Alaa Hayek

Mots Clés

Patrick Rothfuss

Keywords: Exponential stability, analytic semigroup, polynomial stability, Kelvin-Voigt, thermo-elastic, telegraph, networks, improved Kirchhoff law, dynamic boundary condition. iii Stabilité exponentielle, semi-groupe analytique, stabilité polynomiale, Kelvin-Voigt

Solving a problem for which you know there is an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it's not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley. "

YOKO OGAWA.

Stabilization of some coupled systems involving (thermo-)viscoelastic/elastic transmission problems or telegraph equations in bounded domains or in networks by

Introduction

Mathematical models play an important role in designing, understanding and analyzing many physical systems. For instance, the mathematical modeling of the vibration of flexible structures, the regulation of blood flow in cardiovascular networks, the flow of gas in pipes and electromagnetism was set up over years.

Recently, modeling has progressed tremendously which provoked the mathematicians' curiosity in the stabilization of simple and coupled systems of partial differential equations arising in several mathematical models.

In general, the stability of a coupled system of partial differential equations is influenced by several properties. Among these properties we have: the nature of the partial differential equations, the set of boundary conditions, the type of the damping, the kind of the medium and the type of the coupling. In this PhD thesis we study the stabilization of three coupled systems that arise in three different mathematical models. What characterizes the problems studied in this thesis is the diversity in the properties of the considered coupled systems:

• Partial differential equations: wave equation, heat equation, telegraph equation.

• Boundary conditions: Dirichlet, Neumann, standard Kirchhoff condition, dynamic boundary conditions.

• Dissipation mechanisms: visco-elastic damping, thermo-elastic damping, boundary damping.

• Mediums: bounded domains, networks.

• Coupling mechanisms: by displacement, by velocity, other..

In what follows, we will start by a general introduction in which we introduce briefly some of the properties of the coupled systems of partial differential equations investigated in this thesis. To do this, first we begin in section 0.1 by introducing the wave equation used in Chapter 1 (in multi-dimentional space) and the generalized telegraph equation considered in Chapter 2. Next, in section 0.2, we recall the heat equation and explain briefly the thermo-mechanical process leading us to the thermo-elastic system encountered in Chapter 3. Then, due to the fact that the damping mechanism is the soul of the stabilization process of systems of partial differential equations, we focus in section 0.3 on three types of dampings that deals with this thesis. Finally, in section 0.4, we present some types of boundary conditions used in Chapter 2 and Chapter 3.

Interested readers can also go directly to the main introduction in which we present the main results obtained in this thesis.

I) GENERAL INTRODUCTION

Partial differential equations play a relevant role in mathematical physics, mainly those of the second order. These equations are useful to describe a diversity of physical phenomena that include wavelike propagation, diffusion, and transport processes in practically all branches of physics. Namely, in continuous and classical mechanics, it is common to face problems of either the hyperbolic (vibrating strings, stretched membranes) or the parabolic (heat conduction) types.

Vibrations and waves

Vibrations and waves are extremely important phenomena in physics. In nature, oscillations are found everywhere. From the jiggling of atoms to the large oscillations of sea waves. Wave is a flow or transfer of energy in the form of oscillation through a medium -space or mass. In Mathematics, the wave equation is a second order partial differential equation which describes the propagation of oscillations with a constant velocity in some quantity f = f (x 1 , • • • , x n , t) of time variable t and spacial variables (x 1 , • • • , x n ). The wave equation is given by

∂ 2 f ∂ 2 t = s 2 ∆ f , (0.1.1)
where s is the velocity of the wave. The quantity f may be, for example, the pressure in a liquid or gas, or the displacement, along some specific direction, of the particles of a vibrating solid away from their resting positions.

There are different types of waves including electromagnetic waves and mechanical waves.

Mechanical waves:

• A mechanical wave is an oscillation of matter and therefore a transfer of the energy through a medium.

• This kind of waves needs a medium in order to propagate itself.

• Sound waves, water waves, stadium waves, and jump rope waves are examples of mechanical waves; each requires some medium in order to exist. For example, the one-dimensional wave equation of a vibrating string with tension T and mass density m, is

∂ 2 u ∂ 2 t = T m ∂ 2 u ∂ 2 x , (0.1.2)
where u is the displacement along the direction of the particles of the vibrating string away from their resting positions and T m is the speed of the propagation of the wave.

Electromagnetic waves:

• Electromagnetic waves are created by a fusion of electric and magnetic fields. The light you see, the colors around you are visible because of electromagnetic waves.

• Unlike mechanical waves, electromagnetic waves do not need a medium to travel.

• Microwaves, X-ray, radio waves are all examples of waves that are capable to propagate in vacuum.

