Mohamad Sammoury

Ali

Directeur De

Serge Directeur

Ali M Wehbe

Badia Al

Abdellatif

M Fino

Professeur Messieurs Ibrahim Hassan

Maître M Mehrenberger Michel

Invité Président

Examinateur M Mercier

M Nicaise

M T Tebou

Professeur Louis Roder

M Wehbe

Hassan Kamel

Sammoury Né

A L Badia

Tcheugoue Tebou D'

Merci Aux

Ibrahim Michel

E L Amin

Hassan Sahili

Bassam Abaass

Ibrahim Kojok

Hassan Za- Lzali

Ayman Ibrahim

Ayman Mourad Kachmar

Mariam En Particulier

Chiraz Koubeissy

Marwa Kassem

Mohamad Koumaiha

Mohamad Akil

Houssein Gh- Ader

E L Nasser

Bilal A L Dine

Kamel Taki

Abed Attar

Chikh Alwaheb

Mohamad Salah

Fatiha Merabet

Sadjia El Bekkou- Che

Maya Ariche

Zeinab Abbas Bassam

J'ai

« Etude théorique et numérique de la stabilité de certains systèmes distribués avec contrôle frontière de type dynamique »

Keywords: Dynamic boundary control, non exponential stability, polynomial stability, optimality, spectral analysis, frequency domain method, Riesz basis, multiplier method, observability inequality, asymptotic behavior, transfer function, semi-discretization, viscosity terms decay . . . . . . . . . . . 148 5.3 Uniform polynomial energy decay rate . . . . . . . . . . 155 5.4 Convergence results: proof of Theorem 5.1.7 . . . .

This thesis is devoted to the study of the stabilization of some distributed systems with dynamic boundary control. First, we consider the stabilization of the Rayleigh beam equation with only one dynamic boundary control moment or force. We show that the system is not uniformly (exponentially) stable. However, using a spectral method, we establish the optimal polynomial decay rate of the energy of the system. Next, we study the indirect stability of the wave equation with a fractional dynamic boundary control. We show that the decay rate of the energy depends on the nature of the geometry of the domain. Using a frequency approach and a spectral method, we show the non exponential stability of the system and we establish, different polynomial stability results. Finally, we consider the finite difference space discretization of the 1-d wave equation with dynamic boundary control. First, using a spectral approach, we show that the polynomial decay of the discretized energy is not uniform with respect to the mesh size, as the energy of the continuous system. Next, we introduce a viscosity term and we establish the uniform (with respect to the mesh size) polynomial energy decay of our discrete scheme.

Etude théorique et numérique de la stabilité de certains systèmes distribués avec contrôle frontière de type dynamique

Résumé

Cette thèse est consacrée à l'étude de la stabilisation de certains systèmes distribués avec contrôle frontière de type dynamique. Nous considérons, d'abord, la stabilisation de l'équation de la poutre de Rayleigh avec un seul contrôle frontière dynamique moment ou force. Nous montrons que le système n'est pas uniformément (autrement dit exponentiellement) stable; mais par une méthode spectrale, nous établissons le taux polynomial optimal de décroissance de l'énergie du système. Ensuite, nous étudions la stabilisation indirecte de l'équation des ondes avec un amortissement frontière de type dynamique fractionnel. Nous montrons que le taux de décroissance de l'énergie dépend de la nature géométrique du domaine. En utilisant la méthode fréquentielle et une méthode spectrale, nous montrons la non stabilité exponentielle et nous établissons, plusieurs résultats de stabilité polynomiale. Enfin, nous considérons l'approximation de l'équation des ondes mono-dimensionnelle avec un seul amortissement frontière de type dynamique par un schéma de différence finie. Par une méthode spectrale, nous montrons que l'énergie discrétisée ne décroit pas uniformément (par rapport au pas du maillage) polynomialement vers zéro comme l'énergie du système continu. Nous introduisons, alors, un terme de viscosité numérique et nous montrons la décroissance polynomiale uniforme de l'énergie de notre schéma discret avec ce terme de viscosité.

Mots-clés

Contrôle frontière dynamique, non stabilité exponentielle, stabilité polynomiale, optimalité, étude spectrale, méthode fréquentielle, base de Riesz, méthode des multiplicateurs, inegalité d'observabilité, comportement asymptotique, fonction de transfère, semi discrétisation, terme de viscosité.

Finalement, je veux remercier le centre islamique d'orientation et de l'enseignement supérieure representée par son directeur Monsieur Ali ZALZALI, Monsieur Ali SAMMOURY, Monsieur Kamal SAMMOURY et Madame Rafika SAMMOURY RAHHAL d'avoir financer mon projet doctoral durant ces trois années.

Introduction

Cette thèse est consacrée à l'étude de la stabilisation de certains systèmes distribués avec un contrôle frontière de type dynamique. La notion de contrôle dynamique ainsi que le contrôle indirect ont été introduite par Russell dans [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF] et depuis lors, elle a attiré l'attention de beaucoup d'auteurs. En particulier, voir [START_REF] Abbas | The multidimensional wave equation with generalized acoustic boundary conditions ii: Polynomial stability[END_REF][START_REF] Abdallah | Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications[END_REF][START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF]6,7,[START_REF] Bassam | Optimal energy decay rate of Rayleigh beam equation with only one boundary control force[END_REF][START_REF] Rao | Stability and controllability of a wave equation with dynamical boundary control[END_REF]81,[START_REF] Wehbe | Rational energy decay rate for a wave equation with dynamical control[END_REF][START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF].

La thèse est divisée en trois parties. Dans la première partie, nous considérons la stabilisation de l'équation de la poutre de Rayleigh avec un seul contrôle frontière dynamique moment ou force. D'abord, en utilisant la théorie de la décomposition spectrale, nous montrons que le système est fortement stable et par une méthode de perturbation compacte de Russell, nous prouvons que la décroissance de l'énergie du système vers zéro n'est pas exponentielle. Nous passons alors à une décroissance de type polynomiale. Dans le cas d'un seul contrôle frontière dynamique moment, nous faisons une étude spectrale très fine du système non amorti qui nous conduit à un résultat d'observabilité. Ensuite, nous appliquons une méthodologie introduite dans [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF] et nous établissons le taux optimal de décroissance polynomiale de l'énergie de type 1 t . Dans le cas d'un seul contrôle frontière dynamique force, nous donnons le développement asymptotique des valeurs propres et des fonctions propres des systèmes amorti et non amorti. Nous montrons après que le système de vecteurs propres du problème amorti forme une base de Riesz. Finalement, en appliquant une méthode introduite dans [START_REF] Loreti | Optimal energy decay rate for partially damped systems by spectral compensation[END_REF], nous établissons le taux optimal de décroissance polynomiale de l'énergie de type 1 √ t .

La deuxième partie est consacrée à l'étude de la stabilisation indirecte de l'équation des ondes avec un amortissement frontière de type dynamique fractionnel dans un domaine borné de R N , N ≥ 2. D'abord, en utilisant un critère général d'Arendt et Batty dans [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF], nous montrons la stabilité forte du système. Ensuite, nous prouvons que le système n'est pas exponentiellement stable dans le cas où le domaine est un disque de R 2 . Alors, nous cherchons à établir une décroissance de l'énergie de type polynomiale pour des données initiales régulières en employant une méthode fréquentielle combinée avec une méthode de multiplicateur par morceaux. Nous constatons alors que le taux de décroissance polynomiale de l'énergie depend de la nature géométrique du domaine. Plus précisément, si le domaine est Lipschitzien et vérifie la condition d'optique géométrique, nous établissons un taux de type 1 . De plus, si le domaine est presque étoilé et de classe C 1,1 , nous établissons un taux de type 1 t et nous conjecturons que ce taux est optimal. Plus tard, nous nous intéressons à démontrer qu'une telle décroissance polynomiale semble être aussi établie même si les conditions géométriques précédentes ne sont pas satisfaites. Pour cela, nous considérons le système dans un carré de R 2 . Nous montrons d'abord que l'énergie ne décroit pas exponentiellement vers zéro. Finalement, en appliquant une méthode basée sur l'analyse de Fourier, une inégalité d'Ingham et une méthode d'interpolation, nous établissons un taux de décroissance polynomiale de l'énergie de type 1 t pour des donnés initiales assez régulières. Nous conjecturons que ce taux de décroissance est optimal.

Dans la troisième partie, nous passons à un autre sujet qui traite la stabilisation de l'approximation de l'équation des ondes mono-dimensionnelle avec un seul amortissement frontière de type dynamique par un schéma de différence finie. Premièrement, nous montrons que l'énergie discrétisée ne décroit pas uniformément (par rapport au pas du maillage) polynomialement vers zéro comme celle du système continu. Deuxièment, nous introduisons un terme du viscosité numérique dans le schéma d'approximation qui nous conduit à une décroissance uniforme (par rapport au pas du maillage) polynomiale de l'énergie comme celle du système continu. Finalement, nous montrons la convergence du schéma discrétisé vers l'équation des ondes initiale.

Aperçu de la thèse

Cette thèse est divisée en cinq chapitres. Dans le premier chapitre, nous rappelons quelques définitions et théorèmes concernant la théorie de semigroupe et l'analyse spectrale. Ainsi, nous présentons et discutons les méthodes utilisées dans cette thèse pour obténir notre résultats de la stabilité.

Le deuxième chapitre est consacré à la stabilisation de l'équation de la poutre de Rayleigh amortie par un seul contrôle frontière dynamique moment: (1, t) + αη(t) = 0, t > 0, (0.0.1) où γ est le coefficient de moment d'inertie et α > 0 est le coefficient de contrôle dynamique moment. L'amortissement est appliqué indirectement, via une équation différentielle ordinaire en η, à l'extrémité droite de la poutre. Ce type de contrôle indirect à été introduit par Russell dans [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF] et depuis lors, il a retenu l'attention de plusieurs auteurs. Dans le cas d'un amortissement statique, quand η(t) est constante, la stabilisation du système (0.0.1) a été largement étudiée par des approches différentes (voir [START_REF] Lagnese | Boundary stabilization of thin plates[END_REF][START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF]). Cependant, dans le cas des contrôles dynamiques, Wehbe dans [START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF], a considéré l'équation de la poutre de Rayleigh avec deux contrôles dynamiques frontières. D'abord, par une méthode de perturbation compacte, il a prouvé que l'équation de la poutre de Rayleigh n'est pas uniformément stable. Ensuite, par une méthode spectrale, il a établi le taux de décroissance optimal de l'énergie pour des données initiales régulières. Le fait de la présence de deux contrôles frontière dynamique ensemble dans la démonstration, montre que le cas général, quand la poutre de Rayleigh est amortie par un seul contrôle dynamique frontière, reste un problème ouvert. Alors, dans ce chapitre, nous considérons l'équation de poutre de Rayleigh avec un seul contrôle frontière dynamique moment. D'abord, nous écrivons le système (0.0.1) sous forme d'une équation d'évolution du premier ordre

             y tt -γy xxtt + y xxxx = 0, 0 < x < 1, t > 0, y(0, t) = y x (0, t) = 0, t > 0, y xx (1, t) + η(t) = 0, t > 0, y xxx (1, t) -γy xtt (1, t) = 0, t > 0, η t (t) -y xt
U t (t) + A α U (t) = 0, t > 0, U (0) = U 0 ∈ H, (0.0.2)
où U = (y, y t , η), H est un espace de Hilbert convenable, A α = A 0 + αB, A 0 est un opérateur non borné maximal monotone du domaine D(A 0 ) ⊂ H et B est un opérateur borné monotone. Nous montrons après que le problème (0.0.2) est fortement stable dans l'espace de Hilbert H en utilisant la théorie de décomposition spectrale, et par une méthode de perturbation compacte de Russell, nous prouvons que la décroissance de l'énergie E du problème (0.0.2) vers zéro n'est pas exponentielle. Alors, une décroissance de type polynomiale est espérée. Pour cela, nous faisons une étude très fine du spectre σ(A α ) de l'opérateur A α pour α ≥ 0. Plus précisément, nous montrons que pour α ≥ 0, il existe k α ∈ N * suffisamment large tel que le spectre σ(A α ) de l'opérateur A α est donné comme suit:

σ(A α ) = σ α,0 ∪ σ α,1 , avec σ α,0 = {κ α,j } j∈J , σ α,1 = {λ α,k } k∈Z |k|≥k 0
, σ α,0 ∩ σ α,1 = ∅ et J est un ensemble fini. De plus, λ α,k est simple et elle satisfait le développement asymptotique suivant:

λ α,k = i kπ √ γ + π 2 √ γ + D k + E k 2 + α π 2 k 2 + o( 1 k 2 ), avec D = 2γ -1 -2 √ γ tanh(γ -1 2 ) 2γ 3 2 π et E = 2(-1) k γ 3 2 cosh(γ -1 2 )π 2 + 4γ -2 - √ γ tanh(γ - 1
2 ) 2γ

H : C + = {λ ∈ C; (λ) > 0} -→ L(H), H(λ) = -αB * (λ + A 0 ) -1 B
est bornée dans un sous espace de C + . En combinant l'inégalité d'observabilité (0.0.3) et la bornitude de la fonction de transfert H (voir [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF]), nous déduisons qu'il existe une constante c > 0 telle que pour toute donnée initiale U 0 ∈ D(A 0 ), l'énergie E associée au problème amorti (0.0.2) satisfait:

E(t) ≤ c 1 + t U 0 2 D(A 0 ) , ∀t > 0. (0.0.4)
Finalement, en utilisant l'étude spectrale précédente de σ(A α ) et un théorème de Borichev et Tomilov dans [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], nous montrons que le taux de décroissance obtenu dans (0.0.4) est optimal dans le sens que pour tout > 0, la décroissance de l'énergie E ne peut pas atteindre un taux de type 1 t 1+ .

Dans le troisième chapitre, nous continuons l'étude éffectuée dans le deuxième chapitre en considérant l'équation de la poutre de Rayleigh amortie par un seul contrôle frontière dynamique force:

        
y tt -γy xxtt + y xxxx = 0, 0 < x < 1, t > 0, y(0, t) = y x (0, t) = y xx (1, t) = 0, t > 0, y xxx (1, t) -γy xtt (1, t) -ξ(t) = 0, t > 0, ξ t (t) -y t (1, t) + βξ(t) = 0, t > 0, (0.0.5) où γ est le coefficient de moment d'inertie et β > 0 est le coefficient de contrôle dynamique force. L'amortissement est appliqué indirectement, via une équation différentielle ordinaire en ξ, à l'extrémité droite de la poutre. D'abord, nous commençons par la formulation du système (0.0.5) sous forme d'une équation d'évolution du premier ordre U t (t) + Ãβ U (t) = 0, t > 0, U (0) = U 0 ∈ H, (0.0.6) où U = (y, y t , ξ), H est un espace de Hilbert convenable, Ãβ = Ã0 + β B, Ã0 est un opérateur non borné maximal monotone du domaine D( Ã0 ) ⊂ H et B est un opérateur borné monotone. Ensuite, en appliquant la théorie de décomposition spectrale dans [START_REF] Benchimol | A note on weak stabilizability of contraction semigroups[END_REF], nous montrons, comme dans [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF], que la poutre de Rayleigh est fortement stable pour toute donnée initiale si et seulement si γ > γ 0 où γ 0 est la solution de l'équation √ γ 0 sinh -1 ( √ γ 0 π) = 1. En utilisant Mathematica, nous obtenons l'estimation γ 0 0.45001246517627713. Nous savons que la poutre de Rayleigh n'est pas exponentiellement stable ni avec un seul contrôle force direct (voir [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF]) ni avec deux contrôles dynamiques (voir [START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF]). Nous nous intéressons donc à la décroissance polynomiale optimale de l'énergie du système pour des données initiales régulières dans D( Ã0 ). Pour cela, en utilisant une approximation explicite, nous donnons d'abord le développement asymptotique des valeurs propres et des fonctions propres des systèmes amorti et non amorti. Ensuite, nous appliquons le théorème 1.2.10 dans [START_REF] Abdallah | Stabilisation et approximation de certains systèmes distribués par amortissement dissipative et de signe indéfini[END_REF] (qui est une version modifiée du théorème de Bari voir [43, Chaptire 6, théorème 2.3]) et nous prouvons que les vecteurs propres normalisés U k du système amorti forment une base de Riesz dans H. Plus précisément, nous démontrons que ces vecteurs propres sont quadratiquement liés avec les vecteurs propres normalisés U 0 k de l'opérateur Ã0 par l'inégalité suivante:

+∞ k=max{k 0 ,k β } || U k -U 0 k || H < +∞.
Finalement, en appliquant le théorème 2.4 donné dans [START_REF] Loreti | Optimal energy decay rate for partially damped systems by spectral compensation[END_REF], nous déduisons qu'il existe une constante c > 0 telle que pour toute donnée initiale U 0 ∈ D( Ã0 ), l'énergie E associée au problème (0.0.6) satisfait

E(t) ≤ c √ t U 0 2 D( A 0 )
(0.0.7) et le taux obtenu ci-dessus est optimal dans le sens que pour tout > 0, la décroissance de l'énergie E ne peut pas atteindre un taux

1 t 1 2 + .
Le quatrième chapitre est consacré à l'étude de la stabilité de l'équation des ondes avec un amortissement frontière de type dynamique fractionnel. D'abord, soit Ω un domaine borné dans R d , d ≥ 2, avec frontière lipschitzienne Γ = Γ 0 ∪ Γ 1 ; Γ 0 et Γ 1 sont deux sous ensembles de Γ tel que Γ 0 ∩ Γ 1 = ∅ et Γ 1 = ∅. Dans [START_REF] Fourrier | Analysis of existence, regularity and stability of solutions to wave equations with dynamic boundary conditions[END_REF][START_REF] Fourrier | Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions[END_REF][START_REF] Graber | Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions[END_REF], N. Fourier, I. Lasiecka et P. Graber ont étudié la stabilité du problème suivant (avec Γ 0 ∩ Γ 1 = ∅):

                       u tt -∆u -k Ω ∆u t + c Ω u t = 0, in Ω × R * + , u = 0, on Γ 0 × R * + , u -w = 0, on Γ 1 × R * + , w tt -k Γ ∆ T (αw t + w) + ∂ ν (u + k Ω u t ) + c Γ w t = 0, in Γ 1 × R * + , w = 0, on ∂Γ 1 × R * + , u(•, •, 0) = u 0 , u t (•, •, 0) = u 1 , in Ω, w(•, 0) = w 0 , w t (•, 0) = w 1 , on Γ 1 ,
où ∂ ν désigne la dérivée normale sur Γ 1 , ν est le vecteur unitaire normal dérigé vers l'extérieur de la frontière et ∆ T représente l'opérateur de Laplace-Beltrami sur Γ. Dans le système précédent, deux types de dissipation apparaissent: interne (si c Ω > 0) et frontière (si k Γ > 0), interne de type fractionnel (si k Ω > 0) et frontière de type fractionnel visco-élastique (si k Γ α > 0). La première description physique de ce modèle est donnée dans [START_REF] Morse | Theoretical Acoustics. International series in pure and applied physics[END_REF]. Dans [START_REF] Fourrier | Analysis of existence, regularity and stability of solutions to wave equations with dynamic boundary conditions[END_REF][START_REF] Fourrier | Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions[END_REF], Fourier et Lasiecka ont démontré que le système précédent est exponentiellement stable si un de ces trois conditions suivants est satisfait: si k Ω > 0 (un amortissement interne de type visco-élastique), ou c Ω > 0 et c Γ > 0 (deux amortisse-ment interne et frontière de type fractionnel) ou c Ω > 0 et k Γ α > 0 (un amortissement interne de type fractionnel et un amortissement frontière de type visco-élastique). Le premier cas correspond à un amortissement direct, tandis que les autres cas correspondent à un phénomène d'un suramortissement. Alors, le cas d'un seul amortissement frontière de type dynamique fractionnel reste un problème ouvert. Dans ce chapitre, nous nous intéressons à ce cas, i.e. lorsque k

Ω = c Ω = α = 0 et k Γ = c Γ = 1.
Plus précisément, nous considérons le système suivant:

                       u tt -∆u = 0, in Ω × R * + , u = 0, on Γ 0 × R * + , u -w = 0, on Γ 1 × R * + , w tt -∆ T w + ∂ ν u + w t = 0, in Γ 1 × R * + , w = 0, on ∂Γ 1 × R * + , u(•, 0) = u 0 , u t (•, 0) = u 1 ,
in Ω, w(•, 0) = w 0 , w t (•, 0) = w 1 , in Γ 1 .

(0.0.8)

Nous montrons que la stabilité du système (0.0.8) dépend de la nature géométrique du domaine Ω. Nous commençons par la formulation du système (0.0.8) sous la forme d'une équation d'évolution du premier ordre. Si Γ 0 = ∅, nous définissons d'abord l'espace

H 1 Γ 0 (Ω) = u ∈ H 1 (Ω); u = 0 sur Γ 0
et nous introduisons l'espace de Hilbert H H = {(u, v, w, z) ∈ H1 Γ 0 (Ω) × L2 (Ω) × H 1 0 (Γ 1 ) × L 2 (Γ 1 ) : γu = w sur Γ 1 }, où γ désigne l'opérateur du trace définit de H 1 (Ω) dans H 1 2 (Γ), muni du produit scalaire (u 1 , v 1 , w 1 , z 1 ), (u 2 , v 2 , w 2 , z 2 )

H =(∇u 1 , ∇u 2 ) L 2 (Ω) + (v 1 , v 2 ) L 2 (Ω) + (∇ T w 1 , ∇ T w 2 ) L 2 (Γ 1 ) + (z 1 , z 2 ) L 2 (Γ 1 ) , ∀(u 1 , v 1 , w 1 , z 1 ), (u 2 , v 2 , w 2 , z 2 ) ∈ H 1 Γ 0 (Ω) × L 2 (Ω) × H 1 0 (Γ 1 ) × L 2 (Γ 1 ), et muni de la norme • H = (•, •)
tion (e tA ) t≥0 par

D(A) =          U = (u, v, w, z) ∈ H; ∆ T w -∂ ν u ∈ L 2 (Γ 1 ) v ∈ H 1 Γ 0 (Ω), ∆u ∈ L 2 (Ω), z ∈ H 1 0 (Γ 1 ), γv = z sur Γ 1          , AU =      v ∆u z ∆ T w -∂ ν u -z      , ∀U =      u v w z      ∈ D(A).
Nous écrivons alors notre système (0.0.8) sous la forme U t (t) = AU (t), t > 0, U (0) = U 0 ∈ H.

(0.0.9)

De plus, nous caractérisons le domaine D(A) de l'opérateur A lorsque le bord du domaine Ω est suffisamment régulier ou dans le cas où Ω est le carré unité de R 2 . Ensuite, nous étudions la stabilité forte du problème (0.0.9) en appliquant un théorème d'Arendt et Batty dans [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF]. Nous distinguons deux cas:

• Si Γ 0 = ∅, nous montrons que le C 0 -semigoupe de contraction (e tA ) t≥0 est fortement stable dans l'espace de Hilbert H.

• Si Γ 0 = ∅, nous montrons que le C 0 -semigroupe de contraction (e tA ) t≥0 est fortement stable dans l'espace de Hilbert H 0 définit par H 0 = (u, v, w, z) ∈ H :

Ω vdx + Γ 1 zdΓ + Γ 1 wdΓ = 0 .
En outre, nous montrons que la décroissance de l'énergie E associée au problème (0.0.9) vers zéro n'est pas exponentielle dans le cas général. Pour ce but, nous considérons notre système (0.0.9) dans le disque unité de R 2 avec Γ 0 = ∅. Puis, nous faisons une étude spectrale de l'opérateur A et nous trouvons une famille de valeurs propres qui s'approche de l'axe imaginaire. Nous passons alors à une stabilité de type polynomiale.

En appliquant une méthode fréquentielle (voir [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF]), nous établissons deux taux de décroissances polynomiales. Dans un premier temps, en supposant que la frontière Γ de notre domaine Ω est Lipschitzienne, Γ 0 = ∅, Γ 0 ∩ Γ 1 = ∅ et en utilisant des résultats de la stabilité exponentielle de l'équation des ondes avec l'amortissement

∂y ∂ν = -y t , on Γ 1 × R * + ,
nous établissons un taux de décroissance polynomiale de l'énergie de type . Dans un deuxième temps, en supposant que le domaine Ω est presque étoilé, la frontière Γ de Ω est de classe C 1,1 , et que Γ 0 ∩ Γ 1 = ∅, nous établissons un taux de décroissance polynomiale de l'énergie de type 1 t . Plus tard, nous voulons montrer que la stabilité polynomiale de l'équation des ondes avec un amortissement frontière de type dynamique fractionnel, reste valable même si les conditions géométriques précédentes ne sont pas satisfaits. Dans ce but, nous considérons le problème (0.0.8) dans le carré unité Ω = (0, 1)

2 avec frontière Γ = Γ 0 ∪ Γ 1 , Γ 1 = {(0, y), y ∈ (0, 1)}, Γ 0 = Γ\Γ 1 et Γ 0 ∩ Γ 1 = ∅.
Plus précisément, nous considérons le système suivant:

                       u tt -∆u = 0, in Ω × R * + , u = 0, on Γ 0 × R * + , u = w, on Γ 1 × R * + , w tt -w yy -u x + w t = 0, on Γ 1 × R * + , w(0) = w(1) = 0, on R * + , u(•, •, 0) = u 0 , u t (•, •, 0) = u 1 , in Ω, w(•, 0) = w 0 , w t (•, 0) = w 1 , on Γ 1 .
(0.0.10)

Dans ce cas, nous démontrons d'abord, que l'énergie ne décroit pas exponentiellement vers zéro. Plus précisément, en utilisant la méthode de séparation de variable, nous étudions le spectre de l'opérateur A et nous trouvons une branche des valeurs propres qui s'approche de l'axe imaginaire. Nous montrons qu'il existe k 1 ∈ N * suffisamment large tel que le spectre σ(A) de l'opérateur A est sous la forme:

σ(A) = σ 0 ∪ σ 1 (0.0.11) où σ 0 = {κ l,j } j∈J , σ 1 = {λ l,k } k∈Z |k|≥k 1 , σ 0 ∩ σ 1 = ∅, (0.0.12)
J est un ensemble fini, l ∈ N * et λ l,k est simple et elle satisfait le comportement asymptotique suivant:

λ l,k = i kπ + l 2 π 2k - 1 π 2 k 2 + o( 1 k 2 ).
(0.0.13) Après, par une étude spectrale et une inégalité d'Ingham, nous établissons un taux de décroissance polynomiale de type 1 t de l'énergie du système mono-dimensionnel avec paramètre associé au système (0.0.10) en gérant la constante de la décroissance. Finalement, en utilisant l'analyse de Fourier et le taux de décroissance obtenu dans le cas mono-dimensionnel, nous montrons un taux de décroissance polynomiale de l'énergie de type 1 t pour des données initiales assez régulières.

Dans le cinquième chapitre, nous passons à un autre sujet qui traite la stabilisation de l'approximation de l'équation des ondes mono-dimensionnelle avec un seul amortissement frontière de type dynamique par un schéma de différence finie. Plus précisément, nous considérons l'approximation de système suivant:

                       y (x, t) -y xx (x, t) = 0, (x, t) ∈]0, 1[×R + * , y(0, t) = 0, t ∈ R + * , y x (1, t) + η(t) = 0, t ∈ R + , η (t) -y (1, t) + βη(t) = 0, t ∈ R + * , y(x, 0) = y 0 (x), x ∈]0, 1[, y (x, 0) = y 1 (x), x ∈]0, 1[, η(0) = η 0 , (0.0.14) où (y 0 , y 1 , η 0 ) ∈ H = H 1 L (0, 1) × L 2 (0, 1) × C avec H 1 L (0, 1) = y ∈ H 1 (0, 1); y(0) = 0 ,
β est une constante positive et désigne la dérivée par rapport au temps t. Le système (0.0.14) se présente dans de nombreux domaines de la mécanique et de l'ingénierie. Ce modèle peut être considéré comme un modèle qui décrit la description des vibrations de structures, de la propagation des ondes acoustiques ou sismiques, etc. Dans [START_REF] Wehbe | Rational energy decay rate for a wave equation with dynamical control[END_REF], Wehbe a démontré la décroissance polynomiale de type 1 t de l'énergie E associée au système (0.0.14). Dans ce chapitre, nous voulons tester si l'énergie du schéma discrétisé admet la même propriété uniformément par rapport au pas du maillage. Dans de nombreuses applications (voir [START_REF] Abdallah | Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications[END_REF][START_REF] Ervedoza | Perfectly matched layers in 1-d: energy decay for continuous and semi-discrete waves[END_REF][START_REF] Marica | Boundary stabilization of numerical approximations of the 1-D variable coefficients wave equation: a numerical viscosity approach[END_REF][START_REF] Tcheugoué Tébou | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF][START_REF] Zuazua | Optimal and approximate control of finite-difference approximation schemes for the 1D wave equation[END_REF]), bien que le système continu est exponentiellement ou polynomialement stable, les systèmes discretisés associés n'héritent pas la même propriété uniformément par rapport au pas du maillage. Dans [START_REF] Tcheugoué Tébou | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF], Tebou et Zuazua ont considéré l'approximation par un schéma de différence finie de l'équation des ondes mono-dimensionnelle avec un contrôle frontière de type statique. D'abord, en utilisant une étude spectrale, ils ont démontré que l'énergie du schéma discrétisé ne décroit pas uniformément (par rapport au pas du maillage) exponentiellement vers zéro comme le système continu. Ensuite, en ajoutant un terme de type viscosité numérique dans le schéma discrétisé et en utilisant une méthode basée sur des inégalités d'observabilités, Tebou et Zuazua ont montré la décroissance uniforme (par rapport au pas du maillage) exponentielle de l'énergie vers zéro. Finalement, ils ont démontré la convergence du schéma discrétisé avec le terme viscosité numérique vers l'équation des ondes d'origine. A cause de la présence du terme dynamique, la méth-ode utilisée dans [START_REF] Tcheugoué Tébou | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF] ne fonctionne pas pour notre système. Dans [START_REF] Abdallah | Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications[END_REF], Abdallah et al., ont considéré l'approximation de l'équation d'évolution du deuxième ordre avec un contrôle borné. D'abord, en introduisant un terme visco numérique dans le schema d'approximation, ils ont démontré la décroissance uniforme (par rapport au pas du maillage) exponentielle ou polynomiale de l'énergie vers zéro. Ensuite, ils ont utilisé le théore me de Trotter-Kato dans [START_REF] Ito | The Trotter-Kato theorem and approximation of PDEs[END_REF] pour démontrer la convergence du schéma numériques avec le terme de viscosité vers le problème d'origine. Notons que notre système ne rentre pas dans le cadre de celle de [START_REF] Abdallah | Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications[END_REF]. Alors, l'étude de la stabilité de l'approximation de l'équation des ondes mono-dimensionnelle avec un contrôle fontière de type dynamique reste un problème ouvert.

Dans un premier temps, par une méthode spectrale, nous montrons que l'énergie du système discrétisé ne décroit pas uniformément (par rapport au pas du maillage) polynomialement vers zéro. Ce résultat est une conséquence de l'existence d'une valeur propre à haute fréquence qui ne satisfait pas une condition suffisante pour la décroissance uniforme (par rapport au pas du maillage) polynomiale de l'énergie discrétisée. Plusieurs remèdes ont été proposés pour surmonter cette difficulté comme la régularisation de Tychonoff [START_REF] Glowinski | Errata: "A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods[END_REF][START_REF] Glowinski | Exact and approximate controllability for distributed parameter systems[END_REF][START_REF] Ramdani | Uniformly exponentially stable approximations for a class of second order evolution equations-application to LQR problems[END_REF][START_REF] Tcheugoué Tébou | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF], un algorithme bi-grille [START_REF] Glowinski | Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation[END_REF][START_REF] Negreanu | A 2-grid algorithm for the 1-d wave equation[END_REF], une méthode mixte d'éléments finis [START_REF] Banks | Exponentially stable approximations of weakly damped wave equations[END_REF][START_REF] Castro | Boundary controllability of a linear semidiscrete 1-d wave equation derived from a mixed finite element method[END_REF][START_REF] Castro | Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square[END_REF][START_REF] Glowinski | A mixed finite element formulation for the boundary controllability of the wave equation[END_REF][START_REF] Münch | A uniformly controllable and implicit scheme for the 1-D wave equation[END_REF], ou le filtrage des valeurs propres à hautes fréquences [START_REF] Infante | Boundary observability for the space semi-discretizations of the 1-D wave equation[END_REF]57]. Dans un deuxième temps, comme dans [START_REF] Tcheugoué Tébou | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF], nous ajoutons un terme de viscosité numérique dans le schéma d'approximation, et en utilisant une méthode de multiplicateur inspiré de [START_REF] Wehbe | Rational energy decay rate for a wave equation with dynamical control[END_REF], nous montrons que l'énergie Ẽ discrétisé décroit uniformément (par rapport au pas du maillage) polynomialement vers zéro. Plus précisément, nous montrons qu'il existe une constante M uniformément bornée par rapport au pas du maillage tel que l'énergie Ẽ satisfait

T 0 Ẽ2 (t)dt ≤ M Ẽ2 (0), ∀T > 0.
Dès lors, en appliquant le théorème 9.1 donné dans [START_REF] Komornik | Exact Controllability and Stabilization: The Multiplier Method[END_REF], nous déduisons que l'énergie Ẽ de système discrétisé satisfait

Ẽ(t) ≤ M M + t Ẽ(0), ∀t > 0.
Finalement, nous démontrons la convergence du schéma discrétisé avec le terme de viscosité vers l'équation des ondes d'origine, en appliquant la même stratégie utilisée dans [START_REF] Tcheugoué Tébou | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF].

Preliminaries

Since the analysis in this thesis depends on the semigroup theory, otherworldly investigation hypotheses (keeping in mind the end goal to present the principle topic of our study), let's review a portion of the central definitions and hypotheses in this chapter which will be used to prove our main results in the next chapters. The vast majority of the evolution equations can be reduced to the form

   U t (t) = AU (t), t > 0, U (0) = U 0 ∈ H, (1.0.1)
where A is the infinitesimal generator of a C 0 -semigroup (T A (t)) t≥0 in a Hilbert space H. Therefore, we begin by presenting a few essential ideas concerning the semigroups which involve some results regarding the existence, uniqueness and regularity of the solution of system (1.0.1). Next, we exhibit and talk about many recent results on the strong, exponential and polynomial stability and Riesz basis in several sections. For more details we refer to [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF][START_REF] Benchimol | A note on weak stabilizability of contraction semigroups[END_REF][START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF][START_REF] Foguel | Powers of a contraction in Hilbert space[END_REF][START_REF] Guo | Riesz basis approach to the stabilization of a flexible beam with a tip mass[END_REF][START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF][START_REF] Nagy | Harmonic Analysis of Operators on Hilbert Space[END_REF]74,[START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF][START_REF] Russell | Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods[END_REF].

Semigroups, Existence and uniqueness of solution

We begin this section by the definition of semigroup.

Definition 1.1.1. Let X be a Banach space and let I : X → X its identity operator.

1) A one parameter family (T (t)) t≥0 , of bounded linear operators from X into X is a semigroup of bounded linear operators on X if (i) T (0) = I;

(ii) T (t + s) = T (t)T (s) for every s, t ≥ 0.

2) A semigroup of bounded linear operators, (T (t)) t≥0 , is uniformly continuous if lim t→0 T (t) -I = 0.

3) A semigroup (T (t)) t≥0 of bounded linear operators on X is a strongly continuous semigroup of bounded linear operators or a

C 0 -semigroup if lim t→0 T (t)x = x.

4) The linear operator A defined by

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

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

Next, we recall the definitions of the resolvent and the spectrum of an operator.

Definition 1.1.2. Let A be a linear unbounded operator in a Banach space X.

1) The resolvent set of A denoted by ρ(A) contained all the complex number λ ∈ C such that (λI -A) -1 exists as an inverse operator in X.

2) The spectrum of A denoted by σ(A) is the set C\ρ(A).

Semigroups, Existence and uniqueness of solution

Remark 1.1.3. From the above definitions, we can split the spectrum σ(A) of A into three disjoint sets, the ponctuel spectrum denoted by σ p (A), the continuous spectrum denoted by σ c (A) and the residual spectrum denoted by σ r (A) where these sets are defined as follows:

• λ ∈ σ p (A) if ker(λI -A) = {0} and in this case λ is called an eigenvalue of A; • λ ∈ σ c (A) if ker(λI -A) = {0} and Im(λI -A) is dense in X but (λI -A) -1 is not a bounded operator; • λ ∈ σ r (A) if ker(λI -A) = {0} but Im(λI -A) is not dense in X.
Some properties of semigroup and its generator operator A are given in the following theorems:

Theorem 1.1.4. (Pazy [75]) Let A be the infinitesimal generator of a C 0 -semigroup of contractions (T A (t)) t≥0 . Then, the resolvent (λI -A) -1 of A contains the open right half-plane, i.e., ρ(A) ⊂ {λ : (λ) > 0} and for such λ we have

(λI -A) -1 L(H) ≤ 1 (λ) .
Theorem 1.1.5. (Kato [50]) Let A be a closed operator in a Banach space X such that the resolvent (I -A) -1 of A exists and is compact.

Then the spectrum σ(A) of A consists entirely of isolated eigenvalues with finite multiplicities.

