Keywords: Finite differences, Finite elements, Damped Timoshenko system, Nonlinear dissipation, Energy method, Strong stability, Optimality, Asymptotic behavior, Lower energy estimates, Regularity, Energy comparison principles

 for the dimension 1 and in this thesis for the dimension 2. Furthermore, we discuss the well-posedness and the decay rate of the energy of the Mindlin-Timoshenko plate equations subject to a nonlinear dissipation acting on the rotation angles' equations. In the second part of the thesis, we perform a numerical study of the one-dimensional Timoshenko system; our approach is based on an adequate use of the finite element and the finite difference methods giving rise to a new scheme which characterizes the asymptotic behavior of the discrete energy.

ACKNOWLEDGMENTS

To all those who have kindly provided the necessary support to make the dream became a fact, I reserve this page as a sign of my deep gratitude.

I would like to express my deepest appreciation to my advisors, Prof. Ahmed Bchatnia and Prof. Makram Hamouda, for their valuable guidance and instructions, constant support and interest throughout this study. I appreciate honest and insightful feedback offered by Prof. Ahmed Bchtnia. Prof. Makram Hamouda constructive comments and warm encouragement led me through the writing. My advisors consistently allowed this paper to be my own work, but steered me in the right the direction whenever they thought I needed it.

I would like to thank my thesis committee members, starting by Prof. Amel Atallah for accepting to be the committee chair of my thesis. I extend a heartfelt appreciation to Prof.

Stephane Gerbi for the time, the ideas and the comments that have been very interesting. I would like to thank Prof. Kais Ammari generously offering his time and good will throughout the review of this document. I also thank Prof. Fathi Mahmoudi for the valuable service to examine my thesis and to be a member of the committee. I would like to express my very great appreciation to the hospitality, advice and comments offered by Prof. Olivier Goubet and my special thanks also are for all the LAMFA members.

I have the honor to be a member of Nonlinear Analysis and Geometry Research Unit and I am particularly thankful to Prof. Sami Baraket, Director, ANLIG, for his financial and emotional support.

I would like to thank my mother for her constant care; I appreciate the special role that she have always played in my life. I would like to thank her for made me learn who to be strong and selfless woman. I never forgot to be grateful to the man who has always the faith in me, who supported and encourage me in each step of my life. The man that I am proud to call him my father Finally, I would like to offer my special thanks to my sister, my brother and my best friends for helping me to change my stress mode and for creating the healthful free time that refresh my efforts.

Introduction Motivation

The most interesting part of the research in mathematics is the fact that a system of partial differential equations (PDE) is able to describe one of the most complicated physical phenomena which is the vibration of the mechanical structure and to model the role of each parameter. Also, the most exciting part of the work is when you realize that dealing with such kind of systems gives you the chance to know more about other fields such that civil engineering, mechanical engineering, aeronautical, biomechanics, etc.

In the recent years, The Timoshenko system has attracted a considerable attention. Indeed, this system is important, not only from the mathematical point of view but mainly from the physical perspective in view of the multiple with applications in mechanics, among many other sciences. The Timoshenko system is considered also as one of the simplest models which describes the transverse vibration of a beam with fixed extremities.

The Timoshenko system with thermoelasticity

The 1D model presented in this work is based on Timoshenko theory (see e.g [START_REF] Timoshenko | The collected papers of Stephen P. Timoshenko[END_REF]). This model is occupying a great place as a dynamical model describing the transverse vibrations of beams. The Timoshenko theory is considered as an improvement of Euler-Bernoulli theory. Indeed, The Euler-Bernoulli beam theory assumed that, even after deformation, the plane cross-sections and the axis of the beam remain plane and perpendicular. Consequently, the transverse shear strain is equal to zero. This why the Timoshenko theory is rather used then Euler-Bernoulli one when the rotational inertia and the transverse shear are significant in the beam model. To be more precise, we give a quick representation in Figure 1.

The transverse vibration of a beam under the Timoshenko approach is mathematically described by the following coupled system    ρϕ tt -K(ϕ x + ψ) x = 0 in (0, L) × (0, +∞)

I ρ ψ tt -(EIψ x ) x + K(ϕ x + ψ) = 0 in (0, L) × (0, +∞) (1)
t : Time variable x : Space variable L : Length of the beam in the equilibrium position ϕ : The transverse displacement of the beam ψ : The rotation angle of the filament ρ : Density (mass per unit length) I ρ : The polar moment of inertia of a cross-section E : Young's modulus of elasticity I : The moment of inertia of a cross section k : The shear modulus of Timoshenko.

Historically, several Mathematicians have studied the Timoshenko type systems theoretically and numerically and among these works, we quote the standard Timoshenko systems, with thermoelasticity and with thermoelasticity with second sound. In 1807, the French mathematician and physicist Joseph Fourier proposed the classical model of thermoelasticity which states that the heat flux is proportional to the gradient of temperature. However, this law involves some physical paradox of infinite speed of heat propagation. Indeed, experiments showed that the heat conduction in some dielectric crystals at low temperatures is free of this paradox and disturbances, which are almost entirely thermal, propagate with a finite speed. This phenomenon in dielectric crystals is called second sound (see [START_REF] Coleman | Stability of equilibrium for a nonlinear hy-perbolic system describing heat propagation by second sound in solids[END_REF]).

Consequently, to overcome the drawback in the Fourier's law but still keeping the essentials of heat conduction process, a number of modifications of the basic assumption on the relation between the heat flux and the temperature has been made, such as: Cattaneo law, Gurtin Pipkin theory, Jeffreys law, Green Naghdi theory and others. The common feature of these theories is that all lead to hyperbolic differential equation and permit transmission of heat flow as thermal waves at finite speed. In 1948, Cattaneo proposed the following law knows "Cattaneo's law":

τ q t + βq + θ x = 0,
where q is the heat flux and β is the coefficient of the thermal conductivity. This law, according to some authors, is considered as the simplest generalization of the Fourier's law. The parameter τ > 0 represents the relaxation time describing the time lag in the response of the heat flux to the gradient of the temperature. Therefore, the heat is transported by a wave propagation process instead of the usual diffusion eliminating thus the physical paradox of infinite speed of heat propagation in Fourier's law, βq + θ x = 0.

Literature overview

The well understanding of this model was the goal of a great number of researchers, thus, an important amount of research has been devoted to the issue of the stabilization of the Timoshenko system by the use of diverse types of dissipative mechanisms aiming to obtain a solution which decays uniformly to the stable state as time goes to infinity. To achieve this goal several energy estimates have been established. For an overview purpose, we mention some known results in this regard. Kim and Renardy [START_REF] Rivera | R: Global stability for damped Timoshenko systems[END_REF], Messaoudi and Mustafa [START_REF] Mustafa | General energy decay rates for a weakly damped Timoshenko system[END_REF], Raposo et al. [START_REF] Raposo | Exponential stability for Timoshenko system with two weak dampings[END_REF], and others, showed that the presence of damping terms on both equations (1) leads to uniform stability result regardless of the values of the damping coefficients. The situation is much different when the damping term is only imposed on the rotation angle equation in the Timoshenko system. In this case, the exponential stability holds if and only if the propagation velocities are equal. It is worth noting that the first result including the linear and nonlinear indirect damping cases and showing polynomial stability for different speeds of propagation was established in [START_REF] Alabau-Boussouira | Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control[END_REF] giving thus optimal results in the nonlinear damping case (and getting as a particular case the exponential decay for the same speeds of propagation

v 1 = k ρ 1 = v 2 = b ρ 2 ; (2)
see [START_REF] Alabau-Boussouira | Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control[END_REF][START_REF] Abdelhak | Stabilization of the Timoshenko Beam by Thermal Effect[END_REF][START_REF] Guesmia | General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping[END_REF][START_REF] Guesmia | Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-Heat systems[END_REF][START_REF] Guesmia | On the control of a viscoelastic damped Timoshenko-type system[END_REF][START_REF] Muñoz Rivera | Stability of Timoshenko systems with past history[END_REF] and the references therein.

Concerning the stabilization via heat effect, Rivera and Racke [START_REF] Rivera | Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability[END_REF] investigated the following system:

       ρ 1 ϕ tt -σ(ϕ x , ψ) x = 0, in (0, L) × IR + , ρ 2 ψ tt -bψ xx + k(ϕ x + ψ) + γθ x = 0, in (0, L) × IR + , ρ 3 θ t -kθ xx + γψ xt = 0, in (0, L) × IR + ,
where ϕ, ψ, θ are of (x, t) functions and model the transverse displacement of the beam, the rotation angle of the filament, and the difference temperature, respectively. Under appropriate conditions on σ, ρ i , b, k, γ, they proved several exponential decay results for the linearized system and non exponential stability result for the case of different wave speeds.

Concerning Timoshenko systems of thermoelasticity with second sound, Messaoudi et al. [START_REF] Messaoudi | Nonlinear damped Timoshenko systems with second sound{global existence and exponential stability[END_REF] studied

             ρ 1 ϕ tt -σ(ϕ x , ψ) x + µϕ t = 0, in (0, L) × IR + , ρ 2 ψ tt -bψ xx + k(ϕ x + ψ) + βθ x = 0, in (0, L) × IR + , ρ 3 θ t + γq x + δψ tx = 0, in (0, L) × IR + , τ 0 q t + q + κθ x = 0, in (0, L) × IR + ,
where ϕ = ϕ(x, t) is the displacement vector, ψ = ψ(x, t) is the rotation angle of the filament, θ = θ(x, t) is the temperature difference, q = q(x, t) is the heat flux vector and ρ 1 , ρ 2 , ρ 3 , b, k, γ, δ, κ, µ, τ 0 are positive constants. The nonlinear function σ is assumed to be sufficiently smooth and satisfy σ ϕx (0, 0) = σ ψ (0, 0) = k, and σ ϕxϕx (0, 0) = σ ϕxψ (0, 0) = σ ψψ (0, 0) = 0.

Several exponential decay results for both linear and nonlinear cases have been established in the presence of the extra frictional damping µϕ t .

Fernández Sare and Racke [START_REF] Racke | Fernàndez Sare: On the stability of damped Timoshenko systems-Cattaneo versus Fourier law[END_REF] considered

             ρ 1 ϕ tt -k(ϕ x + ψ) x = 0, in (0, L) × IR + , ρ 2 ψ tt -bψ xx + k(ϕ x + ψ) + βθ x = 0, in (0, L) × IR + , ρ 3 θ t + γq x + δψ tx = 0, in (0, L) × IR + , τ 0 q t + q + κθ x = 0, in (0, L) × IR + , (3)
and showed that, in the absence of the extra frictional damping (µ = 0), the coupling via Cattaneo's law causes loss of the exponential decay usually obtained in the case of coupling via Fourier's law [START_REF] Rivera | Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability[END_REF]. This surprising property holds even for systems with history of the form

             ρ 1 ϕ tt -k(ϕ x + ψ) x = 0, in (0, L) × IR + , ρ 2 ψ tt -bψ xx + k(ϕ x + ψ) + +∞ 0 g(s)ψ xx (., t -s)ds + βθ x = 0, in (0, L) × IR + , ρ 3 θ t + γq x + δψ tx = 0, in (0, L) × IR + , τ 0 q t + q + κθ x = 0, in (0, L) × IR + . ( 4
)
Precisely, it has been shown that both systems (3) and ( 4) are no longer exponentially stable even for equal-wave speeds k ρ 1 = b ρ 2 . However, no other rate of decay has been discussed. Very recently, Santos et al. [START_REF] Santos | The stability number of the Timoshenko system with second sound[END_REF] considered (3) and introduced a new stability number

χ = τ 0 - ρ 1 kρ 3 ρ 2 - ρ 1 b k - ρ 1 β 2 ρ 1 kρ 3 ,
and used the semi-group method to obtain exponential decay result for χ = 0 and a polynomial decay for χ = 0. Later, in [START_REF] Ayadi | General decay in a Timoshenko-type system with thermoelasticity with second sound[END_REF] (see also [START_REF] Tijani | Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks[END_REF]), Ayadi et al. considered a vibrating nonlinear Timoshenko system with thermoelasticity with second sound. Precisely, they looked into the following system:

                                     ρ 1 ϕ tt -k(ϕ x + ψ) x = 0, in (0, 1) × IR + , ρ 2 ψ tt -bψ xx + k(ϕ x + ψ) + δθ x + α(t)h(ψ t ) = 0, in (0, 1) × IR + , ρ 3 θ t + q x + δψ xt = 0, in (0, 1) × IR + , τ q t + βq + θ x = 0, in (0, 1) × IR + , ϕ x (0, t) = ϕ x (1, t) = ψ(0, t) = ψ(1, t) = q(0, t) = q(1, t) = 0, ∀ t ≥ 0, ϕ(x, 0) = ϕ 0 (x), ϕ t (x, 0) = ϕ 1 (x), ∀ x ∈ (0, 1), ψ(x, 0) = ψ 0 (x), ψ t (x, 0) = ψ 1 (x), ∀ x ∈ (0, 1), θ(x, 0) = θ 0 (x), q(x, 0) = q 0 (x), ∀ x ∈ (0, 1),
and established an explicit and general decay result using a multiplier method for a wide class of relaxation functions without imposing the usual growth conditions on the frictional damping in both cases when χ = 0 and χ = 0.

Main results

As a first step, we consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound.

             ρ 1 ϕ tt -k(ϕ x + ψ) x = 0, in (0, 1) × IR + , ρ 2 ψ tt -bψ xx + k(ϕ x + ψ) + δθ x + a(x)g(ψ t ) = 0, in (0, 1) × IR + , ρ 3 θ t + q x + δψ xt = 0, in (0, 1) × IR + , τ q t + βq + θ x = 0, in (0, 1) × IR + .
(

) 5
associted with the Dirichlet boundary conditions

ϕ(0, t) = ϕ(1, t) = ψ(0, t) = ψ(1, t) = q(0, t) = q(1, t) = 0, ∀ t ≥ 0. ( 6
)
and the initial conditions

U 0 = (ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , θ 0 , q 0 ) such that        ϕ(x, 0) = ϕ 0 (x), ϕ t (x, 0) = ϕ 1 (x), ∀ x ∈ (0, 1), ψ(x, 0) = ψ 0 (x), ψ t (x, 0) = ψ 1 (x), ∀ x ∈ (0, 1), θ(x, 0) = θ 0 (x), q(x, 0) = q 0 (x), ∀ x ∈ (0, 1),
The corresponding nonlinear term satisfies the following assumption:

• (H 0 ) we assume that a is a smooth function which satisfies a(x) ≥ 0, x ∈ (0, 1) , a > 0 on any nonempty subset Γ of (0, 1).

• (H 1 ) g : R → R is a nondecreasing C 0 -function such that for all ∈ (0, 1), there exist positive constants c 1 , c 2 , and an increasing odd function g 0 ∈ C 1 (0, +∞) with g 0 (0) = 0 such that:

   g 0 (|s|) ≤ |g(s)| ≤ g -1 0 (|s|), for all |s| ≤ , c 1 |s| ≤ |g(s)| ≤ c 2 |s|, for all |s| ≥ .
We suppose in addition that Γ satisfies the assumption (HC)

• (HC): Let (ϕ, ψ) be a weak solution of the conservative system, if ψ t ≡ 0 on Γ then (ϕ, ψ) ≡ (0, 0).

Moreover, we define

• The energy E and the function space H

E(t) := 1 2 1 0 (ρ 1 ϕ 2 t + ρ 2 ψ 2 t + bψ 2 x + k(ϕ x + ψ) 2 + ρ 3 θ 2 + τ q 2 )dx, (7)
H = H 1 0 (0, 1) × L 2 (0, 1) × H 1 0 (0, 1) × (L 2 (0, 1)) 3 .
Our main results for the first part will be sated in the following :

Theorem. (Strong stability)

Assume that the hypotheses (H 0 ),(H 1 ) and (HC) hold Then, for all U 0 ∈ H, we have

E(t) → 0, as t → ∞. ( 8
)
where,

E(t) = E(t, (ϕ, ψ, θ -θ 0 (0), q)) = 1 2 1 0 ρ 1 ϕ t 2 + ρ 2 ψ t 2 + bψ x 2 + k(ϕ x + ψ) 2 + ρ 3 (θ -θ 0 (0)) 2 + τ q 2 dx.
Now, we introduce some basic functions that will be used throughout the following Theorem in which we find a lower bound of the energy that depend explicitly of the function K and it describes the decay rate.

• The function Ψ is given by Ψ

(x) = √ xg( √ x). ( 9
)
• The function Ψ is nondecreasing on [0, r 2 0 ] for r 0 > 0 sufficiently small, given by

Ψ(x) = Ψ(x) x , ∀x > 0, Ψ(0) = 0, (10)

Theorem. (Lower bound)

We assume that the hypotheses (H 0 ),(H 1 ) and (HC) hold. Let U 0 ∈ H. the energy E satisfies the following lower estimate

1 γ 2 K -1 (σ(t -T 0 ) 2 ≤ E(t), ∀ t ≥ T 0 , ( 11
)
where K is given by

K(χ) = γ √ E(T 0 ) χ 1 Ψ(y) dy, ∀ χ ∈ (0, γ E(T 0 )), ( 12
)
with γ as defined by

γ = 4 ρ 2 E (0), ( 13
)
and σ is a positive constant given by σ = αa ρ 2 + βr 0 τ C 1 , where α a is defined by

α a = a L ∞ (0,1) . ( 14
)
Moreover, if lim χ→0 + K(χ) = +∞, then

lim t→∞ K -1 (σ(t -T 0 )) = 0.
Later, we use the previous result to obtain an explicit lower bound of the energy giving in the following Theorem

Theorem. (Explicit lower estimate)

Assume that (H 0 ), (H 1 ) and (H 2 ) hold. Then, for all non vanishing smooth initial data, there exist T 0 > 0 and T 1 > 0 such that the energy E satisfies the following lower estimate

1 γ 2 C 2 σ Ψ -1 1 t -T 0 2 ≤ E(t), ∀ t ≥ T 1 + T 0 . ( 15
)
We end the first part of this work by obtaining an optimal result formulated as follows:

Corollary. (Optimality)

Assume that g = g 0 , where g 0 is given by

g 0 (s) = 1 s exp( -1 s 2 ), ∀ s ∈]0, r 2 0 ], (16)
with r 0 > 0, sufficiently small, so that g is then extended on R as an odd, convex and C 1 function which satisfies the hypotheses of the previous Theorem. Then the energy associated with system [START_REF] Alabau-Boussouira | Control of Partial Differential Equations[END_REF] with the damping term given by [START_REF] Brandts | A note on uniform superconvergence for the Timoshenko beam using mixed finite elements[END_REF] satisfies

c (ln(t)) -1 ≤ E(t) ≤ c (ln(t)) -1 .
Hence, the estimate ( 15) is optimal for the damping function g as given in [START_REF] Brezis | Analyse Fonctionelle, Théorie et Applications[END_REF].

