Serge Nicaise
email: serge.nicaise@univ-valenciennes.fr

Farah Trad
email: farah.trad@uphf.fr

Ali Wehbe
email: ali.wehbe@ul.edu

Indirect stabilization of locally weakly coupled second order evolution systems of hyperbolic type

Keywords: AMS (MOS) subject classification 35B35, 93D15 Coupled evolution equations, stabilization of PDEs, polynomial stability, frequency domain approach, multiplier technique

), we are interested in polynomial energy decay rates. Our main contributions concern abstract strong and polynomial stability properties based on stability properties of two auxilliary problems: the sole damped equation and the second equation with a damping related to the coupling operator. The main novelty is that the polynomial energy decay rates are obtained in different important situations not covered before, let us mention the case of a coupling operator not partially coercive, and not necessarily bounded. The main tools are the use of the frequency domain approach combined with new multipliers technique based on the solutions of the resolvent equations of the two auxilliary problems mentioned before. Finally our abstract framework is illustrated by several concrete examples not treated before.

Introduction

Indirect damping of reversible systems occurs in many applications in engineering and mechanics. Indeed, it arises whenever it is impossible or too expensive to damp all the components of the state, hence it is important to study stabilization properties of coupled systems with a reduced number of feedbacks. The notion of indirect damping mechanisms has been introduced by D.L. Russell in [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF], and since then it attracted the attention of many authors.

In this paper, we consider the following coupled evolution equations (1.1)

   u 1,tt + A 1 u 1 + BB u 1,t + P u 2 = 0, u 2,tt + A 2 u 2 + P u 1 = 0, u 1 (0) = u 1,0 , u 1,t (0) = u 1,1 , u 2 (0) = u 2,0 , u 2,t (0) = u 2,1 ,
where A 1 and A 2 are positive selfadjoint operators (whose restrictions to an infinite dimensional Hilbert space H have a compact resolvent). Here, the coupling operator P and the damping operator B are not necessarily bounded (as usual P and B are the adjoint of P and B respectively). Since only the first equation is damped, the second equation is indirectly damped via the coupling operator P . According to [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF], system (4.11) cannot be exponentially stable, but under some assumptions on the operators A 1 , A 2 , P and B, system (4.11) decays polynomially, see for instance [START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF][START_REF] Alabau-Boussouira | Indirect boundary stabilization of weakly coupled hyperbolic systems[END_REF][START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF][START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF]. In the literature, these assumptions are relatively strong and with a few exceptions, P is bounded and coercive (meaning that P is global). Hence different open problems remain in different important cases, in particular when A 1 = A 2 , when P is unbounded and not assumed to be partially coercive in the sense of [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF]Assumption (A1)]. Here we mainly concentrate in the following cases:

1. A 1 = A 2 , when P is bounded (and B bounded or not) or P and B are both unbounded, 2. A 1 = A 2 , when P and B are both bounded.

Our main tools are the use of the frequency domain approach combined with new multipliers technique based on the solution of the resolvent equation of two auxilliary problems, namely the sole damped equation

u tt + A 1 u + BB u t = 0, u(0) = u 0 , u t (0) = u 1 ,
and the second equation with a damping B 2 B 2 u t related to the coupling operator P , namely

u tt + A 2 u + B 2 B 2 u t = 0, u(0) = u 0 , u t (0) = u 1 .
This allows to treat different important situations not covered before, let us mention the case of a coupling operator not partially coercive, and not necessarily bounded, and further the case when these two auxilliary problems are not exponentially decaying. The stabilization of systems of two second-order equations coupled through displacements (also called weak coupling) when only one equation is effectively damped by a possibly unbounded feedback has been initiated and studied in [START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF][START_REF] Beyrath | Indirect linear locally distributed damping of coupled systems[END_REF][START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF][START_REF] Alabau-Boussouira | Indirect boundary stabilization of weakly coupled hyperbolic systems[END_REF], and further studied by many authors, for instance [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF][START_REF] Liu | Frequency domain approach for the polynomial stability of a system of partially damped wave equations[END_REF][START_REF] Khodja | Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force[END_REF][START_REF] Aloui | Stabilization of two coupled wave equations on a compact manifold with boundary[END_REF][START_REF] Alabau-Boussouira | Indirect stabilization of weakly coupled systems with hybrid boundary conditions[END_REF]. In [START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF][START_REF] Alabau-Boussouira | Indirect boundary stabilization of weakly coupled hyperbolic systems[END_REF], Alabau considers coupled equations with a bounded coupling operator and an unbounded damping operator, using a generalized integral inequality based on higher-order energies and under some assumptions on B and P , it is shown that the energy decays polynomially with explicit polynomial decay rate for sufficiently smooth initial data.

Then different examples such as coupled wave equations or coupled Kirchhoff plates with only one boundary control are considered. Using a direct method based on the multiplier method and using the generalized integral inequality from [START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF], a similar result was obtained in [START_REF] Beyrath | Indirect linear locally distributed damping of coupled systems[END_REF] for coupled wave equations with only one localized internal control. In [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF][START_REF] Alabau-Boussouira | Indirect stabilization of weakly coupled systems with hybrid boundary conditions[END_REF], F. Alabau et al. studied the indirect stabilization of a system of two evolution equations coupled through displacements when the damping is coercive. Using the method initiated in [START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF][START_REF] Alabau-Boussouira | Indirect boundary stabilization of weakly coupled hyperbolic systems[END_REF], they established a polynomial energy decay rate depending on the smoothness of the initial data. Furthermore, the authors in [START_REF] Liu | Frequency domain approach for the polynomial stability of a system of partially damped wave equations[END_REF] proved that in one-dimensional domain the behavior of two wave equations with an internal coupling and a boundary damping is sensitive to the arithmetic property of the ratio of the wave propagation speeds of the two waves. On the other hand the authors in [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF] proved the polynomial stability when the two waves are locally coupled where the coupling and damping regions satisfy the piecewise multiplier geometric condition (PMGC in short, introduced by K. Liu in [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF]), implying that they have a non empty (even very small) intersection in dimension ≥ 2, in the same setting they also prove a polynomial energy decay rate of weakly coupled plate equations. In [START_REF] Aloui | Stabilization of two coupled wave equations on a compact manifold with boundary[END_REF], the authors extend for two coupled wave equations the results from [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF] by replacing the PGMC by the geometric control condition (GCC in short, introduced C. Bardos, G. Lebeau, and J. Rauch in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]) under an additional uniqueness continuation property (that holds if the two regions satisfy the PGMC). These two last papers generalize the case of constant coupling investigated in [START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF][START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF][START_REF] Alabau-Boussouira | Indirect boundary stabilization of weakly coupled hyperbolic systems[END_REF].

The stabilization of systems of two second-order equations coupled by velocities (also called strong coupling) was studied by many authors, we mention the following references [START_REF] Afilal | Stability of coupled second order equations[END_REF][START_REF] Khodja | Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force[END_REF][START_REF] Ammari | Stabilization of coupled systems[END_REF][START_REF] Kassem | Local indirect stabilization of n-d system of two coupled wave equations under geometric conditions[END_REF][START_REF] Gerbi | Exact controllability and stabilization of locally coupled wave equations: theoretical results[END_REF][START_REF] Akil | Stabilization of coupled wave equations with viscous damping on cylindrical and non-regular domains: cases without the geometric control condition[END_REF]. Sufficient conditions for the exponential stabilizability of coupled abstract second order equations are obtained in [START_REF] Afilal | Stability of coupled second order equations[END_REF], to illustrate the usefulness of their results some examples are presented. In [START_REF] Khodja | Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force[END_REF], the authors consider some stabilization problems for a system of two coupled one-dimensional wave equations set on a finite interval (0, 1), using only one internal or boundary control. They show that the internal damping applied to only one of the equations never gives exponential stability if the wave speeds are different. If the wave speeds are the same, they obtain necessary and sufficient conditions for expential stability using a compact perturbation argument. In [START_REF] Ammari | Stabilization of coupled systems[END_REF], the authors characterize the stabilization of some coupled infinite-dimensional systems, by using the methodology introduced in [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF] (where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined with a boundedness property of the transfer function of the associated open loop system) and a result from [START_REF] Mehrenberger | Observability of coupled systems[END_REF] to check this observability estimate for a coupled wave/plate system. Moreover in [START_REF] Kassem | Local indirect stabilization of n-d system of two coupled wave equations under geometric conditions[END_REF] the authors studied the indirect stabilization of a system of wave equations coupled by velocities in the case where the two waves propagate at different speeds under the PMGC. While in [START_REF] Gerbi | Exact controllability and stabilization of locally coupled wave equations: theoretical results[END_REF], the authors have proved that the system is exponentially stable when the coupling region is a subset of the damping region and satisfies the GCC. Further, in [START_REF] Akil | Stabilization of coupled wave equations with viscous damping on cylindrical and non-regular domains: cases without the geometric control condition[END_REF], the authors studied the direct and indirect stability of locally coupled wave equations with local viscous damping on cylindrical and non-regular domains without any geometric control condition. If only one equation is damped, they proved that the energy of the system decays polynomially with the rate t -1 2 if the two waves have the same speed of propagation, and with the rate t - 1 3 if the two waves do not propagate at the same speed.

Other settings have been considered where the situation is much more involved. Indeed the effectiveness of indirect damping mechanisms depends in a complex way on the assumptions on all the operators involved (in our case. on the operators A 1 , A 2 , P , and B in system (1.1)). Let us mention the following examples. Loreti and Rao studied in [START_REF] Loreti | Optimal energy decay rate for partially damped systems by spectral compensation[END_REF] the stability of the following abstract system of coupled equations

(1.2) u 1,tt + Au 1 + A γ u 1,t + αu 2 = 0, u 2,tt + Au 2 + αu 1 = 0,
where γ ≤ 0, α ∈ R and A is a selfadjoint coercive operator with a compact resolvent in a separable Hilbert space H. They proved that system (1.2) is not exponentially stable and an optimal energy decay rate of type t τ (γ) is obtained, where

τ (γ) = 1 1+γ , if -1 2 ≤ γ ≤ 0, -1 τ , if γ ≤ -1 2 .
equations, similar to (1.2) but coupled by velocities

(1.3) u 1,tt + aAu 1 + A γ u 1,t + αu 2,t = 0, u 2,tt + Au 2 -αu 1,t = 0,
where a > 0, γ ≤ 0, α ∈ R is the coupling parameter and A is a selfadjoint operator with compact resolvent in a separable Hilbert space H and with simple eigenvalues only. They proved the exponential stability of the system, when the fractional order damping becomes viscous (i.e., when γ = 0) and the waves propagate with equal speeds; otherwise they establish an optimal polynomial decay rate.

Let us finally mentioned some papers with other type of dampings where some stability issues are considered: strong coupled systems with a fractional damping or with a Kelvin-Voigt damping [START_REF] Akil | Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary[END_REF][START_REF] Akil | The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization[END_REF][START_REF] Wehbe | Stability results of an elastic/viscoelastic transmission problem of locally coupled waves with non smooth coefficients[END_REF][START_REF] Akil | Indirect stability of a multidimensional coupled wave equations with one locally boundary fractional damping[END_REF][START_REF] Akil | Stability results of locally coupled wave equations with local Kelvin-Voigt damping: cases when the supports of damping and coupling coefficients are disjoint[END_REF] and weak coupled systems with Kelvin-Voigt damping [START_REF] Oquendo | Optimal decay for coupled waves with Kelvin-Voigt damping[END_REF] .

The paper is organized as follows: in section 2 we fix the exact general setting and briefly discuss well-posedness of the problem (1.1). Section 3 is devoted to the proof of different strong stability results, while section 4 provides some polynomial decay results. Finally in section 5 different illustrative examples are treated.

Let us finish this introduction with some notation used in the remainder of the paper: The usual norm and semi-norm of H s (Ω) (s ≥ 0) are denoted by • s,Ω and | • | s,Ω , respectively. For s = 0 we drop the index s. Similarly (•, •) (resp. • ) denotes the Euclidean inner product (resp. norm) in C k , for some k ∈ N * . For two Hilbert spaces X and Y and a bounded and linear operator D ∈ L(X, Y ), D always means its dual operator. For any open set ω of R d , with d ≥ 1, I ω means the characteristic function of ω. By A B, we mean that there exists a constant C > 0 independent of A, B, a real parameter λ, the space variable x and the time variable t such that A ≤ CB. By A B, we mean that B A and by A ∼ B, we mean that both A B and A B hold.

Abstract setting and well-posedness

In this section we describe a general abstract setting of coupled hyperbolic type equations introduced in [START_REF] Alabau-Boussouira | Indirect boundary stabilization of weakly coupled hyperbolic systems[END_REF][START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF][START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF] in a more restrictive setting that will be used later on. It is motivated by the examples (and other ones) given in section 5 which all enter in this setting.

Let us fix three Hilbert spaces H, V 1 and V 2 with respective inner products

(•, •) H , (•, •) V1 and (•, •) V2 such that V 1 and V 2 are densely and compactly embedded into H with V 2 is continuously embedded in V 1 .
The associated norms will be denoted by • H , • V1 and • V2 respectively. Identifying H with its dual H we have the standard diagram

V 2 → V 1 → H = H → V 1 → V 2 .
For i = 1 or 2, the duality pairing between V i and V i will be denoted by •, • i , so that

u, v i = (u, v) H , ∀u, v ∈ H.
The damping operator B is supposed to be a linear and bounded operator from a Hilbert space U (which will be identified to its dual space whose norm will be denoted by

• U ) into V 1 . As a consequence B ∈ L(V 1 , U ). If B ∈ L(H), then B ∈ L(U, V 1 ) with U = H, and
Bu, v 1 = (Bu, v) H , ∀u ∈ H, v ∈ V 1 .
If B ∈ L(H), we will say that B is bounded, otherwise we will say that B is unbounded. In the same manner, we assume that P ∈ L(V 2 , V 1 ). As before if P ∈ L(H), then P ∈ L(V 2 , V 1 ) and we will say that P is bounded, otherwise we will say that P is unbounded.

For i = 1 or 2, we also suppose given two sesquilinear and symmetric forms a i :

V i × V i → C such that (2.1) a i (u, u) u 2 Vi , ∀u ∈ V i , the associated bounded operator A i from V i to V i is then defined by A i u, v i = a i (u, v), ∀u, v ∈ V i .
As usual, for i = 1 or 2, we may introduce the unbounded operator

A i from H into itself defined by D(A i ) = {u ∈ V i |A i u ∈ H}, and
A i u = A i u, ∀u ∈ D(A i ),
that is a positive, selfadjoint operator with a compact resolvent. For future uses, for i = 1 or 2, λ i is the smallest positive eigenvalue of the operator A i . An existence and uniqueness result of problem (1.1) is proved using semigroup theory, under the assumption that (2.2)

P L(V2,V 1 ) < 1,
where

P L(V2,V 1 ) is the norm of P when V 1 (resp. V 2 ) is equipped with the norm a 1 (•, •) 1/2 (resp. a 2 (•, •) 1/2 ), namely P L(V2,V 1 ) = sup u∈V 2 ,u =0 v∈V 1 ,v =0 | P u, v 1 ] a 2 (u, u) 1/2 a 1 (v, v) 1/2 .
In the case when P ∈ L(H), this condition reduces to

(2.3) P L(H) < λ 1 λ 2 ,
as one easily shows that

P L(V2,V 1 ) ≤ P L(H) √ λ 1 λ 2 .
Indeed let us show that this system enters into the framework of [18, §1.2.1]. First on V 1 × V 2 , let us introduce the sesquilinear, continuous and symmetric form

(2.4) a((u 1 , u 2 ) , (ũ 1 , ũ2 ) ) = a 1 (u 1 , ũ1 ) + a 2 (u 2 , ũ2 ) + P u 2 , ũ1 1 + P u 1 , ũ2 2 , ∀(u 1 , u 2 ) , (ũ 1 , ũ2 ) ∈ V 1 × V 2 .
This form defines an inner product on V 1 × V 2 due to the assumption (2.2). Indeed for any

(u 1 , u 2 ) ∈ V 1 × V 2 , one has | P u 2 , u 1 1 + P u 1 , u 2 2 | = 2| P u 2 , u 1 1 | ≤ 2 P L(V2,V 1 ) a 2 (u 2 , u 2 ) 1/2 a 1 (u 1 , u 1 ) 1/2 ≤ P L(V2,V 1 ) (a 1 (u 1 , u 1 ) + a 2 (u 2 , u 2 )).
This estimate then implies

(2.5) a((u 1 , u 2 ) , (u 1 , u 2 ) ) ≥ (1 -P L(V2,V 1 ) )(a 1 (u 1 , u 1 ) + a 2 (u 2 , u 2 )), ∀(u 1 , u 2 ) ∈ V 1 × V 2 .
Due to the assumption (2.2), we deduce that the associated norm is equivalent to the natural one of V 1 × V 2 . Now let us introduce the operators A and B as follows

A : V 1 × V 2 → V 1 × V 2 : (u 1 , u 2 ) → A(u 1 , u 2 ) , B : V 1 × V 2 → U × H : (u, v) → (B u, 0) ,
where

A(u 1 , u 2 ) , (ũ 1 , ũ2 ) V 1 ×V 2 -V1×V2 = a((u 1 , u 2 ) , (ũ 1 , ũ2 ) ), ∀(u 1 , u 2 ) , (ũ 1 , ũ2 ) ∈ V 1 × V 2 .
Finally by introducing the vectorial unknown u = (u 1 , u 2 ) , we see that problem (1.1) is equivalent to

(2.6) u tt + Au + BB u t = 0, in H × H, u(0) = (u 1,0 , u 2,0 ) u t (0) = (u 1,1 , u 2,1 ) .
As A is a (unbounded) selfadjoint and positive operator on

H × H and B is bounded from V 1 × V 2 into U ,
U t = AU, in H, U (0) = (u 1,0 , u 1,1 , u 2,0 , u 2,1 ) ,
where the Hilbert space

H = V 1 × H × V 2 × H is equipped with the inner product (U, Ũ ) H = a 1 (u 1 , ũ1 ) + a 2 (u 2 , ũ2 ) + (v 1 , ṽ1 ) H + (v 2 , ṽ2 ) H + P u 2 , ũ1 1 + P u 1 , ũ2 2 , for all U = (u 1 , v 1 , u 2 , v 2 ) , Ũ = (ũ 1 , ṽ1 , ũ2 , ṽ2
) ∈ H (that is indeed an inner product due to assumption (2.2)). The energy of the system (2.7) is defined by

E(t) = 1 2 (U, U ) H = 1 2 u 1,t 2 H + a 1 (u 1 , u 1 ) + u 2,t 2
H + a 1 (u 2 , u 2 ) + P u 2 , u 1 1 + P u 1 , u 2 2 ,
that is non negative by the assumption (2.2). The unbounded operator A is defined by

(2.8) D(A) =    U = (u 1 , v 1 , u 2 , v 2 ) ∈ V 1 × V 1 × V 2 × V 2 | A 1 u 1 + BB v 1 + P u 2 ∈ H, A 2 u 2 + P u 1 ∈ H    , and
AU = (v 1 , -(A 1 u 1 + BB v 1 + P u 2 ), v 2 , -(A 2 u 2 + P u 1 )) , ∀U = (u 1 , v 1 , u 2 , v 2 ) ∈ D(A).
By Theorem 1.2.1 from [START_REF] Ammari | Stabilization of elastic systems by collocated feedback[END_REF], A is maximal dissipative, therefore thanks to Lumer-Phillips' theorem (see [START_REF] Liu | Semigroups associated with dissipative systems[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]), we deduce that A generates a C 0 -semigroup of contractions e tA t≥0 in H and therefore Problem (2.7) is well-posed. Thus, the solution of the evolution equation (2.7) admits the following representation

U (t) = e tA U 0 , t ≥ 0,
which leads to the well-posedness of problem (2.7). Hence, semi-group theory allows us to show the next existence and uniqueness result.