Theorem 1.1.6. (Pazy [75]) Let (T (t)) t≥0 be a C 0 -semigroup on a Hilbert space H. Then there exist two constants ω ≥ 0 and M ≥ 1 such that

T (t) L(H) ≤ M e ωt , ∀t ≥ 0.
If ω = 0, the semigroup (T (t)) t≥0 is called uniformly bounded and if moreover M = 1, then it is called a C 0 -semigroup of contractions. For the existence of solution of problem (1.0.1), we typically use the following Lumer-Phillips and Hille-Yosida theorems from [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]:

(i) A generates a C 0 -semigroup (T A (t)) t≥0 , satisfying T A (t) ≤ M e ωt , ∀t ≥ 0;
(ii) A is closed, densely defined, and for every λ > ω one has λ ∈ ρ(A) and

(λ -ω) n (λ -A) -n ≤ M, ∀n ∈ N;
(iii) A is closed, densely defined, and for every λ ∈ C with (λ) > ω, one has λ ∈ ρ(A) and

(λ -A) -n ≤ M ( (λ) -ω) n , ∀n ∈ N.
Consequently, A is maximal dissipative operator on a Hilbert space H if and only if it generates a C 0 -semigroup of contractions (T A (t)) t≥0 on H. Thus, the existence of solution is justified by the following corollary which follows from Lumer-Phillips theorem.

Corollary 1.1.9. Let H be a Hilbert space and let A be a linear operator defined from D(A) ⊂ H into H. If A is maximal dissipative operator then the initial value problem (1.0.1) has a unique solution Finally, we also recall the following theorem concerning a perturbations by a bounded linear operators (see Theorem 1.1 in Chapter 3 of [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]): Theorem 1.1.10. Let X be a Banach space and let A be the infinitesimal generator of a C 0 -semigroup (T A (t)) t≥0 on X, satisfying T A (t) L(H) ≤ M e ωt for all t ≥ 0. If B is a bounded linear operator on X, then the operator A + B becomes the infinitesimal generator of a C 0 -semigroup (T A+B (t)) t≥0 on X, satisfying T A+B (t) L(H) ≤ M e (ω+M B )t for all t ≥ 0.

U (t) = T A (t)U 0 such that U ∈ C([0, +∞), H), for each initial datum U 0 ∈ H. Moreover, if U 0 ∈ D(A), then U ∈ C([0, +∞), D(A)) ∩ C 1 ([0, +∞), H).

Strong and exponential stability

First, we start by introducing the notion of the strong and the exponential stabilities.

Definition 1.2.1. Assume that A is the generator of a strongly continuous semigroup of contractions (T A (t)) t≥0 on a Hilbert space H. We say that the semigroup

(T A (t)) t≥0 is (i) Strongly (asymptotically) stable if for all initial data U 0 ∈ H we have T A (t)U 0 H ----→ t→+∞ 0. (1.2.1)
(ii) Exponentially stable if there exist two positive constants C and w such that

T A (t)U 0 H ≤ Ce -wt U 0 H , ∀t > 0, ∀U 0 ∈ H. (1.2.2)
Next, in order to show the strong stability of our system, we apply the next theorem due to Arendt and Batty in [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF]. [69, pp. 9-10] and Foguel [START_REF] Foguel | Powers of a contraction in Hilbert space[END_REF].

Now, we recall two results which gives necessary and sufficient conditions for which a semigroup is exponentially stable.

Theorem 1.2.4. [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]) Let (T A (t)) t≥0 be a C 0 -semigroup on a Hilbert space H and A be its infinitesimal generator. Then, the C 0semigroup of contractions (T A (t)) t≥0 is exponentially stable if and only if

(i) iR ⊆ ρ(A), (ii) sup ω∈R (iω -A) -1 < ∞.
Theorem 1.2.5. Let (T A (t)) t≥0 be a C 0 -semigroup on a Hilbert space H and A be its infinitesimal generator. Then (T A (t)) t≥0 is exponentially stable if and only if there exists t 0 > 0 such that

T A (t 0 ) L(H) < 1. (1.2.3)
Since the studies systems in this thesis do not achieve the exponential stability, we present the used methodologies used to prove this objective. The first one is based on the following compact perturbation theory of Russell in [START_REF] Russell | Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods[END_REF]:

Theorem 1.2.6. Assume that A is skew-adjoint. Then, it does not exist two compacts operators B, C, and t 0 > 0 such that

T A+B (t 0 ) L(H) < 1 and T A+C (-t 0 ) L(H) < 1,
where

(T A+B (t)) t≥0 (resp. (T A+C (t)) t≥0 ) designate the C 0 -semigroup gen- erated by A + B (resp. A + C).
Our strategy is to split our operator A as the sum of two operators A 0 and B where A 0 ( respectively B) is a skew-adjoint operator (respectively compact operator). Next, we show that for all t > 0 we have

T A 0 +B (t) L(H) = T A 0 -B (-t) L(H) .
Therefore, by combining the result of Theorem 1.2.6 with the one of 1.2.5, we deduce that the C 0 -semigroup of contraction (T A (t)) t≥0 cannot be exponentially stable.

The second one, is a classical method based on the spectrum analysis. Indeed, we show that the spectrum of the operator A approaches asymptotically the imaginary axis, i.e. we show the existence of a sequence of eigenvalues of A whose real part is close to the imaginary axis. Then, using the eigenvectors associated to these eigenvalues, we show that the resolvent of A is not bounded on the imaginary axis. Thus, from Theorem 1.2.4, we deduce that decay of the energy of system (1.0.1) to zero is not exponential.

Polynomial stability

As we have already said in the previous section, the energies of our systems in this thesis have no uniform (exponentially) decay rate, therefore we look for a polynomial one. In general, polynomial stability results are obtained using different methods like: multipliers method, frequency domain approach, Riesz basis approach, Fourier analysis or a combination of them (see [START_REF] Komornik | Exact Controllability and Stabilization: The Multiplier Method[END_REF]58,[START_REF] Littman | Stabilization of a hybrid system of elasticity by feedback boundary damping[END_REF]). In this section, we discuss the used methods in our work to established a polynomial energy decay of system (1.0.1). First, we say that the C 0 -semigroup (T A (t)) t≥0 generated by A is polynomially stable if there exists two positive constants β and C such that

T A (t) L(H) ≤ Ct -β , t > 0. (1.3.1)
We start by a methodology introduced in [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF] and applied at the first order Cauchy problem. This requires, on one hand, to establish an observability inequality of solution of the undamped system associated to (1.0.1), and on the other hand to verify the boundedness property of the transfer function. First, we decompose the operator A as A 0 + BB * where A 0 : D(A) -→ H is an unbounded operator, B ∈ L(H) and where B * designates the adjoint operator associated to B. We rewrite problem (1.0.1) as

   U t (t) = A 0 U (t) + BB * U (t), t > 0, U (0) = U 0 ∈ H.
Next, we introduce the transfer function H by

H : C + = {λ ∈ C; (λ) > 0} -→ L(C) (1.3.2) H(λ) = B * (λ + A 0 ) -1 B.
Moreover, we define the set C ω = {λ ∈ C; (λ) = ω} where ω > 0 and we denote by (P ) the following proposition:

(P ) : The transfer function H defined in (1.3.2) is bounded on C ω .

Furthermore, we introduce the Banach's spaces X 1 and Y 1 such that

D(A) ⊂ Y 1 ⊂ H ⊂ X 1 , ∀U ∈ D(A), U D(A) ∼ U Y 1 ,
and such that [Y 1 , X 1 ] θ = H
where θ ∈]0, 1[. Now, we are ready to present the theorem which gives a polynomial decay of energy of system (1.0.1).

Theorem 1.3.1. Assume that the proposition (P ) holds. If for all U 0 ∈ X 1 , there exists T > 0 such that

T 0 B * Φ(t) 2 H ≥ C U 0 2 X 1
for some constant C, where Φ is the solution of the conservative system associated to (1.0.1), then there exists a constant C such that for all t > 0 and for all U 0 ∈ D(A), the energy E of (1.0.1) satisfies

E(t) ≤ C (1 + t) θ 1-θ U 0 2 D(A) .
The second method is a frequency domain approach method given in [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF]Theorem 2.4]. It is based on the boundedness of the resolvent of A on the imaginary axis.

Theorem 1.3.2. Let (T A (t)) t≥0 be a bounded C 0 -semigroup of contractions on a Hilbert space H generated by A such that the following condi-tion (H 2 ) holds:

(H 2 ) : iR ⊂ ρ(A).
Then, the following conditions are equivalent:

(H 3 ) : sup |β|≥1 1 |β| l (iβI -A) -1 L(H) < +∞.
(H 4 ) : there exists a constant C > 0 such that for all U 0 ∈ D(A) we have

E(t) ≤ c t 2 l U 0 2 D(A) , ∀t > 0,
where E is the energy of system (1.0.1).

The third one is based on a multiplier method and it consists to determine an integral inequality for the energy of our system. One of these methods is given by the following theorem from [START_REF] Komornik | Exact Controllability and Stabilization: The Multiplier Method[END_REF]:

Theorem 1.3.3. Let E : R + → R + be a non-increasing function and assume that there are two constants α > 0 and T > 0 such that

∞ t E α+1 (s)ds ≤ T E α (0)E(t), ∀t ∈ R + .
Then we have

E(t) ≤ E(0) T + αt T + αT -1 α , ∀t ≥ T.
In this thesis, we use the following corollary deduced from the above theorem:

Corollary 1.3.4. Let E : R + → R + be a non-increasing function and assume that there are two constants α > 0 and M > 0 such that

T S E α+1 (s)ds ≤ M E α (0)E(T ), ∀0 ≤ S ≤ T < +∞.
Then we have

E(t) ≤ E(0) (α + 1)M M + αt -1 α , ∀t ≥ 0.
The fourth method is based on the spectral analysis of the operator A. Indeed, it requires firstly to determine the asymptotic behavior of the eigenvalues associated to A and secondly to prove that the set of the generalized eigenvectors of A form a Riesz basis in the Hilbert space H. This method is given in [START_REF] Loreti | Optimal energy decay rate for partially damped systems by spectral compensation[END_REF]. We also refer to [START_REF] Littman | On the spectral properties and stabilization of acoustic flow[END_REF]

α k ≥ 0, β k > 0 such that (λ k,n ) ≤ - β k µ α k k,n and | (λ k,n )| ≥ µ k,n ∀n ≥ 1.
Then, for any U 0 ∈ D(A θ ) with θ > 0, there exists a constant M > 0 independent of U 0 such that

T A (t)U 0 2 ≤ A θ U 0 2 H M t 2θδ ∀t > 0,
where the decay rate δ is given by

δ := min 1≤k≤K 1 α k = 1 α l . (1.3.3)
Moreover, if there exists two constants c 1 > 0, c 2 > 0 such that

(λ l,n ) ≥ - c 1 µ α l l,n and | (λ l,n )| ≤ c 2 µ l,n ∀n ≥ 1,
then the decay rate δ given in (1.3.3) is optimal.

Remark 1.3.6. The benefit of the last method given in Theorem 1.3.5 is the optimality of decay rate δ in the sense that for any > 0, we cannot expect a decay rate of type 1 t 2θδ+ . But, because of the essential difficulty intervening from the determination of the spectrum of the system, this method is obviously limited to one-dimensional problem. However, the polynomial decay rate of energy of the first three methods cannot probably be optimal, but these methods are well applied in the multidimensional problem.

Riesz basis

Since Theorem 1.3.5 consists that a family of eigenvectors of A must form a Riesz basis in the Hilbert space H, in this section, we give the basic definitions and theorems needed for Riesz basis generation. We refer to [START_REF] Avdonin | The method of moments in controllability problems for distributed parameter systems[END_REF][START_REF] Gohberg | Introduction to the Theory of Linear Nonselfadjoint Operators[END_REF][START_REF] Guo | Riesz basis approach to the stabilization of a flexible beam with a tip mass[END_REF]. Definition 1.4.1. (i) A non-zero element ϕ in a Hilbert space H is called a generalized eigenvector of a closed linear operator A, corresponding to an eigenvalue λ of A, if there exists n ∈ N * such that (λI -A) n ϕ = 0 and (λI -A) n-1 ϕ = 0.

If n = 1, then ϕ is an eigenvector.

(ii) The root subspace of A corresponding to an eigenvalue λ is defined by

N λ (A) = ∞ n=1 ker ((λI -A) n ) .
(iii) The closed subspace spanned by all the generalized eigenvectors of A is called the root subspace of A.

C 1 n∈N |α n | 2 ≤ n∈N α n ϕ n 2 ≤ C 2 n∈N |α n | 2 .
In this case, each element f ∈ Span{ϕ n , n ∈ N} is written as

f = n∈N < f, ψ n > H ϕ n ,
where

Ψ = {ψ n } n∈N is biorthogonal to Φ.
The following two theorems give the necessary and the sufficient conditions so that a family {φ n } n∈N forms a Riesz basis.

∞ n=1 ϕ n -ψ n 2 < ∞.
Then {ψ n } n∈N also forms a Riesz basis of H.

Typically, the comprehension of the number of generalized eigenfunctions corresponding to low eigenvalues seems difficult. The application of the above classical Bari's theorem seems also difficult even if the behavior of the high eigenvalues and their corresponding multiplicities are clearly known. Consequently, in case the behavior of low eigenvalues is vague, we suggest using Theorem 6. energy by using an observability inequality. For that purpose, we give the asymptotic expansion of eigenvalues and eigenfunctions of the undamped underling system. Next, using the real part of the asymptotic expansion of eigenvalues of the damped system, we prove that the obtained energy decay rate is optimal.

Introduction

In [START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF], Wehbe considered a Rayleigh beam clamped at one end and subjected to two dynamical boundary controls at the other end, namely

y tt -γy xxtt + y xxxx = 0, 0 < x < 1, t > 0,(2.1.1)
y(0, t) = y x (0, t) = 0, t > 0,(2.1.2)

y xx (1, t) + aη(t) = 0, t > 0,(2.1.3) y xxx (1, t) -γy xtt (1, t) -bξ(t) = 0, t > 0,(2.1.4)
where γ > 0 is the coefficient of moment of inertia, a > 0 and b > 0 are constants, η and ξ denote respectively the dynamical boundary control moment and force. The damping of the system is made via the indirect damping mechanism at the right extremity of the beam that involves the following two first order differential equations:

η t (t) -y xt (1, t) + αη(t) = 0, t > 0, (2.1.5) ξ t (t) -y t (1, t) + βξ(t) = 0, t > 0, (2.1.6)
where α > 0 and β > 0 are constants. The notion of indirect damping mechanisms has been introduced by Russell in [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF] and since that time, it retains the attention of many authors. In [START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF], Wehbe considered the Rayleigh beam equation with two dynamical boundary controls moment and force, i.e. under the conditions a > 0 and b > 0. The lack of uniform stability was proved by a compact perturbation argument of Gibson [START_REF] Gibson | A note on stabilization of infinite-dimensional linear oscillators by compact linear feedback[END_REF] and a polynomial energy decay rate of type 1 t is obtained by a multiplier method usually used for nonlinear problems. Finally, using a spectral method, he proved that the obtained energy decay is optimal in the sense that for any ε > 0, we cannot expect a decay rate of type 1 t 1+ε . But in [START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF] the effect of each control separately on the stability of the Rayleigh beam equation is not investigated. Indeed, the multiplier method exploits in an explicit way the presence of the two boundary controls. Furthermore, the lack of one of this two controls yield this method ineffective. Then, the important and interesting case when the Rayleigh beam equation is damped by only one dynamical boundary control (a = 0 and b > 0 or a > 0 and b = 0) remained open. The aim of this chapter is to fill this gap by considering a clamped Rayleigh beam equation subject to only one dynamical boundary control moment.

The stabilization of the Rayleigh beam equation retains the attention of many authors. Rao [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF] studied the stabilization of Rayleigh beam equation subject to a positive internal viscous damping. Using a constructive approximation, he established the optimal exponential energy decay rate. In [START_REF] Lagnese | Boundary stabilization of thin plates[END_REF], Lagnese studied the stabilization of system (2.1.1)-(2.1.4) with two static boundary controls (the case a > 0, b > 0, η(t) = y xt (1, t) and ξ(t) = y t (1, t)). He proved that the energy decays exponentially to zero for all initial data. Rao in [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF] extended the results of [START_REF] Lagnese | Boundary stabilization of thin plates[END_REF] to the case of one boundary feedback. In the case of one control moment (the case a > 0, b = 0 and η(t) = y xt (1, t)), using a compact perturbation theory due to Gibson [START_REF] Gibson | A note on stabilization of infinite-dimensional linear oscillators by compact linear feedback[END_REF], he established an exponential stability of system (2.1.1)-(2.1.4).

In this chapter, we consider the Rayleigh beam equation (2.1.1)-(2.1.4) with only one dynamical boundary control moment η, i.e. when a = 1, b = 0 and η is solution of (2.1.5). Using an explicit approximation of the characteristic equation, we give the asymptotic behavior of eigenvalues and eigenfunctions of the associated undamped system with the help of Rouché's theorem. Then to prove the polynomial energy decay, we apply the methodology given in [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF]. This requires, on one hand, to establish an observability inequality of solution of the undamped system and on the other hand, to verify the boundedness property of the transfer function. This attend to establish a polynomial energy decay rate of type 1 t for smooth initial data. Finally, using the frequency domain approach given by Theorem 2.4 in [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], we prove that the obtained energy decay rate is optimal in the sense that for any ε > 0, we cannot expect a decay rate of type 1 t 1+ε .

Let us briefly outline the content of this chapter. Section 2.2 considers the well-posedness property and the strong stability of the problem by the semigroup approach (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF], [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF] and [START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF]). Section 2.3 is divided into two subsections. In subsection 2.3.1, we propose an explicit approximation of the characteristic equation determining the eigenvalues of the corresponding undamped system. In subsection 2.3.2, we give an asymptotic expansion of eigenvalues and eigenfunctions of the corresponding operator. Then, we establish a polynomial energy decay rate for smooth initial data. In section 2.4, we prove that the obtained energy decay rate is optimal. In section 2.5, we give some open problems.

Well-posedness and strong stability

In this section, we study the existence, uniqueness and the asymptotic behavior of the solution of Rayleigh beam equation with only one dynamical boundary control moment:

                   y tt -γy xxtt + y xxxx = 0, 0 < x < 1, t > 0, y(0, t) = y x (0, t) = 0, t > 0, y xx (1, t) + η(t) = 0, t > 0, y xxx (1, t) -γy xtt (1, t) = 0, t > 0, η t (t) -y xt (1, t) + αη(t) = 0, t > 0.
(2.2.1)

Let y and η be smooth solutions of system (2.2.1), we define their associated energy by

E(t) = 1 2 1 0 (|y t | 2 + γ|y xt | 2 + |y xx | 2 )dx + |η(t)| 2 , t ≥ 0. (2.

2.2)

A direct computation gives

d dt E(t) = -α|η(t)| 2 ≤ 0, t ≥ 0.
(2.2.3)

Thus the system (2.2.1) is dissipative in the sense that the energy E is a nonincreasing function of the time variable t. We start our study by formulating the problem in an appropriate Hilbert space. We first introduce the following spaces:

V = y ∈ H 1 (0, 1); y(0) = 0 , y 2 V = 1 0 (|y| 2 + γ|y x | 2 )dx, (2.2.4) W = y ∈ H 2 (0, 1); y(0) = y x (0) = 0 , y 2 W = 1 0 |y xx | 2 dx (2.2.5)
and the energy space

H = W × V × C, (2.2.6)
endowed with the usual inner product

((y 1 , z 1 , η 1 ), (y 2 , z 2 , η 2 )) H = (y 1 , y 2 ) W + (z 1 , z 2 ) V + η 1 η 2 , ∀(y 1 , z 1 , η 1 ), (y 2 , z 2 , η 2 ) ∈ H.
Identify L 2 (0, 1) with its dual so that we have the following continuous embedding:

W ⊂ V ⊂ L 2 (0, 1) ⊂ V ⊂ W . ( 2
Φ > V ×V = (y, Φ) V , ∀y, Φ ∈ V. (2.2.11)
Then, by means of Lax-Milgram's theorem (see [START_REF] Brezis | Analyse fonctionnelle[END_REF]), we see that A (respectively C) is the canonical isomorphism from W into W (respectively from V into V ). On the other hand, using the usual trace theorems and Poincaré's inequality, we easily check that the operator B is continuous for the corresponding topology. Therefore, using the operators A, B and C and the continuous embedding (2.2.7), we formulate the variational equation (2.2.8) as

Cy tt + Ay + Bη = 0 in W .
Assume that Ay + Bη ∈ V , then we obtain

y tt + C -1 (Ay + Bη) = 0 in V. (2.2.12)
Next, we introduce the linear unbounded operator A 0 by

D(A 0 ) = {(y, z, η) ∈ H; z ∈ W and Ay + Bη ∈ V } , ( 2

.2.13)

A 0 U =     -z C -1 (Ay + Bη) -z x (1)     , ∀U = (y, z, η) ∈ D(A 0 ) (2.2.14)
and the linear bounded operator B by

BU =     0 0 η     , ∀U = (y, z, η) ∈ H.
(2.2.15)

Then, denoting U = (y, y t , η) the state of system (2.2.1) and defining A α = A 0 + αB with D(A α ) = D(A 0 ), we can formulate the system (2.2.1) into a first-order evolution equation

   U t (t) + A α U (t) = 0, t > 0, U (0) = U 0 ∈ H. (2.2.16)
It is easy to show that -A 0 is a maximal dissipative operator and -B is a dissipative operator in the energy space H. Therefore, the operator -A α generates a C 0 -semigroup (e -tAα ) t≥0 of contractions in the energy space H following Lumer-Phillips' theorem (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]).

U (t) = e -tAα U 0 such that U ∈ C 1 ([0, ∞[, H) ∩ C 0 ([0, ∞[, D(A 0 )).
Moreover, we characterize the space D(A 0 ) by the following proposition:

Proposition 2.2.2. Let U = (y, z, η) ∈ H. Then U ∈ D(A 0 ) if and only if the following conditions hold:

       y ∈ W ∩ H 3 (0, 1), z ∈ W, y xx (1) + η = 0.
(2.2.17)

In particular, the resolvent (I + A 0 ) -1 of -A 0 is compact on the energy space H and the solution of the system (2.2.1) satisfies

y ∈ C 0 ([0, ∞[, H 3 (0, 1) ∩ W ) ∩ C 1 ([0, ∞[, W ) ∩ C 2 ([0, ∞[, V ). (2.2.18)
The proof is same as in Rao [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF]Proposition 2.3] (see also Wehbe [START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF]) so we omit the details here. Moreover, since the resolvent of the bounded operator B is compact, we deduce that the one of the unbounded operator -A α is also compact. Now we investigate the strong stability of the problem (2.2.16) by the following theorem: Theorem 2.2.3. For any γ > 0, the semigroup of contractions (e -tAα ) t≥0 is strongly asymptotically stable on the energy space H, i.e. for any U 0 ∈ H, we have

lim t→+∞ e -tAα U 0 2 H = 0. (2.2.

19)

Proof: The proof is same as in Rao [79, Theorem 3.1], it is based on the spectral decomposition theory of Sz-Nagy-Foias [START_REF] Nagy | Harmonic Analysis of Operators on Hilbert Space[END_REF], Foguel [START_REF] Foguel | Powers of a contraction in Hilbert space[END_REF] and Benchimol [START_REF] Benchimol | A note on weak stabilizability of contraction semigroups[END_REF]. In order to prove (2.2. [START_REF] Benchimol | A note on weak stabilizability of contraction semigroups[END_REF]) and the fact that -A α has compact resolvent, it is sufficient to show that there is no spectrum in imaginary axis. We omit the details here. Further, since A 0 is skew adjoint operator and B is compact, then using a compact perturbation method of Russell [START_REF] Russell | Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods[END_REF] we deduce that the problem (2.2.16) is not uniformly stable (see also Rao [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF], and Wehbe [START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF]).

Polynomial stability for smooth initial data

Our main result in this section is the following polynomial-type decay estimate:

Theorem 2.3.1. (Polynomial energy decay rate) Let γ > 0. For all initial data U 0 ∈ D(A 0 ), there exists a constant c > 0 independent of U 0 , such that the solution of the problem (2.2.16) satisfies the following estimate:

E(t) ≤ c 1 + t U 0 2 D(A 0 ) , ∀t > 0. (2.3.1)
In order to prove (2.3.1), we need first to analyze the spectrum of the operator A 0 . Next, we will apply a method introduced by Ammari and Tucsnak in [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF], where the polynomial stability for the damped problem is reduced to an observability inequality of the corresponding undamped problem (via the spectral analysis), combined with the boundedness property of the transfer function of the associated undamped system.

Spectral analysis of the conservative operator

First, since A 0 is closed with a compact resolvent, its spectrum σ(A 0 ) consists entirely of isolated eigenvalues with finite multiplicities (see [START_REF] Kato | Perturbation theory for linear operators[END_REF]). Moreover, as the coefficients of A 0 are real then the eigenvalues appear by conjugate pairs. Further, the eigenvalues of A 0 are on the imaginary axis.

Proposition 2.3.2. Let λ be an eigenvalue of A 0 and let U = (y, z, η) ∈ D(A 0 ), U = 0, an associated eigenvector. Then λ is simple and we have η = 0.

Proof: First, a straightforward computation shows that 0 ∈ σ(A 0 ) and is simple. An associated eigenvector being (-x 2 2 , 0, 1), thus its last component η = 1 does not vanish. Next, let λ = iµ ∈ σ(A 0 ), µ ∈ R * and U = (y, z, η) an associated eigenvector. Assume that η = 0. Using equation (2.2.15), we get that BU = 0. Thus, we obtain

A α U = (A 0 + αB)U = A 0 U = iµU. (2.3.2)
Therefore λ = iµ is also an eigenvalue of A α and it is a contradiction with Theorem 2.2.3 since γ > 0.

Later, assume that there exists λ ∈ σ(A 0 ) such that λ is not simple. As A 0 is a skew-adjoint operator, we deduce that there correspond at least two independent eigenvectors U 1 = (y 1 , z 1 , η 1 ) and U 2 = (y 2 , z 2 , η 2 ). Then,

U 3 = η 2 U 1 -η 1 U 2 = (y 3 , z 3 , η 3
) is also an eigenvector associated to λ with η 3 = 0, hence the contradiction with the first part of the proof. Now, in order to get a better knowledge of the spectrum we compute the characteristic equation. Let λ = iµ, µ ∈ R * , be an eigenvalue of A 0 and U = (y, z, η) ∈ D(A 0 ) be an associated eigenfunction. Then we have

       z = -iµy, Ay + Bη = iµCz, z x (1) = -iµη. ( 2
y xx Φ xx dx -µ 2 1 0 yΦ + γy x Φ x dx + y x (1)Φ x (1) = 0, ∀Φ ∈ W.
Equivalently, the function y is determined by the following system:

             y xxxx + γµ 2 y xx -µ 2 y = 0, y(0) = y x (0) = 0, y xx (1) + y x (1)
= 0, y xxx (1) + γµ 2 y x (1) = 0.

(2.3.4)

We have found that λ = iµ = 0 is an eigenvalue of A 0 if and only if there is a non trivial solution of (2.3.4). The general solution of the first equation of (2.3.4) is given by

y(x) = 4 i=1 c i e r i (µ)x , (2.3.5)
where

r 1 (µ) = -γµ 2 + µ √ γ 2 µ 2 + 4 2 , r 2 (µ) = -r 1 (µ), (2.3.6
)

r 3 (µ) = -γµ 2 -µ √ γ 2 µ 2 + 4 2 , r 4 (µ) = -r 3 (µ).
Here and below, for simplicity we denote r i (µ) by r i . Thus the boundary conditions in (2.3.4) may be written as the following system:

M (µ)C(µ) =        1 1 1 1 r 1 r 2 r 3 r 4 g 1 (µ) g 2 (µ) g 3 (µ) g 4 (µ) h 1 (µ) h 2 (µ) h 3 (µ) h 4 (µ)               c 1 c 2 c 3 c 4        = 0, (2.3.7)
where g i (µ) = r i (r 2 i + γµ 2 ) e r i and h i (µ) = r i (r i + 1) e r i for i = 1, 2, 3, 4. Consequently (2.3.4) admits a non-trivial solution if and only if f (µ) := det M (µ) = 0. Finally, we have found that λ = iµ is an eigenvalue of A 0 if and only if µ satisfies the characteristic equation f (µ) = 0.

Proposition 2.3.3. (Spectrum of A 0 )

There exists k 0 ∈ N * , sufficiently large, such that the spectrum σ(A 0 ) of A 0 is given by:

σ(A 0 ) = σ 0 ∪ σ 1 , (2.3.8)
where

σ 0 = iκ 0 j j∈J 0 , σ 1 = λ 0 k = iµ k k∈Z |k|≥k 0 , σ 0 ∩ σ 1 = ∅, (2.3.9)
J 0 is a finite set and κ 0 j , µ k ∈ R. Moreover, µ k satisfies the following asymptotic behavior:

µ k = α k - F 1 F 0 1 kπ + O( 1 k 2 ), |k| → ∞, (2.3.10)
where

α k = kπ √ γ + π 2 √ γ , (2.3.11)
and where

   F 0 = 2γ (2.3.12)
Proof: The proof is decomposed into two steps.

Step 1. First, we start by the expansion of r 1 and r 3 when |µ| → ∞.

After some computations we find

r 1 = 1 √ γ + O( 1 µ 2 ) (2.3.13)
and

r 3 = i √ γµ + i 1 2γ 3 2 µ + O( 1 µ 3 ). (2.3.14)
This gives

r 2 1 e r 1 = e 1 √ γ √ γ + O( 1 µ 2 ), (2.3.15) r 2 2 e r 2 = e -1 √ γ √ γ + O( 1 µ 2 ), (2.3.16) r 2 3 e r 3 = -γe i √ γµ µ 2 + O(1) (2.3.17)
and

r 2 4 e r 4 = -γe -i √ γµ µ 2 + O(1). (2.3.18)
Next, using (2.3.13)-(2.3.18), we find the asymptotic behavior of

g 1 (µ) = √ γe 1 √ γ µ 2 + O(1), (2.3.19) g 2 (µ) = - √ γe -1 √ γ µ 2 + O(1), (2.3.20) g 3 (µ) = - ie i √ γµ µ √ γ + O( 1 µ ) (2.3.21)
and

g 4 (µ) = ie -i √ γµ µ √ γ + O( 1 µ ). (2.3.22)
Similarly, we get

h 1 (µ) = 1 √ γ + 1 e 1 √ γ √ γ + O( 1 µ ), (2.3.23
)

h 2 (µ) = 1 √ γ -1 e -1 √ γ √ γ + O( 1 µ ), (2.3.24)
h 3 (µ) = -γµ 2 + i( √ γ - 1 2 √ γ )µ e i √ γµ + O(1) (2.3.25)
and

h 4 (µ) = -γµ 2 + i( 1 2 √ γ - √ γ)µ e -i √ γµ + O(1). ( 2
M (µ) =                1 1 1 1 P µ 1 P µ 2 P µ 3 P µ 4 P µ 5 P µ 6 P µ 7 P µ 8 P µ 9 P µ 10 P µ 11 P µ 12                , ( 2.3.27)
where

P µ 1 = 1 √ γ + O( 1 µ 2 ), P µ 2 = - 1 √ γ + O( 1 µ 2 ), P µ 3 = i √ γµ + i 1 2γ 3 2 µ + O( 1 µ 2 ), P µ 4 = -i √ γµ -i 1 2γ 3 2 µ + O( 1 µ 3 ), P µ 5 = √ γe 1 √ γ µ 2 + O(1), P µ 6 = - √ γe -1 √ γ µ 2 + O(1), P µ 7 = - ie i √ γµ µ √ γ + O( 1 µ ), P µ 8 = ie -i √ γµ µ √ γ + O( 1 µ
),

P µ 9 = 1 γ + 1 √ γ e 1 √ γ + O( 1 µ ), P µ 10 = 1 γ - 1 √ γ e -1 √ γ + O( 1 µ
),

P µ 11 = -γµ 2 + i( √ γ - 1 2 √ γ )µ e i √ γµ + O(1)
and

P µ 12 = -γµ 2 + i( 1 2 √ γ - √ γ)µ e -i √ γµ + O(µ).
Again after some computations, we find the following asymptotic development of f (µ) = det(M (µ)):

f (µ) = µ 5 f 0 (µ) + µ 4 f 1 (µ) + O(µ 3 ),
where

f 0 (µ) = -2iF 0 √ γ cos( √ γµ) and f 1 (µ) = 2i √ γF 1 sin( √ γµ), (2.3.28)
with F 0 and F 1 are given by (2.3.12). For convenience we set

S(µ) = f (µ) µ 5 = f 0 (µ) + f 1 (µ) µ + O( 1 µ 2 ), (2.3.29)
that has the same root as f , except 0.

Step 2. We look at the roots of S. Is is easy to see that the roots of f 0 are given by

α k = kπ √ γ + π 2 √ γ , k ∈ Z.
Then, with the help of Rouché's theorem, there exists k 0 ∈ N * large enough, such that for all |k| ≥ k 0 , the large roots of S (denoted by µ k ) are close to α k . More precisely, there exists k 0 ∈ N * large enough, such that the splitting of σ(A 0 ) given by (2.3.8)-(2.3.9) holds and we have

µ k = α k + o(1) = kπ √ γ + π 2 √ γ + o(1), |k| → ∞. (2.3.30)
Equivalently, we can write

µ k = kπ √ γ + π 2 √ γ + l k , lim |k|→∞ l k = 0. (2.3.31) It follows that cos( √ γµ k ) = -(-1) k sin( √ γl k ) = -(-1) k √ γl k + o(l 2 k ) (2.3.32) and sin( √ γµ k ) = (-1) k cos( √ γl k ) = (-1) k (1 - √ γl 2 k 2 ) + o(l 2 k ). ( 2
= S(µ k ) = 2i √ γ(-1) k (F 0 √ γl k + F 1 kπ ) + o(l 2 k ) + O( 1 k 2 ).
This implies that

l k = - F 1 F 0 1 kπ + O( 1 k 2 ). (2.3.34)
Finally, inserting the previous identity in (2.3.31), we directly get (2.3.10).

Eigenvectors of A 0 . According to the decomposition of the spectrum σ(A 0 ) of A 0 , a set of eigenvectors associated with σ(A 0 ) is given as follows:

{Φ j = (y j , z j , η j )} j∈J 0 {U k = (y k , z k , η k )} k∈Z |k|≥k 0 , (2.3.35) Φ j ∈ D(A 0 ), ∀j ∈ J 0 , U k ∈ D(A 0 ), ∀k ∈ Z, |k| ≥ k 0 ,
where

Φ j =     y j -iκ 0 j y j y j,x (1)     and U k =     y k -iµ k y k y k,x (1)     .
(2.3.36) Now, for |k| ≥ k 0 and µ = µ k , we give a solution up to a factor of problem (2.3.4) and some appropriated asymptotic behavior.

   y k,x (1) = -(-1) k kπ + O(1) = 0, y k W ∼ |k| 2 and y k V ∼ |k|, |k| → ∞.
(2.3.37)

Moreover, we deduce

U k H ∼ |k| 2 , |k| → ∞. (2.3.38) Proof: For µ = µ k , |k| ≥ k 0 , solving (2.3.4
) amounts to find a solution C(µ k ) = 0 of the system (2.3.7) of rank three. For clarity, we divide the proof into two steps.

Step 1. Estimate of y k,x [START_REF] Abbas | The multidimensional wave equation with generalized acoustic boundary conditions ii: Polynomial stability[END_REF]. For simplicity of notation we write

C(µ k ) = (c 1 , c 2 , c 3 , c 4
). Since we search C(µ k ) up to a factor we choose c 3 = 1, the possibility of this choice will be justify later. Therefore (2.

3.7) becomes        c 1 + c 2 + c 4 = -1, r 1 c 1 + r 2 c 2 + r 4 c 4 = -r 3 , r 1 (r 1 + 1)e r 1 c 1 + r 2 (r 2 + 1)e r 2 c 2 + r 4 (r 4 + 1)e r 4 c 4 = -r 3 (r 3 + 1)e r 3 .
Next, using Cramer's rule we obtain

c 1 = α 1 α 3 , c 2 = α 2 α 3 , c 4 = α 4 α 3 , (2.3.39)
where

α 1 =2r 1 r 3 (1 -r 1 )e -r 1 + r 3 (r 2 3 -r 1 )(e r 3 + e -r 3 ) (2.3.40) + r 2 3 (1 -r 1 )(e r 3 -e -r 3 ), α 2 =2r 1 r 3 (1 + r 1 )e r 1 -r 3 (r 2 3 + r 1 )(e r 3 + e -r 3 ) (2.3.41) -r 2 3 (1 + r 1 )(e r 3 -e -r 3 ), α 3 =2r 1 r 3 (1 -r 3 )e -r 3 + r 1 (r 2 1 -r 3 )(e r 1 + e -r 1 ) (2.3.42) + r 2 1 (1 -r 3 )(e r 1 -e -r 1 )
and where

α 4 =2r 1 r 3 (1 + r 3 )e r 3 -r 1 (r 2 1 + r 3 )(e r 1 + e -r 1 ) (2.3.43) -r 2 1 (1 + r 3 )(e r 1 -e -r 1
).