The Mindlin Timoshenko system

As an extension of the Timoshenko system, we consider a 2D model which was subsequently developed by Reissner [START_REF] Reissner | The effect of transverse shear deformation on the bending of elastic plates[END_REF] and Mindlin [START_REF] Mindlin | Thickness-shear and flexural vibrations of crystal plates[END_REF]. The Reissner-Mindlin-Timoshenko plate theory can be viewed as a two-dimensional generalization of the thin beam model. Indeed, this model takes into account the shear deformations and it assume that the filaments of the plate must remain perpendicular to its mid-plane (see Figure 2 and Figure 3). The Reissner-Mindlin-Timoshenko model is actually a hyperbolic system of three coupled second-order partial differential equations in space dimension two:

ρ 1 w tt -K(ψ + w x ) x + (ϕ + w y ) y = 0, ( 17
)
ρ 2 ψ tt -Dψ xx -D 1 -µ 2 ψ yy -D 1 + µ 2 ϕ xy + K(ψ + w x ) = 0, ( 18
)
ρ 2 ϕ tt -Dϕ yy -D 1 -µ 2 ϕ xx -D 1 + µ 2 ψ xy + K(ϕ + w y ) = 0, (19)
where Ω ⊂ R 2 is a bounded domain, the functions w, ψ and ϕ depend on (t, x, y) ∈ [0, +∞)× Ω and denote the transverse displacement of the plate and the rotational angles of a filament of the plate. The parameter ρ is the (constant) mass per unit of surface area, h is the (uniform) plate thickness, µ is the Poisson's ratio (0

< µ < 1/2), D = Eh 3 12(1-µ 2 )
is the modulus of flexural rigidity, K = kEh 2(1+µ) is the shear modulus where E is the Young's modulus and k is the shear correction.

Literature overview

Concerning the stabilization results, the most well known ones are due to Lagnese [START_REF] Lagnese | Boundary Stabilization of Thin Plates[END_REF], where, he considered a bounded domain Ω having a Lipschitz boundary Γ such that Γ = Γ 0 ∪ Γ 1 and Γ 0 and Γ 1 are relatively open, disjoint subsets of Γ with Γ 1 = ∅. He considered the equations ( 17)-( 18)- [START_REF] Cowsar | A priori estimates for mixed finite element methods for the wave equation[END_REF] with the following boundary conditions.

w = ψ = ϕ = 0, on Γ 0 , ( 20
)
K ∂w ∂x + ψ, ∂w ∂y + ϕ • v = m 1 , on Γ 1 , ( 21
)
D ∂ψ ∂x + µ ∂ϕ ∂y , 1 -µ 2 ∂ϕ ∂x + ∂ψ ∂y • v = m 2 on Γ 1 , ( 22
)
D 1 -µ 2 ∂ϕ ∂x + ∂ψ ∂y , ∂ϕ ∂y + µ ϕ ∂y + µ ∂ψ ∂x • v = m 3 on Γ 1 , (23)
where, v := (v 1 , v 2 ) is the unit exterior normal to Γ and m 1 , m 2 , m 3 denote the linear boundary dissipations given by

{m 1 , m 2 , m 3 } = -F {w t , ψ t , ϕ t }, with F = [f ii ] a 3 × 3 matrix of real L ∞ (Γ 1 )
functions such that F is symmetric and positive semi definite on Γ 1 . Lagnese [START_REF] Lagnese | Boundary Stabilization of Thin Plates[END_REF] proved that the undamped system is exponential stable, without any restriction on the coefficients of the system. Munoz Rivera and Portillo Oquendo [START_REF] Muñoz Rivera | Asymptotic behavior on a Mindlin-Timoshenko plate with viscoelastic dissipation on the boundary[END_REF] considered the system (17)- [START_REF] Cowsar | A priori estimates for mixed finite element methods for the wave equation[END_REF] with a boundary dissipation of memory type and they proved the exponential stability when the kernels have exponential behavior and polynomial one for the kernels of polynomial type. Second, let us further mention some other interesting results as e.g, in [START_REF] Sare | on the stability of Mindlin-Timoshenko Plates Department of Mathematics and Statistics[END_REF], the Reissner-Mindlin Timoshenko system ( 17)- [START_REF] Cowsar | A priori estimates for mixed finite element methods for the wave equation[END_REF] with frictional dissipations acting on the equations for the rotation angles is not exponentially stable independent of any relations between the constants of the system. This is not the case in [START_REF] Campelo | Stability to the dissipative Reissner-Mindlin-Timoshenko acting on displacement equation[END_REF] where we have the existence of a critical number that exponentially stabilizes the system ( 17)- [START_REF] Cowsar | A priori estimates for mixed finite element methods for the wave equation[END_REF] with the frictional dissipation acting only on the equation of the transverse displacement. For a recent result [START_REF] Campelo | Stability to the dissipative Reissner-Mindlin-Timoshenko acting on displacement equation[END_REF], where the Reissner-Mindlin-Timoshenko system is damped by two feedback laws acting on rotation angles, we have the existence of a critical number that stabilizes the system exponentially. Finally, it is clear, from the cited papers, that the linear dissipations have been widely studied in literature. However, there has been less focus on the interaction of nonlinear damping terms within the 2D system (the Reissner-Mindlin Timoshenko system). For recent results on coupled PDE dynamics with interface on Reissner-Mindlin Timoshenko plate equations, we refer the reader to the works by Grobbelaar-Van Dalsen ( [START_REF] Grobbelaar-Van Dalsen | On a structural acoustic model which incorporates shear and thermal effects in the structural component[END_REF][START_REF] Grobbelaar-Van Dalsen | On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid[END_REF][START_REF] Grobbelaar-Van Dalsen | Strong stabilization of models incorporating the thermoelastic Reissner-Mindlin plate equations with second sound[END_REF][START_REF] Grobbelaar-Van Dalsen | On the dissipative effect of a magnetic field in a Mindlin-Timoshenko plate model[END_REF][START_REF] Grobbelaar-Van Dalsen | The role of magnetic fields in the strong stabilization of a hybrid magneto-elastic structure[END_REF] and the references therein), Giorgi and Naso [START_REF] Giorgi | Mathematical models of Reissner-Mindlin thermoviscoelastic plates[END_REF], and Avalos and Toundykov [START_REF] Avalos | Boundary stabilization of structural acoustic interactions with interface on a Reissner-Mindlin plate[END_REF][START_REF] Avalos | On stability and trace regularity of solutions to Reissner-Mindlin-Timoshenko equations, Modern aspects of the theory of partial differential equations[END_REF]. The results presented above precise some of the necessary conditions that lead to the exponential stabilization of the model (boundary conditions, waves speed and damping term). Here, we prove that upper energy decay rate as introduced in [START_REF] Alabau-Boussouira | Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control[END_REF] for a nonlinearly damped hyperbolic system coupled by velocities is extended to our plate model. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method in [START_REF] Alabau-Boussouira | Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control[END_REF][START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF].

Main results

The second step is the study of the following Mindlin Timoshenko model subject to a nonlinear dissipation acting on the equations for the rotation angles.

ρ 1 w tt -K(ψ + w x ) x -K(ϕ + w y ) y = 0, in Ω × R + , ( 24
)
ρ 2 ψ tt -Dψ xx -D 1 -µ 2 ψ yy -D 1 + µ 2 ϕ xy + K(ψ + w x ) + χ 1 (ψ t ) = 0, in Ω × R + , ( 25
)
ρ 2 ϕ tt -Dϕ yy -D 1 -µ 2 ϕ xx -D 1 + µ 2 ψ xy + K(ϕ + w y ) + χ 2 (ϕ t ) = 0, in Ω × R + . ( 26
)
We consider :

• The initial data given by w(x, y, 0) = w 0 (x, y), w t (x, y, 0) = w 1 (x, y), in Ω, [START_REF] Avalos | On stability and trace regularity of solutions to Reissner-Mindlin-Timoshenko equations, Modern aspects of the theory of partial differential equations[END_REF] ψ(x, y, 0) = ψ 0 (x, y), ψ t (x, y, 0) = ψ 1 (x, y), in Ω, [START_REF] Avalos | Boundary stabilization of structural acoustic interactions with interface on a Reissner-Mindlin plate[END_REF] ϕ(x, y, 0) = ϕ 0 (x, y), ϕ t (x, y, 0) = ϕ 1 (x, y), in Ω,

• The speeds of propagation

v 2 1 := K ρ 1 ; and v 2 2 := D ρ 2
• The rectangular configuration Ω ⊂ R 2 given by

Ω := [0, L 1 ] × [0, L 2 ], with L 1 , L 2 > 0.
The boundary is given by

Γ 1 := {(x, y)| 0 < x < L 1 ; y = 0, y = L 2 }, Γ 2 := {(x, y)| 0 < y < L 2 ; x = 0, x = L 1 }.
Moreover,

• The boundary conditions for the system (( 24)-( 25) and ( 26)),

w = ψ = ϕ = 0, Γ × R + , ( 30
)
satisfying

∂Ω = Γ := Γ 1 ∪ Γ 2 .
we assume that the following assumptions on the damping functions χ i are satisfied.

• (H 0 ) : The functions χ i ∈ C(R) are monotone increasing for i = 1, 2,

       ∃ a strictly increasing odd function g ∈ C 1 (R) such that, c 1 g(|s|) ≤ |χ i (s)| ≤ c 2 g -1 (|s|), for all |s| ≤ 1 for i = 1, 2, c 1 |s| ≤ |χ i (s)| ≤ c 2 |s|, for all |s| ≥ 1 for i = 1, 2.
where g -1 denotes the inverse function of g.

• (H 1 ) : we assume that H is strictly convex in [0, r 2 0 ] for r 0 ∈ (0, 1] defined by

H(x) = √ xg( √ x).
First, we exploit the semi-group theory to show that the problem [START_REF] Sare | Stability of Timoshenko systems with past history[END_REF][START_REF] Fernàndez Sare | On the stability of damped Timoshenko systems-Cattaneo versus Fourier's law[END_REF][START_REF] Sare | on the stability of Mindlin-Timoshenko Plates Department of Mathematics and Statistics[END_REF] subject to corresponding initial conditions as well as Dirichlet boundary conditions is well-posed. To this end, we transform Equations [START_REF] Sare | Stability of Timoshenko systems with past history[END_REF][START_REF] Fernàndez Sare | On the stability of damped Timoshenko systems-Cattaneo versus Fourier's law[END_REF][START_REF] Sare | on the stability of Mindlin-Timoshenko Plates Department of Mathematics and Statistics[END_REF] into the Cauchy problem

   dU dt = AU + BU, for t > 0, U (0) = U 0 , ( 31
)
on the Hilbert space

H := H 1 0 (Ω) × L 2 (Ω) × H 1 0 (Ω) × L 2 (Ω) × H 1 0 (Ω) × L 2 (Ω),
where A is the differential operator

A =              0 I 0 0 0 0 K ρ 1 ∆ 0 K ρ 1 ∂ x 0 K ρ 1 ∂ y 0 0 0 0 I 0 0 -k ρ 1 ∂ x 0 A 1 0 D ρ 2 1+µ 2 ∂ x ∂ y 0 0 0 0 0 0 I -k ρ 2 ∂ y D ρ 2 1+µ 2 ∂ x ∂ y 0 A 2 0 0             
where the differential operators A i (i = 1, 2) are defined by

A 1 = D ρ 2 ∂ 2 x + 1 -µ 2 ∂ 2 y - K ρ 2 I, A 2 = D ρ 2 1 -µ 2 ∂ 2 x + ∂ 2 y - K ρ 2 I.
Here I denotes the identity operator and

D(A) = (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) × (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) × (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω),
and B is the damping nonlinear operator given by

B              u 1 u 2 u 3 u 4 u 5 u 6              =              0 0 0 χ 1 (u 4 ) 0 χ 2 (u 6 )              .

Theorem. (Existence and uniqueness)

Assume that (H 0 ) is satisfied. Then for all initial data U 0 ∈ H, the Cauchy problem has a unique solution U ∈ C([0, ∞); H). Moreover, the operator A + B generates a continuous semi-group (T (t)) t≥0 on H. Moreover, for all initial data U 0 ∈ D(A), the solution

U ∈ L ∞ ([0, ∞); D(A)) ∩ W 1,∞ ([0, ∞); H).
We further show that the problem [START_REF] Sare | Stability of Timoshenko systems with past history[END_REF][START_REF] Fernàndez Sare | On the stability of damped Timoshenko systems-Cattaneo versus Fourier's law[END_REF][START_REF] Sare | on the stability of Mindlin-Timoshenko Plates Department of Mathematics and Statistics[END_REF] is strongly stable ( the energy was shown to vanish as t → ∞ ).

Theorem. (Strong stability)

Assume (H 0 ) is satisfied. We have in addition that Υ(U 0 ) is relatively compact in H. Then for all U 0 ∈ H, the energy

E(t, w, ψ, ϕ) := 1 2 Ω [ρ 1 |w t | 2 + ρ 2 |ψ t | 2 + ρ 2 |ϕ t | 2 + K|ψ + w x | 2 + K|ϕ + w y | 2 + D|ψ x | 2 + D|ϕ y | 2 + 1 -µ 2 D|ψ y + ϕ x | 2 + 2Dµψ x ϕ y ]dxdy. ( 32
)
satisfies lim t→∞ E(t) = 0.
Next, we obtain a upper energy decay rate for a nonlinearly damped hyperbolic systems coupled by velocities. This result is stated as follows

Theorem. (Upper energy estimate)

We assume that v 2 1 = v 2 2 and the assumptions (H 0 ) and (H 1 ) hold. Let E be a given decreasing, absolutely continuous function on [0, +∞) and the parameter β such that

E(0) 2L(H (r 2 0 )) ≤ β. ( 33
)
Then, the total energy of (24)-( 26) defined by [START_REF] Grobbelaar-Van Dalsen | Strong stabilization of models incorporating the thermoelastic Reissner-Mindlin plate equations with second sound[END_REF] decays as

E(t) ≤ βL   1 ψ -1 0 t σ   , ∀ t ≥ σ H (r 2 0 ) , ( 34
)
where, the function L is given by

L(y) =    H (y) y if y > 0, 0 if y = 0.
Furthermore, if lim sup x→0 + Λ H (x) < 1, then E satisfies the following decay rate

E(t) ≤ β (H ) -1 σ t , ∀ t ≥ 0. ( 35
)
Here, β and σ are respectively given by

β = max c 3 , E(0) 2L(H (r 2 0 )) , ( 36
)
σ = 2 α 3 1 c 3 + 1 c 6 H (r 2 0 )(α 2 (c 4 + c 5 ) + 1 , ( 37
)
where k > 0 is a constant which does not depend on E(0). Now, we present numerical results of the discrete Timoshenko system.

Numerical study

The goal of this part is to design a numerical scheme to the free-undamped Timoshenko system based on a combination between the finite element and finite difference methods, we design a discretization scheme for the Timoshenko system under consideration. We first adapt this numerical scheme to the corresponding linear and nonlinear systems. Interestingly, this scheme reaches to reproduce the most important properties of the discrete energy. We show, for the discrete energy, the positivity, the energy conservation property and we present cases where the discrete energy of the Timoshenko system behave differently with the presence the damping terms.

• For instance, we consider the following undamped Timoshenko problem,

   ϕ tt -(ϕ x + ψ) x = 0, (x, t) ∈ (0, L) × R + , ψ tt -(ψ x ) x + (ϕ x + ψ) = 0, (x, t) ∈ (0, L) × R + .
We verify that the discrete energy in this case is conservative as was expected.

• Then, in order to verify the numerical exponential decay of the energy, we consider the following linear system in which the theoretical conditions of the speed equality (2

) (v 1 = v 2 = 1) is clearly satisfied.    ϕ tt -(ϕ x + ψ) x = 0, (x, t) ∈ (0, L) × R + , ψ tt -ψ xx + (ϕ x + ψ) + µψ t = 0, (x, t) ∈ (0, L) × R + ,
where µ represents the damping coefficient.

• Finally, we deal with the following nonlinear damped Timoshenko system and we numerically reproduce the known analytical results established on the decay rate of the energy associated with each type of dissipation.

   ϕ tt -(ϕ x + ψ) x = 0, (x, t) ∈ (0, L) × R + , ψ tt -bψ xx + (ϕ x + ψ) + g(ψ t ) = 0, (x, t) ∈ (0, L) × R + .
For g(s) = |s|s, we obtain the polynomial decay rate of the discrete energy. However, for g(s) = exp(-1 s 2 ) we have the logarithmic decay that is in agreement with the theoretical results already established.

Chapter 1

Lower Bound and optimality for a nonlinearly damped Timoshenko system with thermoelasticity of second sound

The model

In this chapter we are concerned with the following nonlinearly damped Timoshenko system in a one-dimensional bounded domain with thermoelasticity where the heat flux is given by Cattaneo's law:

             ρ 1 ϕ tt -k(ϕ x + ψ) x = 0, in (0, 1) × IR + , ρ 2 ψ tt -bψ xx + k(ϕ x + ψ) + δθ x + a(x)g(ψ t ) = 0, in (0, 1) × IR + , ρ 3 θ t + q x + δψ xt = 0, in (0, 1) × IR + , τ q t + βq + θ x = 0, in (0, 1) × IR + . (1.1)
We associate to (1.1) the following Dirichlet boundary conditions:

ϕ(0, t) = ϕ(1, t) = ψ(0, t) = ψ(1, t) = q(0, t) = q(1, t) = 0, ∀ t ≥ 0. (1.2)
Moreover, the initial conditions for the system (1.1) are given by

       ϕ(x, 0) = ϕ 0 (x), ϕ t (x, 0) = ϕ 1 (x), ∀ x ∈ (0, 1), ψ(x, 0) = ψ 0 (x), ψ t (x, 0) = ψ 1 (x), ∀ x ∈ (0, 1), θ(x, 0) = θ 0 (x), q(x, 0) = q 0 (x), ∀ x ∈ (0, 1), (1.3)
where t ∈ (0, ∞) denotes the time variable and x ∈ (0, 1) is the space variable, the function ϕ is the displacement vector, ψ is the rotation angle of the filament, the function θ is the temperature difference, q = q(x, t) ∈ R is the heat flux, and ρ 1 , ρ 2 , ρ 3 , b, k, δ and β are positive constants.

Assumption and preliminaries

We formulate the following assumptions which will be used for establishing our results:

(H 0 ) we assume that a is a smooth function which satisfies a(x) ≥ 0, x ∈ (0, 1) , a > 0 on any nonempty subset Γ of (0, 1).