Theorem 2.1 For any U 0 ∈ H, the Problem (2.7) admits a unique weak solution

U ∈ C (R + ; H) . Moreover, if U 0 ∈ D (A) , then U ∈ C (R + ; D (A)) ∩ C 1 (R + ; H) .
Note that the dissipativeness of A here means that (2.9)

(AU, U ) H = -B v 1 2 U , ∀U = (u 1 , v 1 , u 2 , v 2 ) ∈ D(A).

Some strong stability results

As V 1 × V 2 is compactly emmbedded into H × H, A has a compact resolvent, hence its spectrum is reduced to its point spectrum. Therefore one simple way to prove the strong stability of (1.1) is to combine a theorem due to Arendt & Batty and Lyubich & Vũ (see [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF][START_REF] Lyubich | Asymptotic stability of linear differential equations in banach spaces[END_REF]). with Corollary 1.2.4 of [START_REF] Ammari | Stabilization of elastic systems by collocated feedback[END_REF] (see [18, Such a result combined with some specific tools, like the unique continuation principle or the multiplier method, can be used in an efficient way for some particular examples (see section 5 below for the details). Nevertheless let us present two strong stability results in our abstract framework. The first one is inspired from [53, §6] and is based on an orthonormal basis argument combined with the asymptotic behavior of the system (3.1) below. Before stating it, let us recall some notation. First, let us denote by {λ n } n∈N * , the set of eigenvalues enumerated in increasing order (and not repeated according to their multiplicity) of the operator A 1 . For each n ∈ N * , we also denote by

ϕ n, , = 1, • • • , m(n) (m(n) ∈ N * is the multiplicity of λ n ) the orthonormal eigenvectors associated with λ n , i.e., A 1 ϕ n, = λ n ϕ n, , ∀ = 1, • • • , m(n), n ∈ N * .
Now consider the single damped equation (that corresponds to the first identity from (1.1) without the coupling term) (3.1)

u tt + A 1 u + BB u t = 0, in H, u(0) = u 0 , u t (0) = u 1 ,
It may be written as a first order system in V 1 × H, equipped with the inner product

((u, v) , (ũ, ṽ) ) V1×H = a 1 (u, ũ) + (v, ṽ) H , ∀(u, v) , (ũ, ṽ) ∈ V 1 × H,
by setting U = (u, u t ) and introducing the maximal dissipative operator

A w on V 1 × H by D(A w ) = {(u, v) ∈ V 2 1 | A 1 u + BB v ∈ H}, and
A w (u, v) = (v, -(A 1 u + BB v) , ∀U = (u, v) ∈ D(A w ).
We then say that system (3.1) is strongly stable in V 1 × H if and only if iR ⊂ ρ(A w ).

Theorem 3.3 In addition to the assumptions from section 2, we assume that P = b I for a non zero constant b ∈ C and A 2 = aA 1 , for some positive constant a. Assume furthermore that system (3.1) is strongly stable in V 1 × H. Then there exists a discrete set S of (0, ∞) such that for all b = 0 such that |b| 2 ∈ S, system (1.1) is strongly stable, in the sense that the semigroup (e tA ) t≥0 generated A is strongly stable in H.

Proof. We apply Lemma 3.1 by showing that any non zero eigenvector u of A satisfies B u = 0.

For that purpose, we look for the eigenvalues and eigenvectors of A. So let us fix λ ∈ R and consider u = (u

1 , u 2 ) ∈ V 1 × V 2 such that A(u 1 , u 2 ) = λ 2 (u 1 , u 2 ) ,
or equivalently

(3.2) A 1 u 1 + bu 2 = λ 2 u 1 , aA 1 u 2 + bu 1 = λ 2 u 2 .
We now write u 1 and u 2 in the orthonormal basis {ϕ n, }, namely we write

u 1 = m(n) =1 a n, ϕ n, , u 2 = m(n) =1 b n, ϕ n, ,
with a n, , b n, ∈ C. Inserting these expansions in (3.2), we find

n∈N * m(n) =1 (-λ 2 + λ n )a n, + bb n, ϕ n, = 0, n∈N * m(n) =1 (-λ 2 + aλ n )b n, + ba n, ϕ n, = 0.
By the orthornomal property of the basis {ϕ n, } =1,••• ,m(v),n∈N * , we find

(-λ 2 + λ n )a n, + bb n, = 0, ∀ = 1, • • • , m(n), n ∈ N * , (-λ 2 + aλ n )b n, + ba n, = 0, ∀ = 1, • • • , m(n), n ∈ N * .
As b = 0, the first identity is equivalent to

b n, = b -1 (λ 2 -λ n )a n, , ∀ = 1, • • • , m(n), n ∈ N * ,
and inserting this expression in the second one, we obtain

(λ 2 -λ n )(λ 2 -aλ n ) -|b| 2 a n, = 0, ∀ = 1, • • • , m(n), n ∈ N * .
Hence for each n ∈ N * , = 1, • • • , m(n), a non trivial solution (a n, , b n, ) exists if and only if

(3.3) (λ 2 -λ n )(λ 2 -aλ n ) -|b| 2 = 0.
This equation being a second order equation in λ 2 , (3.3) holds if and only if

(3.4) λ 2 = µ n,± = (1 + a)λ n ± (1 -a) 2 λ 2 n + 4|b| 2 2 .
Note that the condition (2.3) here reduces to

|b| < √ aλ 1 ,
which implies that µ n,-> 0, for all n ∈ N * . For each n ∈ N * , the operator A then has two different eigenvalues µ n,± with m(n) eigenvectors

(3.5) u n, ,± = (1, b -1 (µ n,± -λ n )) ϕ n, .
Remark that the first component of u n, is an eigenvector of A 1 of eigenvalue λ n . Now we notice that µ n,+ = µ m,+ and µ n,-= µ m,-, if n = m since the mappings

(0, ∞) → R : x → (1 + a)x + (1 -a) 2 x 2 + 4 b| 2 2 , (0, ∞) → R : x → (1 + a)x -(1 -a) 2 x 2 + 4|b| 2 2 ,
are strictly increasing. We now impose that

(3.6) µ n,+ = µ m,-, ∀m, n ∈ N * : m = n,
which guarantees that µ n,+ and µ n,-are all different. Under this assumption, we deduce that the set of eigenvalues of A is ∪ n∈N * {µ n,+ , µ n,-}, the eigenvectors associated with µ n,+ (resp. µ n,-) being u n, ,+ (resp. u n, ,-), for all = 1,

• • • , m(n), n ∈ N * . Indeed the set ∪ n∈N * {u n, ,+ , u n, ,-} m(n) =1
is complete in H × H since we readily check that the sole

(u 1 , u 2 ) ∈ H × H orthogonal to ∪ n∈N * {u n, ,+ , u n, ,-} m(n)
=1 is (0, 0). Now applying Corollary 1.2.4 of [START_REF] Ammari | Stabilization of elastic systems by collocated feedback[END_REF] to the system (3.1), we have

(1 + a)λ n + (1 -a) 2 λ 2 n + 4|b| 2 = (1 + a)λ m -(1 -a) 2 λ 2 m + 4|b| 2 ,
or equivalently

(1 -a) 2 λ 2 n + 4|b| 2 = (1 + a)(λ m -λ n ) -(1 -a) 2 λ 2 m + 4|b| 2 ,
Taking the square of this identity, this implies

(1-a) 2 λ 2 n +4|b| 2 = (1+a) 2 (λ m -λ n ) 2 +(1-a) 2 λ 2 m +4|b| 2 -2(1+a)(λ m -λ n ) (1 -a) 2 λ 2 m + 4|b| 2 .
Isolating the last term of this right-hand side and again taking the square, one gets

4(1 + a) 2 (λ m -λ n ) 2 (1 -a) 2 λ 2 m + 4|b| 2 = (1 + a) 2 (λ m -λ n ) 2 + (1 -a) 2 (λ 2 m -λ 2 n ) 2 .
As λ m -λ n = 0, we have found

|b| 2 = (1 + a) 2 (λ m -λ n ) 2 + (1 -a) 2 (λ 2 m -λ 2 n ) 2 16(1 + a) 2 (λ m -λ n ) 2 - (1 -a) 2 λ 2 m 4 .
Now if we define the discrete set S by

S = (1 + a) 2 (λ m -λ n ) 2 + (1 -a) 2 (λ 2 m -λ 2 n ) 2 16(1 + a) 2 (λ m -λ n ) 2 - (1 -a) 2 λ 2 m 4 | m, n ∈ N * , m > n ∩ (0, ∞),
we can conclude that for |b| 2 ∈ S, (3.6) always holds.

Our second stability result can be formulated as follows.

Theorem 3.4 In addition to the assumptions from section 2, assume that A 2 = A 1 . If P is bounded, we further assume that

(3.9) P u H + P u H BB u H , ∀u ∈ H, if B ∈ L(H), BB u V 1 , ∀u ∈ V 1 , else,
and that there exists a bounded and linear operator D ∈ L(H) such that

(3.10) | (P u, u) H | D u 2 H , ∀u ∈ H.
On the contrary if P is unbounded, we assume that B is also unbounded, that

(3.11) P u V 1 + P u V 1 BB u V 1 , ∀u ∈ V 1 ,
that there exists a Hilbert space U p (which will be identified to its dual space whose norm will be denoted by • Up ) and a bounded and linear operator D ∈ L(U p , V 1 ) such that

(3.12) | P u, u 1 | D u 2 Up , ∀u ∈ V 1 .
We finally assume that the system

(3.13) u tt + A 1 u + DD u t = 0, in V 1 , u(0) = u 0 , u t (0) = u 1 , is strongly stable in V 1 × H. Then system (1.1
) is strongly stable, in the sense that the semigroup (T (t)) t≥0 generated A is strongly stable in H.

Proof. Let us fix λ ∈ R and consider u = (u 1 , u 2 ) ∈ V 1 × V 2 such that A(u 1 , u 2 ) = λ 2 (u 1 , u 2 ) ,
or equivalently

(3.14) A 1 u 1 + P u 2 = λ 2 u 1 , A 1 u 2 + P u 1 = λ 2 u 2 .
and suppose that it also satisfies B u 1 = 0. Then by the assumption (3.9) or (3.11), we have P u 1 = 0 and P u 1 = 0. Hence by (3.10) or (3.12), we also get D u 1 = 0. The second identity of (3.14) then implies that

A 1 u 2 = λ 2 u 2 .
Hence we can distinguish two cases:

1. If u 2 = 0, then the first identity of (3.14) becomes

A 1 u 1 = λ 2 u 1 .
Recalling that D u 1 = 0, by Corollary 1.2.4 of [START_REF] Ammari | Stabilization of elastic systems by collocated feedback[END_REF] applied to the system (3.13) we deduce that u 1 = 0. This means that in this case, there is no nontrivial eigenvector u of A satisfying B u = (0, 0) . 2. If u 2 = 0, then it is an eigenvector of A 1 of eigenvalue λ 2 . Coming back to the first identity of (3.14), we get

A 1 u 1 -λ 2 u 1 = -P u 2 ,
for which a nontrivial u 1 exists if and only if

P u 2 , ϕ 1 = 0,
for all eigenvectors ϕ of A 1 of eigenvalue λ 2 . In particular taking ϕ = u 2 , we get

P u 2 , u 2 1 = 0,
which by (3.10) or (3.12) yields D u 2 = 0. and by Corollary 1.2.4 of [START_REF] Ammari | Stabilization of elastic systems by collocated feedback[END_REF] contradicts the hypothesis that the system (3.13) is strongly stable.

In conclusion only the first case is possible and we conclude by Lemma 3.1.

Some polynomial stability results

As System (1.1) is not exponentially stable, we are interested to study a polynomial energy decay rates in the case A 1 = A 2 , but also in the case

A 1 = A 2 .

Technical Lemmas

In this subsection we will state some Lemmas that we will be used to prove our stability results. Namely for the exponential decay of the semigroup we use the following result (see [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF] or [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF]):

Lemma 4.1 Let (e tL ) t≥0 be a bounded C 0 semigroup on a Hilbert space H. Then it is exponentially stable, i.e., it satisfies

||e tL U 0 || H ≤ C e -ωt ||U 0 || H , ∀ U 0 ∈ H, ∀t ≥ 0,
for some positive constants C and ω if and only if

(4.1) iR ⊂ ρ(L), and
(4.2) sup ξ∈R (ıξ -L) -1 < ∞.
On the contrary the polynomial decay is based on the following result stated in Theorem 2.4 of [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] (see also [START_REF] Bátkai | Polynomial stability of operator semigroups[END_REF][START_REF] Batty | Non-uniform stability for bounded semi-groups on banach spaces[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF] for weaker variants). Lemma 4.2 Let (e tL ) t≥0 be a bounded C 0 semigroup on a Hilbert space H such that its generator L satisfies (4.1) and let be a fixed positive real number. Then the following properties are equivalent

||e tL U 0 || H t -1 ||U 0 || D(L) , ∀ U 0 ∈ D(L), ∀t > 1, (4.3) sup ξ∈R 1 1 + |ξ| (ıξ -L) -1 < ∞. (4.4) If (4.
3) holds, we say that the semigroup (e tL ) t≥0 is polynomially stable with a decay in t -1 .

Polynomial energy decay rate in the case

A 1 = A 2
In this subsection we obtain some polynomial energy decay rates under the assumption that A 1 = A 2 (in the sense that V 1 = V 2 and a 1 = a 2 ). Our first stability result concerns the case when P is bounded, while B may be either bounded or not. Theorem 4.3 Assume that A 1 = A 2 , that P ∈ L(H), and that (3.10) holds for some D ∈ L(H). Assume also that iR ⊂ ρ(A) and that system (3.1) is strongly stable in V 1 × H such that its associated resolvent satisfies

(4.5) (iξI -A w ) -1 L(V1×H) 1 + |ξ| 1 , ∀ξ ∈ R,
for some 1 ≥ 0.

Assume furthermore that there exists a bounded and linear operator B 2 ∈ L(H) such that the system

(4.6) u tt + A 2 u + B 2 B 2 u t = 0, in H, u(0) = u 0 , u t (0) = u 1 ,
is strongly stable in V 1 × H and that its associated operator A w,2 satisfies

(4.7) (iξ -A w,2 ) -1 L(V2×H) 1 + |ξ] 2 , ∀ξ ∈ R,
for some 2 ≥ 0. We further assume that

(4.8) B 2 u H D u H , ∀ u ∈ H, If (4.9) 2 2 + 1 < 1, then A satisfies (4.4) with (4.10) = 4 2 + 2 B + 2,
where B = 0 if B ∈ L(H) and B = 1 else. Therefore the semigroup (e tA ) t≥0 generated by A is polynomially stable with a decay in t -1 .