The mostly used equation which models the propagation of electromagnetic waves is the telegraph equation.

Telegraph equation

In many industrial applications, like microwave communication systems, transmission lines (including coaxial cables) are used to transmit signals from one point to another. In fact, these signals are transmitted along cables by electromagnetic waves which consist of variations in the magnitude and direction of the electric and magnetic fields produced by the line voltages and currents. The electromagnetic fields are the solution of the 3D Maxwell's equation. But, since a cable is a thin structure whose transverse dimensions are much smaller than the longitudinal one, one would like to use a simplified 1D model because 3D Maxwell's equations can be very complicated to solve: complex geometry due to defaults, 3D mesh for the thin cable... In such a situation, electrical engineers use the well-known telegraph equation for perfect coaxial cables (homogeneous with circular cross-section), where the electric unknowns are reduced to an electric potential V(x, t) and an electric current I(x, t), where x denotes the abscissa along the cable, t is time and in absence of source (0.1.3) where the capacitance C, the inductance L, the conductance G and the resistance R can be expressed it terms of the geometry of the cross-section.

         C ∂V ∂t + GV - ∂I ∂x = 0, L ∂I ∂t + RI - ∂V ∂x = 0,
∂ 2 V ∂ 2 x = LC ∂ 2 V ∂ 2 t + (LG + RC) ∂V ∂t + RGV. (0.1.4)
Similarly, differentiating in space (resp. time) the first (resp. second) equation in (0.1.3), we get a partial differential equation with only one dependent variable I,

∂ 2 I ∂ 2 x = LC ∂ 2 I ∂ 2 t + (LG + RC) ∂I ∂t + RGI. (0.1.5)
Further, in the case of loss-less coaxial cable (i.e., R = G = 0), equations (0.1.4) and (0.1.5) will be two exact wave equations

           ∂ 2 V ∂ 2 t = s 2 ∂ 2 V ∂ 2 x , ∂ 2 I ∂ 2 t = s 2 ∂ 2 I ∂ 2 x , where s = 1 √ LC
is the propagation speed of the wave along the cable.

In the case of more general and realistic situation namely, when each cross section is heterogeneous (physical characteristics of the medium vary in the cross section) and the cable has a variable cross section and material properties along the cable, another model is derived. In fact, a 1D limit model for the propagation of electromagnetic waves in thin heterogeneous co-axial cables by means of asymptotic analysis of the original 3D model with respect to a small geometric parameter δ (representing the transverse dimensions of the cables) is derived. The resulting model is a generalized telegraphist's model whose coefficients are determined from the solutions of 2D scalar problems in normalized cross sections. Mathematically, the generalized telegraph equation set in a real interval (model on a cable from Imperiale and Joly, 2014) given by:     

V t + gV + aI x + kW = 0, in (0, L) × (0, ∞), I t + rI + bV x = 0, in (0, L) × (0, ∞), W t + cW = V, in (0, L) × (0, ∞).

(0.1.6)

This generalized telegraph equation is a coupling between the usual telegraph equation where the electric unknowns are V and I representing the electric potential and the electric current respectively with a first order differential equation of parabolic type involving an auxiliary variable W representing the non-local effects. The coefficients a, b, c, r, k and g are all non-negative functions in L ∞ (0, L) that verify some assumptions mentioned in Chapter 2.

Heat transfer

Heat transfer describes the flow of heat (thermal energy) due to temperature differences and the subsequent temperature distribution and changes. One of the various mechanisms of heat transfer is heat conduction. Heat conduction, also called diffusion, is the direct microscopic exchange of kinetic energy of particles through the boundary between two systems. When an object is at a different temperature from another body or its surroundings, heat flows so that the body and the surroundings reach the same temperature, at which point they are in thermal equilibrium. Heat conduction is a diffusive process governed by a parabolic differential equation which lacks second-order derivative with respect to time, characteristic of a wave equation. In Mathematics, the heat equation is a second order partial differential equation which describes the propagation of heat with a in some quantity f = f (x 1 , • • • , x n , t) of time variable t and spacial variables (x 1 , • • • , x n ). The heat equation is given by

∂ f ∂t = ∆ f , (0.2.1)
where, f is the temperature deviation from the reference temperature.