(H 1 ) g : R → R is a nondecreasing C 0 -function such that for all ∈ (0, 1), there exist positive constants c 1 , c 2 , and an increasing odd function g 0 ∈ C 1 (0, +∞) with g 0 (0) = 0 such that:

   g 0 (|s|) ≤ |g(s)| ≤ g -1 0 (|s|), for all |s| ≤ , c 1 |s| ≤ |g(s)| ≤ c 2 |s|, for all |s| ≥ .
In addition, we assume that there exists r 0 > 0 such that the function Ψ, given by,

Ψ(x) = √ xg( √ x), (1.4) is a strictly convex C 1 -function from (0, r 2 0 ] to R. Remark 1.1.
1. The function Ψ defined above is the same function H introduced in [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF].

2. In [START_REF] Alabau-Boussouira | Strong lower energy estimates for nonlinearly damped Timosheko beams and Petrowsky equation[END_REF], Alabau-Boussouira assumed that g is an odd, increasing function and has a linear growth at infinity. Here we will make some hypothesis on the function g 0 and not on g.

Remark 1.2. We note that the Hypothesis (H 1 ) implies that sg(s) > 0, for all s = 0.

Existence and uniqueness

Now, we define the function space associated to the problem (1.1)-(1.3),

H = H 1 0 (0, 1) × L 2 (0, 1) × H 1 0 (0, 1) × (L 2 (0, 1)) 3 .
We rewrite (1.1) as a first-order system. For that purpose, let U = (ϕ, ϕ t , ψ, ψ t , θ, q) T and (1.1) becomes

         d dt U (t) + (A + B)U (t) = 0, t > 0, Theorem 1.1.
Assume that (H 0 ) and (H 1 ) are satisfied. Then for all initial data U 0 ∈ H, the system (1.1) has a unique solution U ∈ C([0, ∞); H), the operator A + B generates a continuous semi-group (T (t)) t≥0 on H. Moreover, for all initial data

U 0 ∈ D(A), the solution U ∈ L ∞ ([0, ∞); D(A)) ∩ W 1,∞ ([0, ∞); H).

The energy

The energy associated with the system (1.1) is defined by

E(ϕ, ψ, θ, q)(t) := 1 2 1 0 (ρ 1 ϕ 2 t + ρ 2 ψ 2 t + bψ 2 x + k(ϕ x + ψ) 2 + ρ 3 θ 2 + τ q 2 )dx. (1.7)
Differentiating (1.7), it is easy to see that

E (t) = -β 1 0 q 2 dx - 1 0 a(x)ψ t g(ψ t )dx ≤ 0; (1.8)
this relationship has been obtained by multiplying, formally, the equations of (1.1), respectively, by ϕ t , ψ t , θ and q, and using the integration by parts with respect to x over (0, 1), the boundary and initial conditions, and the hypotheses (H 0 ) and (H 1 ).

Remark 1.3. As we already mentioned in the introduction, the exponential decay result (1.1) depends on the stability number χ introduced in [START_REF] Santos | The stability number of the Timoshenko system with second sound[END_REF]. So, it is natural to wonder about the effects of the nonlinear dissipation mechanism a(x)g(ψ t ) on the stability result of the system (1.1). We recall that, in [START_REF] Ayadi | General decay in a Timoshenko-type system with thermoelasticity with second sound[END_REF], the authors have considered the same stability number χ and obtained a general decay of the system (1.1) with a dissipation term of the form α(t)h(ψ t ) but no optimality result has been proved. As a consequence, the following questions naturally arise:

• Is the system (1.1) strongly stable?

• What is the limit of the energy E(t) as t → ∞? If we obtain a different equilibrium state (E(t) → constant = 0 as t → ∞), how can we characterize the decay rate of the energy?

• Can we obtain lower estimates for the non constant equilibrium state? These questions will be investigated in the next sections.

Strong stability for Timoshenko system

In this section, we focus on the stability result for the energy associated with the Timoshenko system (1.1). For this purpose, we need to handle some auxiliary steps. We first consider the following conservative Timoshenko system:

   ρ 1 ϕ tt -k(ϕ x + ψ) x = 0, in (0, 1) × IR + , ρ 2 ψ tt -bψ xx + k(ϕ x + ψ) = 0, in (0, 1) × IR + .
(1.9)

Then, we assume the assumption below on the subset Γ ⊂ (0, 1), (HC)

  
Let (ϕ, ψ) be a weak solution of (1.9). If ψ t ≡ 0 on Γ then (ϕ, ψ) ≡ (0, 0).

The assumption (HC) is extracted from [START_REF] Alabau-Boussouira | Strong lower energy estimates for nonlinearly damped Timosheko beams and Petrowsky equation[END_REF] and we note that we proceed as in [START_REF] Alabau-Boussouira | Strong lower energy estimates for nonlinearly damped Timosheko beams and Petrowsky equation[END_REF] to extend the techniques there to our problem. Now, we denote by ω(U 0 ) the ω-limit set of U 0 and we consider Z 0 ∈ ω(U 0 ) such that Z(t) = T (t)Z 0 . Then we formulate the stability result for the energy of (1.1) in the following theorem.

Theorem 1.2. Assume that the hypotheses (H 0 ) and (H 1 ) hold. We suppose in addition that Γ satisfies (HC). Then, for all U 0 = (ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , θ 0 , q 0 ) ∈ H, the energy E, defined by (1.7) and corresponding to the solution of (1.1), satisfies

lim t→∞ E(t, U ) = E ∞ , (1.10
)

where E ∞ is the energy of Z ∈ ω(U 0 ) (see (1.20)-(1.

21) below for the definitions of E ∞ and Z).

Moreover, under the same assumptions we prove that the energy

E(t) defined in (1.23) below, satisfies E(t) → 0, as t → ∞. (1.11)
Before establishing the proof of Theorem 1.2, we will state and prove two lemmas which will be useful later in our analysis. First, the lemma below proves the decreasing character of the second-order energy.

Lemma 1.1. Let E (t) be the energy defined as follows:

E (t) := 1 2 1 0 (ρ 1 ϕ 2 tt + ρ 2 ψ 2 tt + bψ 2 tx + k(ϕ tx + ψ t ) 2 + ρ 3 θ 2 t + τ q 2 t )dx. (1.12)
Then, E is a non-increasing function. We shall call E (t) the second-order energy associated with (1.1).

Proof. We set p = ϕ t , z = ψ t , u = θ t , d = q t , then we have

             ρ 1 p t -k(ϕ x + ψ) x = 0, in (0, 1) × IR + , ρ 2 z t -bψ xx + k(ϕ x + ψ) + δθ x + a(x)g(z) = 0, in (0, 1) × IR + , ρ 3 u + q x + δz x = 0, in (0, 1) × IR + , τ d + βq + θ x = 0, in (0, 1) × IR + .
(1.13)

Differentiating the above equations with respect to time, we obtain

                   ρ 1 p tt -k(p x + z) x = 0, in (0, 1) × IR + , ρ 2 z tt -bz xx + k(p x + z) + δu x + a(x)g (z)z t = 0, in (0, 1) × IR + , ρ 3 u t + d x + δz x = 0, in (0, 1) × IR + , τ d t + βd + u x = 0, in (0, 1) × IR + . p(0, t) = p(1, t) = z(0, t) = z(1, t) = d(0, t) = d(1, t) = 0, ∀ t ≥ 0. (1.14)
We remark that if we formally multiply the equations in (1.14), respectively, by p t , z t , u and d, integrate over (0, 1) and use the integration by parts with respect to x, the boundary conditions, and the hypotheses (H 0 ) and (H 1 ), we obtain the following inequality:

E (t) = -β 1 0 d 2 dx - 1 0 a(x)g (z)z 2 t dx ≤ 0. (1.15)
Thus we deduce that E is non-increasing, hence, we have

E (t) ≤ E (0), ∀ t ≥ 0. (1.16)
Now, we derive the compactness of the orbit of an initial data element in D(A) in the following lemma. Lemma 1.2. For the initial data U 0 = (ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , θ 0 , q 0 ) ∈ D(A), the orbit of U 0 , given by γ(U 0 ) = ∪ t≥0 T (t)U 0 , is relatively compact in H.

Proof. Thanks to the first equation of (1.1), we have

ϕ xx = - ρ 1 k (ϕ tt + ψ x ),
and we get

1 0 ϕ 2 xx dx ≤ 2 ρ 1 k 2 1 0 ϕ 2 tt dx + 1 0 ψ 2 x dx .
(1.17)

Using Lemma 1.1, which shows that E is bounded on R + , we deduce that the set {ϕ tt (t, •); t ≥ 0} is bounded in L 2 (0, 1). In addition, using the fact that E is bounded uniformly on R + , we deduce that the set {ψ x (t, •); t ≥ 0} is also bounded in L 2 (0, 1). Applying the Poincaré inequality and the Rellich-Kondrochov theorem and recalling (1.17), we obtain that the set {ϕ(t, •); t ≥ 0} is relatively compact in H 1 0 (0, 1).

Thanks to (1.15) the energy E is bounded in R + , then, the set {ϕ tx (t, •); t ≥ 0} is bounded in L 2 (0, 1). Furthermore, applying the Poincaré inequality for ϕ t ∈ H 1 0 (0, 1), we obtain

ϕ t H 1 0 (0,1) ≤ (1 + c p ) ϕ tx L 2 (0,1) .
Hence, we obtain that the set {ϕ t (t, •); t ≥ 0} is bounded in H 1 0 (0, 1) which implies, using the Rellich theorem, that the set

{ϕ t (t, •); t ≥ 0} is relatively compact in L 2 (0, 1).
From (1.1) 4 , we have θ x = -τ q t -βq.

Thanks to (1.7), (1.8), (1.12) and (1.15), {θ x (t, •), t ≥ 0} is bounded in L 2 (0, 1). Thus {θ(t, •), t ≥ 0} is bounded in H 1 0 (0, 1). Therefore we conclude that

{θ(t, •), t ≥ 0} is relatively compact in L 2 (0, 1).
We also have from (1.1) 2 ,

bψ xx = ρ 2 ψ tt + k(ϕ x + ψ) + δθ x + a(x)g(ψ t ).
In fact, by (1.17) above and recalling the definitions of E(t) and E (t), we have {ψ(t, •), t ≥ 0} is bounded in H 2 (0, 1) and again applying the Rellich-Kondrochov theorem, we deduce that {ψ(t, •), t ≥ 0} is relatively compact in H 1 0 (0, 1).

Since we have the set {ψ t (t, •), t ≥ 0} is bounded in H 1 0 (0, 1), we easily deduce from the Rellich theorem that

{ψ t (t, •), t ≥ 0} is relatively compact in L 2 (0, 1).
Using the fact that E is bounded, and

q x = -δψ tx -ρ 3 θ t ,
we infer that the set {q x (t, •), t ≥ 0} is bounded in L 2 (0, 1). Applying the Poincaré inequality, we deduce that the set {q(t, •), t ≥ 0} is bounded in H 1 0 (0, 1), and this implies that

{q(t, •), t ≥ 0} is relatively compact in L 2 (0, 1).
Now, we recall the definition of the ω-limit that we borrow e.g. from [START_REF] Alabau-Boussouira | Control of Partial Differential Equations[END_REF].

Definition 1.1. Let X be a Banach space and (T (t)) t≥0 a continuous semi-group on X. Then, the ω-limit set of z 0 , in X, is defined by

ω(z 0 ) = {z ∈ X, ∃(t n ) n ⊂ [0, ∞) such that t n → ∞, as n → ∞, and z = lim n→∞ T (t n )z 0 }.
Now, we are ready to give the proof of Theorem 1.2.

Proof of Theorem 1.2.

We aim to apply the Dafermos strong stabilization technique based on Lasalle invariance principle (see [START_REF] Lassalle | The extent of asymptotic stability[END_REF][START_REF] Lasalle | Asymptotic stability criteria[END_REF]).

For that purpose, let U 0 = (ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , θ 0 , q 0 ) ∈ D(A), and U = (ϕ, p, ψ, z, θ, q) = T (t)U 0 . Then, we define the Liapunov function L for (T (t)) t≥0 on H by,

L(U ) = 1 2 1 0 (ρ 1 p 2 + ρ 2 z 2 + bψ 2 x + k(ϕ x + ψ) 2 + ρ 3 θ 2 + τ q 2 )dx, ∀ U ∈ H.
Now, let ω(U 0 ) be the ω-limit of U 0 (see Definition 3.1). Thanks to the Lasalle invariance principle, we have, for each W 0 in ω(U 0 ), the function t → L(T (t)W 0 ) is constant. In particular, let Z 0 ∈ ω(U 0 ) be given and set Z(t) = (w, r, z, p, u, η)(t) = T (t)Z 0 . Since L(Z(•)) is constant, we deduce that (w, z, u, η) is a solution of a conservative system. Then, the dissipation inequality will be equal to zero which yields

-β 1 0 η 2 dx - 1 0 a(x)pg(p)dx = 0 ⇒ η ≡ 0 and a(x)g(p)p = 0, ∀ x ∈ (0, 1), ∀ t ∈ R + .
Hence, the conservative system can be written as follows:

                   ρ 1 w tt -k(w x + z) x = 0, in (0, 1) × IR + , ρ 2 z tt -bz xx + k(w x + z) = 0, in (0, 1) × IR + , ρ 3 u t + δp x = 0, in (0, 1) × IR + , u x = 0, in (0, 1) × IR + , z t = 0, on {x ∈ Ω, a(x) = 0} ⊃ Ω. (1.18)
We use the hypothesis (HC) to infer that p = 0 in (0, 1) × R + and hence we obtain, from (1.18) 3 , that u t = 0 in (0, 1) × R + . So (1.18) reduces to

                   ρ 1 w tt -k(w x + z) x = 0, in (0, 1) × IR + , ρ 2 z tt -bz xx + k(w x + z) = 0, in (0, 1) × IR + , u = c = θ 0 (0), in (0, 1) × IR + , p = 0, in (0, 1) × IR + , z t = 0 {x ∈ Ω, a(x) = 0} ⊃ Ω. (1.19)
Using again the assumption (HC), we have (w, z) = (0, 0). This allows us to identify Z(t) the element of ω(U 0 ) in the form

Z(t) = (0, 0, θ 0 (0), 0). (1.20)
Hence, we conclude that,

lim t→∞ E(t, U 0 ) = E(Z) := E ∞ , ∀ U 0 ∈ D(A). (1.21) Indeed, since D(A) is dense in H, we obtain lim t→∞ E(t, U ) = E ∞ , ∀ U ∈ C([0, +∞), H). (1.22)
Moreover, seeing that E is the energy of the difference between the solution U ∈ H and Z = (0, 0, θ 0 (0), 0) ∈ ω(U 0 ), we obtain

E(t) = E(t, (ϕ, ψ, θ-θ 0 (0), q)) = 1 2 1 0 ρ 1 ϕ t 2 +ρ 2 ψ t 2 +bψ x 2 +k(ϕ x +ψ) 2 +ρ 3 (θ-θ 0 (0)) 2 +τ q 2 dx.
(1.23) Thanks to the dissipation inequality (1.4) and (1.8), we obtain the time derivative of the energy as follows:

E (t, U ) = -β 1 0 q 2 (t, x)dx - 1 0 a(x)ψ 2 t (t, x) Ψ(ψ 2 t (t, x))dx.
We assume that E (t, U ) = 0, ∀ t ≥ 0 and that the hypothesis (HC) holds, we end up with the following system:

                               ρ 1 w tt -k(w x + z) x = 0, in (0, 1) × IR + , ρ 2 z tt -bz xx + k(w x + z) = 0, in (0, 1) × IR + , u -θ 0 (0) = 0, in (0, 1) × IR + p = 0, in (0, 1) × IR + , u t = 0, in (0, 1) × IR + , u x = 0, in (0, 1) × IR + , z t = 0 {x ∈ (0, 1), a(x) = 0} ⊃ Ω. (1.24)
Then for this case the set ω(U 0 -Z) = {(0, 0, 0, 0)}. As before, applying the Dafermos strong stabilization technique, we obtain that lim t→∞ E(t) = 0.

(1.25)

This concludes the proof of Theorem 1.2.

A straightforward consequence of the stabilization result given by Theorem 1.2 is stated in the following lemma.

Lemma 1.3. For any r 0 > 0, there exists T 0 > 0 such that

E(t) ≤ r 2 0 γ 2 , ∀ t ≥ T 0 . (1.26)
Proof. Using the strong stability result given by Theorem 1.2, the energy E(t) converges to 0 as t goes to ∞. Then, the energy E is uniformly bounded on R + . In particular, we choose the initial data (ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , θ 0 , q 0 ) such that E(0) ≤

r 2 0 γ 2
, where γ is given by

γ = 4 ρ 2 E (0). (1.27)
Hence, we deduce (1.26).

Remark 1.4. The obtaining of the expression of γ is explained in the analysis in Section 1.4.2 below.

Lower energy estimates

The aim of this section is to establish the lower bound estimate of the energy of the onedimensional nonlinearly damped Timoshenko system of thermoelasticity and also to prove that the method based on the comparison principle, expressed through the energy of the solutions, can be extended to our case. First, we define (as in [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF]) the function Λ as follows:

Λ(x) = Ψ(x) xΨ (x) , (1.28)
where Ψ is given by (1.4). We introduce the following assumption (which is the hypothesis (H2) in [?]):

(H 2 )                          ∃ r 0 > 0 such that the function Ψ : [0, r 2 0 ] → R defined by (1.4) is strictly convex on [0, r 2 0 ], and either 0 < lim inf x→0 Λ(x) ≤ lim sup x→0 Λ(x) < 1, or there exists µ > 0 such that 0 < lim inf x→0 Ψ(µx) µx z 1 x 1 Ψ(y)
dy , and lim sup x→0 Λ(x) < 1, for some z 1 ∈ (0, z 0 ] and for all z 0 > 0.

Then, we state in the sequel our main result.