Remark 4.4 Note that in [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF], the authors consider (4.11)

   u 1,tt + A 1 u 1 + BB u 1,t + δP u 2 = 0, u 2,tt + A 2 u 2 + P u 1 = 0, u 1 (0) = u 1,0 , u 1,t (0) = u 1,1 , u 2 (0) = u 2,0 , u 2,t (0) = u 2,1 ,
where δ is a fixed positive parameter. But this system can be reduced to (1.1) by setting u 1 = αũ 1 and u 2 = β ũ2 with two positive real numbers α, β chosen such that α 2 β 2 = δ and setting P = √ δP . In the setting of (1.1), the assumption (A1) of [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF] guarantees that (3.10) holds with D = D = Π P (with Π P ∈ H such that Π P H = 1). Furthermore assumption (A2) (resp. (A3)) from [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF] implies that system (3.1) (resp. (4.6) with B 2 = D = Π P ) is exponentially stable in V 1 × H. Hence with the sole additional assumption that iR ⊂ ρ(A), Theorem 4.3 allows to recover Theorem 2.4 (i) of [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF] without any assumption on the size of P L(H) . Note that in their PDE examples, their assumptions (A2) (A3) are deduced under the assumption that the support of the coupling coefficient and the damping coefficient satisfy the PGMC, which implies that the damping region and the coupling region have a very small non empty intersection (see [14, p. 551]) and leading to iR ⊂ ρ(A).

Consider now a new variant of Theorem 4.3 with an additional assumption on the coupling operator P , in the case where P and B are both bounded. Theorem 4.5 Let the assumptions of Theorem 4.3 be satisfied except (4.9). If we assume that B, P ∈ L(H) and (4.12)

P u H + P u H D u H B u H , ∀u ∈ H, then A satisfies (4.4) with (4.13) = max{ 1 + 1, 4 1 -2, 4 2 + 2}.
Therefore the semigroup (e tA ) t≥0 generated by A is polynomially stable with a decay in t -1 . In that case, 1 = 2 = 0, which yields a decay of the energy in t -1 for initial data in D(A) instead of t -1/2 in Theorem 4.2 of [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF].

We consider now the stability result when the coupling and damping operators are unbounded.

Theorem 4.7 Assume that A 1 = A 2 , that P = DD , with D ∈ L(U, V 1 )
. Assume also that iR ⊂ ρ(A) and that system (3.1) is strongly stable in V 1 × H such that its associated resolvent satisfies (4.5) for some 1 ≥ 0. Assume furthermore that there exists a bounded and linear operator

B 2 ∈ L(U, V 1
) such that the system (4.6) is strongly stable in V 1 ×H and that its associated operator A w,2 satisfies (4.7) for some 2 ≥ 0.

In addition we assume that

B 2 u U D u U , ∀ u ∈ V 1 . (4.14) D u U B u U , ∀ u ∈ V 1 , (4.15) and (4.16) B u 2 U u H u V1 , ∀ u ∈ V 1 , then A satisfies (4.4) with (4.17) = max{ 1 + 1, 2 1 -1, 2 2 + 3}.
Therefore the semigroup (e tA ) t≥0 generated by A is polynomially stable with a decay in t -1 .

We need to prove that (4.4) holds with an appropriate > 0, this is done by a contradiction argument, by assuming that there exists a sequence

{(λ n , U n = (u 1,n , v 1,n , u 2,n , v 2,n ))} n≥1 ⊂ R × D (A), with λ n > 0 satisfying (4.18) λ n → +∞, as n → ∞, U n H = 1, ∀n ≥ 1, and
(4.19) λ n ( iλ n U n -AU n ) = (f 1,n , g 1,n , f 2,n , g 2,n ) := F n → 0 in H.
From now on, for simplicity, we drop the index n. Detailing (4.19), we obtain

iλu 1 -v 1 = λ -f 1 in V 1 , (4.20) iλv 1 + A 1 u 1 + BB v 1 + P u 2 = λ -g 1 in H, (4.21) iλu 2 -v 2 = λ -f 2 in V 2 , (4.22) iλv 2 + A 2 u 2 + P u 1 = λ -g 2 in H.
-λ 2 u 1 + A 1 u 1 + iλBB u 1 + P u 2 = λ -(iλf 1 + BB f 1 + g 1 ) in V 1 , (4.24) -λ 2 u 2 + A 2 u 2 + P u 1 = λ -(iλf 2 + g 2 ) in H. (4.25)
Note that the equality (4.25) is in V 1 in the case where P is unbounded. In addition, from the property (4.18), and identities (4.20), and (4.22), we deduce that (4.26)

u 1 H = O λ -1 and u 2 H = O λ -1 .
Furthermore combining (4. [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF]) and (2.9), we have

(4.27) B u 1 2 U = o λ --2 .
For clarity, we now divide the proof of Theorem 4. Proof. We take the duality pairing (4.24) with u 1 in V 1 -V 1 and take the real part to get

a 1 (u 1 , u 1 ) = λ 2 u 1 2 H + (P u 2 , u 1 ) H + λ -(iλf 1 + g 1 , u 1 ) H + λ -(B f 1 , B u 1 ) U .
First using Cauchy-Schwarz inequality, and the boudedness property of P and B , we find

a 1 (u 1 , u 1 ) λ 2 u 1 2 H + u 1 H u 2 H + λ -+1 F H u 1 H + λ -f 1 V1 u 1 V1 .
Using the estimates (4.18), (4. [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF]) and (4.28), we obtain (4.29). The proof is thus complete. Let us go on with the estimate of the norm of u 2 in V 2 .

Lemma 4.9 Assume that P ∈ L(H) and that the estimate

(4.30) λ u 2 H = o(1)
holds, then the solution

(u 1 , v 1 , u 2 , v 2 ) ∈ D(A) of system (4.20)-(4.23) satisfies the following esti- mate (4.31) a 2 (u 2 , u 2 ) = o(1).
Proof. As before we take the inner product of (4.25) with u 2 in H, that yields

-λ 2 u 2 2 H + a 2 (u 2 , u 2 ) + (P u 1 , u 2 ) H = λ -(iλf 2 + g 2 , u 2 ) H .
Again by Cauchy-Schwarz inequality and the boundedness property of P , we then get

a 2 (u 2 , u 2 ) λ 2 u 2 2 H + ( u 1 H + λ -+1 F H ) u 2 H .
The

λ 2 u 1 2 H λ 1-1 + o λ 1-+1 , hence (4.28) holds if (4.33) ≥ 1 + 1. Proof. Inspired from [1, Proposition 2.2], let us consider (ũ 1 , ṽ1 ) ∈ D(A w ) solution of -iλũ 1 -ṽ1 = 0 in V 1 , -iλṽ 1 + A 1 ũ1 + BB ṽ1 = u 1 in H, that,
λ ũ1 H + ũ1 V1 O(λ 1-1 ).
Taking the inner product in H of the identity (4.35) with u 1 , we get

u 1 2 H = (-λ 2 ũ1 + A 1 ũ1 -iλBB ũ1 , u 1 ) H = -λ 2 ũ1 + A 1 ũ1 -iλBB ũ1 , u 1 1 = -λ 2 u 1 + A 1 u 1 + iλBB u 1 , ũ1 1 .
Using (4.24), we then obtain

u 1 2 H = -P u 2 + λ -(iλf 1 + BB f 1 + g 1 ) , ũ1 1 .
Hence using the property BB f 1 ,

ũ1 1 = (B f 1 , B ũ1 ) U , we get (4.37) u 1 2 H = -(P u 2 , ũ1 ) H + λ -(iλf 1 + g 1 , ũ1 ) H + λ -(B f 1 , B ũ1 ) U .
Using Cauchy-Schwarz inequality, and the properties of P and B , we get

(4.38) u 1 2 H u 2 H + λ -g 1 H + λ -+1 f 1 H ũ1 H + λ -f 1 V ũ1 V1 .
The estimations (4.26) and (4.36) yield

u 1 2 H λ 1-3 + λ 1--1 F H .
Using the property (4. [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF] we conclude that (4.32) holds. Consequently (4.28) holds as soon as

1 < 1 and 1 -+ 1 ≤ 0.
Note that the two previous Lemmas yield the appropriated convergence properties of u 1 and v 1 . To transfer these information on u 2 and v 2 , we need some more assumptions. Lemma 4.11 Assume that the estimate (4.32) holds with 0 ≤ 1 < 1, that A 2 = A 1 and that P ∈ L(H) satisfies (3.10) for some D ∈ L(H). Then the solution

(u 1 , v 1 , u 2 , v 2 ) ∈ D(A) of System (4.20)-(4.23) satisfy (4.39) λ 2 D u 2 2 H λ 2 | (P u 2 , u 2 ) H | λ 1 -1 + o λ 1-+1 + o λ B +1-2 + o λ 2-.
Proof. Taking the duality pairing of (4.24) (resp. inner product of (4.25

)) with u 2 in V 1 -V 1 (resp. u 1 in H), we get -λ 2 (u 1 , u 2 ) H + a 1 (u 1 , u 2 ) + iλ BB u 1 , u 2 1 + (P u 2 , u 2 ) H = λ -iλf 1 + BB f 1 + g 1 , u 2 1 , -λ 2 (u 2 , u 1 ) H + a 1 (u 2 , u 1 ) + (P u 1 , u 1 ) H = λ -(iλf 2 + g 2 , u 1 ) H
Taking the real part of both identities and the difference, one gets

(P u 2 , u 2 ) H = (P u 1 , u 1 ) H + λ (B u 1 , B u 2 ) (4.40) + λ -( iλf 1 + BB f 1 + g 1 , u 2 1 -(iλf 2 + g 2 , u 1 ) H ) .
Using Cauchy-Schwarz inequality and the boundedness property of P and the fact that

B ∈ L(V 1 , U ), we get λ 2 | (P u 2 , u 2 ) H | λ 2 u 1 2 H + λ 3 B u 1 |u 2 V2 (4.41) + λ -+3 F H ( u 1 H + u 2 H ) + λ -+2 F H u 2 V2 .
The assumption (4.32)

λ 2 | (P u 2 , u 2 ) H | λ 2 u 1 2 H + λ 3 B u 1 U u 2 H + λ -+3 F H ( u 1 H + u 2 H ).
Using again the assumption (4.32) as well as (4.18), (4.19) , (4.26), and (4.27), we arrive at (4.39) with B = 0.

We are now ready to prove Theorem 4.3.

Proof. of Theorem 4.3 We first notice that (4.9) implies 0 ≤ 1 < 1 and (4.10) implies that ≥ 1 + 1, hence Lemma 4.10 can be applied, therefore (4.32) holds and in particular the property (4.28) is valid. We now notice that the assumption (4.8) and the properties (4.39) yield

(4.42) λ B 2 u 2 H = O(λ 1 -1 2 ) + o λ 1 -+1 2 + o λ 2 B +2- 4 + o λ 1-2 . Now we consider (ũ 2 , ṽ2 ) ∈ D(A 2 ) solution of -iλũ 2 -ṽ2 = 0 in V 2 , (4.43) -iλṽ 2 + A 2 ũ2 + B 2 B 2 ṽ2 = u 2 in
u 2 2 H = (-λ 2 ũ2 + A 2 ũ2 -iλB 2 B 2 ũ2 , u 2 ) H = (-λ 2 u 2 + A 2 u 2 + iλB 2 B 2 u 2 , ũ2 ) H Using (4.25) we find (4.48) u 2 2 H = (λ -(iλf 2 + g 2 ) -P u 1 + iλB 2 B 2 u 2 , ũ2 ) H .
Using Cauchy-Schwarz inequality and the boundedness property of P and B 2 , we find

(4.49) u 2 2 H (λ -+1 F H + u 1 H + λ B 2 u 2 H ) ũ2 H .
The properties (4.19), (4.32), and (4.39) then lead to

u 2 2 H o λ -+1 + O λ 1 -3 2 + o λ 1 --1 2 + O(λ 1 -1 2 ) + o λ 1 -+1 2 + o λ 2 B +2- 4 + o λ 1-2 ũ2 H .
By the estimate (4.47) we conclude that

λ 2 u 2 2 H = o λ 2-+1 + O λ 2 2 + 1 -3 2 + o λ 2 2 + 1 --1 2 + O(λ 2 2 + 1 -1 2 ) + o λ 2 2 + 1 -+1 2 + o λ 4 2 +2 B +2- 4 + o λ 2+1-2 .
This allows to conclude that (4.30) holds, as soon as

2 2 + 1 -3 < 0 and 2 2 + 1 -1 < 0,
as well as

2 -+ 1 ≤ 0, 2 2 + 1 --1 ≤ 0, 2 2 + 1 -+ 1 ≤ 0, 4 2 + 2 B + 2 -≤ 0, 2 2 + 2 -≤ 0.
The first constraints reduce to (4.9) while, due to the constraint 0 ≤ 1 < 1, we readily check that the second ones reduce to (4.10). The property (4.28) and (4.30) combined with Lemmas 4.8 and 4.9 show that U H = o(1), which is in contradiction with (4.18). The constraint on 1 in Lemma 4.10 can be improved under the additional assumptions of Theorem 4.5 on P and between P and B . First we will prove Lemma 4.8 in the new setting Lemma 4.12 Assume that A 2 = A 1 and that P ∈ L(H) satisfies (3.10) and (4.12) for some D ∈ L(H). Then the solution (u 1 , v 1 , u 2 , v 2 ) ∈ D(A) of system (4.20)-(4.23) satisfy the following estimate

(4.50) λ D u 2 H = o λ 1 2 -4 ,
as soon as ≥ 2.

Proof. We start with the identity (4.40) from the proof of Lemma 4.11, but here the absolute value of the first term of its right-hand side is estimated with the help of the assumption (4.12), while for the other terms, we use Cauchy-Schwarz inequality to get

| (P u 2 , u 2 ) H | B u 1 U u 1 H + λ B u 1 U u 2 H + λ -+1 F H ( u 1 H + u 2 H ) + λ -F H u 2 V1 .
Using the property (3.10) and (4.27), we get

D u 2 2 H o λ -2 -2 + o λ -2 u 2 H + λ -+1 F H ( u 1 H + u 2 H ) + λ -F H u 2 V1 .
With the help of (4.18), (4.19) and (4.26), we finally arrive to

λ 2 D u 2 2 H = o λ -2 + o λ -2 +1 + o(λ -+2 ).
The estimation (4.50) directly folllows as soon as ≥ 2.

Lemma 4.13 Under the assumptions of Lemma 4.12, assume further that condition (4.5) holds with 1 ≥ 0, then the solution

(u 1 , v 1 , u 1 , v 2 ) ∈ D(A) of System (4.20)-(4.23) satisfy the estimate (4.51) λ 2 u 1 2 H o λ -1 2 -4 + 1 + o(λ 1 -+1 ),
hence (4.28) holds if

(4.52) ≥ max{ 1 + 1, 2(2 1 -1)}.
Proof. The proof is the same as the one of Lemma 4.10 except for the estimation of the first term of the right-hand side of (4.37), where we use the assumption (4.12) and the property (4.50) to get

|(P u 2 , ũ1 ) H | ≤ P u 2 H ũ1 H D u 2 H ũ1 H o λ -1 2 -4 ũ1 H .
Using this estimate in (4.37) and Cauchy-Schwraz's inequality and the properties of P and B , we obtain

u 1 2 H o λ -1 2 -4 ũ1 H + λ -g 1 H + λ -+1 f 1 H ũ1 H + λ -f 1 H ũ1 V1 .
The (λ

-+1 F H + P u 1 H + λ B 2 u 2 H ) ũ2 H .
The properties (4.19), (4.50), and the assumption (4.12) combined with (4.27) then lead to

u 2 2 H o λ -+1 + o λ -+2 2 + o λ 1 2 -4 ũ2 H .
By the estimate (4.47) we conclude that

λ 2 u 2 2 H = o λ 2-+1 + o λ 2 --2 2 + o λ 4 2 +2- 4
This allows to conclude that (4.30) holds, as soon as

2 -+ 1 ≤ 0, 2 --2 ≤ 0, 4 2 + 2 -≤ 0.
As these constraints are equivalent to

≥ 4 2 + 2
we need to chose in the form (4.13) in order to guarantee that (4.52) holds.

Altogether the property (4.28) (that is valid due Lemma 4.13) and (4.30) combined with Lemmas 4.8 and 4.9 show that U H = o(1), which is in contradiction with (4.18). Now we go on with the proof of Theorem 4.7. Again for clarity, we divide the proof into several lemmas. Let us first show that the statements of Lemmas 4.8 and 4.9 remain valid in the setting of this Theorem. Proof. We take the duality pairing (4.24) with u 1 in V 1 -V 1 and take the real part to get

a 1 (u 1 , u 1 ) = λ 2 u 1 2
H + P u 2 , u 1 1 + λ -(iλf 1 + g 1 , u 1 ) H + (B f 1 , B u 1 ) U .
First using Cauchy-Schwarz inequality, and the boundedness property of B , we find

a 1 (u 1 , u 1 ) λ 2 u 1 2
H + P u 2 , u 1 1 + λ -+1 F H u 1 H + λ -f 1 V1 u 1 V1 .
Using the estimates (4.18), (4.19) and (4.54), we can estimate all the terms of this right-hand side except the second one.

For this term, we use (4.15) to obtain

P u 2 , u 1 1 = (D u 2 , D u 1 ) U D u 2 U B u 1 U .
Using the boundedness property of D , we find

D u 2 U u 2 V1 ,
therefore by (4.18), and (4.27) we get (4.56)

P u 2 , u 1 1 o(λ -2 -1 ).
Hence for all ≥ 0, this term is o(1), thus the proof is completed.

