , ξ 0 , f 0 (•, -ρτ 1 ), g 0 (•, -ρτ 2 )) ∈ H.
The Hilbert space H is defined by

H = H 2 Γ 0 (Ω) × L 2 (Ω) × L 2 (Γ 1 ) 2 × L 2 (Γ 1 × (0, 1)) 2 ,
where

H 2 Γ 0 (Ω) = f ∈ H 2 (Ω) | f = ∂ ν f = 0 on Γ 0 .
The Hilbert space H is equipped with the following inner product

(U, U 1 ) H = a(u, u 1 ) + Ω vv 1 dx + Γ 1 ηη 1 dΓ + Γ 1 ξξ 1 dΓ + τ 1 |β 2 | Γ 1 1 0 z 1 z 1 1 dρdΓ + τ 2 |γ 2 | Γ 1 1 0 z 2 z 2 1 dρdΓ, where U = (u, v, η, ξ, z 1 , z 2 ) , U 1 = (u 1 , v 1 , η 1 , ξ 1 , z 1 1 , z 2 1
) ∈ H. Now, we define the linear unbounded operator A : D(A) ⊂ H -→ H by:

D(A) = U = (u, v, η, ξ, z 1 , z 2 ) ∈ D Γ 0 (∆ 2 ) × H 2 Γ 0 (Ω) × (L 2 (Γ 1 )) 2 × (L 2 (Γ 1 ; H 1 (0, 1))) 2 | B 1 u = -η, B 2 u = ξ, z 1 (•, 0) = η, z 2 (•, 0) = ξ on Γ 1 where D Γ 0 (∆ 2 ) = f ∈ H 2 Γ 0 (Ω) | ∆ 2 f ∈ L 2 (Ω), B 1 f ∈ L 2 (Γ 1 ), and B 2 f ∈ L 2 (Γ 1 ) and A         u v η ξ z 1 z 2         =               v -∆ 2 u ∂ ν v -β 1 η -β 2 z 1 (•, 1) v -γ 1 ξ -γ 2 z 2 (•, 1) - 1 τ 1 z 1

INTRODUCTION

Next, we prove that the operator A is m-dissipative in H. Thus, according to Lumer-Phillips theorem (see Theorem 1.2.8), we deduce that the operator A generates a C 0 -semigroup of contractions e tA in H which gives the well-posedness of (Sys8).

In Subsection 5.2.2, we use a general criteria of Arendt-Batty (see Theorem 1.3.3) to show the strong stability of the C 0 -semigroup e tA associated with system (Sys8) in the absence of the compactness of the resolvent of A.

In Subsection 5.2.3, we use Theorem 1.3.6 to prove the lack of exponential stability of system (Sys8).

In Subsection 5.2.4, we use Theorem 1.3.7 to prove under the multiplier geometric control condition MGC (see Definition 1.4.1) that the energy of system (Sys8) decays polynomially with rate t -1 . In other words, for all U 0 ∈ D(A), there exists a constant C > 0 independent of U 0 such that

E(t) ≤ C t U 0 2 D(A) , t > 0.
In Section 5.3, we study the second system (Sys7). We use Theorem 1.3.6 to prove under MGC geometric condition that system (Sys7) is exponentially stable if

β 1 , γ 1 > 0, β 2 , γ 2 ∈ R * , |β 2 | < β 1 and |γ 2 | < γ 1 .
Moreover, we give some instability examples of system (Sys7) in the cases

|β 2 | ≥ β 1 and |γ 2 | ≥ γ 1 .

Haidar Badawi Valenciennes, France

Chapter 1

Preliminaries

As our analysis is based on the semigroup and spectral theories, in this chapter we will recall some well-known results about semigroups, including some theorems about strong, exponential, polynomial, and analytic stability of a C 0 -semigroup. We also recall the definition of the multiplier geometric control condition denoted by MGC . All of the theorems are stated without proofs, but the relevant references are given. The reader may skip this chapter in the first reading, then refer to it as a reference of related. For more details see [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF][START_REF] Istratescu | Inner Product Structures[END_REF][START_REF] Kato | Perturbation Theory for Linear Operators[END_REF][START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF][START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF][START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF][START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF].

Bounded and Unbounded linear operators

We start this chapter by giving some well known results about bounded and unbounded operators. We are not trying to give a complete development, but rather review the basic definitions and theorems, mostly without proof, see [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF][START_REF] Istratescu | Inner Product Structures[END_REF][START_REF] Kato | Perturbation Theory for Linear Operators[END_REF].

Let (E, • E ) and (F, • F ) be two Banach spaces over C, and H will always denote a Hilbert space equipped with the scalar product •, • H and the corresponding norm • H .

A linear operator T : E -→ F is a transformation which maps linearly E in F , that is

T (αu + βv) = αT (u) + βT (v), ∀ u, v ∈ E and α, β ∈ C. Definition 1.1.1. A linear operator T : E -→ F is said to be bounded if there exists C > 0 such that T u F ≤ C u E ∀ u ∈ E.
The set of all bounded linear operators from E into F is denoted by L(E, F ). Moreover, the set of all bounded linear operators from

E into E is denoted by L(E). Definition 1.1.2. A bounded operator T ∈ L(E, F ) is said to be compact if for each sequence (x n ) n∈N ⊂ E with x n E = 1 for each n ∈ N, the sequence (T x n ) n∈N has a subsequence which converges in F .
The set of all compact operators from E into F is denoted by K(E, F ). For simplicity one writes

K(E) = K(E, E). Definition 1.1.3. Let T ∈ L(E, F ), we define • Range of T by R (T ) = {T u : u ∈ E} ⊂ F. • Kernel of T by ker (T ) = {u ∈ E : T u = 0} ⊂ E.
Theorem 1.1.4. (Fredholm alternative, see Theorem 6.6 in [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]). If T ∈ K(E), then

• ker (I -T ) is finite dimensional (I is the identity operator on E) .

• R (I -T ) is closed.

• ker (I -T ) = 0 ⇔ R (I -T ) = E.
Definition 1.1.5. An unbounded linear operator T from E into F is a pair (T, D (T )), consisting of a subspace D (T ) ⊂ E ( called the domain of T ) and a linear transformation.

T : D (T ) ⊂ E -→ F.
If E = F , then we say (T, D (T )) is an unbounded linear operator on E.

Definition 1.1.6. Let T : D (T ) ⊂ E -→ F be an unbounded linear operator.

• The range of T is defined by

R (T ) = {T u : u ∈ D (T )} ⊂ F.
• The kernel of T is defined by

ker (T ) = {u ∈ D (T ) : T u = 0} ⊂ E.
• The graph of T is defined by

G (T ) = {(u, T u) : u ∈ D (T )} ⊂ E × F. Definition 1.1.7. A map T is said to be closed if G (T ) is closed in E × F .
The closedness of an unbounded linear operator T can be characterized as follows

if u n ∈ D (T ) such that u n → u in E and T u n → v in F, then u ∈ D (T ) and T u = v. Definition 1.1.8. Let T : D (T ) ⊂ E -→ F be a closed unbounded linear operator.
• The resolvent set of T is defined by

ρ (T ) = {λ ∈ C : λI -T is bijective from D (T ) onto F } .
• The resolvent of T is defined by

R (λ, T ) = (λI -T ) -1 , ∀λ ∈ ρ (T ) .
• The spectrum set of T is the complement of the resolvent set in C , denoted by σ (T ) = C\ρ (T ) .

Definition 1.1.9. Let T : D (T ) ⊂ E -→ F be a closed unbounded linear operator. We can split the spectrum σ(T ) of T into three disjoint sets, given by

• The point spectrum of T is defined by

σ p (T ) = {λ ∈ C : ker(λI -T ) = {0}} ,
in this case λ is called an eigenvalue of T .

• The continuous spectrum of T is defined by

σ c (T ) = λ ∈ C ker(λI -T ) = 0, R(λI -T ) = F and (λI -T ) -1 is not bounded .
• The residual spectrum of T is defined by If n = 1, then e is called an eigenvector.

σ r (T ) = {λ ∈ C : ker(λI -T ) = 0 and R(λI -T ) is not dense in F } .
Definition 1.1.11. Let T : D (T ) ⊂ E -→ F be a closed unbounded linear operator. We say that T has a compact resolvent, if there exist λ 0 ∈ ρ (T ) such that (λ 0 I -T ) -1 is compact.

Theorem 1.1.12. (see Theorem 6.5.5 in [START_REF] Istratescu | Inner Product Structures[END_REF]). Let (T, D (T )) be a closed unbounded linear operator on H, then the space D (T ) ,

• D(T ) where u D(T ) = T u H + u H , ∀ u ∈ D (T )
is a Banach space .

Theorem 1.1.13. (see Theorem 6.7 in [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF]). Let (T, D (T )) be a closed unbounded linear operator on H, then ρ (T ) is an open set of C.

Semigroups for Cauchy problems

In this section, we introduce some basic concepts concerning semigroups. The majority of evolution equations can be reduced to the form

U t = AU, t > 0, in H, U (0) = U 0 , (1.2.1)
where A is the infinitesimal generator of a C 0 -semigroup S (t) over a Hilbert space H. Let us start by basic definitions and theorems, see [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]. Let (X, • X ) be a Banach space, and H be a Hilbert space equipped with the inner product • S (0) = I (I is the identity operator on X).

• S (t + s) = S (t) S (s) , ∀ t, s ≥ 0. [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 in X. Then there exist a constant M ≥ 1 and ω ≥ 0 such that

S (t) L(X) ≤ M e ωt , ∀t ≥ 0.
If ω = 0 then the corresponding semigroup is uniformly bounded; moreover, if M = 1 then (S (t)) t≥0 is said to be a C 0 -semigroup of contractions. Definition 1.2.4. An unbounded linear operator (A, D (A)) on H, is said to be dissipative if

Au, u H ≤ 0, ∀ u ∈ D (A) .
Definition 1.2.5. An unbounded linear operator (A, D (A)) on X, is said to be m-dissipative if

• A is a dissipative operator. • ∃ λ 0 > 0 such that R (λ 0 I -A) = X.
Theorem 1.2.6. (See Theorem 4.5 in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). Let A be a m-dissipative operator, then

• R (λI -A) = X, ∀ λ > 0. • ]0, ∞[ ⊆ ρ (A) .
Theorem 1.2.7. (Hille-Yosida, see Theorem 3.1 in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). An unbounded linear operator (A, D (A)) on X, is the infinitesimal generator of 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) of A contains R + , and for all λ > 0,

(λI -A) -1 L(X) ≤ λ -1 .
Theorem 1.2.8. (Lumer-Phillips, see Theorem 4.3 in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). 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 0semigroup of contractions if and only if it is a m-dissipative operator.

Theorem 1.2.9. (see Theorem 4.6 in [92]). Let (A, D (A)) be an unbounded linear operator on X. If A is dissipative with R (I -A) = X and X is reflexive, then D (A) = X.

Corollary 1.2.10. Let (A, D (A)) be an unbounded linear operator on H. A is the infinitesimal generator of a C 0 -semigroup of contractions if and only if A is a m-dissipative operator.

Theorem 1.2.11. Let A be a linear operator with dense domain D (A) in a Hilbert space H.

If A is dissipative and 0 ∈ ρ (A), then A is the infinitesimal generator of a C 0 -semigroup of contractions on H.

Theorem 1.2.12. (see Theorem 7.4 in [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]). 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 .

1. For U 0 ∈ D (A), the problem (1.2.1) admits a unique strong solution

U (t) = S(t)U 0 ∈ C 0 (R + , D (A)) ∩ C 1 (R + , H) .
2. For U 0 ∈ H, the problem (1.2.1) admits a unique weak solution

U (t) ∈ C 0 (R + , H) .

Stability of semigroups

In this section, we introduce some definitions about strong, exponential, polynomial and analytic stability of a C 0 -semigroup. Then, we give some results about the stability of C 0semigroup. For more details, 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][START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF][START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF][START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF]. Let (X, • X ) be a Banach space, and H be a Hilbert space equipped with the inner product •, • H and the induced norm • H .

Definition 1.3.1. Assume that A is the generator of a strongly continuous semigroup of contractions (S (t)) t≥0 on X. We say that the C 0 -semigroup (S (t)) t≥0 is

• Strongly stable if lim t→+∞ S (t) u X = 0, ∀ u ∈ X. • Uniformly stable if lim t→+∞ S (t) L(X) = 0.
• Exponentially stable if there exist two positive constants M and such that

S (t) u X ≤ M e -t u X , ∀ t > 0, ∀ u ∈ X.
• Polynomially stable if there exist two positive constants C and α such that

S (t) u X ≤ Ct -α u D(A) , ∀ t > 0, ∀ u ∈ D(A).
Proposition 1.3.2. Assume that A is the generator of a strongly continuous semigroup of contractions (S (t)) t≥0 on X. The following statements are equivalent

• (S (t)) t≥0 is uniformly stable.

• (S (t)) t≥0 is exponentially stable.

To show the strong stability of a C 0 -semigroup we rely on the following result due to Arendt-Batty [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF].

Theorem 1.3.3. (Arendt and Batty). Assume that A is the generator of a strongly continuous semigroup of contractions (S (t)) t≥0 on a reflexive Banach space X. If

• A has no pure imaginary eigenvalues.

• σ (A) ∩ iR is countable.

Then S (t) is strongly stable.

Remark 1.3.4. If the resolvent (I -T ) -1 of T is compact, then σ (T ) = σ p (T ). Thus, the statement of Theorem 1.3.3 lessens to σ p (A) ∩ iR = ∅.
An alternative method based on Arendt and Batty theorem and a contradiction argument, see [82, page 25] is presented in the following Remark.

Remark 1.3.5. Assume that the unbounded linear operator A : D(A) ⊂ H -→ H is the generator of a C 0 -semigroup of contractions (S(t)) t≥0 on a Hilbert space H and suppose that 0 ∈ ρ(A). According to [82, page 25], in order to prove that

iR ≡ {iλ | λ ∈ R} ⊆ ρ(A), (1.3.1)
we need the following steps:

(i) It follows from the fact that 0 ∈ ρ(A) and the contraction mapping theorem that for any real number λ with |λ| < A -1 -1 , the operator iλI

-A = A(iλA -1 -I) is invertible. Furthermore, (iλI -A) -1 is a continuous function of λ in the interval (-A -1 -1 , A -1 -1 ). (ii) If sup { (iλI -A) -1 | |λ| < A -1 -1 } = M < ∞, then by the contraction mapping theorem, the operator iλI -A = (iλ 0 I -A)(I +i(λ-λ 0 )(iλ 0 I -A) -1 ) with |λ 0 | < A -1 -1 is invertible for |λ -λ 0 | < M -1 . It turns out that by choosing |λ 0 | as close to A -1 -1 as we can, we conclude that {λ | |λ| < A -1 -1 + M -1 } ⊂ ρ(A) and (iλI -A) -1 is a continuous function of λ in the interval (-A -1 -1 -M -1 , A -1 -1 + M -1
) .

(iii) Thus it follows from the argument in (ii

) that if (1.3.1) is false, then there is ω ∈ R with A -1 -1 ≤ |ω| < ∞ such that {iλ | |λ| < |ω|} ⊂ ρ(A) and sup { (iλ -A) -1 | |λ| < |ω|} = ∞. It turns out that there exists a sequence {(λ n , U n )} n≥1 ⊂ R × D (A) , with λ n → ω as n → ∞, |λ n | < |ω| and U n H = 1, such that (iλ n I -A)U n = F n → 0 in H, as n → ∞.
Then, we will prove (1.3.1) by showing that U n H → 0 (up to a subsequence) which contradicts U n H = 1.

Next, when the C 0 -semigroup is strongly stable, we look for the necessary and sufficient conditions of exponential stability of a C 0 -semigroup. In case when the C 0 -semigroup is not exponentially stable, we may look for a polynomial one. In fact, exponential and polynomial stability results are obtained using different methods like: multipliers method, frequency domain approach, Riesz basis approach, Fourier analysis or a combination of them. In this thesis we will review only two methods. The following result is a frequency domain approach method which was 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 1.3.6. Assume that A is the generator of a strongly continuous semigroup of contractions (S (t)) t≥0 on H. S (t) is uniformly stable if and only if

• iR ⊂ ρ (A) . • lim sup λ∈R,|λ|→∞ (iλI -A) -1 L(H) < ∞.
Moreover, the following result is a frequency domain approach method which was obtained by Borichev and Tomilov [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] (see also [START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF] and [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF]) .

Theorem 1.3.7. Assume that A is the generator of a strongly continuous semigroup of contractions (S (t)) t≥0 on H. If iR ⊂ ρ (A), then for a fixed > 0 the following conditions are equivalent lim sup λ∈R,|λ|→∞

1 |λ| (iλI -A) -1 L(H) < ∞, (1.3.2)
S (t) U 0 H ≤ C t 1 U 0 D(A) ∀ t > 0, U 0 ∈ D (A), for some C > 0. (1.3.3)
Also, the analytic property of a C 0 -semigroup of contraction (S (t)) t≥0 is characterized in the following theorem due to [START_REF] Arendt | Vector-Valued Laplace Transforms and Cauchy Problems[END_REF].

|λ| (iλI -A) -1 L(H) < ∞.

The multiplier geometric control condition

In this section, we recall the definition of the multiplier geometric control condition denoted by MGC. We say that the partition (Γ 0 , Γ 1 ) of the boundary Γ satisfies the multiplier geometric control condition MGC (see Fig. 1.1 for an illustration) if there exists a point x 0 ∈ R n and a positive constant δ such that

h • ν ≥ δ -1 on Γ 1 and h • ν ≤ 0 on Γ 0 , (1.4.1)
where h(x) = x -x 0 .

x 0 A B Γ 1 Γ 0 Ω v 0 v 1

Chapter 2

Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay

The purpose of this chapter is to investigate the stabilization of locally coupled wave equations with non-smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Using a general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. Finally, using frequency domain approach combined with the multiplier method, we prove a polynomial energy decay rate of order t -1 . This chapter is published in [START_REF] Akil | Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay[END_REF].

Introduction

Description of the chapter

In this chapter, we investigate the stability of local coupled wave equations with singular localized viscoelastic damping of Kelvin-Voigt type and localized time delay. More precisely, we consider the following system:

                         u tt -[au x + b(x)(κ 1 u tx + κ 2 u tx (x, t -τ ))] x + c(x)y t = 0, (x, t) ∈ (0, L) × (0, ∞), y tt -y xx -c(x)u t = 0, (x, t) ∈ (0, L) × (0, ∞), u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0, (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)), x ∈ (0, L), (y(x, 0), y t (x, 0)) = (y 0 (x), y 1 (x)), x ∈ (0, L), u t (x, t) = f 0 (x, t), (x, t) ∈ (0, L) × (-τ, 0), (2.1.1)
where L, τ, a and κ 1 are positive real numbers, κ 2 is a non-zero real number and (u 0 , u 1 , y 0 , y 1 , f 0 ) belongs to a suitable space. We suppose that there exist 0 < α < β < γ < L and a non-zero constant c 0 , such that System (2.1.1) consists in two wave equations with only one singular viscoelastic damping acting on the first equation, the second one is indirectly damped via a singular coupling between the two equations. The notion of indirect damping mechanisms has been introduced by Russell in [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF] and since then, it has attracted the attention of many authors (see for instance [START_REF] Akil | The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization[END_REF], [START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF], [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF], [START_REF] Ammari | Stabilization of coupled systems[END_REF], [START_REF] Cui | Asymptotic stability of wave equations coupled by velocities[END_REF], [START_REF] Abdallah | Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities[END_REF], [START_REF] Liu | Frequency domain approach for the polynomial stability of a system of partially damped wave equations[END_REF] and [START_REF] Zhang | Polynomial decay and control of a 1 -d hyperbolic-parabolic coupled system[END_REF]). The study of such systems is also motivated by several physical considerations like Timoshenko and Bresse systems (see for instance [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], [START_REF] Akil | Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary[END_REF], [START_REF] Mercier | Decay rate of the Timoshenko system with one boundary damping[END_REF] and [START_REF] Najdi | Weakly locally thermal stabilization of Bresse systems[END_REF]). In fact, there are few results concerning the stability of coupled wave equations with local Kelvin-Voigt damping without time delay, especially in the absence of smoothness of the damping and coupling coefficients (see Subsection 2.1.2). This motivates our interest to study the stabilization of system (2.1.1) in this chapter.

b(x) = 1, x ∈ (0, β), 0, x ∈ (β, L), and c(x) = c 0 , x ∈ (α, γ), 0, x ∈ (0, α) ∪ (γ, L).

Previous Literature

The wave is created when a vibrating source disturbs the medium. In order to restrain those vibrations, several damping can be added such as Kelvin-Voigt damping which is originated from the extension or compression of the vibrating particles. This damping is a viscoelastic structure having properties of both elasticity and viscosity. In the recent years, many researchers showed interest in problems involving this kind of damping where different types of stability, depend on the smoothness of the damping coefficients, has been showed (see [START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF], [START_REF] Alves | The asymptotic behavior of the linear transmission problem in viscoelasticity[END_REF], [START_REF] Hassine | Stability of elastic transmission systems with a local Kelvin-Voigt damping[END_REF], [START_REF] Hassine | Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping[END_REF], [START_REF] Huang | On the Mathematical Model for Linear Elastic Systems with Analytic Damping[END_REF], [START_REF] Liu | Spectrum and Stability for Elastic Systems with Global or Local Kelvin-Voigt Damping[END_REF], [START_REF] Liu | Stability of a String with Local Kelvin-Voigt Damping and Nonsmooth Coefficient at Interface[END_REF], [START_REF] Oquendo | Frictional versus Kelvin-Voigt damping in a transmission problem[END_REF] and [START_REF] Rivera | Stability to localized viscoelastic transmission problem[END_REF]). However, time delays appear in several applications such as in physics, chemistry, biology, thermal phenomena not only depend on the present state but also on some past occurrences (see [START_REF] Ernst | Delay-induced multistable synchronization of biological oscillators[END_REF] and [START_REF] Kolmanovskii | Introduction to the theory and applications of functional-differential equations[END_REF]) . In the last years, the control of partial differential equations with time delays have become popular among scientists, since in many cases time delays induce some instabilities see [START_REF] Datko | Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks[END_REF][START_REF] Datko | Two examples of ill-posedness with respect to time delays revisited[END_REF][START_REF] Datko | An example of the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Dreher | Ill-posed problems in thermomechanics[END_REF].

However, let us recall briefly some systems of wave equations with Kelvin-Voigt damping and time delay represented in previous literature.

Coupled wave equations with Kelvin-Voigt damping and without time delay

In 2020, Hayek et al. in [START_REF] Hayek | A transmission problem of a system of weakly coupled wave equations with kelvin-voigt dampings and non-smooth coefficient at the interface[END_REF] studied the stabilization of a multi-dimensional system of weakly coupled wave equations with one or two locally Kelvin-Voigt damping and non-smooth co-efficient at the interface. They established different stability results. In 2021, Hassine and Souayeh in [START_REF] Hassine | Stability for coupled waves with locally disturbed kelvinvoigt damping[END_REF] studied the behavior of a system with coupled wave equations with a partial Kelvin-Voigt damping, by considering the following system

               u tt -(u x + b 2 (x)u tx ) x + v t = 0, (x, t) ∈ (-1, 1) × (0, ∞), v tt -cv xx -u t = 0, (x, t) ∈ (-1, 1) × (0, ∞), u(0, t) = v(0, t) = 0, u(1, t) = v(1, t) = 0, t > 0, u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ (-1, 1), v(x, 0) = v 0 (x), v t (x, 0) = v 1 (x), x ∈ (-1, 1), (2.1.2)
where c > 0 and b 2 ∈ L ∞ (-1, 1) is a non-negative function. They assumed that the damping coefficient is piecewise function in particular they supposed that b 2 (x) = d1 [0,1] (x), where d is a strictly positive constant. So, they took the damping coefficient to be near the boundary with a global coupling coefficient. They showed the lack of exponential stability and that the semigroup loses speed and it decays polynomially with a rate as t -1 12 . In 2021, Akil, Issa and Wehbe in [START_REF] Wehbe | Stability results of an elastic/viscoelastic transmission problem of locally coupled waves with non smooth coefficients[END_REF] studied the localized coupled wave equations, by considering the following system:

                 u tt -(au x + b(x)u tx ) x + c(x)y t = 0, (x, t) ∈ (0, L) × (0, ∞), y tt -y xx -c(x)u t = 0, (x, t) ∈ (0, L) × (0, ∞), u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0, (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)), x ∈ (0, L), (y(x, 0), y t (x, 0)) = (y 0 (x), y 1 (x)), x ∈ (0, L),
where b(x) = b 0 , x ∈ (α 1 , α 3 ), 0, otherwise and c(x) = c 0 , x ∈ (α 2 , α 4 ), 0, otherwise where a > 0, b 0 > 0, c 0 > 0 and 0 < α 1 < α 2 < α 3 < α 4 < L. They generalized the results of Hassine and Souayeh in [START_REF] Hassine | Stability for coupled waves with locally disturbed kelvinvoigt damping[END_REF] by establishing a polynomial decay rate of type t -1 .

Wave equations with time delay and without Kelvin-Voigt damping

The delay equations of hyperbolic type is given by

u tt -∆u(x, t -τ ) = 0. (2.1.3)
with a delay parameter τ > 0. This system is not well posed since there exists a sequence of solutions tending to infinity for any fixed t > 0 while the norm of the initial data remains bounded (see Theorem 1.1 in [START_REF] Dreher | Ill-posed problems in thermomechanics[END_REF]). In 2006, Nicaise and Pignotti in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF] studied the multidimensional wave equation considering two cases. The first case concerns a wave equation with boundary feedback and a delay term at the boundary

                 u tt (x, t) -∆u(x, t) = 0, (x, t) ∈ Ω × (0, ∞), u(x, t) = 0, (x, t) ∈ Γ D × (0, ∞), ∂u ∂ν (x, t) = -µ 1 u t (x, t) -µ 2 u t (x, t -τ ), (x, t) ∈ Γ N × (0, ∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x, t) = f 0 (x, t), (x, t) ∈ Γ N × (-τ, 0). (2.1.4)
The second case concerns a wave equation with an internal feedback and a delayed velocity term (i.e. an internal delay) and a mixed Dirichlet-Neumann boundary condition

                 u tt (x, t) -∆u(x, t) + µ 1 u t (x, t) + µ 2 u t (x, t -τ ) = 0, (x, t) ∈ Ω × (0, ∞), u(x, t) = 0, (x, t) ∈ Γ D × (0, ∞), ∂u ∂ν (x, t) = 0, (x, t) ∈ Γ N × (0, ∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x, t) = f 0 (x, t), (x, t) ∈ Ω × (-τ, 0), (2.1.5)
where Ω is an open bounded domain of R N with a boundary Γ of class C 2 and Γ = Γ D ∪ Γ N , such that Γ D ∩Γ N = ∅. Under the assumption µ 2 < µ 1 , an exponential decay is achieved for the both systems (2.1.4)-(2.1.5). If this assumption does not hold, they found a sequences of delays {τ k } k , τ k → 0, for which the corresponding solutions have increasing energy. Furthermore, we refer to [START_REF] Benhassi | Feedback stabilization of a class of evolution equations with delay[END_REF] for system (2.1.5) in more general abstract setting. In 2010, Ammari et al. in [START_REF] Ammari | Feedback boundary stabilization of wave equations with interior delay[END_REF] studied the wave equation with interior delay damping and dissipative undelayed boundary condition in an open domain Ω of R N , N ≥ 2. The system is described by:

                 u tt (x, t) -∆u(x, t) + au t (x, t -τ ) = 0, (x, t) ∈ Ω × (0, ∞), u(x, t) = 0, (x, t) ∈ Γ 0 × (0, ∞), ∂u ∂ν (x, t) = -κu t (x, t), (x, t) ∈ Γ 1 × (0, ∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x, t) = f 0 (x, t), (x, t) ∈ Ω × (-τ, 0), (2.1.6)
where τ > 0, a > 0 and κ > 0. Under the condition that Γ 1 satisfies the Γ-condition introduced in [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], they proved that system (2.1.6) is uniformly asymptotically stable whenever the delay coefficient is sufficiently small. In 2012, Pignotti in [START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF] considered the wave equation with internal distributed time delay and local damping in a bounded and smooth domain Ω ⊂ R N , N ≥ 1. The considered system is given by the following:

           u tt (x, t) -∆u(x, t) + aχ ω u t (x, t) + κu t (x, t -τ ) = 0, (x, t) ∈ Ω × (0, ∞), u(x, t) = 0, (x, t) ∈ Γ × (0, ∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x, t) = f (x, t), (x, t) ∈ Ω × (-τ, 0), (2.1.7)
where κ ∈ R, τ > 0, a > 0 and ω is the intersection between an open neighborhood of the set Γ 0 = {x ∈ Γ; (x -x 0 ) • ν(x) > 0} and Ω. Moreover, χ ω is the characteristic function of ω. We remark that the damping is localized and it acts on a neighborhood of a part of Ω. She showed an exponential stability results if the coefficients of the delay terms satisfy the following assumption |κ| < κ 0 < a. Several researches were done on wave equation with time delay acting on the boundary see ( [START_REF] Datko | An example of the effect of time delays in boundary feedback stabilization of wave equations[END_REF], [START_REF] Datko | Two questions concerning the boundary control of certain elastic systems[END_REF], [START_REF] Xu | Stabilization of wave systems with input delay in the boundary control[END_REF], [START_REF] Guo | Boundary Output Feedback Stabilization of A One-Dimensional Wave Equation System With Time Delay[END_REF], [START_REF] Gugat | Boundary feedback stabilization by time delay for one-dimensional wave equations[END_REF], [START_REF] Wang | Wave Equation Stabilization by Delays Equal to Even Multiples of the Wave Propagation Time[END_REF], [START_REF] Xie | Exponential stability of 1-d wave equation with the boundary time delay based on the interior control[END_REF]) and different type of stability has been proved.

Wave equations with Kelvin-Voigt damping and time delay

In 2016, Messaoudi et al. in [START_REF] Messaoudi | Well posedness and exponential stability in a wave equation with a strong damping and a strong delay[END_REF] considered the stabilization of the following wave equation with strong time delay

           u tt (x, t) -∆u(x, t) -µ 1 ∆u t (x, t) -µ 2 ∆u t (x, t -τ ) = 0, (x, t) ∈ Ω × (0, ∞), u(x, t) = 0, (x, t) ∈ Γ × (0, ∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x, t) = f 0 (x, t), (x, t) ∈ Ω × (-τ, 0),
where µ 1 > 0 and µ 2 is a non zero real number. Under the assumption that |µ 2 | < µ 1 , they obtained an exponential stability result. In 2016, Nicaise et al. in [START_REF] Nicaise | Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback[END_REF] studied the multidimensional wave equation with localized Kelvin-Voigt damping and mixed boundary condition with time delay

                 u tt (x, t) -∆u(x, t) -div(a(x)∇u t ) = 0, (x, t) ∈ Ω × (0, ∞), u(x, t) = 0, (x, t) ∈ Γ 0 × (0, ∞), ∂u ∂ν (x, t) = -a(x) ∂ut ∂ν (x, t) -κu t (x, t -τ ), (x, t) ∈ Γ 1 × (0, ∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x, t) = f 0 (x, t), (x, t) ∈ Γ 1 × (-τ, 0), (2.1.8) where τ > 0, κ ∈ R, a(x) ∈ L ∞ (Ω) and a(x) ≥ a 0 > 0 on ω such that ω ⊂ Ω is an open neighborhood of Γ 1 .
Under an appropriate geometric condition on Γ 1 and assuming that a ∈ C 1,1 (Ω), ∆a ∈ L ∞ (Ω), they proved an exponential decay of the energy of system (2.1.8).

In 2019, Anikushyn et al. in [START_REF] Demchenko | On a Kelvin-Voigt viscoelastic wave equation with strong delay[END_REF] considered an initial boundary value problem for a viscoelastic wave equation subjected to a strong time localized delay in a Kelvin-Voigt type. The system is given by the following:

                 u tt -c 1 ∆u -c 2 ∆u(x, t -τ ) -d 1 ∆u t -d 2 ∆u t (x, t -τ ) = 0, (x, t) ∈ Ω × (0, ∞), u(x, t) = 0, (x, t) ∈ Γ 0 × (0, ∞), ∂u ∂ν (x, t) = 0, (x, t) ∈ Γ 1 × (0, ∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u(x, t) = f 0 (x, t), (x, t) ∈ Ω × (-τ, 0).
Under appropriate conditions on the coefficients, a global exponential decay rate is obtained. In 2015, Ammari et al. in [START_REF] Ammari | Stability of an abstract-wave equation with delay and a Kelvin-Voigt damping[END_REF] considered the stabilization problem for an abstract equation with delay and a Kelvin-Voigt damping. The system is given by the following:

       u tt (t) + aBB * u t (t) + BB * u(t -τ ) = 0, t ∈ (0, ∞), (u(0), u t (0)) = (u 0 , u 1 ) , B * u(t) = f 0 (t), t ∈ (-τ, 0),
for an appropriate class of operator B and a > 0. Using the frequency domain approach, they obtained an exponential stability result.

Thus, to the best of our knowledge, it seems to us that there is no result in the existing literature concerning the case of coupled wave equations with localized Kelvin-Voigt damping and localized time delay, especially in the absence of smoothness of the damping and coupling coefficients. The goal of the present chapter is to fill this gap by studying the stability of system (2.1.1). This chapter is organized as follows: In Section 2.2, we prove the well-posedness of our system by using semigroup approach. In Section 2.3, by using a general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. Next, in Section 2.4, by using frequency domain approach combining with a specific multiplier method, we prove a polynomial energy decay rate of order t -1 .

Well-posedness of the system

In this section, we will establish the well-posedness of system (2.1.1) by using semigroup approach. To this aim, as in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF], we introduce the following auxiliary change of variable η(x, ρ, t) := u t (x, t -ρτ ), x ∈ (0, β), ρ ∈ (0, 1), t > 0.

(2.2.1)

Then, system (2.1.1) becomes

u tt -(S b (u, u t , η)) x + c(x)y t = 0, (x, t) ∈ (0, L) × (0, ∞), (2.2.2)
y tt -y xx -c(x)u t = 0, (x, t) ∈ (0, L) × (0, ∞), (2.2.3) τ η t (x, ρ, t) + η ρ (x, ρ, t) = 0, (x, ρ, t) ∈ (0, β) × (0, 1) × (0, ∞), (2.2.4)
where S b (u, u t , η)

:= au x + b(x)(κ 1 u tx + κ 2 u tx (x, t -τ ))
. Moreover, from the definition of b(•), we have

S b (u, u t , η) = S 1 (u, u t , η) := au x + κ 1 u tx + κ 2 η x (•, 1, t), in (0, β), au x , in (β, L). (2.2.5)
With the following boundary conditions

u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0, η(0, ρ, t) = 0, (ρ, t) ∈ (0, 1) × (0, ∞), (2.2.6)
and the following initial conditions

     u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ (0, L), y(x, 0) = y 0 (x), y t (x, 0) = y 1 (x), x ∈ (0, L), η(x, ρ, 0) = f 0 (x, -ρτ ), (x, ρ) ∈ (0, β) × (0, 1). (2.2.7)
The energy of system (2.2.2)-(2.2.7) is given by

E(t) = E 1 (t) + E 2 (t) + E 3 (t), (2.2.8)
where

         E 1 (t) = 1 2 L 0 |u t | 2 + a|u x | 2 dx, E 2 (t) = 1 2 L 0 |y t | 2 + |y x | 2 dx and E 3 (t) = τ |κ 2 | 2 β 0 1 0 |η x (•, ρ, t)| 2 dρdx
d dt L 0 |u t | 2 dx + L 0 S b (u, u t , η)u tx dx + L 0 c(•)y t u t dx = 0.
From the above equation and the definition of S b (u, u t , η) and c(•), we deduce that , integrating over (0, β) × (0, 1), using the fact that η x (•, 0, t) = u tx , then taking the real part, we get In the sequel, we make the following assumptions

d dt E 1 (t) = -κ 1 β 0 |u tx | 2 dx - κ 2 β 0 η x (•, 1, t)u tx dx - c 0 γ α y t u t dx . ( 2
d dt E 3 (t) = - |κ 2 | 2 β 0 |η x (•, 1, t)| 2 -|η x (•, 0, t)| 2 dx = - |κ 2 | 2 β 0 |η x (•, 1, t)| 2 -|u tx | 2 dx. ( 2
κ 1 > 0, κ 2 ∈
H := H 1 0 (0, L) × L 2 (0, L) 2 × W,
where

W := L 2 ((0, 1); H 1 L (0, β)) and H 1 L (0, β) := η ∈ H 1 (0, β) | η(0) = 0 .
The space W is a Hilbert space of H 1 L (0, β)-valued functions on (0, 1), equipped with the following inner product

(η 1 , η 2 ) W := β 0 1 0 η 1 x η 2 x dρdx, ∀ η 1 , η 2 ∈ W.
The Hilbert space H is equipped with the following inner product

U, U 1 H = L 0 au x u 1 x + vv 1 + y x y 1 x + zz 1 dx + τ |κ 2 | β 0 1 0 η x (•, ρ)η 1 x (•, ρ)dρdx, (2.2.15)
where

U = (u, v, y, z, η(•, ρ)) , U 1 = (u 1 , v 1 , y 1 , z 1 , η 1 (•, ρ)) ∈ H
. Now, we define the linear unbounded operator A : D(A) ⊂ H -→ H by:

D(A) =    U = (u, v, y, z, η(•, ρ)) ∈ H | y ∈ H 2 (0, L) ∩ H 1 0 (0, L), v, z ∈ H 1 0 (0, L) (S b (u, v, η)) x ∈ L 2 (0, L), η ρ (•, ρ) ∈ W, η(•, 0) = v(•) in (0, β)    and A       u v y z η(•, ρ)       =       v (S b (u, v, η)) x -c(•)z z y xx + c(•)v -τ -1 η ρ (•, ρ)       , (2.2.16) for all U = (u, v, y, z, η(•, ρ)) ∈ D(A). Now, if U = (u, u t , y, y t , η(•, ρ)) , then system (2.2.2)-(2.2.7
) can be written as the following first order evolution equation

U t = AU, U (0) = U 0 , (2.2.17)
where U 0 = (u 0 , u 1 , y 0 , y 1 , f 0 (•, -ρτ )) ∈ H.

Remark 2.2.1. The linear unbounded operator A is injective (i.e. ker(A) = {0}). Indeed, if U ∈ D(A) such that AU = 0, then v, z, η ρ (•, ρ) = 0 and since η(•, 0) = v(•), we get η(•, ρ) = 0. Consequently, (S b (u, v, η)) x = au xx = 0 and y xx = 0. Now, since u(0) = u(L) = y(0) = y(L) = 0, then u = y = 0. Thus, U = (u, v, y, z, η(•, ρ)) = 0.

Proposition 2.2.1. Under the hypothesis (H), the unbounded linear operator A is mdissipative in the energy space H.