Theorem 1.3. Assume that (H 0 ), (H 1 ) and (H 2 ) hold. Then, for all non vanishing smooth initial data, there exist T 0 > 0 and T 1 > 0 such that the energy E of (1.1) satisfies the following lower estimate The proof of Theorem 1.3 relies on the next proposition together with Lemma 1.4 which is proved in [59, Lemma 2.4] and based on the approach of [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF]. We reproduce here the details for the sake of completeness. Proposition 1.1. We assume that the hypotheses of Theorem 1.1 hold. Let U 0 = (ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , θ 0 , q 0 ) ∈ D(A). We denote by U the solution of (1.1) and E its energy and by E the energy given by (1.23) which satisfies lim t→∞ E(t) = 0. We define Ψ by

1 γ 2 C 2 σ Ψ -1 1 t -T 0 2 ≤ E(t), ∀ t ≥ T 1 + T 0 . ( 1
Ψ(x) = Ψ(x) x , ∀x > 0, Ψ(0) = 0, (1.30)
where Ψ is given by (1.4). We assume that Ψ is nondecreasing on [0, r 2 0 ] for r 0 > 0 sufficiently small. Then there exists T 0 ≥ 0, depending on E (0), such that, the energy E satisfies the following lower estimate:

1 γ 2 K -1 (σ(t -T 0 ) 2 ≤ E(t), (1.31)
where K is given by

K(χ) = γ √ E(T 0 ) χ 1 Ψ(y) dy, ∀ χ ∈ (0, γ E(T 0 )), (1.32)
with γ as defined in (1.27), and σ is a positive constant given by σ = αa ρ 2 + βr 0 τ C 1 , where α a is defined by (1.36)

below. Moreover, if lim χ→0 + K(χ) = ∞, then lim t→∞ K -1 (σ(t -T 0 )) = 0.
Proof. We assume that the initial data U 0 ∈ D(A). Then, thanks to the smoothness of the solution of (1.1), we have

2 x 0 ψ t (t, z)ψ tx (t, z)dz = ψ 2 t (t, x) -ψ 2 t (t, 0).
Using the Dirichlet boundary conditions (1.2) at x = 0, we have

ψ 2 t (t, x) = 2 x 0 ψ t (t, z)ψ tx (t, z)dz.
Applying the Cauchy-Schwartz inequality, we have

ψ 2 t (t, x) ≤ 2 1 0 ψ 2 t dx 1 0 ψ 2 tx dx ≤ 4 ρ 2 E(t) E (0), ∀ t ≥ 0, ∀ x ∈ (0, 1).
Using (1.15) and the fact that E (t) ≤ E (0), we deduce that

ψ 2 t (t, x) ≤ γ E(t), ∀ t ≥ 0, ∀ x ∈ (0, 1), (1.33)
where γ is given by

γ = 4 ρ 2 E (0).
Thanks to Theorem 1.1, we have ψ t ∈ W 1,∞ ([0, ∞); L 2 (0, 1)). From (1.33), we get

ψ 2 t (t, •) L ∞ (0,1) ≤ γ E(t), ∀ t ≥ 0. (1.34)
Thanks to the dissipation inequality (1.8) and using (1.4), we have

E (t) = -β 1 0 q 2 (t, x)dx - 1 0 a(x)ψ 2 t (t, x) Ψ(ψ 2 t (t, x))dx.
On the other hand, from the expression of the energy E we have the following relation between E and E :

E (t) = E (t, U ) -ρ 3 θ 0 (0) d dt 1 0 θ(t, x)dx .
Using (1.1) and the boundary conditions (1.2), we have

d dt 1 0
θ(x, t)dx = 0, and hence

E (t) = E (t, U ).
Now, using the Dafermos strong stabilization result, that is lim t→∞ E(t) = 0, we deduce that there exists T 0 ≥ 0 such that ψ 2 t has values in which the function Ψ is increasing. Hence, we have

Ψ(|ψ 2 t (t, •)|) ≤ Ψ(γ E(t)), ∀ t ≥ T 0 , ∀ x ∈ (0, 1).
Using the last inequality we obtain

1 0 a(x)ψ 2 t (t, x) Ψ(ψ 2 t (t, x))dx ≤ 2α a ρ 2 γ E(t)Ψ(γ E(t)), ∀ t ≥ T 0 , ( 1.35)
where α a = a L ∞ (0,1) .

(1.36)

Moreover, using Lemma 1.3, we obtain

E(t) 1 4 ≤ r 0 γ 1/2 , ∀ t ≥ T 0 .
Thanks to the assumption (H 1 ), we distinguish the following two cases.

Case 1. Let g 0 be a linear function on [0, ], the hypothesis (H 1 ) becomes

c 1 |s| ≤ |g(s)| ≤ c 2 |s|, for all s ∈ R.
In particular, for s = γ 2 (E(t))

1 4 , we have

γ 1 2 (E(t)) 1 4 ≤ 1 c 1 g(γ 1 2 (E(t)) 1 4 ) ≤ 1 c 1 Ψ(γ E(t)), ∀ t ≥ T 0 . (1.37)
Case 2. Let g 0 be a nonlinear function on [0, ]. We assume that max(r 0 , g 0 (r 0 )) < . Let 1 = min(r 0 , g 0 (r 0 )), we deduce from the hypothesis (H 1 ) that

g 0 ( 1 ) |s| ≤ g 0 (|s|) |s| |s| ≤ |g(s)| ≤ g -1 0 (|s|) |s| |s| ≤ g -1 0 ( ) 1 |s|,
for all s satisfying 1 ≤ |s| ≤ . Using the fact that |γ 2 (E(t))

1 4 | ≤ r 0 , we infer that

g 0 ( 1 ) γ 1 2 (E(t)) 1 4 ≤ g(γ 1 2 (E(t)) 1 4 ) ≤ Ψ(γ E(t)), ∀ t ≥ T 0 . (1.38)
Now, thanks to (1.37) and (1.38), we deduce that for the two above cases we obtain the following estimates:

β 1 0 q 2 (t, x)dx ≤ 2β τ E(t) ≤ 2βr 0 E(t) τ C 1 γ Ψ(γ E(t)), ∀ t ≥ T 0 , (1.39)
where C 1 is a positive constant. Hence, by (1.35) and (1.38), there exists T 0 ≥ 0 such that the following inequality holds

-E (t) ≤ 2 γ α a ρ 2 + βr 0 τ C 1 E(t)Ψ(γ E(t)), ∀ t ≥ T 0 . (1.40)
Thus, we deduce that

K(γ E(t)) ≤ α a ρ 2 + βr 0 τ C 1 (t -T 0 ), ∀ t ≥ T 0 ,
where K is given by (1.32). The monotony of the function K completes the proof of (1.31).

Now, we will use the following key lemma (see Lemma 2.4 in [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF] for the proof). Here we will define an auxiliary function Λ which classifies the feedback growths around zero. ] to R such that H(0) = H (0) = 0, where r 0 > 0 and sufficiently small and define Λ on (0, r 0 ] by

Λ(x) = H(x) xH (x) . ( 1

.41)

Let z be the solution of the following ordinary differential equation:

z (t) + κH(z(t)) = 0, z(0) = z 0 , ∀ t ≥ 0, (1.42)
where z 0 > 0 and κ > 0 are given. Then z(t) is defined for all t ≥ 0 and it decays to infinity. Moreover assume that either

0 < lim inf x→0 Λ(x) ≤ lim sup x→0 Λ(x) < 1, (1.43)
or that there exists µ > 0 such that for some z 1 ∈ (0, z 0 ). Then there exists T 1 > 0 such that for all R > 0, there exists a constant

0 < lim inf x→0 H(µx) µx
C > 0 such that z(t) ≥ C(H ) -1 R t , ∀ t ≥ T 1 , (1.45)
where T 1 is a positive constant.

We come back to the proof of Theorem 1.3.

Proof of Theorem 1.3.

Let z(t) be the solution of the ordinary differential equation (1.42), where we consider z 0 = γ E(T 0 ), H = Ψ and κ = σ. Hence, we have

K(z(t)) = α a ρ 2 + βr 0 τ C 1 t, ∀ t ≥ 0.
We set z(t) = z(t -T 0 ), then we have

z(t) = K -1 α a ρ 2 + βr 0 τ C 1 (t -T 0 ) , ∀ t ≥ T 0 . (1.46)
Thanks to (1.31), we have

z(t) 2 γ 2 ≤ E(t), ∀ t ≥ T 0 . (1.47)
Recalling (H 2 ) and that Ψ satisfies the hypotheses in Lemma 1.4 and applying (1.45) to Ψ = H for R = 1, we deduce that there exists C γ > 0 depending on γ, and eventually on µ if the second alternative of (H 2 ) holds, such that

(Ψ ) -1 1 t -T 0 ≤ C γ z(t), ∀ t ≥ T 0 + T 1 . (1.48)
Combining (1.46) (1.47) and (1.48), we obtain (1.29).

Examples

In what follows, we give two examples to illustrate the estimates (1.29).

Example 1. Let g(s) = s p , ∀s ∈ (0, r 2 0 ] for p > 1. We have Ψ(s) = s p+1 2 , Ψ is strictly convex, for s ∈ [0, r 2 0 ], and Ψ (s) = p+1 2 s p-1 2 , then Ψ(s) = Ψ(s) s = s p-1 , for p > 1, ∀s ∈]0, r 2 0 ].
Thus, Ψ is nondecreasing on ]0, r 2 0 ]. Since Λ(x) = 2 p+1 < 1, this proves that g satisfies the first assumption of (H 2 ).

Ψ(s) = Ψ(s) s = 1 s exp -1 4 (ln(s)) 2 , ∀ s ∈ (0, r 2 0 ].
Thus, Ψ is nondecreasing on ]0, r 2 0 ]. In addition, Λ(s) = -2 ln(s) , thus, lim s→0 Λ(s) = 0 < 1, and we get also for any µ > 1, lim inf s→0 Ψ(µs) µs

z 1 s 1 Ψ(y) dy > 0.
It is easy to see that Ψ (t) is equivalent to D(t), as t goes to ∞, where

D(t) = exp(-1 4 (ln(t)) 2 ). So, we have D -1 (t) = exp(-2(ln(t) 1 2
)); here we apply the result of the Theorem 1.3 and we obtain the following inequality:

E(t) ≥ c exp(-4(ln(t) 1 2 )).
By these examples we obtain explicit lower bounds which characterize the decay rate of the energy E(t), associated with the solution of (1.1), to the corresponding non-zero equilibrium state energy E ∞ .

Optimality

In this section, we will emphasize on a specific example for the damping term g in (1.1). More precisely, we consider

g(s) = 1 s exp( -1 s 2 ), s ∈]0, r 2 0 ], (1.50)
for some r 0 > 0 such that the function Ψ, defined in (1.4), is C 1 , strictly convex on [0, r 2 0 ]. We note that all the hypotheses of Theorems 1.2, 1.3 and 1.1 are satisfied for g as given in (1.50). In general, it is difficult to obtain a precise decay rate for the energy, as t → ∞. For the Timoshenko system studied in [START_REF] Alabau-Boussouira | Strong lower energy estimates for nonlinearly damped Timosheko beams and Petrowsky equation[END_REF], although simpler than our system (1.1), the author did not give any optimality result but of course several interesting results on the lower bound estimate for the Timoshenko system, involving only the displacement ϕ and the rotation ψ were obtained.

Here, example (1.50) gives the optimality as we will see in the following proposition. Proposition 1.2. Assume that g = g 0 , where g 0 is given by

g 0 (s) = 1 s exp( -1 s 2 ), ∀ s ∈]0, r 2 0 ],
with r 0 > 0, sufficiently small, so that g is then extended on R as an odd, convex and C 1 function which satisfies the hypotheses of Theorem 1.3. Then the energy associated with system (1.1) with the damping term given by (1.50) satisfies

c (ln(t)) -1 ≤ E(t) ≤ c (ln(t)) -1 .
Hence, the estimate (1.29) is optimal for the damping function g as given in (1.50).

Proof. Let g(s) = 1 s exp( -1 s 2 ) , for all s ∈ (0, r 2 0 ]. We have Ψ(s) = exp( -1 s ), Ψ (s) = 1 s 2 exp( -1 s ), ∀s ∈ (0, r 2 0 ], Ψ (s) = 0, if s = 0, and
Ψ(s) = Ψ(s) s = 1 s 2 exp( -1 s 2 ), ∀ s ∈ [0, r 2 0 ].
Thus, Ψ is nondecreasing on [0, r 2 0 ]. On the other hand, Λ(s) = s, ∀s > 0 and lim s→0 + Λ(s) = 0. Therefore, for any µ > 1, s = 0, we have Ψ(µs) µs

z 1 s 1 Ψ(y) dy ≥ exp -1 µs µs z 1 s e 1 y √ y dy ≥ 1 µ exp -1 µs (s(e 1 s -e 1 z 1 )).
Then, g satisfies the second alternative of (H 2 ). Consequently, we have

Ψ (s) = 1 s 2 exp( -1 s ) ∼ 0 + D(s),
where D(s) = exp( -1 s 2 ), for s ∈ (0, r 2 0 ] and D -1 (s) = (-ln(s)) -1 2 .

Then, the energy (1.7) satisfies the following lower estimate

E(t) ≥ c (ln(t)) -1 , (1.51)
where c is an explicit positive constant. We note that in [START_REF] Ayadi | Numerical Solutions For A Timoshenko-Type System With Thermoelasticity With Second Sound in preparation[END_REF] the energy satisfies, for this example, E(t) ≤ c (ln(t)) -1 .

Chapter 2

Stability results for the Mindlin Timoshenko system

The model

We present the dissipative Reissner-Mindlin-Timoshenko system given by : ρ 1 w tt -K(ψ +w x ) x -K(ϕ+w y ) y = 0, in Ω×R + , (2.1)

ρ 2 ψ tt -Dψ xx -D 1 -µ 2 ψ yy -D 1 + µ 2 ϕ xy +K(ψ +w x )+χ 1 (ψ t ) = 0, in Ω×R + , (2.2) ρ 2 ϕ tt -Dϕ yy -D 1 -µ 2 ϕ xx -D 1 + µ 2 ψ xy +K(ϕ +w y )+χ 2 (ϕ t ) = 0, in Ω×R + , (2.3)
where Ω ⊂ R 2 is bounded, ρ is the (constant) mass per unit of surface area, h is the (uniform) plate thickness, µ is Poisson's ratio (0 < µ < 1/2), D = Eh 3 12(1-µ 2 ) is the modulus of flexural rigidity, K = kEh 2(1+µ) is the shear modulus where E is the Young's modulus and k is the shear correction. The speeds of propagation are given by

v 2 1 := K ρ 1 , (2.4) and v 2 2 := D ρ 2 .
(2.5)

The function w, ψ and ϕ depend on (t, x, y) ∈ [0, +∞) × Ω, where w models the transverse displacement of the plate and ψ, ϕ are the rotational angles of a filament of the plate.

We make the following assumptions on the damping functions χ i

(H 0 )              χ ∈ C(R) is monotone increasing. ∃ a strictly increasing odd function g ∈ C 1 (R) such that, c 1 g(|s|) ≤ |χ i (s)| ≤ c 2 g -1 (|s|), for all |s| ≤ 1 for i = 1, 2, c 1 |s| ≤ |χ i (s)| ≤ c 2 |s|, for all |s| ≥ 1 for i = 1, 2.
where g -1 denotes the inverse function of g.

(H 1 )   
we assume that H is strictly convex in [0, r 2 0 ] for r 0 ∈ (0, 1] defined by

H(x) = √ xg( √ x).
We consider the initial data given by w(x, y, 0) = w 0 (x, y), w t (x, y, 0) = w 1 (x, y), in Ω, (2.6)

ψ(x, y, 0) = ψ 0 (x, y), ψ t (x, y, 0) = ψ 1 (x, y), in Ω, (2.7) ϕ(x, y, 0) = ϕ 0 (x, y), ϕ t (x, y, 0) = ϕ 1 (x, y), in Ω, (2.8)
we are considering Ω ⊂ R 2 as the rectangular configuration given by

Ω := [0, L 1 ] × [0, L 2 ], with L 1 , L 2 > 0.
The boundary is given by

Γ 1 := {(x, y)| 0 < x < L 1 ; y = 0, y = L 2 }, Γ 2 := {(x, y)| 0 < y < L 2 ; x = 0, x = L 1 }.
Moreover, the boundary conditions for the system ((2.1)-(2.2) and (2.3)),

w = ψ = ϕ = 0, Γ × R + , ( 2.9)
satisfying

∂Ω = Γ := Γ 1 ∪ Γ 2 .

Existence and uniqueness of the solution

In order to recall the existence and the uniqueness results, we start by rewriting the system ((2.1)-(2.2)-(2.3)) as an evolution equation for

U = (w, w t , ψ, ψ t , ϕ, ϕ t ) ≡ (u 1 , u 2 , u 3 , u 4 , u 5 , u 6 ) .
Then U formally satisfies

   dU dt = AU + BU, for t > 0, U (0) = U 0 , ( 2.10)
where U 0 = (w 0 , w 1 , ψ 0 , ψ 1 , ϕ 0 , ϕ 1 ) and A is the differential operator

A =              0 I 0 0 0 0 K ρ 1 ∆ 0 K ρ 1 ∂ x 0 K ρ 1 ∂ y 0 0 0 0 I 0 0 -k ρ 1 ∂ x 0 A 1 0 D ρ 2 1+µ 2 ∂ x ∂ y 0 0 0 0 0 0 I -k ρ 2 ∂ y D ρ 2 1+µ 2 ∂ x ∂ y 0 A 2 0 0             
where the differential operators A i (i = 1, 2) are defined by

A 1 = D ρ 2 ∂ 2 x + 1 -µ 2 ∂ 2 y - K ρ 2 I, A 2 = D ρ 2 1 -µ 2 ∂ 2 x + ∂ 2 y - K ρ 2 I.
Here I denotes the identity operator and

D(A) = (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) × (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) × (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω),
and B is the damping nonlinear operator given by

B              u 1 u 2 u 3 u 4 u 5 u 6              =              0 0 0 χ 1 (u 4 ) 0 χ 2 (u 6 )             
. Now, we will show that the system is well posed using the semi-group techniques. Let us denote by H the following Hilbert space

H := H 1 0 (Ω) × L 2 (Ω) × H 1 0 (Ω) × L 2 (Ω) × H 1 0 (Ω) × L 2 (Ω),
with inner product given by

(U, V ) H = ρ 1 Ω u 2 v 2 dxdy + ρ 2 Ω u 4 v 4 dxdy + ρ 2 Ω u 6 v 6 dxdy + K Ω (u 3 + u 1x )(v 3 + v 1x ) dxdy + K Ω (u 5 + u 1y )(v 5 + v 1y ) dxdy + D Ω u 3x v 3x dxdy + D Ω u 5y v 5y dxdy + D 1 -µ 2 Ω (u 3y + u 5x )(v 3y + v 5x )dxdy + Dµ Ω u 3x v 5y dxdy + Dµ Ω u 5y v 3x dxdy.
with norm given by

U 2 H = ρ 1 Ω |u 2 | 2 dxdy + ρ 2 Ω |u 4 | 2 dxdy + ρ 2 Ω |u 6 | 2 dxdy + K Ω |u 3 + u 1x | 2 dxdy + K Ω |u 5 + u 1y | 2 dxdy + D Ω |u 3x | 2 dxdy + D Ω |u 5y | 2 dxdy + D 1 -µ 2 Ω |u 3y + u 5x | 2 dxdy + Dµ Ω u 3x u 5y dxdy + Dµ Ω u 5y u 3x dxdy, where U = (u 1 , u 2 , u 3 , u 4 , u 5 , u 6 ) ; V = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) .
Then as a consequence of Korn and Poincaré inequalities we obtain that . H is equivalent to the usual norm in H (see [START_REF] Haraux | Nonlinear evolution equations-global behavior of solutions[END_REF] ). In the following theorem, we formulate our result on existence and uniqueness of solution. Moreover, for all initial data U 0 ∈ D(A), then the solution

U ∈ L ∞ ([0, ∞); D(A)) ∩ W 1,∞ ([0, ∞); H).
Proof. For U = (w, W, ψ, Ψ, ϕ, Φ) ∈ D(A), using the inner product of the Hilbert space H. We see that

(AU, U ) H = - Ω Φχ 1 (Φ) dxdy - Ω χ 2 (Ψ)Ψ dxdy ≤ 0, (2.11)
from which it follows that A is a dissipative operator. Thanks to the Lax-Milgram theorem (see [START_REF] Brezis | Analyse Fonctionelle, Théorie et Applications[END_REF]) we obtain that the operator A is maximal (see [START_REF] Lagnese | Analysis and Control of Thin Plates[END_REF]). Next, under the assumption (H 0 ), B is a Lipschitz operator. Thus, we obtain the classical existence and regularity result (see [START_REF] Haraux | Systémes dynamiques dissipatifs et applications[END_REF] for the proof) using the theory of maximal nonlinear monotone operator.