Combination between mechanical and thermal processes

Mechanical equipment are affected by various interactions during their operation, the most significant being the mechanical and thermal effects. Mechanical and thermal loads usually occur simultaneously and as a result, the displacement and temperature fields are created in close connection with each other. In fact, Changes in temperatures causes thermal effects on materials. Some of these thermal effects include thermal stress, strain, and deformation. Thermal deformation simply means that as the "thermal" energy (and temperature) of a material increases, so does the vibration of its atoms/molecules and this increased vibration results in what can be considered a stretching of the molecular bonds -which causes the material to expand. In the same manner, if the thermal energy (and temperature) of a material decreases, the material will shrink or contract. Thus, thermo-elasticity is based on temperature changes induced by expansion and compression of the test part. Therefore, both mechanical and thermal fields have to be defined simultaneously taking the relationship between them into account which in practice proves to be rather difficult. Thermo-mechanical processes are described by the basic equations of continuum mechanics and thermodynamics. Mathematically, a linear one-dimensional thermo-elastic system satisfied by a thermoelastic bar (0, L) is represented by the following two equations:

u tt -u xx + αθ x = 0, in (0, L) × (0, ∞), θ t -θ xx + αu tx = 0, in (0, L) × (0, ∞), (0.2.2)
where, u is the displacement, θ is the temperature deviation from the reference temperature and the mechanical-thermal coupling α is a positive constant. In general, during wave propagation, the wave eventually reaches an end of the material. Some of the wave will reflect back to its source. If the reflection reaches the source at the same time a new wave is generated, the two waves will combine and be synchronized in phase. Later, if those two waves reflections return to the source at the same time the next new wave is generated, all three waves will combine. This will continue for as long as waves are generated under these conditions, and the resultant wave will increase in amplitude, theoretically to infinity. This is called resonance. If this wave reinforcement is allowed to continue, the system eventually will either self-destruct or fatigue to failure. Hence, in order to prevent the resonance energy from reaching an infinite value and thus forbidding the structure failure, a dissipation mechanism via a damping is invoked.

Damping

Damping is a phenomenon that dissipates the energy of every vibrating system. Depending on its spatial origin, damping can be classified as material damping, boundary damping and damping due to fluid-structure interactions. Materials with high damping coefficients are used in applications of shock absorption, vibration control, noise reduction, and dissipation of increased heat. Engineers use damping coefficients to compare materials to see which will be the best one for the application. Within this section, we will focus on three types of dampings which are: viscous damping, visco-elastic damping and thermo-elastic damping.

Viscous damping usually models external friction forces such as air resistance acting on the vibrating structures and is thus called "external damping". Most often they damp motion by providing a force or torque opposing motion proportional to the velocity. Visco-elastic damping originates from the internal friction of the material of the vibrating structures and thus called "internal damping". Visco-elastic materials, as their name suggests, combine two different properties which are elasticity and viscosity. The term "viscous" implies that they deform slowly when exposed to an external force. The term "elastic" implies that once a deforming force has been removed the material will return to its original configuration. So, visco-elastic materials exhibit mechanical properties intermediate between those of viscous liquid and those of elastic solid. When a visco-elastic material is subjected to a stress, the response is composed by an elastic deformation (which stores energy) and a viscous flow (which dissipates energy). For example, tall buildings vibrate when dynamically loaded by wind or earthquakes. Visco-elastic materials have the property of absorbing such vibrational energy -damping the vibrations. Visco-elastic dampers are used in some tall buildings, for example in the Columbia Center in Seattle, in which the dampers consist of steel plates coated with a visco-elastic polymer compound -the dampers are fixed to some of the diagonal bracing members. Different models can be used to describe the visco-elastic behavior among them, Maxwell model (i.e., spring and shock absorber in series) and Kelvin-Voigt model (spring and shock absorber in parallel) describe the simpler ones. Thermo-elastic damping is a source of intrinsic material damping due to thermoelasticity present in almost all materials. As the name thermo-elastic suggests, it describes the coupling between the elastic field in the structure (caused by deformation) and the temperature field. The effect of including thermo-elastic damping is that a heat source term, proportional to the rate of stress change, is added to the heat balance equations, see (0.2.2).

In any vibrating structure, the strain field causes a change in the internal energy such that compressed region becomes hotter (assuming a positive coefficient of thermal expansion) and extended region becomes cooler. The mechanism responsible for thermo-elastic damping is the resulting lack of thermal equilibrium between various parts of the vibrating structure. Energy is dissipated when irreversible heat flow which is driven by the temperature gradient causes vibrational energy to be dissipated. So, thermo-elastic damping refers to the process in which part of the vibration energy of a mechanical resonator is dissipated into thermal energy, through irreversible heat conduction accompanying elastic vibrations in the resonator.

In practice, from the research laboratory to the production line, scientists study the influence of such dampings (viscous, visco-elastic or thermo-elastic) on the behavior of the considered physical system. Hence, in this context many questions arise. Among the most frequently asked questions we mention:

1) Is the damping involved in a system of partial differential equations set in a certain medium (bounded domain or a network) strong enough to prove the strong stability of the system? In other words, is the damping able to dissipate the waves propagating in this medium and therefore leading the energy of the solution of the system to converge to zero?