Proof. For all U = (u, v, y, z, η(•, ρ)) ∈ D(A), from (2.2.15) and (2.2.16), we have

(AU, U ) H = L 0 av x u x dx + L 0 (S b (u, v, η)) x vdx + L 0 z x y x dx + L 0 y xx zdx - |κ 2 | β 0 1 0 η xρ (•, ρ)η x (•, ρ)dρdx .
Using integration by parts to the second and fourth terms in the above equation, then using the definition of S b (u, v, η) and the fact that U ∈ D(A), we get

(AU, U ) H = -κ 1 β 0 |v x | 2 dx - κ 2 β 0 η x (•, 1)v x dx - |κ 2 | 2 β 0 1 0 d dρ |η x (•, ρ)| 2 dρdx, the fact that η(•, 0) = v(•) in (0, β), implies that (AU, U ) H = -κ 1 - |κ 2 | 2 β 0 |v x | 2 dx - |κ 2 | 2 β 0 |η x (•, 1)| 2 dx - κ 2 β 0 η x (•, 1)v x dx .
Using Young's inequality in the above equation and the hypothesis (H), we obtain

(AU, U ) H ≤ -(κ 1 -|κ 2 |) β 0 |v x | 2 dx ≤ 0, (2.2.18)
which implies that A is dissipative. Now, let us prove that A is maximal. To this aim, let

F = (f 1 , f 2 , f 3 , f 4 , f 5 (•, ρ)) ∈ H, we look for U = (u, v, y, z, η(•, ρ)) ∈ D(A) unique solution of -AU = F. (2.2.19)
Equivalently, we have the following system

-v = f 1 , (2.2.20) -(S b (u, v, η)) x + c(•)z = f 2 , (2.2.21) -z = f 3 , (2.2.22) -y xx -c(•)v = f 4 , (2.2.23) τ -1 η ρ (•, ρ) = f 5 (•, ρ), (2.2.24)
with the following boundary conditions

u(0) = u(L) = y(0) = y(L) = 0, η(0, ρ) = 0 and η(•, 0) = v(•) in (0, β). ( 2
S b u, f 1 , τ 1 0 f 5 (x, s)ds -f 1 x + c(•)f 3 = -f 2 ,
(2.2.27)

y xx -c(•)f 1 = -f 4 , (2.2.28) u(0) = u(L) = y(0) = y(L) = 0, (2.2.29)
where

S b u, -f 1 , τ 1 0 f 5 (x, s)ds -f 1 =    au x -κ 1 f 1 x + τ κ 2 1 0 f 5 x (•, s)ds -k 2 f 1 x , in (0, β), au x , in (β, L).
Let (φ, ψ) ∈ H 1 0 (0, L) × H 1 0 (0, L). Multiplying (2.2.27) and (2.2.28) by φ and ψ respectively, integrating over (0, L), then using formal integrations by parts, we obtain

a L 0 u x φ x dx = L 0 f 2 φdx + c 0 γ α f 3 φdx + (κ 1 + κ 2 ) β 0 f 1 x φ x dx -τ κ 2 β 0 1 0 f 5 x (•, s)ds φ x dx (2.2.30) and L 0 y x ψ x dx = L 0 f 4 ψdx -c 0 γ α f 1 ψdx. ( 2
B((u, y), (φ, ψ)) = L(φ, ψ), ∀(φ, ψ) ∈ H 1 0 (0, L) × H 1 0 (0, L), (2.2.32)
where

B((u, y), (φ, ψ)) = a L 0 u x φ x dx + L 0 y x ψ x dx and L(φ, ψ) = L 0 f 2 φ + f 4 ψ dx + c 0 γ α f 3 φ -f 1 ψ dx -τ κ 2 β 0 1 0 f 5 x (•, s)ds φ x dx + (κ 1 + κ 2 ) β 0 f 1 x φ x dx.
It is easy to see that, B is a sesquilinear, continuous and coercive form on (H 1 0 (0, L) × H 1 0 (0, L))

2
and L is an antilinear and continuous form on H 1 0 (0, L) × H 1 0 (0, L). Then, it follows by Lax-Milgram theorem that (2.2.32) admits a unique solution (u, y) ∈ H 1 0 (0, L) × H 1 0 (0, L). By using the classical elliptic regularity, we deduce that system (2.2.27)-(2.2.29) admits a unique solution (u, y)

∈ H 1 0 (0, L) × (H 2 (0, L) ∩ H 1 0 (0, L)) such that (S b (u, v, η)) x ∈ L 2 (0, L) and since ker(A) = {0} (see Remark 2.2.1), we get U = u, -f 1 , y, -f 3 , τ ρ 0 f 5 (•, s)ds -f 1 ∈ D(A)
is a unique solution of (2.2.19). Then, A is an isomorphism and since ρ (A) is open set of C (see Theorem 1.1.13), we easily get R(λI -A) = H for a sufficiently small λ > 0. This, together with the dissipativeness of A, imply that D (A) is dense in H and that A is m-dissipative in H (see Theorems 1.2.6, 1.2.9). The proof is thus complete.

According to Lumer-Phillips theorem (see Theorem 1.2.8), Proposition 2.2.1 implies that the operator A generates a C 0 -semigroup of contractions e tA in H which gives the well-posedness of (2.2.17). Then, we have the following result:

Theorem 2.2.1. Under the hypothesis (H), for all U 0 ∈ H, system (2.2.17) admits a unique weak solution

U (x, ρ, t) = e tA U 0 (x, ρ) ∈ C 0 (R + , H).
Moreover, if U 0 ∈ D(A), then system (2.2.17) admits a unique strong solution

U (x, ρ, t) = e tA U 0 (x, ρ) ∈ C 0 (R + , D(A)) ∩ C 1 (R + , H).

Strong Stability

In this section, we will prove the strong stability of system (2.2.

:= (u n , v n , y n , z n , η n (•, ρ)) ) n≥1 ⊂ R * × D(A), with λ n → ω as n → ∞ and |λ n | < |ω| (2.3.2)
and

U n H = (u n , v n , y n , z n , η n (•, ρ)) H = 1, ∀n ≥ 1, (2.3.3) such that (iλ n I -A)U n = F n := (f 1,n , f 2,n , f 3,n , f 4,n , f 5,n (•, ρ)) → 0 in H, as n → ∞. (2.3.4)
Equivalently, we have

iλ n u n -v n = f 1,n → 0 in H 1 0 (0, L), (2.3.5) iλ n v n -(S b (u n , v n , η n )) x + c(•)z n = f 2,n → 0 in L 2 (0, L), (2.3.6) iλ n y n -z n = f 3,n → 0 in H 1 0 (0, L), (2.3.7) iλ n z n -y n xx -c(•)v n = f 4,n → 0 in L 2 (0, L), (2.3.8) iλ n η n (., ρ) + τ -1 η n ρ (•, ρ) = f 5,n (•, ρ) → 0 in W. (2.3.9)
Then, we will prove condition (2.3.1) by finding a contradiction with (2.3.3) such as U n H → 0. The proof of proposition 2.3.1 has been divided into several Lemmas. Lemma 2.3.1. Under the hypothesis (H), the solution

U n = (u n , v n , y n , z n , η n (•, ρ)) ∈ D(A) of system (2.3.5)-(2.3.9) satisfies the following limits lim n→∞ β 0 |v n x | 2 dx = 0, (2.3.10) lim n→∞ β 0 |v n | 2 dx = 0, (2.3.11) lim n→∞ β 0 |u n x | 2 dx = 0, (2.3.12) lim n→∞ β 0 1 0 |η n x (•, ρ)| 2 dρdx = 0, (2.3.13) lim n→∞ β 0 |η n x (•, 1)| 2 dx = 0, (2.3.14) lim n→∞ β 0 |S 1 (u n , v n , η n )| 2 dx = 0. (2.3.15)
Proof. First, taking the inner product of (2.3.4) with U n in H and using (2.2.18) with the help of hypothesis (H), we obtain

β 0 |v n x | 2 dx ≤ - 1 κ 1 -|κ 2 | (AU n , U n ) H = 1 κ 1 -|κ 2 | (F n , U n ) H ≤ 1 κ 1 -|κ 2 | F n H U n H .
(2.3.16)

Passing to the limit in (2.3.16), then using the fact that U n H = 1 and F n H → 0, we obtain (2.3.10). Now, since v n ∈ H 1 0 (0, L), then it follows from Poincaré inequality that there exists a constant

C p > 0 such that v n L 2 (0,β) ≤ C p v n x L 2 (0,β) . ( 2
β 0 |f 1,n x | 2 dx ≤ L 0 |f 1,n x | 2 dx ≤ a -1 F n 2 H , we deduce that β 0 |u n x | 2 dx ≤ 2 (λ n ) 2 β 0 |v n x | 2 dx + 2 (λ n ) 2 β 0 |f 1,n x | 2 dx ≤ 2 (λ n ) 2 β 0 |v n x | 2 dx + 2 a(λ n ) 2 F n 2 H .
(2.3.18)

Passing to the limit in (2.3.18), then using (2.3.2), (2.3.10) and the fact that F n H → 0, we obtain (2.3.12). Moreover, from (2.3.9) and the fact that

η n (•, 0) = v n (•) in (0, β), we deduce that η n (x, ρ) = v n e -iλ n τ ρ + τ ρ 0 e iλ n τ (s-ρ) f 5,n (x, s)ds, (x, ρ) ∈ (0, β) × (0, 1). (2.3.19)
From (2.3.19), and the fact that ρ ∈ (0, 1) and

β 0 1 0 |f 5,n x (•, s)| 2 dsdx ≤ τ -1 |κ 2 | -1 F n 2 H , we obtain β 0 1 0 |η n x (•, ρ)| 2 dρdx ≤ 2 β 0 |v n x | 2 dx + 2τ 2 β 0 1 0 ρ 0 ρ|f 5,n x (•, s)| 2 dsdρdx ≤ 2 β 0 |v n x | 2 dx + 2τ 2 β 0 1 0 1 0 ρ|f 5,n x (•, s)| 2 dsdρdx = 2 β 0 |v n x | 2 dx + 2τ 2 1 0 ρdρ β 0 1 0 |f 5,n x (•, s)| 2 dsdx = 2 β 0 |v n x | 2 dx + τ 2 β 0 1 0 |f 5,n x (•, s)| 2 dsdx ≤ 2 β 0 |v n x | 2 dx + τ |κ 2 | -1 F n 2 H .
(2.3.20)

Passing to the limit in (2.3.20), then using (2.3.10) and the fact that F n H → 0, we obtain (2.3.13). On the other hand, from (2.3.19), we have

η n x (•, 1) = v n x e -iλ n τ + τ 1 0 e iλ n τ (s-1) f 5,n x (•, s)ds,
consequently, by using the same argument as proof of (2.3.13), we obtain (2.3.14). Next, it is clear to see that

β 0 |S 1 (u n , v n , η n )| 2 dx = β 0 |au n x + κ 1 v n x + κ 2 η n x (•, 1)| 2 dx ≤ 3a 2 β 0 |u n x | 2 dx + 3κ 2 1 β 0 |v n x | 2 dx + 3κ 2 2 β 0 |η n x (•, 1)| 2 dx.
Finally, passing to the limit in the above estimation, then using (2.3.10), (2.3.12) and (2.3.14), we obtain (2.3.15). The proof is thus complete.

Now, we fix a function

g ∈ C 1 ([α, β]) such that g(α) = -g(β) = 1 and set max x∈[α,β] |g(x)| = M g and max x∈[α,β] |g (x)| = M g . (2.3.21) Remark 2.3.1.
To prove the existence of a function g, we need to find an example. For this aim, we can take

g(x) = 1 + 2(α -x) β -α , then g ∈ C 1 ([α, β]), g(α) = -g(β) = 1, M g = 1 and M g = 2 β -α . Also, we can take g(x) = cos (α -x)π α -β . Lemma 2.3.2. Under the hypothesis (H), the solution U n = (u n , v n , y n , z n , η n (•, ρ)) ∈ D(A) of system (2.3.5)-(2.
3.9) satisfies the following inequalities

|z n (β)| 2 + |z n (α)| 2 ≤ M g β α |z n | 2 dx + 2|λ n |M g β α |z n | 2 dx 1 2 + 2M g F n H , (2.3.22) |y n x (β)| 2 + |y n x (α)| 2 ≤ M g β α |y n x | 2 dx + 2(|λ n | + c 0 )M g β α |y n x | 2 dx 1 2 + 2M g F n H (2.3.23)
and the following limits

lim n→∞ |v n (α)| = 0 and lim n→+∞ |v n (β)| = 0, (2.3.24) lim n→∞ |(S 1 (u n , v n , η n )) (α)| = 0 and lim n→∞ (S 1 (u n , v n , η n )) (β -) = 0. (2.3.25)
Proof. First, from (2.3.7), we deduce that x respectively, integrating over (α, β), using the definition of c(•), then taking the real part, we get

iλ n y n x -z n x = f 3,n x . ( 2
2iλ n β α gy n x z n dx - β α g |z n | 2 x dx = 2 β α gf 3,n x z n dx (2.3.27) and 2iλ n β α gz n y n x dx - β α g |y n x | 2 x dx - 2c 0 β α gv n y n x dx = 2 β α gf 4,n y n x dx .
(2.3.28)

Using integration by parts in (2.3.27) and (2.3.28), we obtain

-g |z n | 2 β α = - β α g |z n | 2 dx - 2iλ n β α gy n x z n dx + 2 β α gf 3,n x z n dx and -g |y n x | 2 β α = - β α g |y n x | 2 dx - 2iλ n β α gz n y n x dx + 2c 0 β α gv n y n x + 2 β α gf 4,n y n x dx .
Using the definition of g and Cauchy-Schwarz inequality in the above equations, we obtain

|z n (β)| 2 + |z n (α)| 2 ≤ M g β α |z n | 2 dx + 2|λ n |M g β α |y n x | 2 dx 1 2 β α |z n | 2 dx 1 2 + 2M g β α |f 3,n x | 2 dx 1 2 β α |z n | 2 dx 1 2 (2.3.29)
and

|y n x (β)| 2 + |y n x (α)| 2 ≤ M g β α |y n x | 2 dx + 2|λ n |M g β α |y n x | 2 dx 1 2 β α |z n | 2 dx 1 2 + 2|c 0 |M g β α |y n x | 2 dx 1 2 β α |v n | 2 dx 1 2 + 2M g β α |f 4,n | 2 dx 1 2 β α |y n x | 2 dx 1 2
.

(2.3.30)

Therefore, from (2.3.29), (2.3.30) and the fact that (2.3.23). On the other hand, from (2.3.5), we deduce that

β α |ξ n 1 | 2 dx ≤ L 0 |ξ n 1 | 2 dx ≤ U n 2 H = 1 with ξ n 1 ∈ {v n , y n x , z n } and β α |ξ n 2 | 2 dx ≤ L 0 |ξ n 2 | 2 dx ≤ F n 2 H with ξ n 2 ∈ {f 3,n x , f 4,n }, we obtain (2.3.22) and
iλ n u n x -v n x = f 1,n x . (2.3.31)
Multiplying (2.3.31) and (2.3.6) by 2gv n and 2gS 1 (u n , v n , η n ) respectively, integrating over (α, β), using the definition of c(•) and S b (u n , v n , η n ), then taking the real part, we get

2iλ n β α gu n x v n dx - β α g(|v n | 2 ) x dx = 2 β α gf 1,n x v n dx (2.3.32) and 2iλ n β α gv n S 1 (u n , v n , η n )dx - β α g |S 1 (u n , v n , η n )| 2 x dx + 2c 0 β α gz n S 1 (u n , v n , η n )dx = 2 β α gf 2,n S 1 (u n , v n , η n )dx . (2.3.33)
Using integration by parts in (2.3.32) and (2.3.33), we get

-g |v n | 2 β α = - β α g |v n | 2 dx - 2iλ n β α gu n x v n dx + 2 β α gf 1,n x v n dx and -g |S 1 (u n , v n , η n )| 2 β α = - β α g |S 1 (u n , v n , η n )| 2 dx - 2iλ n β α gv n S 1 (u n , v n , η n )dx - 2c 0 β α gz n S 1 (u n , v n , η n )dx + 2 β α gf 2,n S 1 (u n , v n , η n )dx .
Using the definition of g and Cauchy-Schwarz inequality in the above equations, then using the fact that

         β α |z n | 2 dx ≤ L 0 |z n | 2 dx ≤ U n 2 H = 1, β α |f 1,n x | 2 dx ≤ L 0 |f 1,n x | 2 dx ≤ a -1 F n 2 H and β α |f 2,n | 2 dx ≤ L 0 |f 2,n | 2 dx ≤ F n 2 H ,
we obtain

|v n (β)| 2 + |v n (α)| 2 ≤ M g β α |v n | 2 dx + 2|λ n |M g β α |u n x | 2 dx 1 2 β α |v n | 2 dx 1 2 + 2 √ a M g β α |v n | 2 dx 1 2 F n H (2.3.34)
and

(S 1 (u n , v n , η n )) (β -) 2 + |(S 1 (u n , v n , η n )) (α)| 2 ≤ M g β α |S 1 (u n , v n , η n )| 2 dx + 2|λ n |M g β α |S 1 (u n , v n , η n )| 2 dx 1 2 β α |v n | 2 dx 1 2 + 2|c 0 |M g β α |S 1 (u n , v n , η n )| 2 dx 1 2 + 2M g β α |S 1 (u n , v n , η n )| 2 dx 1 2
F n H .

( Proof. First, multiplying (2.3.6) by z n , integrating over (α, β), using the definition of c(•) and S b (u n , v n , η n ), then taking the real part, we get

iλ n β α v n z n dx - β α (S 1 (u n , v n , η n )) x z n dx + c 0 β α |z n | 2 dx = β α f 2,n z n dx . (2.3.38)
From (2.3.7), we deduce that

z n x = -iλ n y n x -f 3,n x . (2.3.39)
Using integration by parts to the second term in (2.3.38), then using (2.3.39), we get

c 0 β α |z n | 2 dx = iλ n β α S 1 (u n , v n , η n )y n x dx + β α S 1 (u n , v n , η n )f 3,n x dx + [S 1 (u n , v n , η n ) z n ] β α + β α f 2,n z n dx - iλ n β α v n z n dx .
Using Cauchy-Schwarz inequality in the above equation and the fact that

β α |ξ n 1 | 2 dx ≤ L 0 |ξ n 1 | 2 dx ≤ U n 2 H = 1 with ξ n 1 ∈ {y n x , z n } and β α |ξ n 2 | 2 dx ≤ L 0 |ξ n 2 | 2 dx ≤ F n 2 H with ξ n 2 ∈ {f 2,n , f 3,n x }, we obtain c 0 β α |z n | 2 dx ≤ (|λ n | + F n H ) β α |S 1 (u n , v n , η n )| 2 dx 1 2 + |λ n | β α |v n | 2 dx 1 2 + F n H + (S 1 (u n , v n , η n )) (β -) |z n (β)| + |(S 1 (u n , v n , η n )) (α)| |z n (α)|.
Passing to the limit in the above inequality, then using (2.3.2), (2.3.36), (2.3.25), Lemma 2.3.1 and the fact that F n H → 0, we obtain the first limit in (2.3.37). On the other hand, multiplying (2.3.8) by -z n (λ n ) -1 , integrating over (α, β), using the definition of c(•), then taking the imaginary part, we get

- β α |z n | 2 dx + (λ n ) -1 β α y n xx z n dx + c 0 (λ n ) -1 β α v n z n dx = - (λ n ) -1 β α f 4,n z n dx .
Using integration by parts to the second term in the above equation, then using (2.3.39), we obtain

β α |y n x | 2 dx = β α |z n | 2 dx - (λ n ) -1 β α f 3,n x y n x dx - (λ n ) -1 [y n x z n ] β α - c 0 (λ n ) -1 β α v n z n dx - (λ n ) -1 β α f 4,n z n dx .
Using Cauchy-Schwarz inequality in the above equation and the fact that U n H = 1, we get Proof. First, from (2.3.5) and (2.3.7), we get

β α |y n x | 2 dx ≤ β α |z n | 2 dx + c 0 |λ n | -1 β α |v n | 2 dx 1 2 + 2|λ n | -1 F n H + |λ n | -1 |y n x (β)||z n (β)| + |λ n | -1 |y n x (α)||z n (α)|. ( 2
U n = (u n , v n , y n , z n , η n (•, ρ)) ∈ D(A) of system (2.3.5)-(2.3.9) satisfies the following estimations lim n→∞ |u n (β)| 2 = 0 and lim n→∞ |y n (β)| 2 = 0, (2.3.42) lim n→∞ |u n x (β + )| 2 = 0 and lim n→∞ |y n x (β)| 2 = 0, (2.3.43)
lim n→∞ γ β |u n | 2 dx + γ β |u n x | 2 dx + γ β |y n | 2 dx + γ β |y n x | 2 dx = 0, (2.3.44)
|u n (β)| 2 ≤ 2(λ n ) -2 |v n (β)| 2 + 2(λ n ) -2 |f 1,n (β)| 2 and |y n (β)| 2 ≤ 2(λ n ) -2 |z n (β)| 2 + 2(λ n ) -2 |f 3,n (β)| 2 .
Using the fact that

|f 1,n (β)| 2 ≤ β β 0 |f 1,n x | 2 dx ≤ βa -1 F n 2 H and |f 3,n (β)| 2 ≤ β β 0 |f 3,n x | 2 dx ≤ β F n 2
H in the above inequalities, we obtain

|u n (β)| 2 ≤ 2(λ n ) -2 |v n (β)| 2 + 2βa -1 (λ n ) -2 F n 2 H and |y n (β)| 2 ≤ 2(λ n ) -2 |z n (β)| 2 + 2β(λ n ) -2 F n 2 H .
Passing to the limit in the above inequalities, then using (2.3.2), (2.3.24), (2.3.41) and the fact that

F n H → 0, we obtain (2.3.42). Secondly, since S b (u n , v n , η n ) ∈ H 1 (0, L) ⊂ C([0, L]), then we deduce that (S 1 (u n , v n , η n )) (β -) 2 = |au n x (β + )| 2 . (2.3.46)
Thus, from (2.3.25) and (2.3.46), we obtain the first limit in (2.3.43). Moreover, passing to the limit in inequality (2.3.23), then using (2.3.2), the second limit in (2.3.37) and the fact that F n H → 0, we obtain the second limit in (2.3.43). On the other hand, (2.3.5)-(2.3.8) can be written in (β, γ) as the following form

(λ n ) 2 u n + au n xx -iλ n c 0 y n = G 1,n in (β, γ), (2.3.47) (λ n ) 2 y n + y n xx + iλ n c 0 u n = G 2,n in (β, γ), (2.3.48)
where

G 1,n = -f 2,n -iλ n f 1,n -c 0 f 3,n and G 2,n = -f 4,n -iλ n f 3,n + c 0 f 1,n . (2.3.49) Let V n = (u n , u n x , y n , y n x ) , then (2.3.47)-(2.3.48
) can be written as the following

V n x = B n V n + G n , (2.3.50)
where

B n =     0 1 0 0 -a -1 (λ n ) 2 0 a -1 iλ n c 0 0 0 0 0 1 -iλ n c 0 0 -(λ n ) 2 0     = (b ij ) 1≤i,j≤4 and G n =     0 a -1 G 1,n 0 G 2,n     .
The solution of the differential equation (2.3.50) is given by

V n (x) = e B n (x-β) V n (β + ) + x β e B n (s-x) G n (s)ds, (2.3.51)
where e B n (x-β) = (c ij ) 1≤i,j≤4 and e B n (s-x) = (d ij ) 1≤i,j≤4 are denoted by the exponential of the matrices B n (x -β) and B n (s -x) respectively. Now, from (2.3.2), the entries b ij are bounded for all 1 ≤ i, j ≤ 4 and consequently, the entries b ij (x -β) and b ij (s -x) are bounded. In addition, from the definition of the exponential of a square matrix, we obtain

e B n ζ = ∞ k=0 (B n ζ) k k! for ζ ∈ {x -β, s -x}.
Therefore, the entries c ij and d ij are also bounded for all 1 ≤ i, j ≤ 4 and consequently, e B n (x-β) and e B n (s-x) are two bounded matrices. From (2.3.42) and (2.3.43), we directly obtain

V n (β + ) → 0 in (L 2 (β, γ)) 4 , as n → ∞. (2.3.52)
Moreover, from (2.3.49), we deduce that

γ β |G 1,n | 2 dx ≤ 3 L 0 |f 2,n | 2 dx + 3(λ n ) 2 L 0 |f 1,n | 2 dx + 3c 2 0 L 0 |f 3,n | 2 dx (2.3.53) and γ β |G 2,n | 2 dx ≤ 3 L 0 |f 4,n | 2 dx + 3(λ n ) 2 L 0 |f 3,n | 2 dx + 3c 2 0 L 0 |f 1,n | 2 dx. (2.3.54)
Now, since f 1,n , f 3,n ∈ H 1 0 (0, L), then it follows from Poincaré inequality that there exist two constants C 1 > 0 and C 2 > 0 such that

f 1,n L 2 (0,L) ≤ C 1 f 1,n x L 2 (0,L) and f 3,n L 2 (0,L) ≤ C 2 f 3,n x L 2 (0,L) . ( 2
γ β |G 1,n | 2 dx ≤ 3 1 + a -1 (λ n C 1 ) 2 + (c 0 C 2 ) 2 F n 2 H , (2.3.56) and γ β |G 2,n | 2 dx ≤ 3 1 + (λ n C 1 ) 2 + a -1 (c 0 C 2 ) 2 F n 2 H . ( 2
F n H → 0, we obtain G n → 0 in (L 2 (β, γ)) 4 , as n → ∞. ( 2
γ β |v n | 2 dx ≤ 2(λ n ) 2 γ β |u n | 2 dx + 2 γ β |f 1,n | 2 dx ≤ 2(λ n ) 2 γ β |u n | 2 dx + 2C 1 a -1 F n 2 H , γ β |z n | 2 dx ≤ 2(λ n ) 2 γ β |y n | 2 dx + 2 γ β |f 3,n | 2 dx ≤ 2(λ n ) 2 γ β |y n | 2 dx + 2C 2 F n 2 H .
Finally, passing to the limit in the above inequalities, then using (2.3.2), (2.3.44) and the fact that F n H → 0, we obtain (2.3.45). The proof is thus complete. Lemma 2.3.5. Let h ∈ C 1 ([0, L]) be a function. Under the hypothesis (H), the solution

U n = (u n , v n , y n , z n , η n (•, ρ)) ∈ D(A) of system (2.3.5)-(2.3.9) satisfies the following equality L 0 h a -1 |S b (u n , v n , η n )| 2 + |v n | 2 + |z n | 2 + |y n x | 2 dx -h a -1 |S b (u n , v n , η n )| 2 + |y n x | 2 L 0 - 2 L 0 c(•)hv n y n x dx + 2 a L 0 c(•)hz n S b (u n , v n , η n )dx + 2iλ n a β 0 hv n (κ 1 v n x + κ 2 η n x (•, 1))dx = 2 L 0 hf 1,n x v n dx + 2 a L 0 hf 2,n S b (u n , v n , η n )dx + 2 L 0 hf 3,n x z n dx + 2 L 0 hf 4,n y n x dx .
Proof. First, multiplying (2.3.6) and (2.3.8) by 2a -1 hS b (u n , v n , η n ) and 2hy n x respectively, integrating over (0, L), then taking the real part, we get

2iλ n a L 0 hv n S b (u n , v n , η n )dx -a -1 L 0 h |S b (u n , v n , η n )| 2 x dx + 2 a L 0 c(•)hz n S b (u n , v n , η n )dx = 2 a L 0 hf 2,n S b (u n , v n , η n )dx (2.3.59) and 2iλ n L 0 hz n y n x dx - L 0 h |y n x | 2 x dx - 2 L 0 c(•)hv n y n x dx = 2 L 0 hf 4,n y n x dx .
(2.3.60) From (2.3.5) and (2.3.7), we deduce that

iλ n u n x = -v n x -f 1,n x , (2.3.61) iλ n y n x = -z n x -f 3,n x . (2.3.62)
Consequently, from (2.3.61) and the definition S b (u n , v n , η n ), we have

iλ n S b (u n , v n , η n ) =    -a v n x + f 1,n x + iλ n (κ 1 v n x + κ 2 η n x (•, 1)) , in (0, β), -a v n x + f 1,n x , in (β, L).
(

- L 0 h |v n | 2 + a -1 |S b (u n , v n , η n )| 2 x dx + 2iλ n a β 0 hv n (κ 1 v n x + κ 2 η n x (•, 1))dx + 2 a L 0 c(•)hz n S b (u n , v n , η n )dx = 2 L 0 hf 1,n x v n dx + 2 a L 0 hf 2,n S b (u n , v n , η n )dx and - L 0 h |z n | 2 + |y n x | 2 x dx - 2 L 0 c(•)hv n y n x dx = 2 L 0 hf 4,n y n x dx + 2 L 0 hf 3,n x z n dx .
Finally, adding the above equations, then using integration by parts and the fact that v n (0) = v n (L) = 0 and z n (0) = z n (L) = 0, we obtain the desired result. The proof is thus complete. Now, we fix the cut-off functions

χ 1 , χ 2 ∈ C 1 ([0, L]) (see Figure 2.2) such that 0 ≤ χ 1 (x) ≤ 1, 0 ≤ χ 2 (x) ≤ 1, for all x ∈ [0, L] and χ 1 (x) = 1 if x ∈ [0, α], 0 if x ∈ [β, L],
and

χ 2 (x) = 0 if x ∈ [0, β], 1 if x ∈ [γ, L],
and set max

x∈[0,L] |χ 1 (x)| = M χ 1 and max x∈[0,L] |χ 2 (x)| = M χ 2 ,
Lemma 2.3.6. Under the hypothesis (H), the solution

U n = (u n , v n , y n , z n , η n (•, ρ)) ∈ D(A)
of system (2.3.5)-(2.3.9) satisfies the following limits Proof. First, using the result of Lemma 2.3.5 with h = xχ 1 , then using the definition of c(•), S b (u n , v n , η n ) and χ 1 , we get

lim n→∞ α 0 |y n x | 2 dx + α 0 |z n | 2 dx = 0, (2.3.64) lim n→∞ a L γ |u n x | 2 dx + L γ |v n | 2 dx + L γ |y n x | 2 dx + L γ |z n | 2 dx = 0. (2.3.65) 0 α β γ L 1 χ 1 χ 2
α 0 |y n x | 2 dx + α 0 |z n | 2 dx = - α 0 |v n | 2 dx -a -1 α 0 |S 1 (u n , v n , η n )| 2 dx - β α (χ 1 + xχ 1 ) a -1 |S 1 (u n , v n , η n )| 2 + |v n | 2 + |y n x | 2 + |z n | 2 dx - 2c 0 a β α xχ 1 z n S 1 (u n , v n , η n )dx + 2c 0 β α xχ 1 v n y n x dx - 2iλ n a β 0 xχ 1 v n (κ 1 v n x + κ 2 η n x (•, 1)) dx + 2 a β 0 xχ 1 f 2,n S 1 (u n , v n , η n )dx + 2 L 0 xχ 1 f 1,n x v n + f 3,n x z n + f 4,n y n x dx .
Using Cauchy-Schwarz inequality in the above equation and the fact that U n H = 1, we obtain

α 0 |y n x | 2 dx + α 0 |z n | 2 dx ≤ α 0 |v n | 2 dx + a -1 α 0 |S 1 (u n , v n , η n )| 2 dx + 1 + βM χ 1 β α a -1 |S 1 (u n , v n , η n )| 2 + |v n | 2 + |z n | 2 + |y n x | 2 dx + 2|c 0 |β a β α |S 1 (u n , v n , η n )| 2 dx 1 2 + 2|c 0 |β β α |v n | 2 dx 1 2 + 2β|λ n | a κ 1 β 0 |v n x | 2 dx 1 2 + |κ 2 | β 0 |η n x (•, 1)| 2 dx 1 2 + 2β a β 0 |S 1 (u n , v n , η n )| 2 dx 1 2 F n H + 2L 1 √ a + 2 F n H .
Passing to the limit in the above inequality, then using (2.3.2), Lemmas 2.3.1, 2.3.3 and the fact that F n H → 0, we obtain (2.3.64). On the other hand, using the result of Lemma 2.3.5 with h = (x -L)χ 2 , then using Cauchy-Schwarz inequality and the fact that

U n H = 1, we get a L γ |u n x | 2 + L γ |v n | 2 dx + L γ |y n x | 2 dx + L γ |z n | 2 dx ≤ 1 + (L -β)M χ 2 γ β a|u n x | 2 + |v n | 2 + |y n x | 2 + |z n | 2 dx + 2|c 0 |(L -β) γ β |v n | 2 dx 1 2 γ β |y n x | 2 dx 1 2 + 2|c 0 |(L -β) γ β |z n | 2 dx 1 2 γ β |u n x | 2 dx 1 2 + 4L 1 √ a + 1 F n H .
Finally, passing to the limit in the above inequality, then using Lemma 2.3.4 and the fact that

F n H → 0,

Polynomial Stability

In this section, we will prove the polynomial stability of system (2.2.2)-(2.2.7). The main result of this section is the following theorem.

Theorem 2.4.1. Under the hypothesis (H), for all U 0 ∈ D(A), there exists a constant C > 0 independent of U 0 such that the energy of system (2.2.

1 |λ| 2 (iλI -A) -1 L(H) < ∞. ( 2
:= (u n , v n , y n , z n , η n (•, ρ)) ) n≥1 ⊂ R * × D(A) with |λ n | → ∞ as n → ∞ and U n H = (u n , v n , y n , z n , η n (•, ρ)) H = 1, ∀n ≥ 1, (2.4.3) such that (λ n ) 2 (iλ n I-A)U n = F n := (f 1,n , f 2,n , f 3,n , f 4,n , f 5,n (•, ρ)) → 0 in H, as n → ∞. (2.4.4)
For simplicity, we drop the index n. Equivalently, from (2.4.4), we have

iλu -v = λ -2 f 1 , f 1 → 0 in H 1 0 (0, L), (2.4.5) iλv -(S b (u, v, η)) x + c(•)z = λ -2 f 2 , f 2 → 0 in L 2 (0, L), (2.4.6) iλy -z = λ -2 f 3 , f 3 → 0 in H 1 0 (0, L), (2.4.7) iλz -y xx -c(•)v = λ -2 f 4 , f 4 → 0 in L 2 (0, L), (2.4.8) iλη(•, ρ) + τ -1 η ρ (•, ρ) = λ -2 f 5 (•, ρ), f 5 (•, ρ) → 0 in W.
(2.4.9)

Here we will check the condition (2.4.2) by finding a contradiction with (2.4.3) such as U H = o(1). For clarity, we divide the proof into several Lemmas.

Lemma 2.4.1. Under the hypothesis (H), the solution U = (u, v, y, z, η(•, ρ)) ∈ D(A) of system (2.4.5)-(2.4.9) satisfies the following estimations

β 0 |v x | 2 dx = o(λ -2 ), (2.4.10) β 0 |u x | 2 dx = o(λ -4 ), (2.4.11) β 0 1 0 |η x (•, ρ)| 2 dρdx = o(λ -2 ), (2.4.12) β 0 |η x (•, 1)| 2 dx = o(λ -2 ), (2.4.13)
β 0 |S 1 (u, v, η)| 2 dx = o(λ -2 ). (2.4.14)
Proof. First, taking the inner product of (2.4.4) with U in H and using (2.2.18) with the help of hypothesis (H), we obtain

β 0 |v x | 2 dx ≤ - 1 κ 1 -|κ 2 | (AU, U ) H = λ -2 κ 1 -|κ 2 | (F, U ) H ≤ λ -2 κ 1 -|κ 2 | F H U H . (2.4.15)
Thus, from (2.4.15) and the fact that F H = o(1) and U H = 1, we obtain (2.4.10). Now, from (2.4.5), we deduce that

β 0 |u x | 2 dx ≤ 2λ -2 β 0 |v x | 2 dx + 2λ -4 β 0 |f 1 x | 2 dx ≤ 2λ -2 β 0 |v x | 2 dx + 2λ -4 L 0 |f 1 x | 2 dx . (2.4.16)
Therefore, from (2.4.10), (2.4.16) and the fact that f 1 x L 2 (0,L) = o(1), we obtain (2.4.11). Next, from (2.4.9) and the fact that η(•, 0) = v(•) , we get

η(x, ρ) = ve -iλτ ρ + τ λ -2 ρ 0 e iλτ (s-ρ) f 5 (x, s)ds, (x, ρ) ∈ (0, β) × (0, 1).
(2.4.17)

From (2.4.17), we deduce that

β 0 1 0 |η x (•, ρ)| 2 dρdx ≤ 2 β 0 |v x | 2 dx + τ 2 λ -4 β 0 1 0 |f 5 x (•, s)| 2 dsdx. (2.4.18)
Thus, from (2.4.10), (2.4.18) and the fact that f 5 (•, ρ) → 0 in W, we obtain (2.4.12). On the other hand, from (2.4.17), we have

η x (•, 1) = v x e -iλτ + τ λ -2 1 0 e iλτ (s-1) f 5 x (•, s)ds,
consequently, similar to the previous proof, we obtain (2.4.13). Next, it is clear to see that

β 0 |S 1 (u, v, η)| 2 dx = β 0 |au x + κ 1 v x + κ 2 η x (•, 1)| 2 dx ≤ 3a 2 β 0 |u x | 2 dx + 3κ 2 1 β 0 |v x | 2 dx + 3κ 2 2 β 0 |η x (•, 1)| 2 dx.
Finally, from (2.4.10), (2.4.11), (2.4.13) and the above estimation, we obtain (2.4.14). The proof is thus complete. (2.4.19)

0 ε 2ε α α + ε β -3ε β -2ε β -ε β γ L 1 θ 1 θ 2 θ 3
Proof. First, we fix a cut-off function

θ 1 ∈ C 1 ([0, L]) (see Figure 2.3) such that 0 ≤ θ 1 (x) ≤ 1, for all x ∈ [0, L] and θ 1 (x) = 1 if x ∈ [ε, β -ε], 0 if x ∈ {0} ∪ [β, L],
and set max

x∈[0,L] |θ 1 (x)| = M θ 1 .
Multiplying (2.4.6) by λ -1 θ 1 v, integrating over (0, L), then taking the imaginary part, we obtain

L 0 θ 1 |v| 2 dx - λ -1 L 0 θ 1 (S b (u, v, η)) x vdx + λ -1 L 0 c(•)θ 1 zvdx = λ -3 L 0 θ 1 f 2 vdx .
Using integration by parts in the above equation and the fact that v(0) = v(L) = 0, we get

L 0 θ 1 |v| 2 dx = - 1 λ L 0 (θ 1 v + θ 1 v x )S b (u, v, η)dx - 1 λ L 0 c(•)θ 1 zvdx + 1 λ 3 L 0 θ 1 f 2 vdx .
(2.4.20)

Using the definition of c(•), S b (u, v, η) and θ 1 , then using Cauchy-Schwarz inequality, we obtain

λ -1 L 0 (θ 1 v + θ 1 v x )S b (u, v, η)dx = λ -1 β 0 (θ 1 v + θ 1 v x )S 1 (u, v, η)dx ≤ |λ| -1 M θ 1 β 0 |v| 2 dx 1 2 + β 0 |v x | 2 dx 1 2 β 0 |S 1 (u, v, η)| 2 dx 1 2
and

λ -1 L 0 c(•)θ 1 zvdx = c 0 λ -1 β α θ 1 zvdx ≤ |c 0 ||λ| -1 β α |z| 2 dx 1 2 β α |v| 2 dx 1 2
.