Let us go on with the estimate of the norm of

u 2 in V 1 .
Lemma 4.15 Let the assumptions of Theorem 4.7 be satisfied and assume that the estimate

(4.57) λ u 2 H = o(1)
holds, then the solution Proof. As before we take the duality pairing (4.25) with u 2 in V 1 -V 1 and take the real part to get that yields

(u 1 , v 1 , u 2 , v 2 ) ∈ D(A) of System
-λ 2 u 2 2 H + a 2 (u 2 , u 2 ) + P u 1 , u 2 1 = λ -(iλf 2 + g 2 , u 2 ) H .
Again by Cauchy-Schwarz inequality, we then get

a 1 (u 2 , u 2 ) λ 2 u 2 2 H + P u 1 , u 2 1 + λ -+1 F H u 2 H .
The properties (4.19), (4.26) and (4.57) allows to estimate the first and third term of this righthand side. For the second term, we use (4.56), hence for any ≥ 0, (4.58) holds.

As before the two previous Lemmas yield the appropriated convergence properties of u 1 and v 1 , we now transfer these information on u 2 and v 2 via some additional Lemmas.

(4.59) P u 2 , u 2 1 = D u 2 2 U = o λ -2 -1 2 ,
as soon as ≥ 0.

Proof. Taking the duality pairing of (4.24) (resp. (4.25

)) with u 2 in V 1 , V 1 (resp. u 1 ), we get -λ 2 (u 1 , u 2 ) + a 1 (u 1 , u 2 ) + iλ BB u 1 , u 2 1 + P u 2 , u 2 1 = λ -iλf 1 + BB f 1 + g 1 , u 2 1 , -λ 2 (u 2 , u 1 ) H + a 1 (u 2 , u 1 ) + P u 1 , u 1 1 = λ -iλf 2 + g 2 , u 1 1
Taking the real part of both identities and the difference, one gets

D u 2 2 U = D u 1 2 U + λ (B u 1 , B u 2 ) U (4.60) + λ -( iλf 1 + BB f 1 + g 1 , u 2 1 -iλf 2 + g 2 , u 1 1 ) .
But now the first term of its right-hand side is estimated with the help of the assumption (4.15), while for the other terms, we use Cauchy-Schwarz inequality and the fact that

B ∈ L(V 1 , U ) to get (4.61) D u 2 2 U B u 1 2 U + λ B u 1 U B u 2 U + λ -+1 F H ( u 1 H + u 2 H ) + λ -F H u 2 V1 .
First using (4.16) we can deduce that

(4.62) B u 2 U u 2 1 2
H u 2

D u 2 U o(λ --2 ) + o(λ -2 -1 2 ) + o(λ -).
Lemma 4.17 Let the assumptions of Theorem 4.7 be satisfied, then the solution (u 1 , v 1 , u 1 , v 2 ) ∈ D(A) of system (4.20)-(4.23) satisfy the estimate

(4.65) λ 2 u 1 2 H o λ -4 + 1 2 -1 4 + o(λ 1-+1 ) + o(λ -+ 1 2 ).
Hence (4.54) holds if Taking the inner product in H of the identity (4.68) with u 1 , we get

(4.66) ≥ max{ 1 + 1, 2 1 -1}. Proof. Inspired from [1, Proposition 2.2], let us consider (ũ 1 , ṽ1 ) ∈ D(A w ) solution of -iλũ 1 -ṽ1 = 0 in V 1 , -iλṽ 1 + A 1 ũ1 + BB ṽ1 = u 1 in H, that,
u 1 2 H = (-λ 2 ũ1 + A 1 ũ1 -iλBB ũ1 , u 1 ) H = -λ 2 ũ1 + A 1 ũ1 -iλBB ũ1 , u 1 1 = -λ 2 u 1 + A 1 u 1 + iλBB u 1 , ũ1 1 .
Using (4.24), we then obtain

u 1 2 H = -P u 2 + λ -(iλf 1 + BB f 1 + g 1 ) , ũ1 1 .
Hence using the property BB f 1 ,

ũ1 1 = (B f 1 , B ũ1 ) U , we get (4.70) u 1 2 H = -P u 2 , ũ1 1 + λ -(iλf 1 + g 1 , ũ1 ) H + λ -(B f 1 , B ũ1 ) U .
For the first term we use the property (4.59) to get (4.71)

| P u 2 , ũ1 1 | ≤ D u 2 U D ũ1 U = o λ -4 -1 4 D ũ1 U .
The assumption (4.15) then yields (4.72)

| P u 2 , ũ1 1 | = o λ -4 -1 4 B ũ1 U .
For the second factor, we use the fact

(-iλI -A w ) Ũ = (0, u 1 ) ,
where Ũ = (ũ 1 , ṽ1 ) . Hence

B ṽ1 2 U = ((-iλI -A w ) Ũ , Ũ ) H = (u 1 , ṽ1 ) H .
As ṽ1 = iλũ 1 and using Cauchy-Schwarz's inequality, we get

λ 2 B ũ1 2 U λ u 1 H ũ1 H ,
and using (4.26) and (4.69), we get

(4.73) B ũ1 U λ 1 2 -2 .
Using this estimate in (4.72), we find (4.74)

| P u 2 , ũ1 1 | = o λ -4 + 1 2 -1 4 -2 .
For the third term of the right-hand side of (4.70), we use (4.73) and the boundedness of B , to obtain

|(B f 1 , B ũ1 ) U | f 1 V1 λ 1 2 -2 .
Using this estimate and (4.74) in (4.70) and Cauchy-Schwarz's inequality, we obtain

u 1 2 H o λ -4 + 1 2 -1 4 -2 + λ -g 1 H + λ -+1 f 1 H ũ1 H + f 1 V1 λ -+ 1 2 -2 .
The Now taking the inner product in H of (4.79) with u 2 , we find

B 2 u 2 U = o λ -4 -1 4 . Now we consider (ũ 2 , ṽ2 ) ∈ D(A 2 ) solution of -iλũ 2 -ṽ2 = 0 in V 1 , (4.76) -iλṽ 2 + A 2 ũ2 + B 2 B 2 ṽ2 = u 2 in
u 2 2 H = (-λ 2 ũ2 + A 2 ũ2 -iλB 2 B 2 ũ2 , u 2 ) H = -λ 2 ũ2 + A 2 ũ2 -iλB 2 B 2 ũ2 , u 2 1 = -λ 2 u 2 + A 2 u 2 + iλB 2 B 2 u 2 , ũ2 1
Using (4.25) we find

u 2 2 H = λ -((iλf 2 + g 2 ) , ũ2 ) H -P u 1 , ũ2 1 + iλ(B 2 u 2 , B 2 ũ2 ) U .
Using Cauchy-Schwarz inequality and the fact that P = DD , we find

(4.82) u 2 2 H (λ -+1 F H ũ2 H + D u 1 U D ũ2 U + λ B 2 u 2 U B 2 ũ2 U .
The third term of this right-hand side is directly estimated by (4.75) and (4.81) to have

(4.83) λ B 2 u 2 U B 2 ũ2 U = o λ -4 + 2 2 + 3 4 -2 .
For the first term of this right-hand side, the properties (4. [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF]) and (4.80) allow to show that (4.84)

λ -+1 F H ũ2 H = o λ 2--1 .
For the second term, we use the assumption (4.15) and (4.27) to get

(4.85) D u 1 U ≤ B u 1 U = o λ -2 -1 .
As before using and (4.16)

(4.86) D ũ2 U ≤ B ũ2 U ũ2 H ũ2 V1 ,
Now using the above estimation and (4.85)

(4.87) D u 1 U D ũ2 U o(λ 2-2 -5 2 )
Using the estimates (4.83), (4.84) and (4.87) into (4.82), we find

λ 2 u 2 2 H = o λ 2-+1 + o λ 2-2 -1 2 + o λ -4 + 2 2 + 3 4 .
This allows to conclude that (4.57) holds, as soon as

2 -+ 1 ≤ 0, 2 - 2 - 1 2 ≤ 0, 2 2 - 4 + 3 4 ≤ 0.
These constraints are satisfied if and only if

≥ 2 2 + 3.
All together the property (4.54) and (4.57) combined with Lemmas 4.14 and 4.15 show that U H = o(1), which is in contradiction with (4.18).

Polynomial energy decay rate in the case

a 1 = a 2
In this subsection we extend Theorems 4.3 and 4.5 in the case a 1 = a 2 when V 2 is only continuously

embedded into V 1 (written in short V 2 → V 1 ).
Theorem 4.18 Assume that a 1 = a 2 , that V 2 → V 1 , that P ∈ L(H), and that (3.10) holds for some D ∈ L(H). Assume also that iR ⊂ ρ(A) and that system (3.1) is strongly stable in V 1 × H such that its associated resolvent satisfies (4.5), for some 1 ≥ 0.

Assume furthermore that there exists a bounded and linear operator B 2 ∈ L(H) such that (4.8) holds and such that the system (4.6) is strongly stable in V 1 × H and that its associated operator A w,2 satisfies (4.7), for some 2 ≥ 0. We finally assume that

(4.88) |a 1 (w, v) -(A 2 w, v) H | w D(A2) B v U , ∀w ∈ D(A 2 ), v ∈ V 1 .
If (4.9) holds, then A satisfies (4.4) with

(4.89) = 4 2 + 4.
Therefore the semigroup (e tA ) t≥0 generated by A is polynomially stable with a decay in t -1 .

Proof. The proof is essentially the same as the one of Theorem 4.3 the main difference stays on the estimate (4.39) from Lemma 4.11 that here takes the form (4.90)

λ 2 D u 2 2 H λ 2 | (P u 2 , u 2 ) H | λ 1 -1 + o λ 1 -+1 + o λ B +1-2 + o λ 2-2 .
Indeed under the assumption V 2 → V 1 , we can still take the inner product of (4.25)) with u 1 in H, which then becomes

-λ 2 (u 2 , u 1 ) H + a 1 (u 2 , u 1 ) + (P u 1 , u 1 ) H = λ -(iλf 2 + g 2 , u 1 ) H + a 1 (u 2 , u 1 ) -(A 2 u 2 , u 1 ) H .
As in the proof of Lemma 4.11, the identity (4.40) is replaced by

(P u 2 , u 2 ) H = (P u 1 , u 1 ) H + λ (B u 1 , B u 2 ) + λ -( iλf 1 + BB f 1 + g 1 , u 2 1 -(iλf 2 + g 2 , u 1 ) H ) - (a 1 (u 2 , u 1 ) -(A 2 u 2 , u 1 ) H ).
By using the assumption (4.88), (4.41) becomes

λ 2 | (P u 2 , u 2 ) H | λ 2 u 1 2 H + λ 3 B u 1 |u 2 V2 + λ -+3 F H ( u 1 H + u 2 H ) + λ -+2 F H u 2 V2 + u 2 D(A2) B u 1 U .
Now using (4.25), we have Therefore the semigroup (e tA ) t≥0 generated by A is polynomially stable with a decay in t -1 .

u 2 D(A2) = u 2 V + A 2 u 2 H u 2 V + λ 2 u 2 H + λ f 2 H + g 2 H +

Remark 4.20

The same results hold if V 1 → V 2 , replacing (4.88) by

|a 1 (w, v) -(A 1 w, v) H | w D(A1) B v U , ∀w ∈ D(A 1 ), v ∈ V 2 .

Polynomial energy decay rate in the case

A 1 = A 2
In this section we are interested to prove the polynomial stability of system (1.1) in the case A 1 = A 2 , where the two operators B and P are supposed to be bounded.

Theorem 4.21 Assume that P and B ∈ L(H), Assume also that iR ⊂ ρ(A) and that system (3.1) is strongly stable in V 1 × H such that its associated resolvent satisfies (4.5) for some 1 ≥ 0.

We further assume that V 2 is continuously embedded into V 1 and that there exist two bounded and selfadjoint operators M 1 , M 2 ∈ L(H), such that the "localization" operator

L 1 = M 2 1 and L 2 = M 2
2 are also bounded from V 1 into itself and satisfy

(4.92) L 1 L 2 = L 2 .
Furthermore we suppose that for i = 1 and 2,

(4.93) M i u H B u H , ∀u ∈ H.
We also suppose that there exists a "local" sesquilinear and continuous form a

1,loc from V 1 × V 1 into C such that it is non negative, i. e., a 1,loc (u, u) ≥ 0, ∀u ∈ V 1 ,
and

a 1 (u, L 1 u) = a 1,loc (u, u) + r(u), ∀u ∈ V 1 , (4.94) |a 1 (u, L 2 v)| a 1,loc (u, u) 1 2 v V2 , ∀u, v ∈ V 1 , (4.95) where (4.96) |r(u)| B u H u V1 , ∀u ∈ V 1 .
Assume furthermore that there exists a bounded and linear operator B 2 ∈ L(H) such that

(4.97) B 2 u H L 2 u H , ∀u ∈ H.
as well as

(4.98) L 2 u 2 H | (P u, L 2 u) H |, ∀u ∈ H,
and that the system (4.6) is strongly stable in V 2 × H and that its associated operator A w,2 satisfies (4.7) for some 2 ≥ 0. If (4.9) holds, then A satisfies (4.4) with (4.99) = 8 2 + 6.

Therefore the semigroup (e tA ) t≥0 generated by A is polynomially stable with a decay in t -1 .

Let us now give two variants of this result.

Theorem 4.22 In addition to the assumptions of Theorem 4.21, assume that V 1 = V 2 and that there exists a second "local" sesquilinear and continuous form a

1,loc2 from V 1 × V 1 into C such that a 1 (u, L 2 v) = a 1,loc2 (u, v) + r 1 (u, v), ∀u, v ∈ V 1 , (4.100) a 2 (v, L 2 u) = aa 1,loc2 (u, v) + r 2 (u, v), ∀u, v ∈ V 1 , (4.101)
where a is a positive constant and

|r 1 (u, v)| a 1,loc (u, u) 1 2 v H , ∀u, v ∈ V 1 , (4.102) |r 2 (u, v)| B u H v V1 , ∀u, v ∈ V 1 . (4.103) Then A satisfies (4.4) with (4.104) = 4 2 + 4.
Therefore the semigroup (e tA ) t≥0 generated by A is polynomially stable with a decay in t -1 .

(1 + ε -1 ) B u H + ε u V1 u V1 , ∀u ∈ V 1 , ε > 0.
Then A satisfies (4.4) with (4.106) = 16 2 + 14.

Therefore the semigroup (e tA ) t≥0 generated by A is polynomially stable with a decay in t -1 .

Let us start by the proof of Theorem 4.21. As in the proof of Theorem 4.3, we may notice that (4.9) implies 0 ≤ 1 < 1 and (4.99) implies that ≥ 1 + 1, hence Lemma 4.10 can be applied, therefore (4.32) holds and in particular the property (4.28) is valid. It therefore remains to estimate u 2 H , what we do in several lemmas.