Strong stability for Mindlin Timoshenko system

We give the energy functional associated to the solution U of the system (2.10) as follow

E(t, w, ψ, ϕ) := 1 2 Ω [ρ 1 |w t | 2 + ρ 2 |ψ t | 2 + ρ 2 |ϕ t | 2 + K|ψ + w x | 2 + K|ϕ + w y | 2 + D|ψ x | 2 + D|ϕ y | 2 + 1 -µ 2 D|ψ y + ϕ x | 2 +
E (t) = - Ω (ψ t χ 1 (ψ t ) + ϕ t χ 2 (ϕ t )) dxdy, ∀ t ≥ 0. ( 2
+ K Ω (ϕ + w y )w yt dxdy + K Ω (ψ + w x )ψ t dxdy + K Ω (ϕ + w y )ϕ t dxdy + D 2 d dt Ω |ψ x | 2 dxdy + D 2 d dt Ω |ϕ y | 2 dxdy + D 2 1 -µ 2 d dt Ω |ψ y | 2 dxdy + D 2 1 -µ 2 Ω ϕ x ψ yt dxdy + D 2 1 -µ 2 Ω ψ y ϕ xt dxdy + D 2 1 -µ 2 d dt Ω |ϕ x | 2 dxdy + Dµ Ω ϕ y ψ xt dxdy + Dµ Ω ψ x ϕ yt dxdy = - Ω ψ t χ 1 (ψ t ) + ψ t χ 2 (ψ t ) dxdy,
and we arrive at

d dt ρ 1 2 Ω |w t | 2 dxdy + ρ 2 2 d dt |ψ t | 2 dxdy + ρ 2 2 d dt Ω |ϕ t | 2 dxdy + K 2 d dt Ω |ϕ + w y | 2 dxdy + K 2 d dt Ω |ψ + w x | 2 dxdy + D 2 Ω |ψ x | 2 dxdy + D 2 Ω |ϕ y | 2 dxdy + D 2 1 -µ 2 d dt Ω |ϕ x + ψ y | 2 dxdy + 2Dµ Ω ψ x ϕ y dxdy = - Ω χ 1 (ψ t )ψ t -χ 2 (ϕ t )ϕ t dxdy. Therefore, d dt E(t) := Ω χ 1 (ψ t )ψ t -χ 2 (ϕ t )ϕ t dxdy ≤ 0, ∀t ≥ 0,
since χ 1 and χ 2 satisfies the assumption (H 0 ). Hence, we obtain the energy dissipation law E(t) ≤ E(0), ∀t ≥ 0.

We now define the energies

E i (t) := E(t, ∂ (i) t w, ∂ (i) t ψ, ∂ (i) t ϕ). (2.14)
We shall call E 1 (t) the second order energy associated with ((2.1)-(2.2) and (2.3))

E 1 (t, w t , ψ t , ϕ t ) := 1 2 Ω [ρ 1 |w tt | 2 + ρ 2 |ψ tt | 2 + ρ 2 |ϕ tt | 2 + K|ψ t + w tx | 2 + K|ϕ t + w ty | 2 + D|ψ tx | 2 + D|ϕ ty | 2 + 1 -µ 2 D|ψ ty + ϕ tx | 2 + 2Dµψ tx ϕ ty ]. (2.15)
It is not difficult to show that

dE 1 dt = - Ω χ 1 (ψ 2 t )ψ 2 t - Ω χ 2 (ϕ 2 t )ϕ 2 t ≤ 0, (2.16)
hence, E 1 is a decreasing function and E 1 (t) ≤ E 1 (0), ∀t ≥ 0.

To obtain the strong stability result, we start by establishing the compactness of the orbit in the following lemma.

Lemma 2.1. For the initial data U 0 ∈ D(A), the orbit of U 0 given by

Υ(U 0 ) = ∪ t≥0 S(t)U 0 is relatively compact in H.
Proof. The proof is that of to the one dimension case (see Lemma 1.14 [START_REF] Sare | on the stability of Mindlin-Timoshenko Plates Department of Mathematics and Statistics[END_REF]).

Thanks to the definition of E 1 , we deduce that the sets

{w tt (t, •), t ≥ 0} is bounded in L 2 (Ω),
and

{w t (t, •), t ≥ 0} is bounded in H 1 0 (Ω). Proof. Let U 0 ∈ D(A).
The energy E given by (2.15) of the solution U = S(t)U 0 is a Lyapunov function for S(t) on H.

Applying the Lasalle invariance principle (see Proposition 3.3 Appendix B ), we obtain that for each

Z 0 ∈ ω(U 0 ) such that Z(t) = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) = S(t)Z 0 , the function L(Z(.)) is constant. Then,(v 1 , v 3 , v 6
) is a solution of the conservative system

ρ 1 v 1tt -K(v 2 + v 1x ) x -K(v 3 + v 1y ) y = 0, ρ 2 v 3tt -Dv 3xx -D 1 -µ 2 v 3yy -D 1 + µ 2 v 6xy + K(v 3 + v 1x ) = 0, ρ 2 v 5tt -Dv 5yy -D 1 -µ 2 v 5xx -D 1 + µ 2 v 3xy + K(v 5 + v 1y ) = 0. (v 1 (0), v 1t (0), v 2 (0), v 2t (0), v 3 (0), v 3t (0)) = Z 0 in Ω
Now, using the dissipation inequality which will be equal to zero, we obtain

E (t) = - Ω v 3t χ(v 3t ) - Ω v 2t χ 2 (v 2t ) = 0 ⇒ v 2t = 0 and v 3t = 0, ∀(x, y) ∈ Ω, t ∈ R + .
Hence, (v 1 , v 3 , v 6 ) ≡ (0, 0, 0). Then, ω(U 0 ) = {(0, 0, 0, 0, 0, 0)} and since the orbit is relatively compact in H, the energy associated to the initial data U 0 ∈ H converge to 0 as time goes to ∞.

Upper energy estimates

In this section we prove that the optimal-weight convexity method lead us to obtain an explicit decay rate formula of the total energy (2.15) associated to the solution of our problem (2.1)-(2.2)-(2.3). Indeed, this method was originally developed in [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF] where the author considered that her work completed the work of [START_REF] Lasiecka | Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping[END_REF] and improved the results of and [START_REF] Martinez | A new method to obtain decay rate estimates for dissipative systems[END_REF]. We assume a convexity assumptions on the feedback, and we prove the results of asymptotic behavior in higher dimension. We refer the reader to [START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities March[END_REF] for the wave equation and to [START_REF] Alabau-Boussouira | Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control[END_REF] for a one dimension Timoshenko system. In order to give a detailed proof of the main results of this section, we begin by introducing the following basic definitions.

Notations

First, let H be a function given by

H(x) =    H(x), if x ∈ [0, r 2 0 ], +∞, if x ∈ R\[0, r 2 0 ],
(2.17)

Second, we define the function L on [0, +∞) by

L(y) =    H (y) y if y > 0, 0 if y = 0, (2.18)
where H is the convex conjugate function of H, given by

H (y) = sup x∈R {xy -H(x)}. (2.19)
Third, we recall the function Λ H on (0,

r 2 0 ] Λ H (x) = H(x) xH (x) , ( 2.20)
which is an essential tool to classify the feedback growths 0, it has been introduced for the first time in [START_REF] Alabau-Boussouira | A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semidiscretized vibrating damped systems[END_REF] and it has been used to simplify the decay estimate formula given in [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF]. Finally, let η > 0 and M > 0 be fixed and f be a strictly increasing function from [0, η) onto [0, +∞). For any r ∈ (0, η), we define a function K r from (0, r] on [0, +∞) by :

K r (τ ) = r τ dy yf (y) , (2.21)
and ψ r which is a strictly increasing function given by:

ψ r (z) = z + K r (f -1 1 z ) ≥ z, ∀z ≥ 1 f (r) . (2.22) For x ≥ 1 H (r 2
0 ) , we define

ψ 0 (x) = 1 H (r 2 0 ) + H (r 2 0 ) 1/x 1 s 2 (1 -Λ H ((H ) -1 (s)))
ds.

(2.23)

Remark 2.1.

The function L is a strictly increasing continuous function from [0, +∞) onto [0, r 2 0 ).(see the proof in Proposition 2.2 [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF]). We recall here the definition of the function f given in [START_REF] Alabau-Boussouira | Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control[END_REF] by

f (s) = L -1 s 2β , ∀ s ∈ [0, 2βr 2 0 ). (2.24)
where, β = βE(0) is given by (2.77).

We also note that f is a is a nonnegative C 1 and strictly increasing function defined from [0, 2βr 2 0 ) onto [0, +∞) (see [START_REF] Alabau-Boussouira | Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control[END_REF] for the proof).

Main results

The first main result of this section is given by the following Theorem. The method used for the proof is adapted to handle corresponding coupled systems of plate equation in the spirt of the general approach given by [[7] Theorem 2.3].

Theorem 2.3.

We assume that v 2 1 = v 2 2 and the assumptions (H 0 ) and (H 1 ) hold. Let E be a given decreasing, absolutely continuous function on [0, +∞) and the parameter β such that

E(0) 2L(H (r 2 0 )) ≤ β. (2.25)
Then, the total energy of (2.1)-( 2.3) defined by (2.12) decay as

E(t) ≤ βL   1 ψ -1 0 t σ   , ∀ t ≥ σ H (r 2 0 ) . ( 2

.26)

Furthermore, if lim sup x→0 + Λ H (x) < 1, then E satisfies the following simplified decay rate

E(t) ≤ β (H ) -1 σ t , ∀ t ≥ 0. (2.27)
Here, β and σ are respectively given by

β = max c 3 , E(0) 2L(H (r 2 0 )) , (2.28) σ = 2 α 3 1 c 3 + 1 c 6 H (r 2 0 )α 2 (c 4 + c 5 ) + 1 . (2.29)
where k > 0 is a constant which do not depend of E(0).

Before proving the result above, we formulate the following Lemmas which are necessary to formulate the proof. Lemma 2.2. Assume that v 2 1 = v 2 2 and χ 1 , χ 2 satisfies (H 0 ). Let f be a strictly increasing function from [0, η) onto [0, +∞) . Then there exists positive constants α i for i = 1, • • • , 3 independent on f , such that the energy of the solutions of (2.10) satisfies the nonlinear weighted estimate:

T S f (E(t))E(t)dt ≤ α 1 E(S)f (E(S)) + α 2 T S f (E(t)) Ω |χ 1 (ψ t )| 2 + |χ 2 (ϕ t )| 2 dxdydt + α 3 T S f (E(t)) Ω |ψ t | 2 + |ϕ t | 2 dxdydt .
(2.30)

Proof. The main difficulty of the proof of this Lemma is how we can find the suitable multiplier.

In the first step, we multiply the equation (2.1) by f (E(t)) D K ψ x and the equation (2.2) by f (E(t))(ψ + w x ) then we integrate the resulting equations over [S, T ] × Ω, we get

ρ 1 D K T S f (E(t)) Ω w tt ψ x dxdydt -D T S f (E(t)) Ω (ψ + w x ) x ψ x dxdydt (2.31) -D T S f (E(t)) Ω (ϕ + w y ) y ψ x dxdydt = 0 and ρ 2 T S f (E(t) Ω ψ tt (ψ + w x ) dxdydt -D T S f (E(t) Ω ψ xx (ψ + w x ) dxdydt -D 1 -µ 2 T S f (E(t) Ω ψ yy (ψ + w x ) dxdydt -D 1 + µ 2 T S f (E(t) × Ω ϕ xy (ψ + w x ) dxdydt + K T S f (E(t) Ω |ψ + w x | 2 dxdydt + T S f (E(t) Ω χ 1 (ψ t )(ψ + w x ) dxdydt = 0 (2.32)
In the second step, summing the equations (2.31) and (2.32) and using integration by parts with respect to the variable x, we arrive at

K T S f (E(t) Ω |ψ + w x | 2 dxdydt = D T S f (E(t)) Ω (ϕ + w y ) y ψ x dxdydt -ρ 2 T S f (E(t) Ω ψ tt (ψ + w x ) dxdydt -ρ 1 D K T S f (E(t)) Ω w tt ψ x dxdydt +D 1 -µ 2 T S f (E(t)) Ω ψ yy (ψ + w x ) dxdydt (2.33) +D 1 + µ 2 T S f (E(t)) Ω ϕ xy (ψ + w x ) dxdydt - T S f (E(t) Ω χ 1 (ψ t )(ψ + w x ) dxdydt.
We integrate by part with respect to time variable t the above equality (2.33), one sees

K T S f (E(t)) Ω |ψ + w x | 2 dxdydt = - T S f (E(t)) Ω χ 1 (ψ t )(ψ + w x ) dxdydt + ρ 2 - ρ 1 D K T S f (E(t)) Ω ψ tt (w x + ψ) dxdydt + ρ 1 D K T S f (E(t)) Ω |ψ t | 2 dxdydt - ρ 1 D K f (E(t)) Ω ψ t ψ T S + ρ 1 D K T S f (E(t))E (t) Ω ψ t ψ dxdydt - ρ 1 D K T S f (E(t)) Ω (w t ψ x + w x ψ t ) t dxdydt + ρ 1 D K T S f (E(t)) Γ w t ψ t dxdydt +D 1 + µ 2 T S f (E(t) Ω ϕ xy (ψ + w x ) dxdydt +D 1 -µ 2 T S f (E(t)) Ω ψ yy (ψ + w x ) dxdydt + D K T S f (E(t)) Ω (ϕ + w y ) y ψ x dxdydt.
Now, Integrating by parts with respect to space variable and using the boundary condition (2.9) ( w = 0 at the boundary), we get

K T S f (E(t)) Ω |ψ + w x | 2 dxdydt = - T S f (E(t)) Ω χ 1 (ψ t )(ψ + w x ) dxdydt + ρ 2 - ρ 1 D K T S f (E(t)) Ω ψ tt (w x + ψ) dxdydt + ρ 1 D K T S f (E(t)) Ω |ψ t | 2 dxdydt + ρ 1 D K T S f (E(t))E (t) Ω ψ t (ψ + w x ) + w t ψ x dxdydt - ρ 1 D K f (E(t)) Ω ψ t (ψ + w x ) + w t ψ x dxdy T S +D 1 + µ 2 T S f (E(t)) Ω ϕ xy (ψ + w x ) dxdydt +D 1 -µ 2 T S f (E(t)) Ω ψ yy (ψ + w x ) dxdydt +D T S f (E(t)) Ω (ϕ + w y ) y ψ x dxdydt. (2.34)
On the other hand, using Cauchy-Schwartz and Young inequalities, we get

T s f (E(t)) Ω χ 1 (ψ t )(ψ + w x ) ≤ 1 2 T s f (E(t)) Ω |χ 1 (ψ t )| 2 + |ϕ x + ψ| 2 dxdydt. (2.35)
and

| ρ 2 - ρ 1 D K | T S f (E(t)) Ω ψ tt (w x + ψ) dxdydt ≤ |ρ 2 - ρ 1 D K | 2 T S f (E(t)) Ω |ψ tt | 2 dxdydt + T S f (E(t)) Ω |w x + ψ| 2 dxdydt, (2.36)
where c is a generic constant which depend on L 1 , L 2 , ρ 2 , K, D and ρ 1 .

Since E is a nondecreasing function, whereas f is an increasing function, we estimate the fifth term on the right hand side of (2.34) as follows

| ρ 1 D K f (E(t)) Ω ψ t (ψ + w x ) + w t ψ x dxdy T S | ≤ | ρ 1 D K f (E(t))E(t) T S | ≤ cE(S)f (E(S)) ∀ 0 ≤ S ≤ T.
(2.37) Also, we estimate the fourth term on the right hand side of (2.34) as follows,

| ρ 1 D K T S f (E(t))E (t) Ω ψ t (ψ + w x ) + w t ψ x dxdydt| ≤ cE(S)f (E(S)), 0 ≤ S ≤ T.(2.