From the above inequalities, Lemma 2.4.1 and the fact that v and z are uniformly bounded in L 2 (0, L), we obtain

         - λ -1 L 0 (θ 1 v + θ 1 v x )S b (u, v, η)dx = o(λ -2 ), - λ -1 L 0 c(•)θ 1 zvdx = O(|λ| -1 ) = o(1).
(2.4.21)

Inserting (2.4.21) in (2.4.20), then using the fact that v is uniformly bounded in L 2 (0, L) and

f 2 L 2 (0,L) = o(1), we obtain L 0 θ 1 |v| 2 dx = o(1).
Finally, from the above estimation and the definition of θ 1 , we obtain (2.4.19). The proof is thus complete. (2.4.22)

Proof. First, we fix a cut-off function

θ 2 ∈ C 1 ([0, L]) (see figure 2.3) such that 0 ≤ θ 2 (x) ≤ 1, for all x ∈ [0, L] and θ 2 (x) = 0 if x ∈ [0, ε] ∪ [β -ε, L], 1 if x ∈ [2ε, β -2ε],
and set max

x∈[0,L] |θ 2 (x)| = M θ 2 .
Multiplying (2.4.6) and (2.4.8) by θ 2 z and θ 2 v respectively, integrating over (0, L), then taking the real part, we obtain

iλ L 0 θ 2 vzdx - L 0 θ 2 (S b (u, v, η)) x zdx + L 0 c(•)θ 2 |z| 2 dx = λ -2 L 0 θ 2 f 2 zdx (2.4.23)
and

iλ L 0 θ 2 zvdx - L 0 θ 2 y xx vdx - L 0 c(•)θ 2 |v| 2 dx = λ -2 L 0 θ 2 f 4 vdx .
(2.4.24)

Adding (2.4.23) and (2.4.24), then using integration by parts and the fact that v(0) = v(L) = 0 and z(0) = z(L) = 0, we get

L 0 c(•)θ 2 |z| 2 dx = L 0 c(•)θ 2 |v| 2 dx - L 0 (θ 2 z + θ 2 z x )S b (u, v, η)dx - L 0 (θ 2 v + θ 2 v x )y x dx + λ -2 L 0 θ 2 f 2 zdx + λ -2 L 0 θ 2 f 4 vdx .
(2.4.25)

From (2.4.7), we deduce that

z x = -iλy x -λ -2 f 3 x . (2.4.26)
Using (2.4.26) and the definition of S b (u, v, η) and θ 2 , then using Cauchy-Schwarz inequality, we obtain

L 0 (θ 2 z + θ 2 z x )S b (u, v, η)dx = β-ε ε θ 2 z + θ 2 (-iλy x -λ -2 f 3 x ) S 1 (u, v, η)dx ≤ M θ 2 β-ε ε |z| 2 dx 1 2 + |λ| β-ε ε |y x | 2 dx 1 2 +λ -2 β-ε ε |f 3 x | 2 dx 1 2 β-ε ε |S 1 (u, v, η)| 2 dx 1 2 and L 0 (θ 2 v + θ 2 v x )y x dx = β-ε ε (θ 2 v + θ 2 v x )y x dx ≤ M θ 2 β-ε ε |v| 2 dx 1 2 + β-ε ε |v x | 2 dx 1 2 β-ε ε |y x | 2 dx 1 2
.

From the above inequalities, Lemmas 2.4.1, 2.4.2 and the fact that y x , z are uniformly bounded in L 2 (0, L) and f 3 x L 2 (0,L) = o(1), we obtain

- L 0 (θ 2 z + θ 2 z x )S b (u, v, η)dx = o(1) and - L 0 (θ 2 v + θ 2 v x )y x dx = o(1). (2.4.27)
Inserting (2.4.27) in (2.4.25), then using the fact that v, z are uniformly bounded in L 2 (0, L) and

f 2 L 2 (0,L) = o(1), f 4 L 2 (0,L) = o(1), we obtain L 0 c(•)θ 2 |z| 2 dx = L 0 c(•)θ 2 |v| 2 dx + o(1).
Therefore, from the above estimation, Lemma 2.4.2 and the definition of c(•) and θ 2 , we obtain the first estimation in (2.4.22). On the other hand, let us fix a cut-off function θ 3 ∈ C 1 ([0, L]) (see Figure 2.3) such that 0 ≤ θ 3 (x) ≤ 1, for all x ∈ [0, L] and

θ 3 (x) = 0 if x ∈ [0, α] ∪ [β -2ε, L], 1 if x ∈ [α + ε, β -3ε],
Now, multiplying (2.4.8) by -λ -1 θ 3 z, integrating over (0, L), then taking the imaginary part, we obtain

- L 0 θ 3 |z| 2 dx + λ -1 L 0 θ 3 y xx zdx + λ -1 L 0 c(•)θ 3 vzdx = - λ -3 L 0 θ 3 f 4 zdx .
Using integration by parts in the above equation and the fact that z(0) = z(L) = 0, then using (2.4.26), we get

L 0 θ 3 |y x | 2 dx = L 0 θ 3 |z| 2 dx + λ -1 L 0 θ 3 y x zdx - λ -1 L 0 c(•)θ 3 vzdx - λ -3 L 0 θ 3 f 3 x y x dx - λ -3 L 0 θ 3 f 4 zdx . (2.4.28)
From the definition of c(•) and θ 3 , the first estimation of (2.4.22) and the fact that v and y x are uniformly bounded in L 2 (0, L), we obtain

         λ -1 L 0 θ 3 y x zdx = λ -1 β-2ε α θ 3 y x zdx = o |λ| -1 , - λ -1 L 0 c(•)θ 3 vzdx = - c 0 λ -1 β-2ε α θ 3 vzdx = o |λ| -1 .
(2.4.29)

Inserting (2.4.29) in (2.4.28), then using the fact that y x , z are uniformly bounded in L 2 (0, L) and

f 3 x L 2 (0,L) = o(1), f 4 L 2 (0,L) = o(1), we get L 0 θ 3 |y x | 2 dx = L 0 θ 3 |z| 2 dx + o(|λ| -1 ).
Finally, from the above estimation, the first estimation of (2.4.22) and the definition of θ 3 , we obtain the second estimation in (2.4.22). The proof is thus complete.

Lemma 2.4.4. 0 < ε < min α 2 , β-α 4
. Under the hypothesis (H), the solution U = (u, v, y, z, η(•, ρ)) ∈ D(A) of system (2.4.5)-(2.4.9) satisfies the following estimations

|v(γ)| 2 + |v(β -3ε)| 2 + a|u x (γ)| 2 + a -1 | (S 1 (u, v, η)) (β -3ε)| 2 = O(1), (2.4.30) |z(γ)| 2 + |z(β -3ε)| 2 + |y x (γ)| 2 + |y x (β -3ε)| 2 = O(1).
(2.4.31)

Proof. First, we fix a function

g 2 ∈ C 1 ([β -3ε, γ]) such that g 2 (β -3ε) = -g 2 (γ) = 1 and set max x∈[β-3ε,γ] |g 2 (x)| = M g 2 and max x∈[β-3ε,γ] |g 2 (x)| = M g 2 .
From (2.4.5), we deduce that

iλu x -v x = λ -2 f 1 x . ( 2
g 2 u x vdx - γ β-3ε g 2 |v| 2 x dx = 2λ -2 γ β-3ε g 2 f 1 x vdx and 2iλ γ β-3ε g 2 vu x dx + 2iλ a β β-3ε g 2 v (κ 1 v x + κ 2 η x (•, 1)) dx -a -1 β β-3ε g 2 |S 1 (u, v, η)| 2 x dx -a γ β g 2 |u x | 2 x dx + 2c 0 a β β-3ε g 2 zS 1 (u, v, η)dx + 2c 0 γ β g 2 zu x dx = 2 aλ 2 β β-3ε g 2 f 2 S 1 (u, v, η)dx + 2 λ 2 γ β g 2 f 2 u x dx .
Adding the above equations, then using integration by parts, we get

-g 2 |v| 2 γ β-3ε + -a -1 g 2 |S 1 (u, v, η)| 2 β β-3ε + -ag 2 |u x | 2 γ β = - γ β-3ε g 2 |v| 2 dx -a -1 β β-3ε g 2 |S 1 (u, v, η)| 2 dx -a γ β g 2 |u x | 2 dx - 2iλ a β β-3ε g 2 v (κ 1 v x + κ 2 η x (•, 1)) dx - 2c 0 a β β-3ε g 2 zS 1 (u, v, η)dx - 2c 0 γ β g 2 zu x dx + 2 λ 2 γ β-3ε g 2 f 1 x vdx + 2 aλ 2 β β-3ε g 2 f 2 S 1 (u, v, η)dx + 2 λ 2 γ β g 2 f 2 u x dx . |v(γ)| 2 + |v(β -3ε)| 2 + a|u x (γ)| 2 + a -1 |(S 1 (u, v, η)) (β -3ε)| 2 + K(β) ≤ M g 2 γ β-3ε |v| 2 dx + a -1 β β-3ε |S 1 (u, v, η)| 2 dx + a γ β |u x | 2 dx + 2|λ|M g 2 a κ 1 β β-3ε |v x | 2 dx 1 2 + |κ 2 | β β-3ε |η x (•, 1)| 2 dx 1 2 β β-3ε |v| 2 dx 1 2 + 2|c 0 |M g 2 a β β-3ε |S 1 (u, v, η)| 2 dx 1 2 β β-3ε |z| 2 dx 1 2 + 2|c 0 |M g 2 γ β |z| 2 dx 1 2 γ β |u x | 2 dx 1 2 + 2M g 2 λ 2 γ β-3ε |f 1 x | 2 dx 1 2 γ β-3ε |v| 2 dx 1 2 + 2M g 2 aλ 2 β β-3ε |f 2 | 2 dx 1 2 β β-3ε |S 1 (u, v, η)| 2 dx 1 2 + 2M g 2 λ 2 γ β |f 2 | 2 dx 1 2 γ β |u x | 2 dx 1 2
.

where

K(β) = g 2 (β) (a|u x (β + )| 2 -a -1 | (S 1 (u, v, η)) (β -)| 2 ). Moreover, since S b (u, v, η) ∈ H 1 (0, L) ⊂ C([0, L]), then we obtain | (S 1 (u, v, η)) (β -)| 2 = |au x (β + )| 2 and consequently K(β) = 0. (2.4.33)
Inserting (2.4.33) in the above inequality, then using Lemma 2.4.1 and the fact that u x , v, z are uniformly bounded in L 2 (0, L) and

f 1 x L 2 (0,L) = o(1), f 2 L 2 (0,L) = o(1)
, we obtain (2.4.30). Next, from (2.4.7), we deduce that

iλy x -z x = λ -2 f 3 x . ( 2
g 2 y x zdx - γ β-3ε g 2 |z| 2 x dx = 2λ -2 γ β-3ε g 2 f 3 x zdx (2.4.35) and 2iλ γ β-3ε g 2 zy x dx - γ β-3ε g 2 |y x | 2 x dx - 2c 0 γ β-3ε g 2 vy x dx = 2λ -2 γ β-3ε g 2 f 4 y x dx .
(2.4.36)

Adding (2.4.35) and (2.4.36), then using integration by parts, we obtain

-g 2 |z| 2 + |y x | 2 γ β-3ε = - γ β-3ε g 2 (|z| 2 + |y x | 2 )dx + 2c 0 γ β-3ε g 2 vy x dx + 2λ -2 γ β-3ε g 2 f 3 x zdx + 2λ -2 γ β-3ε g 2 f 4 y x dx . |z(γ)| 2 + |z(β -3ε)| 2 + |y x (γ)| 2 + |y x (β -3ε)| 2 ≤ M g 2 γ β-3ε |z| 2 + |y x | 2 dx + 2|c 0 |M g 2 γ β-3ε |v| 2 dx 1 2 γ β-3ε |y x | 2 dx 1 2 + 2λ -2 M g 2 γ β-3ε |f 3 x | 2 dx 1 2 γ β-3ε |z| 2 dx 1 2 + γ β-3ε |f 4 | 2 dx 1 2 γ β-3ε |y x | 2 dx 1 2
.

Finally, from the above inequality, the fact that v, y x , z are uniformly bounded in L 2 (0, L) and

f 3 x L 2 (0,L) = o(1), f 4 L 2 (0,L) = o(1)
L 0 h 2 a -1 |S b (u, v, η)| 2 + |v| 2 + |z| 2 + |y x | 2 dx -h 2 a -1 |S b (u, v, η)| 2 + |y x | 2 L 0 - 2 L 0 c(•)h 2 vy x dx + 2 a L 0 c(•)h 2 zS b (u, v, η)dx + 2iλ a β 0 h 2 v n (κ 1 v x + κ 2 η x (•, 1))dx = 2 λ 2 L 0 h 2 f 1 x vdx + 2 aλ 2 L 0 h 2 f 2 S b (u, v, η)dx + 2 λ 2 L 0 h 2 f 3 x zdx + 2 λ 2 L 0 h 2 f 4 y x dx .
Proof. See the proof of Lemma 2.3.5.

Let 0 < ε < min α 2 , β-α 4 , we fix the cut-off functions θ 4 , θ 5 ∈ C 1 ([0, L]) (see Figure 2.4) such that 0 ≤ θ 4 (x) ≤ 1, 0 ≤ θ 5 (x) ≤ 1, for all x ∈ [0, L] and θ 4 (x) = 1 if x ∈ [0, α + ε], 0 if x ∈ [β -3ε, L],
and . Under the hypothesis (H), the solution U = (u, v, y, z, η(•, ρ)) ∈ D(A) of system (2.4.5)-(2.4.9) satisfies the following estimations

θ 5 (x) = 0 if x ∈ [0, α + ε], 1 if x ∈ [β -3ε, L], 0 α α + ε β -3ε β γ L 1 θ 4 θ 5
α+ε 0 |v| 2 dx + α+ε 0 |y x | 2 dx + α+ε 0 |z| 2 dx = o(1), (2.4.37) a L β |u x | 2 dx + L β-3ε |v| 2 dx + L β-3ε |y x | 2 dx + L β-3ε |z| 2 dx = o(1). (2.4.38)
Proof. First, using the result of Lemma 2.4.5 with h 2 = xθ 4 , we obtain

α+ε 0 |v| 2 dx + α+ε 0 |y x | 2 dx + α+ε 0 |z| 2 dx = -a -1 α+ε 0 |S 1 (u, v, η)| 2 dx - β-3ε α+ε (θ 4 + xθ 4 ) a -1 |S 1 (u, v, η)| 2 + |v| 2 + |y x | 2 + |z| 2 dx + 2 L 0 xc(•)θ 4 vy x dx - 2 a L 0 xc(•)θ 4 zS b (u, v, η)dx - 2iλ a β 0 xθ 4 v (κ 1 v x + κ 2 η x (•, 1)) dx + 2 λ 2 L 0 xθ 4 f 1 x vdx + 2 aλ 2 L 0 xθ 4 f 2 S b (u, v, η)dx + 2 λ 2 L 0 xθ 4 f 3 x zdx + 2 λ 2 L 0 xθ 4 f 4 y x dx .
From the above equation, Lemmas 2.4.1-2.4.3 and the fact that v, y x , z are uniformly bounded in L 2 (0, L) and

f 1 x L 2 (0,L) = o(1), f 3 x L 2 (0,L) = o(1), f 4 L 2 (0,L) = o(1), we obtain α+ε 0 |v| 2 dx + α+ε 0 |y x | 2 dx + α+ε 0 |z| 2 dx = 2 L 0 xc(•)θ 4 vy x dx - 2 a L 0 xc(•)θ 4 zS b (u, v, η)dx + 2 aλ 2 L 0 xθ 4 f 2 S b (u, v, η)dx - 2iλ a β 0 xθ 4 v (κ 1 v x + κ 2 η x (•, 1)) dx + o(1).
(2.4.39)

Using the definition of c(•), S b (u, v, η) and θ 4 , then using Cauchy-Schwarz inequality, we obtain

                                                                                               2 L 0 xc(•)θ 4 vy x dx = 2c 0 β-3ε α xθ 4 vy x dx ≤ 2|c 0 |(β -3ε) β-3ε α |v| 2 dx 1 2 β-3ε α |y x | 2 dx 1 2 , 2 a L 0 xc(•)θ 4 zS b (u, v, η)dx = 2c 0 a β-3ε α xθ 4 zS 1 (u, v, η)dx ≤ 2|c 0 | a (β -3ε) β-3ε α |z| 2 dx 1 2 β-3ε α |S 1 (u, v, η)| 2 dx 1 2 , 2 aλ 2 L 0 xθ 4 f 2 S b (u, v, η)dx = 2 aλ 2 β-3ε 0 xθ 4 f 2 S 1 (u, v, η)dx ≤ 2(β -3ε) aλ 2 β-3ε 0 |f 2 | 2 dx 1 2 β-3ε 0 |S 1 (u, v, η)| 2 dx 1 2 , 2iλ a β 0 xθ 4 v (κ 1 v x + κ 2 η x (•, 1)) dx = 2iλ a β-3ε 0 xθ 4 v (κ 1 v x + κ 2 η x (•, 1)) dx ≤ 2|λ|(β -3ε) a κ 1 β-3ε 0 |v x | 2 dx 1 2 + |κ 2 | β-3ε 0 |η x (•, 1)| 2 dx 1 2 β-3ε 0 |v| 2 dx 1 2
.

From the above inequalities, Lemmas 2.4.1-2.4.3 and the fact that v, y x are uniformly bounded in L 2 (0, L) and f 2 L 2 (0,L) = o(1), we obtain

                             2 L 0 xc(•)θ 4 vy x dx = o(1), - 2 a L 0 xc(•)θ 4 zS b (u, v, η)dx = o(|λ| -1 ), 2 aλ 2 L 0 xθ 4 f 2 S b (u, v, η)dx = o(|λ| -3 ), - 2iλ a β 0 xθ 4 v (κ 1 v x + κ 2 η x (•, 1)) dx = o(1).
(2.4.40)

Therefore, by inserting (2.4.40) in (2.4.39), we obtain (2.4.37). On the other hand, using the result of Lemma 2.4.5 with h = (x -L)θ 5 , then using the definition of S b (u, v, η) and θ 5 , Lemmas 2.4.1-2.4.3, and the fact that u x , v, y x , z are uniformly bounded in L 2 (0, L) and

f 1 x L 2 (0,L) = o(1), f 2 L 2 (0,L) = o(1), f 3 x L 2 (0,L) = o(1), f 4 L 2 (0,L) = o(1), we obtain a L β |u x | 2 dx + L β-3ε |v| 2 dx + L β-3ε |y x | 2 dx + L β-3ε |z| 2 dx = I + o(1), (2.4.41)
where

I := 2 L 0 (x -L)c(•)θ 5 vy x dx - 2a -1 L 0 (x -L)c(•)θ 5 zS b (u, v, η)dx .
Moreover, using the definition of c(•), S b (u, v, η) and θ 5 , we get

I = 2c 0 β-3ε α+ε (x -L)θ 5 vy x dx + 2c 0 γ β-3ε (x -L)vy x dx - 2c 0 a β-3ε α+ε (x -L)θ 5 zS 1 (u, v, η)dx - 2c 0 γ β-3ε (x -L)zu x dx - 2c 0 a β β-3ε (x -L)z(κ 1 v x + κ 2 η x (•, 1))dx .
Using Cauchy-Schwarz inequality, Lemmas 2.4.1-2.4.3 and the fact that z is uniformly bounded in L 2 (0, L), we obtain From (2.4.5) and (2.4.7), we deduce that

                     2c 0 β-3ε α+ε (x -L)θ 5 vy x dx = o(1), - 2c 0 a β-3ε α+ε (x -L)θ 5 zS 1 (u, v, η)dx = o(|λ| -1 ), - 2c 0 a β β-3ε (x -L)z(κ 1 v x + κ 2 η x (•, 1))dx = o(|λ| -1 ). ( 2
u x = iλ -1 v x + iλ -3 f 1 x and y x = iλ -1 z x + iλ -3 f 3 x . (2.4.44)
Substituting (2.4.44) in (2.4.43), then using the fact that v, z are uniformly bounded in L 2 (0, L) and

f 1 x L 2 (0,L) = o(1), f 3 x L 2 (0,L) = o(1)
, we obtain

I = 2c 0 i λ γ β-3ε (x -L)vz x dx - 2c 0 i λ γ β-3ε (x -L)zv x dx + o(1).
Using integration by parts to the second term in the above equation, we obtain

I = 2c 0 i λ γ β-3ε zvdx - 2c 0 i λ [(x -L) zv] γ β-3ε
+ o(1).

(2.4.45) Furthermore, by using Cauchy-Schwarz inequality, we get

2ic 0 λ γ β-3ε zvdx ≤ 2|c 0 ||λ| -1 γ β-3ε |z| 2 dx 1 2 γ β-3ε |v| 2 dx 1 2
(2.4.46) and

2ic 0 λ [(x -L) zv] γ β-3ε ≤ 2|c 0 ||λ| -1 [(L -γ) |z(γ)||v(γ)| + (L -β + 3ε)|z(β -3ε)| |v(β -3ε)|] .
         2c 0 i λ γ β-3ε zvdx = O |λ| -1 = o(1), - 2c 0 i λ [(x -L)zv] γ β-3ε = O |λ| -1 = o(1).
( (2.4.51)

         β 0 |u x | 2 dx = o(λ -4 ), β 0 1 0 |η x (•, ρ)| 2 dρdx = o(λ -2 ),
Step 2. From Lemma 2.4.6 and (2.4.51), we deduce that

         ε 0 |v| 2 dx = o(1), α+ε 0 |y x | 2 dx = o(1), α 0 |z| 2 dx = o(1), L β |u x | 2 dx = o(1), L β-ε |v| 2 dx = o(1), L β-3ε |y x | 2 dx = o(1) and L β-2ε |z| 2 dx = o(1).
Step

Conclusion

We have studied the stabilization of a locally coupled wave equations with non smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. We proved the strong stability of the system by using Arendt-Batty criteria. Finally, we established a polynomial energy decay rate of order t -1 .

Chapter 3

Stability results of coupled wave models with locally memory in a past history framework via non-smooth coefficients on the interface

In this chapter, we investigate the stabilization of locally coupled wave equations with local viscoelastic damping of past history type acting only in one equation via non-smooth coefficients. First, using a general criteria of Arendt-Batty, we prove the strong stability of our system. Second, using a frequency domain approach combined with the multiplier method, we establish the exponential stability of the solution if the two waves have the same speed of propagation. In the case of different propagation speeds, we prove that the energy of our system decays polynomially with rate t -1 . Finally, we show the lack of exponential stability if the speeds of wave propagation are different with a global damping and a global coupling. This chapter is published in [START_REF] Akil | Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface[END_REF].

Introduction

In this chapter, we investigate the indirect stability of coupled elastic wave equations with localized past history damping. More precisely, we consider the following system:

                         u tt -au x -b(x) ∞ 0 g(s)u x (x, t -s)ds x + c(x)y t = 0, (x, t) ∈ (0, L) × (0, ∞), y tt -y xx -c(x)u t = 0, (x, t) ∈ (0, L) × (0, ∞), u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0, (u(x, -s), u t (x, 0)) = (u 0 (x, s), u 1 (x)), (x, s) ∈ (0, L) × (0, ∞), (y(x, 0), y t (x, 0)) = (y 0 (x), y 1 (x)), x ∈ (0, L), (3.1.1)
where L and a are positive real numbers. We suppose that there exist a non-zero constant c 0 and positive constants α, β, γ, and b 0 such that 0 < α < β < γ < L, and define The general integral term represents a history term with the relaxation function g that is supposed to satisfy the following hypotheses: The notion of indirect damping mechanisms has been introduced by Russell in [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF] and since this time, it retains the attention of many authors. In particular, the fact that only one equation of the coupled system is damped refers to the so-called class of "indirect" stabilization problems initiated and studied in [START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF][START_REF] Alabau | 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, see for instance [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wavetype 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] Zhang | Polynomial decay and control of a 1 -d hyperbolic-parabolic coupled system[END_REF] and the rich references therein. In 1996, Liu and Zheng in [START_REF] Liu | On the exponential stability of linear viscoelasticity and thermoviscoelasticity[END_REF] studied the one-dimensional linear thermoviscoelastic system

b(x) = b 0 , x ∈ (0, β), 0, x ∈ (β, L), (b(•)) c(x) = c 0 , x ∈ (α, γ), 0, x ∈ (0, α) ∪ (γ, L). (c(•)) α β γ L 0 b 0 c 0 b(x) c(x)
         g ∈ L 1 ([0, ∞)) ∩ C 1 ([0, ∞)) is a positive function such that g(0) := g 0 > 0, ∞ 0 g(s)ds := g, b(x) := a -b(x) g > 0,
   u tt -αu xx + ∞ 0 g(s)u xx (•, t -s)ds + γcθ x = 0, in (0, π) × (0, ∞), θ t + γu xt -θ xx = 0, in (0, π) × (0, ∞), (3.1.3)
where α > 0, γ ≥ 0 and c > 0; and proved that the system is exponential stable. In 2008, Rivera et al. in [START_REF] Rivera | Stability of Timoshenko systems with past history[END_REF] studied the stability of 1-dimensional Timoshenko system with past history acting only in one equation, they showed that the system is exponential stable if and only if the equations have the same wave speeds of propagation. In case that the wave speeds of the equations are different, they proved that the solution of the system decays polynomially to zero. In 2011, Guesmia in [START_REF] Guesmia | Asymptotic stability of abstract dissipative systems with infinite memory[END_REF] studied the asymptotic stability of the following abstract linear dissipative integrodifferential equation with infinite memory

u tt (t) + Au(t) - ∞ 0 g(s)Bu(t -s)ds = 0, ∀ t > 0. (3.1.4)
where A : D(A) -→ H and B : D(B) -→ H are self-adjoint linear positive definite operators with domains D(A) ⊂ D(B) ⊂ H such that the embeddings are dense and compact, H is a Hilbert space, and g : R + -→ R + is the convolution kernel function. He showed that the stability of the system holds for a relatively large class of convolution kernels and he provided a relation between the decay rate of the solution and the growth of the kernel at infinity. In 2012, Matos et al. in [START_REF] Matos | Polynomial decay to a class of abstract coupled systems with past history[END_REF] studied the stability of the abstract coupled wave equations with past history, by considering:

                   u tt + A 1 u - ∞ 0 g(s)A 2 u(t -s)ds + βv = 0, v tt + Bv + βu = 0, in L 2 (R + , H), u(-t) = u 0 (t), t ≥ 0, v(0) = v 0 , u t (0) = u 1 , v t (0) = v 1 , (3.1.5)
where A 1 , A 2 and B are self-adjoint positive-definite operators with the domain

D(A 1 ) ⊆ D(A 2 ) ⊂ H and D(B) ⊂ H with compact embeddings in H, g : [0, ∞) -→ [0, ∞) is a
smooth and summable function and β is a small positive constant. They showed that the abstract setting is not strong enough to produce exponential stability and they proved that the solution decays polynomially to zero. In 2014, Fatori et al. in [START_REF] Fatori | The Timoshenko system with history and Cattaneo law[END_REF] studied a fully hyperbolic thermoelastic Timoshenko system with past history where the thermal effects are given by Cattaneo's law, they established the exponential stability of the solution if and only if the coefficients of their System satisfy the next relation

χ 0 := τ -ρ 1 ρ 3 κ ρ 2 -bρ 1 κ -τ ρ 1 δ 2 ρ 3 κ = 0.
In the case χ 0 = 0, they established optimal polynomial stability rates. In the same year, Santos et al. in [START_REF] Santos | Asymptotic behavior to Bresse system with past history[END_REF] studied the stability of 1-dimensional Bresse system with past history acting in the shear angle displacement, they showed the exponential decay of the solution if and only if the wave speeds are the same. Otherwise, they showed that the Bresse system is polynomial stable with optimal decay rate. In 2014, Jin et al. in [START_REF] Jin | Coupled second order evolution equations with fading memory: Optimal energy decay rate[END_REF] studied the stability of the abstract Cauchy problem for a system of coupled equations with fading memory

               u tt (t) + Au(t) - t 0 g(t -s)Au(s)ds + αu(t) + βBv(t) = f (u(t)), t > 0, v tt (t) + Av(t) + βBu(t) = 0, t > 0, u(0) = u 0 , u t (0) = u 1 , v(0) = v 0 , v t (0) = v 1 , (3.1.6)
where α ≥ 0, β ≥ 0, A is a positive self-adjoint linear operator in a Hilbert space H, B is a symmetric linear operator in H, f : D( √ A) -→ H denotes external forces, and g is the memory kernel. If β > 0, they established a polynomial decay rate of order t -1 of the full energy, while if β = 0, they proved the same decay rate but only on the energy of u. In 2015, Guesmia in [START_REF] Guesmia | Asymptotic behavior for coupled abstract evolution equations with one infinite memory[END_REF] studied the asymptotic behavior for coupled abstract evolution equations with one infinite memory H -→ H is a self-adjoint bounded operator, H is a real Hilbert space, and g : R + -→ R + is the convolution kernel. He proved under a boundedness condition on the past history data that the stability of the system holds for convolution kernels having much weaker decay rates than the exponential one. In 2017, Alabau-Boussouira et al. in [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities[END_REF] studied the energy decay of the coupled wave equations

   u tt (t) + Au(t) - ∞ 0 g(s)Bu(t -s)ds + Bv(t) = 0, ∀ t > 0, v tt (t) + Av(t) + Bu(t) = 0, ∀ t > 0, ( 3
     u tt -∆u + ρ(x, u t ) + α(x)v t = 0, in Ω × (0, ∞), v tt -∆v -α(x)u t = 0, in Ω × (0, ∞), u = v = 0, on Γ × (0, ∞), (3.1.8)
where Ω is a bounded subset of R n , Γ is a smooth boundary of Ω, ρ(x, u t ) is a nonlinear damping, and α ∈ C(Ω) is positive on a subset of positive measure (but may vanish in some parts of Ω). They proved that the total energy of the whole system (3.1.8) decays as fast as the damped single equation. Also, they gave a one-step general explicit decay formula for arbitrary nonlinearity. In 2018, Abdallah, Ghader and Wehbe in [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] studied the stability of a 1-dimensional Bresse system with infinite memory type control and /or with heat conduction given by Cattaneo's law acting in the shear angle displacement. In the absence of thermal effect, under the same speed propagation, they established the exponential stability of the system. However, in the case of different speed propagation, they established a polynomial energy decay rate. In 2018, Cavalcanti et al. in [START_REF] Cavalcanti | Exponential stability for the wave model with localized memory in a past history framework[END_REF] studied the asymptotic stability of the multidimensional damped wave equation, by considering:

ρ(x)u tt -∆u + ∞ 0 g(s)div[a(x)∇u(•, t -s)]ds + b(x)u t = 0, in Ω × (0, ∞), (3.1.9)
where Ω is an open bounded and connected set of R n , n ≥ 2, ρ(x) is constant, a(x) ≥ 0 is a smooth function, b(x) ≥ 0 is a bounded function acting effectively in a region A of Ω where a = 0. Considering that the well-known geometric control condition (ω, T 0 ) holds and supposing that the relaxation function g is bounded by a function that decays exponentially to zero, they proved that the solution to the corresponding partial viscoelastic model decays exponentially to zero, even in the absence of the frictional dissipative effect. Moreover, they proved by removing the frictional damping term b(x)u t and by assuming that ρ is not constant, that localized viscoelastic damping is strong enough to assure that the system is exponentially stable. In 2019, Hao and Wang in [START_REF] Hao | General stability result of abstract thermoelastic system with infinite memory[END_REF] studied the stability of the abstract thermoelastic system with infinite memory

             u tt + Au + Bu t - ∞ 0 g(s)Au(t -s)ds -A α θ = 0, t > 0, θ t + kA β θ + A α u t = 0, t > 0, u(-t) = u 0 (t), t ≥ 0, u t (0) = u 1 , θ(0) = θ 0 , (3.1.10)
where α ∈ [0, 1), β ∈ (0, 1], A : D(A) -→ H and B : D(B) -→ H are self-adjoint linear positive definite operators with domains D(A) ⊂ D(B) ⊂ H such that the embeddings are dense and compact, and H is a real Hilbert space. They obtained the stability result and provided a direct relationship between the decay rate of the energy and the decay rate of kernel function g. In 2019, Hassan and Messaoudi in [START_REF] Hassan | General decay rate for a class of weakly dissipative second-order systems with memory[END_REF] studied the stability of an abstract class of weakly dissipative second-order system with finite memory

     u tt + Au - t -∞ g(t -s)A α u(s)ds = 0, t > 0, u(-t) = u 0 (t), t ≥ 0, u t (0) = u 1 . (3.1.11)
where A : D(A) ⊂ H -→ H is a positive definite self-adjoint operator on H, H is a real separable Hilbert space, g is the convolution kernel, and α ∈ [0, 1]. They established a new general decay rate for the solution of the system under approbritae conditions on the memory kernel g. In 2019, Jin et al. in [START_REF] Jin | Asymptotic behavior for coupled systems of second order abstract evolution equations with one infinite memory[END_REF] studied the stability of an abstract Cauchy problem for a system of coupled equations with one infinite memory, by considering:

             u tt (t) + A 1 u(t) - ∞ 0 g(s)Au(t -s)ds + Bv(t) = 0, t > 0, v tt (t) + A 2 v(t) + Bu(t) = 0, t > 0, u(-t) = u 0 (t), ∀t ≥ 0, u t (0) = u 1 , v(0) = v 0 , v t (0) = v 1 , (3.1.12)
where A, A 1 and A 2 are positive self-adjoint linear operators in a Hilbert space H, B is a positive self-adjoint bounded linear operator in H, and g is the memory kernel. They established a polynomial energy decay rate of order t -1 . In 2011, Almeida et al. in [START_REF] Almeida | Lack of exponential decay of a coupled system of wave equations with memory[END_REF] studied the stability of coupled wave equations with past history effective only in one equation, by considering the following system:

                   u tt -∆u + ∞ 0 g(s)∆u(•, t -s)ds + αv = 0, in Ω × (0, ∞), v tt -∆v + αu = 0, in Ω × (0, ∞), u = v = 0, on Γ × (0, ∞) u(x, 0), v(x, 0)) = (u 0 (x), v 0 (x)) in Ω, u t (x, 0), v t (x, 0)) = (u 1 (x), v 1 (x)) in Ω, (3.1.13)
where Ω is an open bounded set of R n with smooth boundary Γ and α > 0. They showed that the dissipation given by the memory effect is not strong enough to produce exponential decay. They proved that the solution of the system (3.1.13) decays polynomially with rate t -1 2 . Also, in 2020, Cordeiro et al. in [START_REF] Cordeiro | Optimal polynomial decay for a coupled system of wave with past history[END_REF] established the optimality of the decay rate.

But to the best of our knowledge, it seems that no result in the literature exists concerning the case of coupled wave equations with localized past history damping, especially in the absence of smoothness of the damping and coupling coefficients. The goal of the present chapter is to fill this gap by studying the stability of system (3.1.1). This chapter is organized as follows: In Section 3.2, we prove the well-posedness of our system by using semigroup approach. In Section 3.3, following a general criteria of Arendt Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. Next, in Section 3.4, by using the frequency domain approach combining with a specific multiplier method, we establish exponential stability of the solution if the two waves have the same speed of propagation (i.e. a = 1). In the case a = 1, we prove that the energy of our system decays polynomially with the rate t -1 . Finally, in Section 3.5, we show the lack of exponential stability in case that the speeds of wave propagation are different with a global damping and a global coupling (i.e., when a = 1 and b(x) = c(x) = 1).

Well-posedness of the system

In this section, we will establish the well-posedness of system (3.1.1) by using semigroup approach. To this aim, as in [START_REF] Dafermos | Asymptotic stability in viscoelasticity[END_REF], we introduce the following auxiliary change of variable ω(x, s, t) := u(x, t) -u(x, t -s), (x, s, t) ∈ (0, β) × (0, ∞) × (0, ∞).

(3.2.1)

Then, system (3.1.1) becomes

u tt -S b(•) (u, ω) x + c(•)y t = 0, (x, t) ∈ (0, L) × (0, ∞), (3.2.2)
y tt -y xx -c(•)u t = 0, (x, t) ∈ (0, L) × (0, ∞), (3.2.3) ω t (•, s, t) + ω s (•, s, t) -u t = 0, (x, s, t) ∈ (0, β) × (0, ∞) × (0, ∞), (3.2.4)
where

S b(•) (u, ω) :=    S b 0 (u, ω) := b 0 u x + b 0 ∞ 0 g(s)ω x (x, s)ds, in (0, β), au x , in (β, L). (S b(•) (u, ω))
With the following boundary conditions

     u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0, ω(•, 0, t) = 0, (x, t) ∈ (0, β) × (0, ∞), ω(0, s, t) = 0, (s, t) ∈ (0, ∞) × (0, ∞), (3.2.5)
and the following initial conditions

       u(•, -s) = u 0 (•, s), u t (•, 0) = u 1 (•), (x, s) ∈ (0, L) × (0, ∞), y(•, 0) = y 0 (•), y t (•, 0) = y 1 (•), x ∈ (0, L), ω 0 (•, s) := ω(•, s, 0) = u 0 (•, 0) -u 0 (•, s), (x, s) ∈ (0, β) × (0, ∞). (3.2.6)
The energy of system (3.2.2)-(3.2.6) is given by

E(t) = E 1 (t) + E 2 (t) + E 3 (t), (3.2.7)
where

         E 1 (t) = 1 2 L 0 |u t | 2 + b(•)|u x | 2 dx, E 2 (t) = 1 2 L 0 |y t | 2 + |y x | 2 dx and E 3 (t) = b 0 2 β 0 ∞ 0 g(s)|ω x (•, s, t)| 2 dsdx.
Lemma 3.2.1. Under the hypotheses (H). Let U = (u, u t , y, y t , ω) be a regular solution of system (3.2.2)-(3.2.6). Then, the energy E(t) satisfies the following estimation

d dt E(t) = b 0 2 β 0 ∞ 0 g (s)|ω x (•, s, t)| 2 dsdx. (3.2.8)
Proof. First, multiplying (3.2.2) by u t , integrating over (0, L), using integration by parts with (3.2.5), using the definition of S b(•) (u, ω), b(•) and c(•), then taking the real part, we obtain

d dt E 1 (t) = - b 0 β 0 ∞ 0 g(s)ω x (•, s, t)u tx dsdx - c 0 γ α y t u t dx . ( 3
ω xt (•, s, t) + ω xs (•, s, t) -u tx = 0 in (0, β) × (0, ∞) × (0, ∞). (3.2.11)
Multiplying (3.2.11) by b 0 g(s)ω x (•, s, t), integrating over (0, β) × (0, ∞), then taking the real part, we get

d dt E 3 (t) = - b 0 2 β 0 ∞ 0 g(s) d ds |ω x (•, s, t)| 2 dsdx + b 0 β 0 ∞ 0 g(s)ω x (•, s, t)u tx dsdx .
Using integration by parts with respect to s in the above equation with the help of (3.2.5) and the hypotheses (H), we obtain

d dt E 3 (t) = b 0 2 β 0 ∞ 0 g (s)|ω x (•, s, t)| 2 dsdx + b 0 β 0 ∞ 0 g(s)ω x (•,
H := H 1 0 (0, L) × L 2 (0, L) 2 × W g , where
W g := L 2 g ((0, ∞); H 1 L (0, β)) and H 1 L (0, β) := ω ∈ H 1 (0, β) | ω(0) = 0 .
The space W g is a Hilbert space of H 1 L (0, β)-valued functions on (0, ∞), equipped with the following inner product

(ω 1 , ω 2 ) Wg := β 0 ∞ 0 g(s)ω 1 x ω 2 x dsdx, ∀ ω 1 , ω 2 ∈ W g .
The Hilbert space H is equipped with the following inner product

U, U 1 H = L 0 b(•)u x u 1 x + vv 1 + y x y 1 x + zz 1 dx + b 0 β 0 ∞ 0 g(s)ω x (•, s)ω 1 x (•, s)dsdx, (3.2.13)
where U = (u, v, y, z, ω(•, s)) ∈ H and U 1 = (u 1 , v 1 , y 1 , z 1 , ω 1 (•, s)) ∈ H. Now, we define the linear unbounded operator A : D(A) ⊂ H -→ H by:

D(A) =      U = (u, v, y, z, ω(•, s)) ∈ H | y ∈ H 2 (0, L) ∩ H 1 0 (0, L), v, z ∈ H 1 0 (0, L) S b(•) (u, ω) x ∈ L 2 (0, L), ω s (•, s) ∈ W g , ω(•, 0) = 0 in (0, β)      and A       u v y z ω(•, s)       =        v S b(•) (u, ω) x -c(•)z z y xx + c(•)v -ω s (•, s) + v        , (3.2.14) for all U = (u, v, y, z, ω(•, s)) ∈ D(A). Now, if U = (u, u t , y, y t , ω(•, s)) , then system (3.2.2)-(3.2.6
) can be written as the following first order evolution equation

U t = AU, U (0) = U 0 , (3.2.15)
where U 0 = (u 0 (•, 0), u 1 , y 0 , y 1 , ω 0 (•, s)) ∈ H.