First, we study the behavior of α 1 . Inserting (2.3.13) and (2.3.14) (with

µ = µ k ) in (2.
3.40) we find after some computations

α 1 = -2iγ 3/2 cos( √ γµ k )µ 3 k + i(1 + 2 √ γ + 2γ) sin( √ γµ k )µ 2 k (2.3.44) + O(µ k ). Now,
       cos( √ γµ k ) = (-1) k F 1 √ γ F 0 kπ + O( 1 k 2 ), sin( √ γµ k ) = (-1) k + O( 1 k 2 ), (2.3.45)
where F 0 and F 1 are given by (2.3.12). Inserting (2.3.10) and (2.3.45) in (2.3.44), we find again after some computations

α 1 = -i(-1) k 2F 1 γ 3/2 + F 0 -1 -2 √ γ + 2γ π 2 F 0 γ k 2 + O(k) = -2i(-1) k π 2 tanh 1 √ γ -1 √ γ k 2 + O(k). (2.3.46)
Similarly, long computations left to the reader yields

α 2 = 2i(-1) k π 2 tanh 1 √ γ + 1 √ γ k 2 + O(k), (2.3.47) α 3 = -2i(-1) k π 2 √ γ k 2 + O(k) (2.3.48)
and

α 4 = -2i(-1) k π 2 √ γ k 2 + O(k). (2.3.49)
Remark that α 3 = 0 provided we have chosen k 0 large enough; for this reason our choice

c 3 = 1 is valid. Substituting (2.3.46)-(2.3.49) into (2.3.39), we obtain                        c 1 = tanh( 1 √ γ ) -1 + O( 1 k ), c 2 = -tanh( 1 √ γ ) -1 + O( 1 k ), c 3 = 1, c 4 = 1 + O( 1 k
).

(2.3.50)

Finally, we have found that a solution (2.3.7) has the form

C(µ k ) = C 0 + O( 1 |µ k | ), (2.3.51)
where

C 0 = (-1 + tanh( 1 √ γ ), -1 -tanh( 1 √ γ ), 1, 1).
Note that the corresponding solution y k of (2.3.4) is given by (2.3.5).

From equation (2.3.5), we have

y k,x (1) = r 1 c 1 e r 1 + r 2 c 2 e r 2 + r 3 c 3 e r 3 +
y k,x (1) = -(-1) k 2kπ + O(1) = 0. (2.3.53)
Step 2. Estimates of y k W and y k V . We start with

y k 2 W = 1 0 |y k,xx | 2 dx = 4 i=1 4 j=1 c i r 2 i 1 0 e r i x e r j x dx c j r 2 j = C k G k C k T , ( 2.3.54)
where

G k = 1 0 e (r i +r j )x dx 1≤i,j≤4
and

C k = (c i r 2 i ) i=1,...4 .
First, since r 2 = -r 1 ∈ R (for |k| large enough) and r 3 = -r 4 ∈ iR, we directly find

1 0 e (r 1 +r 2 )x dx = 1 0 e (r 2 +r 1 )x dx = 1 0 e (r 3 +r 3 )x dx = 1 0 e (r 4 +r 4 )x dx = 1.
(2.3.55)

In addition, using the identity

G k = G 0 + O( 1 k ), (2.3.56)
where

G 0 =           √ γ 2 (e 2 √ γ -1) 1 0 0 1 √ γ 2 (1 -e -2 √ γ ) 0 0 0 0 1 0 0 0 0 1           (2.
C k = (0, 0, -γµ 2 k , -γµ 2 k ) + O(1). ( 2
y k 2 W = γ 2 |µ k | 4 + O(|µ k | 3 ) ∼ |k| 4 , |k| → ∞. (2.3.59)
Similarly, we easily prove that

1 0 |y k | 2 dx ∼ 1, 1 0 |y k,x | 2 dx ∼ |µ k | 2 ∼ |k| 2 , |k| → ∞.
Therefore, we deduce that

y k V ∼ |k|, |k| → ∞. ( 2
U k H ∼ |k| 2 , |k| → ∞.
This completes the proof.

Observability inequality and boundedness of the transfer function

First, since B is a self-ajdoint operator and BB * = B, we rewrite the problem (2.2.16) as follows

   U t (t) + (A 0 + αBB * ) U (t) = 0, t > 0, U (0) = U 0 ∈ H. (2.3.61)
We will establish an observability inequality for the undamped problem corresponding to (2.3.61) by the following lemma:

Lemma 2.3.5. Let γ > 0. There exist T > 0 and C T > 0 such that the solution U of the problem    U t (t) + A 0 U (t) = 0, t > 0, U (0) = U 0 ∈ H, (2.3.62)
satisfies the following observability inequality:

T 0 B * U (t) 2 H dt ≥ C T U 0 2 (D(A 0 )) , (2.3.63)
where (D(A 0 )) is the dual of D(A 0 ) with respect to the scalar product in H.

Proof: Let U 0 ∈ D(A 0 ), then we can write

U 0 = j∈J 0 U j 0 Φ j + |k|≥k 0 U k 0 U k , (2.3.64)
where

Φ j j∈J 0 U k k∈Z |k|≥K 0 denotes the set of normalized eigenvectors of A 0 such that Φ j = ( y j , z j , η j ) = 1 Φ j H Φ j , ∀j ∈ J 0 (2.3.65)
and

U k = ( y k , z k , η k ) = 1 U k H U k , ∀|k| ≥ k 0 . (2.3.66)
From (2.3.64) we obtain

U (t) = j∈J 0 U j 0 e iκ j t Φ j + |k|≥k 0 U k 0 e iµ k t U k . (2.3.67)
Consequently, we have

η(t) = y x (1, t) = - j∈J 0 U j 0 e iκ j t y j,x (1) - |k|≥k 0 U k 0 e iµ k t y k,x (1), ∀t > 0.
The spectral gap is satisfied by the eigenvalues of A 0 because they are simple and for k large enough, we have

µ k+1 -µ k ≥ π 4 √ γ , in other words, there exists d > 0, such that min λ,λ ∈σ(A 0 ) λ =λ |λ -λ | ≥ d > 0.
Thus, using Ingham's inequality (see [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF]), we deduce that there exist T > 0 and c T > 0 such that

T 0 B * U (t) 2 H dt = T 0 |η(t)| 2 dt (2.3.68) = T 0 |y x (1, t)| 2 dt ≥ c T   j∈J 0 |U j 0 | 2 | y j,x (1)| 2 + |k|≥k 0 |U k 0 | 2 | y k,x (1)| 2   .
On the other hand, using (2.3.37)-(2.3.38) and (2.3.66) we get

|k|≥k 0 |U k 0 | 2 | y k,x (1)| 2 ∼ |k|≥k 0 |U k 0 | 2 |k| 2 .
(2.3.69)

Therefore, we deduce from (2.3.68), Proposition 2.3.2 and equation (2.3.10) that

T 0 B * U (t) 2 H dt ≥ c T   j∈J 0 |U j 0 | 2 | y j,x (1)| 2 + |k|≥k 0 |U k 0 | 2 | 1 |k| 2   ∼ c T U 0 2 D(A 0 ) .
The proof of lemma is completed. Next, we introduce the transfer function H by

H : C + = {λ ∈ C; (λ) > 0} -→ L(C) (2.3.70) λ -→ H(λ) = -αB * (λ + A 0 ) -1 B.
Let ω > 0, we define the set Proof: First, since A 0 generate a C 0 -semigroup of contractions, we deduce (see Corollary I.3.6 in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]) that there exists c ω > 0 such that

C ω = {λ ∈ C; (λ) = ω}.
(λ + A 0 ) -1 H ≤ c ω , ∀λ ∈ C ω .
Next, combining this estimate with the boundedness of the operators B and B * , we deduce the boundedness of the function H on C ω .

Proof of the Theorem 2.3.1. The polynomial energy estimate (2.3.1) is obtained by application of Theorem 2.4 in [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF] on the first order prob-

lem with Y 1 = D(A 0 ), X 1 = (D(A 0 )) and θ = 1 2 .

Optimal polynomial decay rate

The aim of this section is to prove the following optimality result:

Theorem 2.4.1. (Optimal decay rate)

The energy decay rate (2.3.1) is optimal in the sense that for any > 0,

we can not expect the decay rate 1 t 1+ for all initial data U 0 ∈ D(A 0 ).

To prove this theorem, we need the asymptotic behavior of the eigenvalues of the operator A α . Let λ = α be an eigenvalue of A α and U = (y, z, η) be an associated eigenfunction, then we obtain A α U = λU . Equivalently, we have the following system:

               y xxxx -γλ 2 y xx + λ 2 y = 0, y(0) = y x (0) = 0, y xxx (1) -γλ 2 y x (1) = 0, y xx (1) + λ λ -α y x (1) = 0. (2.4.1)
The general solution of the first equation of (2.4.1) is given by

y(x) = 4 i=1 c i e R i (λ)x , (2.4.2)
where

R 1 (λ) = γλ 2 -λ √ γ 2 λ 2 -4 2 , R 2 (λ) = -R 1 (λ), (2.4.3) R 3 (λ) = γλ 2 + λ √ γ 2 λ 2 -4 2 , R 4 (λ) = -R 3 (λ).
Here and below, for simplicity we denote R i (λ) by R i . Thus the boundary conditions in (2.4.1) may be written as the following system:

N (λ) C(λ) =        1 1 1 1 R 1 R 2 R 3 R 4 g 1 (λ) g 2 (λ) g 3 (λ) g 4 (λ) h 1 (λ) h 2 (λ) h 3 (λ) h 4 (λ)               c 1 c 2 c 3 c 4        = 0, (2.4.4)
where we have set

g i (λ) = R i (R 2 i -γλ 2 ) e R i and h i (λ) = R i R i + λ λ-α e R i for i = 1, .., 4.
Since A α is closed with a compact resolvent, its spectrum consists entirely of isolated eigenvalues with finite multiplicities. Further as the coefficients of A α are real, the eigenvalues appear by conjugate pairs.

Proposition 2.4.2. There exists a positive constant c such that any eigenvalue

λ of A α satisfies 0 < (λ) ≤ c.
Proof: Obviously, we already know that the real part of any eigenvalue of A α is positive, so we only have to prove that it is upper bounded. Let λ = α be an eigenvalue of A α and U = (y, -λy, y x (1)) an associated eigenvector such that U H = 1. Multiplying the first equation of the system (2.4.1) by y and integrating by parts yields

y 2 W + λ 2 y 2 V + λ λ -α |y x (1)| 2 = 0. (2.4.5) Next, set λ = u + iv, u ∈ R * + and v ∈ R. A straightforward computation gives λ λ -α = u(u -α) + v 2 (u -α) 2 + v 2 + i αv (u -α) 2 + v 2 , (2.4.6)
then the imaginary part of the equation (2.4.5) gives

2u y 2 V - α (u -α) 2 + v 2 |y x (1)| 2 v = 0.
(2.4.7)

Assume that v = 0 then

|λ| 2 y 2 V = (u 2 + v 2 ) y 2 V = α 2u u 2 + v 2 (u -α) 2 + v 2 |y x (1)| 2 . If u = (λ) is not bounded and since |y x (1)| 2 ≤ U 2
H = 1, it follows from the previous identity that for u large

|λ| 2 y 2 V = O( 1 u ).
Consequently (2.4.5) implies

y 2 W + |y x (1)| 2 = O( 1 u ), then U 2 H = y 2 W + |λ| 2 y 2 V + |y x (1)| 2 = O( 1 u ),
which is not possible. Therefore, for u large enough, we deduce from (2.4.7) that (λ) = v = 0. Finally, taking the real part of the equation (2.4.5) with v = 0, we obtain

y 2 W + u 2 y 2 V + u u -α |y x (1)| 2 = 0.
Hence the contradiction with U 2 H = 1 if u is large enough. In the following proposition we study the spectrum of A α :

Proposition 2.4.3. (Spectrum of A α ) There exists k 1 ∈ N * sufficiently large such that the spectrum σ(A α ) of A α is given by: σ(A α ) = σ 0 ∪ σ 1 , (2.4.8)
where

σ 0 = {κ j } j∈J , σ 1 = {λ k } k∈Z |k|≥k 0 , σ 0 ∩ σ 1 = ∅ (2.4.9)
and J is a finite set. Moreover, λ k is simple and satisfies the following asymptotic behavior

λ k = i kπ √ γ + π 2 √ γ + D k + E k 2 + α π 2 k 2 + o( 1 k 2 ), (2.4.10)
where

D = 2γ -1 -2 √ γ tanh(γ -1 2 ) 2γ 3 2 π (2.4.11) and E = 2(-1) k γ 3 2 cosh(γ -1 2 )π 2 + 4γ -2 - √ γ tanh(γ -1 2 ) 2γ 3 2 π .
(2.4.12)

Proof: The proof is divided into three steps.

Step 1 furnishes an asymptotic development of the characteristic equation for large λ.

Step 2 uses Rouché's theorem to localize high frequency eigenvalues. In step 3, we perform a limited development stopped when a non zero real part appear.

Step 1. First, we start by the expansion of R 1 and

R 3 when |λ| → ∞ R 1 = 1 √ γ + 1 2γ 5 2 λ 2 + O( 1 λ 4 ) (2.4.13)
and

R 3 = λ √ γ - 1 2λγ 3 2 + O( 1 λ 3 ). (2.4.14)
Next, using (2.4.13) and (2.4.14), we find the asymptotic behavior of

g 1 (λ) = - √ γλ 2 - 1 2γ 2 + 1 2γ 3 2 e 1 √ γ + O( 1 λ ), (2.4.15) g 2 (λ) = √ γλ 2 - 1 2γ 2 - 1 2γ 3 2 e -1 √ γ + O( 1 λ ), (2.4.16
)

g 3 (λ) = - λ √ γ + 1 2γ 2 e √ γλ + O( 1 λ ) (2.4.17)
and

g 4 (λ) = λ √ γ + 1 2γ 2 e -√ γλ + O( 1 λ ). (2.4.18)
Similarly, we get

h 1 (λ) = 1 √ γ 1 + 1 √ γ + α λ e 1 √ γ + O( 1 λ 2 ), (2.4.19) h 2 (λ) = 1 √ γ -1 + 1 √ γ - α λ e -1 √ γ + O( 1 λ 2 ), (2.4.20) h 3 (λ) = γλ 2 + (- 1 2 √ γ + √ γ)λ + 1 -12γ + 8γ 2 √ γα 8γ 2 e √ γλ (2.4.21) + O( 1 λ )
and

h 4 (λ) = γλ 2 + ( 1 2 √ γ - √ γ)λ + 1 -12γ -8γ 2 √ γα 8γ 2 e -√ γλ (2.4.22) + O( 1 λ ).
Combining (2.4.13)-(2.4.22) and (2.4.4), we can write the system (2.4.4) as follows:

N (λ) C(λ) = 0,
where N (λ) is given by

N (λ) =                    1 1 1 1 P 1 + O( 1 λ 2 ) P 2 + O( 1 λ 2 ) P 3 + O( 1 λ 4 ) P 4 + O( 1 λ 4 ) e 1 √ γ P 5 + O( 1 λ ) e -1 √ γ P 6 + O( 1 λ ) e √ γλ P 7 + O( 1 λ 2 ) e -√ γλ P 8 + O( 1 λ 2 ) e 1 √ γ P 9 + O( 1 λ ) e -1 √ γ P 10 + O( 1 λ ) e √ γλ P 11 + O( 1 λ 2 ) e -√ γλ P 12 + O( 1 λ 2 )                    , with P 1 = 1 √ γ + 1 2γ 5 2 λ 2 , P 2 = - 1 √ γ - 1 2γ 5 2 λ 2 , P 3 = λ √ γ - 1 2λγ 3 2 , P 4 = -λ √ γ+ 1 2λγ 3 2 , P 5 = - √ γλ 2 - 1 2γ 2 + 1 2γ 3 2 P 6 = √ γλ 2 - 1 2γ 2 - 1 2γ 3 2 , P 7 = - λ √ γ + 1 2γ 2 , P 8 = λ √ γ + 1 2γ 2 , P 9 = 1 √ γ 1 + 1 √ γ + α λ , P 10 = 1 √ γ -1 + 1 √ γ - α λ , P 11 = γλ 2 + - 1 2 √ γ + √ γ λ + 1 8γ 2 - 3 2γ + √ γα
and

P 12 = γλ 2 + 1 2 √ γ - √ γ λ + 1 8γ 2 - 3 2γ - √ γα.
Then, after some computations, we find the following asymptotic development of f (λ) = det N (λ)

f (λ) = λ 5 f 0 (λ) + λ 4 f 1 (λ) + λ 3 f 2 (λ) + O(λ 2 ), (2.4.23)
where

f 0 (λ) = -4γ 2 cosh( 1 √ γ ) cosh( √ γλ), (2.4.24) f 1 (λ) = l 1 (γ) sinh( √ γλ), (2.4.25)
with

l 1 (γ) = 2 √ γ (1 -2γ) cosh( 1 √ γ ) + 2 √ γ sinh( 1 √ γ ) (2.4.26)
and where

f 2 (λ) = 8 -4αγ 3 2 cosh( 1 √ γ ) cosh( √ γλ) + l 2 (γ) sinh( √ γλ), (2.4.27) with l 2 (γ) = (10 - 1 2γ ) cosh( 1 √ γ ) + (4 √ γ - 4 √ γ ) sinh( 1 √ γ ). (2.4.28)
As the real part of λ is bounded, then the functions f i are bounded for i ∈ {0, 1, 2}. For convenience, we set

S(λ) = f (λ) λ 5 = f 0 (λ) + f 1 (λ) λ + f 2 (λ) λ 2 + O( 1 λ 3 ). (2.4.29)
Step 2. We look at the roots of S. It is easy to see that the roots of f 0 are simple and given by

z k = iα k , k ∈ Z, (2.4.30)
where α k is defined in (2.3.11). Then, with the help of Rouché's theorem there exists k 1 large enough such that for all |k| ≥ k 1 , the large eigenvalues of σ(A α ) (denoted by λ k ) are simple and close to z k . More precisely, there exists k 1 ∈ N * large enough, such that the splitting of σ(A α ) given by (2.4.8)-(2.4.9) holds and we have

λ k = iα k + o(1), |k| → ∞. (2.4.31)
Equivalently, we can write

λ k = iα k + k , lim |k|→∞ k = 0. ( 2
0 = S(λ k ) = f 0 (λ k ) + f 1 (λ k ) λ k + f 2 (λ k ) λ 2 k + O( 1 λ 3 k ) = -4γ 2 cosh( 1 √ γ ) cosh( √ γλ k ) + l 1 (γ) sinh( √ γλ k ) λ k (2.4.33) + 8 λ 2 k - 4αγ 3 2 cosh( 1 √ γ ) cosh( √ γλ k ) λ 2 k + l 2 (γ) sinh( √ γλ k ) λ 2 k + O( 1 λ 3 k ).
On the other hand, using (2.4.32) we find cosh(

√ γλ k ) = i(-1) k √ γ k + O( 3 k ) (2.4.34) and sinh( √ γλ k ) = i(-1) k + O( 2 k ). ( 2
= λ k yields f 0 (λ k ) = -4iγ 5 2 (-1) k cosh( 1 √ γ ) k + O( 3 k ) (2.4.36)
and

f 1 (λ k ) = i(-1) k l 1 (γ) + O( 2 k ). (2.4.37)
Similarly, we get

f 2 (λ k ) = 8 -4iαγ 3 2 cosh( 1 √ γ ) + √ γi(-1) k l 2 (γ) k + O(
f 1 (λ k ) λ k = (-1) k l 1 (γ) α k + O( k k ) (2.4.39)
and

f 2 (λ k ) λ 2 k = - 8 α 2 k + 4αiγ 3 2 cosh( 1 √ γ )(-1) k α 2 k + O( k k ). ( 2
= -4iγ 5 2 (-1) k cosh( 1 √ γ ) k + (-1) k l 1 (γ) α k - 8 α 2 k (2.4.41) + 4αiγ 3 2 cosh( 1 √ γ )(-1) k α 2 k + O( k k ). Therefore k = - 8 α 2 k -(-1) k l 1 (γ) α k 4iγ 5 2 (-1) k cosh( 1 √ γ ) + α γα 2 k + O( k k ). ( 2
k = i D k + E k 2 + α π 2 k 2 + o( 1 k 2 ) (2.
β k = -(λ k ), |k| ≥ k 1 .
Next, using (2.4.10), we have

(iIβ k + A α )U k = (iIβ k + λ k )U k = α π 2 k 2 + o( 1 k 2 ) U k , ∀|k| ≥ k 1 . 58 2.5 Open problems Therefore β 2-2l k (iβ k I + A α )U k H ∼ α π 2 × 1 k 2 1+
, ∀|k| ≥ k 1 .

Thus, we deduce

lim k→+∞ β 2-2l k (iβ k I + A α )U k H = 0.
Finally, thanks to Theorem 2.4 in [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], we deduce that the trajectory e -tAα U 0 decays slower that 1

t 1 2-2l
on the time t → +∞. Then we cannot expect the energy decay rate 1 t 1+ .

Open problems

The extension of the results of this chapter to space dimensions greater than or equal to 2 are widely open problems. We maybe apply a multiplier method, but we can not show the optimality using the spectrum study, since it is difficult to analyze it in the multidimensional problem. The exact controllability of Rayleigh beam equation and the stabilization of the numerical approximation schemes with static or dynamic boundary control moment, are also and open problems. Moreover, a Rayleigh beam equation with only one dynamical boundary control force is also an open problem. We deal this case in the next chapter.

Rayleigh beam equation with only one dynamical boundary control force

Introduction

In [START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF], Wehbe considered a Rayleigh beam clamped at one end and subjected to two dynamical boundary controls at the other end, namely

y tt -γy xxtt + y xxxx = 0, 0 < x < 1, t > 0,(3.1.1) y(0, t) = y x (0, t) = 0, t > 0,(3.1.2) y xx (1, t) + aη(t) = 0, t > 0,(3.1.3) y xxx (1, t) -γy xtt (1, t) -bξ(t) = 0, t > 0,(3.1.4)
where γ > 0 is the coefficient of moment of inertia, a > 0 and b > 0 are constants, η and ξ denote respectively the dynamical boundary control moment and force. The damping of the system is made via the indirect damping mechanism at the right extremity of the beam that involves the following two first order differential equations:

η t (t) -y xt (1, t) + αη(t) = 0, t > 0, (3.1.5) ξ t (t) -y t (1, t) + βξ(t) = 0, t > 0, (3.1.6)
where α > 0 and β > 0. The notion of indirect damping mechanisms has been introduced by Russell in [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF] and since that time, it retains the attention of many authors. The lack of uniform stability was proved by a compact perturbation argument of Gibson [START_REF] Gibson | A note on stabilization of infinite-dimensional linear oscillators by compact linear feedback[END_REF] and a polynomial energy decay rate of type 1 t is obtained by a multiplier method usually used for nonlinear problems. Finally, using a spectral method, he proved that the obtained energy decay is optimal in the sense that for any ε > 0, we cannot expect a decay rate of type 1 t 1+ε . But in [START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF] the effect of each control separately on the stability of the Rayleigh beam equation is not investigated. Indeed, the multiplier method exploits in an explicit way the presence of the two boundary controls. Furthermore, the lack of one of this two controls yield this method ineffective. Then, the important and interesting case when the Rayleigh beam equation is damped by only one dynamical boundary control (a = 0 and b > 0 or a > 0 and b = 0) remained open. In chapter 2, we have considered a Rayleigh beam equation damped at one end and subjected to one dynamic boundary control moment at the other end, i.e. when a > 0, b = 0 and η solution of (3.1.5). First, we applied a methodology introduced in [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF] to establish an energy decay rate of polynomial type 1 t . Next, using the analysis of the spectrum of our dissipative operator and from a frequency domain approach given in [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], we have proved that the obtained energy decay rate is optimal. In this chapter, we consider the second case, a Rayleigh beam equation subject to only one dynamical boundary control force, i.e. when a = 0 and b > 0. Rao in [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF] studied the stabilization of system (3.1.1)-(3.1.4) with a = 0, b > 0 and ξ(t) = y 1 (1, t). He first proved the lack of exponential stability of the system (2.1.1)-(2.1.4). Next, he proved that the Rayleigh beam equation can be strongly stabilized by only one control force if and only if the inertia coefficient γ is large enough, but he did not studied the decay rate of the energy of the system. In [START_REF] Bassam | Optimal energy decay rate of Rayleigh beam equation with only one boundary control force[END_REF], Bassam and al. studied the decay rate of energy of system (3.1.1)-(3.1.4) with a = 0, b > 0 and ξ(t) = y t (1, t). First, using an explicit approximation, they gave the asymptotic expansion of eigenvalues and eigenfunctions of the undamped system corresponding to (3.1.1)-(3.1.4), then they established the optimal polynomial energy decay rate of type 1 t via an observability inequality of solution of the undamped system and the boundedness of the transfer function associated with the undamped problem.

In this chapter, we consider the Rayleigh beam equation (3.1.1)-(3.1.4) with only one dynamical boundary control force, i.e. when a = 0, b = 1 and ξ solution of (3.1.6). Here, we prefer to use a Riesz basis approach. First, we give the asymptotic expansion of the eigenvalues and the eigenfunctions of the damped and undamped systems. Next, we show that the system of eigenvectors of high frequencies of the damped problem is quadratically closed to the system of eigenvectors of high frequencies of the undamped one. This yields, from Theorem 1.2.10 given in [START_REF] Abdallah | Stabilisation et approximation de certains systèmes distribués par amortissement dissipative et de signe indéfini[END_REF] (see also [START_REF] Guo | Riesz basis approach to the stabilization of a flexible beam with a tip mass[END_REF]Theorem 6.3]) that the system of generalized eigenvectors of the damped problem forms a Riesz basis of the energy space. Finally, by applying Theorem 2.1 given in [START_REF] Loreti | Optimal energy decay rate for partially damped systems by spectral compensation[END_REF], we establish the optimal energy decay rate of polynomial type 1 √ t .

The plan of this chapter is as follows: In section 3.2 we transform our system into an evolution equation, we deduce the well-posedness property of the problem by the semigroup approach and we recall the condition to reach the strong stability of our system (see [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF]). In section 3.3, we propose an explicit approximation of the characteristic equation determining the eigenvalues of the damped and undamped system. Then, we give an asymptotic expansion of eigenvalues and eigenfunctions of the corresponding operators. In section 3.4, we show that the system of eigenvectors of the damped problem forms a Riesz basis and we establish the optimal polynomial energy decay rate of type 1 √ t . In the last section, we give some open problems.

Well-posedness and strong stability

In this section, we study the existence, uniqueness and the asymptotic behavior of the solution of Rayleigh beam equation with only one dynamical boundary control force:

                   y tt -γy xxtt + y xxxx = 0, 0 < x < 1, t > 0, y(0, t) = y x (0, t) = 0, t > 0, y xx (1, t) = 0, t > 0, y xxx (1, t) -γy xtt (1, t) -ξ(t) = 0, t > 0, ξ t (t) -y t (1, t) + βξ(t) = 0, t > 0. (3.2.1)
First, let y and ξ be smooth solutions of system (3.2.1). We define its associated energy by

E(t) = 1 2 1 0 (|y t | 2 + γ|y xt | 2 + |y xx | 2 )dx + |ξ(t)| 2 , t ≥ 0. (3.2.2)
A direct computation gives

d dt E(t) = -β|ξ(t)| 2 ≤ 0, t ≥ 0.
Then the system (3.2.1) is dissipative in the sense that its energy E(t) is a nonincreasing function of the time variable t. We next introduce the following spaces:

V = y ∈ H 1 (0, 1); y(0) = 0 , y 2 V = 1 0 (|y| 2 + γ|y x | 2 )dx, (3.2.3) W = y ∈ H 2 (0, 1); y(0) = y x (0) = 0 , y 2 W = 1 0 |y xx | 2 dx (3.2.4)
and the energy space

H = W × V × C, (3.2.5)
endowed with the usual inner product

((y 1 , z 1 , η 1 ), (y 2 , z 2 , η 2 )) H = (y 1 , y 2 ) W + (z 1 , z 2 ) V + η 1 η 2 , ∀(y 1 , z 1 , η 1 ), (y 2 , z 2 , η 2 ) ∈ H.
Identify L 2 (0, 1) with its dual so that we have the following continuous embedding:

W ⊂ V ⊂ L 2 (0, 1) ⊂ V ⊂ W . (3.2.6)
Multiplying the first equation of (3.2.1) by Φ ∈ W and integrating by parts, we transform (3.2.1) into a variational equation:

1 0 (y tt Φ + γy xtt Φ x )dx + 1 0 y xx Φ xx dx + ξΦ(1) = 0. (3.2.7)
According, we define the following linear operators

A ∈ L(W, W ), B ∈ L(C, V ) and C ∈ L(V, V ) by: < Ay, Φ > W ×W = (y, Φ) W , ∀y, Φ ∈ W, (3.2.8) < Bξ, Φ > V ×V = ξΦ(1), ∀ξ ∈ C, ∀Φ ∈ V (3.2.9) and < Cy, Φ > V ×V = (y, Φ) V , ∀y, Φ ∈ V. (3.2.10)
Assume that Ay ∈ V , then we can formulate the variational equation (3.2.7) as

y tt + C -1 Ay + C -1 Bξ = 0. (3.2.11)
Later, we introduce the linear unbounded operator A 0 and the linear bounded operator B as follows:

D( A 0 ) = {(y, z, ξ) ∈ H; z ∈ W and Ay ∈ V } , ( 3
.2.12)

A 0 U =     -z C -1 Ay + C -1 Bξ -z(1)     , U = (y, z, ξ) ∈ D( A 0 ) (3.2.13)
and

BU =     0 0 ξ     , U = (y, z, ξ) ∈ H. (3.2.14)
Then, denoting by U = (y, y t , ξ) the state of system (3.2.1) and define

A β = A 0 + β B with D( A β ) = D( A 0 )
, we can formulate system (3.2.1) into an evolution equation

   U t (t) + A β U (t) = 0, t > 0, U (0) = U 0 ∈ H. (3.2.15)
It is easy to prove that -A β is a maximal dissipative operator in the energy space H, therefore it generates a C 0 -semigroup (e -t A β ) t≥0 of contractions in the energy space H following Lumer-Phillips' theorem (see Pazy [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). Thus, we have the following results concerning the existence and uniqueness of the solution of the problem (3.2.15):

Theorem 3.2.1. For any initial data U 0 ∈ H, the problem (3.2.15) has a unique weak solution U (t) = e -t A β U 0 such that U ∈ C 0 ([0, ∞[, H). Moreover, if U 0 ∈ D( A 0 ), then the problem (3.2.15) has a strong solution U (t) = e -t A β U 0 such that U ∈ C 1 ([0, ∞[, H) ∩ C 0 [0, ∞[, D( A 0 ) .
In addition, it is easy to show that an element U = (y, z, ξ) ∈ D( A 0 ) if and only if y ∈ H 3 (0, 1) ∩ W, z ∈ W and y xx (1) = 0. In particular, the resolvent (I + A 0 ) -1 of -A 0 is compact in the energy space H (compare with Proposition 2.2.2). This implies with the compactness of B that the resolvent (I + A β ) -1 of -A β is also compact in the Hilbert space H. Consequently, the spectrum of A β (respectively A 0 ) consists entirely of isolated eigenvalues with finite multiplicities (see [START_REF] Kato | Perturbation theory for linear operators[END_REF]). Moreover, since the coefficients of A β (respectively A 0 ) are real, their eigenvalues appear by conjugate pairs. Now, we investigate the strong stability of the problem (3.2.15). Theorem 4.2 of [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF] shows that the semigroup of contractions (e -t A β ) t≥0 is strongly asymptotically stable in the energy space H, i.e. for any U 0 ∈ H, we have lim

t→+∞ e -t A β U 0 2 H = 0 if γ ≥ γ 0 where √ γ 0 sinh -1 ( √ γ 0 π) = 0. Using a numerical program we find γ 0 0.45001246517627713.
Moreover, from Theorem 4.3 of [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF] there exists an infinite numbers of 0 < γ < γ 0 such that the operator A β has eigenvalues on the imaginary axis and therefore for which problem (3.2.15) is not stable. Further, we know that the Rayleigh beam is not uniformly exponentially stable neither with one boundary direct control force (see [START_REF] Rao | A compact perturbation method for the boundary stabilization of the Rayleigh beam equation[END_REF]) nor with two dynamical boundary control (see [START_REF] Wehbe | Optimal energy decay rate for Rayleigh beam equation with dynamical boundary controls[END_REF]). Then, we look for the optimal polynomial energy decay rate for smooth initial data.

Spectral analysis of the operator

A β for β ≥ 0
In this section, we study the eigenvalues and the eigenvectors of the operator A β for β ≥ 0. First, let λ = β be an eigenvalue of the operator A β and U = (y, z, ξ) be an associated eigenfunction, then we have A β U = λU . Equivalently, λ and y verify the following system:

           y xxxx -γλ 2 y xx + λ 2 y = 0, y xxx (1) -γλ 2 y x (1) - λ λ -β y(1) = 0, y(0) = y x (0) = y xx (1) = 0. (3.3.1)
The general solution of the system (3.3.1) is

y = 4 i=1 c i (λ)e R i (λ)x , (3.3.2)
where R i (λ), i = 1, .., 4 are given by

R 1 (λ) = γλ 2 -λ √ γ 2 λ 2 -4 2 , R 2 (λ) = -R 1 (λ), (3.3.3) R 3 (λ) = γλ 2 + λ √ γ 2 λ 2 -4 2 , R 4 (λ) = -R 3 (λ).
Next, using the boundary conditions, we may write the system (3.3.1) as follows:

M β (λ) • C(λ) = 0, (3.3.4)
where

M β (λ) =     1 1 1 1 R1(λ) R2(λ) R3(λ) R4(λ) R 2 1 (λ)e R 1 (λ) R 2 2 (λ)e R 2 (λ) R 2 3 (λ)e R 3 (λ) R 2 4 (λ)e R 4 (λ) T 1,β (λ) T 2,β (λ) T 3,β (λ) T 4,β (λ)     , (3.3.5) C(λ) =        c 1 (λ) c 2 (λ) c 3 (λ) c 4 (λ)       
, where λ) ,

T i,β (λ) = R i (λ) 3 -γλ 2 R i (λ) - λ λ -β e R i (
for i = 1, 2, 3, 4.
Remark 3.3.1. First, like we did in Proposition 2.4.2, we find that the real part of any eigenvalue λ of A β is bounded, i.e.

∃c > 0, ∀λ ∈ σ( A β ), 0 < (λ) ≤ c.
Next, let λ 0 be an eigenvalue of A 0 and U 0 = (y 0 , z 0 , ξ 0 ) ∈ D( A 0 ) an associated eigenvector. Then, as we did in Proposition 2.3.2, we can easily prove that λ 0 is simple and ξ 0 = 0.

Next, we study the asymptotic behavior of the eigenvalues of the operators A β for β ≥ 0 in the following proposition:

Proposition 3.3.2. (Spectrum of A β ) Let β ≥ 0. Then there exists k β ∈ N * sufficiently large such that the spectrum σ(A β ) of A β is given by σ(A β ) = σ β,0 ∪ σ β,1 , (3.3.6)
where

σ β,0 = {κ β,j } j∈J β , σ β,1 = {λ β,k } k∈Z |k|≥k β , σ β,0 ∩ σ β,1 = ∅, (3.3.7)
where J β is a finite set and λ β,k is simple and satisfies the following asymptotic behavior:

λ β,k = i   α k - ( 1 2 √ γ + tanh( 1 √ γ )) γ 3 2 α k + 2(-1) k γ 5 2 cosh( 1 √ γ )α 2 k (3.3.8) + E α 3 k + F α 4 k + β π 4 cosh( 1 √ γ ) × 1 k 4 + o( 1 k 4 ), with α k = kπ √ γ + π 2 √ γ , (3.3.9) E = 1 3γ 7 2 tanh( 1 √ γ ) 3 + 1 γ 7 2 tanh( 1 √ γ ) (3.3.10) - 1 γ 2 1 + 1 γ + 1 2γ 2 tanh( 1 √ γ ) 2 + 1 γ 2 + 1 γ 4 , F = (-1) k γ 3 cosh( 1 √ γ ) (2 + 6 γ + 1 γ 2 ) tanh( 1 √ γ )- (3.3.11) 1 γ 3 2 (1 + tanh( 1 √ γ ) 2 ) .
Proof: The proof uses the same strategy than the one from Proposition 2.4.3. In Step 1 we furnishe an asymptotic development of the characteristic equation for large λ.

Step 2 uses Rouché's theorem to localize high frequency eigenvalues. In step 3, we perform a limited development stopped when a non zero real part appear. For the sake of completeness, we give the details. For simplicity, we denote R i (λ) by R i .