38)

Thanks to the above inequalities (

, we obtain

K 2 T S f (E(t)) Ω |ψ + w x | 2 dxdydt ≤ K 2 T S f (E(t)) Ω |χ 1 (ψ t )| 2 + 1 2 | ρ 1 D K -ρ 2 | 2 T S f (E(t)) Ω |ψ tt | 2 dxdydt + ρ 1 D K T S f (E(t)) Ω |ψ t | 2 dxdydt + D 1 + µ 2 T S f (E(t)) Ω ϕ xy (ψ + w x ) dxdydt + D 1 -µ 2 T S f (E(t)) Ω ψ yy (ψ + w x ) dxdydt + D T S f (E(t)) Ω (ϕ + w y ) y ψ x dxdydt + cE(S)f (E(S)), ∀ 0 ≤ S ≤ T. (2.39)
Similarly, we multiply the equations (2.1) and (2.3), respectively, by f (E(t)) D K ϕ y and f (E(t))(ϕ+ w y ) then, we integrate the resulting equations over [S, T ]× Ω as follows

ρ 1 D K T S f (E(t)) Ω w tt ϕ y dxdydt -D T S f (E(t)) Ω (ψ + w x ) x ϕ y dxdydt (2.40) -D T S f (E(t)) Ω (ϕ + w y ) y ϕ y dxdydt = 0,
and

ρ 2 T S f (E(t)) Ω ϕ tt (ϕ + w y )dxdydt + K T S f (E(t)) Ω |ϕ + w y | 2 dxdydt -D 1 -µ 2 T S f (E(t)) Ω ϕ xx (ϕ + w y ) dxdydt -D 1 + µ 2 T S f (E(t)) Ω ψ xy (ϕ + w y ) + T S f (E(t)) Ω χ 2 (ϕ t )(ϕ + w y ) -D T S f (E(t)) Ω ϕ yy (ϕ + w y ) dxdydt = 0 (2.41)
By adding the equation (2.40) and (2.41), we obtain the following inequality

K 2 T S f (E(t)) Ω |ϕ + w y | 2 dxdydt ≤ 1 2 | ρ 2 - ρ 1 D K | 2 T S f (E(t)) Ω |ϕ tt | 2 dxdydt + ρ 1 D K T S f (E(t)) Ω |ϕ t | 2 dxdydt + D 1 -µ 2 T S f (E(t)) Ω ϕ xy (ψ + w x ) dxdydt + D 1 -µ 2 T S f (E(t)) Ω (ϕ + w y ) y ψ x dxdydt + D T S f (E(t)) Ω (ψ + w x ) ψ yy dxdydt + cE(S)f (E(S)) ∀ 0 ≤ S ≤ T. ( 2.42)
Thanks to (2.39) and (2.42), we deduce that

K 2 T S f (E(t)) Ω |ϕ + w y | 2 + |ψ + w x | 2 dxdydt (2.43) ≤ 1 2 | ρ 2 - ρ 1 D K | 2 T S f (E(t)) Ω |ϕ tt | 2 + |ψ tt | 2 dxdydt + ρ 1 D K T S f (E(t)) Ω |ϕ t | 2 + |ψ t | 2 dxdydt + K 2 T S f (E(t)) Ω |χ 1 (ψ t )| 2 dxdy +D 1 -µ 2 T S f (E(t)) Ω (ψ + w x ) ψ yy + ϕ xx (ϕ + w y ).
Now, we multiply the equation (2.1) by f (E(t))w and the equation (2.2) by f (E(t))ψ then, we integrate over [S, T ] × Ω, we get

ρ 1 f (E(t)) Ω w t w dxdy T S -ρ 1 T S f (E(t))E (t) Ω ww t dxdydt - K T S f (E(t)) Ω (ψ + w x ) x w dxdydt -ρ 1 T S f (E(t)) Ω |w t | 2 dxdydt 54 -K T S f (E(t)) Ω (ϕ + w y ) y w dxdydt = 0. (2.44)
and

T S f (E(t) Ω χ 1 (ψ t )ψ dxdydt -ρ 2 T S f (E(t))E (t) Ω ψ t ψdxdydt -ρ 2 T S f (E(t)) Ω |ψ t | 2 dxdydt -D 1 -µ 2 T S f (E(t) Ω ψ yy (ψ + w x ) dxdydt -D 1 + µ 2 T S f (E(t) Ω ϕ xy ψ dxdydt -D T S f (E(t)) Ω ψ xx ψ dxdydt +K T S f (E(t)) Ω (ψ + w x )ψ dxdydt + ρ 2 f (E(t)) Ω ψψ t dxdy T S = 0. (2.45)
In addition, we multiply the equation (2.3) by f (E(t))ϕ and we integrate over [S, T ] × Ω, we get

-ρ 2 T S f (E(t)) Ω |ϕ t | 2 dxdydt + ρ 2 f (E(t)) Ω ϕ t ϕ dxdy T S +ρ 2 T S f (E(t))E (t) Ω ϕ t ϕ dxdydt -D 1 -µ 2 T S f (E(t) Ω ϕ xx ϕ dxdydt -D T S f (E(t) Ω ϕ yy ϕ dxdydt -D 1 + µ 2 T S f (E(t) Ω ψ xy ϕ dxdydt +K T S f (E(t) Ω (ϕ + w y )ϕ dxdydt + T S f (E(t) Ω χ 2 (ϕ t )ϕ t dxdydt = 0. (2.46)
Next, by summing (2.44), (2.45) and (2.46), we find

T S f (E(t)) Ω ρ 1 |w t | 2 + ρ 2 |ϕ t | 2 + ρ 2 |ψ t | 2 dxdydt = -ρ 2 T S f (E(t))E (t) Ω ψ t ψ dxdydt +ρ 1 f (E(t)) Ω w t w T S -ρ 1 T S f (E(t))E (t) Ω ww t dxdydt +ρ 2 T S f (E(t))E (t) Ω ϕ t ϕ dxdydt + ρ 2 f (E(t) Ω ψψ t T S + ρ 2 f (E(t)) Ω ϕ t ϕ T S -K T S f (E(t)) Ω (ϕ + w y ) y w + (ψ + w x ) x w dxdydt +K T S f (E(t)) Ω (ϕ + w y )ϕ + (ψ + w x )ψ dxdydt (2.47) -D 1 -µ 2 T S f (E(t) Ω ψ yy ψ + ϕ xx ϕ dxdydt +D T S f (E(t) Ω |ψ x | 2 + |ϕ y | 2 dxdydt 55 -D 1 + µ 2 T S f (E(t) Ω ψ xy ϕ + ϕ xy ψ dxdydt + T S f (E(t) Ω χ 1 (ψ t )ψ+χ 2 (ϕ t )ϕ dxdydt.
A simple computation yields

D 1 -µ 2 T S f (E(t) Ω |ψ y + ϕ x | 2 dxdydt = D 1 -µ 2 T S f (E(t)) Ω |ψ y | 2 + |ϕ x | 2 dxdydt +D(1 -µ) T S f (E(t)) Ω ψ y ϕ x dxdydt (2.48)
using the inequalities (2.43), (2.47), together with the equality (2.48) we find

T S f (E(t))E(t)dt ≤ T S f (E(t)) Ω ρ 1 |w t | 2 + ρ 2 |ϕ t | 2 + ρ 2 |ψ t | 2 + D|ψ x | 2 + D|ϕ y | 2 + K T S f (E(t) Ω |ϕ + w y | 2 + |ψ + w x | 2 dxdydt + T S f (E(t)) Ω 2Dµ[∂ y ∂ x (ψϕ)] + 2Dµψ x ϕ y + D 1 -µ 2 |ψ y + ϕ x | 2 dxdy.dt
The equalities (2.47) and (2.34), together with the use of the definition (2.12) of E, we remain to the desired estimate (2.30).

Lemma 2.3. We assume that the hypotheses of Theorem 2.1 hold. In addition, we have the estimation (2.30). Then, there exists fixed real number σ > 0 such that we have the following inequality :

T S E(t)f (E(t))dt ≤ σE(S), (2.49)
where σ is a positive constant given by (2.29).

The proof of this Lemma is as follows

Proof. We start by choosing a parameter ε 1 sufficiently small, such that ε 1 = g( r 0 √ 2 ). We define the subset, we proceed

Ω t 0 = {(x, y) ∈ Ω such that |ψ t (t, x, y)| ≤ ε 1 and |ϕ t (t, x, y)| ≤ ε 1 }, for fixed t ≥ 0.
Thanks to the assumption (H 0 ), there exist constants c 3 > 0 and c 4 > 0 such that

c 3 g(|ν|) ≤ |χ 1 (ν)| ≤ c 4 g -1 (|ν|), |ν| ≤ ε 1 ,
and there exist constants c 5 and c 6 such that

c 6 g(|ν|) ≤ |χ 2 (ν)| ≤ c 5 g -1 (|ν|), |ν| ≤ ε 1 .
In particular, for ν = ψ t (t, x, y) we have

|χ 1 (ψ t (t, x, y))| 2 ≤ (c 4 g -1 (|ψ t (t, x, y)|) 2 , ∀ (x, y) ∈ Ω t 0 .
Similarly, for ν = ϕ t (t, x, y) we have

|χ 2 (ϕ t (t, x, y))| 2 ≤ (c 5 g -1 (|ϕ t (t, x, y)|) 2 , ∀ (x, y) ∈ Ω t 0 .
Since g -1 is a monotone increasing function on R, we get

|χ 1 (ψ t (t, x, y))| 2 ≤ (c 2 4 g -1 ( 1 )) 2 = c 4 r 2 0 2 , ∀ (x, y) ∈ Ω t 0 ,
and

|χ 2 (ϕ t (t, x, y))| 2 ≤ (c 2 5 g -1 ( 1 )) 2 = c 4 r 2 0 2 , ∀ (x, y) ∈ Ω t 0 .
Hence,

|Ω t 0 | -1 Ω t 0 |c -1 4 χ 1 (ψ t (t, x, y))| 2 dxdy ∈ [0, r 2 0 2 ], (2.50)
and

|Ω t 0 | -1 Ω t 0 |c -1 5 χ 2 (ϕ t (t, x, y))| 2 dxdy ∈ [0, r 2 0 2 ]. (2.51)
On the other hand, H is a strictly convex function on [0, r 2 0 ], then applying the Jensen's inequality to the terms (2.50) and (2.51) and using the definition of the function

H(x) = √ xg( √ x)
, we obtain these inequalities

H |Ω t 0 | -1 Ω t 0 |c -1 4 χ 1 (ψ t (t, x, y))| 2 dxdy ≤ |Ω t 0 | -1 Ω t 0 H |c -1 4 χ 1 (ψ t (t, x, y)| 2 (2.52) = |Ω t 0 | -1 Ω t 0 1 c 4 χ 1 (ψ t (t, x, y)g 1 c 4 χ 1 (ψ t (t, x, y)) ,
and

H |Ω t 0 | -1 Ω t 0 |c -1 5 χ 2 (ϕ t (t, x, y)| 2 dxdy ≤ |Ω t 0 | -1 Ω t 0 H |c -1 5 χ 2 (ϕ t (t, x, y)| 2 (2.53) = |Ω t 0 | -1 Ω t 0 1 c 5 χ 2 (ϕ t (t, x, y)g 1 c 5 χ 2 (ϕ t (t, x, y)) .

57

Now, using the fact that g is an increasing function, we have

g 1 c 4 χ 1 (|ψ t (t, x, y)|) ≤ ψ t (t, x, y) on Ω t 0 , (2.54)
and

g 1 c 5 χ 2 (|ϕ t (t, x, y)| ≤ ϕ t (t, x, y), on Ω t 0 . ( 2.55)
Knowing that H(x 2 ) = xg(x) and using the inequalities (2.54) and (2.55), we have

H |Ω t 0 | -1 Ω t 0 |c -1 4 χ 1 (ψ t (t, x, y))| 2 dxdy ≤ Ω t 0 c -1 4 ψ t χ 1 (ψ t ) dxdy. ( 2.56)
and

H |Ω t 0 | -1 Ω t 0 |c -1 5 χ 2 (ϕ t (t, x, y)| 2 dxdy ≤ |Ω t 0 | -1 Ω t 0 c -1 5 ϕ t χ 2 (ϕ t ) dxdy, (2.57)
Since χ i are increasing functions for i = 1, 2, then

|Ω t 0 | -1 c -1 5 Ω t 0 χ 2 (ϕ t (t, x, y))ϕ t (t, x, y)dxdy ≤ |Ω t 0 | -1 c -1 4 Ω t 0 χ 1 (ε 1 )ε 1 dxdy ≤ c -1 4 ε 1 g -1 (ε 1 ) = H( r 2 0 2 ),
and

|Ω t 0 | -1 c -1 4 Ω t 0 |χ 1 (ψ t (t, x, y))||ψ t (t, x, y)|dxdy ≤ |Ω t 0 | -1 c -1 4 Ω t 0 χ 1 (ε 1 )ε 1 dxdy ≤ c -1 4 ε 1 g -1 (ε 1 ) = H( r 2 0 2 ).
Then, we have

|Ω t 0 | -1 c -1 4 Ω t 0 |χ 1 (ψ t (t, x, y))||ψ t (t, x, y)|dxdy ∈ [0, H r 2 0 2 ], (2.58)
and

|Ω t 0 | -1 c -1 5 Ω t 0 χ 2 (ϕ t (t, x, y))ϕ t (t, x, y)dxdy ∈ [0, H r 2 0 2 ]. (2.59)
Consequently, we get

H -1 |Ω t 0 | -1 Ω t 0 c -1 4 χ 1 (ψ t (t, x, y))ψ t (t, x, y) dxdy ∈ [0, r 2 0 2 ], (2.60)
and f (E(t))

H -1 |Ω t 0 | -1 Ω t 0 c -
Ω t 0 |χ 1 (ψ t )| 2 + Ω t 0 |χ 2 (ϕ t )| 2 ≤ c 2 4 T S f (E(t))H -1 |Ω t 0 |c -1 4 Ω t 0 χ 1 (ψ t (t, x, y))ψ t (t, x, y) +c 2 5 T S f (E(t))H -1 |Ω t 0 |c -1 5 Ω t 0 χ 2 (ϕ t (t, x, y))ϕ t (t, x, y) .
Indeed, applying the following Young's inequality

AB ≤ H (A) + H(B), (2.62)
for A = f (E(t)) and first for

B = B 1 (t) = H -1 |Ω t 0 | -1 c -1 4 Ω t 0 χ 1 (ψ t (t, x, y))ψ t (t, x, y) , second for B = B 2 (t) = H -1 |Ω t 0 | -1 c -1 5 Ω t 0 χ 2 (ϕ t (t, x, y))ϕ t (t, x, y) .
Hence, we have

T S
f (E(t))

Ω t 0 |χ 1 (ψ t )| 2 + Ω t 0 |χ 2 (ϕ t )| 2 ≤ c 2 4 |Ω t 0 | T S H (f (E(t)))dt + |Ω t 0 | |Ω t 0 | -1 c 4 Ω t 0 χ 1 (ψ t (t, x, y))ψ t (t, x, y) + c 2 5 |Ω t 0 | T S H (f (E(t)))dt + |Ω t 0 | |Ω t 0 | -1 c 5 Ω t 0 χ 2 (ϕ t (t, x, y))ϕ t (t, x, y) .
Taking into account the dissipation relation (2.13), it is clear that f (E(t))

T S Ω t 0 c 4 ψ t χ 1 (ψ t ) + c 5 ϕ t χ 2 (ϕ t ) ≤ max(c 4 , c 5 ) (E(S) -E(T )) (2.
Ω t 0 |χ 1 (ψ t )| 2 + Ω t 0 |χ 2 (ϕ t )| 2 ≤ (c 2 4 + c 2 5 )|Ω t 0 | T S H (f (E(t)) + T S Ω t 0 c 4 ψ t χ 1 (ψ t ) + c 5 ϕ t χ 2 (ϕ t ) ≤ (c 2 4 + c 2 5 )|Ω t 0 | T S H (f (E(t))
+ max(c 4 , c 5 ) E(S), ∀ 0 ≤ S ≤ T.

For 0 ≤ S ≤ T , we have

T S f (E(t)) Ω\Ω t 0 |χ 1 (ψ t )| 2 + |χ 2 (ϕ t )| 2 ≤ T S f (E(t)) Ω\Ω t 0 c 4 |ψ t ||χ 1 (ψ t )| + c 5 |ϕ t ||χ 2 (ϕ t )| ≤ max(c 5 , c 4 ) T S f (E(t))(-E (t))
≤ max(c 5 , c 4 ) E(S)f (E(S)).

Finally, using the inequalities (2.64) and (2.63), we get ∀ 0 ≤ S ≤ T.

T S f (E(t)) Ω |χ 1 (ψ t )| 2 + |χ 2 (ϕ t )| 2 dxdydt ≤ (c 2 4 + c 2 5 )|Ω t 0 | T S H (f (E(t)) + max(c 4 , c 5 ) E(S)(f (E(S))) + max(c 4 , c 5 ) E(S).
Now, the second step of the proof start by choosing a sufficiently small parameter ε 2 = min(r 0 , g(r 1 )) where, r 1 is a positive constant such that H(r 2 1 ) = max( c 3 c 4 , c 5 c 6 )H(r 2 0 ). Thanks to (H 0 ) there exist constants, still denoted by c 3 > 0 and c 4 > 0 to avoid too many notion, such that

   c 3 |v| ≤ |χ 1 (v)| ≤ c 4 |v| ∀ |v| ≥ ε 2 c 3 |g(|v|)| ≤ |χ 1 (v)| ≤ c 4 g -1 (|v|) ∀ |v| ≤ ε 2 .
(2.64)

Besides, there exist constants denoted also by c 5 > 0 and c 6 > 0 to avoid too many notion such that

   c 6 |v| ≤ |χ 2 (v)| ≤ c 5 |v| ∀ |v| ≥ ε 2 , c 6 |g(|v|)| ≤ |χ 2 (v)| ≤ c 5 g -1 (|v|) ∀ |v| ≤ ε 2 . (2.65)
In this step, we need to define two sets G t 0 and G t 1 , as following:

G t 0 = {(x, y) ∈ Ω, |ψ t (t, x, y)| ≤ 2 }, ∀ t ≥ 0, and
G t 1 = {(x, y) ∈ Ω, |ϕ t (t, x, y)| ≤ 2 } ∀ t ≥ 0.
Then, for all t ≥ 0, we have

|G t 0 | -1 G t 0 |ψ t (t,
g(|ψ t |) ≤ c -1 3 χ 1 (ψ t ) on G t 1 , g(|ϕ t |) ≤ c -1 6 χ 2 (ϕ t ) on G t 1 .
Using the same arguments as in (2.52), we obtain

H |G t 0 | -1 G t 0 |ψ t | 2 dxdy ≤ |G t 0 | -1 G t 0 H(|ψ t (t, x, y)| 2 )dxdy = |G t 0 | -1 G t 0 |ψ t |g(|ψ t (t, x, y)|) dxdy ≤ 1 c 6 G t 0 G t 0 |ϕ t ||χ 2 (ϕ t )| dxdy, (2.68)
and

H |G t 1 | -1 G t 0 |ϕ t | 2 dxdy ≤ |G t 1 | -1 G t 1 H(|ϕ t (t, x, y)| 2 ) = |G t 1 | -1 G t 1 |ϕ t |g(|ϕ t (t, x, y)|) ≤ 1 c 3 |G t 1 | G t 1 |ψ t ||χ 1 (ψ t )|. (2.69)
Hence, since H is an increasing function, we have

T S f (E(t)) G t 1 |ϕ t | 2 ≤ T S f (E(t))|G t 1 |H -1 |G t 1 | -1 c -1 6 G t 1 |ϕ t ||χ 2 (ϕ t )| ,
and

T S f (E(t)) G t 0 |ψ t | 2 ≤ T S f (E(t))|G t 0 |H -1 |G t 0 | -1 c -1 4 G t 1 |ψ t ||χ 1 (ψ t )| .
Now, we apply Young's inequality (2.62), for A = f (E(t)) and first for

B = B 3 (t) = H -1 |G t 0 | -1 c -1 3 G t 1 |ψ t ||χ 1 (ψ t )| ,
then, second for

B = B 4 (t) = H -1 |G t 1 | -1 c -1 6 G t 1 |ϕ t ||χ 2 (ϕ t )| .
Hence,

T S f (E(t)) G t 0 |ψ t | 2 ≤ |G t 0 | T S H (f (E(t))) + H H -1 |G t 0 | -1 1 c 3 G t 0 χ 1 (ψ t )ψ t , (2.70)
and