Proposition 3.2.1. Under the hypotheses (H), the unbounded linear operator A is mdissipative in the energy space H.

Proof. For all U = (u, v, y, z, ω(•, s)) ∈ D(A), from (3.2.13) and (3.2.14), we have

(AU, U ) H = L 0 b(•)v x u x dx + L 0 S b(•) (u, ω) x vdx + L 0 z x y x dx + L 0 y xx zdx + b 0 β 0 ∞ 0 g(s)v x ω x (•, s)dsdx - b 0 β 0 ∞ 0 g(s)ω xs (•, s)ω x (•, s)dsdx .
Using integration by parts to the second and fourth terms in the above equation, then using the fact that U ∈ D(A) , we obtain

(AU, U ) H = - b 0 β 0 ∞ 0 g(s)ω xs (•, s)ω x (•, s)dsdx = - b 0 2 β 0 ∞ 0 g(s) d ds |ω x (•, s)| 2 dsdx.
Using integration by parts with respect to s in the above equation and the fact that ω(•, 0) = 0 in (0, β) with the help of hypotheses (H), we get

(AU, U ) H = b 0 2 β 0 ∞ 0 g (s)|ω x (•, s)| 2 dsdx ≤ 0, (3.2.16)
which implies that A is dissipative. Now, let us prove that A is maximal. To this aim, let

F = (f 1 , f 2 , f 3 , f 4 , f 5 (•, s)) ∈ H, we want to find U = (u, v, y, z, ω(•, s)) ∈ D(A) unique solution of -AU = F. (3.2.17)
Equivalently, we have the following system

-v = f 1 , (3.2.18) -S b(•) (u, ω) x + c(•)z = f 2 , (3.2.19) -z = f 3 , (3.2.20) -y xx -c(•)v = f 4 , (3.2.21) ω s (•, s) -v = f 5 (•, s), (3.2.22)
with the following boundary conditions

u(0) = u(L) = y(0) = y(L) = 0, ω(•, 0) = 0 in (0, β) and ω(0, s) = 0 in (0, ∞). ( 3
ω(x, s) = s 0 f 5 (x, ξ)dξ -sf 1 , (x, s) ∈ (0, β) × (0, ∞). (3.2.24)
Since v = -f 1 ∈ H 1 0 (0, L) and f 5 (•, s) ∈ W g , then from (3.2.22) and (3.2.24) we get ω s (•, s) ∈ W g and ω(•, s) ∈ H 1 L (0, β) a.e. in (0, ∞). Now, to obtain that ω(•, s) ∈ W g , it is sufficient to prove that

∞ 0 g(s) ω x (•, s) 2 L 2 0,β ds < ∞ where • L 2 0,β := • L 2 (0,β)
. For this aim, let 1 , 2 > 0 , under the hypotheses (H), we have

2 1 g(s) ω x (•, s) 2 L 2 0,β ds ≤ - 1 m 2 1 g (s) ω x (•, s) 2 L 2 0,β ds. (3.2.25)
Using integration by parts in (3.2.25), we obtain

2 1 g(s) ω x (•, s) 2 L 2 0,β ds ≤ 1 m 2 1 g(s) d ds ω x (•, s) 2 L 2 0,β ds + g( 1 ) ω x (•, 1 ) 2 L 2 0,β -g ( 2 ) ω x (•, 2 ) 2 L 2 0,β .
Moreover, from Young's inequality, we have

1 m 2 1 g(s) d ds ω x (•, s) 2 L 2 0,β ds = 2 m 2 1 g(s) β 0 ω x (•, s)ω sx (•, s)dx ds ≤ 1 2 2 1 g(s) ω x (•, s) 2 L 2 0,β ds + 2 m 2 2 1 g(s) ω sx (•, s) 2 L 2 0,β ds.
(3.2.26) Inserting (3.2.26) in the above inequality, we get

2 1 g(s) ω x (•, s) 2 L 2 0,β ds ≤ 4 m 2 2 1 g(s) ω sx (•, s) 2 L 2 0,β ds + 2 m g( 1 ) ω x (•, 1 ) 2 L 2 0,β - 2 m g ( 2 ) ω x (•, 2 ) 2 L 2 0,β .
Using the fact that ω s (•, s) ∈ W g , ω(•, 0) = 0 in (0, β) and the hypotheses (H) in the above inequality, (in particular (3.1.2)) we obtain, as

1 → 0 + and 2 → ∞, that ∞ 0 g(s) ω x (•, s) 2 L 2 0,
S b(•) u, s 0 f 5 (•, ξ)dξ -sf 1 x + c(•)f 3 = -f 2 , (3.2.27) y xx -c(•)f 1 = -f 4 , (3.2.28) u(0) = u(L) = y(0) = y(L) = 0, (3.2.29)
where

S b(•) u, s 0 f 5 (•, ξ)dξ -sf 1 =    b 0 u x + b 0 ∞ 0 g(s) s 0 f 5 x (x, ξ)dξ -sf 1 x , in (0, β), au x , in (β, L).
Let (φ, ψ) ∈ H 1 0 (0, L) × H 1 0 (0, L). Multiplying (3.2.27) and (3.2.28) by φ and ψ respectively, integrating over (0, L), using formal integrations by parts, then using the definition of S b(•) (u, ω), b(•) and c(•), we obtain

L 0 b(•)u x φ x dx = L 0 f 2 φdx + c 0 γ α f 3 φdx -b 0 β 0 ∞ 0 g(s) s 0 f 5 x (•, ξ)dξ -sf 1 x φ x dsdx (3.2.30) and L 0 y x ψ x dx = L 0 f 4 ψdx -c 0 γ α f 1 ψdx. ( 3
B((u, y), (φ, ψ)) = L(φ, ψ), ∀(φ, ψ) ∈ H 1 0 (0, L) × H 1 0 (0, L), (3.2.32)
where

B((u, y), (φ, ψ)) = L 0 b(•)u x φ x dx + L 0 y x ψ x dx and L(φ, ψ) = L 0 f 2 φ + f 4 ψ dx + c 0 γ α f 3 φ -f 1 ψ dx -b 0 β 0 ∞ 0 g(s) s 0 f 5 x (•, ξ)dξ -sf 1 x φ x dsdx.
It is easy to see that, B is a sesquilinear, continuous and coercive form on (H 1 0 (0, L) × H 1 0 (0, L))

2
and L is an antilinear and continuous form on H 1 0 (0, L) × H 1 0 (0, L). Then, it follows by Lax-Milgram theorem that (3.2.32) admits a unique solution (u, y) ∈ H 1 0 (0, L) × H 1 0 (0, L). By taking test-functions (φ, ψ) ∈ (D(0, L)) 2 , we see that (3.2.27)-(3.2.29) hold in the distributional sense, from which we deduce that y ∈ H 2 (0, L) ∩ H 1 0 (0, L), while (S b(•) (u, ω)) x ∈ L 2 (0, L). Consequently, U ∈ D(A) is a unique solution of (3.2.17). Then, A is an isomorphism and since ρ (A) is open set of C (see Theorem 1.1.13), we easily get R(λI -A) = H for a sufficiently small λ > 0. This, together with the dissipativeness of A, imply that D (A) is dense in H and that A is m-dissipative in H (see Theorems 1.2.6, 1.2.9). The proof is thus complete.

According to Lumer-Philips theorem (see Theorem 1.2.8), Proposition 3.2.1 implies that the operator A generates a C 0 -semigroup of contractions e tA in H which gives the well-posedness of (3.2.15). Then, we have the following result: Theorem 3.2.1. Under the hypotheses (H), for all U 0 ∈ H, system (3.2.15) admits a unique weak solution U (x, s, t) = e tA U 0 (x, s) ∈ C 0 (R + , H).

Moreover, if U 0 ∈ D(A), then the system (3.2.15) admits a unique strong solution

U (x, s, t) = e tA U 0 (x, s) ∈ C 0 (R + , D(A)) ∩ C 1 (R + , H).

Strong Stability

This section is devoted to the proof of the strong stability of the C 0 -semigroup e tA t≥0 . To obtain the strong stability of the C 0 -semigroup e tA t≥0 , we use the theorem of Arendt and Batty in [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF] (see Theorem 1.3.3). Proof. From Proposition 3.2.1, we have 0 ∈ ρ(A). We still need to show the result for λ ∈ R . To this aim, suppose that there exists a real number λ = 0 and

U = (u, v, y, z, ω(•, s)) ∈ D(A) such that AU = iλU. (3.3.1)
Equivalently, we have the following system

v = iλu, (3.3.2) S b(•) (u, ω) x -c(•)z = iλv, (3.3.3) z = iλy, (3.3.4) y xx + c(•)v = iλz, (3.3.5) -ω s (•, s) + v = iλω(•, s). (3.3.6)
From the above system and by using Holmgren uniqueness theorem, we deduce that u = y = 0 in (γ, L). The proof is thus complete.

Lemma 3.3.2. Under the hypotheses (H), for all λ ∈ R, we have

R(iλI -A) = H.
Proof. From Proposition 3.2.1, we have 0 ∈ ρ(A). We still need to show the result for λ ∈ R . For this aim, let

F = (f 1 , f 2 , f 3 , f 4 , f 5 (•, s)) ∈ H, we want to find U = (u, v, y, z, ω(•, s)) ∈ D(A) solution of (iλI -A)U = F. (3.3.40)
Equivalently, we have the following system

iλu -v = f 1 , (3.3.41) iλv -S b(•) (u, ω) x + c(•)z = f 2 , (3.3.42)
iλy -z = f 3 , (3.3.43) iλz -y xx -c(•)v = f 4 , (3.3.44) iλω(•, s) + ω s (•, s) -v = f 5 (•, s), (3.3
         -λ 2 u -S b(•) (u, ω) x + iλc(•)y = F 1 := f 2 + c(•)f 3 + iλf 1 , -λ 2 y -y xx -iλc(•)u = F 2 := f 4 -c(•)f 1 + iλf 3 , u(0) = u(L) = y(0) = y(L) = 0, (3.3.48)
where here S b(•) (u, ω) takes the form Let (φ, ψ) ∈ H 1 0 (0, L) × H 1 0 (0, L). Multiplying the first equation of (3.3.48) and the second equation of (3.3.48) by φ and ψ respectively, integrating over (0, L), then using formal integrations by parts, we obtain

S b(•) (u, ω) =    b 0 u x + b 0 iλ ∞ 0 g(s)(1 -e -iλs )f 1 x ds + b 0 ∞ 0 g(s) s 0 f 5 x (x, ξ)e iλ(ξ-s) dξds, in (0, β),
-λ 2 L 0 uφdx + L 0 b(•)u x φ x dx + b 0 iλ β 0 ∞ 0 g(s)(1 -e -iλs )f 1 x φ x dsdx + b 0 β 0 ∞ 0 g(s) s 0 e iλ(ξ-s) f 5 x (•, ξ)φ x dξdsdx + iλc 0 γ α yφdx = L 0 F 1 φdx (3.3.49)
and

-λ 2 L 0 yψdx + L 0 y x ψ x dx -iλc 0 γ α uψdx = L 0 F 2 ψdx, (3.3
B((u, y), (φ, ψ)) = L(φ, ψ), ∀(φ, ψ) ∈ V := H 1 0 (0, L) × H 1 0 (0, L), (3.3.51)
where

B((u, y), (φ, ψ)) = B 1 ((u, y), (φ, ψ)) + B 2 ((u, y), (φ, ψ)) with          B 1 ((u, y), (φ, ψ)) = L 0 b(•)u x φ x dx + L 0 y x ψ x dx, B 2 ((u, y), (φ, ψ)) = -λ 2 L 0 (uφ + yψ)dx -iλc 0 L 0 (uψ -yφ)dx (3.3.52)
and

L(φ, ψ) = L 0 (F 1 φ + F 2 ψ)dx - b 0 iλ β 0 ∞ 0 g(s)(1 -e -iλs )f 1 x φ x dsdx -b 0 β 0 ∞ 0 g(s) s 0 e iλ(ξ-s) f 5 x (•, ξ)dξ φ x dsdx.
Let V be the dual space of V. Let us define the following operators

B : V -→ V (u, y) -→ B(u, y)
and

B i : V -→ V (u, y) -→ B i (u, y) , i ∈ {1, 2}, (3.3.53) such that (B(u, y))(φ, ψ) = B((u, y), (φ, ψ)), ∀(φ, ψ) ∈ V, (B i (u, y))(φ, ψ) = B i ((u, y), (φ, ψ)), ∀(φ, ψ) ∈ V, i ∈ {1, 2}. (3.3.54)
We need to prove that the operator B is an isomorphism. For this aim, we divide the proof into three steps:

Step 1. In this step, we want to prove that the operator B 1 is an isomorphism. For this aim, it is easy to see that B 1 is sesquilinear, continuous and coercive form on V. Then, from (3.3.54) and Lax-Milgram theorem, the operator B 1 is an isomorphism.

Step 2. In this step, we want to prove that the operator B 2 is compact. For this aim, from (3.3.52) and (3.3.54) we have

|B 2 ((u, y), (φ, ψ))| (u, y) (L 2 (0,L)) 2 (φ, ψ) (L 2 (0,L)) 2 , (3.3.55)
and consequently, using the compact embedding from V into (L 2 (0, L)) 2 , we deduce that B 2 is a compact operator.

Therefore, from the above steps, we obtain that the operator B = B 1 + B 2 is a Fredholm operator of index zero. Now, following Fredholm alternative, we still need to prove that the operator B is injective to obtain that the operator B is an isomorphism.

Step 3. In this step, we want to prove that the operator B is injective (i.e. ker(B) = {0}).

For this aim, let ( u, y) ∈ ker(B) which gives

B(( u, y), (φ, ψ)) = 0, ∀(φ, ψ) ∈ V.
Equivalently, we have

L 0 b(•) u x φ x dx + L 0 y x ψ x dx -λ 2 L 0 ( uφ + yψ)dx -iλ L 0 c(•)( uψ -yφ)dx = 0, ∀(φ, ψ) ∈ V.
Thus, we find that

       -λ 2 u -( b(•) u x ) x + iλc(•) y = 0, -λ 2 y -y xx -iλc(•) u = 0, u(0) = u(L) = y(0) = y(L) = 0.
Therefore, the vector U defined by U = ( u, iλ u, y, iλ y, (1 -e -iλs ) u) belongs to D(A) and satisfies iλ U -A U = 0, and consequently U ∈ ker(iλI -A). Then, according to Lemma 3.3.1, we obtain U = 0 and consequently u = y = 0 and ker(B) = {0}.

Finally, from Step 3 and Fredholm alternative, we deduce that the operator B is isomorphism. It is easy to see that the operator L is a antilinear and continuous form on V. Consequently, (3.3.51) admits a unique solution (u, y) ∈ V. By using the classical elliptic regularity, we deduce that U ∈ D(A) is a unique solution of (3.3.40). The proof is thus complete.

Proof of Theorem 3.3.1. From Lemma 3.3.1, we obtain the the operator A has no pure imaginary eigenvalues (i.e. σ p (A) ∩ iR = ∅). Moreover, from Lemma 3.3.2 and with the help of the closed graph theorem of Banach, we deduce that σ(A) ∩ iR = ∅. Therefore, according to Theorem 1.3.3, we get that the C 0 -semigroup (e tA ) t≥0 is strongly stable. The proof is thus complete.

Remark 3.3.1. We mention [START_REF] Akil | Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary[END_REF] for a direct approach of the strong stability of Timoshenko system in the absence of compactness of the resolvent.

Exponential and Polynomial Stability

In this section, under the hypotheses (H), we show the influence of the ratio of the wave propagation speed on the stability of system (3.2.2)-(3.2.6). Our main result in this part are the following theorems.

Theorem 3.4.1. Assume that a = 1 and the hypotheses (H) hold. Then, the C 0 -semigroup e tA is exponentially stable; i.e., for all U 0 ∈ H, there exist constants M ≥ 1 and > 0 independent of U 0 such that e tA U 0 H ≤ M e -t U 0 H , t > 0.

Theorem 3.4.2. Assume that a = 1 and the hypotheses (H) hold. Then, for all U 0 ∈ D(A), there exists a constant C > 0 independent of U 0 such that

E(t) ≤ C t U 0 2 D(A) , t > 0.
Since iR ⊂ ρ(A) (see Section 3. We will prove condition (H 1 ) by a contradiction argument. For this purpose, suppose that (H 1 ) is false, then there exists (λ n , U n := (u n , v n , y n , z n , ω n (•, s))

) n≥1 ⊂ R * × D(A) with |λ n | → ∞ as n → ∞ and U n H = (u n , v n , y n , z n , ω n (•, s)) H = 1, ∀n ≥ 1, (3.4.1) such that (λ n ) (iλ n I -A)U n = F n := (f 1,n , f 2,n , f 3,n , f 4,n , f 5,n (•, s)) → 0 in H, as n → ∞. (3.4.2)
For simplicity, we drop the index n. Equivalently, from (3.4.2), we have

iλu -v = λ -f 1 → 0 in H 1 0 (0, L), (3.4.3) iλv -S b(•) (u, ω) x + c(•)z = λ -f 2 → 0 in L 2 (0, L), (3.4.4
) (3.4.9)

iλy -z = λ -f 3 → 0 in H 1 0 (0, L), (3.4.5) iλz -y xx -c(•)v = λ -f 4 → 0 in L 2 (0, L), (3.4.6) iλω(•, s) + ω s (•, s) -v = λ -f 5 (•, s) → 0 in W g . ( 3
- b 0 2 β 0 ∞ 0 g (s)|ω x (•, s)| 2 dsdx = -(AU, U ) H = 1 λ (F, U ) H ≤ 1 |λ| F H U H . (3.4.10)
Thus, from (3.4.10), (H) and the fact that F H = o(1) and U H = 1, we obtain the first estimation in (3.4.8). From hypotheses (H), we obtain

β 0 ∞ 0 g(s)|ω x (•, s)| 2 dsdx ≤ - 1 m β 0 ∞ 0 g (s)|ω x (•, s)| 2 dsdx. ( 3
iλω x (•, s) + ω sx (•, s) -iλu x = λ -f 5 x (•, s) -λ -f 1 x in (0, β) × (0, ∞).
(3.4.12)

Multiplying (3.4.12) by λ -1 g(s)u x , integrating over (0, β) × (0, ∞), then taking the imaginary part, we obtain

β 0 ∞ 0 g(s)|u x | 2 dsdx = i β 0 ∞ 0 g(s)ω x (•, s)u x dsdx + λ -1 β 0 ∞ 0 g(s)ω xs (•, s)u x dsdx - λ -( +1) β 0 ∞ 0 g(s)f 5 x (•, s)u x dsdx + λ -( +1) β 0 ∞ 0 g(s)f 1 x u x dsdx .
Using integration by parts with respect to s in the above equation, then using hypotheses (H) and the fact that ω(•, 0) = 0 in (0, β), we get

g β 0 |u x | 2 dx = i β 0 ∞ 0 g(s)ω x (•, s)u x dsdx + 1 λ β 0 ∞ 0 -g (s)ω x (•, s)u x dsdx - 1 λ +1 β 0 ∞ 0 g(s)f 5 x (•, s)u x dsdx + gλ -( +1) β 0 f 1 x u x dx .
(3.4.13) Using Young's inequality and Cauchy-Schwarz inequality in (3.4.13) with the help of hypotheses (H), we obtain

g β 0 |u x | 2 dx ≤ g 2 β 0 |u x | 2 dx + 1 2 β 0 ∞ 0 g(s)|ω x (•, s)| 2 dsdx + |λ| -1 √ g 0 β 0 ∞ 0 -g (s)|ω x (•, s)| 2 dsdx 1 2 β 0 |u x | 2 dx 1 2 + |λ| -( +1) g β 0 ∞ 0 g(s)|f 5 x (•, s)| 2 dsdx 1 2 β 0 |u x | 2 dx 1 2 + g|λ| -( +1) β 0 |f 1 x | 2 dx 1 2 β 0 |u x | 2 dx 1 2
.

From the above inequality, (3.4.8) and the fact that u x is uniformly bounded in L 2 (0, L) and f 1 x → 0 in L 2 (0, L), f 5 (•, s) → 0 in W g , we obtain the first estimation in (3.4.9). Now, by using Cauchy-Schwarz inequality, we obtain

β 0 S b 0 (u, ω) 2 dx = β 0 b 0 u x + b 0 ∞ 0 g(s)ω x (•, s)ds 2 dx ≤ 2( b 0 ) 2 β 0 |u x | 2 + 2b 2 0 β 0 ∞ 0 g(s)|ω x (•, s)|ds 2 dx ≤ 2( b 0 ) 2 β 0 |u x | 2 + 2b 2 0 g β 0 ∞ 0 g(s)|ω x (•, s)| 2 dsdx.
Finally, from the above inequality, (3.4.8) and the first estimation in (3.4.9), we obtain the second estimation in (3.4.9). The proof is thus complete. Proof. First, we fix a cut-off function

0 ε α α + ε α + 2ε β -3ε β -2ε β -ε β γ L 1 b 0 c 0 h 1 h 2 h 3 b(x) c(x)
h 1 ∈ C 1 ([0, β]) (see Figure 3.2) such that 0 ≤ h 1 (x) ≤ 1, for all x ∈ [0, β] and h 1 (x) = 1 if x ∈ [ε, β -ε], 0 if x ∈ {0, β}, (h 1 )
and set max

x∈[0,β] |h 1 (x)| = M h 1 .
From (3.4.4), we deduce that

iλv -S b 0 (u, ω) x + c(•)z = λ -f 2 in (0, β).
∞ 0 g(s)ω(•, s)ds and integrate over (0, β), using integration by parts with the help of the properties of h 1 (i.e. h 1 (0) = h 1 (β) = 0), then using the definition of c(•), we obtain

-iλ β 0 h 1 v ∞ 0 g(s)ω(•, s)dsdx = β 0 S b 0 (u, ω) h 1 ∞ 0 g(s)ω(•, s) x dsdx + c 0 β α h 1 z ∞ 0 g(s)ω(•, s)dsdx -λ - β 0 h 1 f 2 ∞ 0 g(s)ω(•, s)dsdx.
(3.4.15)

From (3.4.7), we deduce that

-iλω(•, s) = -ω s (•, s) + v + λ -f 5 (•, s) in (0, β) × (0, ∞).
Inserting the above equation in the left hand side of (3.4.15), then using the definition of c(•) and h 1 , we get

g β 0 h 1 |v| 2 dx = β 0 h 1 v ∞ 0 g(s)ω s (•, s)dsdx -λ - β 0 h 1 v ∞ 0 g(s)f 5 (•, s)dsdx + β 0 S b 0 (u, ω)h 1 ∞ 0 g(s)ω(•, s)dsdx + β 0 S b 0 (u, ω)h 1 ∞ 0 g(s)ω x (•, s)dsdx + c 0 β α h 1 z ∞ 0 g(s)ω(•, s)dsdx -λ - β 0 h 1 f 2 ∞ 0 g(s)ω(•, s)dsdx. (3.4.16)
Using integration by parts with respect to s with the help of ω(•, 0) = 0 in (0, β) and hypotheses (H), Cauchy-Schwarz inequality, Poincaré inequality, v is uniformly bounded in L 2 (0, L), and (3.4.9), we get

β 0 h 1 v ∞ 0 g(s)ω s (•, s)dsdx = β 0 h 1 v ∞ 0 -g (s)ω(•, s)dsdx ≤ √ g 0 β 0 |v| 2 dx 1 2 β 0 ∞ 0 -g (s)|ω(•, s)| 2 dsdx 1 2 √ g 0 β 0 |v| 2 dx 1 2 β 0 ∞ 0 -g (s)|ω x (•, s)| 2 dsdx 1 2 = o |λ| -2 .
(3.4.17)

Using the definition of h 1 , Cauchy-Schwarz inequality, Poincaré inequality, (3.4.8) and the fact that v, z are uniformly bounded in L 2 (0, L) and On the other hand, we have

f 2 L 2 (0,L) = o(1), f 5 → 0 in W g , we get                                                                                1 λ β 0 h 1 v ∞ 0 g(s)f 5 (•, s)dsdx 1 |λ| g β 0 |v| 2 dx 1 2 β 0 ∞ 0 g(s)|f 5 x (•, s)| 2 dsdx 1 2 = o(1) |λ| , c 0 β α h 1 z ∞ 0 g(s)ω(•, s)dsdx ≤ |c 0 | g β α |z| 2 dx β α ∞ 0 g(s)|ω(•, s)| 2 dsdx 1 2 |c 0 | g β α |z| 2 dx β 0 ∞ 0 g(s)|ω x (•, s)| 2 dsdx 1 2 = o(1) |λ| 2 , 1 λ β 0 h 1 f 2 ∞ 0 g(s)ω(•, s)dsdx 1 |λ| g β 0 |f 2 | 2 dx 1 2 β 0 ∞ 0 g(s)|ω x (•, s)| 2 dsdx 1 2 = o (1)
     |S b 0 (u, ω)||h 1 |g(s)|ω(•, s)| ≤ 1 2 |h 1 ||S b 0 (u, ω)| 2 g(s) + 1 2 |h 1 ||ω(•, s)| 2 g(s), |S b 0 (u, ω)||h 1 |g(s)|ω x (•, s)| ≤ 1 2 |h 1 ||S b 0 (u, ω)| 2 g(s) + 1 2 |h 1 ||ω x (•, s)| 2 g(s).
Then from the above inequalities, the definition of S b(•) (u, ω) and h 1 , Poincaré inequality and estimations (3.4.8) and (3.4.9), we obtain

                                   β 0 S b 0 (u, ω)h 1 ∞ 0 g(s)ω(•, s)dsdx ≤ M h 1 2 g β 0 |S b 0 (u, ω)| 2 dx + C p β 0 ∞ 0 g(s)|ω x (•, s)| 2 dsdx = o(|λ| -), β 0 S b 0 (u, ω)h 1 ∞ 0 g(s)ω x (•, s)dsdx ≤ g 2 β 0 |S b 0 (u, ω)| 2 dx + β 0 ∞ 0 g(s)|ω x (•, s)| 2 dsdx = o(|λ| -), ( 3
β 0 h 1 |v| 2 dx = o |λ| -2 .
Finally, from the above estimation and the definition of h 1 , we obtain the desired result (3.4.14). The proof is thus complete.

. Under the hypotheses (H), the solution the solution U = (u, v, y, z, ω(•, s)) ∈ D(A) of ( 3 Proof. First, we fix a cut-off function h 2 ∈ C 1 ([0, L]) (see Figure 3.2) such that 0 ≤ h 2 (x) ≤ 1, for all x ∈ [0, L] and

h 2 (x) = 0 if x ∈ [0, α] ∪ [β -ε, L], 1 if x ∈ [α + ε, β -2ε], ( h 2 )
From (3.4.6), iλ -1 h 2 y xx is uniformly bounded in L 2 (0, L). Multiplying (3.4.4) by iλ -1 h 2 y xx , using integration by parts over (0, L) and over (α, β -ε), the definitions of c(•) and h 2 , and using the fact that

f 2 L 2 (0,L) = o(1), we get L 0 h 2 vy x dx + L 0 h 2 v x y x dx - i λ L 0 h 2 S b(•) (u, ω) x y xx dx - ic 0 λ β-ε α (h 2 zy x + h 2 z x y x ) dx = o(1)
|λ| .

(3.4.21)

From (3.4.3) and (3.4.5), we obtain

v x = iλu x -λ -f 1 x and - i λ z x = y x + iλ -( +1) f 3 x .
Inserting the above equations in (3.4.21) and taking the real part, we get

c 0 L 0 h 2 |y x | 2 dx + iλ L 0 h 2 u x y x dx - i λ L 0 S b(•) (u, ω) x h 2 y xx dx = - L 0 h 2 vy x dx + 1 λ L 0 h 2 f 1 x y x dx + i c 0 λ β-ε α h 2 zy x dx - ic 0 λ +1 β-ε α h 2 f 3 x y x dx + o(1) |λ| . (3.4.22)
Using the fact that y x is uniformly bounded in

L 2 (0, L), f 1 x L 2 (0,L) = o(1) and f 3 x L 2 (0,L) = o(1), we get λ - L 0 h 2 f 1 x y x dx = o(1) |λ| and - ic 0 λ -( +1) β-ε α h 2 f 3 x y x dx = o (
c 0 β-ε α h 2 |y x | 2 dx + iλ β-ε α h 2 u x y x dx - i λ β-ε α S b 0 (u, ω) x h 2 y xx dx = o(1). (3.4.25) From (3.4.4), iλ -1 h 2 S b(•) (u, ω)
x is uniformly bounded in L 2 (0, L). Multiplying (3.4.6) by

iλ -1 S b(•) (u, ω)
x , using integration by parts over (0, L) and over (α, β -ε), the definitions of c(•), h 2 and S b(•) (u, ω), and the fact that f 4 L 2 (0,L) = o(1), we get

L 0 h 2 zS b(•) (u, ω)dx + L 0 h 2 z x S b(•) (u, ω)dx - i λ L 0 h 2 y xx S b(•) (u, ω) x dx + ic 0 λ β-ε α (h 2 v + h 2 v x ) S b 0 (u, ω)dx = o(|λ| -).
(3.4.26)

From (3.4.5), we have

z x = iλy x -λ -f 3
x . Using the above equation and the definition of S b(•) (u, ω) and h 2 , we get

L 0 h 2 z x S b(•) (u, ω)dx = β-ε α h 2 iλy x -λ -f 3 x S b 0 (u, ω)dx = iλ b 0 β-ε α h 2 y x u x dx + iλb 0 β-ε α h 2 y x ∞ 0 g(s)ω x (•, s)dsdx -b 0 λ - β-ε α h 2 f 3 x u x dx -λ -b 0 β-ε α h 2 f 3 x ∞ 0 g(s)ω x (•, s)dsdx.
(3.4.27)

From (3.4.7), we have

iλω x (•, s) = ω xs (•, s) + iλu x + λ -f 1 x -λ -f 5 x (•, s) in (0, β) × (0, ∞).
From the above equation and by using integration by parts with respect to s, we get

iλb 0 β-ε α h 2 y x ∞ 0 g(s)ω x (•, s)dsdx = b 0 β-ε α h 2 y x ∞ 0 -g (s)ω x (•, s)dsdx + iλb 0 g β-ε α h 2 y x u x dx + b 0 gλ - β-ε α h 2 y x f 1 x dx -b 0 λ - β-ε α h 2 y x ∞ 0 g(s)f 5 x (•, s)dsdx.
L 0 h 2 z x S b(•) (u, ω)dx = iλa β-ε α h 2 y x u x dx + b 0 β-ε α h 2 y x ∞ 0 -g (s)ω x (•, s)dsdx + b 0 gλ - β-ε α h 2 y x f 1 x dx -b 0 λ - β-ε α h 2 y x ∞ 0 g(s)f 5 x (•, s)dsdx -b 0 λ - β-ε α h 2 f 3 x u x dx -λ -b 0 β-ε α h 2 f 3 x ∞ 0 g(s)ω x (•, s)dsdx. (3.4.29)
Using Cauchy-Schwarz inequality, the facts that y x , u x is uniformly bounded in L 2 (0, L), and estimation (3.4.8), F H = o(1), we get

                   b 0 β-ε α h 2 y x ∞ 0 -g (s)ω x (•, s)dsdx = o |λ| -2 , b 0 gλ - β-ε α h 2 y x f 1 x = o |λ| -, -b 0 λ - β-ε α h 2 y x ∞ 0 g(s)f 5 x (•, s)dsdx = o |λ| -, -b 0 λ - β-ε α h 2 f 3 x u x dx = o(|λ| -), -λ -b 0 β-ε α h 2 f 3 x ∞ 0 g(s)ω x (•, s)dsdx = o |λ| -3 2 .
Inserting the above estimations in (3.4.29), we get

L 0 h 2 z x S b(•) (u, ω)dx = iλa β-ε α h 2 y x u x dx + o |λ| -2 . (3.4.30) From (3.4.3), we have iλ -1 v x = -u x -iλ -( +1) f 1
x . Then from the above equation and the definition of S b(•) (u, ω) and h 2 , we get

i c 0 λ β-ε α h 2 v x S b 0 (u, ω)dx = - β-ε α u x S b 0 (u, ω)dx -iλ -( +1) β-ε α f 1 x S b 0 (u, ω)dx. (3.4.31)
Using Cauchy-Schwarz inequality, the definition of h 2 , the fact that u x is uniformly bounded in L 2 (0, L) and f 1 x = o(1), and estimation (3.4.9), we get

- β-ε α u x S b 0 (u, ω)dx = o |λ| -2 and -iλ -( +1) β-ε α f 1 x S b 0 (u, ω)dx = o |λ| -3 2 -1
Inserting the above estimations in (3.4.31), we get

i c 0 λ β-ε α h 2 v x S b 0 (u, ω)dx = o |λ| -2 . (3.4.32)
Now, using the definition of h 2 and S b(•) (u, ω), (3.4.9), and the fact that v and z are uniformly bounded in L 2 (0, L), we get

           L 0 h 2 zS b(•) (u, ω)dx = β-ε α h 2 zS b 0 (u, ω)dx = o |λ| -2 , i c 0 λ β-ε α h 2 vS b 0 (u, ω)dx = o |λ| -3 2 . ( 3
iλa β-ε α h 2 y x u x dx - i λ β-ε α h 2 y xx S b 0 (u, ω) x dx = o |λ| -2 . (3.4.34)
Now, adding (3.4.25) and (3.4.34) and using the fact that ≥ 0, we get

β-ε α h 2 |y x | 2 dx = iλ(a -1) c 0 β-ε α h 2 u x y x dx + o(1).
Using the definition of h 2 in the above equation, we get the desired estimation (3.4.20). The proof is thus complete.

. Under the hypotheses (H), the solution U = (u, v, y, z, ω(•, s)) ∈ D(A) of ( 3 Proof. First, we fix a cut-off function h 3 ∈ C 1 ([0, L]) (see Figure 3.2) such that 0 ≤ h 3 (x) ≤ 1, for all x ∈ [0, L] and

h 3 (x) = 0 if x ∈ [0, α + ε] ∪ [β -2ε, L], 1 if x ∈ [α + 2ε, β -3ε], ( h 3 )
Multiplying (3.4.6) by -iλ -1 h 3 z, using integration by parts over (0, L), the fact that z is uniformly bounded in L 2 (0, L) and f 4 = o(1), and the definition of c(•), we get

L 0 h 3 |z| 2 dx - i λ L 0 h 3 zy x dx - i λ L 0 h 3 z x y x dx + i c 0 λ β-2ε α+ε h 3 vzdx = o |λ| -( +1) . (3.4.36)
From (3.4.5), we have

- i λ z x = -y x + iλ -( +1) f 3
x . Inserting the above equation in (3.4.36), we get

L 0 h 3 |z| 2 dx = L 0 h 3 |y x | 2 dx -iλ -( +1) L 0 h 3 f 3 x y x dx + i λ L 0 h 3 zy x dx -i c 0 λ β-2ε α+ε h 3 vzdx + o |λ| -( +1) .
(3.4.37)

Using the fact that f 3 x L 2 (0,L) = o(1), y x and z are uniformly bounded in L 2 (0, L), and the definition of h 3 , we get

         -iλ -( +1) L 0 h 3 f 3 x y x dx = o |λ| -( +1) , i λ L 0 h 3 zy x dx = o(1)
and

-i c 0 λ β-2ε α+ε h 3 vzdx = o(1). (3.4.38)
Using (3.4.20) and the definition of h 3 , we get It is easy to see the existence of χ(x). For example, we can take

L 0 h 3 |y x | 2 dx ≤ 3 β-2ε α+ε |y x | 2 dx ≤ 3|a -1| |c 0 | |λ| β-ε α |u x ||y x |dx + o(1). ( 3
χ(x) = 1 (γ -β + 3ε) 2 -2x 2 + 4(β -3ε)x + γ 2 -(β -3ε) 2 -2γ(β -3ε) , to get χ(β -3ε) = -χ(γ) = 1, χ ∈ C 1 ([β -3ε, γ]), M χ = 1 and M χ = 4 γ-β+3ε .
. Under the hypotheses (H), the solution U = (u, v, y, z, ω(•, s)) ∈ D(A) of (3.4.3)-(3.4.7) satisfies the following estimations

|v(γ)| 2 +|v(β -3ε)| 2 = O(|λ|), |z(γ)| 2 +|z(β -3ε)| 2 +|y x (γ)| 2 +|y x (β -3ε)| 2 = O(1). (3.4.40)
Proof. First, deriving Equation (3.4.3) with respect to x, we obtain

iλu x -v x = λ -f 1 x in (β -3ε, γ).
Multiplying the above equation by 2χv, integrating over (β -3ε, γ), then taking the real part, we obtain

2iλ γ β-3ε χu x vdx - γ β-3ε χ(|v| 2 ) x dx = 2λ - γ β-3ε χf 1 x vdx . (3.4.41)
Using integration by parts in (3.4.41), we obtain

-χ|v| 2 γ β-3ε = - γ β-3ε χ |v| 2 dx - 2iλ γ β-3ε χu x vdx + 2λ - γ β-3ε χf 1 x vdx . (3.4.42)
Using the definition of χ and Cauchy-Schwarz inequality in (3.4.42), we obtain

|v(γ)| 2 + |v(β -3ε)| 2 ≤ M χ γ β-3ε |v| 2 dx + 2|λ|M χ γ β-3ε |u x | 2 dx 1 2 γ β-3ε |v| 2 dx 1 2 + 2|λ| -M χ γ β-3ε |f 1 x | 2 dx 1 2 γ β-3ε |v| 2 dx 1 2
.