2) How is the decay of the energy of the solution of the system varying with respect to the different localizations of the damping?

3) What is the optimal decay rate obtained and in which geometric situations? On the other hand, in addition to the significant role of the damping, the type of the medium and the type of the boundary conditions also contribute in the stabilization process of a physical system. For this aim, we will discuss briefly in the following section some boundary conditions used in this thesis.

Boundary conditions on networks

From the theoretical point of view the most challenging current problems are probably in the control questions for flows on networks. Probably, this is due to the fact that the recent applications, like electrical circuits, arterial networks, networks of open channels, traffic flows on networks, computer networks, chemical and biological networks, all involve systems of partial differential equations set on networks or graphs. Typical examples are the telegrapher equations for electrical lines, the shallow water (Saint-Venant) equations for open channels, the Euler equations for gas flow in pipelines or the Aw-Rascle equations for road traffic (see [START_REF] Bastin | Stability and Boundary Stabilization of 1-D Hyperbolic Systems[END_REF][START_REF] Gugat | Gas Flow in Fan-Shaped Networks: Classical Solutions and Feedback Stabilization[END_REF][START_REF] Leugering | Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[END_REF][START_REF] Valein | Stabilization of the wave equation on 1-d networks[END_REF][START_REF] Zhang | Exponential and Super Stability of a Wave Network[END_REF]Kramar, Mugnolo, and Nicaise, 2020b). For a spatial network, the dynamics is typically described first at the level of individual edges, followed by a model for the junctions. So, once a reduced model is obtained, the next question is to impose the correct coupling conditions between the different branches (edges). In this case some transmission conditions that translate some physical preservation have to be imposed at the junctions. In fact, a particular choice for the coupling conditions determines the particular physics of the underlying problem. Hence, different coupling conditions give rise to different network

   ∑ j∈I (v) q j (v) = 0, p j (v) = p ℓ (v); j, ℓ ∈ I(v), (0.4.1)
where v is the node of the junction of the connected edges adjacent to v.

The first equation of (0.4.1) describes the mass conservation (resp. charge conservation) for fluid flow models (resp. electrical circuits); on the contrary the second equation means that the pressure (resp. voltage) is continuous at the junction.

But since such transmission conditions do not take into account the geometry of the 3D structure, another coupling conditions called the improved Kirchhoff's circuit laws characterizing (up to a certain extent) the 3D properties and the physical parameters of the junction was derived (see [START_REF] Beck | Mathematical modeling of electrical networks[END_REF]Joly and[START_REF] Joly | Construction and analysis of improved Kirchoff conditions for acoustic wave propagation in a junction of thin slots[END_REF].

In this sense, the interesting 1D variables used are the electrical potential and the electrical current present in the limit star-like graph made out of one knot (assume x=0 is the node at the knot) and L + 1 branches. Such a reduced model was derived using asymptotic analysis of Maxwell's equations. The so-called improved Kirchhoff's law is a dynamic boundary condition made of L + 1 equations and given by

         N ∑ k=1 Z ℓk I k,t (0, t) = 1 δ (V 0 (0, t) -V ℓ (0, t)), in R * + , ∀ℓ ∈ {1, ...., N}, V 0,t (0, t) = -1 δY N ∑ ℓ=0 I ℓ (0, t) in R * + , (0.4.2)
where Y is a positive constant and Z = (Z ℓk ) N×N is a symmetric, positive definite matrix. The two effective coefficients Y and Z integrate the complex structure of the knot area. The cable 0 th plays a privileged role because it is chosen as the reference from which the electrical potential of each cable is expressed. Further, if δ = 0 then, the improved Kirchhoff's circuit rules become the usual Kirchhoff's law given in (0.4.1).

On the other hand, concerning the terminals of a network also several boundary conditions can be considered such as the well-known Dirichlet and Neumann boundary conditions. In addition to these conditions and in the presence of disturbances, such as a blockage, the response of the system is to return it to it's equilibrium. Hence, boundary control conditions are imposed at the terminals to control the dynamic system. For example, in the network of large arteries, the process of auto-regulation can be modeled using boundary controls (at the root of the network-the heart and the peripheral nodes-the ends of the large arteries) deriving the system back to a dynamic equilibirium in a minimum time. In fact, the terminal boundary condition widely used in the modelling of arterial networks in cardio-vascular systems (in analogy with electrical circuits) is the pure resistance condition given by