4.107) a 1,loc (u 1 , u 1 ) = o λ -2 -1 .
Proof. We first take the inner product of (4.24) with L 1 u 1 to get

-λ 2 (u 1 , L 1 u 1 ) H + a 1 (u 1 , L 1 u 1 ) + iλ(B u 1 , B L 1 u 1 ) H + (P u 2 , L 1 u 1 ) H (4.108) = λ -(iλf 1 + BB f 1 + g 1 , L 1 u 1 ) H .
We now estimate all terms of this identity except the second one of the left-hand side: By our assumption on L 1 (in particular (4.93)), we have

(u 1 , L 1 u 1 ) H = M 1 u 1 2 H B u 1 2 H .
Using the estimate (4.27) (since the assumption B ∈ L(H) yields U = H), we find

(4.109) λ 2 (u 1 , L 1 u 1 ) H = o λ -.
Since M 1 is selfadjoint, we have

(P u 2 , L 1 u 1 ) H = (M 1 P u 2 , M 1 u 1 ) H .
Using Cauchy-Schwarz's inequality and the fact that P ∈ L(H) and M 1 ∈ L(H), we get

|(P u 2 , L 1 u 1 ) H | ≤ u 2 H M 1 u 1 H .
Using (4.26) and again the estimate (4.27), we find (4.110)

|(P u 2 , L 1 u 1 ) H | = o λ -2 -2 .
Using the fact that the operator M 1 is bounded and combined to our assumption (4.93), we get

(4.111) L 1 u 1 H B u 1 H ,
and therefore using Cauchy-Schwarz's inequality and the fact that B ∈ L(H), we obtain

λ|(B u 1 , B L 1 u 1 ) H | λ B u 1 2
H , By (4.27), we conclude that

(4.112) λ|(B u 1 , B L 1 u 1 ) H | = o λ --1 .
For the right-hand side, using Cauchy-Schwarz's inequality, (4.111) and (4.27), we obtain

(4.113) λ -|(iλf 1 + BB f 1 + g 1 , L 1 u 1 ) H | = (λ -+1 + λ -o λ -2 -1 = o λ -3 2 .
Inserting the estimates (4.109), (4.110), (4.112), and (4.113) into (4.108) yields

|a 1 (u 1 , L 1 u 1 )| = o λ -+ o λ -2 -2 + o λ -3 2 .
Hence assuming that ≥ 4, we find

(4.114) |a 1 (u 1 , L 1 u 1 )| = o λ -2 -2 .
At this stage, we use the assumption (4.94) to write

(4.115) a 1,loc (u 1 , u 1 ) ≤ |a 1 (u 1 , L 1 u 1 )| + |r(u 1 )|,
and using (4.96), we get

a 1,loc (u 1 , u 1 ) ≤ |a 1 (u 1 , L 1 u 1 )| + B u 1 H .
By the estimates (4.27) and (4.114), we arrive at

(4.116) a 1,loc (u 1 , u 1 ) = o λ -2 -1 .
L 2 u 2 2 H = o λ -m( ) + o λ -2 -1 , with m( ) = 4 + 1 2 .
Proof. Take the inner product of (4.24) with L 2 u 2 to get

-λ 2 (u 1 , L 2 u 2 ) H + a 1 (u 1 , L 2 u 2 ) + iλ(B u 1 , B L 2 u 2 ) H + (P u 2 , L 2 u 2 ) H (4.117) = λ -(iλf 1 + BB f 1 + g 1 , L 2 u 2 ) H .
The goal is to obtain an estimate of L 2 u 2 H , so in view of the assumption (4.98), we need to estimate all the terms of this identity except the last one of this left-hand side. First, we notice that the assumption (4.95) yields

|a 1 (u 1 , L 2 u 2 )| a 1,loc (u 1 , u 1 ) 1 2 u 2 V2 ,
and by (4.107), we find

(4.118) |a 1 (u 1 , L 2 u 2 )| = o λ -m( ) .
By our assumptions on L 2 , we may write

(u 1 , L 2 u 2 ) H = (u 1 , L 1 L 2 u 2 ) H = (L 1 u 1 , L 2 u 2 ) H ,
and by Cauchy-Schwarz's inequality, (4.111) and (4.27), we find (4.119)

λ 2 |(u 1 , L 2 u 2 ) H | = o λ -2 +1 L 2 u 2 H .
Using Cauchy-Schwarz's inequality, the boudedness of B L 2 , and (4.26), we have

λ|(B u 1 , B L 2 u 2 ) H | B u 1 H ,
and again by (4.27), we obtain

(4.120) λ|(B u 1 , B L 2 u 2 ) H | = o λ -2 -1 .
For the right-hand side, Cauchy-Schwarz's inequality directly yields

(4.121) λ -|(iλf 1 + BB f 1 + g 1 , L 2 u 2 ) H | = o λ -+1 L 2 u 2 H .
Using the estimates (4.118) to (4.121) into (4.117), we obtain

| (P u 2 , L 2 u 2 ) H | = o λ -m( ) + o λ -2 -1 + o λ -2 +1 L 2 u 2 H .
The assumption (4.98) then leads to

L 2 u 2 2 H = o λ -m( ) + o λ -2 -1 + o λ -2 +1 L 2 u 2 H .
By Young's inequality, we get

L 2 u 2 2 H = o λ -m( ) + o λ -2 -1 + o λ -+2 = o λ -m( ) + o λ -2 -1 .
Proof. of Theorem 4.21 By the assumption (4.97) and using the previous Lemma, we obtain

(4.122) B 2 u 2 2 H = o λ -m( ) + o λ -2 -1 .
Now we proceed as in the proof of Theorem 4.

u 2 2 H (λ -+1 F H + u 1 H + λ B 2 u 2 H ) ũ2 H .
Hence using the estimates (4.32), (4.47) and (4.122), we here find

u 2 2 H = o(λ -+1 ) + O λ 1 -3 2 + o λ 1 --1 2 + o λ 1-m( ) 2 + o λ -4 + 1 2 O(λ 2-2 ).
Consequently

λ 2 u 2 2 H = o(λ -+1 ) + O λ 1 -3 2 + o λ 1 --1 2 + o λ 1-m( ) 2 + o λ -4 + 1 2 O(λ 2 ) (4.123) = o(λ -+ 2 +1 ) + O λ 2 + 1 -3 2 + o λ 2+ 1 --1 2 + o λ 2+1-m( ) 2 + o λ 2-4 + 1 2 .
This allows to conclude that (4.30) holds, as soon as (4.9) holds, as well as

(4.124) 2 -+ 1 ≤ 0, 2 2 + 1 --1 ≤ 0, 2 2 + 2 -m( ) ≤ 0, 4 2 + 2 -≤ 0.
Since m( ) = 4 + 1 2 and since the condition (4.9) implies that 1 < 1 and 2 2 < 1, (4.124) holds with the choice (4.99) of .

The properties (4.28) and (4.30) combined with Lemmas 4.8 and 4.9 show that U H = o(1), which is in contradiction with (4.18).

Let us go on with the proof of Theorem 4.22. Proof. of Theorem 4. [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] The proof follows the same lines as the previous one, except for the estimation of a 1 (u 1 , L 2 u 2 ). Indeed to estimate this term, we first take the inner product of (4.25) with L 2 u 1 to find

(4.125) -λ 2 (u 2 , L 2 u 1 ) H + a 2 (u 2 , L 2 u 1 ) + (P u 1 , L 2 u 1 ) H = λ -(iλf 2 + g 2 , L 2 u 1 ) H .
Now we estimate all terms except the second one of this right-hand side. First, by the assumptions on L 2 and Cauchy-Schwarz's inequality, we have

|(u 2 , L 2 u 1 ) H | = |(M 2 u 2 , M 2 u 1 ) H | u 2 H M 2 u 1 H ,
By our assumption (4.93), we get

|(u 2 , L 2 u 1 ) H | = |(M 2 u 2 , M 2 u 1 ) H | u 2 H B u
λ 2 |(u 2 , L 2 u 1 ) H | = o λ -2 .
As P and M 2 are bounded, and using Cauchy-Schwarz's inequality, we have

|(P u 1 , L 2 u 1 ) H | u 1 H M 2 u 1 H ,
By our assumption (4.93) and (4.26) and (4.27) we obtain (4.127)

|(P u 1 , L 2 u 1 ) H | = o λ -2 -2 .
For the right-hand side, using Cauchy-Schwarz's inequality and our assumption (4.93), we get

(4.128) λ -(iλf 2 + g 2 , L 2 u 1 ) H = o λ -3 2 .
The estimates (4.126) to (4.128) into the identity (4.125) lead to

(4.129) a 2 (u 2 , L 2 u 1 ) = o λ -2 .
Now the two assumptions (4.100) and (4.101) imply that

a 1 (u 1 , L 2 u 2 ) = a -1 a 2 (u 2 , L 2 u 1 ) -r 2 (u 1 , u 2 ) + r 1 (u 1 , u 2 ).
Using the assumptions (4.102) and (4.103), we obtain

|a 1 (u 1 , L 2 u 2 )| |a 2 (u 2 , L 2 u 1 )| + a 1,loc (u 1 , u 1 ) 1 2 u 2 H + B u 1 H u 2 V1 .
Assuming that ≥ 4, and using (4.27), (4.107), (4.129) we find

|a 1 (u 1 , L 2 u 2 )| = o λ -2 + o λ -4 -3 2 .
This shows that (4.118) holds with m( ) = min{ 2 , 4 + 3 2 }. The remainder of the proof is the same as before and leads to the estimate (4.123). This allows to conclude that (4.30) holds, as soon as (4.9) holds, as well as (4.124). Since here m( ) = min{ 2 , 4 + 3 2 } and since the condition (4.9) implies that 1 < 1 and 2 2 < 1, (4.124) holds with the choice (4.99) of .

As before with this choice the property (4.28) is also valid, hence with the help of Lemmas 4.8 and 4.9 we obtain that U H = o(1), which is in contradiction with (4.18).

We end this section by the proof of Theorem 4.24. Proof. of Theorem 4. [START_REF] Batty | Non-uniform stability for bounded semi-groups on banach spaces[END_REF] The proof is the same as the one of Theorem 4.21, the sole difference is in the estimate of the second term of the right-hand side of (4.115), where we here use (4.105) to find

a 1,loc (u 1 , u 1 ) ≤ |a 1 (u 1 , L 1 u 1 )| + (1 + ε -1 ) B u 1 H + ε u 1 V1 , ∀
for all ε > 0.

As before, Lemma 4.10 can be applied, therefore (4.32) holds and in particular the property (4.28) is valid, hence by Lemma 4.8, u 1 V1 = o(1). This property and the estimates (4.27), and (4.114) lead to

a 1,loc (u 1 , u 1 ) = o λ -2 -2 + (1 + ε -1 )o λ -2 -1 + εo(1). Choosing ε = λ -4 -1 2 , we find (4.130) a 1,loc (u 1 , u 1 ) = o λ -4 -1 2 .
As in the proof of Lemma 4.26, this leads to

(4.131) |a 1 (u 1 , L 2 u 2 )| = o λ -m( ) .
with m( ) = 8 + 1 4 , instead of (4.118). The remainder of the proof is the same as before and leads to the estimate (4.123). This allows to conclude that (4.30) holds, as soon as (4.9) holds, as well as (4.124). Since here m( ) = 8 + 1 4 , one shows as before that (4.124) holds with the choice (4.106) of .

The properties (4.28) and (4.30) combined with Lemmas 4.8 and 4.9 yield U H = o(1), which is in contradiction with (4.18).

Applications

We will apply the previous Theorems to different concrete second order systems. In each of the following subsections, we first show how the different problems can be formulated in the abstract setting from section 2 and satisfies the assumptions stated there. Therefore all these systems are well posed due to Theorem 2.1. We will also check their strong stability and the assumptions of the Theorems from section 4 to obtain some polynomial stability results.

Internal stabilization of locally coupled wave equations

Here, we are interested in the internal stabilization of multidimensional coupled wave equations. Namely, if Ω is a bounded domain of R d , with d ≥ 1, with a Lipschitz boundary Γ, we consider the multidimensional coupled wave equations

u 1,tt -∆u 1 + c(x)u 1,t + b(x)u 2 = 0, in Ω × (0, ∞), (5.1) u 2,tt -a ∆u 2 + b(x)u 1 = 0, in Ω × (0, ∞), (5.2)
u 1 = u 2 = 0, on Γ × (0, ∞), (5.3)
with the following initial conditions (5.4)

u 1 (0) = u 1,0 , u 1,t (0) = u 1,1 , u 2 (0) = u 2,0 , u 2,t (0) = u 2,1 , in Ω,
where a is a positive real constant, the damping function c ∈ L ∞ (Ω) is such that

Ω |θ| 2 dx ≤ C 2 0 Ω |∇θ| 2 dx, ∀ θ ∈ H 1 0 (Ω) .
As mentioned in the introduction, problem (5.1)-(5.4) was extensively studied in the literature in some particular cases where some polynomial stability results were obtained. In comparison with the literature, we assume that the coupling function b is not necessarily non-negative, not necessarily real valued with no restriction on the size of b ∞ . Further we get some polynomial decay rates, under the assumption that two associated wave equations with local viscous dampings (see below for the details) are either exponentially or polynomially stable.

Let us now show that problem (5.1)-(5.4) enters in the abstract framework from section 2. Indeed we first take

H = U = L 2 (Ω) and V 1 = V 2 = H 1 0 (Ω)
with their usual inner products and norms. Then we define the sesquilinear, symmetric, continuous and coercive form a 1 by (5.9)

a 1 (u, v) = Ω ∇u • ∇v dx, ∀u, v ∈ V 1 .
Its associated unbounded operator A 1 from

D(A 1 ) = {u ∈ V 1 : ∆u ∈ L 2 (Ω)} into H is defined by A 1 : D(A 1 ) -→ H : u -→ -∆u,
and is therefore a positive selfadjoint operator with a compact resolvent. We further take a 2 = aa 1 , and hence A 2 = aA 1 .

In addition we define the operators (5.10)

P : L 2 (Ω) → L 2 (Ω) : u → b u, and B : L 2 (Ω) → L 2 (Ω) : u → √ c u,
that are clearly bounded from H into itself. Note further that B is clearly selfadjoint.

In this setting, we simply notice that condition (2.3) reduces to the assumption (5.7) since the first eigenvalue λ 1 of A 1 equals 1 C 2 0 . Therefore 2.1 can be invoked to obtain an existence and uniqueness result for system (5.1)-(5.4).

Strong stability

Stability results for system (5.1)-(5.4) can be deduced from the results from section 3. As an illustration let us state the one corresponding to Theorem 3.4. Before let us give a useful technical result. Proof. From its definition, we have

| (P u, u) H | 2 = ( Ω b|u| 2 dx) 2 ,
and as b does not change of sign, we get

| (P u, u) H | 2 = ( Ω | b||u| 2 dx) 2 ,
which proves that (3.10) holds with the above choice of D. Proof. We simply need to check that the assumptions of Theorem 3.4 are satisfied. First we notice that (5.12) guarantees that (3.9) holds, while the previous Lemma shows that (3.10) holds with the choice of (5.11) of D. Finally this choice of D implies that (3.13) is nothing else than the wave equation with a viscosity damping with coefficient | b|, namely

u tt -∆u + | b|u t = 0, in Ω × (0, ∞), u ( 0) = u 0 , u t (0) = u 1 ,
that, due to (5.13), is strongly stable in H by using the unique continuation principle.

In this result the condition (5.12) is quite strong because it implies that the support of b is included into the support of c. This can be significantly improved by using the specificity of the considered problem. Namely we have the following result. (5.17)

             v = iλu, in Ω, ∆u -by = iλv, in Ω, z = iλy, in Ω, a∆y -bu = iλz, in Ω.
From the first equation in (5.17) and the second property from (5.16), we deduce (5.18) u ≡ 0 in ω.

Eliminating v and z from (5.17), and using the first propery from (5.16), we obtain the following system Thus using Calderón theorem for a 2 system (see [START_REF] Hayek | A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface[END_REF]Theorem 2.5]), we deduce that u = y = 0 and thus, by the first and third identity of (5.17), v = z = 0, hence U = 0. The conclusion then follows from Lemma 3.1.

λ 2 u + ∆u -by = 0 in Ω, (5.19)
Remark 5.4 Theorem 5.3 allows to recover Corollary 5.2 since (5.12) and (5.13) imply (5.14) with ω = ω b .

Polynomial stability

This subsection is devoted to prove the polynomial energy decay rate of system (5.1)-(5.4) by applying Theorem 4.3, Theorem 4.5 or Theorem 4.22.

One of the ingredients is the exponential or polynomial stability of the wave equation in Ω with Dirichlet boundary condition with a local viscous interior damping. More precisely, we say that a non negative function β ∈ L ∞ (ω) satisfies the exponentially stability property (in short ESP) if the system (5.22)

     u tt -∆u + βu t = 0
in Ω × (0, +∞), u = 0 on Γ × (0, +∞), u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in Ω, is exponentially stable. Since this property is equivalent to an observability property

E y (0) = 1 2 (|y(•, 0)| 2 1,Ω + y t (•, 0)| 2 Ω ) ≤ C T 0 Ω α|y t | 2 dxdt,
for all y solution ot the wave equation in Ω with Dirichlet boundary condition, see [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF]Propositions 1 and 2], this property implies that the system (5.23)

     u tt -∆u + γu t = 0 in Ω × (0, +∞), u = 0 on Γ × (0, +∞), u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in Ω,
is exponentially stable for any non negative function γ ∈ L ∞ (ω) such that γ ≥ a 0 β a.e. in Ω, for some positive real number a 0 . As the converse implication is also valid, this means that if β, γ ∈ L ∞ (ω) are such that γ ∼ α a.e. in Ω, then β satisfies the ESP if and only if γ satisfies the ESP. Now we want to consider a similar notion for the system (5.24)

     u tt -a∆u + αu t = 0 in Ω × (0, +∞), u = 0
on Γ × (0, +∞), u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in Ω, But we notice that the change of variable s = √ at allows to transform it into (5.25)

           ũss -∆ũ + α √ a ũs = 0 in Ω × (0, +∞), ũ = 0 on Γ × (0, +∞), ũ(x, 0) = u 0 (x) and ũs (x, 0) = 1 √ a u 1 (x) in Ω,
where ũ(x, s) = u(x, s √ a ). This system is nothing else than (5.22) with β = α √ a and due to the previous equivalence, the exponential stability of (5.25) (or equivalently of (5.24)) is independent of a. This means that (5.24) is exponenttially stable if and only if α satisfies the ESP.

In the manner we say that a non negative function α ∈ L ∞ (ω) satisfies the a-polynomialy stability property (in short a-PSP) with decay rate t -2/ , for some > 0 if the resolvent set of the associated operator A w (in H 1 0 (Ω) × L 2 (Ω)) associated with (5.24) contains iR and satisfies (4.5) with 1 = . Note that in that last case, by Lemma 4.2, this implies that its associated energy E w (t) satisfies

E w (t) 1 t 2/ ||U 0 || 2 D(Aw) , ∀t > 0, U 0 ∈ D(A w ).
In its full generality, we cannot prove the equivalence between the a-PSP and the 1-PSP, but in all examples below the a-PSP holds for all a > 0, so for shortness we drop the a dependence. for some 0 < a 1 < L and 0 < a 2 < L, see Figure 1, since the multiplier method with the muliplier m(x 1 , x 2 ) = (x 1 -L, x 2 -L) applies due to the results from [START_REF] Grisvard | Contrôlabilté exacte des solutions de l'équation des ondes en présence de singularités[END_REF]. (p, q) × (0, 1) ⊂ ω, for some 0 ≤ p < q < 1, see Figure 2, then = 2 if p > 0 due to [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF], while = 3/2 if p = 0 and α depends only on the first variable due to [START_REF] Stahn | Optimal decay rate for the wave equation on a square with constant damping on a strip[END_REF]. By symmetry, the same results hold if (0, 1) × (p, q) ⊂ ω, Proof. First, as by assumption (5.14) holds, by Theorem 5.3, iR ⊂ ρ(A). Second as b does not change of sign, by Lemma 5.1, (3.10) holds with D defined by (5.11). Taking B 2 = D, system (4.6) is nothing else than (5.24) with α = | b|, therefore by our assumption on b, it is exponentially stable. Similarly the assumption on c guarantees that system (3.1), that here takes the form (5.24) with α = c is exponentially stable. Therefore we can apply Theorem 4.3 with 1 = 2 = 0, that implies that A satisfies (4.4) with = 2 (because here B = 0) and yields (5.28).