(3.4.43) Thus, from (3.4.43) and the fact that u x , v are uniformly bounded in L 2 (0, L) and f 1 x L 2 (0,L) = o(1), we obtain the first estimation in (3.4.40). From (3.4.5), (3.4.6) and the definition of c(•), we have

iλy x -z x = λ -f 3 x in (β -3ε, γ) and iλz -y xx -c 0 v = λ -f 4 in (β -3ε, γ).
Multiplying the above equations by 2χz and 2χy x respectively, integrating over (β -3ε, γ), taking the real part, then using the fact that y x , z are uniformly bounded in L 2 (0, L) and

f 2 L 2 (0,L) = o(1) and f 3 x L 2 (0,L) = o(1), we obtain 2iλ γ β-3ε χy x zdx - γ β-3ε χ(|z| 2 ) x dx = o(|λ| -) (3.
-χ(|z| 2 + |y x | 2 ) γ β-3ε = - γ β-3ε χ (|z| 2 + |y x | 2 )dx + 2c 0 γ β-3ε χvy x dx + o(|λ| -).
Using the definition of χ and Cauchy-Schwarz inequality in the above equation, we obtain

|z(γ)| 2 + |z(β -3ε)| 2 + |y x (γ)| 2 + |y x (β -3ε)| 2 ≤ M χ γ β-3ε (|z| 2 + |y x | 2 )dx + 2c 0 M χ γ β-3ε |v| 2 dx 1 2 γ β-3ε |y x | 2 dx 1 2
+ o(|λ| -).

(3.4.46)

Finally, from (3.4.46) and the fact that v, y x , z are uniformly bounded in L 2 (0, L), we obtain the second estimation in (3.4.40). The proof is thus complete.

L 0 θ |v| 2 + a -1 S b(•) (u, ω) 2 + |z| 2 + |y x | 2 dx + 2a -1 L 0 c(•)θzS b(•) (u, ω)dx - 2 L 0 c(•)θvy x dx = o |λ| -2 .
(3.4.47)

Proof. First, from (3.4.3), we deduce that

iλu x = -v x -λ -f 1 x . ( 3
L 0 θvS b(•) (u, ω)dx = 2iλ L 0 θvu x dx - 2iλa -1 b 0 g β 0 θvu x dx + 2iλa -1 b 0 β 0 θv ∞ 0 g(s)ω x (•, s)dsdx .
Inserting (3.4.48) in the above equation and using the fact that v is uniformly bounded in

L 2 (0, L), f 1 x L 2 (0,L) = o(1), we get 2iλa -1 L 0 θvS b(•) (u, ω)dx = - L 0 θ |v| 2 x dx + 2a -1 b 0 g β 0 θvv x dx + 2iλa -1 b 0 β 0 θv ∞ 0 g(s)ω x (•, s)dsdx + o(|λ| -).
Now, inserting the above equation in (3.4.49), we obtain

- L 0 θ |v| 2 + a -1 S b(•) (u, ω) 2 x dx + 2a -1 L 0 c(•)θzS b(•) (u, ω)dx = - 2a -1 b 0 g β 0 θvv x dx - 2iλa -1 b 0 β 0 θv ∞ 0 g(s)ω x (•, s)dsdx + o |λ| -.
(3.4.50)

From (3.4.7), we deduce that

iλω x (•, s) = ω xs (•, s) -v x -λ -f 5 x (•, s) in (0, β) × (0, ∞). (3.4.51)
Inserting (3.4.51) in the right hand side of (3.4.50), then using integration by parts with respect to s with the help of hypotheses (H) and the fact that ω(•, 0) = 0, we get

- L 0 θ |v| 2 + a -1 S b(•) (u, ω) 2 x dx + 2a -1 L 0 c(•)θzS b(•) (u, ω)dx = - 2b 0 a β 0 θv ∞ 0 -g (s)ω x (•, s)dsdx - 2b 0 aλ β 0 θv ∞ 0 g(s)f 5 x (•, s)dsdx + o |λ| -. (3.4.52)
Using Cauchy-Schwarz inequality, the fact that v is uniformly bounded in L 2 (0, L), the definition of g and (3.4.8), we obtain

         - 2a -1 b 0 β 0 θv ∞ 0 -g (s)ω x (•, s)dsdx = o |λ| -2 , 2a -1 λ -b 0 β 0 θv ∞ 0 g(s)f 5 x (•, s)dsdx = o |λ| -.
(3.4.53)

Inserting (3.4.53) in (3.4.52), then using integration by parts and the fact that θ(0) = θ(L) = 0, we obtain

L 0 θ |v| 2 + a -1 S b(•) (u, ω) 2 dx + 2a -1 L 0 c(•)θzS b(•) (u, ω)dx = o |λ| -2 . (3.4.54)
Next, multiplying (3.4.6) by 2hy x , integrating over (0, L), taking the real part, then using the fact that y x is uniformly bounded in L 2 (0, L) and Let 0 < ε < min α, β-α

f 4 L 2 (0,L) = o(1), we obtain 2iλ L 0 θzy x dx - L 0 θ(|y x | 2 ) x dx - 2 L 0 c(•)θvy x dx = o(|λ| -). ( 3

5

, we fix cut-off functions h 4 , h 5 ∈ C 1 ([0, L]) (see Figure 3.3) such that 0 ≤ h 4 (x) ≤ 1, 0 ≤ h 5 (x) ≤ 1, for all x ∈ [0, L] and

h 4 (x) = 1 if x ∈ [0, α + 2ε], 0 if x ∈ [β -3ε, L],
and

h 5 (x) = 0 if x ∈ [0, α + 2ε], 1 if x ∈ [β -3ε, L],
and set max

x∈[0,L] |h 4 (x)| = M h 4 and max x∈[0,L] |h 5 (x)| = M h 5 . 0 α α + 2ε β -3ε β γ L c 0 b 0 1 h 4 h 5 b(x) c(x)
|v| 2 + |y x | 2 + |z| 2 dx ≤ K 1 |a -1||λ| β-ε α |u x ||y x |dx + o(1), (3.4.59)
a L β |u x | 2 dx + L β-3ε |v| 2 + |y x | 2 + |z| 2 dx ≤ K 2 |a -1||λ| β-ε α |u x ||y x |dx + o(1), (3.4.60)
where

K 1 = 4 |c 0 | 1 + (β -3ε) M h 4 and K 2 = 4 |c 0 | (1 + (L -α + 2ε)M h 5 ).
Proof. First, using the result of Lemma 3.4.6 with θ = xh 4 and the definition of S b(•) (u, ω) and c(•), we obtain

α+2ε 0 |v| 2 + |y x | 2 + |z| 2 dx = -a -1 α+2ε 0 S b 0 (u, ω) 2 dx - β-3ε α+2ε (h 4 + xh 4 ) |v| 2 + a -1 S b 0 (u, ω) 2 + |y x | 2 + |z| 2 dx - 2a -1 c 0 β-3ε α xh 4 zS b 0 (u, ω)dx + 2c 0 β-3ε α xh 4 vy x dx + o |λ| -2 .
Using Cauchy-Schwarz inequality in the above equation, we obtain

α+2ε 0 |v| 2 + |y x | 2 + |z| 2 dx ≤ a -1 α+2ε 0 S b 0 (u, ω) 2 dx + 1 + (β -3ε)M h 4 β-3ε α+2ε |v| 2 + a -1 S b 0 (u, ω) 2 + |y x | 2 + |z| 2 dx + 2c 0 (β -3ε)a -1 β-3ε α |z| 2 dx 1 2 β-3ε α |S b 0 (u, ω)| 2 dx 1 2 + 2c 0 (β -3ε) β-3ε α |v| 2 dx 1 2 β-3ε α |y x | 2 dx 1 2 + o |λ| -2 .
Thus, from the above inequality, Lemmas 3.4.1-3.4.4 and the fact that y x , z are uniformly bounded in L 2 (0, L), we obtain (3.4.59). Next, using the result of Lemma 3.4.6 with θ = (x -L)h 5 and the definition of S b(•) (u, ω) and c(•), we obtain

a L β |u x | 2 dx + L β-3ε |v| 2 + |z| 2 + |y x | 2 dx = -a -1 β β-3ε S b 0 (u, ω) 2 dx - β-3ε α+2ε (h 5 + (x -L)h 5 ) |v| 2 + a -1 S b 0 (u, ω) 2 + |y x | 2 + |z| 2 dx - 2a -1 c 0 β-3ε α+2ε (x -L)h 5 zS b 0 (u, ω)dx + 2c 0 β-3ε α+2ε (x -L)h 5 vy x dx - 2a -1 b 0 c 0 β β-3ε (x -L)z -gu x + ∞ 0 g(s)ω x (•, s)ds dx - 2c 0 γ β-3ε (x -L)zu x dx + 2c 0 γ β-3ε (x -L)vy x dx .
Using Cauchy-Schwarz inequality in the above equation, Lemmas 3.4.1-3.4.4 and the fact that y x , z are uniformly bounded in L 2 (0, L), we obtain

a L β |u x | 2 dx + L β-3ε |v| 2 + |z| 2 + |y x | 2 dx ≤ 4 |c 0 | 1 + (L -α -2ε)M h 5 |a -1||λ| β-ε α |u x ||y x |dx + I + o(1), (3.4.61)
where

I := 2c 0 γ β-3ε (x -L)vy x dx - 2c 0 γ β-3ε (x -L)zu x dx . (3.4.62)
From (3.4.3) and (3.4.5), we have

u x = iλ -1 v x + iλ -( +1) f 1 x and y x = iλ -1 z x + iλ -( +1) f 3 x . (3.4.63)
Inserting (3.4.63) in (3.4.62), then using the fact that v, z are uniformly bounded in L 2 (0, L) and

f 1 x L 2 (0,L) = o(1), f 3 x L 2 (0,L) = o(1)
, we obtain

I = 2c 0 iλ -1 γ β-3ε (x -L)vz x dx - 2c 0 iλ -1 γ β-3ε (x -L)zv x dx +o(|λ| -( +1) ). (3.4.64)
Using integration by parts to the second term in (3.4.64), we obtain Using Cauchy-Schwarz inequality, (3.4.66) and the fact that v, z are uniformly bounded in L 2 (0, L), we obtain

I = 2c 0 iλ -1 γ β-3ε zvdx - 2c 0 iλ -1 [(x -L)zv] γ β-3ε + o(|λ| -( +1) ). ( 3
         2c 0 iλ -1 γ β-3ε zvdx = O |λ| -1 = o(1), - 2c 0 iλ -1 [(x -L)zv] γ β-3ε = O |λ| -1 2 = o(1).
Inserting the above estimations in (3.4.65), we get

I = o(1).
Finally, from the above estimation and (3.4.61), we obtain the desired estimation (3.4.60). The proof is thus complete.

Proof of Theorem 3.4.1. The proof of Theorem 3.4.1 is divided into three steps.

Step 1. Under the hypotheses (H), by taking a = 1 and = 0 in Lemmas 3.4.1-3.4.4, we obtain

         β 0 ∞ 0 g(s)|ω x (•, s)| 2 dsdx = o(1), β 0 |u x | 2 dx = o(1), β-ε ε |v| 2 dx = o(1), β-2ε α+ε |y x | 2 dx = o(1) and β-3ε α+2ε |z| 2 dx = o (1).
(3.4.67)

Step 2. Using the fact that a = 1 and (3.4.67) in Lemma 3.4.7, we obtain

         ε 0 |v| 2 dx = o(1), L β-ε |v| 2 dx = o(1), L β |u x | 2 dx = o(1), α+ε 0 |y x | 2 dx = o(1), L β-2ε |y x | 2 dx = o(1), α+2ε 0 |z| 2 dx = o(1) and L β-3ε |z| 2 dx = o(1). (3.4.68)
Step 3. According to Step 1 and Step 2, we obtain U H = o(1), which contradicts (H 1 ). Therefore, (H 1 ) holds, and so by Theorem 1.3.6, we deduce that system (3.2.2)-(3.2.6) is exponentially stable.

Proof of Theorem 3.4.2. The proof of Theorem 3.4.2 is divided into three steps.

Step 1. Under the hypotheses (H) and a = 1, using the fact that y x is uniformly bounded in L 2 (0, L) and (3.

β 0 ∞ 0 g(s)|ω x (•, s)| 2 dsdx = o(λ -2 ), β 0 |u x | 2 dx = o(λ -2 ), β-ε ε |v| 2 dx = o(|λ| -1 ). (3.4.70)
Step 2. Using the fact that a = 1, y x is uniformly bounded in L 2 (0, L) and (3.4.70) in Lemma 3.4.7, we obtain

α+2ε 0 |v| 2 + |y x | 2 + |z| 2 dx = o(1), (3.4.71)
a L β |u x | 2 dx + L β-3ε |v| 2 + |y x | 2 + |z| 2 dx = o(1). ( 3
       ε 0 |v| 2 dx = o(1), L β-ε |v| 2 dx = o(1), L β |u x | 2 dx = o(1), α+ε 0 |y x | 2 dx = o(1), L β-2ε |y x | 2 dx = o(1), α+2ε 0 |z| 2 dx = o(1) and L β-3ε |z| 2 dx = o (1). (3.4.73)
Step 3. According to Step 1 and Step 2, we obtain U H = o(1), which contradicts (H 1 ). This implies that lim sup λ∈R, |λ|→∞

1 λ 2 (iλI -A) -1 H < ∞.
Finally, according to Theorem 1.3.7, we obtain the desired result. The proof is thus complete.

Lack of exponential stability with global past history damping in case of different speed propagation waves ( a = 1)

This section is independent from the previous ones, here we prove the lack of exponential stability with global past history damping and global coupling. For this aim, we consider the following system:

                     u tt -au xx + ∞ 0 g(s)u xx (x, t -s)ds + y t = 0, (x, t) ∈ (0, L) × (0, ∞), y tt -y xx -u t = 0, (x, t) ∈ (0, L) × (0, ∞), u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0, (u(x, -s), u t (x, 0)) = (u 0 (x, s), u 1 (x)), (x, s) ∈ (0, L) × (0, ∞), (y(x, 0), y t (x, 0)) = (y 0 (x), y 1 (x)), x ∈ (0, L), (3.5.1)
and the general integral term represent a history term with the relaxation function g that is supposed to satisfy the following hypotheses

         g ∈ L 1 ([0, ∞)) ∩ C 1 ([0, ∞)) is a positive function such that g(0) := g 0 > 0, ∞ 0
g(s)ds := g, a := a -g > 0, and g (s) ≤ -mg(s), for some m > 0, ∀s ≥ 0.

(H G ) Now, as in [START_REF] Dafermos | Asymptotic stability in viscoelasticity[END_REF], we introduce the following auxiliary change of variable

ω(x, s, t) := u(x, t) -u(x, t -s), (x, s, t) ∈ (0, L) × (0, ∞) × (0, ∞). (3.5.2)
Then system (3.5.1) becomes

u tt -au xx - ∞ 0 g(s)ω xx (•, s, t) + y t = 0, (x, t) ∈ (0, L) × (0, ∞), (3.5.3)
y tt -y xx -u t = 0, (x, t) ∈ (0, L) × (0, ∞), (3.5
.4) ω t (•, s, t) + ω s (•, s, t) -u t = 0, (x, s, t) ∈ (0, L) × (0, ∞) × (0, ∞), (3.5.5)
with the following boundary conditions

     u(0, t) = u(L, t) = y(0, t) = y(L, t) = 0, t > 0, ω(•, 0, t) = 0, (x, t) ∈ (0, L) × (0, ∞), ω(0, s, t) = ω(L, s, t) = 0, (s, t) ∈ (0, ∞) × (0, ∞), (3.5.6)
and the following initial conditions

     u(•, -s) = u 0 (•, s), u t (•, 0) = u 1 (•), (x, s) ∈ (0, L) × (0, ∞), y(•, 0) = y 0 (•), y t (•, 0) = y 1 (•), x ∈ (0, L), ω 0 (•, s) := ω(•, s, 0) = u 0 (•, 0) -u 0 (•, s), (x, s) ∈ (0, L) × (0, ∞). (3.5.7)
The energy of system (3.5.3)-(3.5.7) is given by

E G (t) = 1 2 L 0 |u t | 2 + a|u x | 2 + |y t | 2 + |y x | 2 dx + 1 2 L 0 ∞ 0 g(s)|ω x (•, s, t)| 2 dsdx. (3.5.8)
Under the hypotheses (H G ) and by letting U = (u, v, y, z, ω) be a regular solution of system (3.5.3)-(3.5.7), then we get with the help of (3.5.6) that

d dt E G (t) = 1 2 L 0 ∞ 0 g (s)|ω x (•, s, t)| 2 dsdx ≤ 0,
which implies that the system (3.5.3)-(3.5.7) is dissipative in the sense that its energy is nonincreasing with respect to time. Now, we define the following Hilbert space H G by

H G = H 1 0 (0, L) × L 2 (0, L) 2 × L 2 g ((0, ∞); H 1 0 (0, L))
, and it is equipped with the following inner product y,y t ,ω(•,s)) , then system (3.5.3)-(3.5.7) can be written as the following first order evolution equation

U, U 1 H G = L 0 au x u 1 x + vv 1 + y x y 1 x + zz 1 dx + L 0 ∞ 0 g(s)ω x (•, s)ω 1 x (•, s)dsdx, where U = (u, v, y, z, ω(•, s)) ∈ H G and U 1 = (u 1 , v 1 , y 1 , z 1 , ω 1 (•, s)) ∈ H G . We define the linear unbounded operator A G : D(A G ) ⊂ H G -→ H G by: D(A G ) =            U = (u, v, y, z, ω(•, s)) ∈ H G | y ∈ H 2 (0, L) ∩ H 1 0 (0, L), v, z ∈ H 1 0 (0, L) au x + ∞ 0 g(s)ω x (•, s)ds x ∈ L 2 (0, L), ω s (•, s) ∈ L 2 g ((0, ∞); H 1 0 (0, L)), ω(•, 0) = 0.            and A G       u v y z ω(•, s)       =         v au x + ∞ 0 g(s)ω x (•, s)ds x -z z y xx + v -ω s (•, s) + v         . Now, if U = (u, u t ,
U t = A G U, U (0) = U 0 , (3.5.9)
where

U 0 = (u 0 (•, 0), u 1 , y 0 , y 1 , ω 0 (•, s)) ∈ H G .
Theorem 3.5.1. Under the hypotheses (H G ). If a = 1, then for any 0 < < 2, we can not expect the energy decay rate t -2 2-for every U 0 ∈ D(A G ).

Proof. 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] (see also Theorem 1.3.6), it is sufficient to show the existence of sequences (λ n

) n ⊂ R * + with λ n → ∞, (U n ) n ⊂ D(A G ) and (F n ) n ⊂ H G such that (iλ n I -A G ) U n = F n is bounded in H G and lim n→∞ λ -2+ n U n H G = ∞.
(3.5.10)

For this aim, take

F n = 0, 0, 0, sin nπx L , 0 and U n = (u n , iλ n u n , y n , iλ n y n , ω n ) such that                λ n = nπ L - L 2nπ(a -1) such that n 2 > L 2 2π 2 (a -1) , u n (x) = A n sin nπx L , y n (x) = B n sin nπx L , ω n (x, s) = A n (1 -e -iλns ) sin nπx L , (3.5.11)
where A n and B n are complex numbers depending on n and determined explicitly in the sequel. Note that this choice is compatible with the boundary conditions. So, its is clear that

λ n > 0, lim n→∞ λ n = ∞, F n is uniformly bounded in H and U n ∈ D(A G ). Next, detailing iλ n U n -A G U n = F n , we get iA n L 2 λ n + λ 2 n L 2 -π 2 n 2 B n = -L 2 , n 2 π 2 (a -g λn ) -λ 2 n L 2 A n + iL 2 λ n B n = 0, (3.5.12)
where g λn = ∞ 0 g(s)e -iλns ds. From the first equation of (3.5.12), we get

A n = i λ n + i(L 2 λ 2 n -π 2 n 2 )B n L 2 λ n . (3.5.13)
Inserting (3.5.13) in the second equation of (3.5.12), we get

B n = (λ 2 n L 2 -(a -g λn )n 2 π 2 ) L 2 -n 4 (a -g λn )π 4 + L 2 π 2 n 2 λ 2 n (a + 1 -g λn ) + L 4 (λ 2 n -λ 4 n )
.

Consequently, the solution of (3.5.12) is given by

A n = i λ n + i(L 2 λ 2 n -π 2 n 2 )B n L 2 λ n and B n = B 1,n 1 + B 2,n λ n g λn + B 3,n , (3.5.14)
where

         B 1,n = L 2 (n 2 π 2 -L 2 λ 2 n ) , B 2,n = L 4 λ 3 n n 2 π 2 (λ 2 n L 2 -n 2 π 2 ) B 3,n = (-π 4 an 4 + L 2 n 2 λ 2 n (a + 1)π 2 + L 4 (λ 2 n -λ 4 n )) λ n n 2 π 2 (n 2 π 2 -L 2 λ 2 n )
. Now, inserting λ n given in (3.5.11) in the above equation, then using asymptotic expansion, we get

B 1,n = a -1 + O(n -2 ), B 2,n = 1 -a L πn + O(n -1 ), B 3,n = O(n -1 ). (3.5.15)
On the other hand, using hypotheses (H G ) and integration by parts, we obtain

λ n g λn = -ig 0 -i ∞ 0 g (s)e -iλns ds.
It is clear from Riemann-Lebesgue Lemma that the second term in the above equation goes to zero as λ n → ∞. Thus, we obtain

λ n g λn = -ig 0 + o(1). (3.5.16)
Substituting (3.5.15) and (3.5.16) in (3.5.14), we get

A n = O(1) and B n = - i(a -1) 2 g 0 L + o(1) nπ.
Therefore, from the above equation and (3.5.16), we get

z n (x) = iλ n B n sin nπx L = (a -1) 2 g 0 L 2 + o(1) n 2 π 2 sin nπx L . Consequently, L 0 |z n | 2 dx 1 2 ∼ L 2 (a -1) 2 g 0 L 2 + o(1) n 2 π 2 . Since U n H ≥ L 0 |z n | 2 dx 1 2 ∼ L 2 (a -1) 2 g 0 L 2 + o(1) n 2 π 2 ∼ λ 2 n ,
then for all 0 < < 2, we have

λ -2+ n U n H 1 ∼ λ n → ∞ as n → ∞,
hence, we get (3.5.10). Consequently, we cannot expect the energy decay rate t -2 2-. The proof is thus complete. Remark 3.5.1. In [START_REF] Almeida | Lack of exponential decay of a coupled system of wave equations with memory[END_REF] and [START_REF] Cordeiro | Optimal polynomial decay for a coupled system of wave with past history[END_REF], the authors proved the lack of exponential stability of a coupled wave equations system with past history damping by taking a particular relaxation function g(s) = e -µs such that s ∈ R + and µ > 1.

Conclusion and Future Works

We have studied the stabilization of a locally coupled wave equations with local viscoelastic damping of past history type acting only in one equation via non-smooth coefficients. We proved the strong stability of the system by using Arendt-Batty criteria. We established the exponential stability of the solution if the waves have the same speed propagation (i.e. a = 1). In the case a = 1, we proved that the energy of our system decays polynomially with the rate t -1 . Lack of exponential stability result has been proved in case that the speeds of wave propagation are different with a global damping and a global coupling (i.e. a = 1 and b = c = 1). According to Theorem 3.5.1, we can conjecture that the energy decay rate t -1 is optimal but this question remains open. Moreover, it would be interesting to 1. study system (3.1.1) in the multidimensional case, 2. obtain the decay rates of system (3.1.1) for a much wider class of relaxation functions g, like in [START_REF] Guesmia | Asymptotic stability of abstract dissipative systems with infinite memory[END_REF][START_REF] Guesmia | Asymptotic behavior for coupled abstract evolution equations with one infinite memory[END_REF] and using the recent results from [START_REF] Rozendaal | Optimal rates of decay for operator semigroups on hilbert spaces[END_REF],

3. study system (3.1.1) with local internal past history damping, in other words, by only assuming that b is positive on a non empty subinterval of (0, L) that could be away from the boundary. The Bresse system is a model for arched beams (see Fig. 4.2 for an illustration), see [START_REF] Lagnese | Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures[END_REF]Chap. 6]. It can be expressed by the equations of motion:

α β L 0 d 0 d(x)
     ρ 1 ϕ tt = Q x + lN, ρ 2 ψ tt = M x -Q, ρ 1 w tt = N x -lQ, (4.1.5) where N = k 3 (w x -lϕ) + d(x)(w tx -lϕ t ) is the axial force, Q = k 1 (ϕ x + ψ + lw)
is the shear force, and M = k 2 ψ x is the bending moment. The functions ϕ, ψ, and w are respectively the vertical, shear angle, and longitudinal displacements. Here

ρ 1 = ρA, ρ 2 = ρI, k 1 = kGA, k 3 = EA, k 2 = EI and l = R -1
, in which ρ is the density of the material, E the modulus of the elasticity, G the shear modulus, k the shear factor, A the cross-sectional area, I the second moment of area of the cross section, R the radius of the curvature, and l the curvature.

There are several publications concerning the stabilization of Bresse system with different kinds of damping (see [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], [START_REF] Afilal | On the exponential and polynomial stability for a linear Bresse system[END_REF], [START_REF] Boussouira | Stability to weak dissipative Bresse system[END_REF], [START_REF] De Lima | Stability of thermoelastic Bresse systems[END_REF], [START_REF] Arwadi | On the stabilization of the Bresse beam with Kelvin-Voigt damping[END_REF], [START_REF] Fatori | Stability conditions to Bresse systems with indefinite memory dissipation[END_REF], [START_REF] Fatori | The optimal decay rate for a weak dissipative Bresse system[END_REF], [START_REF] Fatori | Rates of decay to weak thermoelastic Bresse system[END_REF], [START_REF] Guesmia | Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements[END_REF], [START_REF] Guesmia | Bresse system with infinite memories[END_REF] [79], [START_REF] Najdi | Weakly locally thermal stabilization of Bresse systems[END_REF], [START_REF] Noun | Stabilisation faible interne locale de système élastique de Bresse[END_REF] and [START_REF] Wehbe | Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks[END_REF]). We note that by neglecting w (l → 0) in (4.1.5), the Bresse system reduces to the following conservative Timoshenko system:

ρ 1 ϕ tt -k 1 (ϕ x + ψ) x = 0, ρ 2 ψ tt -k 2 ψ xx + k 1 (ϕ x + ψ) = 0.
There are also several publications concerning the stabilization of Timoshenko system with different kinds of damping (see [START_REF] Akil | The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization[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] and [START_REF] Wehbe | Stabilization of the uniform Timoshenko beam by one locally distributed feedback[END_REF]).

In the recent years, many researchers showed interest in problems involving Kelvin-Voigt damping where different types of stability, depending on the smoothness of the damping coefficients, has been showed (see [START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF], [START_REF] Alves | The asymptotic behavior of the linear transmission problem in viscoelasticity[END_REF], [START_REF] Hassine | Stability of elastic transmission systems with a local Kelvin-Voigt damping[END_REF], [START_REF] Hassine | Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping[END_REF], [START_REF] Huang | On the Mathematical Model for Linear Elastic Systems with Analytic Damping[END_REF], [START_REF] Liu | Spectrum and Stability for Elastic Systems with Global or Local Kelvin-Voigt Damping[END_REF], [START_REF] Liu | Stability of a String with Local Kelvin-Voigt Damping and Nonsmooth Coefficient at Interface[END_REF], [START_REF] Oquendo | Frictional versus Kelvin-Voigt damping in a transmission problem[END_REF] and [START_REF] Rivera | Stability to localized viscoelastic transmission problem[END_REF]). Moreover, there is a number of new results concerning systems with local Kelvin-Voigt damping and non-smooth coefficients at the interface (see [START_REF] Akil | Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay[END_REF], [START_REF] Wehbe | Stability results of an elastic/viscoelastic transmission problem of locally coupled waves with non smooth coefficients[END_REF], [START_REF] Ghader | Optimal polynomial stability of a string with locally distributed kelvin-voigt damping and nonsmooth coefficient at the interface[END_REF], [START_REF] Ghader | Stability results for an elastic-viscoelastic wave equation with localized kelvin-voigt damping and with an internal or boundary time delay[END_REF], [START_REF] Ghader | A transmission problem for the Timoshenko system with one local Kelvin-Voigt damping and non-smooth coefficient at the interface[END_REF], [START_REF] Hayek | A transmission problem of a system of weakly coupled wave equations with kelvin-voigt dampings and non-smooth coefficient at the interface[END_REF] and [START_REF] Wehbe | Stability of n-d transmission problem in viscoelasticity with localized kelvin-voigt damping under different types of geometric conditions[END_REF]).

Among this vast literature let us recall some specific results on the Bresse systems.

In 2017, Guesmia in [START_REF] Guesmia | Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements[END_REF] studied the stability of Bresse system with one infinite memory in the longitudinal displacement (i.e. third equation) under Dirichlet-Neumann-Neumann boundary conditions, he established some stability results under a smallness condition on l and on ∞ 0 g(s)ds, where l is the curvature and g is the memory kernel. In [START_REF] Afilal | On the exponential and polynomial stability for a linear Bresse system[END_REF] studied the stability of Bresse system with global frictional damping in the longitudinal displacement, by considering the following system on (0, 1) × (0, ∞):

     ρ 1 ϕ tt -k 1 (ϕ x + ψ + lw) x -lk 3 (w x -lϕ) = 0, ρ 2 ψ tt -k 2 ψ xx + k 1 (ϕ x + ψ + lw) = 0, ρ 1 w tt -k 3 (w x -lϕ) + lk 1 (ϕ x + ψ + lw) + δw t = 0, (4.1.6)
with the initial conditions (4.1.3) where L = 1 and under mixed boundary conditions of the form:

ϕ(0, t) = ψ x (0, t) = w x (0, t) = 0, in (0, ∞), ϕ x (1, t) = ψ(1, t) = w(1, t) = 0, in (0, ∞),
where δ is a positive real number, they assumed that:

l = π 2 + mπ, ∀ m ∈ N. (4.1.7)
They proved under (4.1.7), the strong stability of system (4.1.6) provided that the curvature l satisfies:

l 2 = ρ 2 k 3 + ρ 1 k 2 ρ 2 k 3 π 2 + mπ 2 + ρ 1 k 1 ρ 2 (k 1 + k 3 ) , ∀m ∈ Z. ( 4
the strong stability of system (4.1.6) on (0, L) × (0, ∞) under Dirichlet-Neumann-Neumann boundary conditions provided that:

k 1 ρ 1 -ρ 2 (k 3 + k 1 )l 2 ≥ 0 or 0 < ρ 2 (k 3 + k 1 )l 2 -k 1 ρ 1 = ρ 1 ρ 2 (k 3 + k 1 ) k 3 k 3 ρ 1 n 2 + k 2 ρ 2 m 2 π 2 L 2 , (4.1.10)
for all m ∈ N and n ∈ N . Also, they established under (4.1.9) and (4.1.10) the exponential stability of system (4.1.6) on (0, L) × (0, ∞) if and only if

ρ 1 ρ 2 = k 1 k 2 and k 1 = k 3 . (4.1.11)
Moreover, they used the previous results (i.e. strong and exponential stability of (4.1.6) on (0, L) × (0, ∞)) to obtain under (4.1.9), (4.1.10) and (4.1.11) the exponential stability of Bresse system with indefinite memory in the longitudinal displacement under Dirichlet-Neumann-Neumann boundary conditions.

In 2019, El Arwadi and Youssef in [START_REF] Arwadi | On the stabilization of the Bresse beam with Kelvin-Voigt damping[END_REF] studied the stabilization of the Bresse beam with three global Kelvin-Voigt damping under fully Dirichlet boundary conditions, they established an exponential energy decay rate. In 2020, Gerbi et al. in [START_REF] Gerbi | Stabilization of non-smooth transmission problem involving Bresse systems[END_REF] studied the stabilization of non-smooth transmission problem involving Bresse systems with fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions, by considering system (4.1.5) on (0, L) × (0, ∞) with

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 ,
where D 1 , D 2 and D 3 are bounded positive functions over (0, L). They established: Analytic stability in the case of three global Kelvin-Voigt dampings (i.e.

D i ∈ L ∞ (0, L), D i ≥ d 0 > 0 in (0, L), i = 1, 2, 3).
Exponential stability in the case of three local Kelvin-Voigt dampings with smooth coefficients at the interface (i.e.

D i ∈ W 1,∞ (0, L), D i ≥ d 0 > 0 in ∅ = ω := (α, β) ⊂ (0, L), i = 1, 2, 3).
Polynomial energy decay rate of order t -1 in the case of three local Kelvin-Voigt dampings with non-smooth coefficients at the interface (i.e.

D i ∈ L ∞ (0, L), D i ≥ d i 0 > 0 in (α i , β i ) ⊂ (0, L), i = 1, 2, 3, and 3
i=1 (α i , β i ) = ω).
Polynomial stability energy decay rate of order t -1

But to the best of our knowledge, it seems that no result in the literature exists concerning the case of Bresse system with only one discontinuous local internal Kelvin-Voigt damping on the axial force, especially under fully Dirichlet boundary conditions and without any condition on the curvature l. The goal of the present chapter is to fill this gap by studying the stability of system (4.1.1)-(4.1.3). This chapter is organized as follows: In Section 4.2, we prove the well-posedness of our system by using semigroup approach. In Section 4.3, following a general criteria of Arendt Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. Finally, in Section 4.4, by using the frequency domain approach combining with a specific multiplier method, we prove that the energy of our system decays polynomially with the rates:

       t -1 if k 1 ρ 1 = k 2 ρ 2 , t -1 2 if k 1 ρ 1 = k 2 ρ 2 .

Well-posedness of the system

In this section, we will establish the well-posedness of system (4.1.1)-(4.1.3) by using semigroup approach. The energy of system (4.1.1)-(4.1.3) 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 + k 2 |ψ x | 2 + k 3 |w x -lϕ| 2 dx.
Let (ϕ, ϕ t , ψ, ψ t , w, w t ) be a regular solution of system (4.

E (t) = - L 0 d(x)|w tx -lϕ t | 2 dx = -d 0 β α |w tx -lϕ t | 2 dx ≤ 0. (4.2.1)
From (4.2.1), system (4.1.1)-(4.1.3) is dissipative in the sense that its energy is non-increasing with respect to time. Now, we define the following Hilbert space H by:

H := H 1 0 (0, L) × L 2 (0, L) 3 .
The Hilbert space H is equipped with the following inner product and norm

(U, U 1 ) H = L 0 k 1 (v 1 x + v 3 + lv 5 )( v 1 x + v 3 + l v 5 ) + ρ 1 v 2 v 2 + k 2 v 3 x v 3 x + ρ 2 v 4 v 4 + k 3 (v 5 x -lv 1 )( v 5 x -l v 1 )dx + ρ 1 v 6 v 6 dx
and

U 2 H = L 0 k 1 |v 1 x + v 3 + lv 5 | 2 + ρ 1 |v 2 | 2 + k 2 |v 3 x | 2 + ρ 2 |v 4 | 2 + k 3 |v 5 x -lv 1 | 2 + ρ 1 |v 6 | 2 dx. (4.2.2) Where U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ H and U = ( v 1 , v 1 , v 2 , v 3 , v 4 , v 5 , v 6
) ∈ H. Now, we define the linear unbounded operator A : D(A) ⊂ H -→ H by:

D(A) =    U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ H | v 1 , v 3 ∈ H 2 (0, L) ∩ H 1 0 (0, L) v 2 , v 4 , v 6 ∈ H 1 0 (0, L), k 3 v 5 x + d(x)(v 6 x -lv 2 ) x ∈ L 2 (0, L)    (4.2.3) and A         v 1 v 2 v 3 v 4 v 5 v 6         =               v 2 k 1 ρ 1 (v 1 x + v 3 + lv 5 ) x + lk 3 ρ 1 (v 5 x -lv 1 ) + ld(x) ρ 1 (v 6 x -lv 2 ) v 4 k 2 ρ 2 v 3 xx - k 1 ρ 2 (v 1 x + v 3 + lv 5 ) v 6 1 ρ 1 k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) x - lk 1 ρ 1 (v 1 x + v 3 + lv 5 )               , (4.2.4
)

for all U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A).
In this sequel, • will denote the usual norm of L 2 (0, L).

Remark 4.2.1. From Poincaré inequality, we deduce that there exists a positive constant c 1 such that

k 1 v 1 x + v 3 + lv 5 2 + k 2 v 3 x 2 + k 3 v 5 x -lv 1 2 ≤ c 1 v 1 x 2 + v 3 x 2 + v 5 x 2 ,
for all (v 1 , v 3 , v 5 ) ∈ (H 1 0 (0, L)) 3 . Moreover, we can show by a contradiction argument that there exists a positive constant c 2 such that

c 2 v 1 x 2 + v 3 x 2 + v 5 x 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 ,
for all (v 1 , v 3 , v 5 ) ∈ (H 1 0 (0, L)) 3 . Therefore, the norm defined in (4.2.2) is equivalent to the usual norm of H. Now, if U = (ϕ, ϕ t , ψ, ψ t , w, w t ) , then system (4.1.1)-(4.1.3) can be written as the following first order evolution equation

U t = AU, U (0) = U 0 , (4.2.5)
where U 0 = (ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , w 0 , w 1 ) ∈ H. Proof. For all U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A), we have

(AU, U ) H = - L 0 d(x) v 6 x -lv 2 2 dx = -d 0 β α v 6 x -lv 2 2 dx ≤ 0. (4.2.6)
which implies that A is dissipative. Let us prove that A is maximal. To this aim, let

F = (f 1 , f 2 , f 3 , f 4 , f 5 , f 6 ) ∈ H, we look for U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A) unique solution of -AU = F. (4.2.7)
Detailing (4.2.7), we obtain

-v 2 = f 1 , (4.2.8) -k 1 v 1 x + v 3 + lv 5 x -lk 3 (v 5 x -lv 1 ) -ld(x)(v 6 x -lv 2 ) = ρ 1 f 2 , (4.2.9) -v 4 = f 3 , (4.2.10) -k 2 v 3 xx + k 1 (v 1 x + v 3 + lv 5 ) = ρ 2 f 4 , (4.2.11) -v 6 = f 5 , (4.2.12) -k 3 v 5 x -lv 1 + d(x)(v 6 x -lv 2 ) x + lk 1 (v 1 x + v 3 + lv 5 ) = ρ 1 f 6 , (4.2.13)
with the following boundary conditions respectively, integrating over (0, L), then using formal integrations by parts, we obtain

v 1 (0) = v 1 (L) = v 3 (0) = v 3 (L) = v 5 (0) = v 5 (L) = 0. ( 4
-k 1 v 1 x + v 3 + lv 5 x -lk 3 (v 5 x -lv 1 ) = ρ 1 f 2 + ld(x)(-f 5 x + lf 1 ), (4.2.15) -k 2 v 3 xx + k 1 (v 1 x + v 3 + lv 5 ) = ρ 2 f 4 , (4.2.16) -k 3 v 5 x -lv 1 + d(x)(-f 5 x + lf 1 ) x + lk 1 (v 1 x + v 3 + lv 5 ) = ρ 1 f 6 . (4.2.17) Let (φ 1 , φ 2 , φ 3 ) ∈ (H 1 0 (0, L)) 3 .
B((v 1 , v 3 , v 5 ), (φ 1 , φ 2 , φ 3 )) = L((φ 1 , φ 2 , φ 3 )), ∀(φ 1 , φ 2 , φ 3 ) ∈ H 1 0 (0, L) 3 , (4.2.18)
where

B((v 1 , v 3 , v 5 ), (φ 1 , φ 2 , φ 3 )) = k 1 L 0 (v 1 x + v 3 + lv 5 )φ 1 x dx -lk 3 L 0 (v 5 x -lv 1 )φ 1 dx + k 2 L 0 v 3 x φ 2 x dx + k 1 L 0 (v 1 x + v 3 + lv 5 )φ 2 dx + k 3 L 0 (v 5 x -lv 1 )φ 3 x dx + lk 1 L 0 (v 1 x + v 3 + lv 5 )φ 3 dx and L((φ 1 , φ 2 , φ 3 )) = ρ 1 L 0 f 2 φ 1 dx + l L 0 d(x)(-f 5 x + lf 1 )φ 1 dx + ρ 2 L 0 f 4 φ 2 dx + L 0 d(x)(f 5 x -lf 1 )φ 3 x dx + ρ 1 L 0 f 6 φ 3 dx.
It is easy to see that B is a sesquilinear and continuous form on (H 1 0 (0, L))

3 × (H 1 0 (0, L)) 3 and

L is an antilinear and continuous form on (H 1 0 (0, L)) 3 . In fact, from Remark 4.2.1, we deduce that there exists a positive constant c such that

B((v 1 , v 3 , v 5 ), (v 1 , v 3 , v 5 )) = k 1 v 1 x + v 3 + lv 5 2 + k 2 v 3 x 2 + k 3 v 5 x -lv 1 2 ≥ c ( v 1 x 2 + v 3 x 2 + v 5 x 2 ) = c (v 1 , v 3 , v 5 ) 2 (H 1 0 (0,L)) 3 . (4.2.19)
Thus, B is a coercive form on (H 1 0 (0, L))

3 × (H 1 0 (0, L)) 3 . Then, it follows by Lax-Milgram theorem that (4.2.18) admits a unique solution (v 1 , v 3 , v 5 ) ∈ (H 1 0 (0, L)) 3 . By taking test-functions (φ 1 , φ 2 , φ 3 ) ∈ (D(0, L)) 3 , we see that (4.2.15)-(4.2.17) hold in the distributional sense, from which we deduce that (v

1 , v 3 ) ∈ (H 2 (0, L) ∩ H 1 0 (0, L)) 2 , while [k 3 v 5 x + d(x)(v 6 x -lv 2 )] x ∈ L 2 (0, L). Consequently, U = (v 1 , -f 1 , v 3 , -f 3 , v 5 , -f 5 ) ∈ D(A)
is the unique solution of (4.2.7). Then, A is an isomorphism and since ρ (A) is open set of C (see Theorem 1.1.13), we easily get R(λI -A) = H for a sufficiently small λ > 0. This, together with the dissipativeness of A, imply that D (A) is dense in H and that A is m-dissipative in H (see Theorems 1.2.6, 1.2.9). The proof is thus complete.