Step 1. First, we start by the expansion of R 1 and

R 3 when |λ| → ∞ R 1 = 1 √ γ + 1 2γ 5 2 λ 2 + O( 1 λ 4 ) (3.3.12)
and

R 3 = λ √ γ - 1 2λγ 3 2 + O( 1 λ 3 ). (3.3.13)
Using the expansions (3.3.12) and (3.3.13), we find the following asymp-totic behavior:

R 2 1 e R 1 = 1 γ + ( 1 2γ 7 2 + 1 γ 3 ) 1 λ 2 e 1 √ γ + O( 1 λ 4 ), (3.3.14) R 2 1 e R 1 = 1 γ + ( 1 2γ 7 2 + 1 γ 3 ) 1 λ 2 e 1 √ γ + O( 1 λ 4 ), (3.3.15) R 2 3 e R 3 = γλ 2 - λ 2 √ γ + 1 8γ 2 - 1 γ -( 1
8γ 5 2 + 1 48γ 7 2 ) 1 λ e √ γλ (3.3.16) + O( 1 λ 2 )
and

R 2 4 e R 4 = γλ 2 + λ 2 √ γ + 1 8γ 2 - 1 γ + ( 1
8γ 5 2 + 1 48γ 7 2 ) 1 λ e -√ γλ (3.3.17) + O( 1 λ 2 ).
Similarly, we get

T β,1 (λ) = - √ γλ 2 - 1 2γ 2 + 1 2γ 3 2 -1 - β λ e 1 √ γ + O( 1 λ 2 ), (3.3.18)
T β,2 (λ) = √ γλ 2 - 1 2γ 2 - 1 2γ 3 2 -1 - β λ e -1 √ γ + O( 1 λ 2 ), (3.3.19)
T β,3 (λ) = - λ √ γ + 1 2γ 2 -1 + 1 2γ 3 2 - 1 2γ 5 2 - 1 8γ 7 2 -β 1 λ (3.3.20) + -β 2 + β 2γ 3 2 - 1 8γ 3 + 7 8γ 4 + 1 48γ 5 1 λ 2 e √ γλ + O( 1 λ 3 )
and

T β,4 (λ) = λ √ γ + 1 2γ 2 -1 + - 1 2γ 3 2 + 1 2γ 5 2 + 1 8γ 7 2 -β 1 λ (3.3.21) + -β 2 - β 2γ 3 2 - 1 8γ 3 + 7 8γ 4 + 1 48γ 5 1 λ 2 e -√ γλ + O( 1 λ 3 ).
Combining (3.3.12)-(3.3.21) and (3.3.5), we can write

M β (λ) =                      1 1 1 1 q 1 + O( 1 λ 4 ) q 2 + O( 1 λ 4 ) q 3 + O( 1 λ 2 ) q 4 + O( 1 λ 2 ) q 5 e 1 √ γ + O( 1 λ 4 ) q 6 e -1 √ γ + O( 1 λ 4 ) q 7 e √ γλ + O( 1 λ 2 ) q 8 e -√ γλ + O( 1 λ 2 ) q 9 e 1 √ γ + O( 1 λ 2 ) q 10 e -1 √ γ + O( 1 λ 2 ) q 11 e √ γλ + O( 1 λ 3 ) q 12 e -√ γλ + O( 1 λ 3 )                     
, where

q 1 = 1 √ γ + 1 2γ 5 2 λ , q 2 = - 1 √ γ - 1 2γ 5 2 λ 2 , q 3 = λ √ γ- 1 2λγ 3 2 , q 4 = -λ √ γ+ 1 2λγ 3 2 , q 5 = 1 γ + ( 1 2γ 7 2 + 1 γ 3 ) 1 λ 2 , q 6 = 1 γ + (- 1 2γ 7 2 + 1 γ 3 ) 1 λ 2 , q 7 = γλ 2 - λ 2 √ γ + 1 8γ 2 - 1 γ -( 1
8γ 5 2 + 1 48γ 7 2 ) 1 λ , q 8 = γλ 2 + λ 2 √ γ + 1 8γ 2 - 1 γ + ( 1 8γ 5 2 + 1 48γ 7 2 ) 1 λ , q 9 = - √ γλ 2 - 1 2γ 2 + 1 2γ 3 2 -1 - β λ , q 10 = √ γλ 2 - 1 2γ 2 - 1 2γ 3 2 -1 - β λ , q 11 = - λ √ γ + 1 2γ 2 -1 + 1 2γ 3 2 - 1 2γ 5 2 - 1 8γ 7 2 -β 1 λ + -β 2 + β 2γ 3 2 - 1 8γ 3 + 7 8γ 4 + 1 48γ 5 1 λ 2 and q 12 = λ √ γ + 1 2γ 2 -1 + - 1 2γ 3 2 + 1 2γ 5 2 + 1 8γ 7 2 -β 1 λ + -β 2 + β 2γ 3 2 - 1 8γ 3 + 7 8γ 4 + 1 48γ 5 1 λ 2 .
Then, after long computations, we find the following asymptotic devel- 1), (3.3.22) where

opment of f β (λ) = det(M β (λ)): f β (λ) = λ 5 f 0 (λ)+λ 4 f 1 (λ)+λ 3 f 2 (λ)+λ 2 f β,3 (λ)+λf β,4 (λ)+O(
f 0 (λ) = L 0 (γ) cosh( √ γλ), L 0 (γ) = 4γ 2 cosh( 1 √ γ ), (3.3.23
)

f 1 (λ) = L 1 (γ) sinh( √ γλ), (3.3.24) L 1 (γ) = -2 √ γ cosh( 1 √ γ ) + 2 √ γ sinh( 1 √ γ ) , f 2 (γ) = -8 + L 2 (γ) cosh( √ γλ), (3.3.25) L 2 (γ) = 1 2γ -8 cosh( 1 √ γ ) + 4 √ γ + 4γ √ γ sinh( 1 √ γ ), f β,3 (λ) = L β,3c (γ) cosh( √ γλ) + L 3s (γ) sinh( √ γλ), (3.3.26) L β,3c (γ) = 4βγ 3 2 sinh( 1 √ γ ), L 3s (γ) = -2 + 3 2γ 2 sinh( 1 √ γ ) (3.3.27) - 1 12γ 5 2 + 1 2γ √ γ + 4 √ γ cosh( 1 √ γ )
and where

f β,4 (λ) = L 4c (γ) cosh( √ γλ) + L β,4s (γ) sinh( √ γλ), (3.3.28) L 4c (γ) = 1 3γ 7 2 - 1 2γ 5 2 + 1 2γ 3 2 - 10 √ γ sinh( 1 √ γ ) (3.3.29) + 2 γ + 13 2γ 2 + 1 2γ 3 cosh( 1 √ γ ), L β,4s (γ) = -4 √ γβ cosh( 1 √ γ ) + 1 √ γ sinh( 1 √ γ ) . (3.3.30)
Since the real part of λ is bounded, the functions f i , i ∈ {0, 1, 2, 3, 4} are also bounded. For convenience we set

S β (λ) = f β (λ) λ 5 =f 0 (λ) + f 1 (λ) λ + f 2 (λ) λ 2 + f β,3 (λ) λ 3 (3.3.31) + f β,4 (λ) λ 4 + O( 1 λ 5 ).
Step 2. Large eigenvalues of A β . We look at the roots of S β . It is easy to see that the roots of f 0 are simple and given by

z k = iα k = i kπ √ γ + π 2 √ γ .
Then, with the help of Rouché's theorem, there exists k β ∈ N * large enough, such that ∀|k| ≥ k β the large eigenvalues of A β (denoted by λ β,k ) are simple and close to z k , i.e.

λ β,k = iα k + o β (1), |k| → ∞. (3.3.32)
Equivalently we can write

λ β,k = iα k + ζ β,k , lim |k|→∞ ζ β,k = 0. ( 3
= S β (λ β,k ) =L 0 (γ) cosh( √ γλ β,k ) + L 1 (γ) sinh( √ γλ β,k ) λ β,k (3.3.34) + -8 + L 2 (γ) cosh( √ γλ β,k ) λ 2 β,k + L β,3c (γ) cosh( √ γλ β,k ) λ 3 β,k + L 3s (γ) sinh( √ γλ β,k ) λ 3 β,k + L 4c (γ) cosh( √ γλ β,k ) λ 4 β,k + L β,4s (γ) sinh( √ γλ β,k ) λ 4 β,k + O( 1 λ 5 β,k
).

On the other hand, using (3.3.33) we obtain cosh(

√ γλ β,k ) = i(-1) k sinh( √ γζ β,k ) (3.3.35) = i(-1) k   √ γζ β,k + γ γζ 3 β,k 9 + o(ζ 4 β,k )   , sinh( √ γζ β,k ) = i(-1) k cosh( √ γζ β,k ) (3.3.36) = i(-1) k 1 + γζ 2 β,k 2 + γ 2 ζ 4 β,k 6 + o(ζ 4 β,k ) and 1 λ β,k = 1 iα k 1 - ζ β,k iα k + o( ζ 2 β,k α 2 k ) = - i α k + ζ β,k α 2 k + o( ζ 2 β,k α 2 k
).

(3.3.37)

Similarly, we get

1 λ 2 β,k = - 1 α 2 k -2i ζ 2 β,k α 3 k + o( ζ 2 β,k α 2 k ), (3.3.38)
1 λ 3 β,k = i α 3 k -3 ζ β,k α 4 k + o( ζ 2 β,k α 2 k ) (3.3.39)
and

1 λ 4 β,k = 1 α 4 k + 4i ζ β,k α 5 k + o( ζ 2 β,k α 2 k ). ( 3
0 =iL 0 (γ) √ γζ β,k + iγ √ γL 0 (γ) 6 ζ 3 β,k + L 1 (γ) α k + iL 1 (γ) α 2 k ζ β,k (3.3.41) + γL 1 (γ) 2α k ζ 2 β,k + 8(-1) k α 2 k + 16i(-1) k α 3 k ζ β,k - iL 2 (γ) α 2 k ζ β,k - √ γL β,3c (γ) α 3 k ζ β,k - L 3s (γ) α 3 k + iL β,4s (γ) α 4 k + o(ζ 4 β,k ) + o( ζ 2 β,k α 2 k ) + o( 1 α 4 k ).
Next, using (3.3.41) we find the first development of ζ k,β given by

ζ β,k = iL 1 (γ) √ γL 0 (γ)α k + e β,
ζ β,k = iL 1 (γ) √ γL 0 (γ)α k + 8i(-1) k √ γL 0 (γ)α 2 k + e β,
where

Q 1 = 1 3γL 3 0 (γ) - √ γL 3 1 (γ) -3L 0 (γ)L 2 1 (γ) (3.3.46) +3 √ γL 0 (γ)L 1 (γ)L 2 (γ) -3 √ γL 2 0 (γ)L 3s (γ)
and where

e β,3 = O β ( 1 α 4 k
). Then, substituting (3.3.45) into (3.3.44) yields

ζ β,k = iL 1 (γ) √ γL 0 (γ)α k + 8i(-1) k √ γL 0 (γ)α 2 k + iQ 1 α 3 k + e 3,
e β,3 = iQ 2 α 4 k + Q β,3 α 4 k + o( 1 α 4 k ), (3.3.48)
where

Q 2 = 4(-1) k γL 3 0 (γ) 2 √ γL 0 (γ)L 2 (γ) - √ γL 2 1 (γ) -6L 0 (γ)L 1 (γ) (3.3.49)
and where

Q β,3 = L β,3c (γ)L 1 (γ) -L 0 (γ)L β,4s (γ) √ γL 2 0 . (3.3.50)
Then, substituting (3.3.48) into (3.3.47) yields

ζ β,k = iL 1 (γ) √ γL 0 (γ)α k + 8i(-1) k √ γL 0 (γ)α 2 k + iQ 1 α 3 k + i Q 2 α 4 k + Q β,3 α 4 k + o( 1 k 4 ). ( 3
ζ β,k = i   - ( 1 2 √ γ + tanh( 1 √ γ )) γ 3 2 α k + 2(-1) k γ 5 2 cosh( 1 √ γ )α 2 k + E α 3 k + F α 4 k   + β π 4 cosh( 1 √ γ ) × 1 k 4 + o( 1 k 4 ),
where E and F are given by (3.

Φ β,k = (y β,k , -λ β,k y β,k , y β,k (1)) (3.3.52)
is the eigenvector associated with the eigenvalue λ β,k of high frequency and by {Φ β,j,l } m β,j l=1 the Jordan chain of root vectors associated with the eigenvalue λ β,j of low frequency (Φ 0,j,l are in fact eigenvectors of A 0 ) . Thus we obtain a system of root vectors of

A β {Φ β,k , |k| ≥ k β } ∪ {Φ β,j,l , 1 ≤ l ≤ m β,j , j ∈ J β } . ( 3
       y β,k (1) = - 2 cosh( 1 √ γ ) + o(1) = 0, y β,k W ∼ |k| 2 , y β,k V ∼ |k|, |k| → ∞. (3.3.54)
and we deduce that

Φ β,k H ∼ |k| 2 , |k| → ∞. ( 3
       c 1 (λ k ) + c 2 (λ k ) + c 3 (λ k ) = -1, R 1 (λ k )c 1 (λ k ) + R 2 (λ k )c 2 (λ k ) + R 3 (λ k )c 3 (λ k ) = -R 4 (λ k ), R 2 1 (λ k )e R 1 (λ k ) c 1 (λ k ) + R 2 2 (λ k )e R 2 (λ k ) c 2 (λ k ) + R 2 3 (λ k )e R 3 (λ k ) = -R 2 4 (λ k )e R 4 (λ k ) .
Next, using Cramer's rule, we obtain

c 1 (λ k ) = b 1 b 4 , c 2 (λ k ) = b 2 b 4 , c 3 (λ k ) = b 3 b 4 , (3.3.56)
where

b 1 =2R 1 (λ k )R 3 (λ k ) 2 sinh(R 3 (λ k )) -2R 3 (λ k ) 3 cosh(R 3 (λ k )) (3.3.57) + 2R 1 (λ k ) 2 R 3 (λ k )e -R 1 (λ k ) , b 2 =2R 1 (λ k )R 3 (λ k ) 2 sinh(R 3 (λ k )) + 2R 3 (λ k ) 3 ) cosh(R 3 (λ k )) (3.3.58) -2R 1 (λ k ) 2 R 3 (λ k )e R 1 (λ k ) , b 3 =2R 1 (λ k ) 2 R 3 (λ k ) sinh(R 1 (λ k )) -2R 1 (λ k ) 3 cosh(R 1 (λ k )) (3.3.59) + R 1 (λ k )R 3 (λ k ) 2 e -R 3 (λ k )
and where

b 4 =2R 1 (λ k ) 2 R 3 (λ k ) sinh(R 1 (λ k )) + 2R 1 (λ k ) 3 cosh(R 1 (λ k )) (3.3.60) -R 1 (λ k )R 3 (λ k ) 2 e R 3 (λ k ) .
First, we study the behavior of b 1 . Inserting (3.3.12) and (3.3.13) (with λ = λ k ) in (3.3.57) we find after some computations

b 1 = -2γ 3 2 λ 3 k cosh( √ γλ k ) + (1 + 2 √ γ)λ 2 k sinh( √ γλ k ). (3.3.61)
Now, using the asymptotic behavior (3.3.8) we find

           cosh( √ γλ k ) = i(-1) k (1 + 2 √ γ tanh( 1 √ γ )) 2γ 3 2 λ k + O( 1 λ 2 k ), sinh( √ γλ k ) = i(-1) k + O( 1 λ 2 k
).

(3.3.62)

Then, inserting (3.3.62) in (3.3.61) we find again after some computations

b 1 = 2 √ γi(-1) k 1 -tanh( 1 √ γ ) λ 2 k + O(λ k ). (3.3.63)
Similarly, long computations left to the reader yield

b 2 = 2i(-1) k √ γ 1 + tanh( 1 √ γ ) λ 2 k + O(λ k ), (3.3.64) b 3 = -2 √ γi(-1) k λ 2 k + O(λ k ) (3.3.65)
and

b 4 = -2 √ γi(-1) k λ 2 k + O(λ k ). ( 3
                       c 1 (λ k ) = -1 + tanh( 1 √ γ ) + O( 1 |λ k | ), c 2 (λ k ) = -1 -tanh( 1 √ γ ) + O( 1 |λ k | ), c 3 (λ k ) = 1 + O( 1 |λ k | ), c 4 (λ k ) = 1. (3.3.67)
Finally, we have found that a solution of (3.3.4) has the form

C(λ k ) = C 0 + O( 1 |λ k | ), (3.3.68)
where

C 0 = (-1 + tanh( 1 √ γ ), -1 -tanh( 1 √ γ ), 1, 1). (3.3.69)
Note that the corresponding solution y k of (3.3.1) is given by

y k = 4 i=1 c i (λ k )e R i (λ k ) . (3.3.70)
Step 2. Estimate of y k [START_REF] Abbas | The multidimensional wave equation with generalized acoustic boundary conditions ii: Polynomial stability[END_REF]. From equation (3.3.70), we have

y k (1) = c 1 (λ k )e R 1 (λ k ) + c 2 (λ k )e R 2 (λ k ) + c 3 (λ k )e R 3 (λ k ) + c 4 (λ k )e R 4 (λ k ) ,
where we recall that for i ∈ {1, 2, 3, 4} R i (λ k ) are given by (3.3.3) and c i satisfy (3.3.67). Therefore using the series expansions (3.3.8) and (3.3.12)-(3.3.13) for λ = λ k we easily find

y k (1) = - 2 cosh( 1 √ γ ) + o(1) = 0. (3.3.71)
Step 3. Estimates of y k W and y k V . We start with

y k 2 W = 1 0 |y k,xx | 2 dx = 4 i=1 4 j=1 c i (λ k )R i (λ k ) 2 1 0 e R i (λ k )x e R j (λ k )x dx c j (λ k )R j (λ k ) 2 = C k G k C k T , ( 3.3.72)
where

G k = 1 0 e (R i (λ k )+R j (λ k ))x dx 1≤i,j≤4
and where

C k = (c i (λ k )R i (λ k ) 2 ) i=1,..,4 .
First, using (3.3.8), then from (3.3.12) and (3.3.13) we can write R 1 (λ k ) and R 3 (λ k ) as follows

R 1 (λ k ) = q 1 + ir 1 , (3.3.73)
where

q 1 = 1 √ γ + O( 1 λ 2 k ), r 1 = - γ 7 2 β cosh( 1 √ γ )λ 7 k + O( 1 λ 8 k ) and R 3 (λ k ) = q 3 + ir 3 , (3.3.74)
where

q 3 = γ 5 2 β λ 4 k + O( 1 λ 6 k ), r 3 = √ γλ k + O(1).
Then, the fact that R 2 (λ k ) = -R 1 (λ k ) and R 4 (λ k ) = -R 3 (λ k ) and using the asymptotic behavior (3.3.73)-(3.3.74) we directly find

1 0 e (R 1 (λ k )+R 2 (λ k ))x dx = 1 0 e (R 2 (λ k )+R 1 (λ k ))x dx = 1 0 e (R 3 (λ k )+R 3 (λ k ))x dx = 1 0 e (R 4 (λ k )+R 4 (λ k ))x dx = 1 + O( 1 λ 4 k ).
Moreover, using the asymptotic behavior (3.3.73)-(3.3.74) we find that G k is given as follows:

G k = G 0 + O( 1 λ k ), (3.3.75)
where

G 0 =           √ γ 2 (e 2 √ γ -1) 1 0 0 1 √ γ 2 (1 -e -2 √ γ ) 0 0 0 0 1 0 0 0 0 1           (
y k 2 W = γ 2 |λ k | 4 + O(|λ k | 3 ) ∼ |k| 4 , |k| → ∞. (3.3.78)
Similarly, we easily prove that

y k 2 L 2 (0,1) ∼ 1, y k,x 2 L 2 (0,1) ∼ |k| 2 , |k| → ∞.
Therefore, we deduce that

y k V ∼ |k|, |k| → ∞. ( 3

Riesz basis and optimal energy decay rate

Our main result is the following optimal polynomial-type decay estimation:

Theorem 3.4.1. (Optimal energy decay rate) Assume that β > 0 and that γ ≥ γ 0 . Then, for all initial data U 0 ∈ D( A 0 ), there exists a constant c > 0 independent of U 0 , such that the energy of the problem (3.2.15) satisfies the following estimation:

E(t) ≤ c √ t U 0 2 D( A 0 ) . (3.4.1)
Moreover, the energy decay rate (3.4.1) is optimal.

First, we prove that the set of the generalized eigenvectors associated with A β forms a Riesz basis of H by the following theorem:

Theorem 3.4.2. The set of generalized eigenvectors associated with σ( A β ) forms a Riesz basis of H.

Proof: First, since A 0 is a skew-adjoint operator, its set of normalized eigenvectors form an orthonormal basis in H. Next, we prove the following property:

+∞ k=max{k 0 ,k β } Φ β,k -Φ 0,k H < +∞ (3.4.2)
where

Φ β,k = ( y β,k , z β,k , ξ β,k ) = 1 Φ 0,k H Φ β,k , ∀|k| ≥ k β (3.4.3)
and where

Φ 0,k = ( y 0,k , z 0,k , ξ 0,k ) = 1 Φ 0,k H Φ 0,k , ∀|k| ≥ k 0 . (3.4.4)
We first estimate

Φ β,k -Φ 0,k 2
H = y β,k -y 0,k 2 W + z β,k -z 0,k 2 V + | ξ β,k -ξ 0,k | 2 . (3.4.5)
For clarity, we divide the proof into several steps.

Step 1. Estimate of y β,k -y 0,k from (3.4.3) and (3.4.4) we obtain

y β,k -y 0,k W ∼ 1 |k| 2 y β,k -y 0,k W .
(3.4.6)

Next, using (3.3.70) we obtain

y β,k,xx -y 0 0,k,xx ∼ 1 |k| 2 R 2 1 (λ β,k )c 1 (λ β,k )e R 1 (λ β,k )x -R 2 1 (λ 0,k )c 1 (λ 0,k )e R 1 (λ 0,k )x + 1 |k| 2 R 2 1 (λ β,k )c 2 (λ β,k )e -R 1 (λ β,k )x -R 2 1 (λ 0,k )c 2 (λ 0,k )e -R 1 (λ 0,k )x + 1 |k| 2 R 2 3 (λ β,k )c 3 (λ β,k )e R 3 (λ β,k )x -R 2 3 (λ 0,k )c 3 (λ 0,k )e R 3 (λ 0,k )x + 1 |k| 2 R 2 3 (λ β,k )c 4 (λ β,k )e -R 3 (λ β,k )x -R 2 3 (λ 0,k )c 4 (λ 0,k )e -R 3 (λ 0,k )x .
For simplicity we denote c i (λ β,k ) by c β,k i and c i (λ 0,k ) by c 0,k i for i ∈ {1, 2, 3, 4}. Then, a direct computation gives

y β,k -y 0,k 2 W J 1 + J 2 + J 3 + J 4 (3.4.7)
where

J 1 = 1 |k| 4 1 0 |R 2 1 (λ β,k ) -R 2 1 (λ 0,k )| 2 |c β,k 1 | 2 |e R 1 (λ β,k )x | 2 dx (3.4.8) + 1 |k| 4 1 0 |R 2 1 (λ 0,k )| 2 |c β,k 1 -c 0,k 1 | 2 |e R 1 (λ β,k )x | 2 dx + 1 |k| 4 1 0 |R 2 1 (λ 0,k )| 2 |c 0,k 1 | 2 |e R 1 (λ β,k )x -e R 1 (λ 0,k )x | 2 dx, J 2 = 1 |k| 4 1 0 |R 2 1 (λ β,k ) -R 2 1 (λ 0,k )| 2 |c β,k 2 | 2 |e -R 1 (λ β,k )x | 2 dx (3.4.9) + 1 |k| 4 1 0 |R 2 1 (λ 0,k )| 2 |c β,k 2 -c 0,k 2 | 2 |e -R 1 (λ β,k )x | 2 dx + 1 |k| 4 1 0 |R 2 1 (λ 0,k )| 2 |c 0,k 2 | 2 |e -R 1 (λ β,k )x -e -R 1 (λ 0,k )x | 2 dx, J 3 = 1 |k| 4 1 0 |R 2 3 (λ β,k ) -R 2 3 (λ 0,k )| 2 |c β,k 3 | 2 |e R 3 (λ β,k )x | 2 dx (3.4.10) + 1 |k| 4 1 0 |R 2 3 (λ 0,k )| 2 |c β,k 3 -c 0,k 3 | 2 |e R 3 (λ β,k )x | 2 dx + 1 |k| 4 1 0 |R 2 3 (λ 0,k )| 2 |c 0,k 3 | 2 |e R 3 (λ β,k )x -e R 3 (λ 0,k )x )| 2 dx
and

J 4 = 1 |k| 4 1 0 |R 2 3 (λ β,k ) -R 2 3 (λ 0,k )| 2 |e -R 3 (λ β,k )x | 2 dx (3.4.11) + 1 |k| 4 1 0 |R 3 (λ 0,k )| 2 |c β,k 4 -c 0,k 4 | 2 |e -R 3 (λ β,k )x | 2 dx + 1 |k| 4 1 0 |R 2 3 (λ 0,k )| 2 |e -R 3 (λ β,k )x -e -R 3 (λ 0,k )x | 2 dx.
                   R 1 (λ β,k ) 2 -R 1 (λ 0,k ) 2 ∼ 1 |k| 4 , e R 1 (λ β,k ) -e R 1 (λ 0,k ) ∼ 1 |k| 2 , c β,k 1 -c 0,k 1 ∼ 1 |k| (3.4.12)
and

                   R 3 (λ β,k ) 2 -R 3 (λ 0,k ) 2 ∼ 1 |k| 3 , e R 3 (λ β,k )x -e R 3 (λ 0,k )x ∼ 1 |k| , c β,k 3 -c 0,k 3 ∼ 1 |k| . (3.4.13) Since c β,k 1 ∼ c 0,k 1 ∼ e R 1 (λ β,k )x ∼ R 2 1 (λ 0,k
) ∼ 1 and using (3.4.12), then from (3.4.8) we obtain

J 1 ∼ 1 |k| 4 1 0 1 |k| 8 dx + 1 |k| 4 1 0 1 |k| 2 dx + 1 |k| 4 1 0 1 |k| 4 dx ∼ 1 |k| 6 . (3.4.14)
Similarly, we get

J 2 ∼ 1 |k| 8 . (3.4.15)
In the same way, since c β,k 2 and using (3.4.13) then from (3.4.10) we obtain

3 ∼ c 0,k 3 ∼ e R 3 (λ β,k ) ∼ 1, R 1 (λ β,k ) 2 ∼ |k|
J 3 ∼ 1 |k| 4 1 0 1 |k| 6 dx + 1 |k| 4 1 0 |k| 2 dx + 1 |k| 4 1 0 |k| 2 dx ∼ 1 |k| 2 . (3.4.16)
Similarly, we get

J 4 ∼ 1 |k| 2 .
(3.4.17)

Finally, using (3.4.14)-(3.4.17), from (3.4.7) we deduce

y β,k -y 0,k 2 W 1 |k| 2 .
(3.4.18)

Step 2. Estimates

z β,k -z 0,k 2 V and | ξ β,k -ξ 0,k | 2 . First, since Φ 0,k H ∼ |k|
z k -z 0 k 2 V ∼ 1 |k| 4 z k -z 0 k 2 V . (3.4.19)
Then, using (3.3.52) we obtain

z β,k -z 0,k 2 V ∼ 1 |k| 4 λ β,k y β,k -λ 0,k y 0,k 2 V (3.4.20) ≤ 1 |k| 4 |λ β,k -λ 0,k | 2 y β,k 2 V + |λ 0,k | 2 |k| 4 y β,k -y 0,k 2 V . Now, since |λ β,k -λ 0,k | ∼ 1 |k| 4 and y β,k V ∼ |k| 2 we get 1 |k| 4 |λ β,k -λ 0,k | 2 y β,k 2 V ∼ 1 |k| 8 . ( 3
Φ β,k -Φ 0,k 2 H 1 |k| 2
and consequently

∞ k=max{k 0 ,k β } Φ β,k -Φ 0,k 2 H < +∞.
Therefore, using a clarified form of Guo's theorem given by Theorem 1.2.10 in [START_REF] Abdallah | Stabilisation et approximation de certains systèmes distribués par amortissement dissipative et de signe indéfini[END_REF] (see also [START_REF] Guo | Riesz basis approach to the stabilization of a flexible beam with a tip mass[END_REF]Theorem 6.3]), we deduce that the set of generalized eigenvectors associated with σ( A β ) forms a Riesz basis in H.

Proof of Theorem 3.4.1: First, using (3.3.8) we have (λ k ) ∼ 1 k 4 . Next, from Theorem 3.4.2 we know that the set of generalized eigenvectors associated with σ( A β ) form a Riesz basis of H. Then, applying Theorem 2.1 given in [START_REF] Loreti | Optimal energy decay rate for partially damped systems by spectral compensation[END_REF] (see also [START_REF] Littman | Stabilization of a hybrid system of elasticity by feedback boundary damping[END_REF], [START_REF] Loreti | Optimal energy decay rate for partially damped systems by spectral compensation[END_REF] and [START_REF] Wehbe | Rational energy decay rate for a wave equation with dynamical control[END_REF]), we deduce the optimal polynomial energy decay rate (3.4.1) for smooth initial data.

Open problems

The extensions of our results to multidimensional space are open problems. We cannot use the same strategy since it is based on the spectrum study and it is difficult to analyze it in these type of space. We maybe can apply a multiplier method like these given in [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF][START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF]. The exact controllability of Rayleigh beam equation and the stabilization of the numerical approximation schemes with static or dynamic boundary control force are also an open problems.

Indirect Stability of the wave equation with a dynamic boundary control

Abstract: In this chapter, we consider a damped wave equation with a dynamic boundary control. First, using Arendt and Batty's theorem (see [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF]) we show the strong stability of our system. Next, we show that our system is not uniformly stable in general, since it is the case of the unit disk. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach. In a first step, by giving some sufficient conditions on the boundary of our domain and by using the exponential decay of the wave equation with a standard damping, we prove a polynomial decay in 1 t 1 4 of the energy. In a second step, under appropriated condition on the boundary of our system named by multiplier control conditions, we establish a polynomial decay in 1 t of the energy. Later, we show that such a polynomial decay seems to be also available even if the previous conditions is not satisfied. For this aim, we consider our system on the unit square of the plane. Using a spectral analysis, we show that the decay rate to zero of the energy is not exponential. Then, using a method based on a Fourier analysis, a specific analysis of the obtained 1-d problem combining Ingham's inequality and an interpolation method, we establish a polynomial decay in 1 t of the energy for sufficiently smooth initial data.

Introduction

Let Ω be a bounded domain of R d , d ≥ 2, with a Lipschitz boundary Γ = Γ 0 ∪ Γ 1 , with Γ 0 and Γ 1 open subsets of Γ such that Γ 0 ∩ Γ 1 = ∅ and Γ 1 is non empty. In [START_REF] Fourrier | Analysis of existence, regularity and stability of solutions to wave equations with dynamic boundary conditions[END_REF][START_REF] Fourrier | Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions[END_REF][START_REF] Graber | Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions[END_REF], N. Fourrier, I. Lasiecka and P. Graber studied the following problem (under the assumption that Γ 0 ∩ Γ 1 = ∅):

                       u tt -∆u -k Ω ∆u t + c Ω u t = 0, in Ω × R * + , u = 0, on Γ 0 × R * + , u -w = 0, on Γ 1 × R * + , w tt -k Γ ∆ T (αw t + w) + ∂ ν (u + k Ω u t ) + c Γ w t = 0, on Γ 1 × R * + , w = 0, on ∂Γ 1 × R * + , u(•, •, 0) = u 0 , u t (•, •, 0) = u 1 , in Ω, w(•, 0) = w 0 , w t (•, 0) = w 1 , on Γ 1 , (4.1.1)
where ∂ ν means the normal derivative on Γ 1 , ν is the unit outward normal vector along the boundary and ∆ T denotes the Laplace-Beltrami operator on Γ. In system (4.1.1), two types of dissipation appear: internal (if c Ω > 0) and boundary (if k Γ > 0) frictional ones and internal (if k Ω > 0) and boundary (if k Γ α > 0) viscoelastic ones. A physical description of this model is first described in [START_REF] Morse | Theoretical Acoustics. International series in pure and applied physics[END_REF]. In [START_REF] Fourrier | Analysis of existence, regularity and stability of solutions to wave equations with dynamic boundary conditions[END_REF][START_REF] Fourrier | Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions[END_REF], it is shown that system (4.1.1) is exponentially stable if one of the following three conditions is satisfied: k Ω > 0 (interior viscoelastic damping), or c Ω > 0 and c Γ > 0 (internal and boundary frictional damping) or c Ω > 0 and k Γ α > 0 (internal frictional damping and boundary viscoelastic damping). The first case corresponds to a direct damping, while the other cases correspond to a phenomenon of overdamping. This phenomenon was the motivation of these authors to study the balance between the competiting dampings. On the contrary, in this chapter, we are interested in the important case where only a boundary frictional damping occurs, i.e. k Ω = c Ω = α = 0 and k Γ = c Γ = 1. More precisely, we consider the following problem:

                       u tt -∆u = 0, in Ω × R * + , u = 0, on Γ 0 × R * + , u -w = 0, on Γ 1 × R * + , w tt -∆ T w + ∂ ν u + w t = 0, in Γ 1 × R * + , w = 0, on ∂Γ 1 × R * + , u(•, 0) = u 0 , u t (•, 0) = u 1 , in Ω, w(•, 0) = w 0 , w t (•, 0) = w 1 , on Γ 1 . (4.1.2)
In this case, the damping term is the term w t in the fourth equation of (4.1.2) and therefore the system in Ω is only damped indirectly. The notion of the indirect damping mechanisms has been introduced by Russell in [START_REF] Russell | Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods[END_REF][START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF], and since that time it retains the attention of many authors, because several models from acoustic theory enter in this framework.

There are many results concerning the wave equation with different models of damping: In [START_REF] Cavalcanti | Geometrically constrained stabilization of wave equations with Wentzell boundary conditions[END_REF], M. Cavalcanti and al. considered the following wave equation with Wentzell boundary conditions:

     ∆u + a(x)g(u t ) = 0, in Ω×]0, T [, βu tt + b(x)g 0 (u t ) + ∂ ν u + u = γ∆ T u, on Γ×]0, T [, u(0, x) = u 0 (x), u t (0, x) = u 1 (x), in Ω, (4.1.3)
where Ω denotes a bounded class-C 2 domain in R 3 with boundary Γ; the feedback maps g, g 0 are continuous monotone increasing, both vanishing at 0; a(x) ∈ L ∞ (Ω) and b(x) ∈ C 1 (Γ) are non-negative functions localizing the effect of the feedbacks to some subset of the domain and its boundary; the constants β, γ are non-negative. The system (4.1.3) can be seen as a hybrid system described by two potentially independent (but coupled) evolutions: one in Ω and another one on the boundary Γ, or else as single wave equation with higher-order tangential boundary conditions. M. Cavalcanti and al. have shown that system (4.1.3) for β > 0 (Wentzell dynamic boundary conditions) or for β = 0 (Wentzell static boundary conditions), can be exponentially stable under appropriated conditions. In [START_REF] Aissa | Stabilization of a wave-wave system[END_REF], N. Aissa and D. Hamroun considered the following system of coupled wave equations:

                             u tt -∆u = d • ∇u t , in Ω × R * + , u = 0, on Γ 0 × R * + , ∂ ν u = -(-∂ xx ) 1 2 v t , on Γ 1 × R * + , v tt -v xx -(-∂ xx ) 1 2 (u t ) = 0, on Γ 1 × R * + , v(0, t) = v(1, t) = 0, in R * + , u(0) = u 0 , u t (0) = u 1 , in Ω, v(0) = v 0 , v t (0) = v 1 , on Γ 1 , (4.1.4)
where Ω is a square in R 2 , Γ 1 =]0, 1[×{0}, Γ 0 = ∂Ω\Γ 1 and d ∈ C 1 (R 2 , R). This system can be seen as a hybrid system of equation arising in the control of noise where the dissipation is located both on Ω and Γ 1 . Using the multiplier method and under some hypothesis on d, N. Aissa and D. Hamroun proved that the energy of system (4.1.4) decays exponentially to 0. The most popular model is the wave equation with acoustic boundary conditions, that takes the following form:

                       u tt -∆u = 0, in Ω × R * + , u = 0, on Γ 0 × R * + , ∂ ν u = w t , on Γ 1 × R * + , mw tt + dw t + kw + ρu t = 0, on Γ 1 × R * + , u(•, 0) = u 0 , u t (•, 0) = u 1 , in Ω, w(•, 0) = w 0 , on Γ 1 . (4.1.5)
In [START_REF] Beale | Spectral properties of an acoustic boundary condition[END_REF], Beale showed that this problem is governed by a C 0 -semigroup of contraction, while in [82], the authors obtained, under some geometrical conditions, a polynomial stability.

In [START_REF] Micu | Asymptotics for the spectrum of a fluid/structure hybrid system arising in the control of noise[END_REF], S. Micu and E. ZuaZua considered the following simple model arising in the control of noise consisting of two coupled hyperbolic equations of dimensions two and one respectively:

                                 u tt -∆u = 0, in Ω × R * + , ∂ ν u = 0, on Γ 0 × R * + , ∂u ∂y = -w t , on Γ 1 × R * + , w tt -w xx + w t + u t = 0, on Γ 1 × R * + , w x (0, t) = w x (1, t) = 0, for t > 0, u(0) = u 0 , u t (0) = u 1 , in Ω, w(0) = w 0 , w t (0) = w 1 , on Γ 1 , (4.1.6)
where Γ 1 = {(x, 0); x ∈ (0, 1)} and Γ 0 = Γ\Γ 1 . This system is nothing else than system (4.1.5) where the Dirichlet boundary conditions on Γ 0 have been replaced by the Neumann ones. Using separation of variables method, they studied the asymptotic behavior of the eigenvalues and eigenfunctions of system (4.1.6). Since there exists a sequence of eigenvalues which approach the imaginary axis, E. ZuaZua and S. Micu proved that the decay rate of the energy of (4.1.6) is not exponential in the energy space. Later, they proved that system (4.1.6) can be exponentially stable in a subspace of the energy space. This subspace is generated by the eigenfunctions corresponding to a sequence of eigenvalues with uniformly bounded negative real parts. For a generalization of system (4.1.5) and polynomial decay rates, we refer to [START_REF] Abbas | The multidimensional wave equation with generalized acoustic boundary conditions ii: Polynomial stability[END_REF], while an abstract framework is extensively studied in [START_REF] Mugnolo | Abstract wave equations with acoustic boundary conditions[END_REF]. For other related problems we refer to [START_REF] Aissa | Stabilization of a wave-wave system[END_REF][START_REF] Cavalcanti | Geometrically constrained stabilization of wave equations with Wentzell boundary conditions[END_REF][START_REF] Gal | Oscillatory boundary conditions for acoustic wave equations[END_REF][START_REF] Gerbi | Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions[END_REF][START_REF] Littman | Stabilization of a hybrid system of elasticity by feedback boundary damping[END_REF].