Obviously the previous proof is valid if we replace the exponential decay assumptions by polynomial ones, therefore the following result holds.

Theorem 5.7 In addition to the assumptions from this subsection, assume that a = 1, that b does not change of sign in Ω, that (5.14) holds, that c satisfies the PSP with decay rate t -2/ 1 , for some 1 > 0, and that | b| satisfies the PSP with decay rate t -2/ 2 , for some 2 > 0 with 1 and 2 satisfying (4.9). Then for all initial data U 0 ∈ D(A) the energy of the system (5.1)-( 5.3) satisfies the following decay rate estimate

(5.29) E(t) 1 t 2 U 0 2 D(A) , ∀t > 0, U 0 ∈ D(A), with = 4 2 + 2.
To remove the condition (4.9), we may apply Theorem 4.5, that in our case yields the following result.

Theorem 5.8 In addition to the assumptions from this subsection, assume that a = 1, that b does not change of sign in Ω, that (5.12) holds and that (5.13) holds, that c satisfies the PSP with decay rate t -2/ 1 , for some 1 > 0 and that | b| satisfies the PSP with decay rate t -2/ 2 , for some 2 > 0. Then (5.29) holds with given by (4.13).

Proof. We only need to check (4.12). But in view of the definition (5.11) of D, we clearly have

D u H

B u H , ∀u ∈ H due to (5.12).On the other hand, since for any u ∈ H, one has

P u 2 H = Ω |b| 2 |u| 2 dx = Ω ( b) 2 |u| 2 dx + Ω ( b) 2 |u| 2 dx,
as b does not change of sign in Ω. Using condition (5.6), we then get

P u 2 H b ∞ Ω | b||u| 2 dx.

This proves that

P u H = P u H D u H , ∀u ∈ H.
Now we are interested to prove the polynomial stability of system (5.1) -(5.4) in the case where the two waves propagate at different speeds (a = 1). Theorem 5.9 Let a = 1. In addition to the assumptions from this section, assume that b does not change of sign in Ω, that (5.14) holds. Assume that there exists an open set ω of Ω such that

(5.30) ω ⊂ ω ⊂ ω,
where ω is the open set from (5.14), and that c and I ω satisfy the ESP. Then for all initial data U 0 ∈ D(A) the energy of the system (5.1)-( 5.3) satisfies the following decay rate estimate. Then for all initial data U 0 ∈ D(A) the energy of the system (5.1)-( 5.3) satisfies the following estimate decay rate:

E(t) 1 √ t U 0 2 D(A) , ∀t > 0.
Proof. The proof is based on Theorem 4.22. We therefore need to check all its assumptions. First, as by assumption (5.

a 1 (u, L 1 u) = Ω ∇u • ∇(η 1 u) dx = Ω η 1 |∇u| 2 dx + Ω u∇u • ∇η 1 dx ∀ u ∈ H 1 0 (Ω).
This yields (4.94) with Therefore (4.96) holds since by Cauchy-Schwarz's inequality, for any u ∈ H 1 0 (Ω) we have

|r(u)| u L 2 (ω) u H 1 0 (Ω) B u L 2 (Ω) u H 1 0 (Ω)
, due to the assumption (5.14). Similarly we have (5.32)

a 1 (u, L 2 v) = Ω η 2 ∇u • ∇v dx + Ω v∇u • ∇η 2 dx, ∀u, v ∈ H 1 0 (Ω).
Hence by Cauchy-Schwarz's inequality, we obtain

|a 1 (u, L 2 v)| ∇u L 2 (supp η2) v H 1 0 (Ω) , ∀u, v ∈ H 1 0 (Ω).
As the assumption (5.31) yields

∇u 2 L 2 (supp η2) ≤ a 1,loc (u, u), ∀u, ∈ H 1 0 (Ω),
this estimate in the previous one shows that (4.95) holds. If we come back to (5.32), the identity (4.100) holds with

a 1,loc2 (u, v) = Ω η 2 ∇u • ∇v dx, r 1 (u, v) = Ω v∇u • ∇η 2 dx, ∀ u, v ∈ H 1 0 (Ω).
with the following initial conditions (5.39)

u i (0) = u i,0 , u i,t (0) = u i,1 , in Ω, i ∈ {1, 2, 3, 4},
where the damping functions c 1 , c 2 ∈ L ∞ (Ω) are such that c i (x) ≥ 0 for almost all x ∈ Ω, i = 1, 2, while the coupling matrix function

Q = p 13 p 14 p 23 p 24 belongs to L ∞ (Ω, R 2×2
) is different from 0 (but is not assumed to be symmetric) and satisfies

(5.40) Q ∞ < 1 C 2 0 ,
where C 0 is the Poincaré's constant from subsection 5.1 and

Q ∞ = sup x∈Ω Q(x) 2 ,
• 2 being the matrix norm induced by the Euclidean norm of C 2 .

To the best of our knowledge, problem (5.34)-(5.39) was not studied in the literature. Again we show that problem (5.34)-(5.39) enters in the abstract framework from section 2. First we take

H = U = L 2 (Ω) 2 and V 1 = V 2 = H 1 0 (Ω)
2 with their usual inner products and norms. Then we define the sesquilinear, symmetric, continuous and coercive form a 1 by

a 1 (u, v) = i=1;2 Ω ∇u i • ∇v i dx, ∀u = (u 1 , u 2 ) , v = (v 1 , v 2 ) ∈ V 1 .
Its associated unbouded operator A 1 from

D(A 1 ) = {u = (u 1 , u 2 ) ∈ H 1 0 (Ω) 2 : ∆u 1 , ∆u 2 ∈ L 2 (Ω)} into H is defined by A 1 : D(A 1 ) -→ H : (u 1 , u 2 ) -→ -(∆u 1 , ∆u 2 )
and is therefore a positive selfadjoint operator with a compact resolvent. We further take a 2 = a 1 , and hence A 2 = A 1 .

In addition we define the operators

P : L 2 (Ω) × L 2 (Ω) → L 2 (Ω) × L 2 (Ω) : (u 1 , u 2 ) → Q(u 1 , u 2 ) , and
B : L 2 (Ω) × L 2 (Ω) → L 2 (Ω) × L 2 (Ω) : (u 1 , u 2 ) → ( √ c 1 u 1 , √ c 2 u 2 ) ,
that are clearly bounded from H into itself. Note further that B is clearly selfadjoint, while P is selfadjoint if and only if Q is symmetric almost everywhere in Ω.

In this setting, we simply notice that condition (2.3) reduces to the assumption (5.40) since the first eigenvalue λ 1 of A 1 equals 1 C 2 0 . Therefore 2.1 can be invoked to obtain an existence and uniqueness result for system (5.34)- (5.39).

where H 1 Γ D (Ω) is the closed subspace of H 1 (Ω) made of functions that are zero on Γ D (in the sense of trace), namely H 1 Γ D (Ω) = {v ∈ H 1 (Ω) : γ 0 u = 0 on Γ D }. where γ 0 is the trace operator.

Let us now show that problem (5.59)-(5.64) enters in the abstract framework from section 2. Indeed we first take H = L 2 (Ω), U = L 2 (Γ N ) and V 1 = V 2 = H 1 Γ D (Ω) with their usual inner products and norms. Then we define the sesquilinear, symmetric, continuous and coercive form a 1 by (5.9). Its associated unbounded operator A 1 is defined as in subsection 5.1 and is a positive and selfadjoint operator with a compact resolvent. We further take a 2 = aa 1 , and hence A 2 = aA 1 .

In addition we define the operators In this setting, as in the previous subsections, condition (2.3) reduces to the assumption (5.65) since the first eigenvalue λ 1 of A 1 equals 1 C 2 m . Therefore Theorem 2.1 can be invoked to obtain an existence and uniqueness result for system (5.59)-(5.63).

Strong Stability

In this setting we state two strong stability results that are deduced from section 3. The first one concerns the case when b is a non-zero constant of C for which the application of Theorem 3.3 directly yields a discrete set S of (0, ∞) such that if |b| 2 ∈ S, then system (5.59)-(5.64) is strongly stable. Contrary to the previous subsection, we cannot combine Lemma 3.1 with a unique continuation argument. We then substitute this argument by a multiplier one, see [START_REF] Akil | Indirect stability of a multidimensional coupled wave equations with one locally boundary fractional damping[END_REF]Lemma 2.9] or [4, Theorem 3.2] for a similar argument in the case of a strong coupling. We now divide the proof into several steps:

Step 1. Multiplying (5.80) and (5.81) by ȳ and -ū respectively, integrating in Ω, then taking the real part and using Green's formula, we obtain On the other hand, using Green's formula, the fact that u = 0 on Γ D and ∂ ν u = 0 on Γ N , we obtain Step 3. Multiply (5.80) by -dū, integrating the result in Ω, then using Green's Formula, the fact that u = 0 on Γ and ∂u ∂ν = 0 on Γ N , we obtain By the interpolation inequality (see [START_REF] Grisvard | Elliptic problem in nonsmooth domains[END_REF]Theorem 1.4.3.3])

u H 1 (ω) ≤ ε u H 2 (ω) + Kε -1 u L 2 (ω) ,
valid for all ε > 0 and some K > 0 depending only on ω, we find

|r(u)| ≤ ∆u L 2 (Ω) (1 + ε -1 ) u L 2 (ω) + ε u H 2 (Ω) .
This yields the assumption (4.105) since u V1 = ∆u L 2 (Ω) ∼ u H 2 (Ω) (because the boundary of Ω is smooth enough) and since

u L 2 (ω) √ cu L 2 (Ω) .
Then we can apply Theorem 4.24, and deduce that A satisfies (4.4) with = 14.

(3. 7 )

 7 B ϕ n, = 0. With the above characterization of the eigenvectors of A, for all = 1, • • • , m(n), n ∈ N * , we have B u n, ,+ = B u n, ,-= (B ϕ n, , 0) , which is different from zero due to (3.7). In conclusion under the assumption (3.6), Lemma 3.1 guarantees that the semigroup (e tA ) t≥0 generated A is strongly stable in H. Let us now characterize the assumption (3.6) in terms of b. Fix m, n ∈ N * : m > n, then by the definition (3.4) of µ n,± , (3.8) µ n,+ = µ m,- if and only if

Remark 4 . 6

 46 Theorem 4.5 allows to improve Theorem 4.2 of [10] in the case A 1 = A 2 by taking D = B 2 = B , since there P = αI, with a positive constant α and B satisfies B u H u H , ∀u ∈ H.

  (4.23) Eliminating v 1 and v 2 from (4.20) and (4.22), (4.21) and (4.23) become

  3 and Theorem 4.5 into several lemmas. Let us first show that the main points are to show that λ u 1 H = o(1) and λ u 2 H = o(1). Indeed the next two Lemmas show that these properties imply u 1 V1 = o(1) and u 2 V2 = o(1) respectively. Let us start with the estimate of the norm of u 1 in V 1 . Lemma 4.8 Assume that P ∈ L(H) and that the estimate

( 4 . 28 ) λ 2 u 1 2 H

 4282 = o(1) holds, then the solution (u 1 , v 1 , u 2 , v 2 ) ∈ D(A) of system (4.20)-(4.23) satisfies the following estimate (4.29) u 1 V1 = o(1).

Lemma 4 . 14 u 1 2 H

 4142 Let the assumptions of Theorem 4.7 be satisfied and assume that the estimate (4.54) λ 2 = o(1) holds, then the solution (u 1 , v 1 , u 2 , v 2 ) ∈ D(A) of System (4.20)-(4.23) satisfies the following estimate (4.55) u 1 V1 = o(1).

  (4.20)-(4.23) satisfies the following estimate (4.58) a 1 (u 2 , u 2 ) = o(1).

Lemma 4 . 16

 416 Let the assumptions of Theorem 4.7 be satisfied. Then the solution (u 1 , v 1 , u 2 , v 2 ) ∈ D(A) of system (4.20)-(4.23) satisfy the following estimate

Remark 4 . 23

 423 Theorem 4.22 allows to recover Theorem 2 of [58] by taking M 1 = M 2 = I, r 1 = r 2 = 0, since there P = αI, B = √ εI with two positive constant α and ε. In that case, 1 = 2 = 0 which yields = 4.Let us end with the last stability result.

Theorem 4 . 24

 424 Under all assumptions of Theorem 4.21, except that we replace (4.96) by (4.105) |r(u)|

Lemma 4 . 25

 425 Let the assumptions of Theorem 4.21 be satisfied. Assume that condition (4.5) holds with 0 ≤ 1 < 1, then for ≥ 4 the solution (u 1 , v 1 , u 2 , v 2 ) ∈ D(A) of system (4.20)-(4.23) satisfies the following estimate

(

Lemma 4 . 26

 426 Let the assumptions of Theorem 4.21 be satisfied. Then for ≥ 4 the solution (u 1 , v 1 , u 2 , v 2 ) ∈ D(A) of System (4.20)-(4.23) satisfies the following estimate

(5. 5 ),

 5 c(x) ≥ 0 for almost all x ∈ Ω, while the coupling function b ∈ L ∞ (Ω, C) is different from 0, such that (5.6) | b| | b| a.e. in Ω, where C 0 denotes Poincaré's constant. More precisely, C 0 is the smallest positive constant such that (5.8)

Lemma 5 . 1

 51 Assume that b ∈ L ∞ (Ω) is such that b does not change of sign in Ω, then (3.10) holds with the following choice of D (5.11) D : H → H : u → | b|u, that is clearly bounded and selfadjoint.

Corollary 5 . 2

 52 In addition to the assumptions from this subsection, assume that a = 1, that (5.12) |b(x)| c(x), a.e. in Ω, that b does not change of sign in Ω and that (5.13) ∃ an open set ω b ⊂ Ω such that | b(x)| 1 a.e. in ω b .Then (5.1)-(5.4) is strongly stable.

Theorem 5 . 3

 53 In addition to the assumptions from this subsection, assume that(5.14) ∃ an open set ω ⊂ Ω such that c(x) 1 a.e. in ω, and | b| 1 a.e. in ω.Then iR ⊂ ρ(A) and therefore (5.1)-(5.4) is strongly stable. Proof. In order to prove the result, we use directly Lemma 3.1 to the operator A associated with (5.1)-(5.4). To this end, let λ ∈ R * and U = (u, v, y, z) ∈ D(A), such that (5.15) AU = iλU and satisfying B U = 0. By using the definitions of B and B, we get (5.16) √ cv = 0 in Ω and v ≡ 0 in ω, due to the first condition from (5.14). Next, detailing (5.15), we get

λ 2 y

 2 + a∆y -bu = 0, in Ω, (5.20) u = y = 0, on Γ. (5.21) Using equations (5.18) and (5.19), we deduce that u = y = 0 in ω.

Remark 5 .5 1 .

 51 Using the results from[START_REF] Zuazua | Exponential decay for the semilinear wave equation with locally distributed damping[END_REF][START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF], a non negative function α ∈ L ∞ (ω) satisfies the ESP if there exists an open subset ω of Ω such that(5.26) α(x) 1 a.e. in ω, and if one of the following assumptions holds: a. The boundary of Ω is of class C ∞ and ω satisfies the Geometric Control Condition (GCC)[START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF], namely if there exists T > 0 such that every geodesic traveling at speed one issued from Ω at time t = 0 intersects ω before time T . b. The boundary of Ω is of class C 2 and ω is a neighbourhood of Γ(x 0 ) for some x 0 ∈ R d , where(5.27)Γ(x 0 ) = {x ∈ Γ; m(x) • ν(x) > 0}.where m(x) = (x -x 0 ), and ν(x) is the unit outward normal vector at x ∈ Γ. c. Ω is convex or the boundary of Ω is of class C 1,1 and ω contains G, where G satisfies the condition (g, G) from [45, p. 1583]. d. If Ω is the square (0, L) 2 of R 2 , with L > 0 and (0, a 1 ) × (0, L) ∪ (0, L) × (a 2 , L) ⊂ ω,

Figure 1 :

 1 Figure 1: The square domain with a control domain (in grey) yielding the ESP 2. Similarly, a non negative function α ∈ L ∞ (ω) satisfies the PSP with decay rate t -2/ , for some > 0 specified below, if it satisfies (5.26) and one of the following assumptions holds: a. Ω is the unit square of R 2 and(p, q) × (0, 1) ⊂ ω,

Figure 2 :

 2 Figure 2: A square domain with a control region (a vertical strip in grey) not satisfying the GCC but yielding the PSP.