According to Lumer-Philips theorem (see Theorem 1.2.8), Proposition 4.2.1 implies that the operator A generates a C 0 -semigroup of contractions e tA in H which gives the well-posedness of (4.2.5). Then, we have the following result: Theorem 4.2.1. For all U 0 ∈ H, system (4.2.5) admits a unique weak solution

U (t) = e tA U 0 ∈ C 0 (R + , H).
Moreover, if U 0 ∈ D(A), then the system (4.2.5) admits a unique strong solution

U (t) = e tA U 0 ∈ C 0 (R + , D(A)) ∩ C 1 (R + , H).

Strong Stability

In this section, we will prove the strong stability of system (4.1.1)-(4.1.3). The main result of this section is the following theorem. Proof. From Proposition 4.2.1, we have 0 ∈ ρ(A). We still need to show the result for λ ∈ R * . For this aim, suppose that λ = 0 and let

U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A) such that AU = iλU. (4.3.1)
Equivalently, we have the following system

v 2 = iλv 1 , (4.3.2) k 1 (v 1 x + v 3 + lv 5 ) x + lk 3 (v 5 x -lv 1 ) + ld(x)(v 6 x -lv 2 ) = iλρ 1 v 2 , (4.3.3) v 4 = iλv 3 , (4.3.4) k 2 v 3 xx -k 1 (v 1 x + v 3 + lv 5 ) = iλρ 2 v 4 , (4.3.5) v 6 = iλv 5 , (4.3.6) k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) x -lk 1 (v 1 x + v 3 + lv 5 ) = iλρ 1 v 6 . ( 4
ρ 1 λ 2 v 1 + k 1 (v 1 x + v 3 + lv 5 ) x = 0 in (α, β), (4.3.11) ρ 2 λ 2 v 3 + k 2 v 3 xx -k 1 (v 1 x + v 3 + lv 5 ) = 0 in (α, β), (4.3.12) ρ 1 λ 2 v 5 -lk 1 (v 1 x + v 3 + lv 5 ) = 0 in (α, β). (4.3.13)
Deriving (4.3.13) with respect to x, we get

ρ 1 λ 2 v 5 x -lk 1 (v 1 x + v 3 + lv 5 ) x = 0 in (α, β).
Inserting (4.3.11) in the above equation, we get

ρ 1 λ 2 (v 5
x + lv 1 ) = 0 in (α, β) and consequently as λ = 0, we get v 5

x + lv 1 = 0 in (α, β).

ρ 1 λ 2 v 1 + k 1 (v 1 x + v 3 + lv 5 ) x + lk 3 (v 5 x -lv 1 ) = 0 in (0, α) ∪ (β, L), (4.3.17) ρ 2 λ 2 v 3 + k 2 v 3 xx -k 1 (v 1 x + v 3 + lv 5 ) = 0 in (0, α) ∪ (β, L), (4.3.18) ρ 1 λ 2 v 5 + k 3 (v 5 x -lv 1 ) x -lk 1 (v 1 x + v 3 + lv 5 ) = 0 in (0, α) ∪ (β, L). (4.3.19) Let V = (v 1 x , v 1 xx , v 3 x , v 3 xx , v 5
x , v 5 xx ) . From (4.3.15), (4.3.16) and the regularity of v i , i ∈ {1, 3, 5}, we have V (α) = 0. Now, by deriving system (4.3.17)-(4.3.19) with respect to x in (0, α), we deduce that

V x = A λ V in (0, α), (4.3.20)
where

A λ =          0 1 0 0 0 0 l 2 k 3 -λ 2 ρ 1 k 1 0 0 -1 0 -l(1 + k 3 k 1 ) 0 0 0 1 0 0 0 k 1 k 2 k 1 -ρ 2 λ 2 k 2 0 lk 1 k 2 0 0 0 0 0 0 1 0 l( k 1 k 3 + 1) l k 1 k 3 0 l 2 k 1 -ρ 1 λ 2 k 3 0          . (4.3.21)
Proof. From Proposition 4.2.1, we have 0 ∈ ρ(A). We still need to show the result for λ ∈ R * . For this aim, let

F = (f 1 , f 2 , f 3 , f 4 , f 5 , f 6 ) ∈ H, we want to find U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A) solution of (iλI -A)U = F. (4.3.28)
Detailing (4.3.28), we obtain

iλv 1 -v 2 = f 1 , (4.3.29) iλv 2 - k 1 ρ 1 v 1 x + v 3 + lv 5 x - lk 3 ρ 1 (v 5 x -lv 1 ) - ld(x) ρ 1 (v 6 x -lv 2 ) = f 2 , (4.3.30) iλv 3 -v 4 = f 3 , (4.3.31) iλv 4 - k 2 ρ 2 v 3 xx + k 1 ρ 2 (v 1 x + v 3 + lv 5 ) = f 4 , (4.3.32)
iλv 5 -v 6 = f 5 , (4.3.33)

iλv 6 - 1 ρ 1 k 3 v 5 x -lv 1 + d(x)(v 6 x -lv 2 ) x + lk 1 ρ 1 (v 1 x + v 3 + lv 5 ) = f 6 , (4.3.34)
with the following boundary conditions

v 1 (0) = v 1 (L) = v 3 (0) = v 3 (L) = v 5 (0) = v 5 (L) = 0. (4.3.35) Inserting v 2 = iλv 1 -f 1 , v 4 = iλv 3 -f 3 and v 6 = iλv 5 -
-λ 2 v 1 - k 1 ρ 1 v 1 x + v 3 + lv 5 x - lk 3 ρ 1 (v 5 x -lv 1 ) - iλld(x) ρ 1 (v 5 x -lv 1 ) = g 1 , (4.3.36) -λ 2 v 3 - k 2 ρ 2 v 3 xx + k 1 ρ 2 (v 1 x + v 3 + lv 5 ) = g 2 , (4.3.37) -λ 2 v 5 - 1 ρ 1 k 3 (v 5 x -lv 1 ) + iλd(x)(v 5 x -lv 1 ) x + lk 1 ρ 1 (v 1 x + v 3 + lv 5 ) = g 3 , (4.3.38) where            g 1 := iλf 1 + f 2 + ld(x) ρ 1 (-f 5 x + lf 1 ) ∈ H -1 (0, L), g 2 := iλf 3 + f 4 ∈ H -1 (0, L), g 3 := iλf 5 + f 6 + ρ -1 1 d(x)(-f 5 x + lf 1 ) x ∈ H -1 (0, L). (4.3.39)
For all U = (v 1 , v 3 , v 5 ) ∈ H := (H 1 0 (0, L)) 3 , we define the linear operator L : H -→ H := (H -1 (0, L))

3 by:

LU =         - k 1 ρ 1 v 1 x + v 3 + lv 5 x - lk 3 ρ 1 (v 5 x -lv 1 ) - iλld(x) ρ 1 (v 5 x -lv 1 ) - k 2 ρ 2 v 3 xx + k 1 ρ 2 (v 1 x + v 3 + lv 5 ) - 1 ρ 1 k 3 (v 5 x -lv 1 ) + iλd(x)(v 5 x -lv 1 ) x + lk 1 ρ 1 (v 1 x + v 3 + lv 5 )         . (4.3.40)
Let us prove that the operator L is an isomorphism. For this aim, take the duality bracket

•, • H ,H of (4.3.40) with Ψ := (ρ 1 ψ 1 , ρ 2 ψ 2 , ρ 1 ψ 3 ) ∈ H, we obtain LU, Ψ H ,H = -k 1 v 1 x + v 3 + lv 5 x -lk 3 (v 5 x -lv 1 ) -iλld(x)(v 5 x -lv 1 ), ψ 1 H -1 (0,L),H 1 0 (0,L) + -k 2 v 3 xx + k 1 (v 1 x + v 3 + lv 5 ), ψ 2 H -1 (0,L),H 1 0 (0,L) + -k 3 (v 5 x -lv 1 ) + iλd(x)(v 5 x -lv 1 ) x + lk 1 (v 1 x + v 3 + lv 5 ), ψ 3
H -1 (0,L),H 1 0 (0,L) . Consequently, we obtain

LU, Ψ H ,H = k 1 L 0 (v 1 x + v 3 + lv 5 )ψ 1 x dx -lk 3 L 0 (v 5 x -lv 1 )ψ 1 dx -iλl L 0 d(x)(v 5 x -lv 1 )ψ 1 dx + k 2 L 0 v 3 x ψ 2 x dx + k 1 L 0 (v 1 x + v 3 + lv 5 )ψ 2 dx + k 3 L 0 (v 5 x -lv 1 )ψ 3 x dx + iλ L 0 d(x)(v 5 x -lv 1 )ψ 3 x dx + lk 1 L 0 (v 1 x + v 3 + lv 5 )ψ 3 dx,
defines a continuous sesquilinear form which is coercive on H. Indeed, from Remark 4.2.1, we deduce that there exists a positive constant c such that

LU, U H ,H = k 1 v 1 x + v 3 + lv 5 2 + k 2 v 3 x 2 + k 3 v 5 x -lv 1 2 ≥ c ( v 1 x 2 + v 3 x 2 + v 5 x 2 ) = c (v 1 , v 3 , v 5 ) 2 H = c U 2
H . Therefore, by using Lax-Milgram theorem, we deduce that L is an isomorphism from H onto H . Now, let U = (v 1 , v 3 , v 5 ) and G = (g 1 , g 2 , g 3 ) , then system (4.3.36)-(4.3.38) can be transformed into the following form:

(I -λ 2 L -1 )U = L -1 G. ( 4
I -λ 2 L -1 is injective. Let V = (v 1 , v 3 , v 5 ) ∈ H such that V -λ 2 L -1 V = 0 ⇐⇒ λ 2 V -LV = 0. (4.3.42)
Equivalently, we have

-λ 2 v 1 - k 1 ρ 1 v 1 x + v 3 + lv 5 x - lk 3 ρ 1 (v 5 x -lv 1 ) - iλld(x) ρ 1 (v 5 x -lv 1 ) = 0, (4.3.43) -λ 2 v 3 - k 2 ρ 2 v 3 xx + k 1 ρ 2 (v 1 x + v 3 + lv 5 ) = 0, (4.3.44) -λ 2 v 5 - 1 ρ 1 (k 3 + iλd(x))v 5 x -l(k 3 + iλ)v 1 x + lk 1 ρ 1 (v 1 x + v 3 + lv 5 ) = 0. (4.3.45) It is easy to see that if V = (v 1 , v 2 , v 3
) is a solution of (4.3.43)-(4.3.45), then the vector W defined by W = (v 1 , iλv 1 , v 3 , iλv 3 , v 5 , iλv 5 ) belongs to D(A) and satisfies iλW -AW = 0. Thus, by using Lemma 4.3.1, we obtain W = 0 and consequently I -λ 2 L -1 is injective. Thanks to Fredholm's alternative, (4.3.41) admits a unique solution U ∈ H and

v 1 , v 3 ∈ H 2 (0, L), [k 3 v 5 x + d(x)(iλv 5 x -f 5 x -l(iλv 1 -f 1 ))] x ∈ L 2 (0, L).
Finally, by setting

v 2 = iλv 1 -f 1 , v 4 = iλv 3 -f 3 and v 6 = iλv 5 -f 5 ,

Polynomial Stability

In this section, we will prove the polynomial stability of system (4.1.1)-(4.1.3) with different rates. The main results of this section are the following theorems.

Theorem 4.4.1. If k 1 ρ 1 = k 2 ρ 2 ,
then, for all U 0 ∈ D(A), there exists a constant C > 0 independent of U 0 such that

E(t) ≤ C t U 0 2 D(A) , t > 0. Theorem 4.4.2. If k 1 ρ 1 = k 2 ρ 2 ,
then, for all U 0 ∈ D(A), there exists a constant C > 0 independent of U 0 such that

E(t) ≤ C √ t U 0 2 D(A) , t > 0.
Since

:= (v 1,n , v 2,n , v 3,n , v 4,n , v 5,n , v 6,n ) ) n≥1 ⊂ R * × D(A) with |λ n | → ∞ as n → ∞ and U n H = (v 1,n , v 2,n , v 3,n , v 4,n , v 5,n , v 6,n ) H = 1, ∀n ≥ 1, (4.4.1) such that (λ n ) (iλ n I -A)U n = F n := (f 1,n , f 2,n , f 3,n , f 4,n , f 5,n , f 6,n ) → 0 in H as n → ∞. (4.4.2)
For simplicity, we drop the index n. Equivalently, from (4.4.2), we have

iλv 1 -v 2 = λ -f 1 , (4.4.3) iλρ 1 v 2 -k 1 (v 1 x + v 3 + lv 5 ) x -lk 3 (v 5 x -lv 1 ) -ld(x)(v 6 x -lv 2 ) = ρ 1 λ -f 2 , (4.4.4) iλv 3 -v 4 = λ -f 3 , (4.4.5) iλρ 2 v 4 -k 2 v 3 xx + k 1 (v 1 x + v 3 + lv 5 ) = ρ 2 λ -f 4 , (4.4.6) iλv 5 -v 6 = λ -f 5 , (4.4.7) iλρ 1 v 6 -k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) x + lk 1 (v 1 x + v 3 + lv 5 ) = ρ 1 λ -f 6 . ( 4
λ 2 ρ 1 v 1 + k 1 (v 1 x + v 3 + lv 5 ) x + lk 3 (v 5 x -lv 1 ) + ld(x)(v 6 x -lv 2 ) = h 1 , (4.4.9) λ 2 ρ 2 v 3 + k 2 v 3 xx -k 1 (v 1 x + v 3 + lv 5 ) = h 2 , (4.4.10) λ 2 ρ 1 v 5 + k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) x -lk 1 (v 1 x + v 3 + lv 5 ) = h 3 . (4.4.11)
where

h 1 = -ρ 1 λ -f 2 -iρ 1 λ -+1 f 1 , h 2 = -ρ 2 λ -f 4 -iρ 2 λ -+1 f 3 and h 3 = -ρ 1 λ -f 6 -iρ 1 λ -+1 f 5 .
Here we will check the condition (H) by finding a contradiction with (4.4.1) by showing U H = o(1). For clarity, we divide the proof into several Lemmas. From the above system and the fact that ∈ {2, 4}, U H = 1 and F H = o(1), we remark that

v 1 = O |λ| -1 , v 3 = O |λ| -1 , v 5 = O |λ| -1 , v 1 xx = O (|λ|) , v 3 xx = O (|λ|) and k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) x = O (|λ|) .
(4.4.12)

Also, from Poincaré inequality and the fact that F H = o(1), we remark that Proof. First, taking the inner product of (4.4.2) with U in H and using (4.2.6), we get

f 1 f 1 x = o(1), f 3 f 3 x = o(1) and f 5 f 5 x = o(1). (4.4.13) Lemma 4.4.1. If k 1 ρ 1 = k 2 ρ 2 and = 2 or k 1 ρ 1 = k 2 ρ 2 and = 4 . Then, the solution U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A) of
L 0 d(x) v 6 x -lv 2 2 dx = d 0 β α v 6 x -lv 2 2 dx = -(AU, U ) H = λ -(F, U ) H ≤ λ -F H U H . ( 4
(v 5 x -lv 1 ) -(v 6 x -lv 2 ) = λ -(f 5 x -lf 1 ).
From the above equation, we obtain

β α v 5 x -lv 1 2 dx ≤ 2 λ 2 β α v 6 x -lv 2 2 dx + 2 λ 2 +2 β α f 5 x -lf 1 2 dx ≤ 2 λ 2 β α v 6 x -lv 2 2 dx + 4 λ 2 +2 f 5 x 2 + 4l 2 λ 2 +2 f 1 2 . ( 4
β α |v 6 x | 2 dx = β α |v 6 x -lv 2 + lv 2 | 2 dx ≤ 2 β α |v 6 x -lv 2 | 2 dx + 2l 2 β α |v 2 | 2 dx.
From the above estimation, the first estimation in (4.4.14) and the fact that v 2 is uniformly bounded in L 2 (0, L), we get the third estimation in (4.4.14). From (4.4.7), we deduce that

β α v 5 x 2 dx ≤ 2 λ 2 β α v 6 x 2 dx + 2 λ 2 +2 β α f 5 x 2 dx.
Finally, from the above estimation, the third estimation in (4.4.14) and the fact that f 5 x = o(1), we obtain the fourth estimation in (4.4.14). The proof is thus complete.

For all 0 < ε < β -α 10 , we fix the following cut-off functions (4.4.17)

f j ∈ C 2 ([0, L]), j ∈ {1, • • • , 5} such that 0 ≤ f j (x) ≤ 1, for all x ∈ [0, L] and f j (x) = 1 if x ∈ [α + jε, β -jε], 0 if x ∈ [0, α + (j -1)ε] ∪ [β + (1 -j)ε, L]. q 1 , q 2 ∈ C 1 ([0, L]) such that 0 ≤ q 1 (x) ≤ 1, 0 ≤ q 2 (x) ≤ 1, for all x ∈ [0, L] and q 1 (x) = 1 if x ∈ [0, γ 1 ], 0 if x ∈ [γ 2 , L], and
q 2 (x) = 0 if x ∈ [0, γ 1 ], 1 if x ∈ [γ 2 , L], with 0 < α < γ 1 < γ 2 < β < L. Lemma 4.4.2. If k 1 ρ 1 = k 2 ρ 2 and = 2 or k 1 ρ 1 = k 2 ρ 2 and = 4 . Then, the solution U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A) of
Proof. First, multiplying (4.4.8) by -iλ -1 f 1 v 6 and integrating over (α, β), then using the fact that v 6 is uniformly bounded in L 2 (0, L) and f 6 = o(1), we obtain

ρ 1 β α f 1 |v 6 | 2 dx = - i λ β α f 1 k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) x v 6 dx + ilk 1 λ β α f 1 (v 1 x + v 3 + lv 5 )v 6 dx + o(1) |λ| +1 ,
using the fact that (v 1 x + v 3 + lv 5 ), v 6 are uniformly bounded in L 2 (0, L), we get

ilk 1 λ β α f 1 (v 1 x + v 3 + lv 5 )v 6 dx = o(1),
consequently, as ∈ {2, 4}, we obtain

ρ 1 β α f 1 |v 6 | 2 dx = i λ β α -f 1 k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) x v 6 dx :=I 1 +o(1). (4.4.18)
Using integration by parts and the fact that f 1 (α) = f 1 (β) = 0, then using the definition of d(x), we get

I 1 = i λ β α f 1 k 3 (v 5 x -lv 1 ) + d 0 (v 6 x -lv 2 ) v 6 x dx + i λ β α f 1 k 3 (v 5 x -lv 1 ) + d 0 (v 6
x -lv 2 ) v 6 dx, using Lemma 4.4.1 and the fact that v 6 is uniformly bounded in L 2 (0, L), ∈ {2, 4}, we get

I 1 = o(1) |λ| 2 +1 . ( 4
ρ 1 β α f 1 |v 6 | 2 dx = o(1).
From the above estimation and the definition of f 1 , we obtain the first estimation in (4.4.17). Next, from (4.4.7), we deduce that

β-ε α+ε |λv 5 | 2 dx ≤ 2 β-ε α+ε |v 6 | 2 dx + 2λ -2 β-ε α+ε |f 5 | 2 dx.
Finally, from the above inequality, the first estimation in (4.4.17) and the fact that f 5 = o(1), ∈ {2, 4}, we obtain the second estimation in (4.4.17). The proof is thus complete. Proof. First, multiplying (4.4.8) by p 2 v 1 x , integrating over (α + ε, β -ε), using the fact that v 1

x is uniformly bounded in L 2 (0, L) and f 6 = o(1), we get

iλρ 1 β-ε α+ε f 2 v 6 v 1 x dx :=I 2 + β-ε α+ε -f 2 k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) x v 1 x dx :=I 3 + lk 1 β-ε α+ε f 2 v 1 x 2 dx + lk 1 β-ε α+ε f 2 (v 3 + lv 5 )v 1 x dx = o(λ -), using the fact that v 1 x is uniformly bounded in L 2 (0, L), v 3 = O(|λ| -1 ), v 5 = O(|λ| -1 ) (see (4.4.12)), we get lk 1 β-ε α+ε f 2 (v 3 + lv 5 )v 1 x dx = o(1),
consequently, as ∈ {2, 4}, we obtain

lk 1 β-ε α+ε f 2 v 1 x 2 dx + I 2 + I 3 = o(1). (4.4.21)
Now, using integration by parts and the definition of f 2 , then using Lemma 4.4.2 and the fact that v 1 = O(|λ| -1 ), we get

I 2 = -iλρ 1 β-ε α+ε f 2 v 6 x v 1 dx -iλρ 1 β-ε α+ε f 2 v 6 v 1 dx = -iλρ 1 β-ε α+ε f 2 v 6 x v 1 dx + o(1). (4.4.22)
Now, it is easy to see that

-iλρ 1 β-ε α+ε f 2 v 6 x v 1 dx = -iλρ 1 β-ε α+ε f 2 (v 6 x -lv 2 + lv 2 )v 1 dx = -iλρ 1 β-ε α+ε f 2 (v 6 x -lv 2 )v 1 dx -iλρ 1 l β-ε α+ε f 2 v 2 v 1 dx,
using Lemma 4.4.1 and the fact that v 1 = O(|λ| -1 ), we get

-iλρ 1 β-ε α+ε f 2 v 6 x v 1 dx = -iλρ 1 l β-ε α+ε f 2 v 2 v 1 dx + o(|λ| -2 ). Inserting v 2 = iλv 1 -λ -f 1 in the above equation, we get -iλρ 1 β-ε α+ε f 2 v 6 x v 1 dx = lρ 1 β-ε α+ε f 2 |λv 1 | 2 dx + iλ -+1 lρ 1 β-ε α+ε f 2 f 1 v 1 dx + o(|λ| -2 ), using the fact that v 1 = O(|λ| -1 ) and f 1 = o(1), ∈ {2, 4}, we get -iλρ 1 β-ε α+ε f 2 v 6 x v 1 dx = lρ 1 β-ε α+ε f 2 |λv 1 | 2 dx + o(|λ| -2 ). ( 4
I 2 = lρ 1 β-ε α+ε f 2 |λv 1 | 2 dx + o(1). ( 4
.4.24)

I 3 = β-ε α+ε f 2 k 3 (v 5 x -lv 1 ) + d 0 (v 6 x -lv 2 ) v 1 x dx + β-ε α+ε f 2 k 3 (v 5 x -lv 1 ) + d 0 (v 6 x -lv 2 ) v 1 xx dx, (4.4
lk 1 β-ε α+ε f 2 v 1 x 2 dx + lρ 1 β-ε α+ε f 2 |λv 1 | 2 dx = o(1). ( 4
= (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A) of (4.
Proof. First, take = 2 in (4.4.9) and multiply it by ρ -1 1 f 3 v 3 x , integrating over (α + 2ε, β -2ε), using the definition of d(x) and fact that v 3

x is uniformly bounded in L 2 (0, L), f 1 = o(1), f 2 = o(1), we obtain x is uniformly bounded in L 2 (0, L), we get

k 1 ρ 1 β-2ε α+2ε f 3 v 3 x 2 dx = -λ 2 β-2ε α+2ε f 3 v 1 v 3 x dx - k 1 ρ 1 β-2ε α+2ε f 3 v 1 xx v 3 x dx - lk 1 ρ 1 β-2ε α+2ε f 3 v 5 x v 3 x dx - lk 3 ρ 1 β-2ε α+2ε f 3 (v 5 x -lv 1 )v 3 x dx - ld 0 ρ 1 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 3 x dx + o(|λ| -1 ). ( 4
         - lk 1 ρ 1 β-2ε α+2ε f 3 v 5 x v 3 x dx = o(1), - lk 3 ρ 1 β-2ε α+2ε f 3 (v 5 x -lv 1 )v 3 x dx = o(λ -2 ) and - ld 0 ρ 1 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 3 x dx = o(|λ| -1
k 1 ρ 1 β-2ε α+2ε f 3 v 3 x 2 dx = -λ 2 β-2ε α+2ε f 3 v 1 v 3 x dx - k 1 ρ 1 β-2ε α+2ε f 3 v 1 xx v 3 x dx + o(1). (4.4.31)
Now, taking = 2 in (4.4.10), we deduce that

λ 2 ρ 2 v 3 + k 2 v 3 xx -k 1 (v 1 x + v 3 + lv 5 ) = -ρ 2 λ -2 f 4 + iρ 2 λ -1 f 3 . (4.4.32)
Multiplying (4.4.32) by ρ -1 2 f 3 v 1 x , integrating over (α + 2ε, β -2ε), we obtain

λ 2 β-2ε α+2ε f 3 v 1 x v 3 dx + k 2 ρ 2 β-2ε α+2ε f 3 v 1 x v 3 xx dx - k 1 ρ 2 β-2ε α+2ε f 3 v 1 x (v 1 x + v 3 + lv 5 )dx = o(|λ| -1 ). (4.4.33)
Using integration by parts to the first two terms in the above equation, we get

-λ 2 β-2ε α+2ε f 3 v 1 v 3 x dx - k 2 ρ 2 β-2ε α+2ε f 3 v 1 xx v 3 x dx = λ 2 β-2ε α+2ε f 3 v 1 v 3 dx + k 2 ρ 2 β-2ε α+2ε f 3 v 1 x v 3 x dx + k 1 ρ 2 β-2ε α+2ε f 3 v 1 x (v 1 x + v 3 + lv 5 )dx + o(|λ| -1 ). (4.4.34)
Using Lemma 4.4.3 and the fact that v 3 x , (v 1 x + v 3 + lv 5 ) are uniformly bounded in L 2 (0, L) and

v 3 = O(|λ| -1 ), we get            λ 2 β-2ε α+2ε f 3 v 1 v 3 dx = o(1), k 2 ρ 2 β-2ε α+2ε f 3 v 1 x v 3 x dx = o(1) and k 1 ρ 2 β-2ε α+2ε f 3 v 1 x (v 1 x + v 3 + lv 5 )dx = o (1).
2 ρ 2 = k 1 ρ 1 , we get -λ 2 β-2ε α+2ε f 3 v 1 v 3 x dx - k 1 ρ 1 β-2ε α+2ε f 3 v 1 xx v 3 x dx = o(1).
Inserting the above estimation in (4.4.31), then using the definition of f 3 , we obtain the first estimation in (4.4.28). Next, multiplying (4.4.32) by f 4 v 3 , integrating over (α+3ε, β -3ε), using integration by parts and the definition of f 4 and the fact that

v 3 = O(|λ| -1 ), f 3 = o(1) and f 4 = o(1), we get ρ 2 β-3ε α+3ε f 4 λv 3 2 dx = k 2 β-3ε α+3ε f 4 |v 3 x | 2 dx + k 2 β-3ε α+3ε f 4 v 3 x v 3 dx + k 1 β-3ε α+3ε f 4 (v 1 x + v 3 + lv 5 )v 3 dx + o(λ -2 ).
From the above estimation, the first estimation in (4.4.28) and the fact that (v 1 x + v 3 + lv 5 ) is uniformly bounded in L 2 (0, L) and v 3 = O(|λ| -1 ), we obtain

ρ 2 β-3ε α+3ε f 4 λv 3 2 dx = o(1).
Finally, from the above estimation and the definition of f 4 , we obtain the second estimation desired. The proof is thus complete. Proof. For clarity, we divide the proof into five steps:

Step 1: In this step, we will prove that:

lρ 1 β-2ε α+2ε f 3 λv 1 2 dx -lk 1 β-2ε α+2ε f 3 v 1 x 2 dx - lk 1 β-2ε α+2ε f 3 v 3 v 1 x dx - l 2 k 1 β-2ε α+2ε f 3 v 5 v 1 x dx = o(λ -2 ).
(4.4.37)

For this aim, take = 4 in (4.4.9) and multiply it by lf 3 v 1 , integrating over (α + 2ε, β -2ε), using the fact that v 1 = O(|λ| -1 ), f 1 = o(1) and f 2 = o(1), then taking the real part, we get

lρ 1 β-2ε α+2ε f 3 λv 1 2 dx + lk 1 β-2ε α+2ε f 3 (v 1 x + v 3 + lv 5 ) x v 1 dx :=I 4 + l 2 k 3 β-2ε α+2ε f 3 (v 5 x -lv 1 )v 1 dx + l 2 d 0 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 1 dx = o(λ -4 ). (4.4.38)
Using integration by parts and the definition of f 3 , we obtain

I 4 = - lk 1 β-2ε α+2ε f 3 (v 1 x + v 3 + lv 5 )v 1 dx - lk 1 β-2ε α+2ε f 3 (v 1 x + v 3 + lv 5 )v 1 x dx = - lk 1 2 β-2ε α+2ε f 3 v 1 2 x dx - lk 1 β-2ε α+2ε f 3 v 3 v 1 dx - l 2 k 1 β-2ε α+2ε f 3 v 5 v 1 dx -lk 1 β-2ε α+2ε f 3 v 1 x 2 dx - lk 1 β-2ε α+2ε f 3 v 3 v 1 x dx - l 2 k 1 β-2ε α+2ε f 3 v 5 v 1 x dx . (4.4.39)
Using integration by parts and the fact that f 3 (α + 2ε) = f 3 (β -2ε) = 0, then using Lemma 4.4.3, we obtain Step 2: In this step, we will prove that:

- lk 1 2 β-2ε α+2ε f 3 v 1 2 x dx = lk 1 2 β-2ε α+2ε f 3 v 1 2 dx = o(λ -2 ). ( 4
- lk 1 β-2ε α+2ε f 3 v 3 v 1 dx = o(λ -2 ), - l 2 k 1 β-2ε α+2ε f 3 v 5 v 1 dx = o(λ -2 ). ( 4
I 4 = -lk 1 β-2ε α+2ε f 3 v 1 x 2 dx - lk 1 β-2ε α+2ε f 3 v 3 v 1 x dx - l 2 k 1 β-2ε α+2ε f 3 v 5 v 1 x dx + o(λ -2 ).
         l 2 k 3 β-2ε α+2ε f 3 (v 5 x -lv 1 )v 1 dx = o(λ -4 ), l 2 d 0 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 1 dx = o(|λ| -3 ).
2lρ 1 β-2ε α+2ε f 3 λv 1 2 dx = iλρ 1 β-2ε α+2ε f 3 v 6 v 1 dx - d 0 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 1 xx dx + o(λ -2 ). (4.4.44)
For this aim, multiplying (4.4.8) by f 3 v 1 x , integrating over (α + 2ε, β -2ε), using the fact that v 1

x is uniformly bounded in L 2 (0, L) and f 6 = o(1), then taking the real part, we get Using integration by parts and the fact that f 3 (α + 2ε) = f 3 (β -2ε) = 0, we obtain

iλρ 1 β-2ε α+2ε f 3 v 6 v 1 x dx :=I 5 + - β-2ε α+2ε f 3 k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) x v 1 x dx :=I 6 + lk 1 β-2ε α+2ε f 3 |v 1 x | 2 dx + lk 1 β-2ε α+2ε f 3 v 3 v 1 x dx + l 2 k 1 β-2ε α+2ε f 3 v 5 v 1 x dx = o(λ -4 ).
I 5 = - iλρ 1 β-2ε α+2ε f 3 v 6 x v 1 dx - iλρ 1 β-2ε α+2ε f 3 v 6 v 1 dx . (4.4.47)
Now, it is easy to see that

-iλρ 1 β-2ε α+2ε f 2 v 6 x v 1 dx = -iλρ 1 β-2ε α+2ε f 3 (v 6 x -lv 2 + lv 2 )v 1 dx = -iλρ 1 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 1 dx - iλρ 1 l β-2ε α+2ε f 3 v 2 v 1 dx ,
using Lemma 4.4.1 and the fact that v 1 = O(|λ| -1 ), we get

-iλρ 1 β-2ε α+2ε f 3 v 6 x v 1 dx = -iλρ 1 l β-2ε α+2ε f 3 v 2 v 1 dx + o(λ -2 ).
Inserting v 2 = iλv 1 -λ -4 f 1 in the above estimation, then using the fact that v 1 = O(|λ| -1 ) and f 1 = o(1), we get

-iλρ 1 β-2ε α+2ε f 3 v 6 x v 1 dx = lρ 1 β-2ε α+2ε f 3 |λv 1 | 2 dx + o(λ -2 ),
Inserting the above estimation in (4.4.47), we obtain

I 5 = lρ 1 β-2ε α+2ε f 3 |λv 1 | 2 dx - iλρ 1 β-2ε α+2ε f 3 v 6 v 1 dx + o(λ -2 ). (4.4.48)
Now, Using integration by parts and the fact that f 3 (α + 2ε) = f 3 (β -2ε) = 0, then using the definition of d(x), we obtain

I 6 = β-2ε α+2ε f 3 k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) v 1 x dx + β-2ε α+2ε f 3 k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) v 1 xx dx = k 3 β-2ε α+2ε f 3 (v 5 x -lv 1 )v 1 x dx - d 0 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 1 x dx + k 3 β-2ε α+2ε f 3 (v 5 x -lv 1 )v 1 xx dx + d 0 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 1 xx dx ,
consequently, by using Lemma 4.4.1 with = 4 and the fact that v 1

x is uniformly bounded in L 2 (0, L), v 1 xx = O(|λ|), we get Step 3: In this step, we will prove that:

I 6 = d 0 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 1 xx dx + o(λ -2 ). ( 4
iλρ 1 β-2ε α+2ε f 3 v 6 v 1 dx = o(λ -2 ). ( 4

.4.50)

For this aim, take = 4 in (4.4.8) and multiply it by f 3 v 1 , integrating over (α + 2ε, β -2ε), using the fact that v 1 = O(|λ| -1 ), f 6 = o(1), then taking the real part, we get

iλρ 1 β-2ε α+2ε f 3 v 6 v 1 dx + - β-2ε α+2ε f 3 k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) x v 1 dx :=I 7 + lk 1 2 β-2ε α+2ε f 3 |v 1 | 2 x dx + lk 1 β-2ε α+2ε f 3 (v 3 + lv 5 )v 1 dx = o(|λ| -5 ), ( 4
lk 1 2 β-2ε α+2ε f 3 v 1 2 x dx = o(λ -2 ) and lk 1 β-2ε α+2ε f 3 (v 3 + lv 5 )v 1 dx = o(λ -2 ).
Consequently, (4.4.51) implies

iλρ 1 β-2ε α+2ε f 3 v 6 v 1 dx + I 7 = o(λ -2 ). (4.4.52)
Using integration by parts and the fact that f 3 (α + 2ε) = f 3 (β -2ε) = 0, then using Lemma 4.4.1 and the fact that v 1

x is uniformly bounded in L 2 (0, L), v 1 = O(|λ| -1 ), we obtain

I 7 = β-2ε α+2ε f 3 k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) v 1 dx + β-2ε α+2ε f 3 k 3 (v 5 x -lv 1 ) + d(x)(v 6 x -lv 2 ) v 1 x dx = o(λ -2 ).
Therefore, from the above estimation and (4.4.52), we obtain (4.4.50).

Step 4: In this step, we will prove that:

d 0 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 1 xx dx = - d 0 ρ 1 k 1 λ 2 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 1 dx + o(λ -2 ). (4.4.53)
For this aim, take = 4 in (4.4.9) and multiply it by

d 0 k 1 f 3 (v 6
x -lv 2 ), integrating over (α + 2ε, β -2ε) and taking the real part, then using Lemmas 4.4.1 and the fact that

f 1 = o(1), f 2 = o(1), we get                          d 0 ρ 1 k 1 λ 2 β-2ε α+2ε f 3 v 1 (v 6 x -lv 2 )dx + d 0 β-2ε α+2ε f 3 v 1 xx (v 6 x -lv 2 )dx + d 0 β-2ε α+2ε f 3 v 3 x (v 6 x -lv 2 )dx + d 0 l β-2ε α+2ε f 3 v 5 x (v 6 x -lv 2 )dx + d 0 lk 3 k 1 β-2ε α+2ε f 3 (v 5 x -lv 1 )(v 6 x -lv 2 )dx + ld 2 0 k 1 β-2ε α+2ε f 3 |v 6 x -lv 2 | 2 dx = o(|λ| -5 ),
consequently, by using Lemma 4.4.1 and the fact that v 3

x is uniformly bounded in L 2 (0, L), we get

d 0 β-2ε α+2ε f 3 v 1 xx (v 6 x -lv 2 )dx = - d 0 ρ 1 k 1 λ 2 β-2ε α+2ε f 3 v 1 (v 6 x -lv 2 )dx + o(λ -2 ).
Thus, from the above estimation, we obtain (4.4.53).