W b = -R T W f
where, W f is the forward characteristics information leaving the outlet of an arterial domain, W b is the backward characteristic information reflected by the terminal model and -1 ≤ R T ≤ 1 is the terminal reflection coefficient. R t = 1 corresponds to a complete reflection of the characteristic (complete blockage in the terminal site); R t = 0 means that there is no reflected characteristic at the terminal site, and R T = 1 represents an outflow with an open end (free end), see [START_REF] Cascaval | Flow optimization in vascular networks[END_REF] In reality, a controlling mechanism specifies what resistance values are applied in order to regulate the desired equilibrium. Beside the boundary conditions stated above there are still various dynamic, mixed or control boundary conditions used in the control and stabilization of dynamic systems on flow networks (see [START_REF] Bressan | Flows on networks: recent results and perspectives[END_REF]Kramar, Mugnolo, and Nicaise, 2020a) but within this section we just focus on the boundary conditions that will be used within the thesis. Now, we will introduce the main introduction followed by a thesis overview in which we give a brief explanation of the problems and the results obtained.

II) MAIN INTRODUCTION

In this thesis we study the stabilization of three coupled systems concerned with (thermo-) viscoelastic/elastic transmission problems and telegraph equations in boun-ded domains or in networks. This PhD thesis is divided into three parts. In part 1, we give a general introduction followed by a main introduction and a thesis overview in which we give a brief mathematical description of the problems and the results obtained in each chapter.

In part 2, we introduce Chapter 1, in which we consider a viscoelastic/elastic transmission problem in a bounded domain. Mainly, in this part we study the stabilization of a system of weakly coupled wave equations with one or two Kelvin-Voigt dampings and non-smooth coefficient at the interface. First, using semigroup theory, we prove the well-posedness of the system. Second, in the absence of the compactness of the resolvent and using a unique continuation result combined with a general criteria of Arendt and Batty, 1988, we show the strong stability of our system in different situations. Then, we establish a non-uniform stability result for the case when only one equation is damped with a globally distributed damping. Next, we study the energy decay rate in several distinguished cases. We prove that the corresponding semigroup is analytic when the Kelvin-Voigt dampings are globally distributed. Then, when only one non-smooth local Kelvin-Voigt damping is effective and under some assumptions (some of which are related to the decay rate of a wave equation with a frictional damping), we prove that the energy of the system decays polynomially with different decay rates according to these conditions. The results obtained are established by using a frequency domain approach combined with a multiplier technique. The method used give different quantitative results related to the asymptotic behavior of the energy depending on some geometric situations.

In part 3, we investigate the stabilization of two coupled systems involving a thermo-elastic/elastic transmission problem and telegraph equations on networks. This part is composed of two chapters, Chapter 2 and Chapter 3.

Chapter 2 concerns with the existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped networks. At the central interior node of the network, we consider the improved Kirchhoff conditions. In such node conditions, not only function values but also time derivatives appear. That is at the junction, the conditions considered are not system of algebraic equations but instead a system of ordinary differential equations. At the exterior nodes, we imposed a dissipative boundary condition. First, we provide a well-posedness result. Next, in the absence of the compactness of the resolvent, we prove that our system is strongly stable using a general criteria in Arendt and Batty, 1988, combined with a new uniqueness result. Finally, using a frequency domain approach combined with a multiplier technique and the construction of a new multiplier satisfying some ordinary differential inequalities, we show that the energy of the system decays exponentially to zero, under some regularity assumptions on two coefficient functions of the system. Chapter 3 concerns with the stability of a transmission problem of a thermoelastic system on networks. Mainly, we focus on networks composed of both elastic and thermo-elastic materials. On the thermo-elastic edges, we consider a system of two wave equations coupled by velocity, such that one wave equation is coupled to a heat equation with a thermal effect. On the purely elastic edges, we consider only a system of two conservative wave equations. Using a general criteria of [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF], we prove the strong stability of the system under some conditions related to the coupling coefficient and the geometry of the network. Then, we prove the exponential stability of the system under the condition that the two waves propagate with the same speed on each thermo-elastic edge. Otherwise, we establish a polynomial decay rate.

Thesis Overview

Let us now briefly outline the content of this thesis.

Chapter 1: In this chapter, we consider the following system of coupled wave equations with viscoelastic dampings:

           Z tt -div(∇Z + D∇Z t ) + BZ = 0, in Ω × R * + , Z = 0, on Γ × R * + , Z(0) = Z 0 , Z t (0) = Z 1 in Ω, (0.4.3)
where

• Z = (u, y) with u = u(x, t) and y = y(x, t), Z t = (∂ t u, ∂ t y) denotes the time derivative of Z. • D(x) = b(x) 0 0 c(x)
• B(x) = 0 α(x) α(x) 0 with α : Ω → R is the coupling function satisfying α ∈ L ∞ (Ω), ||α|| ∞ < 1 C 0 , (0.4.6)
where C 0 denotes the Poincaré constant.