In a first step, using Arendt and Batty's Theorem (see [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF]) and with help of Holmgren's theorem, we show the strong stability of system (4.1.2), but for the simple example like the case when Ω is the unit disc of R 2 and Γ 0 = ∅, we show that our system is not uniformly stable, since the corresponding spatial operator has a sequence of eigenvalues that approach the imaginary axis. Hence, we are interested in proving a weaker decay of the energy, for that purpose, we will apply a frequency domain approach (see [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF]) based on the growth of the resolvent on the imaginary axis. More precisely, we will give sufficient conditions that guarantee the polynomial decay of the energy of our system (for sufficiently smooth initial data). We actually obtain two different decay rates. In the first case, we will use the exponential decay of the wave equation with the standard damping

∂y ∂ν = -y t , on Γ 1 × R * + ,
and establish a polynomial energy decay rate of type 1

t 1 4
. In the second case, under a stronger geometrical conditions on Γ 0 and Γ 1 , we establish a polynomial energy decay rate of type 1 t .

In a second step, we want to show that such a polynomial decay seems to be also available even if the previous geometrical assumption is not satisfied. Therefore, we consider the case of the unit square of the plane where Γ 1 is only one edge of the boundary. In this case, using the separation of variables method, we study the spectrum of system (4.1.2) and we show that the decay of the energy to zero is not uniform. Then, using a method based on a Fourier analysis (compare with [START_REF] Nicaise | Polynomial stabilization of the wave equation with Ventcel's boundary conditions[END_REF] where a similar method was used for the wave equation with Ventcel 's boundary conditions), a specific analysis of the obtained 1-d problem combining Ingham's inequality and an interpolation method from [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF], we establish a polynomial energy decay rate of type 1 t for sufficiently smooth initial data.

This chapter is organized as follows: Section 4.2 deals with the well-posedness of the problem obtained by using semigroup theory. We further characterize the domain of the associated operator in some particular cases and obtain the strong stability. In section 4.3, we show that our system is not uniformly stable in the unit disc. Section 4.4 is devoted to the proof of the polynomial decay in the general setting by using the frequency domain approach. In section 4.5, we show that the energy of our system is not uniform stable in the unit square. In section 4.6, we obtain the polynomial stability result for a 1-d model with a parameter associated with (4.1.2). This result is then used in section 4.7 to show for the unit square a polynomial decay in 1/t of the energy for sufficiently smooth initial data.

Let us finish this section with some notations used in the remainder of the chapter. For a bounded domain D, the usual norm and semi-norm of H s (D) (s ≥ 0) are denoted by • s,D and | • | s,D , respectively. For s = 0, we will drop the index s.

Well-posedness and strong stability

In this section, we study the existence, uniqueness and the asymptotic behavior of the solution of system (4.1.2). If Γ 0 is non empty, we introduce the space H 1 Γ 0 (Ω) as follows:

H 1 Γ 0 (Ω) = u ∈ H 1 (Ω); u = 0 on Γ 0 , (4.2.1)
which is a Hilbert space with the norm

u 1,Ω = ∇u Ω . (4.2.2)
Next, we introduce the Hilbert space

H =    (u, v, w, z) ∈ H 1 Γ 0 (Ω) × L 2 (Ω) × H 1 0 (Γ 1 ) × L 2 (Γ 1 ); γu = w on Γ 1 ,    (4.2.3) endowed with the product (u 1 , v 1 , w 1 , z 1 ), (u 2 , v 2 , w 2 , z 2 ) H =(∇u 1 , ∇u 2 ) Ω + (v 1 , v 2 ) Ω (4.2.4) + (∇ T w 1 , ∇ T w 2 ) Γ 1 + (z 1 , z 2 ) Γ 1 , ∀(u 1 , v 1 , w 1 , z 1 ), (u 2 , v 2 , w 2 , z 2 ) ∈ H 1 Γ 0 (Ω) × L 2 (Ω) × H 1 0 (Γ 1 ) × L 2 (Γ 1 ),
and the associated norm

• H = (•, •) 1 2
H , γ being the usual trace operator from H 1 (Ω) into H 1 2 (Γ). For simplicity, we will denote γu by u. If Γ 0 is empty, we define H is the same manner, in this case we equip it with its natural norm: (u, v, w, z) 2 := (u, v, w, z) 2 H + u 2 Ω + w 2 Γ . The energy of the solution of (4.1.2) is defined by

E(t) = 1 2 (u, u t , w, w t ) 2 H . (4.2.5)
For smooth solution, a direct computation gives

d dt E(t) = -w t 2 Γ 1 . (4.2.6)
Then, system (4.1.2) is dissipative in the sense that its energy is a nonincreasing function of the time variable t. We can now introduce the unbounded operator A on H with domain

D(A) =              U = (u, v, w, z) ∈ H; ∆ T w -∂ ν u ∈ L 2 (Γ 1 ), v ∈ H 1 Γ 0 (Ω), ∆u ∈ L 2 (Ω), z ∈ H 1 0 (Γ 1 ), v = z on Γ 1              , ( 4.2.7)
defined by

AU =        v ∆u z ∆ T w -∂ ν u -z        , ∀U =        u v w z        ∈ D(A). (4.2.8)
Then, denoting (u, u t , w, w t ) the state of system (4.1.2), we can rewrite system (4.1.2) into a first-order evolution equation

   U t (t) = AU (t), t > 0, U (0) = U 0 ∈ H, (4.2.9)
where U 0 = (u 0 , v 0 , w 0 , z 0 ) ∈ H. It is easy to show that A is a maximal dissipative operator, therefore its generates a C 0 -semigroup (e tA ) t≥0 of contractions on the energy space H following Lumer-Phillips' theorem (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). Hence, semigroup theory allows to show the next existence and uniqueness results: Theorem 4.2.1. For any initial data U 0 ∈ H, the problem (4.2.9) has a unique weak solution U (t) = e tA U 0 such that U ∈ C 0 ([0, +∞[, H). Moreover, if U 0 ∈ D(A), then the problem (4.2.9) has a strong solution

U (t) = e tA U 0 such that U ∈ C 1 ([0, +∞[, H) ∩ C 0 ([0, +∞[, D(A)).
Now, we characterize the domain D(A) of A in two different cases: either Γ is smooth enough and Γ 0 ∩ Γ 1 = ∅ or Ω is the unit square. We start with the first situation:

Proposition 4.2.2. If the boundary Γ of Ω is C 1,1 and if Γ 0 ∩ Γ 1 = ∅, then D(A) = H 2 (Ω) ∩ H 1 Γ 0 (Ω) × H 1 Γ 0 (Ω) × H 2 (Γ 1 ) ∩ H 1 0 (Γ 1 ) × H 1 0 (Γ 1 ), with (u, v, w, z) D(A) ∼ u 2,Ω + v 1,Ω + w 2,Γ 1 + z 1,Γ 1 , ∀(u, v, w, z) ∈ D(A).
In particular, the resolvent (I -A) -1 of A is compact on the energy space H.

Proof:

The proof is based on a bootstrap argument. Let us fix U = (u, v, w, z) ∈ D(A), and set h = ∆ T w -∂ ν u -z, that belongs to L 2 (Γ 1 ).

Then by definition, u ∈ H 1 (Ω) with ∆u ∈ L 2 (Ω). Hence, by a result of Lions and Magenes (see the end of subsection 1.5 of [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]), we will have

∂ ν u ∈ H -1 2 (Γ 1 ) (as Γ 0 and Γ 1 are disjoint) with ∂ ν u -1 2 ,Γ 1 u 1,Ω + ∆u Ω . (4.2.10) Therefore w ∈ H 1 (Γ 1 ) satisfies ∆ T w = h + ∂ ν u + z ∈ H -1 2 (Γ 1 ). ( 4
u 2,Ω + v 1,Ω + w 2,Γ 1 + z 1,Γ 1 (u, v, w, z) D(A) , ∀(u, v, w, z) ∈ D(A),
reminding that U D(A) = U H + AU H . The converse inclusion and estimate being trivial, the proof is complete.

Corollary 4.2.3. If the boundary Γ of Ω is C 2,1 and if Γ 0 ∩ Γ 1 = ∅, then D(A 2 ) = H 3 (Ω) ∩ H 1 Γ 0 (Ω) × H 2 (Ω) ∩ H 1 Γ 0 (Ω) × H 3 (Γ 1 ) ∩ H 1 0 (Γ 1 ) × H 2 (Γ 1 ) ∩ H 1 0 (Γ 1 , with (u, v, w, z) D(A 2 ) ∼ u 3,Ω + v 2,Ω + w 3,Γ 1 + z 2,Γ 1 , ∀(u, v, w, z) ∈ D(A 2 ). Proof: First, U = (u, v, w, z) belongs to D(A 2 )

if and only if U ∈ D(A) and AU ∈ D(A).

Hence by the previous result we will have ∆u ∈ H 1 (Ω),

and h = ∆ T -∂ ν u -z ∈ H 1 0 (Γ 1 )
. Next, as the previous characterization yields u ∈ H 2 (Ω), we know that ∂ ν u belongs to H 1 2 (Γ 1 ) and coming back to (4.2.11), we deduce that

∆ T w = h + ∂ ν u + z ∈ H 1 2 (Γ 1 ).
A shift theorem will lead to w ∈ H 5 2 (Γ 1 ). Then coming back to (4.2.13), the improved regularity on ∆u and w, combined with a shift theorem give u ∈ H 3 (Ω). Again coming back to (4.2.11), we deduce that ∆ T w = h + ∂ ν u + z ∈ H 1 (Γ 1 ), and therefore w ∈ H 3 (Γ 1 ). This proves the result (for shortness we have skipped the estimates). Proof: The difficulty stays on the fact that Ω has a non smooth boundary and that Γ 0 ∩ Γ 1 is not empty. But we take advantage of the particular geometry.

Let us start with the characterization of D(A). Let U = (u, v, w, z) be in D(A). Then by a localization argument and Proposition 4.2.2, we directly see that u (resp. w) belongs to

H 2 (Ω \ W ) (resp. H 2 (Γ 1 \ W )),
where W is any neighborhood of the corners. Hence it remains to improve the regularity of u and w near the corners. But in a small neigborhood V of the corner (1, 0) (or (1, 1)), as u is solution of a homogeneous Dirichlet problem with ∆u ∈ L 2 , it is wellknown (see Theorem 3.2.1.2 of [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] for instance) that u ∈ H 2 (V ). Hence, the main difficulty is to show the regularity of u and w in a neighborhood V of the corner (0, 0) (or (0, 1)). By symmetry, it suffices to look at the case of the corner (0, 0). Now fix a cut-off function η ∈ D(R 2 ) such that η = 1 in the disc of center (0, 0) and radius 1/4 and equal to 0 outside the disc of center (0, 0) and radius 1/2. Then we easily check that ηU belongs to D(A 0 ), the operator A 0 being our operator A but defined in the quater plane Q = {(x, y) ∈ R 2 ; x, y > 0}, with Γ 1 = {(0, y) ∈ R 2 ; y > 0} and Γ 0 = {(x, 0) ∈ R 2 ; x > 0}. -w(-y) if y < 0, defined in the line {(0, y) ∈ R 2 ; y ∈ R}.

Now the first statement holds if we show that

D(A 0 ) ⊂ H 2 (Q) × H 1 (Q) × H 2 (Γ 1 ) × H 1 0 (Γ 1 ). ( 4

Now we denote by Ã0 our operator

A but defined in the half-plane R 2 + , with Γ 1 = {(0, y) ∈ R 2 ; y ∈ R}. Then by Proposition 4.2.2, it is clear that D( Ã0 ) = H 2 (R 2 + ) × H 1 (R 2 + ) × H 2 (Γ 1 ) × H 1 (Γ 1 ).
Hence (4.2.16) holds if we can show that (ũ, ṽ, w, z) belongs to D( Ã0 ). The only non trivial properties are to check that ∆ũ belongs to L 2 (R 2 + ) and that wyy -∂ ν ũ belongs to L 2 (Γ 1 ). For the first assertion, we show that ∆ũ(x, y)

=    ∆u(x, y) if y > 0, -∆u(x, -y) if y < 0.
(4.2.17)

Indeed we take ϕ ∈ D(R 2 + ), we clearly have

∆ũ, ϕ = Q u∆d ϕ ,
where

d ϕ ∈ H 1 Γ 0 (Q) is defined by d ϕ (x, y) = ϕ(x, y) -ϕ(x, -y), ∀(x, y) ∈ Q.
Since d ϕ is zero in a neighborhood of (0, 0), we can apply Theorem 1.5.3.6 of [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] and deduce that

∆ũ, ϕ = Q ∆ud ϕ ,
and (4.2.17) follows.

Similarly we show that

wyy (y) =    w yy (y) if y > 0, -w yy (-y) if y < 0. (4.2.18)
Finally for any v ∈ H 1 (R 2 + ), we have

∂ ν ũ, v = R 2 + (∆ũv + ∇ũ • ∇v).
Hence by the previous argument, we have

∂ ν ũ, v = Q (∆ud v + ∇ u • ∇d v ).
where

d v ∈ H 1 Γ 0 (Q).
Hence by the definition of ∂ ν u, we deduce that

∂ ν ũ, v = ∂ ν u, d v ,
which means that

∂ ν ũ(y) =    ∂ ν u(y) if y > 0, -∂ ν u(-y) if y < 0. ( 4
-∂ ν ũ)(y) =    h(y) if y > 0, -h(-y) if y < 0, (4.2.20)
and consequently it belongs to L 2 as well.

For the characterization of D(A 2 ), it suffices to notice that for (u, v, w, z)

∈ D(A 2 0 ), then ∆u ∈ H 1 Γ 0 (Q).
In a neighborhood of the corner (0, 0), we first notice that ∆ũ given by (4.2.17) belongs to H 1 (R 2 + ). Similarly h = w yy -∂ ν u belongs to H 1 0 , and hence wyy -∂ ν ũ given by (4.2.20) belongs to H 1 . This means that (ũ, ṽ, w, z) belongs to D( Ã2 0 ) and we conclude by Corollary 4.2.3. In a neighborhood of the corners (1, 0) or (1, 1), we simply use the same reflexion technique as before (see Lemma 2.4 of [START_REF] Hell | Modification of dimension-splitting methods-overcoming the order reduction due to corner singularities[END_REF]) to get the H 3 regularity of u. Now, we investigate the strong stability of system (4.2.9). But before going on, if Γ 0 is empty, we need to introduce the closed subspace Note that this definition is meaningful because for all U ∈ D(A), AU belongs to H 0 . Hence B also generates a C 0 -semigroup of contractions that is simply the restriction of (e tA ) t≥0 to H 0 . If Γ 0 = ∅, then the semigroup of contractions (e tA ) t≥0 is strongly stable on the space H 0 . Further, for any

H 0 = {(u, v, w, z) ∈ H : Ω vdx + Γ 1 zdΓ + Γ 1 wdΓ = 0}
U 0 = (u 0 , v 0 , w 0 , z 0 ) ∈ H, if α = 1 |Γ 1 | ( Ω v 0 dx + Γ 1 z 0 dΓ + Γ 1 w 0 dΓ) (where |Γ 1 | means the measure of Γ 1 ), then lim t→+∞ e tA U 0 -α(1, 0, 1, 0) H = 0. ( 4

.2.22)

To prove the above theorem, we apply the strategy used in [START_REF] Nicaise | Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback[END_REF]. It is based on the theorem of Arendt and Batty in [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF]. We need to proof the following two lemmas: Lemma 4.2.6. For all λ ∈ R * , we have

ker(iλI -A) = {0}, while ker(A) = {0},
if Γ 0 is non empty, and

ker(A) = Span {(1, 0, 1, 0)}, if Γ 0 is empty, but ker(iλI -B) = {0}, ∀λ ∈ R. (4.2.23) Proof: Let U = (u, v, w, z) ∈ D(A) and let λ ∈ R, such that AU = iλU. (4.2.24)
First, by detailing (4.2.24) we get

             v = iλu, ∆u = iλv, z = iλw, ∆ T w -∂u ∂ν -z = iλz. (4.2.25)
Next, a straightforward computation gives

(AU, U ) H = - Γ 1 |z| 2 dx. ( 4
       ∆u + λ 2 u = 0, in Ω, u = 0, on Γ, ∂ ν u = 0, on Γ 1 .
(4.2.28)

Therefore, using Holmgren's theorem, we deduce that u = 0 and consequently, U = 0. Case 2: λ = 0. The system (4.2.25) becomes

             v = 0, in Ω, ∆u = 0, in Ω, z = 0, in Γ 1 , ∆ T w -∂ ν u = 0, on Γ 1 . (4.2.29)
By integrating by parts and using the boundary conditions u = 0 on Γ 0 and w = 0 on ∂Γ 1 , we have

0 = Ω ∆uū = - Ω |∇u| 2 + Γ 1 ∂ ν uū = - Ω |∇u| 2 - Γ 1 |∇ T u| 2 .
Hence u is constant in the whole domain Ω. Therefore if Γ 0 is non empty we deduce that u = w = 0 and directly conclude that ker(iλI -A) = {0}. On the other hand, if Γ 0 is empty, then u = w constant is allowed and we find that (1, 0, 1, 0) is the sole eigenvector of A of eigenvalue 0. But since (1, 0, 1, 0) does not belongs to H 0 , 0 is not an eigenvalue of B and consequently we deduce that (4.2.23) holds.

Lemma 4.2.7. If

Γ 0 = ∅, for all λ ∈ R, we have R(iλI -A) = H, while if Γ 0 = ∅, for all λ ∈ R, we have R(iλI -B) = H 0 .
Proof: We give the proof in the case Γ 0 = ∅, the proof of the second statement is fully similar by using (4.2.23). Let F = (f, g, h, k) ∈ H, then we look for U = (u, v, w, z) ∈ D(A) such that iλU -AU = F, (4.2.30) or equivalently

             iλu -v = f, iλv -∆u = g, iλw -z = h, iλz -∆ T w + ∂ ν u + z = k. (4.2.31)
From the first and the third identities of (4.2.31) and the fact that w = u on Γ 1 , we get

   -∆u -λ 2 u = g + iλf, in Ω, -λ 2 u -∆ T u + ∂ ν u + iλu = k + (iλ -1)h, on Γ 1 . (4.2.32)
Next, we define the space V by

V = u ∈ H 1 Γ 0 (Ω) : u ∈ H 1 0 (Γ 1 ) ,
endowed with the norm

u 2 V = ∇u 2 Ω + ∇ T u 2 Γ 1 .
Multiplying the first equation of (4.2.32) by ũ ∈ V , integrating in Ω and using the second equation of the same problem, and formal integration by parts, we get formally the following identity:

a λ (u, ũ) = L λ (ũ), (4.2.33)
where a λ is a bilinear form from V × V into C × C given by

a λ (u, ũ) = Ω (∇u • ∇ũ -λ 2 uũ)dx (4.2.34) + Γ 1 (∇ T u • ∇ T ũ + (iλ -λ 2 )uũ)dΓ,
and L λ is a linear form from V into C defined by

L λ (ũ) = Ω (g + iλf )ũdx + Γ 1 (k + (iλ -1)h)ũdΓ. (4.2.35)
Now, we introduce the operator A λ : V → V by

< A λ u, ũ > V ,V = a λ (u, ũ), ∀ũ ∈ V.
For λ, λ ∈ R, we have

|< (A λ -A λ )u, ũ > V ,V | = |a λ (u, ũ) -a λ (u, ũ)| ≤ Ω (λ 2 -λ 2 )uũdx + Γ 1 (i(λ -λ ) + (λ 2 -λ 2 ))uũdΓ ≤ C λ,λ ,Ω u V ( ũ L 2 (Ω) + ũ L 2 (Γ 1 ) ) ≤ C λ,λ ,Ω u V ũ H 1/2+ε Γ 0 (Ω) .
This implies that

A λ -A λ ∈ L V ; H 1/2+ε Γ 0 (Ω)
and thus A λ -A λ is a compact operator from V into V . On the other hand, since Γ 0 = ∅, then, it is easy to see that the operator A 0 is an isomorphism and consequently, it is a Fredholm operator of index zero. It follows, from the compactness of A λ -A λ , that A λ is also a Fredholm operator of index zero for all λ. Therefore, A λ is surjective if and only if it is injective. Using Lemma 4.2.6 we deduce the injectivity of the operator A λ (compare with Proposition 3.3 in [START_REF] Nicaise | Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback[END_REF]). This means that A λ is an isomorphism for all λ ∈ R and therefore problem (4.

e tB U 0 H = 0, ∀ U 0 ∈ H 0 .
But, for U 0 ∈ H and α given as in the second statement of Theorem 4.2.5, we notice that

U 0 := U 0 -α(1, 0, 1, 0)
belongs to H 0 . The conclusion then follows by noticing that e tA (1, 0, 1, 0) = (1, 0, 1, 0). The proof is thus completed (compare with Theorem 4.3.2 of [START_REF] Fourrier | Analysis of existence, regularity and stability of solutions to wave equations with dynamic boundary conditions[END_REF]).

Non-uniform stability result

In this section we show that the uniform stability (i.e. exponential stability) of problem (4.2.9) does not hold in general, since it is already the case for the unit disk D of R 2 and Γ 0 = ∅ as shown below. This result is due to the fact that a subsequence of eigenvalues of A which is close to the imaginary axis. First, let U = (u, v, w, z) ∈ D(A) such that AU = λU . Equivalently we have

             v = λu, in D, ∆u = λv, in D, z = λw, on ∂D, ∆ T w -∂ ν u -z = λz, on ∂D.
Next, by eliminating v and z from the above system and using the fact that u = w on Γ 1 we get the following system:

   ∆u -λ 2 u = 0, in D, ∆ T u -∂ ν u -λ(λ + 1)u = 0, on ∂D. (4.3.1) A radial solution u(r, θ) = f (r) of (4.3.1) is a solution of      f (r) + 1 r f (r) -λ 2 f (r) = 0, r ∈ (0, 1), f (1) + (λ 2 + λ)f (1) = 0. (4.3.2)
If λ = 0, the general solution of the first equation of (4.3.2) is given by

f (r) = c J J 0 (iλr) + c Y Y 0 (iλr), c J , c Y ∈ C,
where J 0 (resp. Y 0 ) is the Bessel of first (resp. second) kind. Since u is regular in D, necessarily we have c Y = 0 and c J = 0. Therefore, using the second equation of (4.3.2), we find that if λ ∈ C and λ = 0 satisfies

-iλJ 1 (iλ) + (λ 2 + λ)J 0 (iλ) = 0, (4.3.3)
then λ is an eigenvalue of A where J 1 is the Bessel function which satisfies J 1 = -J 0 . Our goal is to find large eigenvalues which are closed to the imaginary axis and to give their expansion. For that reason, we fix c > 0 large enough and we consider the solution of (4.3.3) which are in the strip

S = {λ ∈ C; -c ≤ |λ| ≤ c}.
For convenience, we set

φ(λ) = 1 λ 2 iπλ 2 (-iλJ 1 (iλ) + (λ 2 + λ)J 0 (iλ)), thus (4.3.
3) is equivalent to the following characteristic equation:

φ(λ) = 0. ( 4
λ k = i kπ - π 4 + 9 8kπ + 9 32πk 2 - 1 π 2 k 2 + o( 1 k 2 ). ( 4

.3.5)

Proof: For clarity, the proof is divided into two steps.

Step 1. First, using the asymptotic expansions of Bessel's functions (see equation 10.7.3 [74] for instance) for λ ∈ S we have

iπλ 2 J 0 (iλ) =i cos( π 4 + iλ) + 1 8λ cos( π 4 -iλ) (4.3.6) + 9i 128λ 2 cos( π 4 + iλ) + O( 1 |λ| 3 ), |λ| → ∞ and iπλ 2 (-iλJ 1 (iλ)) =λ cos( π 4 -iλ) - 3
λ k = λ 0 k + o(1) = ikπ -i π 4 + o(1), |k| → ∞.
Equivalently, we have

λ k = ikπ -i π 4 + k and lim |k|→∞ k = 0. (4.3.9)
Step 2. Asymptotic behavior of k : First, using (4.3.9) we obtain cos(

π 4 + iλ k ) = -i(-1) k k + o( k ), (4.3.10) cos( π 4 -iλ k ) =(-1) k + o( k ) (4.3.11)
and

1 λ k = - i kπ + o( 1 k ). (4.3.12)
Next, by inserting (4.3.10)-(4.3.12) in the identity φ(λ k ) = 0 and keeping only the terms of order 1 k , we find after a simplification

(-1) k k + o( k ) - 9i(-1) k 8kπ + o( 1 k ) = 0, thus k = 9i 8kπ + o( 1 k ).
Later, from the above equality we can write

λ k = ikπ -i π 4 + 9i 8kπ + k k , lim |k|→∞ k = 0. That implies cos( π 4 + iλ k ) = 9(-1) k 8kπ - i(-1) k k k + o( 1 k 2 ), (4.3.13) cos( π 4 -iλ k ) =(-1) k - 81(-1) k 128π 2 k 2 + o( 1 k 2 ), (4.3.14) 1 λ k = - i kπ - i 4k 2 π + o( 1 k 2 ) (4.3.15)
and

1 λ 2 k = - 1 k 2 π 2 + o( 1 k 2 ). ( 4
k k k + (-1) k k 2 π 2 - 9i(-1) k 32k 2 π + o( 1 k 2 ) = 0, thus k = 9i 32kπ - 1 kπ 2 + o( 1 k ).
Finally, we find

λ k = i(kπ - π 4 + 9 8kπ + 9 32k 2 π ) - 1 k 2 π 2 + o( 1 k 2 ).
As (4.3.5) shows that the eigenvalues λ k of A approach the imaginary axis as k goes to infinity, system (4.2.9) in the unit disc is clearly not uniformly stable. The asymptotic behavior of λ k in (4.3.5) can be numerically validated. Namely, from (4.3.5) we have

-lim k→+∞ k 2 π 2 (λ k ) = 1.
The table below confirms this behavior.

Polynomial energy decay rate

In this section we study the polynomial decay rate of the energy of problem (4.2.9) under appropriated conditions. First, we consider the following auxiliary problem:

       ϕ tt (x, t) -∆ϕ(x, t) = 0, x ∈ Ω, t > 0, ϕ(x, t) = 0, x ∈ Γ 0 , t > 0, ∂ ν ϕ(x, t) = -ϕ t (x, t), x ∈ Γ 1 , t > 0. (4.4.1)
Next, we denote by (H 1 ) the following condition:

(H 1 ) : the problem (4.4.1) is uniformly stable in

H 1 Γ 0 (Ω) × L 2 (Ω),
or equivalently there exist two positive constants C and w such that for any (ϕ 0 , ϕ

1 ) ∈ H 1 Γ 0 × L 2 (Ω), the solution ϕ of (4.4.1) with initial conditions ϕ(•, 0) = ϕ 0 , ϕ t (•, 0) = ϕ 1 , satisfies |ϕ(•, t)| 2 1,Ω + ϕ t (•, t) 2 Ω ≤ Ce -wt (|ϕ 0 | 2 1,Ω + ϕ 1 2 Ω ), ∀t ≥ 0.
Alternatively, we recall the multiplier control condition MCC by the following definition:

m • ν ≤ 0 on Γ 0 and m • ν ≥ m 0 on Γ 1 , with m(x) = x -x 0 ∈ R d .
Remark 4.4.2. In [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF], Bardos and al.,

proved that (H 1 ) holds if Γ is smooth (of class C ∞ ), Γ 0 ∩ Γ 1 =
∅ and under a geometric control condition named by GCC. We say that Γ satisfies the geometric control condition GCC, if every ray of geometrical optics, starting at any point x ∈ Ω at time t = 0, hits Γ 1 in finite time T . For less regular domains, namely of class C 2 , (H 1 ) holds if the vector field assumptions described in [START_REF] Lagnese | Decay of solutions of wave equations in a bounded region with boundary dissipation[END_REF] (see (i), (ii), (iii) of Theorem 1 in [START_REF] Lagnese | Decay of solutions of wave equations in a bounded region with boundary dissipation[END_REF]) hold. Moreover, in Theorem 1.2 of [56] the authors prove that (H 1 ) holds for smooth domains under weaker geometric conditions than in [START_REF] Lagnese | Decay of solutions of wave equations in a bounded region with boundary dissipation[END_REF] (without (ii) of Theorem 1). It is easy to see that the multiplier control condition MCC implies that the vector field assumptions described in [START_REF] Lagnese | Decay of solutions of wave equations in a bounded region with boundary dissipation[END_REF] are satisfied and therefore the condition (H 1 ) holds if MCC holds.

Next, we present our main result of this section by the following theorem:

Theorem 4.4.3. Assume that Γ 0 = ∅ and Γ 0 ∩ Γ 1 = ∅.
1. Assume that the boundary Γ of Ω is Lipschitz and that the condition (H 1 ) holds. Then for all initial data U 0 ∈ D(A), there exists a constant c > 0 independent of U 0 , such that the solution of the problem (4.2.9) satisfies the following estimation:

E(t) ≤ c t 1 4 U 0 2 D(A) , ∀t > 0. (4.4.2)
2. Assume that the boundary Γ of Ω is C 1,1 and that the multiplier control condition MCC on Γ 1 holds. Then for all initial data U 0 ∈ D(A), there exists a constant c > 0 independent of U 0 , such that the solution of problem (4.2.9) satisfies the following estimation:

E(t) ≤ c t U 0 2 D(A) , ∀t > 0. (4.4.3)
In order to prove our results, we will use Theorem 2.4 of [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF]. A C 0semigroup of contractions (e tA ) t≥0 in a Hilbert space H satisfies (4.4.2) (respectively (4.4.3)) if

(H 2 ) : iR ⊂ ρ(A), (H 3 ) : sup |β|≥1 1 |β| l (iβI -A) -1 L(H)
< +∞ hold with l = 8 (respectively with l = 2). As condition (H 2 ) was already checked in Theorem 4.2.5, we now prove that condition (H 3 ) holds, using an argument of contradiction. For this aim, we suppose that there exists a sequence β n ∈ R such that β n ----→ n→+∞ +∞, and a sequence

U n = (u n , v n , w n , z n ) ∈ D(A) such that U n H = 1 (4.4.4)
and

β l n (iβ n I -A)U n H ----→ n→+∞ 0. (4.4.5)
For simplification, we denote

β n by β, U n = (u n , v n , w n , z n ) by U = (u, v, w, z) and F n = β l n (iβ n I -A)U n = (f 1,n , f 2,n , f 3,n , f 4,n ) by F = (f 1 , f 2 , f 3 , f 4 )
. Next, by detailing (4.4.5) we obtain

             β l (iβu -v) = f 1 -→ 0 in H 1 Γ 0 (Ω), β l (iβv -∆u) = f 2 -→ 0 in L 2 (Ω), β l (iβw -z) = f 3 -→ 0 in H 1 0 (Γ 1 ), β l (iβz -∆ T w + ∂ ν u + z) = f 4 -→ 0 in L 2 (Γ 1 ). (4.4.6)
Later, by eliminating v and z from system (4.4.6) and since u = w on Γ 1 we obtain Next, using the third equation of system (4.4.6) and using (4.4.9), we get

         β 2 u + ∆u = - f 2 + iβf 1 β l , β 2 u + ∆ T u -∂ ν u -iβu = - f 4 + (1 + iβ)f 3 β l . ( 4
Γ 1 |w| 2 dΓ = o(1) β l+2 . ( 4
. Let u ∈ H 1 (Ω) such that ∆u ∈ L 2 (Ω). Then u ∈ H 1 (Γ) ⇐⇒ ∂ ν u ∈ L 2 (Γ) (4.4.11)
and in this case we have

∆u Ω + u 1,Γ ∼ ∂ ν u Γ + ∆u Ω . ( 4
.4.12)

Proof: First, we denote by h = ∆u and we set

h =    h in Ω, 0 in R d \Ω. Moreover, we consider O a smooth domain such that Ω ⊂ O. Next, let w ∈ H 1 0 (O) be a solution of ∆ w = h in O.
Then w ∈ H 2 (O) and we have

w 2,O h O h Ω . ( 4

.4.13)

Consequently v = u -w ∈ H 1 (Γ) and satisfies ∆v = 0 in Ω. On the other hand, using Lemma 1 of [START_REF] Costabel | A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains[END_REF], we deduce that

v ∈ H 1 (Γ) ⇐⇒ ∂ ν v ∈ L 2 (Γ) (4.4.14) and v 1,Γ ∼ ∂ ν v Γ . ( 4
u ∈ H 1 (Γ) ⇐⇒ ∂ ν u ∈ L 2 (Γ).
Now, to prove estimate (4.4.12), we notice that

u 1,Γ = v + w 1.Γ ≤ v 1,Γ + w 1,Γ .
Hence, using (4.4.15) we get

u 1,Γ ∂ ν v Γ + w 1,Γ ∂ ν u Γ + ∂ ν w Γ + w 1,Γ .
Finally, by using trace result theorem and (4.4.13) we obtain

u 1,Γ ∂ ν u Γ + w 2,Ω ∂ ν u Γ + h Ω .
The converse inequality is proved similarly.

Lemma 4.4.6. Assume that the boundary Γ of Ω is Lipschitz, Γ 0 = ∅ and l ≥ 1. Then, the solution (u, v, w, z) ∈ D(A) of system (4.4.6) satisfies the following estimation:

Γ 1 |∂ ν u| 2 dΓ = O(β 2 ). ( 4

.4.16)

Proof: First, since u ∈ H 1 0 (Γ 1 ) and since u = 0 on Γ 0 , we have u ∈ H 1 (Γ). Next, as ∆u ∈ L 2 (Ω) and the boundary Γ of Ω is Lipschitz, then using (4.4.12) and Poincaré's inequality we obtain

∂ ν u Γ ∆u Ω + u 1,Γ (4.4.17)
∆u

Ω + ∇ T u Γ 1 .
Next, using the first equation of system (4.4.7) we have

∆u Ω β 2 u Ω + o(1) β l-1 . ( 4.4.18)
Moreover, since by the first equation of (4.4.6) and by (4.4.4), we have βu and ∇ T u are uniformly bounded in L 2 (Ω) and in L 2 (Γ 1 ) respectively. Finally, combining (4.4.17)-(4.4.18) with l ≥ 1, we deduce (4.4.16).

Lemma 4.4.7. Assume that the boundary

Γ of Ω is C 1,1 , Γ 0 ∩ Γ 1 = ∅,
l ≥ 1 and that the multiplier control condition MCC on Γ 1 holds. Then, the solution (u, v, w, z) ∈ D(A) of system (4.4.6) satisfies the following estimation:

Γ 1 |∂ ν u| 2 dΓ = O(1). ( 4.4.19)
Proof: First, we define the cut off function η ∈ C 2 (Ω) by

η(x) =    1 x ∈ Γ 1 , 0 x ∈ Ω\O α , ( 4.4.20)
where O α is a neighborhood of Γ 1 given by

O α = x ∈ Ω; inf y∈Γ 1 |x -y| ≤ α (4.4.21)
and where α is a positive constant small enough such that Γ 0 ∩ O α = ∅.

Next, multiplying the first equation of system (4.4.7) by 2ηm • ∇u we get

2β 2 Ω ηu(m • ∇u)dx + 2 Ω η∆u(m • ∇u)dx = o(1) β l-1 . (4.4.22)
On the other hand, by integrating by parts we obtain

2β 2 Ω ηu(m • ∇u)dx = -d Ω η|βu| 2 dx - Ω (m • ∇η)|βu| 2 dx (4.4.23) + Γ 1 (m • ν)|βu| 2 dΓ.
Moreover, since U ∈ D(A) then using Proposition 4.2.2 we have ηu ∈ H 2 (Ω). Then, using Green's formula we can easily check that

2 Ω η∆u(m • ∇u)dx =(d -2) Ω η|∇u| 2 dx (4.4.24) -2 Ω (∇u • ∇η)(m • ∇u)dx + 2 Γ 1 ∂ ν u(m • ∇u)dΓ - Γ 1 (m • ν)|∇u| 2 dΓ + Ω (m • ∇η)|∇u| 2 dx. The fact that -→ ∇u = ∂ ν u - → ν + ∇ T u on Γ 1 ,
(m • ν)|∂ ν u| 2 dΓ + Γ 1 (m • ν)|βu| 2 dΓ + (d -2) Ω η|∇u| 2 dx = d Ω η|βu| 2 dx + Ω (m • ∇η)|βu| 2 dx + 2 Ω (∇u • ∇η)(m • ∇u)dx -2 Γ 1 ∂ ν u(m • ∇ T u)dΓ + Γ 1 (m • ν)|∇ T u| 2 dΓ - Ω (m • ∇η)|∇u| 2 dx + o(1) β l-1 .
Later, using the multiplier control condition MCC on Γ 1 we get

m 0 Γ 1 |∂ ν u| 2 dΓ + m 0 Γ 1 |βu| 2 dΓ + (d -2) Ω η|∇u| 2 dx ≤ d Ω η|βu| 2 dx + Ω (m • ∇η)|βu| 2 dx + 2 Ω (∇u • ∇η)(m • ∇u)dx -2 Γ 1 ∂ ν u(m • ∇ T u)dΓ + Γ 1 (m • ν)|∇ T u| 2 dΓ - Ω (m • ∇η)|∇u| 2 dx + o(1) β l-1 .
Its follows that

m 0 Γ 1 |∂ ν u| 2 dΓ ≤d Ω η|βu| 2 dx + Ω (m • ∇η)|βu| 2 dx + 2 Ω (∇u • ∇η)(m • ∇u)dx -2 Γ 1 ∂ ν u(m • ∇ T u)dΓ + Γ 1 (m • ν)|∇ T u| 2 dΓ - Ω (m • ∇η)|∇u| 2 dx + o(1) β l-1 .
Thus, applying Cauchy-Schwarz's and Young's inequalities we obtain

(m 0 -) Γ 1 |∂ ν u| 2 dΓ ≤ R 2 + R Γ 1 |∇ T u| 2 dΓ + C 1 Ω |βu| 2 dx (4.4.25) + C 2 Ω |∇u| 2 dx + o(1) β l-1 , where is a positive constant, R = m ∞ , C 1 = C(R, η ∞ ) and C 2 = C( η ∞ , ∇η ∞ , R
). Now, since U ∈ D(A), we have u = w and therefore ∇ T u = ∇ T w on Γ 1 . Thus, from (4.4.4) we deduce that ∇ T u (respectively ∇u) is uniformly bounded on Γ 1 (respectively in Ω). Further, using the second equation of system (4.4.6) we deduce that βu is uniformly bounded in Ω. Finally, setting = m 0 2 in (4.4.25) and taking l ≥ 1, we get directly (4.4.19).