Step 5: In this step, we conclude the proof of (

f 3 λv 1 2 dx = d 0 ρ 1 k 1 λ 2 β-2ε α+2ε f 3 (v 6 x -lv 2 )v 1 dx + o(λ -2 ) ≤ d 0 ρ 1 k 1 λ 2 β-2ε α+2ε f 3 |v 6 x -lv 2 ||v 1 |dx + o(λ -2 ) = β-2ε α+2ε d 0 √ ρ 1 k 1 √ l |λ| f 3 |v 6 x -lv 2 | lρ 1 |λ| f 3 |v 1 | dx + o(λ -2 ) ≤ ρ 1 d 2 0 2k 2 1 l λ 2 β-2ε α+2ε f 3 |v 6 x -lv 2 | 2 dx =o(λ -2 ) + lρ 1 2 β-2ε α+2ε f 3 |λv 1 | 2 dx + o(λ -2 ), consequently, we obtain 3lρ 1 2 β-2ε α+2ε f 3 λv 1 2 dx = o(λ -2 ).
Finally, from the above estimation and the definition of f 3 , we obtain (4.4.36). The proof is thus complete.

Lemma 4.4.6.

If k 1 ρ 1 = k 2 ρ 2 and = 4. Then, the solution U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A) of system (4.
Proof. First, take = 4 in (4.4.9) and multiply it by k -1 1 f 4 v 3 x , integrating over (α + 3ε, β -3ε), using the definition of d(x) and the fact that v 3

x is uniformly bounded in L 2 (0, L), f 1 = o(1), f 2 = o(1), then taking the real part, we obtain

λ 2 ρ 1 k 1 β-3ε α+3ε f 4 v 1 v 3 x dx + β-3ε α+3ε f 4 v 1 xx v 3 x dx + β-3ε α+3ε f 4 |v 3 x | 2 dx + l β-3ε α+3ε f 4 v 5 x v 3 x dx + lk 3 k 1 β-3ε α+3ε f 4 (v 5 x -lv 1 )v 3 x dx + ld 0 k 1 β-3ε α+3ε f 4 (v 6 x -lv 2 )v 3 x dx = o(|λ| -3 ),
consequently, from Lemmas 4.4.1, 4.4.5 with = 4 and the fact that v 3

x is uniformly bounded in L 2 (0, L), we obtain

β-3ε α+3ε f 4 v 1 xx v 3 x dx + β-3ε α+3ε f 4 |v 3 x | 2 dx = o(1). ( 4

.4.55)

Now, take = 4 in (4.4.10) and multiply it by k -1 2 f 4 v 1 x , integrating over (α + 3ε, β -3ε) and integrating by parts, using the fact that v 1

x is uniformly bounded in L 2 (0, L) and f 3 = o(1), f 4 = o(1), then taking the real part, we obtain Finally, from the above estimation and the definition of f 5 , we obtain the second estimation in (4.4.54). The proof is thus complete. Proof. First, multiplying (4.4.9) by 2hv 1 x , integrating over (0, L), taking the real part, then using Lemma 4.4.1, the fact that v 1

- λ 2 ρ 2 k 2 β-3ε α+3ε f 4 v 3 x v 1 dx - λ 2 ρ 2 k 2 β-3ε α+3ε f 4 v 3 v 1 dx - β-3ε α+3ε f 4 v 3 x v 1 xx dx - β-3ε α+3ε f 4 v 3 x v 1 x dx - k 1 k 2 β-3ε α+3ε f 4 (v 1 x + v 3 + lv 5 )v 1 x dx = o(|λ| -3 ),
x is uniformly bounded in L 2 (0, L), v 1 = O(|λ| -1 ), f 1 = o(1) and f 2 = o(1), we obtain x is uniformly bounded in L 2 (0, L), v 5 = O(|λ| -1 ), we obtain Proof. First, take h = xq 1 + (x -L)q 2 in (4.4.7), then using the definition of d(x) and the fact that 0 < α < γ 1 < γ 2 < β < L, we obtain Finally, according to Theorem 1.3.7, we obtain the desired result. The proof is thus complete.

                                      
γ 1 0 ρ 1 λv 1 2 + k 1 v 1 x 2 + ρ 2 λv 3 2 + k 2 v 3 x 2 + ρ 1 |λv 5 | 2 dx + k 3 α 0 |v 5 x | 2 dx + L γ 2 ρ 1 λv 1 2 + k 1 v 1 x 2 + ρ 2 λv 3 2 + k 2 v 3 x 2 + ρ 1 λv 5 2 dx + k 3 L β |v 5 x | 2 dx = - γ 2 γ 1 (q 1 + xq 1 ) ρ 1 λv 1 2 + k 1 v 1 x 2 + ρ 2 λv 3 2 + k 2 v 3 x 2 + ρ 1 λv 5 2 + k -1 3 k 3 v 5 x + d 0 (v 6 x -lv 2 ) 2 dx - γ 2 γ 1 (q 2 + (x -L)q 2 ) ρ 1 λv 1 2 + k 1 v 1 x 2 + ρ 2 λv 3 2 + k 2 v 3

Conclusion

We have studied the stabilization of a Bresse system with one discontinuous local internal viscoelastic damping of Kelvin-Voigt type acting on the axial force under fully Dirichlet boundary conditions. We proved the strong stability of the system by using Arendt-Batty criteria. We proved that the energy of our system decays polynomially with the rates:

       t -1 if k 1 ρ 1 = k 2 ρ 2 , t -1 2 if k 1 ρ 1 = k 2 ρ 2 .
For further purposes, we need a weaker version of it. Indeed as D(Ω) is dense in E(∆ Finally, by adding (5.2.15), (5.2.16) and (5.2.17), we obtain the desired result. The proof is thus complete.

In the sequel, we make the following assumptions

β 1 , γ 1 > 0, β 2 , γ 2 ∈ R * ,
H = H 2 Γ 0 (Ω) × L 2 (Ω) × L 2 (Γ 1 ) 2 × L 2 (Γ 1 × (0, 1)) 2 ,
where

H 2 Γ 0 (Ω) = f ∈ H 2 (Ω) | f = ∂ ν f = 0 on Γ 0 .
The Hilbert space H is equipped with the following inner product (U, U 1 ) H = a(u, u 1 ) +

Ω vv 1 dx + Γ 1 ηη 1 dΓ + Γ 1 ξξ 1 dΓ + τ 1 |β 2 | Γ 1 1 0 z 1 z 1 1 dρdΓ + τ 2 |γ 2 | Γ 1 1 0 z 2 z 2 1 dρdΓ, (5.2.18)
where U = (u, v, η, ξ, z 1 , z 2 ) , U 1 = (u 1 , v 1 , η 1 , ξ 1 , z 1 1 , z 2 1 ) ∈ H. Now, we define the linear unbounded operator A : D(A) ⊂ H -→ H by:

D(A) = U = (u, v, η, ξ, z 1 , z 2 ) ∈ D Γ 0 (∆ 2 ) × H 2 Γ 0 (Ω) × (L 2 (Γ 1 )) 2 × (L 2 (Γ 1 ; H 1 (0, 1))) 2 | B 1 u = -η, B 2 u = ξ, z 1 (•, 0) = η, z 2 (•, 0) = ξ on Γ 1 where D Γ 0 (∆ 2 ) = f ∈ H 2 Γ 0 (Ω) | ∆ 2 f ∈ L 2 (Ω), B 1 f ∈ L 2 (Γ 1 ), and B 2 f ∈ L 2 (Γ 1 )
and

A         u v η ξ z 1 z 2         =               v -∆ 2 u ∂ ν v -β 1 η -β 2 z 1 (•, 1) v -γ 1 ξ -γ 2 z 2 (•, 1) - 1 τ 1 z 1 ρ - 1 τ 2 z 2 ρ              
, ∀ U = (u, v, η, ξ, z 1 , z 2 ) ∈ D(A).

(5.2. [START_REF] Ammari | Stabilization of coupled systems[END_REF] such that (Bu)(ϕ) = b(u, ϕ), ∀ϕ ∈ V, (B i u)(ϕ) = b i (u, ϕ), ∀ϕ ∈ V, i ∈ {1, 2}.

(5.2.77)

We need to prove that the operator B is an isomorphism. For this aim, we divide the proof into two steps:

Step 1. In this step, we prove that the operator B 2 is compact. For this aim, let us define the following Hilbert space

H s Γ 0 (Ω) := {ϕ ∈ H s (Ω) | ϕ = ∂ ν ϕ = 0 on Γ 0 } with s ∈ 3 2 , 2 .
Now, from (5.2.74) and a trace theorem, we get

|b 2 (u, ϕ)| u L 2 (Ω) ϕ H 2 (Ω) + ∂ ν u L 2 (Γ 1 ) ∂ ν ϕ L 2 (Γ 1 ) + u L 2 (Γ 1 ) ϕ L 2 (Γ 1 ) u H s (Ω) ϕ H 2 (Ω) ,
for all s ∈ 3 2 , 2 . As V is compactly embedded into H s Γ 0 (Ω) for any s ∈ 3 2 , 2 , B 2 is indeed a compact operator. This compactness property and the fact that B 1 is an isomorphism imply that the operator B = B 1 + B 2 is a Fredholm operator of index zero. Now, following Fredholm alternative, we simply need to prove that the operator B is injective to obtain that it is an isomorphism.

Step 2. In this step, we prove that the operator B is injective (i.e. ker(B) = {0}). For this aim, let u ∈ ker(B) which gives b(u, ϕ) = 0, ∀ϕ ∈ V.

Equivalently, we have

a(u, ϕ) -λ 2 Ω uϕdx + iλC iλ Γ 1 ∂ ν u∂ ν ϕdΓ + iλK iλ Γ 1 uϕdΓ = 0, ∀ϕ ∈ V.
Thus, we find that

           -λ 2 u + ∆ 2 u = 0 in D (Ω), u = ∂ ν u = 0 on Γ 0 B 1 u = -iλC iλ ∂ ν u on Γ 1 , B 2 u = iλK iλ u on Γ 1 .
Therefore, the vector U defined by U = (u, iλu, iλC iλ ∂ ν u, iλK iλ u, iλC iλ ∂ ν ue -iλτ 1 ρ , iλK iλ ue -iλτ (5.2.90)

By using the trace theorem of interpolation type (see Theorem 1.4.4 in [START_REF] Liu | Semigroups associated with dissipative systems[END_REF] and Theorem 1.5.1.10 in [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]), we obtain

∂ ν ϕ n 2 L 2 (Γ 1 )
ϕ n H 2 (Ω) ϕ n H 1 (Ω) , (5.2.91)

ϕ n 2 L 2 (Γ 1 )
ϕ n H 1 (Ω) ϕ n L 2 (Ω) .

(5.2.92)

Now, it follows from Theorem 4.17 in [START_REF] Adams | Sobolev spaces[END_REF] that

ϕ n H 1 (Ω) ϕ n 1 2
H 2 (Ω) ϕ n L 2 (Ω) . Inserting the above inequality in (5.2.91) and (5.2.92), we get

∂ ν ϕ n 2 L 2 (Γ 1 )
ϕ n L 2 (Ω) ,

(5.2.93)

ϕ n 2 L 2 (Γ 1 ) ϕ n 1 2 H 2 (Ω) ϕ n 3 2 L 2 (Ω) , (5.2.94)
Now, we notice that a(ϕ n , ϕ n )

1 2 = a(ϕ n , ϕ n ) + Γ 1 |∂ ν ϕ n | 2 dΓ + Γ 1 |ϕ n | 2 dΓ 1 2 = µ 2 n ϕ n L 2 (Ω) = µ 2 n .
Since the norm defined on the left-hand side of the above equation is equivalent to the usual norm of H 2 (Ω), then we get ϕ n H 2 (Ω) µ 2 n . Inserting the above inequality and (5.2.84) in (5.2.93) and (5.2.94), we obtain

∂ ν ϕ n 2 L 2 (Γ 1 )
|µ n | 3 , (5.2.95)

ϕ n 2 L 2 (Γ 1 )
|µ n |.

(5.2.96)

Finally, from the above inequalities and (5.2.90), we obtain

F n 2 H |µ n | 3 + |µ n | µ 4 n → 0 as n → ∞.
The proof is thus complete.

Proof. First, taking the inner product of (5.2.100) with U in H and using (5. The above inequality, (5.2.107) and the fact that g 1 → 0 in L 2 (Γ 1 × (0, 1)) lead to the first estimation in (5.2.108). Now, from (5.2.110), we deduce that z 1 (•, 1) = ηe -iλτ 1 + τ 1 λ 2 1 0 g 1 (•, s)e iλτ 1 (s -1)ds on Γ 1 , consequently, by using Cauchy-Schwarz inequality, we get Therefore, from the above inequality, (5.2.107) and the fact that g 1 → 0 in L 2 (Γ 1 × (0, 1)), we get the second estimation in (5.2.108). The same argument as before yielding (5.2.109), the proof is complete.

Lemma 5.2.7. Under the hypothesis (H), the solution U = (u, v, η, ξ, z From the equivalence between the norm a(u, u) and the usual norm of H 2 (Ω), there then exists a positive constant C eq such that Inserting the above inequality in (5.2.123) and taking ε 1 = C tr C eq , we obtain (5.2.114). Finally, the case when B 1 u, B 2 u ∈ L 2 (Γ 1 ) can be easily obtained by the standard density arguments as in Lemma 3.1 in [START_REF] Rao | Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear boundary feedback[END_REF]. The proof is thus complete.

Lemma 5.2.9. Under the hypotheses (H) and (1.4.1), the solution U = (u, v, η, ξ, z

ρ 1 =Figure 1 :

 11 Figure 1: The circular arch

  is a positive function such that g(0) := g 0 > 0, ∞ 0 g(s)ds := g, b(x) := a -b(x) g > 0, and g (s) ≤ -mg(s), for some m > 0, ∀s ≥ 0. Moreover, from the definition of b(x), we have b(x) := a -b(x) g = b 0 := a -b 0 g, in (0, β), a, in (β, L).

Definition 1 . 1 . 10 .

 1110 Let T : D (T ) ⊂ E -→ F be a closed unbounded linear operator and let λ be an eigevalue of A. A non-zero element e ∈ E is called a generalized eigenvector of T associated with the eigenvalue value λ, if there exists n ∈ N * such that (λI -T ) n e = 0 and (λI -T ) n-1 e = 0.

Definition 1 . 4 . 1 .

 141 Let Ω ⊂ R n , n ≥ 2 be a bounded open set with the boundary Γ = Γ 0 ∪ Γ 1 .

Figure 1 . 1 :

 11 Figure 1.1: An example where the MGC boundary condition holds.

Figure 2 Figure 2 . 1 :

 221 Figure 2.1 describes system (2.1.1).

Figure 2 . 2 :

 22 Figure 2.2: Geometric description of the functions χ 1 and χ 2 .

Figure 2 . 3 : 4 .

 234 Figure 2.3: Geometric description of the functions θ 1 , θ 2 and θ 3 .

Lemma 2 . 4 . 3 . 4 .

 2434 Let 0 < ε < min α 2 , β-α Under the hypothesis (H), the solution U = (u, v, y, z, η(•, ρ)) ∈ D(A) of system (2.4.5)-(2.4.9) satisfies the following estimations β-2ε α |z| 2 dx = o(1) and β-3ε α+ε |y x | 2 dx = o(1).

  .4.32) Multiplying (2.4.32) and (2.4.6) by 2g 2 v and 2a -1 g 2 S b (u, v, η) respectively, integrating over (β -3ε, γ), using the definition of c(•) and S b (u, v, η), then taking the real part, we obtain 2iλ γ β-3ε

Figure 2 . 4 :

 24 Figure 2.4: Geometric description of the functions θ 4 and θ 5 .

  L)zu x dx + o(1).(2.4.43)

( 2 . 4 . 47 )From Lemma 2 . 4 . 4 ,

 2447244 we deduce that |v(β -3ε)| = O(1), |v(γ)| = O(1), |z(β -3ε)| = O(1) and |z(γ)| = O(1). (2.4.48) Using the fact that v, z are uniformly bounded in L 2 (0, L) in (2.4.46) and inserting (2.4.48) in (2.4.47), we obtain

|z| 2

 2 dx = o(1) and β-3ε α+ε |y x | 2 dx = o(1).

Figure 3 . 1 :

 31 Figure 3.1: Geometric description of the functions b(x) and c(x).

  and g (s) ≤ -mg(s), for some m > 0, ∀s ≥ 0.(H)Remark that, the last assumption in (H) implies that g(s) ≤ g 0 e -ms , ∀s ≥ 0. (3.1.2) Moreover, from the definition of b(•) (see Figure 3.1), we have b(x) := a -b(x) g = b 0 := a -b 0 g, in (0, β), a, in (β, L). ( b(•))

.1. 7 )

 7 where A : D(A) -→ H, A : D( A) -→ H, and B : D(B) -→ H are self-adjoint linear positive definite operators with domains D(A) ⊂ D(B) ⊂ H and D( A) ⊂ H such that the embeddings are dense and compact, B :

Theorem 3 . 3 . 1 .Lemma 3 . 3 . 1 .

 331331 Assume that the hypotheses (H) hold. Then, the C 0 -semigroup of contractions e tA t≥0 is strongly stable in H; i.e., for all U 0 ∈ H, the solution of (3.2.15) satisfies lim t→+∞ e tA U 0 H = 0.According to Theorem 1.3.3, to prove Theorem 3.3.1, we need to prove that the operator A has no pure imaginary eigenvalues and σ(A) ∩ iR is countable. The proof of Theorem 3.3.1 has been divided into the following two Lemmas. Under the hypotheeis (H), we have ker(iλI -A) = {0}, ∀λ ∈ R.

  (3.3.2),(3.3.4) and(3.3.37), we obtainu = v = 0 in (γ, L) and y = z = 0 in (γ, L). (3.3.38) Finally, from (3.3.10) (3.3.11), (3.3.15), (3.3.20), (3.3.31) and (3.3.38), we obtain U = 0. (3.3.39)

. 45 ) 0 f 5

 4505 with the following boundary conditions u(0) = u(L) = y(0) = y(L) = 0, ω(•, 0) = 0 in (0, β) and ω(0, s) = 0 in (0, ∞). (3.3.46) From (3.3.41), (3.3.45) and (3.3.46), we have ω(x, s) = 1 iλ (iλu -f 1 )(1 -e -iλs ) + s (x, ξ)e iλ(ξ-s) dξ, (x, s) ∈ (0, β) × (0, ∞). (3.3.47) Due the definition of S b(•) (u, ω), inserting (3.3.41), (3.3.43) and (3.3.47) in (3.3.42) and (3.3.44), we obtain the following system

  au x , in (β, L), and b 0 := a -b 0 ∞ 0 g(s)e -iλs ds.

  .50) where b(x) = b 0 , x ∈ (0, β), a, x ∈ (β, L). Adding (3.3.49) and (3.3.50), we get

  3), according to Theorem 1.3.6 and Theorem 1.3.7, to prove Theorem 3.4.1 and Theorem 3.4.2, we still need to prove the following condition lim sup

.4. 7 )Lemma 3 . 4 . 1 . 0 ∞ 0 g 0 ∞ 0 g 0 S

 734100000 Here we will check the condition (H 1 ) by finding a contradiction with (3.4.1) by showing U H = o(1). For clarity, we divide the proof into several lemmas. Under the hypotheses (H), the solution U = (u, v, y, z, ω(•, s)) ∈ D(A) of system (3.4.3)-(3.4.7) satisfies the following estimations β (s)|ω x (•, s)| 2 dsdx = o (1) |λ| and β (s)|ω x (•, s)| 2 dsdx = o (1) |λ| , (3.4.8) β 0 |u x | 2 dx = o(|λ| -) and β b 0 (u, ω) 2 dx = o(|λ| -).

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

 3233425 Figure 3.2: Geometric description of the functions h 1 , h 2 and h 3 .

  .4.3)-(3.4.7) satisfies the following estimation β-2ε α+ε |y x | 2 dx ≤ |a -1||λ| |c 0 | β-ε α |u x ||y x |dx + o(1). (3.4.20)

1 ) 0 h 2 h 2

 1022 |λ| +1 . (3.4.23) Using Cauchy-Schwarz inequality, the definition of h 2 , y x and z are uniformly bounded in L 2 (0, L), and estimation (3.4.14), we get -L vy x dx = o |λ| -zy x dx = O |λ| -1 = o(1). (3.4.24) Inserting (3.4.23) and (3.4.24) in (3.4.22), then using the definition of h 2 , we get

( 3 . 4 . 28 )

 3428 Inserting(3.4.28) in(3.4.27), then using the fact that b 0 = a -b 0 g, we get

  .4.33) Inserting (3.4.30), (3.4.32) and (3.4.33) in (3.4.26), using the definition of h 2 , then taking the real part, we get

  .4.3)-(3.4.7) satisfies the following estimation β-3ε α+2ε |z| 2 dx ≤ 3|a -1||λ| |c 0 | β-ε α |u x ||y x |dx + o(1). (3.4.35)

.4. 39 )

 39 Inserting (3.4.38) and (3.4.39) in (3.4.37) and using the definition of h 3 , we get the desired estimation (3.4.35). The proof has been completed. Now, we fix a function χ ∈ C 1 ([β -3ε, γ]) by χ(β -3ε) = -χ(γ) = 1, and set max x∈[β-3ε,γ] |χ(x)| = M χ and max x∈[β-3ε,γ] |χ (x)| = M χ . (χ) Remark 3.4.1.

  χvy x dx = o(|λ| -). (3.4.45) Adding (3.4.44) and (3.4.45), then using integration by parts, we obtain

Lemma 3 . 4 . 6 .

 346 Let θ ∈ C 1 ([0, L]) be a function with θ(0) = θ(L) = 0. Under the hypotheses (H), the solution U = (u, v, y, z, ω(•, s)) ∈ D(A) of (3.4.3)-(3.4.7) satisfies the following estimation

.4. 48 ) 1 L 0

 4810 Multiplying(3.4.4) by 2a -1 θS b(•) (u, ω), integrating over (0, L), taking the real part, then using (3.4.9) and the fact thatf 2 L 2 (0,L) = o(1), we get 2iλa -θvS b(•) (u, ω)dx -a -1 )θzS b(•) (u, ω)dx = o(|λ| -).(3.4.49)Using the definition of S b(•) (u, ω) and the fact that b 0 = a -b 0 g, we obtain 2iλa -1

.4. 55 )From ( 3 . 4 . 5 ) 0 θ(|z| 2 + 0 θ (|z| 2 + 2 L 0 c

 55345020220 , we deduce that iλy x = -z x -λ -f 3x .(3.4.56)Inserting(3.4.56) in(3.4.55), then using the fact that z is uniformly bounded in L 2 (0, L) andf 3 x L 2 (0,L) = o(1), we obtain -L |y x | 2 ) x dx -2 L 0 c(•)θvy x dx = o(|λ| -). (3.4.57)Using integration by parts in (3.4.57) and the fact that θ(0) = θ(L) = 0, we obtainL |y x | 2 ) x dx -(•)θvy x dx = o(|λ| -). (3.4.58) Finally, adding (3.4.54) and (3.4.58), we obtain the desired estimation (3.4.47). The proof is thus complete.

Figure 3 . 3 : 5 . 3 . 4 . 7 . 5 .

 3353475 Figure 3.3: Geometric description of the functions h 4 and h 5 .

.4. 65 )From Lemma 3 . 4 . 5 ,

 65345 we deduce that |v(γ)| = O( |λ|), |v(β -3ε)| = O( |λ|), |z(γ)| = O(1) and |z(β -3ε)| = O(1). (3.4.66)

  4.9) in estimation (3.4.20), we get β-2ε α+ε |y x | 2 dx = o(|λ| -2 +1 ) and β-3ε α+2ε |z| 2 dx = o(|λ| -2 +1 ). Taking = 2 in the above estimations, we obtain β-2ε α+ε |y x | 2 dx = o(1) and

  .4.72) Using (3.4.69) and (3.4.70) in (3.4.71) and (3.4.72), we obtain

Figure 4 . 1 :

 41 Figure 4.1: Geometric description of the function d(x).

Figure 4 . 2 :

 42 Figure 4.2: The circular arch

Proposition 4 . 2 . 1 .

 421 The unbounded linear operator A is m-dissipative in the Hilbert space H.

Theorem 4 . 3 . 1 .

 431 The C 0 -semigroup of contractions e tA t≥0 is strongly stable in H; i.e., for all U 0 ∈ H, the solution of (4.2.5) satisfies lim t→∞ e tA U 0 H = 0. According to Theorem 1.3.3, to prove Theorem 4.3.1, we need to prove that the operator A has no pure imaginary eigenvalues and σ(A) ∩ iR is countable. The proof of Theorem 4.3.1 has been divided into the following two Lemmas. Lemma 4.3.1. For all λ ∈ R, iλI -A is injective i.e. ker(iλI -A) = {0}, ∀λ ∈ R.

( 4 .

 4 3.14) Now, adding (4.3.10) and (4.3.14), we obtain v 5 x = 0 in (α, β) and consequently v 1 = 0 in (α, β). (4.3.15) Inserting (4.3.15) in (4.3.11), we get v 3 x = 0 in (α, β). (4.3.16) Now, system (4.3.2)-(4.3.7) can be written in (0, α) ∪ (β, L) as the following:

  (4.4.3)-(4.4.8) satisfies the following estimations β-ε α+ε |v 6 | 2 dx = o(1) and β-ε α+ε |λv 5 | 2 dx = o(1).

Lemma 4 . 4 . 3 . 1 x 2

 44312 If k 1 ρ 1 = k 2 ρ 2 and = 2 or k 1 ρ 1 = k 2 ρ 2 and = 4 . Then, the solution U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A) of(4.4.3)-(4.4.8) satisfies the following estimations β-2ε α+2ε v dx = o(1) and β-2ε α+2ε λv 1 2 dx = o(1). (4.4.20)

λv 3 2

 3 4.3)-(4.4.8) satisfies the following estimations dx = o(1).

.4. 29 )

 29 Using Lemma 4.4.1 with = 2, the definition of f 3 and the fact that v 3

( 4 . 4 . 35 )

 4435 Inserting(4.4.35) in (4.4.34), then using the fact that k

Lemma 4 . 4 . 5 .

 445 If k 1 ρ 1 = k 2 ρ 2 and = 4. Then, the solution U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A) of (4.4.3)-(4.4.8) satisfies the following estimation β-3ε α+3ε |λv 1 | 2 dx = o(λ -2 ). (4.4.36)

.4. 40 )

 40 Using the definition of f 3 , Lemmas 4.4.1, 4.4.3 with = 4 and the fact that v 3 = O(|λ| -1 ), v 5 = O(|λ| -1 ), we obtain

( 4 . 4 . 42 )

 4442 Moreover, from Lemmas 4.4.1, 4.4.3 and the fact that = 4, we obtain

( 4 . 4 . 43 )

 4443 Inserting (4.4.42) and (4.4.43) in (4.4.38), we obtain (4.4.37).

f 3

 3 λv 1 2 dx + I 5 + I 6 = o(λ -2 ).(4.4.46)

  .4.51) using (4.4.40), Lemma 4.4.3 and the fact that v 3 = O(|λ| -1 ), v 5 = O(|λ| -1 ), we obtain

λv 3 2

 3 4.3)-(4.4.8) satisfies the following estimations dx = o(1).

f 5 |λv 3 | 2 dx = k 2 β-4ε α+4ε f 5 |v 3 x | 2 dx + k 2 β-4ε α+4ε f 5 v 3 x v 3 dx + k 1 β-4ε α+4ε f 5 (v 1 x + v 3 +

 5322221513 consequently, from Lemmas 4.4.3, 4.4.5 with = 4 and the fact that v 3x , (v 1 x + v 3 + lv 5 ) are uniformly bounded in L 2 (0, L) and v 3 = O(|λ| -1 ), we obtain -4.55) and (4.4.56), then using the definition of f 4 , we obtain the first estimation in (4.4.54). Next, take = 4 in (4.4.10) and multiply it by f 5 v 3 , integrating over (α + 4ε, β -4ε) and integrating by parts, then using the fact that v 3 = O(|λ| -1 ),f 3 = o(1), f 4 = o(1), lv 5 )v 3 dx + o(λ -4 ).From the above estimation, the first estimation in (4.4.54) and the fact that (v 1x + v 3 + lv 5 ) is uniformly bounded in L 2 (0, L) and v 3 = O(|λ| -1 ), we obtain ρ 2 β-4ε α+4ε f 5 |λv 3 | 2 dx = o(1).

Lemma 4 . 4 . 7 . 1 = k 2 ρ 2 and = 2 or k 1 ρ 1 = k 2 ρ 2 0 h ρ 1 λv 1 2 + k 1 v 1 x 2 + ρ 2 λv 3 2 + k 2 v 3 x 2 + ρ 1 λv 5 2 + k - 1 3k 3 v 5 x

 44711120212232215 Let h ∈ C 1 ([0, L]) such that h(0) = h(L) = 0. Ifk 1 ρ and = 4 , then the solution U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A) of system (4.4.3)-(4.4.8) satisfies the following estimation L + d(x)(v 6x -lv 2 ) 2 dx = o(1).

6 x -lv 2 )v 1 x dx =o |λ| - 2 =

 62 o(|λ| -+1 ).

( 4 . 4 . 57 ) 0 h ρ 2 λv 3 2 + k 2 v 3 Let S := k 3 v 5 x 3 L 0 hv 5 h

 44570235305 Now, multiplying (4.4.10) by 2hv3 x , integrating over (0, L), taking the real part, then using the fact that v 3x is uniformly bounded inL 2 (0, L), v 3 = O(|λ| -1 ), v 5 = O(|λ| -1 ), f 3 = o(1) and f 4 = o(1), we obtain L + d(x)(v 6x -lv 2 ), from Lemma 4.4.1, the definition of d(x) and the fact that v 5x is uniformly bounded in L 2 (0, L), we get S is uniformly bounded in L 2 (0, L). Now, multiplying (4.4.11) by 2k -1 3 hS, integrating over (0, L), taking the real part, then using the fact thatv 3 = O(|λ| -1 ), v 5 = O(|λ| -1 ), f 5 = o(1) and f 6 = o(1), we obtain 2λ 2 ρ 1 k Sdx + k -1 -ρ 1 λ -f 6 -iλ 1-ρ 1 f 5 Sdx =o(|λ| -+1 ) . (4.4.59) Moreover, from the definition of S and d(x), Lemma 4.4.1 and the fact that v 1

2λ 2 ρ 1 k 3 L 0 hv 5 Sdx = λ 2 ρ 1 L 0 h v 5 1 x 6 x -lv 2 )dx =o |λ| - 2 . 0 h ρ 1 λv 5 2 +1 ρ 1 = k 2 ρ 2 1 0ρ 1 λv 1 2 + k 1 v 1 x 2 + ρ 2 λv 3 2 + k 2 v 3 x 2 + ρ 1 |λv 5 | 2 dx + k 3 2 ρ 1 λv 1 2 + k 1 v 1 x 2 + ρ 2 λv 3 2 + k 2 v 3 x 2 + ρ 1 λv 5 2 dx + k 3 L β |v 5 x | 2

 30510516202212122325232212232352 Sdx = -2l(k 1 + k 3 )Inserting the above estimations in (4.4.59) and using the fact that ∈ {2, 4}, we obtain L 57), (4.4.58), (4.4.60) and using the fact that ∈ {2, 4}, then using integration by parts, we obtain (4.4.7). The proof is thus complete. Lemma 4.4.8. The solution U = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) ∈ D(A) of system (4.4.3)-(4.4.8) satisfies the following estimations J(α + 4ε, β -4ε) = o(1) if k and = 2, (4.4.61) J(α + 5ε, β -5ε) = o(1) dx,for all 0 < α < γ 1 < γ 2 < β < L.

x 2 + ρ 1 λv 5 2 + k - 1 3k 3 v 5 x + d 0 (v 6 x -lv 2 ) 2 dx +k -1 3 γ 2 α q 1 |k 3 v 5 x 3 β γ 1 q 2 5 x | 2 |λv 3 |

 22156325312523 + d 0 (v 6 x -lv 2 )| 2 dx + k -1 |k 3 v 5 x + d 0 (v 6 x -lv 2 )| 2 dx.Now, take γ 1 = α + 4ε and γ 2 = β -4ε in the above equation, then using Lemmas 4.4.1-4.4.4 in case of k 1 ρ 1 = k 2 ρ 2 and = 2, we obtain (4.4.61). Finally, take γ 1 = α + 5ε and γ 2 = β -5ε in the above equation, then using Lemmas 4.4.1-4.4.3, 4.4.6 in case of k 1 ρ 1 = k 2 ρ 2 and = 4, we obtain (4.4.62). The proof is thus complete. Proof of Theorem 4.4.1. First, from Lemmas 4.4.1-4.4.4 and the fact that = dx = O(λ -2 ) = o(1), 2 dx = o(1).

( 4 . 4 . 63 )

 4463 Now, from (4.4.61),(4.4.63) and the fact that 0 < ε < β -α 10 , we deduce that U H = o(1), which contradicts (H). This implies that lim supλ∈R, |λ|→∞ 1 λ 2 (iλI -A) -1 H < ∞.

Finally, according to Theorem 1 . 3 . 7 , 5 x | 2 |λv 3 |

 137523 we obtain the desired result. The proof is thus complete. Proof of Theorem 4.4.2. First, from Lemmas 4.4.1, 4.4.2, 4.4.3, 4.4.6 and the fact that = 4, we obtain dx = O(λ -2 ) = o(1), 2 dx = o(1).

( 4 . 4 . 64 )

 4464 Now, from (4.4.62), (4.4.64) and the fact that 0 < ε < β -α 10 , we deduce that U H = o(1), which contradicts (H). This implies that lim sup λ∈R, |λ|→∞ 1 λ 4 (iλI -A) -1 H < ∞.

Thus U n = ϕ n λ n , ϕ n , 1 λ n ∂ ν ϕ n , ϕ n λ n , η n e -iµ 2 n τ 1 ρ , ξ n e -iµ 2 n τ 2 ρ( 5 . 2 . 88 ) 1 + β 2 e -iµ 2 n τ 1 iµ 2 n∂ ν ϕ n and f 4,n = γ 1 + γ 2 e -iµ 2 n τ 2 iµ 2 n 2 H ≥ ϕ n 2 L 2 ( 2 H = f 3,n 2 L 2 (Γ 1 ) + f 4,n 2 L 2 (Γ 1 )≤ (β 1 + 4 n∂ ν ϕ 2 L 2 (Γ 1 ) + (γ 1 + |γ 2 |) 2 µ 4 n ϕ 2 L 2 (Γ 1 ) 1 µ 4 n∂ ν ϕ n 2 L 2 (Γ 1 ) + ϕ n 2 L 2 (

 122528811212222222212211422122422122 2 ρ ) belongs to D(A) and satisfies iλU -AU = 0, and consequently U ∈ ker(iλI -A). Hence Lemma 5.2.2 yields U = 0 and consequently u = 0 and ker(B) = {0}.belongs to D(A) and is a solution of (5.2.80) withF n = (0, 0, f 3,n , f 4,n , 0, 0) , f 3,n = β Ω) = 1, ∀n ∈ N, which means that (5.2.81) holds with M = 1. Moreover, we haveF n |β 2 |) 2µ Γ 1 ) .

3 2 H 2 (Ω) ϕ n 1 2

 3222

1 |η| 2 1 |ξ| 2 1 λ 2 F 1 . 5 . 2 . 6 . 1 0|z 1 | 1 |z 1 |z 2 | 1 |z 2 + τ 1 λ 2 ρ 0 g 1 ( 1 1 0|z 1 | 2 dρdΓ ≤ 2 Γ 1 |η| 2 |g 1

 121212152611112122011112121 2.25), we get(β 1 -|β 2 |) Γ dΓ + (γ 1 -|γ 2 |) Γ dΓ ≤ -(AU, U ) H = 1 λ 2 (F, U ) H ≤ H U H ,from the hypothesis (H), we notice thatβ 1 -|β 2 | > 0 and γ 1 -|γ 2 | > 0,using the fact that F H = o(1) and U H = 1, we obtain the first two estimations in (5.2.107).The last two estimations in (5.2.107) directly follows from the first two estimations in (5.2.107) and the fact thatB 1 u = -η, B 2 u = ξ on ΓLemma Under the hypothesis (H), the solution U = (u, v, η, ξ, z 1 , z 2 ) ∈ D(A) of (5.2.101)-(5.2.106) satisfies the following estimationsΓ 1 2 dρdΓ = o(λ -2) andΓ (•, 1)| 2 dΓ = o(λ -2 2 dρdΓ = o(λ -2 ) and Γ (•, 1)| 2 dΓ = o(λ -2 ).(5.2.109)Proof. First, from (5.2.105) and the fact that z 1 (•, 0) = η(•) on Γ 1 , we obtainz 1 (•, ρ) = ηe -iλτ 1 ρ •, s)e iλτ 1 (s-ρ) ds on Γ 1 × (0, 1). (5.2.110) From (5.2.110), Cauchy-Schwarz inequality and the fact that ρ ∈ (0, 1), we get Γ (•, s)| 2 dsdΓ.

Γ 1 |z 1 (•, 1 )| 2 dΓ ≤ 2 Γ 1 |η| 2 |g 1 (

 1112121 •, s)| 2 dsdΓ.

Γ 1 |∇u| 2

 12 dΓ ≤ C tr u 2 H 2 (Ω) ≤ C tr C eq a(u, u).

  .

	Lemma 2.2.1. Let U = (u, u t , y, y t , η) be a regular solution of system (2.2.2)-(2.2.7). Then,
	the energy E(t) satisfies the following estimation
	d dt	E(t) ≤ -(κ 1 -|κ 2 |)	0	β	|u tx | 2 dx.	(2.2.9)
	Proof. First, multiplying (2.2.2) by u t , integrating over (0, L), using integration by parts with
	(2.2.6), then taking the real part, we obtain
	1
	2

  R * and |κ 2 | < κ 1 .

	(H)
	Under the hypothesis (H) and from Lemma 2.2.1, the system (2.2.2)-(2.2.7) is dissipative in
	the sense that its energy is non-increasing with respect to time (i.e. E (t) ≤ 0). Let us define
	the Hilbert space H by

  According to Theorem 1.3.3, to prove Theorem 2.3.1, we need to prove that the operator A has no pure imaginary eigenvalues and σ(A) ∩ iR is countable. The proof of Theorem 2.3.1 will be achieved from the following proposition.