Figure 3 :Theorem 5 . 6

 356 Figure 3: A partially rectangular domain with a control region yielding the PSP c. For δ > 0 and Ω = (-L, L) d with L > 0, if we take α(x 1 , • • • , x n ) = x 2δ i near x i = 0, positive elsewhere and depending only on x i , for some i ∈ {1, • • • , n}, then = δ δ+1 (that belongs to (0, 1)), using [43, Theorem 1.8] or [28, Theorem 1.2] (and [20, Remark 2.8]), see also [54, Example 6.4]. d. If Ω and ω satisfy the assumptions of [60, §1.1], then such a > 0 exists but is not explicit.

Figure 4 :

 4 Figure 4: Another partially rectangular domain with a control region yielding the PSP.

Figure 5 :

 5 Figure 5: The Bunimovich stadium with a control region yielding the PSP.

a 1 ,

 1 loc (u, v) = Ω η 1 ∇u • ∇v dx, and r(u) = Ω u∇u • ∇η 1 dx.

(5. 67 )

 67 P : L 2 (Ω) → L 2 (Ω) : u → b u, and B : H 1 Γ D (Ω) → L 2 (Γ N ) : u → √ c γ 0 u.

Theorem 5 . 15

 515 In addition to the assumptions from this subsection, assume that a = 1, b is real-valued and does not change of sign, that there exists x 0 ∈ R d such that(5.68) (x -x 0 ) • ν(x) ≤ 0, ∀x ∈ Γ D ,that (5.13) holds, that(5.69) c 1 in Γ N , m x -x 0 ∞ .Then iR ⊂ ρ(A) and therefore (5.59)-(5.64) is strongly stable. Proof. In order to prove the result, we use directly Lemma 3.1 to the operator A associated with (5.59)-(5.64). To this end, let λ ∈ R * and U = (u, v, y, z) ∈ D(A) such that (we recall that 0 ∈ ρ(A), see Remark 3.2) (5.71) AU = iλU and satisfying B U = 0. By using the definitions of B and B and the assumption (5.69), we get (5.72) v = 0 on Γ N , Γ N . Next, detailing (5.71), we get v = iλu, in Ω, (5.74) ∆u -by = iλv, in Ω, (5.75) z = iλy, in Ω, (5.76) a∆y -bu = iλz, in Ω, (5.77) ∂u ∂ν = 0, on Γ N . (5.78) From (5.74) and using (5.72), we get u = 0 on Γ. This implies that ∇u = ( ∂u ∂ν )ν on Γ, and from (5.73), we deduce that (5.79) ∇u = 0 on Γ N and ∇u = ( ∂u ∂ν )ν on Γ D . Eliminating now, v and z from System (5.74)-(5.75), we obtain λ 2 u + ∆u -by = 0, in Ω, (5.80) λ 2 y + ∆y -bu = 0, in Ω, (5.81) ∂u ∂ν = ∂y ∂ν = 0, on Γ N , (5.82) y = 0, on Γ D (5.83) u = 0, on Γ. (5.84)

  dx + Ω b(x)|u| 2 dx = 0. Adding the above two equations, we obtain Ω b(x)|u| 2 dx = Ω b(x)|y| 2 dx, and since b does not change of sign, we get (5.85) Ω |b(x)||u| 2 dx = Ω |b(x)||y| 2 dx.

Step 2 . 2 Ωu 2 Ω 2 Ω 2 Ω 2 Ω 2 Ω 2 (

 22222222 Multiply (5.80) by 2(m • ∇ū), when m(x) := x -x 0 , and integrating in Ω, we get(5.86) 2λ (m• ∇u) dx + ∆u (m • ∇u) dx -2 Ω b(x) y (m • ∇u) dx = 0.Using Green's formula for the second term of this left-hand side, we get (5.87)∆u (m • ∇u) dx = -∇u • ∇ (m • ∇u) dx + 2 Γ ∂u ∂ν (m • ∇u) dΓ .Using Green's formula and (5.79), we get-∇u • ∇(m • ∇u) dx = (d -2) Ω |∇u| 2 dx -Γ (m • ν)|∇u| 2 dΓ = (d -2) Ω |∇u| 2 dx -Inserting this identity in (5.87), and using again equation (5.79), we get (5.88) ∆u(m • ∇u) dx = (d -2) m • ν) dΓ.

(5.89) 2λ 2 Ωu 2 Ωm

 22 (m • ∇u) dx = λ • ∇|u| 2 dx = -dλ 2 Ω |u| 2 dx. Inserting the identities (5.88) and (5.89) in (5.86), we obtain (5.90) d Ω |λu| 2 dx+(2-d) ) y (m • ∇u) dx = 0.

2 Ω

 2 y (du + 2(m • ∇u)) dx . The boundary geometric condition (5.68) then yields (5.92) |∇u| 2 dx ≤ Ω b(x)y (du + 2(m • ∇u)) dx . Theorem 5.26 Let a = 1. In addition to the assumptions from this section, assume that b does not change of sign in Ω, that (5.14) holds. Assume that there exists an open set ω of Ω satisfying (5.30) with the open set ω from (5.14), and that c and I ω satisfy the ESPP. Then for all initial data U 0 ∈ D(A) the energy of the system (5.120)-(5.124) satisfies the following estimate decay rate: ) , ∀t > 0.Proof. First, as by assumption (5.14) holds, by Theorem 5.3, iR ⊂ ρ(A). The assumption on c guarantees that system (3.1), that here takes the form (5.135) with α = c and a = 1, is exponentially stable. Similarly, taking B 2 = I ω , system (4.6) is nothing else than (5.135) with α = I ω , therefore by our assumption on I ω , it is exponentially stable. Now, as in the proof of Theorem 5.9, we define M 1 = √ η 1 I and M 2 = √ η 2 I, where η 1 , η 2 ∈ C ∞ c (ω) (extended by zero outside ω) are fixed cut-off functions such that 0 ≤ η i (x) ≤ 1, i = 1, 2, and satisfying(5.31). From the definition of η 1 and η 2 conditions (4.92), (4.93), (4.97) and (4.98) are satisfied. As for any v ∈ H 2 0 (Ω), and i = 1 or, 2, L i u = η i u, for all u ∈ H 2 0 (Ω), we then havea 1 (u, L i v) = Ω ∆u∆(η i v) dx = Ω η i ∆u∆vdx + Ω ∆u(∆η i v + 2∇η i • ∇v) dx.This yields (4.94) witha 1,loc (u, v) = Ω η 1 ∆u∆v dx, r(u) = Ω ∆u(∆η 1 u + 2∇η 1 • ∇u) dx, ∀u, v ∈ H 2 0 (Ω).Furthermore Cauchy-Schwarz's inequality yields|a 1 (u, L 2 v)| ∆u L 2 (supp η2) v H 2 0 (Ω). As the assumption (5.31) yields ∆u 2 L 2 (supp η2) ≤ a 1,loc (u, u), this proves (4.95). Again by Cauchy-Schwarz inequality, we have |r(u)| ≤ ∆u L 2 (Ω) u L 2 (ω) + u H 1 (ω) .

  Lemma 4.10 Assume that P ∈ L(H) and that condition (4.5) holds with 0 ≤ 1 < 1, then the solution (u 1 , v 1 , u 2 , v 2 ) ∈ D(A) of system (4.20)-(4.23) satisfy the following estimate

	(4.32)

properties (4.19), (4.26) and (4.30) then lead to

(4.31)

. We now concentrate on the behavior of λ u 1 H .

  combined with (4.18), (4.19), (4.26), and (4.27) yields (4.39) with

B = 1. Now if B is a bounded operator from H into itself, then the estimate (4.41) is replaced by

  estimations (4.26) and (4.36) and the property (4.19) yield (4.51). The proof is thus complete as soon as -1 2 -4 + 1 ≤ 0 and 1 -+ 1 ≤ 0. Since these conditions are equivalent to Now, we consider (ũ 2 , ṽ2 ) ∈ D(A 2 ) solution of (4.76)-(4.77) that satisfies (4.47). As the identity (4.48) remains valid, using Cauchy-Schwarz inequality and the boundedness property of B 2 , we

	find
	u 2	2 H
		≥ 2(2 1 -1) and ≥ 1 + 1,
	we arrive to the conclusion.
	Proof. of Theorem 4.5 The proof follows the lines of the proof of Theorem 4.3. First by the
	assumption (4.8) and (4.50) we have
	(4.53)	λ B 2 u 2 H = o λ	1 2 -4 .

  estimations (4.26), (4.69) and (4.71) and the property (4.19) yield (4.65). The proof is thus complete as soon as -4 + 1

	2 -1 4 ≤ 0, 1 -+ 1 ≤ 0, and -1 2 ≤ 0. Since we always have
	1 2 ≤ 1 + 1, these conditions hold under the assumption (4.66).
	Proof. of Theorem 4.7 First by the assumption (4.14) and (4.59) we have
	(4.75)

  3, by considering (ũ 2 , ṽ2 ) ∈ D(A 2 ) solution of (4.76) that satisfies (4.45) and (4.47) (due to the assumption (4.7)). As in that proof, taking the inner product in H of (4.46) with u 2 , the estimate (4.49) remains valid, namely

  [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF]) holds, by Theorem 5.3, iR ⊂ ρ(A). The assumption on c guarantees that system (3.1), that here takes the form (5.22) with β = c, is exponentially stable. Similarly, for the choice B 2 = B 2 = I ω , system (4.6), is nothing else than (5.24) with α = I ω , therefore by our assumption on I ω , it is exponentially stable. Now, letM 1 = √ η 1 I and M 2 = √ η 2 I, where η 1 , η 2 ∈ C ∞ c (ω) (extended by zero outside ω) are fixed cut-off functions such that 0 ≤ η i (x) ≤ 1, i = 1,2, and (5.31) η 2 = 1 in ω, and η 1 = 1 in supp η 2 . This choice of η 1 and η 2 directly guarantees that conditions (4.92), (4.93), (4.97) and (4.98) are satisfied. Now as L 1 u = η 1 u, by Leibniz's rule, we have

D(A) , ∀t > 0, U 0 ∈ D(A).

As before Cauchy-Schwarz's inequality and the assumption (5.31) yield (5.33) |r 1 (u, v)| ∇u L 2 (supp η2) v L 2 (supp η2) a 1,loc (u, u)

, ∀u, v ∈ H 1 0 (Ω), which proves (4.102). Finally as a 2 = aa 1 , (4.101) holds with r 2 (u, v) = ar 1 (v, u). Hence the first estimate in (5.33) yields

(Ω), due to (5.14). This shows that (4.103) holds.

In conclusion as all hypotheses of Theorem 4.22 with 1 = 2 = 0 hold, we deduce that A satisfies (4.4) with = 4.

As a final conclusion, we may notice that Remark 5.5 furnishes explicit examples of functions c and | b| satisfying ESP or PSP, hence leading to explicit decay rates for the solution of our system (5.1)-(5.3) (under the additional assumptions stated in our different theorems). Let us mention two examples not covered in the literature. The first one is a consequence of Remark 5.5.2a and Theorem 5.8, while the second one is a consequence of Remark 5.5.2c and Theorem 5.7.

Example 5.10 Let Ω = (0, 1) × (0, 1) and assume that c and |b| satisfies (5.12) and (5.13). If furthermore ω b ⊃ (p, q) × (0, 1), for some 0 ≤ p < q < 1, then

Example 5.11 Let Ω = (-L, L) d with L > 0 and assume that c(x 1 , • • • , x n ) = x 2δ1 i near x i = 0, positive elsewhere and depending only on x i , for some i ∈ {1, • • • , n} and δ c > 0, while b(x 1 , • • • , x n ) = x 2δ2 j near x j = 0, positive elsewhere and depending only on x j , for some j ∈ {1, • • • , n}, and δ 2 > 0, then we can apply Theorem 5.7 with i = δi δi+1 as soon as δ 1 and δ 2 are chosen such that (4.9) holds. In such a case, (5.29) holds with given by (4.13).

Note finally that Theorem 4.19 could be used to obtain some polynomial decay rate of a system similar to (5.1)- (5.4), where we replace the Dirichlet boundary condition on u 1 by mixed boundary conditions, see subsection 5.4 for such a setting.

Internal stabilization of four locally coupled wave equations

Here we consider the following system of four coupled wave equations in a bounded domain Ω of R d , with d ≥ 1, with a Lipschitz boundary Γ: u 1,tt -∆u 1 + c 1 u 1,t + p 13 u 3 + p 14 u 4 = 0,

in Ω × (0, ∞), (5.34) u 2,tt -∆u 2 + c 2 u 2,t + p 23 u 3 + p 24 u 4 = 0,

in Ω × (0, ∞), (5.35) u 3,tt -∆u 3 + p 13 u 1 + p 23 u 2 = 0, in Ω × (0, ∞), (5.36) u 4,tt -∆u 4 + p 14 u 1 + p 24 u 2 = 0, in Ω × (0, ∞), (5.37)

on Γ × (0, ∞), (5.38)

Strong stability

Let us formulate a strong stability result similar to Theorem 5.3. Theorem 5.12 In addition to the assumptions from this subsection, assume that (5.41) ∃ an open set ω ⊂ Ω such that c i 1 a.e. in ω, i = 1, 2, and Q is invertible a.e. in ω.

Then iR ⊂ ρ(A) and therefore (5.34)-(5.39) is strongly stable.

Proof. We use directly Lemma 3.1 to the operator A associated with (5.34)- (5.39). To this end, (5.15) holds and satisfying B U = 0. By using the definitions of B and B, we get

due to the first condition from (5.41). Next, detailing (5.15) and eliminating v j = iλu j , j = 1, 2, 3, 4, we get

-λ 2 u 4 -∆u 4 + p 14 u 1 + p 24 u 2 = 0, in Ω, (5.46) u j = 0, on Γ, j = 1, 2, 3, 4.

(5.47) and using the second property from (5.42) (5.48)

Using this property in the first two identities above, we deduce that

Using the second condition from (5.41), we deduce that

which, by (5.48) leads to

Since (5.43)-(5.46) can be written as

where M is a 4 × 4 matrix-valued function, we can use Calderón theorem (see [START_REF] Hayek | A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface[END_REF]Theorem 2.5]), to deduce that u j = 0 in Ω, for j = 1, 2, 3, 4. Hence U = 0 and the conclusion then follows from Lemma 3.1.

Polynomial stability

To get polynomial stability results, we assume that Q satisfies the following assumption

where m ∈ L ∞ (Ω) is a non negative function.

This assumption holds if Q is symmetric and positive definitive almost everywhere in Ω, but it also holds for other situations like (5.50)

The assumption (5.49) directly implies that (3.10) holds, with D defined by (5.51)

which is one of the key assumptions to apply Theorem 4.3 or Theorem 4.5. Furthermore the single damped equation (3.1) takes here the form

with the initial conditions (5.55)

that corresponds to two decoupled wave equations in u 1 and u 2 . Hence decay rate results for the single wave equation with internal dampings from the literature can be used to obtain polynomial decay rates for our system (5.34)-(5.39).

Theorem 5.13 In addition to the assumptions from this subsection, assume that (5.41) and (5.49) holds, that c 1 , c 2 and m satisfy the ESP. Then for all initial data U 0 ∈ D(A) the energy of the system (5.34)-(5.39) satisfies the following decay rate estimate

Proof. First, as by assumption (5.41) holds, by Theorem 5.12, iR ⊂ ρ(A). Second, as said above, system (3. 1 = 2 = 0, that implies that A satisfies (4.4) with = 2 (because here B = 0) and yields (5.56).

Again as before to remove the condition (4.9), we may apply Theorem 4.5, that in our case yields the following result.

Theorem 5.14 In addition to the assumptions from this subsection, assume that (5.41) and (5.49) hold, that c 1 and c 2 satisfy the PSP with decay rate t -2/ 1 , for some 1 > 0 and that m satisfies the PSP with decay rate t -2/ 2 , for some 2 > 0. Assume further that Then (5.29) holds with given by (4.13).

Proof. We only need to check (4.12) that is a direct consequence of our assumptions (5.57) and (5.58).

Boundary stabilization of coupled wave equations with compatible boundary conditions

In this subsection, we are interested in the boundary stabilization of multidimensional coupled wave equations with compatible boundary conditions. To the best of our knowledge, this problem was not studied in the literature.

Let Ω be a bounded domain of R d , with d ≥ 1, with a Lipschitz boundary Γ = Γ D ∪ Γ N , with Γ D and Γ N non empty open subsets of Γ such that Γ D ∩ Γ N = ∅. In this setting, we consider the following multidimensional coupled wave equations

in Ω × (0, ∞), (5.59)

in Ω × (0, ∞), (5.60)

on Γ D × (0, ∞), (5.61)

with the following initial conditions

where a is a positive real number and the damping function c ∈ L ∞ (Γ N ) is real-valued and satisfies (5.5), while the coupling function b ∈ L ∞ (Ω, C) is different from 0 and satisfies (compare with (5.7))

where C m is a sort of Poincaré's constant. More precisely, C m is the smallest positive constant such that (5.66)

Using Cauchy-Schwarz to this right-hand side, we get

.

For the above estimate, using (5.85) and Poincaré inequality (5.66), we obtain

Inserting this estimate in (5.92), we get

Under the hypothesis (5.70), we then obtain By Calderón theorem (see [START_REF] Hayek | A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface[END_REF]Theorem 2.5]) we conclude that u = y = 0 in Ω and thus U = 0. Hence, ker (iλI -A) = 0 and the proof is thus complete.