T S f (E(t)) G t 1 |ϕ t | 2 ≤ |G t 1 | T S H (f (E(t))) + H H -1 |G t 1 | -1 c -1 6 G t 1 ϕ t χ 2 (ϕ t ) . (2.71)
On the other hand, thanks to (2.64) and (2.65), we have

|G t 0 | -1 c -1 3 G t 1 ψ t χ 1 (ψ t ) ≤ c 4 |G t 0 | -1 c -1 3 G t 0 ψ t g -1 (ψ t ) ≤ c 4 c 3 ε 1 g -1 (ε 1 ) = c 4 c 3 H(r 2 1 ) ≤ max( c 4 c 3 , c 5 c 6 )H(r 2 1 ) = H(r 2 0 ). (2.72) Also, |G t 1 | -1 c -1 6 G t 1 |ϕ t ||χ 2 (ϕ t )| ≤ c 5 |G t 0 | -1 c -1 6 G t 0 ϕ t g -1 (ϕ t ) ≤ c 5 c 6 ε 1 g -1 (ε 1 ) = c 5 c 6 H(r 2 1 ) ≤ max( c 4 c 3 , c 5 c 6 )H(r 2 1 ) = H(r 2 0 ). (2.73)
Using the fact that H is an increasing function, we have

H -1 |G t 0 | -1 1 c 3 G t 0 χ 1 (ψ t )ψ t ∈ [0, r 2 0 ], H -1 |G t 1 |c -1 6 G t 1 ϕ t χ 2 (ϕ t ) ∈ [0, r 2 0 ].
and H(x) = H(x), ∀ x ∈ [0, r 2 0 ], using the dissipation relation (2.13), we get

T S f (E(t)) G t 0 |ψ t | 2 ≤ |G t 0 | T S H (f (E(t))) + 1 c 3 E(S), ∀ 0 ≤ S ≤ T, (2.74)
and

T S f (E(t)) G t 1 |ϕ t | 2 ≤ |G t 1 | T S H (f (E(t))) + 1 c 6 E(S), ∀ 0 ≤ S ≤ T.
(2.75)

We prove as in (2.64) that:

T S f (E(t)) Ω\G t 0 |ψ t | 2 ≤ c -1 3 E(S)f (E(S)), ∀ 0 ≤ S ≤ T, T S f (E(t)) Ω\G t 1 |ϕ t | 2 ≤ c -1 6 E(S)f (E(S)), ∀ 0 ≤ S ≤ T.
Consequently, we have

T S f (E(t)) Ω |ψ t | 2 + |ϕ t | 2 dxdy ≤ 1 c 3 + 1 c 6 E(S)f (E(S)) + (|G t 0 | + |G t 1 |) T S H (f (E(t))) + 1 c 3 + 1 c 6 E(S), ∀ 0 ≤ S ≤ T.
(2.76)

Finally using the estimates (2.30), (2.76) and (2.63), we obtain

T S E(t)f (E(t)) dt ≤ α 3 1 c 3 + 1 c 6 + α 2 (c 4 + c 5 ) E(S)f (E(S)) + α 3 (|G t 0 | + |G t 1 |) + α 2 (c 2 4 + c 2 5 )|Ω t 0 | T S H (f (E(t))) + α 3 1 c 3 + 1 c 6 E(S), ∀ 0 ≤ S ≤ T.
In order to finish the proof of this lemma, we need to estimate the term f (E(s)), for this, we define the constant β which depend on E(0) as follows

β = max C 3 , E(0) 2L(H (r 2 0 )) , (2.77)
where, L is defined in (2.18) and

C 3 = α 3 (|G t 0 | + |G t 1 |)+α 2 (c 2 4 +c 2 5 )|Ω t 0 |.
Here C 3 is a positive constant which depends on the physical characteristics of the plate (ρ 1 , ρ 2 , b and K) and it do not depend on E(0). Now, using the definition of f given by (2.24), the fact that E is nondecreasing function and the inequality (2.25), we have

E(t) 2β ≤ E(0) 2β ≤ L(H (r 2 0 )) < r 2 0 .
Recall that f is increasing, we obtain,

f (E(S)) ≤ f (E(0)) = L -1 E(0) 2β ≤ H (r 2 0 ), ∀ s ≥ 0.
Thus, the inequality (2.77) reduces to

T S E(t)f (E(t)) dt ≤ α 3 1 c 3 + 1 c 6 + α 2 (c 4 + c 5 ) E(S)f (E(S)) + α 3 1 c 3 + 1 c 6 E(S) + α 3 (|G t 0 | + |G t 1 |) + α 2 (c 2 4 + c 2 5 )|Ω t 0 | T S H (f (E(t))) ≤ α 3 1 c 3 + 1 c 6 (H (r 2 0 ) α 2 (c 4 + c 5 ) + 1) E(S) + C 3 T S H (f (E(t))) ∀ 0 ≤ S ≤ T.
Thanks to the choice in [START_REF] Alabau-Boussouira | A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semidiscretized vibrating damped systems[END_REF] of the function f , we have

C 3 H (f (s)) ≤ β H (f (s)) = sf (s) 2 , ∀ s ∈ [0, βr 2 0 ).
This relation holds for s = E(S), then, we deduce that

T S E(t)f (E(t)) ≤ σE(S) ∀ 0 ≤ S ≤ T, ( 2.78)
where,

σ = 2 α 3 1 c 3 + 1 c 6 (H (r 2 0 )α 2 (c 4 + c 5 ) + 1) . (2.79)
In the third step, putting all the estimates together, we shall deduce the result in Theorem 2.3 by applying Theorem 2.3 [START_REF] Alabau-Boussouira | Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems[END_REF] (see also [START_REF] Alabau-Boussouira | Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control[END_REF][START_REF] Alabau-Boussouira | A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities March[END_REF]) in the following proof Proof. (proof of Theorem2.3) The energy E (2.12) associated to the solution our system ((2.1)-(2.2)) is a nondecreasing function thanks to the dissipative relation (2.13), absolutely continuous function from [0, +∞) on [0, +∞).It satisfies

T S E(t)f (E(t)) ≤ σE(S) ∀ 0 ≤ S ≤ T,
(2.80)

Chapter 3

Discrete Energy behavior of a damped Timoshenko system.

Numerical approximation

This chapter aims to give a validation of the theoretical part carried out in the literature.

For that purpose, we design a numerical scheme based on a combination between the finite element in space and finite difference in time methods. We start by the discretization of the Timoshenko system in 1D case and we define its discrete energy. Firstly, let us recall the 1D undamped Timoshenko system in the continuous case given by

   ρ 1 ϕ tt -k(ϕ x + ψ) x = 0, (x, t) ∈ (0, L) × R + , ρ 2 ψ tt -bψ xx + k(ϕ x + ψ) = 0, (x, t) ∈ (0, L) × R + , ( 3.1)
and for which we associate the following initial conditions:

   ϕ(0, x) = ϕ 0 (x), ψ(0, x) = ψ 0 (x), ∀x ∈ (0, L), ϕ t (0, x) = ϕ 1 (x), ψ t (0, x) = ψ 1 (x), ∀x ∈ (0, L), (3.2)
together with the Dirichlet boundary conditions as below

ϕ = ψ = 0, at x = 0, x = L. ( 3.3)
The energy associated with (3.2)-(3.3) is defined by

E(U, t) := 1 2 L 0 ρ 1 ϕ 2 t + ρ 2 ψ 2 t + bψ 2 x + k(ϕ x + ψ) 2 dx. (3.4)
We study the case ρ 1 = ρ 2 = k = b = 1, then, the linear system considered is

ϕ tt -(ϕ x + ψ) x = 0, (x, t) ∈ (0, L) × R + , (3.5) ψ tt -ψ xx + (ϕ x + ψ) = 0, (x, t) ∈ (0, L) × R + . (3.6)
Taking into account the boundary conditions (3.3), we obtain

dE(t) dt = 0, ∀ t ∈ [0, T ], (3.7)
which states the conservation of the energy and this can be expressed as follows:

E(t) = E(0) := ρ 1 2 L 0 |ϕ 1 (x)| 2 dx + ρ 2 2 L 0 |ψ 1 (x)| 2 dx+ + b 2 L 0 |ψ x (0, x)| 2 dx + k 2 L 0 |ϕ x (0, x) + ψ 0 (x)| 2 dx, ∀ t ≥ 0. (3.8)
This energy conservation property implies that the Timoshenko equations are purely conservative. Therefore, it is important to show that the numerical solution of the Timoshenko equations consistently preserve the property (3.8) as well, that is its corresponding discrete energy will obey the energy conservation property.

The variational form of the problem

First we will drive the variational form of the problem (3.5). Let u and v be in H 1 0 (0, L), respectively, and we use the following notation for convenience

(f, g) = L 0 f (x)g(x)dx.
Integrating over the interval (0, L) and using an integration by parts, we find

(ϕ tt , u) = -(ϕ x , u x ) + (ψ x , u), (x, t) ∈ (0, L) × R + , (3.9) (ψ tt , v) = -(ψ x , v x ) -(ϕ x , v) -(ψ, v), (x, t) ∈ (0, L) × R + . (3.10)
By adding equations (3.9) and (3.10) we end up with the variational form of the problem formulated in the product space H 1 0 (0, L) × H 1 0 (0, L) as follows. Find u, v such that for all t > 0, ∀ u ∈ H 1 0 (0, L) and v ∈ H 1 0 (0, L), we have

(ϕ tt , u) + (ψ tt , v) = (ψ x , u) -(ψ x , v x ) -(ϕ x , v) -(ψ, v) -(ϕ x , u x ). (3.11)
These matrices are given by

M ij = (w i , w j ) =              0 if |i -j| ≥ 2, ∆x 6 if |i -j| = 1, 2∆x 3 if i = j, S ij = (w i , w j ) =             1 2 if i = j -1, - 1 2 if i = j + 1, 0 if not, and
K ij = (w i , w j ) =             0 if |i -j| ≥ 2, -1 ∆x if |i -j| = 1, 2 ∆x if i = j.

Fully-discrete scheme in Finite Differences

Set t n+1 -t n = ∆t for all n ∈ N and introduce (Φ n , Ψ n ) = (Φ(n∆t), Ψ(n∆t)) the discrete solution of the semi-discrete equations (3.12) for n = {1, • • • , N t }, such that N t ∈ N and T = N t • ∆t. We design an explicit scheme using finite differences of system (3.12) is thus, for n ∈ N, we have

M Φ n+1 -2Φ n + Φ n-1 ∆t 2 = -KΦ n + SΨ n (3.13) M Ψ n+1 -2Ψ n + Ψ n-1 ∆t 2 = -KΨ n -SΦ n -M Ψ n . (3.14)
We note

Φ(t) = [ϕ(t, x 0 ), • • • , ϕ(t, x N x )], Ψ(t) = [ψ(t, x 0 ), • • • , ψ(t, x N x )], (3.15)
and the initial data using the explicit Euler for the first step we have

Φ 0 = [ϕ 0 (x 0 ), • • • , ϕ 0 (x N x )], Ψ 0 = [ψ 0 (x 0 ), • • • , ψ 0 (x N x )], (3.16) Φ 1 -Φ -1 2∆t + O(∆t 2 ) = ∂ϕ(t, x) ∂t | t=0 = [ϕ 1 (x 0 ), • • • , ϕ 1 (x N x )], Ψ 1 -Ψ -1 2∆t + O(∆t 2 ) = ∂ψ(t, x) ∂t | t=0 = [ψ 1 (x 0 ), • • • , ψ 1 (x N x )],
Thus,

Φ -1 = Φ 1 -2∆t [ϕ 1 (x 0 ), • • • , ϕ 1 (x N x )], Ψ -1 = Ψ 1 -2∆t [ψ 1 (x 0 ), • • • , ψ 1 (x N x )]. (3.17)

The discrete energy E n+ 1 2

In this section we define the discrete energy and we prove the conservation property.

Definition

Let (Φ n , Ψ n ) be the discrete solutions of the leapfrog scheme (3.13), (3.14). Then, the discrete energy E n+ 1 2 is defined as follows

E n+ 1 2 := M Φ n+1 -Φ n ∆t , Φ n+1 -Φ n ∆t + M Ψ n+1 -Ψ n ∆t , Ψ n+1 -Ψ n ∆t + (KΦ n , Φ n+1 ) + (KΨ n , Ψ n+1 ) (3.18) + (SΨ n , Φ n+1 ) -(SΦ n , Ψ n+1 ) + (M Ψ n , Ψ n+1 ) , ∀ n = {0, • • • , N t -1}.

The conservation property

Using the numerical scheme (3.13),(3.14) as previously described, we prove in the following proposition the conservation property of the discrete energy.

Proposition 3.1.

Let ∆x > 0, ∆t > 0 and (Φ n , Ψ n ) be the solution of the finite difference scheme (3.13),(3.14) associated with initial conditions (3.16). Then, for all n ∈ {1, • • • , N t -1}, we have

E n+ 1 2 = E n-1 2 = E 1 2 . (3.19)
where, the functions a and b are defined as follows

b = N i=1 b i w i (x), a = N i=1 a i w i (x), Proof. For a x = N i=0 a i w i (x) and b = N i=1 b i w i (x) we have 1 0 (a x + b) 2 dx = N i=1 a i w i (x) + b i w i (x) 2 = N i=1 a 2 i (w i (x), w i (x)) + i,k=1,i =k a i a k (w i (x), w k (x)) + N i=1 b 2 i (w i (x), w i (x)) - N i,k=1,i =k a i b k (w i (x), w k (x)) - N i,k=1,i =k a k b i (w k (x), w i (x)) + i,k=1,i =k b i b k (w i (x), w k (x)) = (Ka, a) + 2(Sa, b) + (M b, b) ≥ 0, (3.27)
where, M ik = (w i , w k ), K ik = (w i , w k ) and S ik = (w i , w k ). Now, we prove the positivity of the discrete energy using Lemma 3.1.

Theorem 3.1. Define the matrices M 1 and M 2 as

M 1 = (M - ∆t 2 4 K), (3.28)
and

M 2 = M - ∆t 2 4 (2M + K). ( 3

.29)

If M 1 and M 2 are definite positive, then, E n+ 1 2 ≥ 0 and the discrete explicit scheme (3.13)-(3.14)) is stable.

Proof.

In order to ensure the positivity of the energy E n+ 1 2 and to estimate the term

(SΨ n , Φ n+1 ) -(SΨ n , Φ n+1 ), (3.30)
we use Lemma 3.1 to obtain

(SΨ n , Φ n+1 ) ≥ - 1 2 (M Ψ n , Ψ n ) + (KΦ n+1 , Φ n+1 ) , ( 3.31)
and similarly the estimate of the second term of (3.30) can be obtained as follows

(SΦ n , Ψ n+1 ) ≥ - 1 2 (M Φ n , Φ n ) + (KΨ n+1 , Ψ n+1 ) . (3.32)
Consequently, the total discrete energy E n+ 1 2 of the system (3.20) and (3.21) satisfies the following estimate.

E n+ 1 2 ≥ Φ n+1 -Φ n ∆t 2 M + Ψ n+1 -Ψ n ∆t 2 M - 1 2 K(Φ n+1 -Φ n ), (Φ n+1 -Φ n ) - 1 2 M (Ψ n+1 -Ψ n ), (Ψ n+1 -Ψ n ) + (KΨ n , Ψ n+1 ).
Taking into account the fact that K is a symmetric matrix, we have

1 4 (K(u + v), (u + v)) - 1 4 (K(u -v), (u -v)) = (Ku, v)
Using the above equality we obtain the following result

E n+ 1 2 ≥ M - ∆t 2 2 K Φ n+1 -Φ n ∆t , Φ n+1 -Φ n ∆t (3.33) + (M - ∆t 2 4 (2M + K)) Ψ n+1 -Ψ n ∆t , Ψ n+1 -Ψ n ∆t +(K Ψ n+1 + Ψ n 2 , Ψ n+1 + Ψ n 2 ). E n+ 1 2 ≥ M 1 Φ n+1 -Φ n ∆t , Φ n+1 -Φ n ∆t + M 2 Ψ n+1 -Ψ n ∆t , Ψ n+1 -Ψ n ∆t +(K Ψ n+1 + Ψ n 2 , Ψ n+1 + Ψ n 2 ),
where

M 1 = (M - ∆t 2 4 K), ( 3.34)
and

M 2 = M - ∆t 2 4 (2M + K). ( 3.35)
If M 1 and M 2 are defined positive, then, the discrete energy E n+ 1 2 is positive. This is in fact the well-known CFL condition which can be written as follows:

CFL                ∆t 2 4 sup V =0 (2M + K)(V, V ) (M V, V ) < 1, ∆t 2 4 sup V =0 K(V, V ) (M V, V ) < 1. ( 3
4 sup V =0 (M V, V ) (KV, V ) < 1 2 . (3.38)
Then, we assume ∆t 2 ≤ c min ∆x 2 , 1 , and without loss of generality, we suppose that ∆x < 1, then, the (CF L) becomes ∆t 2 < ∆x 2 .

We infer the Courant-Friedrichs-Lewy (CFL) calculation Corollary 1. Assuming the following CFL condition

(CF L) ∆t ∆x < 1. (3.39)
Then, the matrices M 1 and M 2 are definite positive.

Remark 3.1. The usual method for obtaining the CFL condition is to use the Fourier analysis on a linear wave equation (the discrete version with the whole stability and decreasing of the energy made by Sylvie Benzoni-Gavage [START_REF] Brezis | Analyse Fonctionelle, Théorie et Applications[END_REF]). The method used here is based on the energy technique which allows us to consider

(CF L) ∆t ∆x < 1.
We do not claim that this condition is optimal.

Remark 3.2. We observe that the discrete energy E n+ 1 2 is consistent with the continuous energy.