2)-(2.2.7). The main result of this section is the following theorem. Theorem 2.3.1. Assume that (H) is true. Then, the C 0 -semigroup of contractions e tA t≥0 is strongly stable in H; i.e., for all U 0 ∈ H, the solution of (2.2.17) satisfies lim t→+∞ e tA U 0 H = 0. Proposition 2.3.1. Under the hypothesis (H), we have iR ⊂ ρ(A). (2.3.1) We will prove Proposition 2.3.1 by a contradiction argument. Remark that, it has been proved in Proposition 2.2.1 that 0 ∈ ρ(A). Now, suppose that (2.3.1) is false, then there exists ω ∈ R * such that iω / ∈ ρ(A). According to Remark 1.3.5, let (λ n , U n

  .3.17) Thus, from (2.3.10) and (2.3.17), we obtain (2.3.11). Next, from (2.3.5) and the fact that

  Finally, passing to the limit in(2.3.40), then using (2.3.2), (2.3.11), (2.3.36), the first limit in (2.3.37), (2.3.41), and the fact that F n H → 0, we obtain the second limit in (2.3.37). The proof is thus complete. Lemma 2.3.4. Under the hypothesis (H), the solution

					.3.40)
	Now, passing to the limit in (2.3.22), then using (2.3.2), the first limit in (2.3.37) and the fact
	that F n	H → 0, we get
		lim n→∞	|z n (α)| = 0 and lim n→+∞	|z n (β)| = 0.	(2.3.41)

  .3.58) Therefore, from (2.3.51), (2.3.52),(2.3.58) and as e B n (x-β) , e B n (s-x) are two bounded matrices, we get V n → 0 in (L 2 (β, γ))4 and consequently, we obtain(2.3.44). Next, from (2.3.5) , (2.3.7) and (2.3.55), we deduce that

  we obtain(2.3.65). The proof is thus complete.

Proof of Proposition 2.3.1. From Lemmas 2.3.1-2.3.6, we obtain U n H → 0 as n → ∞, which contradicts U n H = 1. Thus, (2.3.1) holds. The proof is thus complete. Proof of Theorem 2.3.1. From proposition 2.3.1, we have iR ⊂ ρ(A) and consequently σ(A) ∩ iR = ∅. Therefore, according to Theorem 1.3.3, we get that the C 0 -semigroup of contraction (e tA ) t≥0 is strongly stable. The proof is thus complete.

  , we obtain(2.4.31). The proof is thus complete.

	Lemma 2.4.5. Let h 2 ∈ C 1 ([0, L]) be a function. Under the hypothesis (H), the solution
	U = (u, v, y, z, η(•, ρ)) ∈ D(A) of system (2.4.5)-(2.4.9) satisfies the following equality

  3. According to Step 1 and Step 2, we obtain U H = o(1), which contradicts (2.4.3). Thus, (2.4.2) holds. Next, since conditions (2.4.1) and (2.4.2) are proved, then according to Theorem 1.3.7, the proof of Theorem 2.4.1 is achieved. The proof is thus complete.

  Multiplying (4.2.15), (4.2.16) and (4.2.17) by φ 1 , φ 2 and φ 3

  f 5 in (4.3.30), (4.3.32) and (4.3.34) respectively, we obtain

  .3.41) Since I is compact operator from H onto H and L -1 is an isomorphism from H onto H, the operator I -λ 2 L -1 is Fredholm of index zero. Then, by Fredholm's alternative, (4.3.41) admits a unique solution U ∈ H if and only if

  we deduce that U ∈ D(A) is a unique solution of(4.3.28). The proof is thus complete Proof of Theorem 4.3.1. From Lemma 4.3.1, we obtain that the operator A has no pure imaginary eigenvalues (i.e. σ p (A) ∩ iR = ∅). Moreover, from Lemma 4.3.1 and Lemma 4.3.2, iλI -A is bijective for all λ ∈ R and since A is closed, we conclude with the help of the closed graph theorem that iλI -A is an isomorphism for all λ ∈ R, hence that σ(A) ∩ iR = ∅. Therefore, according to Theorem 1.3.3, we get that the C 0 -semigroup (e tA ) t≥0 is strongly stable. The proof is thus complete.

  .4.15) Thus, from(4.4.15) and the fact that F H = o(1) and U H = 1, we obtain the first estimation in(4.4.14). Deriving (4.4.7) with respect to x and multiply (4.4.3) by l, then subtract the resulting equations, we deduce that iλ

  .4.27) Finally, from the above estimation and the definition of f 2 , we obtain (4.4.20). The proof is thus complete. Lemma 4.4.4. If k 1 ρ 1 = k 2 ρ 2 and = 2. Then, the solution U

  2 , L 2 (Ω)) := {f ∈ H 2 (Ω) | ∆ 2 f ∈ L 2 (Ω)} equipped with its natural norm, we deduce that f ∈ E(∆ 2 , L 2 (Ω)) (see Theorem 5.6 in[START_REF] Nicaise | Polygonal Interface Problems[END_REF]) satisfiesB 1 f ∈ H -1 2 (Γ) and B 2 f ∈ H -3 2 (Γ) with Lemma 5.2.1. Let U = (u, u t , η, ξ, z 1 , z 2) be a regular solution of system (5.2.2)-(5.2.10).Then, the energy E(t) satisfies the following estimationd dt E(t) ≤ -(β 1 -|β 2 |) Γ 1 |η| 2 dΓ -(γ 1 -|γ 2 |)Proof. First, multiplying (5.2.2) by u t , integrating over Ω, using (5.2.12) and (5.2.3), then taking the real part, we obtain(B 1 u∂ ν u t -B 2 uu t ) dΓ = 0. (5.2.14) Multiplying (5.2.8) and (5.2.9) by |β 2 |z 1 (•, ρ, t) and |γ 2 |z 2 (•, ρ, t) respectively, integrating over Γ (•, ρ, t)| 2 dρdΓ = -|γ 2 | 2 Γ 1 |z 2 (•, 1, t)| 2 dΓ + |γ 2 | 2 Γ 1 |ξ| 2 dΓ. (5.2.17)

	and
	τ 2 |γ 2 | 2	d dt Γ 1	0	1	|z 2
	a(f, g) =												H -1 2 (Γ),H	1 2 (Γ) -B 2 f, g	H -3 2 (Γ),H	3 2 (Γ) , ∀g ∈ H 2 (Ω). (5.2.13)
													|ξ| 2 dΓ.
													Γ 1
		1 2	d dt Ω	|u t | 2 +	1 2	d dt	a(u, u) -	Γ 1
	Now, from (5.2.4)-(5.2.7), we get
	-
	Γ 1
												=	1 2	d dt Γ 1	|η| 2 dΓ + β 1	Γ 1	|η| 2 dΓ +	β 2	Γ 1	ηz 1 (•, 1, t)dΓ
												+	1 2	d dt Γ 1	|ξ| 2 dΓ + γ 1	Γ 1	|ξ| 2 dΓ +	γ 2	Γ 1	ξz 2 (•, 1, t)dΓ .
	Inserting the above equation in (5.2.14), then using Young's inequality, we obtain
			1 2	d dt Ω	|u t | 2 +	1 2	d dt	a(u, u) +	1 2	d dt Γ 1	|η| 2 dΓ +	1 2	d dt Γ 1	|ξ| 2 dΓ
				= -β 1			|η| 2 dΓ -	β 2	ηz 1 (•, 1, t)dΓ -γ 1	|ξ| 2 dΓ
								Γ 1				Γ 1	Γ 1
						-			γ 2		ξz 2 (•, 1, t)dΓ	(5.2.15)
												Γ 1
				≤ -β 1	Γ 1	|η| 2 dΓ +	|β 2 | 2 Γ 1	|η| 2 dΓ +	|β 2 | 2 Γ 1	|z 1 (•, 1, t)| 2 dΓ
						-γ 1	Γ 1	|ξ| 2 dΓ +	|γ 2 | 2 Γ 1	|ξ| 2 dΓ +	|γ 2 | 2 Γ 1	|z 2 (•, 1, t)| 2 dΓ.
													|β 2 | 2 Γ 1	|η| 2 dΓ	(5.2.16)

Ω ∆ 2 fgdx + B 1 f, ∂ ν g (B 1 u∂ ν u t -B 2 uu t ) dΓ = Γ 1 η(η t + β 1 η + β 2 z 1 (•, 1, t))dΓ + Γ 1 ξ(ξ t + γ 1 ξ + γ 2 z 2 (•, 1, t)

)dΓ 1 × (0, 1), using the fact that z 1 (•, 0, t) = η and z 2 (•, 0, t) = ξ, then taking the real part, we obtain

τ 1 |β 2 | 2 d dt Γ 1 1 0 |z 1 (•, ρ, t)| 2 dρdΓ = -|β 2 | 2 Γ 1 |z 1 (•, 1, t)| 2 dΓ +

  |β 2 | < β 1 and |γ 2 | < γ 1 . (H)Under the hypothesis (H) and from Lemma 5.2.1, system (5.2.2)-(5.2.10) is dissipative in the sense that its energy is non-increasing with respect to time (i.e. E (t) ≤ 0). Let us define the Hilbert space H by

  1 , z 2 ) ∈ D(A) of (5.2.101)-(5.2.106) satisfies the following estimationsΓ 1 |∂ ν u| 2 dΓ = o(λ -2 ) and Γ 1 |u| 2 dΓ = o(λ -2 ).(5.2.111)Proof. First, inserting (5.2.101) in (5.2.103), we obtainiλ∂ ν u = 1 λ 2 (∂ ν f 1 -f 3 ) + (iλ + β 1 )η + β 2 z 1 (•, 1) on Γ 1 . |∂ ν f 1 | 2 + |f 3 | 2 dΓ + (λ 2 + β 2 1 )Using a trace theorem and the fact that a(f 1 , f 1 ) = o(1), we get Thus, from the above estimation, (5.2.107), (5.2.108), (5.2.112), and the fact that f 3 → 0 in L 2 (Γ 1 ), we get the first estimation in (5.2.111). Now, inserting (5.2.101) in (5.2.104), we obtainiλu = 1 λ 2 (f 1 -f 4 ) + (iλ + γ 1 )ξ + γ 2 z 2 (•, 1) on Γ 1 . (|f 1 | 2 + |f 4 | 2 )dΓ + (λ 2 + γ 2 1 )Again by a trace theorem and the fact that a(f 1 , f 1 ) = o(1), we get Finally, from the above estimation, (5.2.107), (5.2.109), (5.2.113) and the fact that f 4 → 0 in L 2 (Γ 1 ), we obtain the second estimation in (5.2.111). The proof is thus complete. Lemma 5.2.8. Under the hypotheses (H) and (1.4.1), for all u ∈ D Γ 0 (∆ 2 ), we have = h L ∞ (Ω) and ε 1 , ε 2 are positive constants explicitly given below. for all ε 2 > 0. Now, from (5.2.116) and (1.4.1), we get Inserting (5.2.118), (5.2.119), (5.2.121) and (5.2.122) in (5.2.115), we obtain |B 1 u| 2 dΓ, using (1.4.1) and taking ε 2 ≥ δ(1 -µ) -1 , we obtain Now, by using a trace theorem, there exists a positive constant C tr such that

	From the above equation, we get Γ 1 |λ∂ ν u| 2 dΓ 1 λ 4 Γ 1 |∂ ν f 1 | 2 dΓ Γ 1 From the above equation, we deduce that Γ 1 |λu| 2 dΓ 1 λ 4 Γ 1 Γ 1 |f 1 | 2 dΓ f 1 Γ 1 f 1 2 |η| 2 dΓ + β 2 2 2 H 2 (Ω) Γ 1 |ξ| 2 dΓ + γ 2 Γ 1 |z 1 (•, 1)| 2 dΓ. 2 Γ 1 |z 2 (•, 1)| 2 dΓ (5.2.113) (5.2.112) H 2 (Ω) -Ω ∆ 2 u(h • ∇u)dx ≤ -1 2 a(u, u) + ε 1 R 2 2 Γ 1 |B 2 u| 2 dΓ + Γ 1 |B 1 u| 2 dΓ 1 2 Γ 1 |∂ ν u| 2 dΓ 1 2 + R 2 ε 2 2 Γ 1 |B 1 u| 2 dΓ, (5.2.114) 2 Γ 1 (h • ν)c(u, u)dΓ ≥ 1 -µ 2δ Γ 1 d(u, u)dΓ. (5.2.122) -Ω ∆ 2 u(h • ∇u)dx ≤ -a(u, u) + 1 2 Γ 0 (h • ν)|∆u| 2 dΓ + ε 1 R 2 2 Γ 1 |B 2 u| 2 dΓ + 1 2ε 1 Γ 1 |∇u| 2 dΓ + 1 2ε 2 -1 -µ 2δ Γ 1 d(u, u)dΓ + Γ 1 |B 1 u| 2 dΓ 1 2 Γ 1 |∂ ν u| 2 dΓ 1 2 + R 2 ε 2 2 Γ 1 where R 1 -Ω ∆ 2 u(h • ∇u)dx ≤ -a(u, u) + ε 1 R 2 2 Γ 1 |B 2 u| 2 dΓ + 1 |∇u| 2 dΓ 2ε 1 Γ 1 + Γ 1 |B 1 u| 2 dΓ 1 2 Γ 1 |∂ ν u| 2 dΓ 1 2 + Γ 1 2 |B 1 u| 2 dΓ. R 2 ε 2 (5.2.123)

a(f 1 , f 1 ) = o(1). a(f 1 , f 1 ) = o(1). Γ 1 |∇u| 2 dΓ ≤ C tr u 2 H 2 (Ω) .

  1 , z 2 ) ∈ D(A) of (5.2.101)-(5.2.106) satisfies the following estimations

					(5.2.124)
	Proof. First, inserting (5.2.101) in (5.2.102), we get
		-λ 2 u + ∆ 2 u =	if 1 λ	+	f 2 λ 2 in Ω.
	Multiplying the above equation by (h • ∇u), integrating over Ω, then taking the real part, we
	obtain
		-λ 2
	=	i λ Ω	f 1 (h • ∇u)dx +	1 λ 2

Ω |λu| 2 dx = o(1) and a(u, u) = o(1). Ω u(h • ∇u)dx + Ω ∆ 2 u(h • ∇u)dx Ω f 2 (h • ∇u)dx (5.2.125)

in the case of one local Kelvin-Voigt damping on the bending moment with non-smooth coefficient at the interface (i.e. D 1 = D

3 = 0, D 2 ∈ L ∞ (0, L) and D 2 ≥ d 0 > 0 in ω).

Proof. We check that T is the Friedrichs extension of the sesquilinear, symmetric and coercive form a(f, g) = a(f, g)+ Γ 1 (∂ ν f ∂ ν g + f g)dΓ, defined in H 2 Γ 0 (Ω) × H 2 Γ 0 (Ω). Indeed, by Friedrichs extension Theorem, we can writeD(T ) = f ∈ H 2 Γ 0 (Ω) : ∃!F f ∈ L 2 (Ω) such that a(f, g) = (F f , g), ∀g ∈ H 2 Γ 0 (Ω)and T f = F f , ∀f ∈ D(T ).

, kind support, and advice during my Ph.D. research.

where k(s) is a constant depending on s. Then, from (3.3.9) and the fact that ω(•, s) ∈ W g (i.e. ω(0, s) = 0), we get ω(•, s) = 0 in (0, β) × (0, ∞). Next, from (3.3.13), (3.3.16) and the definition of c(•), we obtain the following system λ 2 y + y xx = 0, in (0, α), (3.3.17) y(0) = y(α) = y x (α) = 0.

(3.3.18)

Thus, from the above system and by using Holmgren uniqueness theorem, we obtain y = 0 in (0, α). (3.3.19) Therefore, from (3.3.4) and (3.3.19), we obtain y = z = 0 in (0, α). Let V = (u, u x , y, y x ) , then system (3.3.25)- (3.3.27) can be written as the following

where

The solution of the differential equation (3.3.28) is given by V (x) = e B(x-β) V (β). (3.3.29) Thus, from (3.3.29) and the fact that V (β) = 0, we get V = 0 in (β, γ) and consequently u = u x = y = y x = 0 in (β, γ). Chapter 4

On the Stability of Bresse system with one discontinuous local internal Kelvin-Voigt damping on the axial force

In this chapter, we investigate the stabilization of a linear Bresse system with one discontinuous local internal viscoelastic damping of Kelvin-Voigt type acting on the axial force, under fully Dirichlet boundary conditions. First, using a general criteria of Arendt-Batty, we prove the strong stability of our system. Finally, using a frequency domain approach combined with the multiplier method, we prove that the energy of our system decays polynomially with different rates. This chapter is published in [START_REF] Akil | On the stability of Bresse system with one discontinuous local internal Kelvin-Voigt damping on the axial force[END_REF].

Introduction

In this chapter, we investigate the stability of a Bresse system with only one discontinuous local internal Kelvin-Voigt damping on the axial force. More precisely, we consider the following system in (0, L) × (0, ∞):

with the following Dirichlet boundary conditions

and the following initial conditions

where ρ 1 , ρ 2 , k 1 , k 2 , k 3 , l and L are positive real numbers. We suppose that there exist 0 < α < β < L and a positive constant d 0 , such that

The solution of the differential equation (4.3.20) is given by

Thus, from (4.3.22) and the fact that V (α) = 0, we get

From (4.3.23) and the fact that v 1 (0) = v 3 (0) = v 5 (0) = 0, we get v 1 = 0 in (0, α), v 3 = 0 in (0, α) and v 5 = 0 in (0, α).

consequently, from (4.3.2), (4.3.4), (4.3.6) and the fact that λ = 0, we obtain

x ) . From (4.3.26) and the regularity of v i , i ∈ {1, 3, 5}, we have W (β) = 0 and system (4.3.17)- (4.3.19) in (β, L) implies:

where A λ is defined before (see (4.3.21)). Thus, we have

The proof is thus complete.

Lemma 4.3.2. For all λ ∈ R, we have

Chapter 5

Stability and instability results of the Kirchhoff plate equation with delay terms on the boundary or dynamical boundary controls

In this chapter, we consider two models of the Kirchhoff plate equation, the first one with delay terms on the dynamical boundary controls (see system (5.1.1) below), and the second one where delay terms on the boundary control are added (see system (5.1.2) below). For the first system, we prove its well-posedness, strong stability, non-exponential stability, and polynomial stability under a multiplier geometric control condition. For the second one, we prove its well-posedness, strong stability, and exponential stability under the same multiplier geometric control condition. Finally, we give some instability examples of system (5.1.2) for some choices of delays.

Introduction

Let Ω ⊂ R 2 be a bounded open set with boundary Γ of class C 4 consisting of a clamped part Γ 0 = ∅ and a rimmed part Γ 1 = ∅ such that Γ 0 ∩ Γ 1 = ∅. In the first part of this chapter, we study the stability of a Kirchhoff plate equation with delay terms on the dynamical boundary controls, namely we consider

In the second part of this chapter, we study the stability of the Kirchhoff plate equation with delay terms on the boundary controls, by considering:

Here and below, β 1 , γ 1 , τ 1 and τ 2 are positive real numbers, β 2 and γ 2 are non-zero real numbers, ν = (ν 1 , ν 2 ) is the unit outward normal vector along Γ, and τ = (-ν 2 , ν 1 ) is the unit tangent vector along Γ. The constant 0 < µ < 1 2 is the Poisson coefficient and the boundary operators B 1 and B 2 are defined by

where

Moreover, easy computations show that

(5.1.3)

In 1993, Rao in [START_REF] Rao | Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear boundary feedback[END_REF] studied the stabilization of the Kirchhoff plate equation with non-linear boundary controls (in the linear case, it corresponds to system (5.1.2) with β 2 = γ 2 = 0), under a multiplier geometric control condition he established an exponential energy decay rate. Furthermore, in 2005, Rao and Wehbe in [START_REF] Rao | Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent[END_REF] studied the stabilization of the Kirchhoff plate equation with dynamical boundary controls (corresponding to system (5.1.1) with β 2 = γ 2 = 0), under the same mulitplier geometric control condition they established a polynomial energy decay rate of order t -1 . Time delays appear in several applications such as in physics, chemistry, biology, thermal phenomena not only depending on the present state but also on some past occurrences (see [START_REF] Ernst | Delay-induced multistable synchronization of biological oscillators[END_REF][START_REF] Kolmanovskii | Introduction to the theory and applications of functional-differential equations[END_REF]). In the last years, the control of partial differential equations with time delays have become popular among scientists, since in many cases time delays induce some instabilities see [START_REF] Datko | Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks[END_REF][START_REF] Datko | Two examples of ill-posedness with respect to time delays revisited[END_REF][START_REF] Datko | An example of the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Dreher | Ill-posed problems in thermomechanics[END_REF].

In 2006, Nicaise and Pignotti in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF] studied the multidimensional wave equation with boundary feedback and a delay term at the boundary, by considering the following system: (5.1.4) where µ 1 and µ 2 are positive real numbers, and Ω is an open bounded domain of R n with a boundary Γ of class C 2 and Γ = Γ D ∪ Γ N , such that Γ D ∩ Γ N = ∅. Under the assumption µ 2 < µ 1 , an exponential decay is achieved. If this assumption does not hold, they found a sequences of delays {τ k } k , τ k → 0, for which the corresponding solutions have increasing energy. In 2020, Bayili et al. in [START_REF] Bayili | Rational energy decay rate for the wave equation with delay term on the dynamical control[END_REF] studied the multidimensional wave equation with a delay term in the dynamical control, by considering the following system:

where β 1 and β 2 are positive real numbers, and Ω is an open bounded domain in R n with a lipschitz boundary Γ = Γ D ∪ Γ N with meas(Γ D ) = 0 and meas(Γ N = 0). Under the assumption β 2 < β 1 , they showed that the system is not exponentially stable, but they proved that the system has the same decay rate than the one without delay.

But to the best of our knowledge, it seems that there is no result in the existing literature concerning the case of the Kirchhoff plate equation with dynamical boundary controls and time delays (or with boundary controls and time delay). The goal of the present chapter is to fill this gap by studying the stability of systems (5.1.1) and (5.1.2).

In the first part of this chapter, we study the stability of system (5.1.1). In Subsection 5.2.1, we prove the well-posedness of our system by using semigroup approach. In Subsection 5.2.2, following a general criteria of Arendt and Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. In subsection 5.2.3, we prove that the system (5.1.1) is not exponentially stable. Next, in Subsection 5.2.4, by combining the frequency domain approach with a specific multiplier method, we prove under the multiplier geometric control condition (MGC) that the energy of our system decays polynomially with the rate t -1 .

In the second part of this chapter, we study both stability and instability of system (5.1.2). In subsection 5.3.1, we prove the well-posedness and the strong stability of our system. In subsection 5.3.2, we prove under the same (MGC) condition that system (5.1.2) is exponentially stable. Finally, in subsection 5.3.3, if |β 2 | ≥ β 1 and |γ 2 | ≥ γ 1 , we give some instability examples of system (5.1.2) for some particular choices of delays 5.2 Kirchhoff plate equation with delay terms on the dynamical boundary control

Well-posedness of the system

In this section, we will establish the well-posedness of system (5.1.1) by using semigroup approach. To this aim, as in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF], we introduce the following auxiliary variables

(5.2.1)

Then, system (5.1.1) becomes

with the following initial conditions

(5.2.10)

The energy of system (5.2.2)-(5.2.10) is given by

where the sequilinear form a :

(5.2.11)

We first recall the following Green's formula (see [START_REF] Lagnese | Boundary Stabilization of Thin Plates[END_REF]):

(5.2.12)

Remark 5.2.1. From the fact that 2 (u

consequently, from (5.2.11), we get

Hence the sesquilinear form a is coercive on H 2 Γ 0 (Ω), since Γ 0 is non empty. On the other hand, from (5.2.13) (see also Lemma 3.1 and Remark 3.1 in [START_REF] Rao | Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear boundary feedback[END_REF]), we remark that

) is regular enough, then system (5.2.2)-(5.2.10) can be written as the following first order evolution equation

where

Proposition 5.2.1. Under the hypothesis (H), the unbounded linear operator A is mdissipative in the energy space H.

Proof. For all U = (u, v, η, ξ, z 1 , z 2 ) ∈ D(A), from (5.2.18) and (5.2.19), we have

Using (5.2.21) and the fact that U ∈ D(A), we obtain

(5.2.23) Now, by using Young's inequality, we get

(5.2.24)

Inserting (5.2.24) in (5.2.23) and using the hypothesis (H), we obtain

which implies that A is dissipative. Now, let us prove that A is maximal. To this aim, if

Equivalently, we have the following system

with the following boundary conditions

From (5.2.27) and the fact that F ∈ H, we get

(5.2.34)

From (5.2.31), (5.2.32), (5.2.33) and the fact that F ∈ H, we obtain

(5.2.39)

Multiplying the first equation in (5.2.39) by ϕ and integrating over Ω, then using Green's formula, we obtain

where

It is easy to see that, a is a sesquilinear, continuous and coercive form on H 2 Γ 0 (Ω) × H 2 Γ 0 (Ω) and l is an antilinear and continuous form on H 2 Γ 0 (Ω). Then, it follows by Lax-Milgram theorem that (5.2.40) admits a unique solution u ∈ H 2 Γ 0 (Ω). By taking the test function ϕ ∈ D(Ω), we see that the first identity of (5.2.39) holds in the distributional sense, hence ∆ 2 u ∈ L 2 (Ω). Coming back to (5.2.40), and again applying Greens's formula (5.2.13), we find that

and

Further since F ∈ H, we deduce that u ∈ D Γ 0 (∆ 2 ). Consequently, if we define U = (u, v, η, ξ, z 1 , z 2 ) with u ∈ H 2 Γ 0 (Ω) the unique solution of (5.2.40), v = -f 1 , ξ (resp. η) defined by (5.2.37) (resp. (5.2.38)) and z 1 (resp. z 2 ) defined by (5.2.35) (resp. (5.2.36)), U belongs to D(A) is the unique solution of (5.2.26). Then, A is an isomorphism and since ρ (A) is open set of C (see Theorem 1.1.13), we easily get R(λI -A) = H for a sufficiently small λ > 0. This, together with the dissipativeness of A, imply that D (A) is dense in H and that A is m-dissipative in H (see Theorems 1.2.6, 1.2.9). The proof is thus complete.

According to Lumer-Phillips theorem (see Theorem 1.2.8), Proposition 5.2.1 implies that the operator A generates a C 0 -semigroup of contractions e tA in H which gives the well-posedness of (5.2.22). Then, we have the following result: Theorem 5.2.1. For all U 0 ∈ H, system (5.2.22) admits a unique weak solution

Moreover, if U 0 ∈ D(A), then the system (5.2.22) admits a unique strong solution

Strong Stability

In this section, we will prove the strong stability of system (5.2.2)-(5.2.10). The main result of this section is the following theorem.

Theorem 5.2.2. Under the hypothesis (H), the C 0 -semigroup of contractions e tA t≥0 is strongly stable in H; i.e., for all U 0 ∈ H, the solution of (5.2.22) satisfies lim t→+∞ e tA U 0 H = 0.

According to Theorem 1.3.3, to prove Theorem 5.2.2, we need to prove that the operator A has no pure imaginary eigenvalues and σ(A) ∩ iR is countable. The proof of these results is not reduced to the analysis of the point spectrum of A on the imaginary axis since its resolvent is not compact. Hence the proof of Theorem 5.2.2 has been divided into the following two Lemmas.

Lemma 5.2.2. For all λ ∈ R, iλI -A is injective i.e., ker(iλI -A) = {0}.

Proof. From Proposition 5.2.1, we have 0 ∈ ρ(A). We still need to show the result for λ ∈ R * . To this aim, suppose that λ = 0 and let U = (u, v, η, ξ, z 1 , z 2 ) ∈ D(A) be such that AU = iλU.

(5.2.41)

Equivalently, we have the following system v = iλu, (5.2.42) -∆ 2 u = iλv, (5.2.43)

(5.2.45)

Thus, we have η = ξ = 0 on Γ 1 .

(5.2.48) Using (5.2.46), (5.2.47) and the fact that z 1 (•, 0) = η, z 2 (•, 0) = ξ on Γ 1 , then using (5.2.48), we obtain z 1 (•, ρ) = ηe -iλτ 1 ρ = 0 on Γ 1 × (0, 1), (5.2.49)

consequently, from (5.2.42) and the fact that λ = 0, we obtain

(5.2.52) Now, from (5.2.48) and the fact that U ∈ D(A), we get

(5.2.54) Using (5.2.52) and the fact that ∇u = ∂ τ uτ + ∂ ν uν on Γ 1 , we obtain

(5.2.55)

Now, from (5.1.3), (5.2.52) and (5.2.55), we get

Holmgren uniqueness theorem (see [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF]) yields

The proof is thus complete.

Lemma 5.2.3. Under the hypothesis (H), for all λ ∈ R, we have

Proof. From Proposition 5.2.1, we have 0 ∈ ρ(A). We still need to show the result for λ ∈ R . For this aim, for

Equivalently, we have the following system

(5.2.64)

)

with the following boundary conditions

From (5.2.65), (5.2.66) and (5.2.67), we deduce that

s)e iλτ 1 (s-ρ) ds on Γ 1 × (0, 1), (5.2.68)

where

(5.2.71) It follows from (5.2.61), (5.2.62), (5.2.67) and (5.2.70) that

(5.2.72) Let ϕ ∈ H 2 Γ 0 (Ω). Multiplying the first equation in (5.2.72) by ϕ, integrating over Ω, then using Green's formula, we obtain

.2.75)

Let V be the dual space of V. Let us define the following operators

Steps 1 and 2 guarantee that the operator B is isomorphism. Furthermore it is easy to see that the operator l is an antilinear and continuous form on V. Consequently, (5.2.73) admits a unique solution u ∈ V. In (5.2.73), by taking test functions ϕ ∈ D(Ω), we see that the first identity of (5.2.72) holds in the distributional sense, hence ∆ 2 u ∈ L 2 (Ω). Coming back to (5.2.73), and again applying Green's formula (5.2.13), we find that

Further since u, ∂ ν u, F iλ and G iλ belong to L 2 (Γ 1 ), we deduce that u ∈ D Γ 0 (∆ 2 ). Consequently, if u ∈ V is the unique solution of (5.2.73) and if we define η and ξ by (5.2.70) and z 1 (resp. z 2 ) by (5.2.68) (resp. (5.2.69)), we deduce that

belongs to D(A) and is the unique solution of (5.2.60). The proof is thus complete.

Proof of Theorem 5.2.2. From Lemma 5.2.2, the operator A has no pure imaginary eigenvalues (i.e. σ p (A) ∩ iR = ∅). Moreover, from Lemma 5.2.2 and Lemma 5.2.3, iλI -A is bijective for all λ ∈ R and since A is closed, we conclude with the help of the closed graph theorem that iλI -A is an isomorphism for all λ ∈ R, hence that σ(A) ∩ iR = ∅. Therefore, according to Theorem 1.3.3, we get that the C 0 -semigroup (e tA ) t≥0 is strongly stable. The proof is thus complete.

Lack of exponential stability

In this section, we will prove that the system (5.2.2)-(5.2.10) is not exponential stable. Let us start with a technical result.

Lemma 5.2.4. Define the linear unbounded operator T : D(T ) -→ L 2 (Ω) by

Then, T is a positive self-adjoint operator with a compact resolvent.

We now need to show that this operator T coincides with the one defined by (5.2.78)-(5.2.79). For that purpose, let us denote by D(T ) the right-hand side of (5.2.78). By Green's formula (5.2.13), we directly see that D(T ) ⊆ D(T ) and that for f ∈ D(T ), T f is indeed given by (5.2.79). Let us then prove the converse inclusion. For this aim, let f ∈ D(T ), then we have

Hence f belongs to E(∆ 2 , L 2 (Ω)) and using Green's formula (5.2.13), we obtain

This proves that D(T ) = D(T ). Finally as H 2 Γ 0 (Ω) is compactly embedded in L 2 (Ω), T has clearly a compact resolvent. The proof is thus complete.

The main result of this section is the following theorem.

Theorem 5.2.3. The C 0 -semigroup (e tA ) t≥0 is not uniformly stable in the energy space H.

Proof. According to Theorem 1.3.6 due to 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], it is sufficient to show that the resolvent of A is not uniformly bounded on the imaginary axis. In other words, it is enough to show the existence of a positive real number M and some sequences

Now, let us choose

(5.2.85)

It is easy to see that

(5.2.87)

Polynomial stability

In this section, we will prove the polynomial stability of system (5.2.2)-(5.2.10). The main result of this section is the following theorem.

Theorem 5.2.4. Under the hypothesis (H) and the multiplier geometric control condition MGC (see Definition 1.4.1), for all U 0 ∈ D(A), there exists a constant C > 0 independent of U 0 such that the energy of system (5.2.2)-(5.2.10) satisfies the following estimation

According

(5.2.100) For simplicity, we now drop the index n. Equivalently, from (5.2.100), we have

(5.2.104)

)

Here we will check the condition (5.2.98) by finding a contradiction with (5.2.99) by showing U H = o(1). For clarity, we divide the proof into several Lemmas.

Lemma 5.2.5. Under the hypothesis (H), the solution U = (u, v, η, ξ, z 1 , z 2 ) ∈ D(A) of (5.2.101)-(5.2.106) satisfies the following estimations

(5.2.107)

Proof. In this proof, we follow the arguments of the proof of Lemma 3.1 in [START_REF] Rao | Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear boundary feedback[END_REF] and Lemma 3.1 in [START_REF] Rao | Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent[END_REF]. First, we assume that

, then as u ∈ D Γ 0 (∆ 2 ) we get u ∈ H 4 (Ω). Now, by the identity (3.5) in [START_REF] Rao | Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear boundary feedback[END_REF] (see also [START_REF] Lagnese | Boundary Stabilization of Thin Plates[END_REF], [START_REF] Rao | Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent[END_REF] and [START_REF] Ammari | Stabilization of a transmission wave/plate equation[END_REF]), we get

where

From (5.2.20), we deduce that

where

Now, since u = ∂ ν u = 0 on Γ 0 , then using the identities (3.5) and (3.6) in [START_REF] Rao | Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent[END_REF], we have

where C 1 is defined in (5.1.3). Consequently, we get

(5.2.118) Now, by using Young's inequality, we get

where R = h L ∞ (Ω) and ε 1 is an arbitrary positive constant fixed below. Now, according to the identity (3.9) in [START_REF] Rao | Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent[END_REF] (see also (3.7) in [START_REF] Rao | Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear boundary feedback[END_REF]), we notice that

(5.2.120) Using (5.2.120), Cauchy-Schwarz and Young's inequalities, (5.2.107), and (5.2.111), we get

Now, by using Green's formula and the fact that u = 0 on Γ 0 , then using (5.2.111), we get

Using the fact that a(u, u) = O(1) and a(f 1 , f 1 ) = o(1), we obtain

Thus, from the above estimations and the fact that f 2 → 0 in L 2 (Ω), we obtain

inserting the above estimation in (5.2.128), we get

The proof is thus complete. Proof of Theorem 5.2.4 From Lemmas 5.2.5, 5.2.6 and 5.2.9, we deduce that

which contradicts (5.2.99).

Kirchhoff plate equation with delay terms on the boundary controls

Wellposedness and strong stability

In this section, we will establish the well-posedness and the strong stability of system (5.1.2). For this aim, as in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF], we introduce the following auxiliary variables

Then, system (5.1.2) becomes

with the following initial conditions

(5.3.8)

The energy of system (5.3.2)-(5.3.8) is given by

where a is defined in (5.2.11). If (u, u t , z 1 , z 2 ) is a regular solution of (5.3.2)-(5.3.8), then similarly to the proof of Lemma 5.2.1, we obtain

(5.3.10)

Hence under the hypothesis (H), system (5.3.2)-(5.3.8) is dissipative in the sense that its energy is non-increasing with respect to time . Let us define the Hilbert space H 0 by

equipped with the following inner product

where

) ∈ H 0 . Now, we define the linear unbounded operator A 0 : D(A 0 ) ⊂ H 0 -→ H 0 by:

Now, if U = (u, u t , z 1 , z 2 ) is solution of (5.3.2)-(5.3.8) and is sufficiently regular, then system (5.3.2)-(5.3.8) can be written as the following first order evolution equation

where U 0 = (u 0 , u 1 , f 0 (•, -ρτ 1 ), g 0 (•, -ρτ 2 )) ∈ H 0 .

Proposition 5.3.1. Under the hypothesis (H), the unbounded linear operator A 0 is mdissipative in the energy space H 0 .

Proof. Similarly to the proof of Lemma 5.2.1, we show that

and that 0 ∈ ρ(A 0 ).

According to Lumer-Phillips theorem (see Theorem 1.2.8), Proposition 5.3.1 implies that the operator A 0 generates a C 0 -semigroup of contractions e tA 0 in H 0 which gives the well-posedness of (5.3.13). Then, we have the following result:

Theorem 5.3.1. For all U 0 ∈ H 0 , system (5.3.13) admits a unique weak solution U (t) = e tA 0 U 0 ∈ C 0 (R + , H 0 ). Moreover, if U 0 ∈ D(A 0 ), then the system (5.3.13) admits a unique strong solution

Theorem 5.3.2. Under the hypotheses (H) and (1.4.1), the C 0 -semigroup of contractions e tA 0 t≥0 is strongly stable in H 0 ; i.e., for all U 0 ∈ H 0 , the solution of (5.3.13) satisfies lim t→+∞ e tA 0 U 0 H 0 = 0.

Proof. Similarly to the proof of Theorem 5.2.2, we can show that ker(iλI

consequently A 0 has no pure imaginary eigenvalues and σ(A 0 ) ∩ iR = ∅, and we conclude by Theorem 1.3.3.

Exponential stability

Theorem 5.3.1. Under the hypotheses (H) and (1.4.1), the C 0 -semigroup e tA 0 is exponentially stable; i.e., for all U 0 ∈ H 0 , there exist constants M ≥ 1 and > 0 independent of U 0 such that e tA 0 U 0 H 0 ≤ M e -t U 0 H 0 , t > 0.

Proof. Since iR ⊂ ρ(A 0 ) (see Section 5.

For simplicity, we drop the index n. Equivalently, from (5.3.18), we have

)

.3.22)

Taking the inner product of (5.3.18) with U in H 0 and using (5.3.14), we get

From the above estimation, (H) and the fact that F H 0 = o(1) and U H 0 = 1, we obtain

From the above equations, (5.3.23) and the fact that F H 0 = o(1), we obtain

and consequently, from (5.3.28), we deduce that which contradicts (5.3.17). The proof is thus complete.

Some instability results

In this subsection, we will give some instability examples of system (5.1.2) in the cases |β 2 | ≥ β 1 and |γ 2 | ≥ γ 1 . This is achieved by distinguishing between the following cases:

)

Theorem 5.3.3. If (IS 1 ) or (IS 2 ) hold, then there exist sequences of delays and solutions of (5.1.2) corresponding to these delays such that their standard energy is constant.

Proof. We seek for a solution of system (5.1.2) in the form u(x, t) = e iλt ϕ(x), with λ = 0.

(

(5.3.32)

Let θ ∈ H 2 Γ 0 (Ω). Multiplying the first equation in (5.3.32) by θ, then using Green's formula, we get Thus, from (5.3.37) and (5.3.38), we get λ 2 -a(ϕ, ϕ) -λ β 2 2 -β 2 1 q ν (ϕ) -λ γ 2 2 -γ 2 1 q(ϕ) = 0, (5.3.39) where q(ϕ) = which becomes arbitrarily small (or large) for suitable choices of the indices n, k, l ∈ N. Therefore, we have found sets of time delays for which system (5.1.2) is not asymptotically stable.

Case 2: If (IS 2 ) holds, then from (5.3.39), we have Let us prove that if the minimum in the right-hand side of (5.3.45) is attained at ϕ, that is β 2 2 -β 2 1 q ν (ϕ) + γ 2 2 -γ 2 1 q(ϕ) + β 2 2 -β 2 1 q ν (ϕ) + γ 2 2 -γ 2 1 q(ϕ)

thus, from (5.3.46), we get f (ε) ≥ f (0) = β 2 2 -β 2 1 q ν (ϕ) + γ 2 2 -γ 2 1 q(ϕ)