Lemma 4.4.8. Assume that Γ 0 = ∅, the boundary Γ of Ω is Lipschitz and l=8. Then, the solution (u, v, w, z) ∈ D(A) of system (4.4.6) satisfies the following estimation:

Γ 1 |∇ T u| 2 dΓ = o(1) β 4 . (4.4.26)
On the other hand, assume that the boundary

Γ of Ω is C 1,1 , Γ 0 ∩ Γ 1 = ∅,
the multiplier control condition MCC on Γ 1 holds and l = 2. Then, the solution (u, v, w, z) ∈ D(A) of system (4.4.6) satisfies the following estimation:

Γ 1 |∇ T u| 2 dΓ = o(1) β 2 . ( 4.4.27)
Proof: Multiplying the second equality of system (4.4.7) by u and integrating by parts and using (4.4.8) we obtain

Γ 1 |∇ T u| 2 dΓ+ Γ 1 ∂ ν uudΓ+iβ Γ 1 |u| 2 dΓ- Γ 1 |βu| 2 dΓ = o( 1) β 3l 2 . (4.4.28)
First, if Γ 0 = 0, the boundary Γ of Ω is Lipschitz and l = 8, then using (4.4.8) and (4.4.16) we get

                   Γ 1 |βu| 2 dΓ = o(1) β 8 , Γ 1 ∂ ν uudΓ = o(1) β 4 , iβ Γ 1 |u| 2 dΓ = o(1) β 9 . ( 4
                   Γ 1 |βu| 2 dΓ = o(1) β 2 , Γ 1 ∂ ν uudΓ = o(1) β 2 , iβ Γ 1 |u| 2 dΓ = o(1) β 3 . (4.4.30)
Finally, substituting (4.4.30) into (4.4.28) with l = 2 we strictly get (4.4.27). Now, we consider the following auxiliary problem:

       -(β 2 + ∆)ϕ u = u, in Ω, ϕ u = 0, on Γ 0 , ∂ ν ϕ u + iβϕ u = 0, on Γ 1 , (4.4.31)
where u is solution of system (4.4.7). Lemma 4.4.9. Assume that the conditions (H 1 ) holds. Then, the solution ϕ u of problem (4.4.31) satisfies the following estimation:

β ϕ u Ω + ∇ϕ u Ω + β ϕ u Γ 1 u Ω . (4.4.32)
Proof: The proof is same as in [START_REF] Abbas | The multidimensional wave equation with generalized acoustic boundary conditions ii: Polynomial stability[END_REF], it is based on a result for Huang and Prüss in [START_REF] Gearhart | Spectral theory for contraction semigroups on Hilbert space[END_REF][START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]. Since the problem (4.4.1) is uniformly stable, then the resolvent of its operator is bounded on the imaginary axis. We omit the details here.

Lemma 4.4.10. Assume that Γ 0 ∩ Γ 1 = ∅ and the condition (H 1 ) holds. Then, the solution ϕ u of system (4.4.31) satisfies the following estimation:

Γ 1 |∇ T ϕ u | 2 dΓ = O(1). ( 4
∂ ν (ηϕ u ) =    ∂ ν ϕ u on Γ 1 , 0 on Γ 0 . ( 4.4.34)
Thus, using the first equation of (4.4.31) and using (4.4.32), we obtain

∂ ν (ηϕ u ) ∈ L 2 (Γ). (4.4.35)
Later, we can assume that the boundary ∂O α of O α defined in (4.4.21) is Lipschitz. Then, using (4.4.35) and applying (4.4.12) we obtain

Γ |∇ T ϕ u | 2 dΓ Ω |∆(ηϕ u )| 2 dx + Γ |∂ ν (ηϕ u )| 2 dΓ = Ω |h| 2 dx + Γ 1 |∂ ν ϕ u | 2 dΓ. (4.4.36)
On the other hand, using (4.4.32) and the third equation of system (4.4.31) we get

Ω |h| 2 dx ≤ η 2 ∞ Ω |∆ϕ u | 2 dx + ∇η 2 ∞ Ω |∇ϕ u | 2 dx + ∆η 2 ∞ Ω |ϕ u | 2 dx β 2 Ω |u| 2 dx (4.4.37)
and

Γ 1 |∂ ν ϕ u | 2 dΓ = β 2 Γ 1 |ϕ u | 2 dx Ω |u| 2 dx. ( 4
Γ of Ω is C 1,1 , Γ 0 ∩ Γ 1 = ∅
and that the multiplier control condition MCC on Γ 1 holds. Then, the solution ϕ u of system (4.4.31) verifies the following estimation:

Γ 1 |∇ T ϕ u | 2 dΓ = O(1) β 2 . ( 4.4.39)
Proof: First, multiplying the first equation of system (4.

-2β 2 Ω ϕ u η(m • ∇ϕ u )dx -2 Ω ∆ϕ u η(m • ∇ϕ u )dx = 2 Ω uη(m • ∇ϕ u )dx.
Then, by taking the real part of the above equation and using (4.4.23)-(4.4.24) for u = ϕ u we obtain

d Ω η|βϕ u | 2 dx + Ω (m • ∇η)|βϕ u | 2 dx - Γ 1 (m • ν)|βϕ u | 2 dΓ (4.4.40) -(d -2) Ω η|∇ϕ u | 2 dx + 2 Ω (∇ϕ u • ∇η)(m • ∇ϕ u )dx -2 Γ 1 ∂ ν ϕ u (m • ∇ϕ u )dΓ + Γ 1 (m • ν)|∇ϕ u | 2 dΓ - Ω (m • ∇η)|∇ϕ u | 2 dx = 2 Ω uη(m • ∇ϕ u )dx.
Next, using the first equation of system (4.4.6) we get u 2 Ω = O(1) β 2 . Since by remark 4.4.2, we claim that the multiplier control condition MCC implies that the condition (H 1 ) holds. Using (4.4.32), we obtain

Γ 1 (m•∇η)|βϕ u | 2 dΓ ≤ R ∇η ∞ Γ 1 |βϕ u | 2 dΓ R ∇η ∞ Ω |u| 2 dΓ = O(1) β 2
where R = m ∞ and therefore

Γ 1 (m • ∇η)|βϕ u | 2 dΓ = O(1) β 2 . (4.4.41)
Similarly, we get 2

Ω (∇ϕ u • ∇η)(m • ∇ϕ u )dx = O(1) β 2 , ( 4.4.42) (d -2)
Ω η|∇ϕ u | 2 dx = O(1) β 2 , ( 4.4.43)
Γ 1 (m • ν)|βϕ u | 2 dΓ = O(1) β 2 , ( 4.4.44
)

Ω (m • ∇η)|∇ϕ u | 2 dx = O(1) β 2 (4.4.45) and 2
Ω uη(m • ∇ϕ u )dx = O(1) β 2 . ( 4.4.46)
Later, inserting (4.4.41)-(4.4.46) into (4.4.40) we obtain

Γ 1 (m • ν)|∇ϕ u | 2 dΓ -2 Γ 1 ∂ ν ϕ u (m • ∇ϕ u )dΓ = O(1) β 2 .
Which implies that

Γ 1 (m • ν)|∇ T ϕ u | 2 dΓ ≤ Γ 1 (m • ν)|∂ ν ϕ u | 2 dΓ (4.4.47) + 2 Γ 1 ∂ ν ϕ u (m • ∇ T ϕ u )dΓ + O(1) β 2 .
Now, using the multiplier control condition MCC on Γ 1 and the third equation of system (4.4.31) we get

m 0 Γ 1 |∇ T ϕ u | 2 dΓ ≤R Γ 1 |βϕ u | 2 dΓ (4.4.48) + 2R Γ 1 |∇ T ϕ u ||βϕ u |dΓ + O(1) β 2 .
Finally, by applying Cauchy-Schwarz's and Young's inequalities and using (4.4.32), we directly deduce (4.4.39). Proof of Theorem 4.4.3:

1. First, multiplying the first equation of system (4.4.7) by ϕ u and applying Green's formula we obtain

Ω u(β 2 + ∆)ϕ u dx + Γ 1 (∂ ν uϕ u -u∂ ν ϕ u ) dΓ (4.4.49) = - Ω f 2 + iβf 1 β l ϕ u dx.
Moreover, using the second equation of system (4.4.7) we have

∂ ν u = f 4 + (1 + iβ)f 3 β l + β 2 u + ∆ T u -iβu. ( 4.4.50)
Then, substituting (4.4.50) into (4.4.49) and using the first equation of problem (4.4.31) and integrating by parts yields

Ω |βu| 2 dx = Ω f 2 + iβf 1 β l-2 ϕ u dx (4.4.51)
+ Γ 1 f 4 + (1 + iβ)f 3 β l-2 ϕ u dΓ + Γ 1 β 4 uϕ u dΓ - Γ 1 (β∇ T u)(β∇ T ϕ u )dΓ.
On the other hand, multiplying the first equation of system (4.4.7) by u and applying Green's formula and using (4.4.50), we get

Ω |∇u| 2 dΩ =β 2 Ω |u| 2 dΩ - Γ 1 |u| 2 dΓ (4.4.52) - Γ 1 |∇ T u| 2 dΓ -iβ Γ 1 |u| 2 dΓ + Γ 1 f 4 -(1 + iβ) f 3 udΓ + Ω f 2 + iβ f 1 udΩ.
Next, under the assumptions Γ 0 = ∅, Γ 0 ∩ Γ 1 = ∅, the conditions (H 1 ) holds and l = 8, using Lemmas 4.4.

Non-uniform stability on the unit square

In this section we prove that the uniform stability (i.e. exponential stability) of (4.2.9) does not hold in the unit square domain Ω = (0, 1) 2 with Γ 1 = {(0, y), y ∈ (0, 1)} and Γ 0 = Γ\Γ 1 . This outcome is due to the existence of a subsequence of eigenvalues of A which is close to the imaginary axis. First, let λ be an eigenvalue of A and U = (u, v, w, z) be an associated eigenfunction, then we obtain AU = λU . Equivalently, we have the following system:

             ∆u = λ 2 u, in Ω, u = 0, on Γ 0 , u = w, on Γ 1 , w yy + u x = (λ 2 + λ)w, on Γ 1 . (4.5.1)
Next, using the separation of variables method and by a straightforward computation, we give a solution of system (4.5.1) by the following proposition: Proposition 4.5.1. A solution (u, w) of system (4.5.1) is given as follows:

   u(x, y) = 2ab sinh( √ λ 2 + l 2 π 2 (1 -x)) sin(lπy), w(y) = 2ab sinh( √ λ 2 + l 2 π 2 ) sin(lπy), (4.5.2)
where a, b ∈ C are two constants and l ∈ N * . Moreover, the eigenvalue λ associated to A verify the following characteristic equation:

λ 2 + λ + √ λ 2 + l 2 π 2 coth( √ λ 2 + l 2 π 2 ) + l 2 π 2 = 0. ( 4.5.3)
Proof: First, using the separation of variables method and the boundary conditions of system (4.5.1), we find that u and w are given as follows:

u(x, y) =X(x)Y (y), (4.5.4
)

w(y) =X(0)Y (y), (4.5.5
)

X(1) = Y (0) = Y (1) = 0. (4.5.6)
Then, substituting (4.5.4)-(4.5.5) in the first equation of system (4.5.1) and dividing by X(x)Y (y) yields

X xx (x) X(x) + Y yy (y) Y (y) = λ 2 . (4.5.7)
Therefore, X and Y are the solutions of the following problems:

   X xx (x) = (λ 2 + l2 )X(x), X(1) = 0 (4.5.8) and    Y yy (y) = -l2 Y (y), Y (0) = Y (1) = 0, (4.5.9)
where l ≥ 0 denotes the constant of separation. Next, using the boundary condition of (4.5.8), we can easily prove that the solution X system (4.5.8) is given by

X(x) = 2a sinh( λ 2 + l2 (1 -x)), (4.5.10)
where a ∈ C is a constant. Similarly, the solution Y of the first equation of (4.5.9) is given by Y (y) = c 1 e i ly + c 2 e -i ly where c 1 , c 2 ∈ C are two constants. The boundary conditions in (4.5.9) imply that c 1 = -c 2 and l = lπ where l ∈ N * . Then, setting b = 2ic 1 , we claim that the solutions X and Y of the systems (4.5.8)-(4.5.9) are given by

X(x) = 2a sinh( √ λ 2 + l 2 π 2 (1 -x))
and Y (y) = b sin(lπy). (4.5.11)

Now, inserting (4.5.11) into (4.5.4) and (4.5.5) we directly get (4.5.2). Finally, using the expression of u and w in (4.5.2) and the last equality of system (4.5.1) we get (4.5.3). Now, since A is closed with compact resolvent (Proposition 4.2.4), the spectrum σ(A) of A consists entirely of isolated eigenvalues with finite multiplicities. Moreover, as the coefficients of A are real then the eigenvalues appears by conjugate pairs. Finally, we study the spectrum of σ(A) of A by the following proposition: Proposition 4.5.2. There exists k 1 ∈ N * sufficiently large such that the spectrum σ(A) of A in the unit square is given by

σ(A) = σ 0 ∪ σ 1 , (4.5.12)
where

σ 0 = {κ l,j } j∈J , σ 1 = {λ l,k } k∈Z |k|≥k 1 , σ 0 ∩ σ 1 = ∅ (4.5.13)
and where J is a finite set. Moreover, λ l,k is simple and satisfies the following asymptotic behavior:

λ l,k = i kπ + l 2 π 2k - 1 π 2 k 2 + o( 1 k 2 ). (4.5.14)
Proof: For clarity, we divided the proof into several steps.

Step 1. Roots of characteristic equation. First, we set

ξ = λ 1 + l 2 π 2 λ 2 . (4.5.15) Then ξ 2 = λ 2 + l 2 π 2 and λ = ξ 1 -l 2 π 2 ξ 2 .
Using the characteristic equation (4.5.3) we get

ξ 2 + ξ 1 - l 2 π 2 ξ 2 + ξ e 2ξ + 1 e 2ξ -1 = 0. ( 4.5.16)
Multiplying (4.5.16) by f 0 (ξ) = e 2ξ -1, then (4.5.16) is equivalent to

0 = f (ξ) = f 0 (ξ) + f 1 (ξ) ξ (4.5.17) = e 2ξ -1 + 1 ξ (e 2ξ + 1) + (e 2ξ -1) 1 - l 2 π 2 ξ 2 .
The real part of ξ is bounded. This is due to the fact that if (ξ) -→ -∞ then f (ξ) -→ -1. Then, with the help of Rouché's theorem, there exists k 1 large enough such that for all |k| ≥ k 1 the large roots of f (denoted by ξ k ) are simple and close to ξ 0 k roots of f 0 (ξ). More precisely we have

ξ k = ξ 0 k + k and lim |k|→∞ k = 0, (4.5.18)
where

ξ 0 k = ikπ, k ∈ Z. (4.5.19)
Step 2. Asymptotic behavior of k and λ l,k . Using equation (4.5.18), we get

e 2ξ k =1 + 2 k + o( k ), (4.5.20)
1 ξ k = 1 ikπ + o( 1 k 2 ), (4.5.21)
1 ξ 2 k = - 1 k 2 π 2 + o( 1 k 3 ) (4.5.22) and 1 - l 2 π 2 ξ 2 k =1 + l 2 2k 2 + o( 1 k 3 ). (4.5.23)
Substituting equations (4.5.20)-(4.5.23) into (4.5.17) and after some computations yields

k = i kπ + o( 1 k ) = i kπ + ˜ k k with ˜ k -→ 0. ( 4.5.24)
Then, using equation (4.5.24) we get after some computations yields

e 2ξ k -1 = 2i kπ - 2 k 2 π 2 + 2 ˜ k k + o( 1 k 2 ). ( 4
˜ k = -1 kπ 2 + o( 1 k ). (4.5.26)
Inserting equation (4.5.26) into (4.5.24) and (4.5.18) we obtain (4.5.14).

Numerical validation.

The asymptotic behavior of λ l,k in (4.5.14) can be numerically validated. For instance, with l = 1, then from (4.5.14) we have lim

k→+∞ k 2 π 2 (λ 1,k ) = 1.
The In addition, figure 4.1 represents some eigenvalues in this case.

Polynomial energy decay rate of 1-d model with a parameter

The aim of this section is to establish a polynomial energy decay rate of 1-d model with a parameter associated with problem (4.1.2) on the unit square domain Ω = (0, 1) 2 with Γ 1 = {(0, y), y ∈ (0, 1)} and Γ 0 = Γ\Γ 1 . First, we fixed a real parameter L = pπ with p ∈ N * and we consider the solution (u L , w L ) of the following wave equation in (4.1.2) with damping at 0:

                   u L tt -u L xx + L 2 u L = 0, in (0, 1), ∀t > 0, u L (1, t) = 0, ∀t > 0, u L (0, t) = w L , ∀t > 0, w L tt + L 2 w L -u x (0, t) + w L t = 0, ∀t > 0, u L (•, 0) = u L 0 , u L t (•, 0) = u L 1 , ∀x ∈ (0, 1), w L (0) = w L 0 , w L t (0) = w L 1 .
(4.6.1)

Next, we introduce the energy associated to (4.6.1) by

E L (t) = 1 2 1 0 |u L t (x, t)| 2 + |u L x (x, t)| 2 + L 2 |u L (x, t)| 2 dx (4.6.2) + 1 2 |w L t (t)| 2 + L 2 2 |w L (t)| 2 .
A simple integration by parts gives

d dt E L (t) = -|w L t (t)| 2 . ( 4.6.3)
Later, we split the solution U L = (u L , w L ) of system (4.6.1) as follows:

U L = U 1 + U 2 , ( 4.6.4)
where U 1 = (u 1 , w 1 ) and where U 2 = (u 2 , w 2 ), (u 1 , w 1 ) is solution of the same problem that (u L , w L ) but without damping and (u 2 , w 2 ) is the remainder (for shortness we do not write the dependence of (u i , w i ), i = 1, 2 with respect to L). This means that they are respective solutions of

                       u 1,tt -u 1,xx + L 2 u 1 = 0, in (0, 1), ∀t > 0, u 1 (1, t) = 0, ∀t > 0, u 1 (0, t) = w 1 , ∀t > 0, w 1,tt + L 2 w 1 -u 1,x (0, t) = 0, ∀t > 0, u 1 (•, 0) = u L 0 , u 1,t (•, 0) = u L 1 , w 1 (0) = w L 0 , w 1,t (0) = w L 1 (4.6.5)
and

                       u 2,tt -u 2,xx + L 2 u 2 = 0, in (0, 1), ∀t > 0, u 2 (1, t) = 0, ∀t > 0, u 2 (0, t) = w 2 , ∀t > 0, w 2,tt + L 2 w 2 -u 2,x (0, t) + w L t = 0, ∀t > 0, u 2 (•, 0) = 0, u 2,t (•, 0) = 0, w 2 (0) = 0, w 2,t (0) = 0. (4.6.6)
The above splitting is quite standard and it is based on the following idea: First, for the problem (4.6.5), we prove an observability inequality for the solution via spectral analysis and Ingham's inequality. Next, by a perturbation argument based on the dependence of the constants with respect to the time T and L, we find the requested observability estimate for the starting problem (4.6.1).

First, the problem (4.6.5) is related to the positive self-adjoint operator A L from H = L 2 (0, 1) × C into itself (with a compact inverse) with domain

D(A L ) =        U 1 = (u 1 , w 1 ) ∈ H 2 (0, 1) × C; u 1 (1) = 0, u 1 (0) = w 1        (4.6.7)
and defined by

A L U 1 = (-u 1,xx + L 2 u 1 , L 2 w 1 -u 1,x (0)
). (4.6.8) Therefore, we can formulate problem (4.6.5) into a second order evolution equation

       U 1,tt (t) + A L U 1 (t) = 0, U 1 (0) = U L 0 , U 1,t (0) = U L 1 , (4.6.9)
where

U L 0 = (u L 0 , w L 0 ) and U L 1 = (u L 1 , w L 1 )
. The spectrum of A L is characterized as follows:

with θ = √ λ 2 -L 2 . Writing {λ 2 k } ∞
k=0 the sequence of these roots in in-creasing order, it forms the set of eigenvalues of A L which are simple and of associated normalized eigenvectors given by

U 1,k = 1 δ k (sin(θ k (1 -x)), sin(θ k )) , δ k = 1 + sin(θ k ) 2 √ 2 (4.6.11)
and θ k = λ 2 k -L 2 . Furthermore, the next gap condition holds:

λ k+1 -λ k ≥ γ L , ∀k ∈ N, (4.6.12)
with γ is a constant independent on k.

Proof: First, let λ 2 be an eigenvalue of A L and U 1 = (u 1 , w 1 ) an associated eigenvector. Then, using Green's formula and the boundary conditions of system (4.6.5) we obtain

A L U 1 , U 1 H = 1 0 |u 1,x | 2 dx + L 2 1 0 |u 1 | 2 dx + L 2 |w 1 | 2 (4.6.13) ≥ L 2 U 1 2 H ,
which clearly implies that the eigenvalues of A L are larger than L 2 . Next, for λ 2 ≥ L 2 , we look for (u 1 , w 1 ) solution of

             -u 1,xx + L 2 u 1 = λ 2 u 1 , in (0, 1), L 2 w 1 -u 1,x (0) = λ 2 w 1 , u 1 (0) = w 1 , u 1 (1) = 0.
(4.6.14)

We easily check that if λ 2 = L 2 , the only solution of problem (4.6.14) is u 1 = w 1 = 0, hence λ 2 = L 2 cannot be an eigenvalue of A L . Now for λ 2 > L 2 , there exists α ∈ R such that the solution of the first equation of system (4.6.14) is given by with

u 1 (x) = α sin(θ(1 -x)), ( 4
θ k = λ 2 k -L 2 .
Later, we check the gap between λ k . We have

λ k+1 -λ k = θ 2 k+1 + L 2 -θ 2 k + L 2 . (4.6.18)
Then, setting ϕ(t, L) = √ t 2 + L 2 and using the mean value's theorem, we deduce that there exists θ c ∈ (θ k , θ k+1 ) such that:

ϕ(θ k+1 , L) -ϕ(θ k , L) = ∂ t ϕ(θ c , L)(θ k+1 -θ k ) = θ c θ 2 c + L 2 (θ k+1 -θ k ). (4.6.19) Since t t 2 L 2 + 1
is an increasing function of the time variable t we obtain Before going on, we recall Lemma 3.3 from [START_REF] Nicaise | Polynomial stabilization of the wave equation with Ventcel's boundary conditions[END_REF] which give a variant version of Ingham's inequality [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF] (see also [START_REF] Haraux | Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire[END_REF]), where the dependence of the constants of equivalence are given with respect to the gape condition. Lemma 4.6.2. Let ξ n , n ∈ Z, be a sequence of real numbers and a positive real number γ such that the following gap condition:

θ c θ 2 c + L 2 ≥ θ 0 L θ 2 0 L 2 + 1 ≥ C L , ( 4
ξ n+1 -ξ n ≥ γ, ∀k ∈ Z holds.
Then, there exists two positive constants c, C independent of γ such that for all function f in the form

f (t) = n∈Z a n e iξnt with a n ∈ C, we have c γ n∈Z |a n | 2 ≤ 4π γ 0 |f (t)| 2 dt ≤ C γ n∈Z |a n | 2 . Now, we set V L = D(A 1 2
L ). We will bound a weak energy of system (4.6.9) with respect to an appropriate boundary term by the following theorem: Theorem 4.6.3. Let E U 1 be a weak energy of U 1 (x, t) = (u 1 (x, t), w 1 (t)) solution of (4.6.9) defined by

E U 1 (t) = 1 2 U 1 (x, t) 2 H + 1 2 U 1,t (x, t) 2 V L , t ≥ 0. (4.6.21)
Then, there exists two positive constants C 1 , C 2 independent on L such that for all T ≥ C 1 L we have Proof: First, let λ 2 k be an eigenvalue of A L and U 1,k = (u 1,k , w 1,k ) the associated eigenvectors already determined in (4.6.11). By the spectral theorem, the solution U 1 of (4.6.9) is given by

C 2 L E U 1 (0) ≤ T 0 |w 1,t (
U 1 (•, t) = +∞ k=0 u k 0 cos(λ k t) + u k 1 sin(λ k t) λ k U 1,k , (4.6.23)
where u k 0 (resp. u k 1 ) is the Fourier coefficients of U L 0 (resp. U L 1 ), i.e.

U L 0 = +∞ k=0 u k 0 U 1,k and U L 1 = +∞ k=0 u k 1 U 1,k .
Writing U 1,k = (u 1,k , w 1,k ), this implies that

w 1 (t) = +∞ k=0 u k 0 cos(λ k t) + u k 1 sin(λ k t) λ k w 1,k . (4.6.24)
Thus we obtain

w 1,t (t) = +∞ k=0 -u k 0 λ k sin(λ k t) + u k 1 cos(λ k t) w 1,k . (4.6.25)
Then according to the gap condition (4.6.12) and using Lemma 4.6.2, we deduce that there exists C 1 , C 3 independent of L such that for all T ≥ C 1 L we have

C 3 L +∞ k=0 λ 2 k |u k 0 | 2 + |u k 1 | 2 |w 1,k | 2 ≤ T 0 |w 1,t (t)| 2 dt. (4.6.26)
Next, using (4.6.10) and (4.6.11) we get

|w 1,k | 2 = 1 δ 2 k × sin 2 (θ k ) = 2 (1 + sin 2 (θ k )) × cos 2 (θ k ) θ 2 k ∼ C θ 2 k ∼ C λ 2 k , (4.6.27)
as θ k ∼ kπ as k goes to infinity. This equivalence in (4.6.26) yields the existence of a positive constant C 2 independent of L such that for T ≥ C 1 L we have

C 2 L +∞ k=0 |u k 0 | 2 + |u k 1 | 2 λ 2 k ≤ T 0 |w 1,t (t)| 2 dt. ( 4.6.28)
We now conclude by identity

+∞ k=0 |u k 0 | 2 + |u k 1 | 2 λ 2 k = U L 0 2 H + U L 1 2 V L .
We go on with an estimate on w 2 .

Theorem 4.6.4. There exists a positive constant C 4 independent of L and T > 0 such that

T 0 |w 2,t (t)| 2 dt ≤ C 2 4 T 2 T 0 |w L t (t)| 2 dt. ( 4
.6.29)

Proof: First, we start by rewriting problem (4.6.6) as follows:

       U 2,tt (t) + A L U 2 (t) = K(t)H 0 , U 2 (0) = 0, U 2,t (0) = 0, (4.6.30)
with H 0 = (0, 1) ∈ H and K(t) = w L t (t). Remark that H 0 is given by

H 0 = ∞ k=0 w 1,k U 1,k .
Indeed, we have

< H 0 , U 1,k > H =< (0, 1), (u 1,k , w 1,k ) > H = w 1,k . (4.6.31)
Next, using the orthonormal basis {U 1,k } +∞ k=0 of H, we can write the solution U 2 = (u 2 , w 2 ) of problem (4.6.30) as follows:

U 2 (t) = +∞ k=0 u 2,k (t)U 1,k .
(4.6.32)

From (4.6.30)-(4.6.32) we deduce that for every fixed k ∈ N, u 2,k is solution of the following problem: We are ready to prove our main results of this section.

       u 2,k,tt (t) + λ 2 k u 2,k (t) = K(t)w 1,k , u 2,k (0) = 0, u 2,k,t (0) = 0. ( 4
Theorem 4.6.5. There exists a positive constant C 9 independent on L such that for all initial data

(U L 0 , U L 1 ) ∈ D(A L ) × V L we have E L (t) ≤ C 9 L 2 t E 1 L (0), (4.6.41)
where E 1 L is given by . Next, since E U 1 (0) = E U L (0) and using (4.6.3) the inequalities (4.6.44) becomes

E 1 L (t) = U L (t) 2 D(A L ) + U L t (t) 2 V L , t ≥ 0. ( 4
E L (0) -E L (T ) ≥ C 6 L T 2 E U L (0). (4.6.45)
On another hand, using interpolation theory we can show that

U L 0 2 H ≥ U L 0 4 V L U L 0 2 D(A L ) and U L 1 2 V L ≥ U L 1 4 H U L 1 2 V L . ( 4
E U L (0) ≥ 1 2 U L 0 4 V L + U L 1 4 H U L 0 2 D(A L ) + U L 1 2 V L = E 2 L (0) 2E 1 L (0) (4.6.47)
where E 1 L (t) is defined in (4.6.42). Later, substituting (4.6.47) into (4.6.45) and using the fact that E L (t) is a decreasing function of variable t we get

E L (T ) ≤ E L (0) -C 7 E 2 L (T ) E 1 L (0) , ( 4
F k = M k + 1 , k ∈ N.
First, we notice that

F k -F k+1 = M (k + 1)(k + 2) ≤ 2 M F 2 k+1 . ( 4
E L (kT ) ≤ M k + 1 E 1 L (0), (4.6.53)
and therefore, for any t > 0, as there exists k ∈ N such that kT ≤ t ≤ (k + 1)T , we deduce that

E L (t) ≤ M C 1 L t E 1 L (0) =              C 1 LE L (0) t if M = ξ 0 , or C 8 L 2 E 1 L (0) t if M = 2 C 7 , (4.6.54)
where C 8 = 4C 3 1 C 6 . On the other hand, from Theorem 4.6.1 we have λ 2 k ≥ L 2 , then we can easily prove the following inequality:

E L (0) ≤ 1 L 2 E 1 L (0). (4.6.55)
Finally, combining (4.6.54) and (4.6.55), we conclude that (4.6.41) holds.

Polynomial energy decay rate on the unit square

In this section, we establish a polynomial decay rate of the energy of system (4.1.2) when our domain Ω is the unit square of the plane in R 2 with Γ 1 = {0} × (0, 1). This case does not satisfies the assumptions of Theorem (4.4.3) since neither the condition (H 1 ) holds nor Γ 0 ∩ Γ 1 = ∅ holds. Nevertheless, combining a Fourier analysis and the results from the previous section we obtain a polynomial decay rate (compare with [START_REF] Nicaise | Polynomial stabilization of the wave equation with Ventcel's boundary conditions[END_REF]). Consequently, we perform the partial Fourier analysis of the solution U = (u, w) of system (4.1.2)

U (x,
E(t) = +∞ p=1 E pπ (t) ≤ C 9 t +∞ p=1 p 2 π 2 E 1 pπ (0), (4.7.5)
where

E 1 pπ (0) = 1 2 U pπ (x, 0) D(A L ) + 1 2 U pπ t (x, 0) V L , (4.7.6)
U pπ (x, 0) = (u pπ 0 (x), w pπ 0 ), U pπ t (0) = (u pπ 1 (x), w pπ 1 ) and where (u pπ i (x), w pπ i ), i ∈ {0, 1} are the initial data of system (4.6.1). By integrating by parts and by using the boundary conditions of system (4.6.1) we obtain

A L U pπ (x, 0) 2 H = 1 0 |u pπ 0,xx (
+ +∞ p=1 1 0 p 2 π 2 |u pπ 1,x (x)| 2 + p 4 π 4 |u pπ 1 (x)| 2 dx + +∞ p=1 p 4 π 4 |w pπ 1 | 2 .
Finally, by Parseval's inequality and the result of Proposition 4.2.4 we deduce that

E 2 (0) ≤ u 0 2 H 3 (Ω) + w 0 2 H 3 (Γ 1 ) + u 1 2 H 2 (Ω) + w 1 2 H 2 (Γ 1 )
(4.7.11)

∼ U 0 D(A 2 ) .

Polynomial stabilization of the finite difference space discretization of the 1-d wave equation with dynamic boundary control

Abstract: In [START_REF] Wehbe | Rational energy decay rate for a wave equation with dynamical control[END_REF], Wehbe has shown that the energy of the wave equation with dynamic boundary control decays polynomially in 1 t to zero as time goes to infinity. In this chapter, we consider its finite difference space discretization scheme and we analyze whether the decay rate is independent of the mesh size. First, we show that the polynomial decay in 1 t of the energy of the classical semi-discrete system is not uniform with respect to the mesh size. Next, we add a suitable vanishing numerical viscosity term which leads to a uniform (with respect to the mesh size) polynomially decay in 1 t of the energy. Finally, we prove the convergence of the scheme towards the original damped wave equation. Our method is essentially based on discrete multiplier techniques.

Introduction

In this chapter, we consider the finite difference space discretization scheme of the following 1-d damped wave equation with dynamic boundary control:

                       y (x, t) -y xx (x, t) = 0, (x, t) ∈]0, 1[×R + * , y(0, t) = 0, t ∈ R + * , y x (1, t) + η(t) = 0, t ∈ R + * , η (t) -y (1, t) + βη(t) = 0, t ∈ R + * , y(x, 0) = y 0 (x), x ∈]0, 1[, y (x, 0) = y 1 (x), x ∈]0, 1[, η(0) = η 0 , (5.1.1)
where (y 0 , y

1 , η 0 ) ∈ H := V × L 2 (0, 1) × C, V = H 1 L (0, 1) = y ∈ H 1 (0, 1); y(0) = 0 ,
β > 0 is a constant and denotes the partial derivative with respect to the time variable t. The concept of dynamical control in the infinite dimensional case is very close to the one of indirect damping proposed by Russel [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF]. System (5.1.1) arises in many areas of mechanics, engineering and technology. This model may be viewed as a model for describing the vibrations of structures, the propagation of acoustic or seismic waves, etc.

The stabilization of the wave equation retains the attention of many authors. In this regard, different types of wave equation with diverse dampings and in various domains was studied: semilinear wave equation with localized damping in unbounded domains [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF][START_REF] Zuazua | Exponential decay for the semilinear wave equation with locally distributed damping[END_REF], wave equation with a nonlinear internal damping [START_REF] Liu | Decay rates for dissipative wave equations[END_REF] and wave equation on general 1-d networks [8, [START_REF] Ammari | Stabilization of star-shaped networks of strings[END_REF][START_REF] Ammari | Remark on stabilization of tree-shaped networks of strings[END_REF][START_REF] Ammari | Stabilization of generic trees of strings[END_REF][START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF][START_REF] Nicaise | Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks[END_REF][START_REF] Valein | Stabilization of the wave equation on 1-D networks[END_REF][START_REF] Xu | Abstract second order hyperbolic system and applications to controlled network of strings[END_REF]. The energy of the damped wave equation (5.1.1) is given by

E(t) = 1 2 1 0 |y (x, t)| 2 dx + 1 0 |y x (x, t)| 2 dx + |η(t)| 2 , t ≥ 0 (5.1.2)
and pursues the following dissipation law:

E (t) = -β|η(t)| 2 , t ≥ 0.
(5.1.3) Equation (5.1.3) shows that the energy decreases as time increases. In [START_REF] Wehbe | Rational energy decay rate for a wave equation with dynamical control[END_REF], Wehbe started by formulate system (5. and where CU = (0, 0, βη), ∀U ∈ H.

Next, using a multiplier method, he has shown a polynomial decay in 1 t of the energy of system (5.1.1), for smooth initial data. Roughly speaking, he showed that the energy of (5.1.1) satisfies the following estimation:

E(t) ≤ M 1 M 1 + t E(0), ∀t > 0, (5.1.5)
where

M 1 = 2 E 1 (0) βE(0) + β + 1 2β + 3 (5.1.6)
and where E 1 denotes the energy of higher order system associated to (5.1.1), i.e. Later, using a spectral method and a Riesz basis approach, Wehbe proved that the obtained decay rate is optimal in the sense that for any > 0, we cannot expect a decay rate of type 1 t 1+ . However as far as numerical approximation schemes are concerned, little is known about the uniform (with respect to the mesh size) decay of the discretized energy. In many applications, although the continuous system is exponentially or polynomially stable, the discrete ones does not inherit the same property uniformly with respect to the mesh size. In [START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF], Tebou and Zuazua considered the approximation scheme of the wave equation with static boundary condition

               y (x, t) -y xx (x, t) = 0, (x, t) ∈]0, 1[×R + * , y(0, t) = 0, t ∈ R + * , y x (1, t) + αy (1, t) = 0, t ∈ R + * , y(x, 0) = y 0 (x), x ∈]0, 1[, y (x, 0) = y 1 (x),
x ∈]0, 1[.