In Najdi, 2016, the author considered system (0.4.3) and proved a polynomial energy decay rate of type t -1/4 provided that the coupling coefficient is a constant real number, the damping coefficient b ∈ C 1,1 (Ω), and the damping region is covering the whole boundary (that represent a quite restrictive assumption). In addition, it was proved that the polynomial energy decay rate of order t -1/4 is optimal in some sense. Also a decay rate of the energy in t -1/2 and an optimality result is proved in Oquendo and Pacheco, 2017 provided that the coupling function is a constant, c = 0 and the visco-elastic coefficient b is a positive constant (that means that the damping acts on one equation but on the whole domain).

In this chapter, our main interest is to improve these previous results. Mainly, we consider system (0.4.3) with a coupling function and a non-smooth locally Kelvin-Voigt damping distributed in a part of the domain and we get a better polynomial decay rate than the one obtained in Najdi, 2016.

We assume that there exist three constants b 0 , c 0 and α 0 and three open sets

ω b , ω α and ω c contained in Ω such that b(x) ≥ b 0 > 0, ∀x ∈ ω b , (0.4.7) α(x) ≥ α 0 > 0, ∀x ∈ ω α , (0.4.8) and c(x) ≥ c 0 > 0, ∀x ∈ ω c . (0.4.9)
First, we study the existence, uniqueness and regularity of the solution of system (0.4.3) using a semigroup approach. We start by defining the energy space

H by H = (H 1 0 (Ω) × L 2 (Ω)) 2 (0.4.10)
equipped with the following inner product: (0.4.11) for all U = (u, v, y, z) , V = ( u, v, y, z) ∈ H.

(U, V) H = Ω (∇u • ∇ u + ∇y • ∇ y) dx + Ω (v v + z z) dx +Re Ω α(x)(u y + y u) dx,
Then, we define the unbounded linear operator A by:

D(A) = (u, v, y, z) ∈ H : div(∇u + b∇v) ∈ L 2 (Ω), div(∇y + c∇z) ∈ L 2 (Ω) and v, z ∈ H 1 0 (Ω) , AU = (v, div(∇u + b(x)∇v) -αy, z, div(∇y + c(x)∇z) -αu), ∀ U = (u, v, y, z) ∈ D(A).
The energy of the system is given by

E(t) = 1 2 Ω (|u t | 2 + | ∇u | 2 +|y t | 2 + | ∇y | 2 ) dx + 2Re Ω αuy dx . (0.4.12)
For a smooth solution, a straightforward computation leads to

E ′ (t) = - Ω (b | ∇v | 2 +c | ∇z | 2 ) dx ≤ 0. (0.4.13)
Then, system (0.4.3) is dissipative in the sense that its energy is non-increasing with respect to time t.

By putting v = u t , z = y t and if U = (u, u t , y, y t ) is a regular solution of system (0.4.3), we can rewrite this system as the following evolution equation: (0.4.14) where U 0 = (u 0 , u 1 , y 0 , y 1 ).

U t = AU, U(0) = U 0 ,
We easily prove that the operator A is maximal dissipative. Then, using Lumer-Phillips Theorem (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] we conclude that A generates a C 0 -semigroup of contractions on H. Hence, semigroup theory leads us to an existence and uniqueness result stated in Chapter 1, Theorem 1.2.2.

Second, we study the strong stability of the system in different geometrical situations. More precisely, we prove that for an initial data in H, the energy converges to zero as t tends to infinity if one of the following geometric situations hold.

(H1) If α = α 0 is a positive constant, then suppose that either ω b or ω c is a non-empty open subset of Ω.

(H2) If α is a non-zero function, then suppose that ω b or ω c is is a non-empty open subset of Ω. Further, assume that either

ω 1 = ω b ∩ ω α satisfies meas(ω 1 ∩ Γ) > 0 or ω 2 = ω c ∩ ω α satisfies meas(ω 2 ∩ Γ) > 0. (H3) ω 3 = ω b ∩ ω c is a non-empty open subset of Ω.
To prove the system strongly stable, we first use a unique continuation result based on a Carleman estimate to derive a new uniqueness result. Then, we use the obtained uniqueness result combined with a general criteria of [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF] to achieve our goal. Later on, we show that strongly stable semigroup associated to system (0.4.3) is analytic when the Kelvin-Voigt dampings are globally distributed. Then, we show that system (0.4.3) is not uniformly stable for the case when the coupling coefficient is a positive constant and only one equation is damped with a globally distributed damping. For this desire, we study the asymptotic behavior of the eigenvalues of A near the imaginary axis. In fact, we

show that there exists k 0 ∈ N * sufficiently large and two sequences λ + k and λ - k satisfying the following asymptotic behavior

λ + k = iµ k - α 2 2µ 4 k + O 1 µ 9 k , ∀|k| ≥ k 0 (0.4.15)
and (0.4.16) where {µ 2 k } ∞ k=1 is the sequence of eigenvalues of the Laplace operator ∆.