Exponential decay rate of the discrete energy

First, we recall that the exponential stability is known for Timoshenko systems on bounded domains when the two linear damping terms ϕ t and ψ t are considered in the right-hand sides of the first and second equations, respectively, see, e.g., [START_REF] Sare | on the stability of Mindlin-Timoshenko Plates Department of Mathematics and Statistics[END_REF]. In [START_REF] Rivera | R: Global stability for damped Timoshenko systems[END_REF], the authors considered the one-dimensional system with a linear damping term as follows:

   ρ 1 ϕ tt -k(ϕ x + ψ) x = 0, (x, t) ∈ (0, L) × R + , ρ 2 ψ tt -bψ xx + k(ϕ x + ψ) + µψ t = 0, (x, t) ∈ (0, L) × R + , (3.40) and they proved that the solution of the system (3.40) is exponentially stable if and only if the wave speeds are equal ( k ρ 1 = b ρ 2 ). Such a result has been the common point in several works [START_REF] Sare | Stability of Timoshenko systems with past history[END_REF][START_REF] Rivera | R: Global stability for damped Timoshenko systems[END_REF][START_REF] Kim | Boundary control of the Timoshenko beam[END_REF][START_REF] Rivera | Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability[END_REF][START_REF] Racke | Fernàndez Sare: On the stability of damped Timoshenko systems-Cattaneo versus Fourier law[END_REF] with different types of dissipation. In this section, we confirm numerically this theoretical result. More precisely, we will prove that the damped Timoshenko system with a linear damping term considered in one equation causes the exponential stability which depends on the parameter µ. Let us consider the following linearly damped Timoshenko system (3.40), where µ represents the damping coefficient. We will design a numerical scheme with finite element methods taking advantage of the discrete formulation carried out for the undamped case (3.5). As we did for the undamped system (3.5), we present here a matrix formulation for the semi-discrete linearly damped problem corresponding to (3.40) which reads as follows:

         M d 2 Φ dt 2 = -KΦ + SΨ, M d 2 Ψ dt 2 = -KΨ -SΦ -M Ψ -µM dΦ dt .
(3.41)

Second, we consider the finite difference scheme applied to (3.41) and we end up with the following formulation: Find (Φ n , Ψ n ) such that

           M Φ n+1 -2Φ n + Φ n-1 ∆t 2 = -KΦ n + SΨ n , M Ψ n+1 -2Ψ n + Ψ n-1 ∆t 2 = -KΨ n -SΦ n -M Ψ n -µM Ψ n+1 -Ψ n-1 2∆t .
(3.42)

Now, we give the following theorem which proves that the total discrete energy of the system (3.42) is decreasing along time.

Theorem 3.2. Let ∆t > 0, ∆x > 0. Then, the discrete energy E n+ 1 2 , associated with the solutions of the discrete equations (3.42) defined by (3.18) verifies

E n+ 1 2 -E n-1 2 2∆t = -µ. Ψ n+1 -Ψ n-1 2∆t 2 M , ∀ n. (3.43)
Proof.

We multiply the first equation of the system (3.42) by Φ n+1 -Φ n-1 2∆t and the second one by we assume that g(ψ t ) = |ψ t |ψ t , then, the system (3.47) can be rewritten as follows:

         M d 2 Φ dt 2 = -KΦ + SΨ, M d 2 Ψ dt 2 = -KΨ -SΦ -M Ψ -M |V |V, (3.48)
where, V = dΨ dt and we approximate the nonlinear damping as

V (t)|V (t)| = Nx i=0
|v i (t)|v i (t)w i (x).

(3.49)

• Case 2: g(ψ t ) = exp(-(ψ t (t, x)) -2 ). This example is one of others that has been taken to illustrate the optimal energy decay rate. More precisely, for this feedback, lower and upper energy estimates have been obtained, see [START_REF] Alabau-Boussouira | Strong lower energy estimates for nonlinearly damped Timosheko beams and Petrowsky equation[END_REF] and [START_REF] Mustafa | General energy decay rates for a weakly damped Timoshenko system[END_REF]. Here, the aim is to present some numerical results completing thus the theoretical results already established. We use the finite element method as follows. First, we have

ψ t = Nx i=1 ψ i w i (x).
Then, the nonlinear damping term can be approximated as

g(ψ t ) Nx i=1
g(ψ i ) w i (x).

the system (3.47) can be rewritten as follows:

         M d 2 Φ dt 2 = -KΦ + SΨ, M d 2 Ψ dt 2 = -KΨ -SΦ -M Ψ -M g( dΨ dt ), (3.50)
we assume that V n = Ψ n+1 -Ψ n-1 2∆t and then, g(V n ) = (g(ψ 0 ), • • • , g(ψ N x )).

Numerical experiments

We consider the following initial conditions ϕ 0 (x) = ψ 0 (x) = x(1 -x), ψ 1 (x) = ϕ 1 (x) = 0, the final time is T = 500, L = 1, ∆t ∆x < 1. We note that the study of the undamped case is just a first step in our numerical approach but it will allow us to reach later on the understanding of the damped case. Thus, it would be interesting to compare and show the difference between the undamped case and the damped one. In this figure the energy is represented by a straight line, it keeps a constant value which is equal to the initial discrete energy E 1/2 = 0.75. Then, from this figure we can see that the discrete Timoshenko system is purely conservative. In addition, we see in Figure 3.5 that E n -E n-1 is equal to zero at almost every time t n and this observation is conformed by the result as in Proposition 3.1. In conclusion, thanks to For final time T = 500. The energy E n+ 1 2 decreases in time and it seems that it does not tend to zeros when µ = 0.5 as shown in Figure 3 Comment 4. Figure 3.8 shows that the solution is small but not enough to produce the total damping of the energy. This interpretation is observed in Figure 3.9.

For final time T = 500. The energy E n+ 1 2 decreases in time and it seems that it does not tend to zeros when µ = 1 as shown in Figure 3.9. For final time T = 500. The energy E n+ 1 2 decreases in time and it seems that it does not tend to zeros when µ = 2 as shown in Figure 3.11. (3.51)

Then there is a constant C ≥ 0, known as the Korn constant of Ω, such that, for all v ∈ H 1 (Ω), where, e denotes the symmetrized gradient given by

e ij v = 1 2 ∂ i v j + ∂ j v i . (3.53)
An important property of dissipative systems is the fact that trajectories decay along a Lyapunov function. If the trajectories are relatively compact in some appropriate spaces, then one can prove that the trajectories asymptotically converge to equilibria . Deformos and Lasalle used this property to study convergence of solutions toward equilibrium and to prove strong stability. Here, we recall these important results as a preliminary.

The ω-limit set Definition 3.1. [START_REF] Dafermos | Asymptotic behavior of solutions of evolution equations[END_REF] Let (T (t)) t≥0 be a continuous semi-group on a Banach X. We recall that the ω-limit set of z 0 , in X, denoted by ω(z 0 ), is defined by ω(z 0 ) = {z ∈ X, ∃(t n ) n ⊂ [0, ∞) such that t n → ∞, and z = lim n→∞ T (t n )z 0 }.

(3.54)

We recall the following basic result on the ω-limit sets.

Proposition 3.2. Let (T (t)) t≥0 be a continuous semi-group on a Banach space X and z 0 ∈ X be given such that the orbit γ(z 0 ) ≡ ∪ t≥0 T (t)z 0 is relatively compact in X. Then d(T (t)z 0 , ω(z 0 )) → as t → ∞, and the ω-limit set of z 0 is a non empty compact, connected subset of X and it is invariant under the action of the semi-group.

A proof is given in [START_REF] Dafermos | Asymptotic behavior of solutions of evolution equations[END_REF](see also [START_REF] Haraux | Systémes dynamiques dissipatifs et applications[END_REF])

Figure 1 :

 1 Figure 1: Timoshenko Beam Theory, Beam, Eulerbernoulli Beam Theory.

Figure 2 :

 2 Figure 2: Deformation of a plate high lighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue).

Figure 3 :

 3 Figure 3: Displacement of the mid-surface (left) and of a normal (right).

. 29 ) 1 . 5 .

 2915 RemarkThe result of Theorem 1.3 holds true without any assumption on the wave speeds corresponding to the first two equations in (1.1).

1

 1

1

 1

Lemma 1 . 4 .

 14 [START_REF] Racke | Fernàndez Sare: On the stability of damped Timoshenko systems-Cattaneo versus Fourier law[END_REF] Lemma 2.4]) Let H be a given strictly convex C 1 -function from [0, r 2 0

63 )≤ 59 Using ( 2 .

 63592 max(c 4 , c 5 ) E(S), ∀ 0 ≤ S ≤ T. 63) and (2.63), we get T S

Figure 3 . 1 :Comment 1 .

 311 Figure 3.1: The initial conditions ϕ 0 and ψ 0 .

Figure 3 . 2 :Comment 2 .

 322 Figure 3.2: The Undamped solution ϕ(t, x), ψ(t, x).

Figure 3 . 3 :

 33 Figure 3.3: Decay rate of the energy for ∆t = 10 -1 ∆x = 2 * ∆x , T = 500

Figure 3 . 4 :Comment 3 .

 343 Figure 3.4: The undamped case: the conservative property of the discrete energy E n

Figure 3 . 5 :

 35 Figure 3.5: The undamped case: the conservative property of the discrete energy E n

Figure 3 .Comment 4 .

 34 3 and Figure3.5 we have the conservative character of the discrete energy E n which is in agreement with the theoretical results. Figure3.6 shows that the solution is small but not enough to produce the total damping of the energy. This interpretation is observed in Figure3.7.

  .

7

 7

Figure 3 . 7 :

 37 Figure 3.7: Final time T=500, the energy tend to 0.2.

Figure 3 . 9 :

 39 Figure 3.9: Final time T=500, the energy tend to 0.1.

Figure 3 . 11 :Comment 6 .

 3116 Figure 3.11: Final time T=500, the energy tend to almost zeros.

Figure 3 . 12 :

 312 Figure 3.12: Final time T = 5.10 4 , the behavior of the linear damped solution ϕ(t, x), ψ(t, x), for g(x) = µ.x.

Figure 3 . 13 :

 313 Figure 3.13: The behavior of the discrete energy, for T = 5.10 4 .

  |v i (x)| 2 + |e ijv (x)| 2 dx, (3.52)

  2Dµψ x ϕ y ] dxdy.(2.12) To show that (2.12) is a decreasing function of the time t, we have the following inequality which obtained is by multiplying the equation (2.1) by w t , (2.2) by ψ t and (2.3) by ϕ t (for more details on the proof in the case of linear damped Reissner-Mindlin-Timoshenko (see Proposition 2.3[START_REF] Campelo | Stability to the dissipative Reissner-Mindlin-Timoshenko acting on displacement equation[END_REF]).

The dissipation relation corresponding to the nonlinear damping is given Proposition 2.1. Let U = (w, w t , ψ, ψ t , ϕ, ϕ t ) be the solution of (2.1)-(2.3). Then, the energy satisfies the following relation

  1 5 χ 2 (ϕ t (t, x, y))ϕ t (t, x, y)dxdy ∈ [0,

	r 2 0 2	].	(2.61)
	Therefore, from the above inequalities (2.57) and (2.56), we deduce that

T S

Hence, the set {w t (t, •), t ≥ 0} is relatively compact in L 2 (Ω).

Next, using the fact that E is bounded uniformly on R + , we deduce that the sets {ψ tt (t, •), t ≥ 0} and {ψ t (t, •), t ≥ 0} are bounded respectively in L 2 (Ω) and in H 1 0 (Ω).

Then, the set {ψ t (t, •), t ≥ 0} is relatively compact in L 2 (Ω).

In addition, {ϕ tt (t, •), t ≥ 0} and {ϕ t (t, •), t ≥ 0} are bounded, respectively, in L 2 (Ω) and in H 1 0 (Ω). Hence, the set {ϕ t (t, •), t ≥ 0} is relatively compact in L 2 (Ω).

On the other hand, using (2.1), we deduce that {∆w(t, •), t ≥ 0} is bounded in L 2 (Ω), using (2.2) and (2.3) we have,

Then, {∆ψ(t, ), t ≥ 0} and {∆ϕ(t, ), t ≥ 0} are bounded in L 2 (Ω). Therefore, the sets {ψ(t, ), t ≥ 0}, {w(t, ), t ≥ 0} and {ϕ(t, ), t ≥ 0} are relatively compact in H 1 0 (Ω). We infer that the orbit Υ(U 0 ) is relatively compact in H Now, we denote by ω(U 0 ) the ω-limit set of U 0 and we consider Z 0 ∈ ω(U 0 ) such that

Then, we formulate the stability result for the energy of (2.15) in the following theorem. Theorem 2.2. Assume the hypotheses of the Theorem (2.1). We have in addition that Υ(U 0 ) satisfies the result of Lemma (2.1).Then for all U 0 ∈ H, the energy (2.15) satisfies

where, σ is a constant given by (2.79) and

Hence, by applying Theorem 2.3 [START_REF] Alabau-Boussouira | Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control[END_REF], we deduce that E satisfies the desired estimate, which concludes the proof.

Semi-discrete finite element scheme

Let N x ∈ N and ∆x = 1 Nx such that the mesh

. where (w i (x)) i is the basis of the shape functions w j (x i ) with compact support in each interval [x j-1 , x j+1 ] which are defined as

Now, we write the semi-discrete problem as follows.

Find the functions ϕ h and ψ h such that

and

The discrete boundary conditions read

Then, for Φ(t) = [ϕ h (x 0 , t), . . . , ϕ h (x Nx , t)] t , and

we have the following matrix formulation for the semi-discrete problem

where M is the mass matrix with M i,j = (w i , w j ), K is the rigidity matrix with K i,j = (w i , w j ) and S is the matrix defined by S i,j = (w i , w j ) (note that S is a skew symmetric matrix).

Proof. This problem reads in terms of Φ and Ψ as:

Now, we will use the energy technique, multiplying (3.20) by Φ n+1 -Φ n-1 2∆t and (3.21) by

, we obtain

and

We can easily see that the equations (3.22) and (3.23) are written as follows

and

Using (3.24) + (3.25) = 0, we have,

Taking into account that K, M are symmetric definite positive matrices and S is skew symmetric matrix and the definition of E n+ 1 2 given by (3.18). We have

Therefore, E n+ 1 2 is a conserved along the discrete flow quantity.

The stability of the discrete problem (CFL)

Our concern in this section is to show that the scheme is stable if the discrete energy is positive. To obtain our stability condition we start by giving the following lemma.

, we obtain

and

The use of the definition of the total discrete energy(3.18) and the two equations (3.44) and (3.45) leads to:

Consequently, the total discrete energy of the system (3.41) is decreasing along time.

More general cases of damping

We consider the nonlinear damped Timoshenko system

• Case 1: g(ψ t ) = |ψ t |ψ t . Comment 7. We have plotted the graph of ln(E(t)) as a function of t for each µ as shown Figure 3.14. Thus, we can precise the decay rate of the discrete energy and based on the results obtained in Figure 3.14, we can write ln(E(t)) = at + b. This confirms that the discrete energy decays like an exponential function in the full damping case. Hence, the discrete solution of the Timoshenko system is exponentially stable.

Conclusions

In this thesis, we studied a nonlinear Timoshenko system with thermoelasticity with second sound as given in system (1.1). For this system we proved the strong stability using the Defermos' Technique which is based on Lasalle invariance principle [START_REF] Alabau-Boussouira | Strong lower energy estimates for nonlinearly damped Timosheko beams and Petrowsky equation[END_REF]. Then, we have used the comparison method developed in [START_REF] Alabau-Boussouira | Strong lower energy estimates for nonlinearly damped Timosheko beams and Petrowsky equation[END_REF] to establish the lower estimate of the energy of the one-dimensional damped Timoshenko system with thermoelasticity (1.1). We have gone further and taken examples of nonlinear damping terms and we have shown that the lower bound of the energy yields the optimality of the decay rates for a specific example of the damping function g. Moreover, we have addressed an important problem in mathematical analysis of beam, namely the problem of determining the decay rate of the discrete energy by taking into account a few dissipative mechanisms. As we already know, Timoshenko system (3.5) has two wave speeds and we have proved numerically that is sufficient to consider only one dissipation mechanism in order to obtain the exponential decay, for the case where the speeds are equal. Nevertheless, other dissipative cases have been considered in this thesis, we look for the behavior of the energy when the system is nonlinearly damped and we deduce an explicit (polynomial and logarithmic) decay rate of the discrete energy. One of the interesting feature of this work is the obtaining of the approximate values of the constants (coefficients and monomial degrees) of the decay rate function in time of the discrete energy associated with the Timoshenko systems (3.5) in an explicit manner.

Appendix Basic inequalities

In this appendix we recall some basic inequalities in oder to make the thesis self-contained. Young's inequality: Let 1 < p < +∞ and p such that 1 p + 1 p = 1 (i.e p = p p-1 ). Let a, b ∈ (0, +∞). Then,

How do convex functions act on integrals? That is what Jensen's inequality tells us. Jensen's inequality: Let (Ω, A, µ) be a probability space, such that µ(Ω) = 1. If g is a real-valued function that is µ -integrable, and if ϕ is a convex function on the real line, then:

Poincaré's inequality:

Let Ω be an open, bounded, and connected subset of R d for some d and let dx denotes the d-dimensional Lebesgue measure on R d . In functional analysis, the Poincaré inequality says that there exist constants C 1 and C 2 such that

for all functions g in the Sobolev space H 1 (Ω).

Korn's inequality

Let Ω be an open, connected domain in n-dimensional Euclidean space R n , n ≥ 2.

Let H 1 (Ω) be the Sobolev space of all vector fields v = (v 1 , • • • , v n ) on Ω that, along with their (first) weak derivatives, lie in the Lebesgue space L 2 (Ω). Denoting the partial derivative with respect to the i th component by ∂ i , the norm in H 1 (Ω) is given by

Lasalle invariance principle

The proof of Defermos' strong stability is based on the Lasalle invariance principle [START_REF] Lassalle | The extent of asymptotic stability[END_REF][START_REF] Lasalle | Asymptotic stability criteria[END_REF].

Proposition 3.3. Let (T (t)) t≥0 be a continuous semi-group on a Banach space X such that the orbit γ(z 0 ) ≡ ∪ t≥0 T (t)z 0 of a point z 0 ∈ X is relatively compact in X. Assume that V : X → R is continuous and is a Lyapunov function for (T (t)) t≥0 , that is V (T (t)z) ≤ V (z) for all z ∈ X and all t ≥ 0. Then V is constant on ω-limit set of z 0 .

Proof. Since (T (t)) satisfies the semigroup property, V is a Lyapunov function and the orbit of z 0 is relatively compact in X, we deduce that t -→ V (T (t)z 0 ) is a non-increasing bounded function. Hence there exists V ∞ ∈ X such that V ∞ = lim t→∞ V (T (t)z 0 ). Let z ∈ w(z 0 ) be given. Then there exists (t n ) n → ∞, such that z = lim n→∞ T (t n )z 0 . Since V is continuous on X, we have V (z) = lim n→∞ V (T (t n )z 0 ) = V ∞ . Hence, V is constant on the w-limit set of z 0 .

Kondrachov embedding theorem

Let Ω ⊆ R n be an open, bounded Lipschitz domain, and let 1 ≤ p < n. Set

Then, the Sobolev space W 1,p (Ω; R) is continuously embedded in the L p space L p (Ω; R) and is compactly embedded in L q (Ω; R) for every 1 ≤ q < p . In symbols, W 1,p (Ω) → L p (Ω), and W 1,p (Ω) ⊂⊂ L q (Ω), for 1 ≤ q < p . On a compact manifold with C 1 boundary, the Kondrachov embedding theorem states that if k > l and k -n/p > l -n/q then the Sobolev embedding W k,p (M ) ⊂ W l,q (M ), is completely continuous.