(5.1.8)

It is surely understood (see [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF][START_REF] Chen | Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain[END_REF][START_REF] Komornik | Rapid boundary stabilization of the wave equation[END_REF][START_REF] Komornik | Exact Controllability and Stabilization: The Multiplier Method[END_REF][START_REF] Komornik | A direct method for the boundary stabilization of the wave equation[END_REF][START_REF] Lagnese | Decay of solutions of wave equations in a bounded region with boundary dissipation[END_REF][START_REF] Quinn | Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping[END_REF][START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF][START_REF] Zuazua | Uniform stabilization of the wave equation by nonlinear boundary feedback[END_REF]) that the energy of (5.1.8) satisfies the exponential decay to zero. First, Tebou and Zuazua shown that the decay rate of the energy of the classical semi-discrete system associated to (5.1.8) is not uniform with respect to the net-spacing size. This is due to the existence of high frequency spurious solutions of the semi-discrete model that propagate very slowly (with group velocity of the order of the mesh size). By adding a suitable vanishing numerical viscosity term, they next proved the uniform (with respect to the mesh size) exponential decay of the energy. Finally, they shown the convergence of the scheme towards the original damped wave equation (without viscosity term). Due to the presence of the dynamic term, the method used in [START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF] does not work for our system. In [START_REF] Abdallah | Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications[END_REF] , Abdallah and al. considered the approximation of second order evolution equations with a bounded damping. First, they damped the spurious high frequency modes by introducing a numerical viscosity term in the approximation scheme. Next, with this viscosity term, they showed the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay and when the spectrum of the spatial operator associated with the undamped problem satisfies the generalized gap condition. Finally, using the Trotter-Kato's theorem [START_REF] Ito | The Trotter-Kato theorem and approximation of PDEs[END_REF], they showed the convergence of the discrete solution to the continuous one. In our work, the damping is not bounded and for this reason there method cannot be applied for our system. Then, the stabilization of the discrete scheme of the wave equation with dynamical boundary control remains to be an open problem. For more details about the stabilization of the discretization scheme of wave equation we refer to [START_REF] Ervedoza | Perfectly matched layers in 1-d: energy decay for continuous and semi-discrete waves[END_REF][START_REF] Marica | Boundary stabilization of numerical approximations of the 1-D variable coefficients wave equation: a numerical viscosity approach[END_REF][START_REF] Tcheugoué Tébou | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF][START_REF] Zuazua | Optimal and approximate control of finite-difference approximation schemes for the 1D wave equation[END_REF].

In this chapter, we consider the finite-difference space semi-discretization scheme of (5.1.1) and we analyze whether the decay rate is independent of the mesh size. Our main purpose in this work is twofold: 1) To scrupulously prove that for the classical finite difference scheme, the polynomial decay of the discretized energy is not uniform with respect to the mesh size.

2) To add a correctly numerical viscous term in the equation in order to achieve an uniformly (with respect to the mesh size) energy decay rate of type (5.1.5).

We now introduce the finite difference scheme we will work on. For this aim, let N be an non-negative integer, set h = 1 N + 1 and consider the subdivision of ]0, 1[ given by

0 = x 0 < x 1 < • • • < x N < x N +1 = 1, x j = jh.
The finite-difference space semi-discretization of system (5.1.1) that we consider is

                             y j (t) - y j+1 (t) -2y j (t) + y j-1 (t) h 2 = 0, t > 0, j = 1, .., N, y 0 (t) = 0, t > 0, y N +1 (t) -y N (t) h + η(t) = 0, t > 0, η (t) -y N +1 (t) + βη(t) = 0, t > 0, y j (0) = y 0 j , j = 1, ..., N, y j (0) = y 1 j , j = 1, ..., N, η(0) = η 0 , ( 5.1.9)
where (y i 0 , y i 1 , .., y i N , y i N +1 ) for i = 0, 1 provides an approximation of the function y i (x) for i = 0, 1 at point x j for j = 0, .., N + 1 and η 0 is the third component of the initial data (y 0 , y 1 , η 0 ) of system (5.1.1). The energy of system (5.1.9) is given by

E h (t) = h 2   N j=1 |y j (t)| 2 + N j=0 y j+1 (t) -y j (t) h 2   + 1 2 |η(t)| 2 , (5.1.10)
for t ≥ 0 and it is a non-increasing function with respect to the time variable t since its derivative is given by

E h (t) = -β|η(t)| 2 , t ≥ 0. (5.1.11)
For simplicity, here and below we eliminate t, i.e. we denote y j (t) (respectively η(t)) by y j for j = 0, .., N + 1 (respectively η).

Note that E h is a natural semi-discrete version of the energy E of system (5.1.1) and that (5.1.11) is the semi-discrete analogue of the energy dis-bilinear form a h as follows:

(A h y h , y h ) C h ×C h = a h (y h , y h ) = h N j=1 y j -y j-1 h y j -y j-1 h (5.1.16) = 1 h y h K 0 h y t h , ∀y h , y h ∈ C h ,
where

K 0 h =           2 -1 0 . . . 0 -1 2 -1 . . . 0 . . . . . . . . . . . . . . . -1 2 -1 0 0 . . . -1 1           ∈ M N ×N (R).
Moreover, for all η ∈ C, we define the operator B h ∈ L(C, C h ) by

B h η = η h V 0 , ( 5.1.17)
where V 0 = (0, .., 1) ∈ C h . We can easily check that the adjoint operator

B * h ∈ L(C h , C) associated to B h is given by B * h y h = y N , ∀y h ∈ C h . ( 5
∀U h = (y h , z h , η) ∈ H h , where (z h , z h ) h = h N j=1 |z j | 2
and we define the bounded operator A β,h in the Hilbert space H h by

A β,h U h = z h , -A h y h -B h η, 1 1 + h (B * h z h -βη) (5.1.20) ∀U h = (y h , z h , η) ∈ H h . Let U h = (y h , y h , η) ∈ D(A β,h
) be a solution of system (5.1.13). Then we have

   U h (t) = A β,h U h (t), t ≥ 0, U h (0) = U 0 h ∈ H h .
(5.1.21)

A direct computation gives

(A β,h U h , U h ) H h = -β|η(t)| 2 .
Furthermore, we introduce the norm in the domain of A β,h by

U h 2 D(A β,h ) = U h 2 H h + A β,h U h 2 H h , U h ∈ H h . (5.1.22)
In the following theorem, we show that the discrete energy E h does not inherit the polynomial decay type of the continuous one E, uniformly with respect to the mesh size h. More precisely, we have the next result:

Theorem 5.1.2. For any h 0 > 0, there do not exists a positive constant M 2 independent of h ∈]0, h 0 [ and of U 0 h ∈ H h , such that the energy E h of system (5.1.21) satisfies the following estimation:

E h (t) ≤ M 2 t U 0 h 2 D(A β,h ) , ∀t > 0. ( 5.1.23)
Result of Theorem 5.1.2 is due to the existence of a sequence of eigenvalues associated to the operator A β,h , for h = 1 N +1 , N large enough, which do not satisfy a necessary condition for (5.1.23). Several remedies have been proposed and analyzed to overcome this difficulties like the Tychonoff regularization [START_REF] Glowinski | Errata: "A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods[END_REF][START_REF] Glowinski | Exact and approximate controllability for distributed parameter systems[END_REF][START_REF] Ramdani | Uniformly exponentially stable approximations for a class of second order evolution equations-application to LQR problems[END_REF][START_REF] Tcheugoué Tébou | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF], a bi-grid algorithm [START_REF] Glowinski | Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation[END_REF][START_REF] Negreanu | A 2-grid algorithm for the 1-d wave equation[END_REF], a mixed finite element method [START_REF] Banks | Exponentially stable approximations of weakly damped wave equations[END_REF][START_REF] Castro | Boundary controllability of a linear semidiscrete 1-d wave equation derived from a mixed finite element method[END_REF][START_REF] Castro | Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square[END_REF][START_REF] Glowinski | A mixed finite element formulation for the boundary controllability of the wave equation[END_REF][START_REF] Münch | A uniformly controllable and implicit scheme for the 1-D wave equation[END_REF], or filtering the high frequencies [START_REF] Infante | Boundary observability for the space semi-discretizations of the 1-D wave equation[END_REF]57]. As in [START_REF] Abdallah | Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications[END_REF][START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF], our goal is to damp the spurious high frequency modes by introducing a numerical viscosity in the approximation scheme. For this aim, we consider the new system with the extra numerical viscosity

                       y j - yj+1 -2yj + yj-1 h 2 -(y j+1 -2y j + y j-1 ) = 0, t > 0, j = 1, .., N, yN+1 -yN h + η = 0, t > 0, η -y N +1 + βη = 0, t > 0, y0 = 0, t > 0, yj(0) = y 0 j , j = 1, ..., N, y j (0) = y 1 j , j = 1, ..., N, η(0) = η 0 .
(5. 1.24) where y 0 j and y 1 j (for j = 0, .., N + 1) are given by y 0 j = y 0 (jh), for j = 0, .., N + 1, (5.1.25)

y 1 0 = 0, y 1 j = 1 h jh (j-1)h y 1 (x)dx, for j = 1, .., N + 1 (5.1.26)
and where (y 0 , y 1 , η 0 ) designates the initial data of system (5.1.1). The natural energy of system (5.1.24) is given by

E h (t) = h 2 N j=1 |y j (t)| 2 + h 2 N j=0 y j+1 (t) -y j (t) h 2 (5.1.27) + (1 + h 2 β) 2 |η(t)| 2 , t ≥ 0
and a direct computation gives

E h (t) = -h 3 N j=0 y j+1 (t) -y j (t) h 2 -β|η(t)| 2 (5.1.28) -h 2 |η (t)| 2 , t ≥ 0.
Hence, E h is a nonincreasing function with respect to the time variable t. Note that, the first equation of system (5.1.24) is the semi-discrete analogue of y -y xx -h 2 y xxt = 0. Now, for system (5.1.24), we prove: i) A decay rate of type (5.1.5) which is uniform with respect to the net-spacing h.

ii) The convergence of its solution towards the one of the original wave equation (5.1.1) as h → 0.

These two results show that the discretization (5.1.24) of system (5.1.1) is a good approximate scheme because it not only warranties the convergence of solutions as h → 0 but it also furnishes a uniform (with respect to the mesh size h) polynomial decay rate of the energy as t → ∞.

The second fact proves that the viscous damping term added in (5.1.24) captures the long time asymptotic properties of system (5.1.1). The suitability of this numerical damping mechanism to restore the uniform polynomial decay is closely connected to the efficiency of the Tychonoff regularization techniques. Before going, we give the following natural convergence result which is a consequence of our choice of discretization in (5.1.25)-(5.1.26):

Lemma 5.1.3. Assume the initial data (y 0 , y 1 , η 0 ) of system (5.1.1) belongs to D(A) where D(A) is given in (5.1.4). Then, the initial discrete energy E h (0) of system (5.1.24) tends to the initial continuous one E(0) as h → 0.

For the uniform polynomial decay result of energy, we introduce the discrete energy E h,1 of higher order system associated to (5.1.24) given by

E h,1 (t) = h 2 N j=1 |y j (t)| 2 + h 2 N j=0 y j+1 (t) -y j (t) h 2 (5.1.29) + (1 + h 2 β) 2 |η (t)| 2 t ≥ 0
and we need the following result: Lemma 5.1.4. Assume the initial data (y 0 , y 1 , η 0 ) of system (5.1.1) belongs to D(A) where D(A) is given in (5.1.4). Then, the energy E h,1 given in (5.1.29) is uniformly bounded with respect to h at t = 0.

We are ready to state our uniform polynomial decay of the energy (5.1.27):

Theorem 5.1.5. Assume the initial data (y 0 , y 1 , η 0 ) of system (5.1.1) belongs to D(A) where D(A) is given in (5.1.4). Then, for h 0 small enough, for all (y 0 j ) j , (y 1 j ) j , j = 1, .., N + 1, in C N +1 and η ∈ C, the energy E h of system (5.1.24) satisfies

E h (t) ≤ M M + t E h (0), ∀t > 0, ( 5
M h = 22 + β + 3 2β + 1 2β (6 + β) E h,1 (0) E h (0) . ( 5
E h (t) = 1 2 U h (t) 2 H h = 1 2 e λt U 0 h 2 H h = 1 2 e -2| (λ)|t U 0 h 2 H h = 1 2 e -2| (λ)|t . (5.2.2)
The fact that U 0 h H h = 1 and |λ| > 1, from (5.1.22) we obtain Taking

U 0 h 2 D(A β,h ) = (1 + |λ| 2 ) U 0 h 2 H h = (1 + |λ| 2 ) < 2|λ| 2 . ( 5
t = - 1 2 (λ)
in the above inequality we obtain

1 2 e -1 < -4M 2 (λ)|λ| 2 , or equivalently 1 8eM 2 < -(λ)|λ| 2 .
Finally, setting

M 3 = 1 8eM 2 , ( 5.2.4)
we deduce (5.2.1).

In order to proof Theorem 5.1.2, thanks to Lemma 5.2.1, we need to find a sequence (λ

N ) N ∈ σ(A β,h ) (with h = 1 N +1 ) such that |λ N | > 1 and (λ N )|λ N | 2 → 0.
For this aim, let λ ∈ σ(A β,h ) be an eigenvalue of A β,h and U h = (y h , z h , η) ∈ H h be an associated eigenvector, for h = 1 N +1 , N large enough. Equivalently, λ and U h verify the following system:

       z h = λy h , -A h y h -B h η = λz h , B * h z h -βη = (1 + h)λη,
where the operators A h , B h and B * h are given in (5.1.16), (5.1.17) and (5.1.18) respectively. Eliminating z h and η from above system, we obtain

A h y h + α h y N B1 = -λ 2 y h , ( 5.2.5)
where

α h = d h λ λ + d h β and d h = 1 1 + h . ( 5
K 0 h y t h + hα h y N V t 0 = -h 2 λ 2 y t h , (5.2.7)
where V 0 is given in (5.1.17). Next, setting

K 0 h = 2I -K 0 h , from (5.2.7) we get K 0 h y t h -hα h y N V t 0 = (2 + h 2 λ 2 )y t h . (5.2.8) Equivalently K h y t h = (2 + h 2 λ 2 )y t h , ( 5

.2.9)

where

K h =           0 1 0 . . . 0 1 0 1 . . . 0 . . . . . . . . . . . . . . . 1 0 1 0 0 . . . 1 1 -hα h           ∈ M N ×N (R).
We have found that λ is an eigenvalue of A β,h (equivalently 2 + h 2 λ 2 is an eigenvalue of K h ) if and only if there is a non trivial solution y h of (5.2.7) (equivalently of (5.2.9)). The following lemma give us the eigenvalue which does not satisfy condition (5.2.1):

Lemma 5.2.2. There exists a sequence of eigenvalues of A

β,h , h = 1 N +1 , equation: sin((N + 1)θ) -sin(N θ) + hα h sin(N θ) = 0.
(5.2.17)

We rewrite (5.2.17) as

f N (θ) = f 0 N (θ) + f 1 N (θ) = 0,
where f 0 N (θ) = sin((N + 1)θ) -sin(N θ) and where f 1 N (θ) = hα h sin(N θ). For clarity, we divide the rest of the proof into three steps.

Step 1. A root of f N . The roots of f 0 N are given by

θ 0 N,k = 2π 2N + 1 (k + 1 2 ), k ∈ Z
and we are interested to a root of f N which is close to

θ 0 N,N -1 = π - 2π 2N + 1 . (5.2.18) For this aim, let θ ∈ ∂D(θ 0 N,N -1 , 1 N 2 ), i.e. θ = θ 0 N,N -1 + 1 N 2 e it where t ∈ [0, 2π]. For N ∈ N * large enough, we can check that |f 0 N (θ)| 1 N and |f 1 N (θ)| 1 N 2 .
Consequently, there exists N 0 ∈ N * , sufficiently large, such that for N ≥ N 0 we have

|f N (θ) -f 0 N (θ)| = |f 1 N (θ)| < |f 0 N (θ)|, ∀θ ∈ ∂D(θ 0 N,N -1 , 1 N 2 ).
Then, with the help of Rouché's theorem, we deduce that f N has a root (denoted by θ N ) which is simple and close to θ 0 N,N -1 for N ≥ N 0 . More precisely, for N ≥ N 0 , θ N is given by

θ N = θ 0 N,N -1 + ζ N , ζ N = O( 1 N 2 ). ( 5

.2.19)

Step 2. Asymptotic behavior of θ N . First, using (5.2.17) we have sin

((N + 1)θ N ) -sin(N θ N ) + hα h sin(N θ N ) = 0. ( 5
((N + 1)θ N ) =(-1) N sin - π 2N + 1 + (N + 1)ζ N (5.2.21) =(-1) N sin - π 2N + 1 cos((N + 1)ζ N ) + (-1) N cos π 2N + 1 sin((N + 1)ζ N ).
On the other hand, using the asymptotic behavior when N → +∞ we get

sin - π 2N + 1 = - π 2N + π 4N 2 + π 3 -6π 48N 3 + O( 1 N 4 ), (5.2.22) cos((N + 1)ζ N ) = 1 - N 2 ζ 2 N 2 -N ζ 2 N + O( 1 N 4 ), (5.2.23) cos - π 2N + 1 = 1 - π 2 8N 2 + π 2 8N 3 + O( 1 N 4 ) (5.2.24) and sin((N + 1)ζ N ) = N ζ N + ζ N - N 3 ζ 3 N 6 + O( 1 N 4 ). ( 5
N ) =(-1) N - π 2N + N ζ N + π 4N 2 + ζ N + π 3 -6π 48N 3 (5.2.26) -(-1) N N 3 ζ 3 N 6 + π 2 ζ N 8N - N πζ 2 N 4 + O( 1 N 4 ).
Similarly, we get

sin(N θ N ) =(-1) N -1 π 2N + N ζ N - π 4N 2 + π 8N 3 (5.2.27) -(-1) N -1 π 3 48N 3 + πN ζ 2 N 4 + N 3 ζ 3 N 6 + π 2 ζ N 8N + O( 1 N 4 ).
Now, using (5.2.6) and (5.2.12) we have

hα h = hλ N d h λ N + d h β = id h 2 -2 cos(θ N ) i 2 -2 cos(θ N ) + d h hβ . ( 5
) = -1 + π 2 2N 2 - πζ N N - π 2 2N 3 + O( 1 N 4 ) (5.2.29) and therefore 2 -2 cos(θ N ) = 2 - π 2 4N 2 + πζ N 2N + π 2 4N 3 + O( 1 N 4 ).
(5.2.30) Thus, inserting (5.2.30) into (5.2.28) and after some computations we obtain

hα h = 1 N - 2 N 2 + 1 4N 3 (16 -β 2 ) + i β 2N 2 - 2β N 3 + O( 1 N 4 ). ( 5
hα h sin(N θ N ) =(-1) N -1 ζ N + π 2N 2 - 2ζ N N - 5π 4N 3 (5.2.32) + i(-1) N -1 βζ N 2N + βπ 4N 3 + O( 1 N 4 ).
Substituting (5.2.26), (5.2.27) and (5.2.32) into (5.2.20) and after a straightforward computation we get

ζ N = N 2 ζ 3 N 6 + π 2 ζ N 2N 2 - ζ N N 2 - 5π 8N 4 + π 4N 3 (5.2.33) + i βζ N 4N 2 + βπ 8N 4 + O( 1 N 5 ). From (5.2.19) we have ζ N ∼ 1 N 2 and therefore ζ N = π 4N 3 - 5π 8N 4 + i βπ 8N 4 + o( 1 N 4 ). ( 5
θ N = π - 2π 2N + 1 + π 4N 3 - 5π 8N 4 + i πβ 8N 4 + o( 1 N 4 ). ( 5

.2.35)

Step 3. Asymptotic behavior of λ N . First, using (5.2.35) we obtain

cos(θ N ) = -1 + π 2 2N 2 - π 2 2N 3 + π 2 8N 4 - π 4 24N 4 + o( 1 N 4 ).
Next, using the above equation and after some computations we get

2 -2 cos(θ N ) = 2 - π 2 4N 2 + π 2 4N 3 - π 2 16N 4 + π 4 192N 4 + o( 1 N 4 ). (5.2.36)
Finally, inserting (5.2.36) into (5.2.12), after some calculations we deduce (5.2.10). Proof of Theorem 5.1.2: First, assume that there exists a positive constant M 2 independent of h such that for all U h 0 ∈ H h inequality (5.1.23) holds. Thanks to Lemma 5.2.1, there exists another constant M 3 independent of h such that equation (5.2.1) holds for any eigenvalue λ of A β,h with |λ| > 1. On the other hand, using (5.2.10) we have |λ N | > 1 and

-(λ)|λ| 2 ∼ 1 N 2
, which leads to a contradiction with (5.2.1).

Uniform polynomial energy decay rate

In this section, we give the proofs of Lemma 5.1.4 and Theorem 5.1.5. The proof of Lemma 5.1.3 is left to the reader. It is based on the discretization choice in (5.1.25)-(5.1.26) and the fact that the initial data (y 0 , y 1 , η 0 ) of system (5.1.1) belongs to D(A). Proof of Lemma 5.1.4: We recall that the energy E h,1 at t = 0 is given by

E h,1 (0) = h 2 N j=1 |y j (0)| 2 + h 2 N j=0 y j+1 (0) -y j (0) h 2 + (1 + h 2 β) 2 |η (0)| 2 = h 2 N j=1 |y j (0)| 2 + h 2 N j=0 y 1 j+1 -y 1 j h 2 (5.3.1) + (1 + h 2 β) 2 |η (0)| 2
with y 0 j , y 1 j , η 0 are given in (5.1.25)-(5.1.26). For x ∈ [0, 1 -h], we introduce the function τ h y defined by τ h y(x) = y(x + h) -y(x) and we set I j = [(j -1)h, jh], for j = 1, .., N + 1. Moreover, we notice that since that the initial data (y 0 , y 1 , η 0 ) of system (5.1.24) belongs to D(A) then we have y 0 ∈ V ∩ H 2 (0, 1) and y 1 ∈ V . For clarity, we divide the proof into three steps.

Step 1. Uniformly boundedness of the first term of E h,1 (0). First, from the first equation of system (5.1.24) we have

y j (0) = y j+1 (0) -2y j (0) + y j-1 (0) h 2 + (y j+1 (0) -2y j (0) + y j-1 (0)) = y 0 j+1 -2y 0 j + y 0 j-1 h 2 + (y 1 j+1 -2y 1 j + y 1 j-1 ), j = 1, .., N. (5.3.2)
Taking the square of above equation, the sum over j and multiplying by h 2 we obtain

h 2 N j=1 |y j (0)| 2 ≤h N j=1 y 0 j+1 -2y 0 j + y 0 j-1 h 2 2 (5.3.3) + h N j=1 y 1 j+1 -2y 1 j + y 1 j-1 2 .
Next, using the definition of τ h given at the beginning of this proof and since (y 0 , y 1 ) ∈ V ∩ H 2 (0, 1) × V , applying Cauchy-Schwarz's inequality we get

|y 0 j+1 -2y 0 j + y 0 j-1 | 2 = I j τ h y 0 x (x)dx 2 ≤ h I j |τ h y 0 x (x)| 2 dx ≤ h 3 I j ∪I j+1 |y 0 xx (x)| 2 dx, j = 1, .., N and y 1 j+1 -2y 1 j + y 1 j-1 2 = I j τ h y 1 x (x)dx 2 ≤ h I j |τ h y 1 x (x)| 2 dx, j = 1, .., N.
Now, multiplying by h and taking the sum over j, we can check that

h N j=1 y 0 j+1 -2y 0 j + y 0 j-1 h 2 2 ≤ N j=1 I j ∪I j+1 |y 0 xx (x)| 2 dx ≤ 2 1 0 |y 0 xx (x)| 2 dx (5.3.4) and h N j=1 y 1 j+1 -2y 1 j + y 1 j-1 2 ≤ h 2 N j=1 I j |τ h y 1 x (x)| 2 dx ≤ h 2 N j=1 I j ∪I j+1 |y 1 x (x)| 2 dx ≤ 2h 2 1 0 |y 1 x (x)| 2 dx. (5.3.5)
Finally, combining (5.3.3), (5.3.4) and (5.3.5), we deduce that the first term of E h,1 (0) is uniformly bounded with respect to the mesh size h.

Step 2. Uniformly boundedness of the second term of E h,1 (0). Since y 1 ∈ V , using Cauchy-Schwarz's inequality we get

y 1 j+1 -y 1 j 2 = I j+1 y 1 x (x)dx 2 ≤ h I j+1 |y 1 x (x)| 2 dx.
Then multiplying by h 2 and taking the sum over j, we obtain

h 2 N j=0 y 1 j+1 -y 1 j h 2 ≤ 1 2 N j=0 I j+1 |y 1 x (x)| 2 dx ≤ 1 2 1 0 |y 1 x (x)| 2 dx. (5.3.6)
Hence, we deduce the uniformly boundedness of the second term of E h,1 (0) with respect to the mesh size h.

Step 3. Uniformly boundedness of E h,1 (0). From the third and the sixth equations of (5.1.24), we have

η (0) = y N +1 (0) -βη(0) = y 1 N +1 -βη 0 . (5.3.7) Next, since y 1 ∈ V ⊂ C([0, 1]
) and using (5.1.26) for j = N + 1 we get

y 1 N +1 = 1 h I N +1 y 1 (x)dx ≤ max x∈[0,1] |y 1 (x)| y 1 V . ( 5

.3.8)

Therefore y 1 N +1 is uniformly bounded with respect to the mesh size h. On the other hand, since η 0 is independent of h, from (5.3.7) we conclude the uniformly boundedness of the third term of E h,1 (0) with respect to the mesh size h. Finally, combining the results of the previous steps with equation (5.3.1), we deduce that E h,1 (0) is uniformly bounded with respect to the mesh size h.

Next, we prove Theorem 5.1.5 using a multiplier method inspired from [START_REF] Wehbe | Rational energy decay rate for a wave equation with dynamical control[END_REF]. Indeed, we need to find a constant M uniformly bounded with respect to the mesh size h and T > 0 such that

T 0 E 2 h (t)dt ≤ M E 2 h (0).
(5.3.9)

In this case, applying Theorem 9.1 in [START_REF] Komornik | Exact Controllability and Stabilization: The Multiplier Method[END_REF] we deduce directly (5.1.30). We now prove (5.3.9) by the following lemma:

Lemma 5.3.1. Let T > 0 and assume the initial data (y 0 , y 1 , η 0 ) of system (5.1.1) belongs to D(A) where D(A) is given in (5.1.4). Then there exists h 0 > 0 small enough such that the energy E h of system (5.1.24) given in (5.1.27) satisfies the following estimation: Proof: For simplicity, in this proof we denote η(t) by η. First, we set T > 0 and we define the following scalars:

T 0 E 2 h (t)dt ≤ M E 2 h (0), ( 5
       Ψ j = j y j+1 -y j-1 2 , for j = 1, ..., N, Ψ N +1 = (N + 1) 2 (y N +1 -y N ).
We notice that the multiplier Ψ j is no more then the discrete form of xy x used in [START_REF] Wehbe | Rational energy decay rate for a wave equation with dynamical control[END_REF] to prove Theorem 2.1. Next, multiplying the first equation of (5.1.24) by Ψ j E h (t), integrating from 0 to T and taking the sum over j we obtain

I 1 -I 2 -I 3 = 0, (5.3.11)
where

I 1 = 1 2 T 0   N j=1
jy j (y j+1 -y j-1 )

  E h (t)dt, (5.3.12)

I 2 = T 0   N j=1 j y j+1 -2y j + y j-1 h 2 y j+1 -y j-1 2   E h (t)dt (5.3.13)
and where

I 3 = 1 2 T 0   N j=1 j(y j+1 -2y j + y j-1 )(y j+1 -y j-1 )   E h (t)dt. (5.3.14)
For clarity, we split the rest of the poof into five steps.

Step 1. Elementary calculations for I 1 and I 2 .

First, integrating by parts in (5.3.12) we obtain

I 1 = 1 2 N j=1
jy j (y j+1 -y j-1 )E h (t) On the other hand, using the second equation of ( Step 2. First estimation of the energy. Substituting (5. Since the initial data (y 0 , y 1 , η 0 ) of system (5. Following ZuaZua in [START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF], from (5.1.27)-(5.1.28) and the definitions of p h and q h in (5.1.32)-(5.1.33), we first rewrite the energies E h , E h,1 and their derivatives as follows: (n) and q h y h (n) in the Hilbert space L 2 (0, 1) by the following lemma:

E h (t) = 1 2 p h y h (t) 2 V + 1 2 q h y h (t)
Lemma 5.4.1. The functions p h y h (n) and q h y h (n) satisfy the following relation:

p h y h (n) -q h y h (n) 2 L 2 (0,1) = h 3 3 N j=0   y (n) j+1 -y (n) j h   2 .
(5.4.5)

Proof: Using (5.1.32)-(5.1.33), we can check that

p h y h (n) -q h y h (n) 2 L 2 (0,1) = 1 0 |p h y h (n) (x) -q h y h (n) (x)| 2 dx = N j=0 (j+1)h jh y j+1 -y j j (x -jh) 2 dx = h 3 3 N j=0   y (n) j+1 -y (n) j h   2 .
Next, we prove the convergence results (5.1.36) by the following two lemmas:

Lemma 5.4.2. Assume the initial data (y 0 , y 1 , η 0 ) of system (5.1.1) belongs to D(A) where D(A) is given in (5.1.4). Then, the convergence results in (5.1.36) hold.

Proof: First, since E h is a decreasing function as time increase and using (5.4.1) we obtain

p h y h (t) 2 V ≤ 2E h (0), ∀t ≥ 0.
Since the initial data (y 0 , y 1 , η 0 ) of system (5.

p h y h (t) 2 V < +∞,
i.e. p h y h is bounded in L ∞ (0, ∞; V ). Thus, p h y h is bounded in L ∞ (0, ∞; L 2 (0, 1)). For the same reason, we conclude that q h y h is bounded in L ∞ (0, ∞; L 2 (0, 1)) and η is bounded in L ∞ (0, ∞; C). Moreover, using (5.4.3) and (5.4.5) for n = 1, we have p h y h (t) 2 L 2 (0,1) ≤ p h y h (t) -q h y h (t) 2 L 2 (0,1) + q h y h (t) Since the initial data (y 0 , y 1 , η 0 ) of system (5.1.1) belongs to D(A), thanks to Lemma 5.1.4, we know E h,1 (0) is uniformly bounded with respect to the mesh size h and therefore p h y h is bounded in L ∞ (0, ∞; L 2 (0, 1)). Moreover, since E h,1 is a decreasing function as time increase, we deduce from (5.4.3) that p h y h is bounded in L ∞ (0, ∞; V ), q h y h is bounded in L ∞ (0, ∞; L 2 (0, 1)) and η is bounded in L ∞ (0, ∞; C). Thus, from the third equation of (5.1.24), we deduce that y N +1 is bounded in L ∞ (0, ∞; C). On the other hand, integrating (5.4. Taking s → ∞ and since the energy E h,1 at t = 0 is uniformly bounded with respect to h, we deduce from the above equation that hp h y h is bounded in L 2 (0, ∞; V ), η and hη are bounded in L 2 (0, ∞; C). Finally, using the third equation of (5.1.24), we claim that y N +1 is bounded in L 2 (0, ∞; C).

From the above boundedness results, we can extract the following subsequences:

                                  
p h y h → y weakly* in L ∞ (0, ∞; V ), p h y h → y weakly* in L ∞ (0, ∞; L 2 (0, 1)), p h y h → y weakly* in L ∞ (0, ∞; V ), p h y h → y weakly* in L ∞ (0, ∞; L 2 (0, 1)), q h y h → y weakly* in L ∞ (0, ∞; L 2 (0, 1)), q h y h → y weakly* in L ∞ (0, ∞; L 2 (0, 1)), y N +1 → y (1, t) weakly in L 2 (0, ∞; C), hp h y h → 0 weakly in L 2 (0, ∞; V ), hη → 0 weakly in L 2 (0, ∞; C).

(5.4.7)

The eighth convergence in (5.4.7) follows from the fourth one and the boundedness of the sequence of hp h y h in L 2 (0, ∞; V ). As for the seventh convergence, it follows from the first one and the boundedness of y N +1 in L 2 (0, ∞; C). Note that in (5.4.7), using (5.4.5) we implicitly claim that the limits of p h y h (n) and q h y h (n) are the same for n = 0 and n = 1. The proof is complete.

Lemma 5.4.3. Assume the initial data (y 0 , y 1 , η 0 ) of system (5.1.1) belongs to D(A) where D(A) is given in (5.1.4). Then, the weakly* limit (y, y , η) of the previous lemma is the unique solution of system (5.1.1).

Proof: For clarity, we divide the proof into several steps.

Step 1. y verifies the first equation of (5.1.1). Applying the same strategy used in section 3. To complete the proof it remains to verify (5.1.37) and (5.1.38). For clarity, we divide the proof into three steps.

Step 1. Proof of (5.1.37). We begin by integrating (5.4.

Thèse de Doctorat

Mohamad Ali Hassan SAMMOURY Etude théorique et numérique de la stabilité de certains systèmes distribués avec contrôle frontière de type dynamique Résumé Cette thèse est consacrée à l'étude de la stabilisation de certains systèmes distribués avec un contrôle frontière de type dynamique. Nous considérons d'abord la stabilisation de l'équation de la poutre de Rayleigh avec un seul contrôle frontière dynamique moment ou force. Ensuite, nous étudions la stabilisation indirecte de l'équation des ondes avec un amortissement frontière de type dynamique fractionnel. Nous montrons que le taux de décroissance de l'énergie dépendent de la nature géométrique du domaine et nous établissons plusieurs résultats de stabilité polynomiale. Enfin, nous considérons l'approximation de l'équation des ondes un-dimensionnelle avec un seul amortissement frontière de type dynamique par un schéma de différence finie. Par une méthode spectrale, nous montrons que l'énergie discrétisée ne décroit pas uniformément (par rapport au pas du maillage) polynomialement vers zéro comme l'énergie du système continu. Nous introduisons, alors, un terme de viscosité numérique et nous montrons la décroissance polynomiale uniforme de l'énergie de notre schéma discret avec ce terme de viscosité.

Abstract

This thesis is devoted to the study of the stabilization of some distributed systems with dynamic boundary control. First, we consider the stabilization of the Rayleigh beam equation with only one dynamic boundary control moment or force. Next, we study the indirect stability of the wave equation with a fractional dynamic boundary control.

We show that the decay rate of the energy depends on the nature of the geometry of the domain and we establish different polynomial stability results. Finally, we consider the finite difference space discretization of the 1-d wave equation with dynamic boundary control. First, using a spectral approach, we show that the polynomial decay of the discretized energy is not uniform with respect to the mesh size, as the energy of the continuous system. Next, we introduce a viscosity term and we establish the uniform (with respect to the mesh size) polynomial energy decay of our discrete scheme.

Mots clés

Contrôle frontière dynamique, Stabilité polynomiale, étude spectrale, Méthode de Différence fini.

Key Words

Boundary dynamical control, Polynomial stability, Spectral study, Finite difference method.

Notations:

  Dans toute la thèse, la notation A B (respectivementA B) signifie l'existence d'une constante positive C 1 (respectivement C 2 ), indépendante de A et B tel que A ≤ C 1 B (respectivement A ≥ C 2 B). La notation A ∼ B désigne que A B et A B sont satisfaites simultanément.

. 0 B

 0 Nous déduisons alors, qu'il existe T > 0 et C T > 0 telle que la solution U du problème non amorti associé à (0.0.2) satisfaitT * U (t) 2 H dt ≥ C T U 0 (D(A 0 )) , (0.0.3) où B * représente l'opérateur adjoint associé à B et (D(A 0 )) est le dual de l'espace D(A 0 ) par rapport au produit scalaire de l'espace de Hilbert H. De plus, nous montrons que la fonction de transfert définit par

20

 20

1. 2

 2 Strong and exponential stability

Proposition 2 . 3 . 4 .

 234 Let |k| ≥ k 0 . Then, a solution y k of the undamped initial value problem (2.3.4) with µ = µ k satisfies the following estimations:

1 0e rx dx = e r r - 1 r

 11 for r = 0 and the asymptotic behavior (2.3.13)-(2.3.14) we find that

Lemma 2 . 3 . 6 .

 236 (Boundedness of H on C ω ) The transfer function H defined in (2.3.70) is bounded on C ω .

Abstract:

  In [92], Wehbe considered a Rayleigh beam equation with two dynamical boundary controls and established the optimal polynomial energy decay rate of type 1 t . The proof exploits in an explicit way the presence of two boundary controls, hence the case of the Rayleigh beam damped by only one dynamical boundary control remained open. In this chapter, we fill this gap by considering a clamped Rayleigh beam equation subject to only one dynamical boundary control force. We use a Riesz basis approach. First, we start by giving the asymptotic expansion of the eigenvalues and the eigenfunctions of the damped and undamped systems. Next, we show that the system of eigenvectors of the damped problem form a Riesz basis. Finally, we deduce the optimal energy decay rate of polynomial type in 1 √ t .

.3. 33 ) 3 .

 333 Step Asymptotic behavior of ζ β,k . First, using (3.3.31) and the identities (3.3.23)-(3.3.30) we have 0

  .3.40) Then, substituting (3.3.35)-(3.3.40) into (3.3.34) and after some computation yields

.3. 53 )Proposition 3 . 3 . 3 .Figure 3 . 1 : 10 problem ( 3 . 3 . 1 )

 533333110331 Figure 3.1: Eigenvalues of A 1 (in blue) and A 0 (in red) with β = 1 and γ = 10

.3. 55 )

 55 Proof: For simplicity, in this proof we denote λ β,k by λ k and y β,k by y k . For β ≥ 0, λ = λ k and |k| ≥ k β , solving (3.3.1) amounts to find a solution C(λ k ) = 0 of system (3.3.4) of rank three. For clarity, we divide the proof to several steps.Step 1. Determination of y k . Since we search C(λ k ) up to factor we choose c 4 (λ k ) = 1, the possibility of this choice will be justify later. Therefore(3.3.4) becomes

Now, using ( 3

 3 .3.8) ,(3.3.67) and the asymptotic behaviors (3.3.12)-(3.3.13) for λ = λ β,k and for λ = λ 0,k we find

2

 2 and using (3.4.3)-(3.4.4) we obtain

Proposition 4 . 2 . 4 .