λ - k = -iµ k - α 2 2µ 4 k + O 1 µ 9 k , ∀|k| ≥ k 0 ,
Due to the fact that the sequences of eigenvalues of A ((0.4.15) and (0.4.16)) are close to the imaginary axis as k tends to infinity, a polynomial decay is hoped. This leads us to our main result in this chapter. Mainly, when the system (0.4.3) is partially damped by one locally Kelvin-Voigt damping distributed in a flexible geometry that covers several situations and under conditions (0.4.7) and (0.4.8), we confirm that there exists a constant C > 0 such that for every initial data U 0 = (u 0 , u 1 , y 0 , y 1 ) ∈ D(A), the energy of system (0.4.3) verifies the following estimate: (0.4.17) This result is achieved by using a frequency domain approach combined with a multiplier technique and using the exponential stability of a wave equation with a frictional damping in

E(t) ≤ C 1 √ t ∥U 0 ∥ 2 D(A) , ∀t > 0.
ω 1 = ω b ∩ ω α , namely      φ tt -∆φ + 1 ω 1 φ t = 0 in Ω × (0, ∞), φ = 0 on Γ × (0, ∞), φ(t = 0) = φ 0 , φ t (t = 0) = φ 1 .
(0.4.18)

After that and under an additional condition on ω α and by using the exponential or polynomial decay of the energy of system (0.4.18), we establish different quantitative results relating the decay rate of the energy of system (0.4.18) and the decay rate of the energy of system (0.4.3). Then, we prove a better polynomial decay rate in the case of two active dampings. Finally, we prove the optimality of the polynomial decay rate t -1/2 in the case where only one of the damping coefficients is effective and is globally distributed.

In the next chapter, we focus on a coupled system involving the telegraph equation on a network. Let S be a star shaped network made of N + 1 cables where, N is a positive integer. Without loss of generality, we assume that all the N + 1 cables are of equal length L > 0. Also, fix different real valued and non-negative functions

a = (a ℓ ) N ℓ=0 , b = (b ℓ ) N ℓ=0 , c = (c ℓ ) N ℓ=0 , k = (k ℓ ) N ℓ=0 , r = (r ℓ ) N ℓ=0
, and g = (g ℓ ) N ℓ=0 in (L ∞ (0, L)) N+1 satisfying the following assumption (0.4.19) where µ > 0 is a positive real number.

a ℓ ≥ µ, b ℓ ≥ µ, c ℓ ≥ µ, k ℓ + g ℓ ≥ µ a.e in (0, L), ∀ℓ = 0, . . . , N,
Chapter 2: In this chapter, we consider the generalized telegraph equation on a star shaped network S:

     V ℓ,t + g ℓ V ℓ + a ℓ I ℓ,x + k ℓ W ℓ = 0, (x, t) ∈ (0, L) × R * + , I ℓ,t + r ℓ I ℓ + b ℓ V ℓ,x = 0, (x, t) ∈ (0, L) × R * + , W ℓ,t + c ℓ W ℓ = V ℓ , (x, t) ∈ (0, L) × R * + .
(0.4.20)

Previously, Nicaise, 2015, has considered the stabilization of the generalized telegraph equation set in a real interval (0, L) with L > 0 (model on a cable from Imperiale and Joly, 2014) with the Dirichlet boundary condition at x = 0, L. He proved that under the condition r ̸ = 0, system (0.4.20) is strongly stable in the energy Hilbert space. Then, under the additional conditions that are r ∈ W 1,∞ (0, L) and r + g > 0 in (0, L), an exponential energy decay rate was established. Otherwise, a polynomial energy decay rate of type t -1 for smooth initial data was attained. Moreover, he showed that the obtained polynomial decay rate is optimal in the case r = g = 0.

Our main interest is to extend the analysis to a networked system. This generalization is very well-motivated by engineering applications since electric power grids are complex networked systems.