Polynomial Stability

In this subsection we look for a polynomial decay rate of our system by applying Theorem 4.3. Again, we use the exponential (or polynomial) stability of the wave equation in Ω with a boundary damping or with an internal damping with mixed boundary conditions. More precisely, we say that a non negative function α ∈ L ∞ (Γ N ) satisfies the boundary exponentially stability property (in short BESP) if the system (5.93)

is exponentially stable. The sole polynomial stability results of this system are due to Nishiyama [START_REF] Nishiyama | Boundary stabilization of the waves in partially rectangular domains[END_REF] and to Phung [START_REF] Phung | Boundary stabilization for the wave equation in a bounded cylindrical domain[END_REF]. In [START_REF] Nishiyama | Boundary stabilization of the waves in partially rectangular domains[END_REF] a decay rate t -1/2 of the energy for partially rectangular domain of R 2 and Γ D = ∅ is obtained. Since this decay rate corresponds to the case 1 = 4 for the behavior (4.5) of the resolvent of the corresponding system (3.1), we cannot apply Theorem 4.3 in such a situation (neither the other Theorems from section 4). On the contrary in [START_REF] Phung | Boundary stabilization for the wave equation in a bounded cylindrical domain[END_REF], a decay rate t -δ of the energy for bounded cylindrical domain of R 3 is obtained with a positive real number δ which is not specified. We therefore restrict ourselves to the notion of BESP.

In the same manner, we say that a non negative function β ∈ L ∞ (Ω satisfies the exponentially stability property with mixed boundary condition (in short ESPM) if the system (5.94)

and if one of the following assumptions holds: a. The boundary of Ω is of class C ∞ and ω N satisfies the Geometric Control Condition (GCC) [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF], namely if there exists T > 0 such that every geodesic traveling at speed one issued from Ω at time t = 0 intersects ω N before time T . b. The boundary of Ω is of class C 2 , ω N = Γ N and there exists a vector field h ∈ C 2 ( Ω) d such that its symmetric gradient ∇h + (∇h) is uniformly positive definite in Ω and

(5.96) h(x) • ν(x) > 0 on Γ N , and h(x) • ν(x) ≤ 0 on Γ D , see [START_REF] Lagnese | Decay of solutions of wave equations in a bounded region with boundary dissipation[END_REF][START_REF] Lasiecka | Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions[END_REF] in the general situation and [START_REF] Komornik | A direct method for the boundary stabilization of the wave equation[END_REF] for α(x) = h(x) • ν(x), where h(x) = (x -x 0 ) for some

Similarly, using [55, Proposition 4.2] and [34, Propositions 1 and 2], a non negative function β ∈ L ∞ (Ω satsifies the ESPM if the boundary of Ω is of class C 2 , there exists a vector field h ∈ C 2 Ω) d such that its symmetric gradient ∇h + (∇h) is uniformly positive definite in Ω and h(x) • ν(x) ≤ 0 on Γ D , and (5.26) holds for a neighborhood ω of Γ N , Theorem 5.17 In addition to the assumptions from this subsection, assume that a = 1, that b does not change of sign in Ω, that (5.13) holds, that system (5.59)-(5.64) is strongly stable, c satisfies the BESP and | b| satisfies the ESPM. Then for all initial data U 0 ∈ D(A) the energy of the system (5.59)-(5.64) satisfies the following decay rate estimate

Proof. By assumption we have iR ⊂ ρ(A). Now as b does not change of sign, by Lemma 5.1, (3.10) holds with D defined by (5.11). Taking B 2 = D, system (4.6) is nothing else than (5.94) with β = | b|, therefore by our assumption on b, it is exponentially stable. Similarly, system (3.1), that here takes the form (5.93) with α = c is exponentially stable. Therefore we can apply Theorem 4.3 with 1 = 2 = 0, that implies that A satisfies (4.4) with = 4 (because here B = 1) and yields (5.97).

Boundary stabilization of coupled wave equations with incompatible boundary conditions

In this subsection, we are interested in the boundary stabilization of multidimensional coupled wave equations with incompatible boundary conditions. In the same setting than in the previous section, we consider the system

in Ω × (0, ∞), (5.99)

with the initial conditions (5.64).

In comparison with the system (5.59)-(5.63), we note that a = 1 and that u 2 satisfies the Dirichlet boundary condition on the whole boundary.

Here the damping function c ∈ L ∞ (Γ N ) is real-valued and satisfies (5.5), while the coupling function b ∈ L ∞ (Ω, C) is different from 0 and satisfies (compare with (5.7) and (5.65))

where C 0 (resp. C m ) is the Poincaré's constant from (5.8) (resp. (5.66)). Problem (5.98)-(5.102) was analyzed in [START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF][START_REF] Alabau-Boussouira | Indirect boundary stabilization of weakly coupled hyperbolic systems[END_REF] usng the multiplier method under the multiplier geometrical assumption (5.96) with h(x) = (x -x 0 ) for some x 0 ∈ R d and a specific choice of the function c. She obtains polynomial decay rate of the energy in the case a = 1, with b ∈ R small enough (with a non explicit bound) and in the case a = 1, when Ω a hypercube of R d , with d ≤ 3, Γ D = ∅, and a non explicit coupling operator. In [START_REF] Liu | Frequency domain approach for the polynomial stability of a system of partially damped wave equations[END_REF] the authors also studied problem (5.98)-(5.102) under the above multiplier geometrical assumption on Γ D and Γ N , and a positive constant c; in the case a = 1, with b ∈ R small enough the multiplier method allows to obtain some polynomial decay rate, while in the case a = 1, it is shown that the decay rate depends on the arithmetic property of a.

Let us now show that problem (5.98)-(5.102) with initial conditions (5.64) enters in the abstract framework from section 2. Indeed we first take

with their usual inner products and norms. Then we define the sesquilinear, symmetric, continuous and coercive form a 1 by (5.9). Its associated unbounded operator A 1 is defined as in subsection 5.1 and is a positive and selfadjoint operator with a compact resolvent. Finally a 2 = a 1 but restricted to H 1 0 (Ω), namely

Its associated unbounded operator A 2 is defined as in subsection 5.1 and is a positive and selfadjoint operator with a compact resolvent.

In addition we define the operators P and B by (5.67).

In this setting, as in the previous subsections, condition (2.3) reduces to the assumption (5.103) since the first eigenvalue

Therefore Theorem 2.1 can be invoked to obtain an existence and uniqueness result for system (5.98)-(5.102) with initial conditions (5.64).

Strong Stability

For shortness we state one strong stability result that corresponds to Theorem 5.15. The proof follows mutatis mutandis the one of Theorem 5.15 and is left to the reader.

Theorem 5.18 In addition to the assumptions from this subsection, assume b is real-valued and does not change of sign, that (5.68) holds for some x 0 ∈ R d , that (5.13) holds, that (5.69) holds, and that

Then iR ⊂ ρ(A) and therefore (5.98)-(5.102) with initial conditions (5.64) is strongly stable.

Polynomial Stability

In this subsection we obtain a polynomial decay rate of our system by applying Theorems 4.18.

Theorem 5. [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF] In addition to the assumptions from this subsection, assume that b does not change of sign in Ω, that (5.13) holds, that (5.69) holds, that system (5.59)-(5.64) is strongly stable, c satisfies the BESP and | b| satisfies the ESP. Then for all initial data U 0 ∈ D(A) the energy of the system (5.98)-(5.102) with initial conditions (5.64) satisfies the decay rate estimate (5.97).

Proof. The proof is exactly the same as the one of Theorem 5.17 applying Theorem 4.18 instead of Theorem 4.3, once we have checked that (4.88). But in our situation, for w ∈ D(A 2 ) and v ∈ H 1 Γ D (Ω), we have

Hence by Green's formula (see [32,

Due to Poincaré inequality, this yields (5.106) ∂w ∂ν

With this regularity in hands, the duality pairing in the right-hand side of (5.105) can be expressed as a L 2 (Γ) inner product, namely the identity (5.105) may be written

recalling that v = 0 on Γ D . Applying Cauchy-Schwarz's inequality and using (5.106), we get

Since our assumption (5.69) implies that

we deduce that (4.88) holds.

Boundary stabilization of wave equations coupled and damped via the boundary

Let Ω be a bounded domain of R d , with d ≥ 1, with a Lipschitz boundary Γ = Γ D ∪ Γ N , with Γ D and Γ N non empty open subsets of Γ such that Γ D ∩ Γ N = ∅. In this setting, we here consider

in Ω × (0, ∞), (5.108)

on Γ N × (0, ∞), (5.111) with the following initial conditions (5.112)

where the damping function c ∈ L ∞ (Γ N ) is real-valued and satisfies c(x) ≥ 0, for almost all x ∈ Γ N , while the coupling function b ∈ L ∞ (Γ N ) is non negative, different from 0 and satisfies

where C b is the smallest positive constant such that (5.114)

To the best of our knowledge, problem (5.59)-(5.63) was not studied in the literature. Problem (5.59)-(5.63) enters in the abstract framework from section 2, by taking H, V 1 = V 2 and a 1 and a 2 as in the previous subsection 5.3 with a = 1, while here U = L 2 (Γ N ). In addition we define the operators B as in (5.67) and take P = DD with

As (5.113) implies that (2.3) holds, Theorem 2.1 allows to to obtain an existence and uniqueness result for system (5.107)- (5.112).

A stability result for system (5.107)-( 5.112) can be deduced from Theorem 3.4.

Theorem 5.20 In addition to the assumptions from this subsection, assume that Then (5.107)-( 5.112) is strongly stable.

Proof. We simply need to check that the assumptions of Theorem 3.4 are satisfied. First we notice that by definition of P , (3.12) trivially holds with U p = L 2 (Γ N ). Further (5.115) guarantees that (3.11) holds. Finally the choice of D implies that (3.13) is nothing else that the wave equation with a viscosity damping with coefficient b, namely (5.117)

that, due to (5.116), is strongly stable in H 1 Γ N (Ω) × L 2 (Ω) by using the unique continuation principle.

Polynomial Stability

This subsection is devoted to the proof of a polynomial energy decay rate with the help of Theorem 4.7.

Before going on, let us recall the following non standard trace inequality, whose proof can be found in the proof of Theorem 1.5.1.10 of [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]. Lemma 5.21 For all u ∈ H 1 (Ω), we have

Theorem 5.22 In addition to the assumption from this subsection. Assume that (5.115) and (5.116) hold, that c and b satisfies the BESP. Then for all initial data U 0 ∈ D(A) the energy of the system (5.107)-(5.112) satisfies the following estimate decay rate

Proof. First, as by assumption, (5.115) and (5. With the choice B 2 = D, system (4.6) is nothing else than (5.93) with α = b, therefore by our assumption on b, it is exponentially stable. Similarly the assumption on c guarantees that system (3.1), that here takes the form (5.93) with α = c is exponentially stable. Furthermore, using Lemma 5.21 we guarantee that (5.113) holds. Therefore we can apply Theorem 4.7 with 1 = 2 = 0, that implies that A satisfies (4.4) with = 3 and yields (5.119).

Internal stabilization of locally coupled plate equations

Let Ω be a bounded domain of R d , with d ≥ 1, with a boundary Γ of calss C 4 . In this domain, we study the internal stabilization of the next coupled plate equations with clamped boundary conditions.

in Ω × (0, ∞), (5.121)

with the following initial conditions (5.124)

where a is a positive real constant, the damping function

where C 1 is the smallest constant such that (5.126)

Problem (5.120)-(5.124) was analyzed in [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF] in the case b ∈ R satisfying (5.125) and a positive constant c, where a decay rate t -1/4 of the semigroup is obtained (that is conformed with Remark 4.23), while in [START_REF] Alabau-Boussouira | Indirect stabilization of locally coupled wave-type systems[END_REF] a decay rate t -1/2 of the semigroup is obtained if the coupling and damping regions satisfy the PMGC and for b small enough (with no specific bound).

Again problem (5.120)-(5.124) enters in the framework of section 2 with the following choices. We take

with their usual inner products and norms. Then we define the sesquilinear, symmetric and coercive form a 1 by

Its associated unbounded operator A 1 from

and is therefore a positive selfadjoint operator with a compact resolvent. We further take a 2 = aa 1 , and hence A 2 = aA 1 .

In addition we define the operators (5.127)

that are clearly bounded from H into itself.

In this setting, we simply notice that condition (2.3) reduces to the assumption (5.125) since the first eigenvalue λ 1 of A 1 equals 1

. Therefore Theorem 2.1 can be used to obtain an existence and uniqueness result for system (5.120)-(5.124).

Strong stability

Stability results for system (5.120)-(5.124) can be deduced from the results from section 3. Let us present one result based on the specificity of the considered problem.

Theorem 5.23 In addition to the assumptions from this subsection, assume that (5.14) holds. Then iR ⊂ ρ(A) and therefore (5.120)-(5.124) is strongly stable.

Proof. In order to prove the result, we use directly Lemma 3.1 to the operator A associated with (5.1)-(5.4). To this end, let λ ∈ R * and U = (u, v, y, z) ∈ D(A), such that

From the first equation in (5.130) and the second property from (5.129), we deduce (5.131) u ≡ 0 in ω.

Eliminating v and z from (5.130), and using the first property from (5.129), we obtain the following system Since u = y = ũ = ỹ = 0 in ω, we can use the Calderón theorem (see [START_REF] Hayek | A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface[END_REF]Theorem 2.5]) to deduce that u = y = 0 in Ω and thus, by the first and third identity of (5.145), v = z = 0, hence U = 0.

The conclusion follows from Lemma 3.1.

Polynomial stability

Here we want to prove some polynomial energy decay rates of System (5.120)-(5.124) by applying Theorem 4.3 or Theorem 4.22. According to our general approach, one of the main ingredient is the exponential (or polynomial stability) of the plate equation with internal damping. More precisely, we say that a non negative function α ∈ L ∞ (ω) satisfies the exponentially stability property for the plate system (in short ESPP) if the system (5.135)

u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in Ω, is exponentially stable. We do not introduce the notion of polynomial stability because no references seem to exist on the subject. We can only mention the recent paper [START_REF] Rousseau | Spectral inequality and resolvent estimate for the bi-Laplace operator[END_REF] that proves some logarithmic decay. (5.136)

Proof. First, as by assumption (5. In that case some exponential stability result for the associated single damped equation

are available in [START_REF] Jaffard | Contrôle interne exact des vibrations d'une plaque rectangulaire[END_REF] (using [34, Propositions 1 and 2]) and [START_REF] Ramdani | Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems[END_REF] for rectangular domains.

Internal stabilization of coupled wave equation and plate equation

Let Ω be a bounded domain of R d , with d ≥ 1, with a boundary Γ of calss C 4 . On this domain, we consider the following system

with the following initial conditions (5.141)

where a is a positive real constant, the damping function c ∈ L ∞ (Ω) is real-valued such that c(x) ≥ 0 for almost all x ∈ Ω, while the coupling function b ∈ L ∞ (Ω, C) is different from 0 and satisfies

where C 0 is the Poincaré constant and C 1 is the positive constant from (5.126). Problem (5.139)-(5.141) with the boundary conditions

instead of (5.140), was analyzed in [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF][START_REF] Alabau-Boussouira | Indirect stabilization of weakly coupled systems with hybrid boundary conditions[END_REF] in the case b ∈ R small enough and a positive constant c, where a decay rate t -1/8 of the semigroup is obtained. The boundary conditions (5.140) are not covered in [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF][START_REF] Alabau-Boussouira | Indirect stabilization of weakly coupled systems with hybrid boundary conditions[END_REF] (see assumption (5) from [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF] or assumption [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF] from [START_REF] Alabau-Boussouira | Indirect stabilization of weakly coupled systems with hybrid boundary conditions[END_REF]), therefore, up to now, the polynomial stability property of (5.139)- (5.141) was an open problem.

As expected, problem (5.139)-(5.141) enters in the abstract framework from section 2. Indeed we first take

with their usual inner products and norms. Then we define the two sesquilinear, symmetric, continuous and coercive form, a 1 by

and a 2 by

that we have already used before. Consequently their associated unbounded operators A 1 and A 2 are positive and selfadjoint operators with a compact resolvent. Finally we define P and B as in (5.10).

In this setting, we simply notice that condition (2.3) reduces to the assumption (5.142) since the first eigenvalue λ 1 of A 1 equals 1 C 2 0 and that of A 2 is equal 1

. Therefore Theorem 2.1 can be invoked to obtain an existence and uniqueness result for system (5.138)-(5.141).

Strong stability

Let us give a strong stability result for our system (5.138)-(5.141).

Theorem 5.28 In addition to the assumptions from this subsection, assume that (5.14) holds. Then iR ⊂ ρ(A) and therefore (5.138)-(5.141) is strongly stable. The assumption on c guarantees that system (3.1), that here takes the form (5.22) with β = c, is exponentially stable.

Similarly system (4.6) is nothing else than (5.135) with α = I ω , therefore by our assumption on I ω , it is exponentially stable. The remaining assumptions from Theorem 4.21 were satisfied in the proof of Theorem 5.9 (with M i defined as in this proof) because the sesquilinear form a 1 and the operator B are the same and H 2 0 (Ω) is continuously embedded into H 1 0 (Ω). From Theorem 4.21, we conclude that A satisfies (4.4) with = 6.