 424 If Ω is the unit square with Γ 1 = {(0, y), y ∈ (0, 1)}, and Γ 0 = Γ\Γ 1 , then the statements of Proposition 4.2.2 and Corollary 4.2.3 are valid.

.2. 16 )

 16 For that purpose, we use a reflexion technique. Let us fix (u, v, w, z) ∈ D(A) and introduce the functionũ(x, y) =    u(x, y)if y > 0, -u(x, -y) if y < 0, defined in the half-plane R 2 + := {(x, y) ∈ R 2 : x > 0}, and similarly w

  of H and the restriction B of A to H 0 , defined by D(B) = D(A) ∩ H 0 , and BU = AU, ∀U ∈ D(B).

.3. 4 )Proposition 4 . 3 . 1 .

 4431 By the following proposition we give the asymptotic behavior of the eigenvalue of high frequency associated to the radial solution f (r) of problem (4.3.2) in S: There exists k 0 ∈ N * and a sequence (λ k ) k≥k 0 of simple roots of φ (that are also simple eigenvalues of A) and satisfying the following asymptotic behavior:

  2 k 2 (λ k ) 1.00495 1.00331 1.00249 1.00199 1.00166 1.00142 1.00125 1.00111 1.001

Definition 4 . 4 . 1 .

 441 We say that the boundary Γ of Ω satisfies the multiplier control condition MCC, if there exists x 0 ∈ R d and a positive constant m 0 > 0 such that

Lemma 4 . 4 . 4 . 1 |z| 2

 44412 .4.7) The solution (u, v, w, z) ∈ D(A) of system (4.4.6) satisfies the following estimation: First, multiplying equation (4.4.5) by U in H, we get Γ dΓ = (iβU -AU, U ) H = o(1) β l . (4.4.9)

.4. 10 )Lemma 4 . 4 . 5 .

 10445 Finally, since u = w on Γ 1 , from (4.4.10) we deduce directly (4.4.8). Before going on, we give a relation between u and ∂ ν u by the following lemma: Let Ω be a bounded domain of R d , d ≥ 1, with Lipschitz boundary

.4. 15 )

 15 As u = v + w and ∂ ν u = ∂ ν v + ∂ ν w and since by (4.4.13) w ∈ H 1 (Γ) and ∂ ν w ∈ L 2 (Γ), using (4.4.14)-(4.4.15) we deduce that

.4. 33 )

 33 Proof: First, let h = ∆(ηϕ u ) = η∆ϕ u + 2∇η • ∇ϕ u + ∆ηϕ u where η is defined in (4.4.20)-(4.4.21). Next, it is easy to check that

4 . 31 )

 431 by 2ηm • ∇ϕ u where η is the cut off function define in (4.4.20)-(4.4.21), we get

  .5.25) Substituting equations (4.5.21)-(4.5.23) and (4.5.25) into (4.5.17) and 123

Figure 4 . 1 :

 41 Figure 4.1: Eigenvalues of A with l = 1

Theorem 4 . 6 . 1 .

 461 The eigenvalues λ 2 of A L are strictly larger then L 2 and are the roots of the transcendental equationtan(θ) = 1 θ , (4.6.10)

1 .

 1 Finally, setting γ = C min k∈N (θ k+1 -θ k ), then from (4.6.18)-(4.6.20) we obtain (4.6.12).

1 . 1 )

 11 into a first order evolution equation U (t) = (A + C)U (t) for t > 0, with U (0) = U 0 ∈ H, where U = (y, y t , η), D(A) = {U = (y, z, η) ∈ H; z ∈ V and y x (1) + η = 0} , (5.1.4) AU = (z, y xx , z(1)), ∀U = (y, z, η) ∈ D(A)

E 1 ( 1 0

 11 |y x (x, t)| 2 dx + |η (t)| 2 , t ≥ 0. (5.1.7)

.1. 31 )

 31 Remark 5.1.6. The constant M of the inequality (5.1.30) exists. Indeed, since the initial data (y 0 , y 1 , η 0 ) of system (5.1.1) belongs to D(A), then (λ) ≤ 0 and we have

  .3.10) where M = sup h∈]0,h 0 [ M h , for h 0 small enough and where M h is given in (5.1.31).

2 y

 2 j (y j+1 -y j-1 )   E h (t)dt. N y N +1 .

j 0 y N +1 -y N h 2 E

 02 (y j+1 -2y j + y j-1 ) (y j+1 -y j-1 ) y j+1 -y j ) -(y j -y j-1 ))((y j+1 -y j ) + (y j -y j-1 )) h (t)dt.

1 . 1 )

 11 belongs to D(A), thanks to Remark 5.1.6, from above equation we deduce (5.3.9) with M = sup h∈]0,h 0 [ M h , for h 0 small enough.

5. 4

 4 Convergence results: proof of Theorem 5.1.7

1 j=0(y j+1 -y j )w j+1 =h 3 N j=0 y= -h 3 N j=0 y 1 0+ h 2 ∞ 0 1 0

 13j=03j=011 2 in[START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF], we show that y satisfies y -y xx = 0.(5.4.8)But for calculations reason needed in the following steps of this proof, we will give the details. First, let w ∈ D([0, 1]×]0, ∞[) with w(0, •) ≡ 0 and we set w h = (w j ) j where w j = w(jh, •). Multiplying the first equation of (5.1.24) by hw j , integrating by parts on (0, ∞) and taking the sum over j, -2y j + y j-1 )w j dt.Using the second equation of (5.1.24) and since w 0 = 0, we haveh N j=1 y j+1 -2y j + y j-1 -2y j + y j-1 )w j =h N j=1 (y j+1 -y j )w j -h N j=1 (y j -y j-1 )w j =h N j=0 (y j+1 -y j )w j -h N -j+1 -y j h w j -w j+1 h + h(y N +1 -y N )w N +1 j+1 -y j h w j+1 -w j h (5.4.11) -h 2 η w N +1 .Next, inserting (5.4.10)-(5.4.11) into (5.4.9)w N +1 (t)dt + h 2 ∞ 0 η (t)w N +1 (t)dt + h 3 ∞ 0 N j=0 y j+1 -y j h w j+1 -w j h .It follows from the definitions of p h and q h in (5.1.32)-(5.1.33) that (5.4.12) is equivalent toy h )(q h w h )dxdt + ∞ 0 (p h y h ) x (p h w h ) x dxdt (5)w N +1 (t)dt + h 2 ∞ 0 η (t)w N +1 (t)dt (p h y h ) x (p h w h ) x dxdt.Now, we recall the following elementary convergence results: for every w ∈ D([0, 1] × (0, ∞)) we have   p h w h → w strongly in L 2 (0, ∞; H 1 (0, 1)), q h w h → w strongly in L 2 (0, ∞; L 2 (0, 1)). (5.4.14) Using the convergence results in (5.4.7) and (5.4.14), passing to the limit Proof of Theorem 5.1.7: First, from the two Lemmas 5.4.2 and 5.4.3, we directly get the convergence results in (5.1.36).

Theorem 1.4.4. (Theorem

  2.1 of Chapter VI in [40]) An arbitrary family {φ n } n∈N of vectors forms a Riesz basis of a Hilbert space H if and only if {φ n } n∈N is complete in H and there corresponds to it a complete biorthogonal sequence {ψ n } n∈N such that for any f ∈ H one has

	|< ψ n , f >| 2 < ∞.	(1.4.1)
	n∈N
	Theorem 1.4.5. (Classical Bari's Theorem)
	Let {ϕ n } n∈N be a Riesz basis of a Hilbert space H and another ω-linearly
	independent family {ψ n } n∈N is quadratically close to {ϕ n } n∈N in the sense
	that

n∈N |< φ n , f >| 2 < ∞,

2 Rayleigh beam equation with only one dynamical boundary control moment

	1.4 Riesz basis
	an integer N ≥ 0 and a sequence of generalized eigenvectors {φ n } ∞ n=N +1
	of A such that	∞
		ϕ n -φ n	2 < ∞,
	n=N +1
	then the set of generalized eigenvectors of A, {φ n } ∞ n=1 , forms a Riesz
	basis of H.
	In this thesis, we use the following theorem from [2, Theorem 1.4.10]
	which clarifies the results of Theorem 1.4.6:
	Theorem 1.4.7. Let A be a densely defined operator in a Hilbert space
	H with a compact resolvent. Let {ϕ n } ∞ n=1 be a Riesz basis of H. If there
	are two integers N 1 , N 2 ≥ 0 and a sequence of generalized eigenvectors
	{φ n } ∞ n=N +1 of A such that
	∞
		ϕ n+N 2 -φ n+N 1	2 < ∞,	(1.4.2)
	n=1
	then the set of generalized eigenvectors (or root vectors) of A, {φ n } ∞ n=1
	forms a Riesz basis of H.
	Abstract: In [92], Wehbe considered a Rayleigh beam equation with two dynamical bound-
	ary controls and established the optimal polynomial energy decay rate of type	1 t	. The proof
	exploits in an explicit way the presence of two boundary controls, hence the case of the
	Rayleigh beam damped by only one dynamical boundary control remained open. In this
	chapter, we fill this gap by considering a clamped Rayleigh beam equation subject to only
	3 of [43] which is a new form of Bari's theorem one dynamical boundary control moment. First, we prove a polynomial decay in 1 t of the
	(see Theorem 2.3 of Chapter VI in [40]).
	Theorem 1.4.6. Let A be a densely defined operator in a Hilbert space H
	with a compact resolvent. Let {ϕ n } ∞ n=1 be a Riesz basis of H. If there are

  inserting (2.3.34) in (2.3.32) and in (2.3.33) we obtain

  r 4 c 4 e r 4 , (2.3.52) where we recall that for i = 1, ..., 4, r i = r i (µ k ) are given by (2.3.6) and c

i for i = 1, ..., 4, satisfy (2.3.50). Therefore using the series expansion (2.3.10), (2.3.13), (2.3.14) and (2.3.50) we easily find

  .4.32)

	Step 3. Asymptotic behavior of k . First, using (2.4.29) and the
	identities (2.4.24)-(2.4.27) we have

  .3.66) Remark that b 4 = 0 provided we have chosen k β large enough, for this reason our choice c 4 (λ k ) = 1 is valid. Substituting (3.3.63)-(3.3.66) into (3.3.56), we deduce

  If Γ 0 is non empty, then the semigroup of contractions (e tA ) t≥0 is strongly stable on the energy space H, i.e. for any U 0 ∈ H,

	we have
	lim t→+∞	e tA U 0 H = 0.	(4.2.21)

Theorem 4.2.5.

  Using Lemmas 4.2.6 and 4.2.7, we directly deduce that the imaginary axis is included in the resolvent set of A. We then conclude (4.2.21) with the help of Arendt-Batty's theorem[START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF].

	Case 2. Γ 0 = ∅. As before using Lemmas 4.2.6 and 4.2.7 and Arendt-
	Batty's theorem, we conclude that the semigroup generated by B is sta-
	ble, in other words
	lim t→+∞

2.33

) has a unique solution u ∈ V . By choosing appropriated test functions in (4.2.33), we see that u satisfies (4.2.32). By defining w = u, z = iλw -h on Γ 1 and v = iλu -f in Ω, we deduce that U = (u, v, w, z) belongs to D(A) and is solution of (4.2.30). This completes the proof.

Proof of Theorem 4.2.5: We distinguish two cases: Case 1. Γ 0 = ∅.

  admits an infinity of simple roots in S denoted by λ k , with |k| ≥ k 0 , k 0 large enough, such that

	Next, from (4.3.6) and (4.3.7) it follows that for λ ∈ S we have
	φ(λ) =i cos(	π 4	+ iλ)		(4.3.8)
	+	9 8		cos(	π 4	-iλ) + i cos(	π 4	+ iλ)	1 λ
	+ -	39 128	i cos(	π 4	+ iλ) +	1 8	cos(	π 4	-iλ)	1 λ 2
	+ O(	1 |λ| 3 ).
	Then, since the roots of the analytic function λ → cos( ikπ -i π 4 , k ∈ Z, using Rouché's theorem, we deduce from (4.3.8) that φ π + iλ) are λ 0 k = 4
											8	i cos(	π 4	+ iλ)	(4.3.7)
			+ O(	1 |λ|	),	|λ| → ∞.

  then by taking the real part of (4.4.22) and using (4.4.23)-(4.4.24) we obtain

	Γ 1

  .4.29) Thus, substituting (4.4.29) into (4.4.28) with l = 8 we directly get (4.4.26). Next, if Γ 0 ∩ Γ 1 = ∅, the multiplier control condition MCC on Γ 1 holds and if l = 2, then using (4.4.8) and (4.4.19) we obtain

  If the boundary Γ of Ω is C 1,1 and under the assumptions Γ 0 ∩Γ 1 = ∅, the geometric control condition GC hold and l = 2, using Lemmas 4.4.4, 4.4.8, 4.4.9 and 4.4.11, then from (4.4.51) and (4.4.52) we get (4.4.53), (4.4.54) and (4.4.55). Which is a contradiction with (4.4.4).

	This implies from the first equation of (4.4.6) that
	|v| 2 dΩ = o(1)	(4.4.54)
	Ω
	and therefore
	U H = o(1).	(4.4.55)
	Which is a contradiction with (4.4.4).
	2.
	4, 4.4.8, 4.4.9 and 4.4.10,
	then from (4.4.51) and (4.4.52) we obtain
		(4.4.53)

Ω |βu| 2 dx = Ω |∇u| 2 dx = o(1).

  table below confirms this behavior.

	k	100	150	200	250	300	350	400	450	500
	-π 2 k 2 (λ 1,k ) 0.999949 0.999977 0.999987 0.999992 0.999994 0.999996 0.999997 0.999998 0.999998
								40
								20
	0.05	0.04		0.03	0.02		0.01
								20
								40

  y, t) = + |u pπ x (x, t)| 2 + p 2 π 2 |u pπ (x, t)| 2 dx (4.7.2) There exists a positive constant C > 0, such that for all initial data U 0 = (u 0 , u 1 , w 0 , w 1 ) ∈ D(A 2 ), the energy of system (4.1.2) in the unit square of R 2 with Γ 1 = {0} × (0, 1), satisfies

	where U pπ (x, t) = (u pπ (x, t), w pπ (t)) is solution of system (4.6.1) with
	L = pπ. Recalling that the energy of system (4.6.1) is given by
	E pπ (t) =	1 2 t (x, t)| 2 + 1 0 |u pπ 1 2 |w pπ t (t)| 2 + p 2 π 2 2	|w pπ (t)| 2 .
	We clearly have
				+∞
		E(t) =	E pπ (t).	(4.7.3)
				p=1
	Using a Fourier synthesis and the result of Theorem 4.6.5, we obtain
	the following polynomial decay of energy of system (4.1.2) on the unit
	square:
	Theorem 4.7.1. E(t) ≤	C t	U 0	2 D(A 2 ) .	(4.7.4)
	Proof: First, combining (4.6.41) and (4.7.3) we obtain
		+∞
				U pπ (x, t) sin(pπy),	(4.7.1)
		p=1

  x)| 2 dx + p 4 π 4 1

	while • D(A L )			A L • H , while by definition we have
	U pπ t (x, 0) 2 V L = A L U pπ t (x, 0), U pπ t (x, 0)
					=	0	1	|u pπ 1,x (x)| 2 dx + p 2 π 2 1 0	|u pπ 1 (x)| 2 dx	(4.7.8)
						+ p 2 π 2 |w pπ 1 | 2 .
	Now, combining (4.7.5), (4.7.6), (4.7.7) and (4.7.8) we get
							E(t) ≤	C 9 t	E 2 (0)	(4.7.9)
	where
	E 2 (0) =	+∞ p=1	0	1	p 2 π 2 |u pπ 0,xx (x)| 2 + p 6 π 6 |u pπ 0 (x)| 2 + 2p 4 π 4 |u pπ 0,x (x)| 2 dx
		+∞					+∞
	+	p 6 π 6 |w pπ 0 | 2 +	p 2 π 2 |u pπ 0,x (0)| 2	(4.7.10)
		p=1					p=1
							0	|u pπ 0 (x)| 2 dx	(4.7.7)
					+ 2p 2 π 2 1 0	|u pπ 0,x (x)| 2 dx + p 4 π 4 |w pπ 0 | 2 + |u pπ 0,x (0)| 2 ,

  5.1.24) we have y N = hη + y N +1 . Thus, from the third one we get

	Then, substituting (5.3.17) into (5.3.16) yields
	1 2	N j=1	jy j (y j+1 -y j-1 ) =	h 2 4	N j=0	y j+1 -y j h	2	-	1 2	N j=0	|y j | 2	(5.3.18)
													+	(N + 1) 2	|y N +1 | 2 -	1 4	|y N +1 | 2
													+	1 2	|η | 2 +	β 2	ηη .
	Inserting (5.3.18) in (5.3.15), we obtain
	I 1 =	1 2	N j=1	jy j (y j+1 -y j-1 )E h (t)	T 0	(5.3.19)
			-	h 2 4		0	T	  j=0 N	y j+1 -y j h	2   E h (t)dt
			+	1 2	0	T	 	N j=0	|y j | 2	  E h (t)dt -	(N + 1) 2	0	T	|y N +1 | 2 E h (t)dt
			+	1 4
		(N + 1) 2	y N y N +1 =	1 2	η y N +1 +	(N + 1) 2	|y N +1 | 2
												=	(N + 1) 2	|y N +1 | 2 +	1 2	|η | 2 +	β 2	ηη .	(5.3.17)

T 0 |y N +1 | 2 E h (t)dt -1 2 T 0 |η | 2 E h (t)dtβ 2 T 0 ηη E h (t)dt

Step 5. Second estimation of the energy. Inserting

  3.14),(5.3.19) and(5.3.21) into(5.3.11) and multiplying by h yields E h (t)dt on the left and right sides of above identity, taking in mind that for h small enough we have 1 + h 2 βStep 3. Estimations of the terms of(5.3.23). First, since E h is a decreasing function with respect to time t and using(5.1.28) we getWe notice that the energy E h,1 is a decreasing function with respect to time t since its derivative of E h,1 is given byE h,1 (t) = -h 3 -β|η (t)| 2 -h 2 |η (t)| 2 .(5.3.25) Thus, from (5.1.28),(5.3.25) and the third equation of (5.1.24) we obtain j+1 -y j ) + (y j -y j-1 )) h j+1 -2y j + y j-1 )(y j+1 -y j-1 ) j+1 -2y j + y j-1 ) 2 + 4 (5.3.24), (5.3.26),

	Further, using Young's inequality and (5.1.27), we check that (5.1.27) and (5.1.29) we get
	h 2 Adding T 0   j=0 N h 3 4 T 0   j=0 |y j | 2 N -h 4 T 0 |y N +1 | 2 E h (t)dt +   E h (t)dt + y j+1 -y j h 2  h 2  E h (t)dt + T 0   N j=0 h 2 T 0 |η | 2 E h (t)dt + y j+1 -y j h 1 2 T 0 |y N +1 | 2 E h (t)dt 2   E h (t)dt = hβ 2 T 0 ηη E h (t)dt (5.3.22) + 1 2 T 0 y N +1 -y N h 2 E h (t)dt + h 2 T 0   N j=1 jy j (y j+1 -y j-1 ) h 3 4 T 0   N j=0 y j+1 -y j h 2   E h (t)dt ≤ -1 4 T 0 E h (t)E h (t)dt = 1 8 E 2 h (0) -E 2 h (T ) ≤ -h 2 N j=1 jy j (y j+1 -y j-1 ) = -h 2 N j=1 hjy j ((y ≤ h 2 2 N j=1 j|y j | 2 + h 2 4 N j=1 j y j+1 -y j h 2 N h 2 j=1 j(y ≤ h 2 N j(y N j(y j+1 -y j-1 ) 2 4 j=1 j=1 1 8 E 2 h (0). (5.3.24) + h 2 4 N j=1 j y j -y j-1 h 2 ≤ h 4 2 N j=0 j y j+1 -y j h 2 + h 4 2 N 2 y j -y j-1 j h j=1   E h (t)dt -h 2 N j=1 jy j (y j+1 -y j-1 )E h (t) T 0 + hI 3 . (1 + h 2 β) 2 T 0 |η| 2 2 ≤ 1 and since -h 4 we obtain T 0 E 2 h (t)dt ≤ h 3 4 T 0   N j=0 y j+1 -y j h 2   E h (t)dt (5.3.23) + 1 2 T 0 |y N +1 | 2 E h (t)dt + h 2 T 0 |η | 2 E h (t)dt + T 0 |η| 2 E h (t)dt + hβ 2 T ηη E h (t)dt N j=0 y j+1 -y j h ≤ h 2 N 2 N j=1 |y j | 2 + 3h 2 N 4 N j=0 y j+1 -y j h 2 + h 2 2 N j=0 j y j+1 -y j h 2 + h 2 2 N j=1 2 y j -y j-1 j h 2 1 2 T 0 |y N +1 | 2 E h (t)dt + h 2 T 0 |η | 2 E h (t)dt+ T 0 |η| 2 E h (t)dt ≤(1 + h 2 ) ≤ -1 β (1 + ≤ 5 2 E h (t). (5.3.28) ≤ 3h 4 N 2 N j=0 y j+1 -y j h 2 3h 2 N N 2 y j+1 -y j + 2 h j=0 Its follows that h 2 T 0   N j=1 jy j (y j+1 -y j-1 )   (E h (t))dt ≤ -5 2 T 0 ≤ 3h 3 2 N j=0 y j+1 -y j h 2 3h N 2 y j+1 -y j + 2 j=0 h E h (t)E h (t)dt ≤ 5 4 E 2 h (0). (5.3.29) ≤ -3 2 E Combining the above inequality together with (5.3.14) we obtain h 2 ) ≤ -1 β (1 + h 2 )E h (0) T 0 E h,1 (t)dt + (1 + β 2 ) 2β E 2 h (0) ≤(1 + h 2 ) E h,1 (0) E 2 h (0) + (1 + β 2 ) 2β E 2 h (0) Now, using the second equation of (5.1.24) and (5.1.28) we get 1 2 T 0 y N +1 -y N h 2 E h (t)dt = 1 2 T 0 hI 3 ≤ -3 2 T 0 E h (t)E h (t)dt + 3 T E 2 h (t)dt 0 |η| 2 E h (t)dt ≤ -1 2β T 0 ≤ 3 4 E 2 h (0) + 3 T E 2 (5.3.32) h (t)dt. 0 E h (t)E h (t)dt βE h (0) = 1 2β 1 + β 2 + (2 + h) E h,1 (0) E h (0) E 2 h (0). (5.3.26) ≤ 1 4β E 2 h (0). (5.3.30) 1 (5.3.27), (5.3.29), (5.3.30), (5.3.31) and (5.3.32) into (5.3.23) with = 6 we get 0 + h 2 T 0   N j=1 jy j (y j+1 -y j-1 )   E h (t)dt + 1 2 T 0 y N +1 -y N h 2 E h (t)dt -h 2 N j=1 jy j (y j+1 -y j-1 )E h (t) T 0 + hI 3 . Finally, using (5.3.28) we obtain Moreover, using Young's inequality, (5.1.28) and (5.3.25) we get hβ 2 T 0 ηη E h (t)dt ≤ hβ 4 T 0 |η| 2 E h (t)dt + hβ 4 T 0 |η | 2 E h (t)dt ≤ -h 4 T 0 E h (t)E h (t)dt -h 4 E h (0) T 0 E h,1 (t)dt ≤ -h 8 E 2 h (t) T 0 + 4E h (0) E 2 h (0) Step 4. Estimation of hI 3 . Setting > 0 and using Young's inequality, hE h,1 (0) -h 2 N j=1 jy j (y j+1 -y j-1 )E h (t) T 0 ≤ h 2 N j=1 T T 0 E 2 h (t)dt ≤ 87 4 + β + 3 2β + h 4 + 1 2β (4 + 2h + hβ) E h,1 (0) E 2 h (0) E h (0) jy j (y j+1 -y j-1 )E h (t) 0 ≤ 5E 2 h (0). ≤ 22 + β + 3 2β + 1 2β (6 + β) E h,1 (0) E 2 h (0) E h (0) (5.3.31) = M h E 2 h (0). (5.3.33)
	≤	h 8	1 + 2	E h,1 (0) E h (0)	E 2 h (0).	(5.3.27)

T 0 |y N +1 | 2 E h (t)dt ≤ 0, T 0 |η | 2 E h (t)dt + (1 + β 2 ) T 0 |η| 2 E h (t)dt T 0 E h,1 (t)E h (t)dt -(1 + β 2 ) β T 0 E h (t)E h (t)dt h (t) + 3 E h (t).

  1.1) belongs to D(A), we claim from Lemma 5.1.3 that E h (0) is uniformly bounded with respect to the mesh size h. Therefore from above equation we get

	sup
	t∈[0,+∞[

  2) from 0 to s we get E ∀s > 0. From Theorem 5.1.5 we know that the energy E h decrease to zero as s tends to ∞. Taking s → ∞ we obtainE h (0) = |η (t)| 2 dt, ∀s > 0.Since E h (0) is uniformly bounded with respect to h, we deduce from above equation that hp h y h is bounded in L 2 (0, ∞; V ), η is bounded in L 2 (0, ∞; C) and hη is bounded in L 2 (0, ∞; C). Furthermore, integrating (5.4.4) from 0 to s we getE h,1 (0) -E h,1 (s) =

	0	∞	hp h y h (t) 2 V dt + β	0	∞	|η(t)| 2 dt	(5.4.6)
	+ h 2 ∞
			0
			0	s	hp h y h (t) 2 V dt + β	0	s	|η (t)| 2 dt
			+		s	|hη (t)| 2 dt, ∀s > 0.
				0
	It follows that
	E h,1 (0) ≥

h (0) =E h (s) + s 0 hp h y h (t) 2 V dt + β s 0 |η(t)| 2 dt + h 2 s 0 |η (t)| 2 dt, s 0 hp h y h (t) 2 V dt + β s 0 |η (t)| 2 dt + s 0 |hη (t)| 2 dt, ∀s > 0.

  (5.4.18) which combined with the weak convergence results in (5.4.7) and the identities (5.4.17)-(5.4.18) we obtainhp h y h → 0 strongly in L 2 (0, ∞; V ), hη → 0 strongly in L 2 (0, ∞; C). (5.4.19) Now, using (5.1.3) and (5.4.16), we can check for all s > 0 that |E h (s) -E(s)| ≤|E h (0) -E(0)| + So that, combining the above inequality with (5.4.19) and Lemma 5.1.3, we deduce that lim -η(t)| 2 =(1 + h 2 β)|η(t)| 2 + |η(t)| 2 (5.4.23) -2 1 + h 2 β|η(t)| 2 . Combining (5.1.2), (5.4.1) and (5.4.21)-(5.4.23), we obtain p h y h (•, t) -y(•, t) 2 V + q h y h (•, t) -y (•, t) 2 L 2 (0,1) + | 1 + h 2 βη(t) -η(t)| 2

	
	
	
		0	s	0	1	|hp h y h | 2 dxdt +	0	s	|hη (t)| 2 dt
	≤|E h (0) -E(0)| +	0	∞		0	1	|hp h y h | 2 dxdt +		0	∞	|hη (t)| 2 dt.
							1	|y (x, t)| 2 dx
							0
										(5.4.22)
			1
	2) over 0 and |hη (t)| 2 dt. (5.4.16) Since E h (t) decreases to zero as t → ∞, if follows from above equation s E h (0) =E h (s) + s 0 hp h y h (t) 2 V dt + β s 0 |η(t)| 2 dt + s 0 that E h (0) = ∞ 0 hp h y h (t) 2 V dt + β ∞ 0 |η(t)| 2 dt + ∞ 0 |hη (t)| 2 dt. (5.4.17) -2 0 1 1 q =2E h (t) + 2E(t) -2 0 (p h y h (x, t)) x y x (x, t)dx -2 0 q h y

Next, the fact that E (t) = -β|η(t)|

2

and since E(t) decreases to zero as t → ∞, we get

E(0) = β ∞ 0 |η(t)| 2 dt, h→0 E h -E C([0,∞[) = 0.

(5.4.20)

Step 2. Strongly convergence in L 2 Loc (0, ∞; H). First, we have

p h y h (•, t) -y(•, t) 2 V = 1 0 |(p h y h (x, t)) x | 2 dx + 1 0 |y x (x, t)| 2 dx (5.4.21) -2 1 0 (p h y h (x, t)) x y x (x, t)dx, q h y h (•, t) -y (•, t) 2 L 2 (0,1) = 1 0 |q h y h (x,

t)| 2 dx + h y h (x, t)y (x, t)dx and | 1 + h 2 βη(t) h (x, t)y (x, t)dx -2 1 + h 2 βη(t)η(t).

H . Si Γ 0 = ∅, nous définissons H de la même manière mais muni de la norme usuelle (u, v, w, z)[START_REF] Abdallah | Stabilisation et approximation de certains systèmes distribués par amortissement dissipative et de signe indéfini[END_REF] := (u, v, w, z) 2 H + u 2 Ω + w 2 Γ .Nous introduisons aussi l'opérateur non borné maximal dissipatif A qui engendre un C 0 -semigroupe de contrac-

2 cosh( 1 √ γ ), F 1 = (1 -2γ) cosh( 1 √ γ ) + 2 √ γ sinh( 1 √ γ ).

W . First, since Φ 0,k H ∼ |k| 2 then

Hence by a standard shift theorem, we deduce that w ∈ H 3 2 (Γ 1 ) with

Thus by (4.2.10), we get

(4.2.12)

Now this improved regularity on w allows to look at u ∈ H 1 (Ω) as the solution of the next boundary value problem:

Again a shift theorem yields w ∈ H 2 (Γ 1 ) with

And by (4.2.14), we deduce that

We have shown that

sipation (5.1.3). It is logic to ask if the energy E h decays polynomially, and uniformly with respect to h, to zero as the time t tends to infinity. For any fixed h > 0, it is easy to see that the energy of (5.1.9) tends exponentially to zero as time goes to infinity. However, as we already said, earlier results obtained by Banks et al. [START_REF] Banks | Exponentially stable approximations of weakly damped wave equations[END_REF], Tebou and Zuazua [START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF] and others [START_REF] Abdallah | Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications[END_REF][START_REF] Ervedoza | Perfectly matched layers in 1-d: energy decay for continuous and semi-discrete waves[END_REF][START_REF] Marica | Boundary stabilization of numerical approximations of the 1-D variable coefficients wave equation: a numerical viscosity approach[END_REF][START_REF] Tcheugoué Tébou | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF][START_REF] Zuazua | Optimal and approximate control of finite-difference approximation schemes for the 1D wave equation[END_REF] lead us to think that the decay rates degenerates as h tends to zero. Before presenting our first result which confirms this issue, we need to introduce some operators. First, using the third equation of (5.1.9) we have

(5.1.12)

Eliminating y N +1 from (5.1.9), we get the following system:

(5.1.13)

The energy of system (5.1.13) is given by Remark 5.1.1. We can check that the solution of system (5.1.13) and the one of (5.1.9) are equivalent via equation (5.1.12). Consequently, the energy of system (5.1.13) given in (5.1.14) is equal to the one of system (5.1.9) given in (5.1.10) via the same equation.

Next, we define C h := C N the subspace which contains the discretized vectors y h = (y 1 , .., y N ) of y and we introduce the operator A h and its then from Lemma 5.1.3 we have E h (0) → E(0) > 0 as h tends to zero. Therefore, E h (0) is uniformly bounded with respect to the mesh size h and there exists a constant α > 0 such that E h (0) ≥ α. Moreover, from Lemma 5.1.4, we know that E h,1 (0) is uniformly bounded with respect to the mesh size h. Thus, the supremum of M h given in (5.1.31) exists.

Theorem 5.1.5 proves that the numerical viscosity term -(y j+1 -2y j + y j-1 ) added in system (5.1.24) is enough to restore the uniform polynomial decay in 1 t of the energy with respect to h. This was already showed for the uniform (with respect to the mesh size h) exponential decay of the energy, in [START_REF] Tcheugoué Tébou | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF] in the case where the damping term is locally distributed in the domain, and in [START_REF] Tebou | Uniform boundary stabilization of the finite difference space discretization of the 1 -d wave equation[END_REF] where the damping term is static and localized on the boundary. In addition, we will use the discretization of the multiplier xy x E(t) used by Wehbe in [START_REF] Wehbe | Rational energy decay rate for a wave equation with dynamical control[END_REF] and then we will deduce estimation (5.1.30) by using a classical result of Haraux in [START_REF] Haraux | Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire[END_REF] (see also [START_REF] Komornik | Exact Controllability and Stabilization: The Multiplier Method[END_REF]). Before showing our convergence results, we require some complementary notations. We set y h = (y j ) j , for j = 1, .., N and we introduce the extension operators defined by

(5.1.32) j = 0, .., N and

(5.1.33)

Further, we can check that

and

y j z j .

(5. 1.35) We are now ready to give our convergence results.

Theorem 5.1.7. Let ( y h , y h , η) denotes the solution of (5.1.24) and assume the initial data (y 0 , y 1 , η 0 ) of system (5.1.1) belongs to D(A) where D(A) is given in (5.1.4). Then as h → 0, we have

where (y, y , η) is the solution of system (5.1.1).

Moreover, the following convergence holds:

where T > 0 and lim

Non uniform polynomial energy decay

In this section, we give the proof of Theorem 5.1.2. This proof relies on the following lemma:

Lemma 5.2.1. Assume that there exists a positive constant M 2 independent of h and of U 0 h ∈ H h such that the energy E h of system (5.1.21) given in (5.1.14) satisfies (5.1.23). Then, there exists another constant M 3 independent of h such that any eigenvalue λ of A β,h with |λ| > 1, must satisfies the following estimation:

(5.2.1)

Proof: Let λ be an eigenvalue of A β,h such that |λ| > 1 and U 0 h ∈ H be an associated eigenvector such that U 0 h H h = 1. Since A β,h is dissipative which has the following expansion:

(5.2.10)

Proof: First, in order to solve (5.2.9), we set y h = (y 1 , .., y N ) and we introduce y N +1 = (1 -hα h )y N (5.2.11)

(5.2.12)

From (5.2.9) we obtain the following system:

We can check that δ = 0 cannot be an eigenvalue of K h . Next, let ξ ∈ C, we look for a solution of system (5.2.13) such that y j = ξ j . Thus

The above equation has two complex roots ξ 1 and ξ 2 which can be writed as ξ 1 = e α 1 +iθ 1 and ξ 2 = e α 2 +iθ 2 , where α i , θ i ∈ R for i = 1, 2. Since the product P = ξ 1 ξ 2 = 1, we have α 1 = -α 2 and θ 1 = -θ 2 . Introducing θ such that iθ = α 1 + iθ 1 , then the general solution y j of the first equation of (5.2.13) has the form Finally, choosing w such that we also have w(1, •) ≡ 0, from above equation we easily derive (5.4.8).

Step 2. Regularity of y. In order to prove that y satisfies the boundary conditions of system (5.1.1), we need to show that y belongs to H 2 (0, 1).

To this end, we demonstrate that y(0) = y 0 and y (0) = y 1 and therefore we obtain our goal since (y 0 , y 1 ) ∈ H 2 (0, 1) × V . First, for T > 0, we have from (5.4.7) that y ∈ L ∞ (0, T ; V ) and y t ∈ L ∞ (0, T ; L 2 (0, 1)).

It follows from Aubin and Simon's theorem (see [START_REF] Aubin | Un théorème de compacité[END_REF][START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]) that y ∈ C([0, T ]; L 2 (0, 1)). On the other hand, we can see that y xx ∈ L ∞ (0, T ; H -1 (0, 1)) where H -1 (0, 1) denotes the dual of H 1 0 (0, 1). Thus, from (5.4.8) we obtain that y tt ∈ L ∞ (0, T ; H -1 (0, 1)), which implies again from Aubin and Simon's theorem that

Multiplying the first equation of (5.1.24) by hv j l, integrating by parts on [0, ∞[, taking the sum over j and after some calculations we find 0 = -hl(0)

From the definitions of p h and q h given in (5.1.32)-(5.1.33), it is easy to check that the above equation is equivalent to 0 = -l(0)

Passing to the limit as h → 0 we get 0 = -l(0)

Integrating by parts over [0, ∞[ and over (0, 1) and using (5.4.8) we obtain

from which we derive that y(0) = y 0 and y (0) = y 1 .

Step 3. The solution (y, y , η) verifies the boundary conditions of system (5.1.1). First, choosing w(1, •) = 0, integrating by parts in (5.4.15) and using (5.4.8) we obtain

Next, using the third equation of (5.1.24) and the convergence results in (5.4.7), we get the third equation of (5.1.1). Finally, since system (5.1.1) has a unique solution, we conclude that the convergence results hold for the whole sequence {h}, and not only for an extracted subsequence.

Next, let s > 0. Integrating the above equation between 0 and s, letting h → 0 and using (5.4.20), we deduce that

Loc (0, ∞; C).

(5.4.24)

Step 3. p h y h → y strongly in C([0, T ], L 2 (0, 1)). First, we have From (5.1.25), we can check that p h y h 0 → y 0 weakly in V , thus 3 p h y h 0 (•) -y 0 (•) 2 L 2 (0,1) → 0, as h → 0.

Whence the desired convergence result from the first one of (5.4. [START_REF] Benchimol | A note on weak stabilizability of contraction semigroups[END_REF]) and the second one in (5.4.24).