We considered system (0.4.20) with the following dissipative boundary condition at the exterior vertices (0.4.21) and the transmission conditions from Beck, 2016, §8.2 or Joly and[START_REF] Joly | Construction and analysis of improved Kirchoff conditions for acoustic wave propagation in a junction of thin slots[END_REF] (called by these authors the improved Kirchhoff conditions) at the interior common vertex

V ℓ (L, t) -α ℓ I ℓ (L, t) = 0, in R * + , α ℓ > 0, ∀ℓ ∈ {0, ...., N},
         N ∑ k=1 Z ℓk I k,t (0, t) = 1 δ (V 0 (0, t) -V ℓ (0, t)), in R * + , ∀ℓ ∈ {1, ...., N}, V 0,t (0, t) = -1 δY N ∑ ℓ=0 I ℓ (0, t) in R * + , (0.4.22)
where, Y and δ are two positive constants and Z = (Z ℓk ) N×N is a symmetric, positive definite matrix. These boundary conditions are derived in Beck, 2016, §8.2 or Joly and[START_REF] Joly | Construction and analysis of improved Kirchoff conditions for acoustic wave propagation in a junction of thin slots[END_REF] In fact, these junction nodes are improved in the sense that instead of zero-order Taylor approximation, a first order Taylor approximation is considered. As a consequence, in the node conditions not only functions appear but also time derivative appears and thus leading to a system of ordinary differential equations.

The geometry of the joint can be described by the angles between the joined edges. This angle enters in the node conditions as a parameter. In addition, the coefficient Y and the matrix Z are defined from the material properties of the medium and from 3D potentials defined in the reference domain.

By setting ν(t) = V 0 (0, t) and η(t) = (η ℓ (t)) N ℓ=1 with η ℓ (t) = N ∑ k=1 Z ℓk I k (0, t), system (0.4.20)-(0.4.22) is completed with the following initial conditions

     V (x, 0) = V 0 (x), I(x, 0) = I 0 (x), W (x, 0) = W 0 (x), x ∈ (0, L), ν(0) = ν 0 , η(0) = η 0 .
(0.4.23)

Our main desire within this chapter is to find the sufficient conditions on the coefficients of the system, i.e., on the functions a ℓ , b ℓ , c ℓ , k ℓ , r ℓ , and g ℓ that guarantee that system (0.4.20)-(0.4.23) is strongly stable and then exponentially stable.

For all p ∈ [1, ∞], denote by L p (S) = L p (0, L) N+1 , 1 ≤ p ≤ ∞ and H 1 (S) = H 1 (0, L) N+1 .
First, we study the existence, uniqueness and regularity of the solution of system (0.4.20) using a semigroup approach. We start by defining the energy space

H = (L 2 (S)) 3 × C N+1 ,
that is a Hilbert space equipped with the following inner product:

⟨(V, I, W, ν, η) ⊤ , (V * , I * , W * , ν * , η * ) ⊤ ⟩ H = N ∑ ℓ=0 L 0 (θ ℓ V ℓ V * ℓ + β ℓ I ℓ I * ℓ + γ ℓ W ℓ W * ℓ ) dx + δY νν * + δ (η, Z -1 η * ) C N where θ = (θ ℓ ) N ℓ=0 , β = (β ℓ ) N ℓ=0 , γ = (γ ℓ ) N
ℓ=0 ∈ L ∞ (S) will be fixed later but are such that (0.4.24) for some µ 0 > 0.

θ ℓ ≥ µ 0 , β ℓ ≥ µ 0 , γ ℓ ≥ µ 0 a.e in Ω, ℓ = 0, . . . , N,
The energy of the system is given by

E(t) = 1 2 ||(V, I, W, ν, η)|| 2 H .
For a strong solution, technical computations gives

E ′ (t) ≤ - 1 2 N ∑ ℓ=0 L 0 (θ ℓ g ℓ |V ℓ | 2 + 2β ℓ r ℓ |I ℓ | 2 + γ ℓ c ℓ |W ℓ | 2 )dx - N ∑ ℓ=0 α ℓ |I ℓ (L)| 2 ≤ 0.
This means that our system is dissipative in the sense that its energy is nonincreasing with respect to t.

Next, we define the unbounded linear operator A : D(A) → H by

D(A) = (V, I, W, ν, η) ∈ H : V ∈ H 1 (S), I ∈ H 1 (S), V ℓ (L) = α ℓ I ℓ (L), ∀ℓ ∈ {0, . . . , N}, ν = V 0 (0), and η = (η ℓ ) N ℓ=1 with η ℓ = N ∑ k=1 Z ℓk I k (0) , and A       V I W ν η       = -g ⊗ V + a ⊗ I x + k ⊗ W, r ⊗ I + b ⊗ V x , c ⊗ W -V , 1 δY N ∑ ℓ=0 I ℓ (0), 1 δ (V ℓ (0) -ν) N ℓ=1 ⊤ ,
where for two vector functions

P = (p ℓ ) N ℓ=0 ∈ L ∞ (S), and Q = (q ℓ ) N ℓ=0 ∈ L 2 (S), we set P ⊗ Q = (p ℓ q ℓ ) N ℓ=0 , while for Q = (q ℓ ) N ℓ=0 ∈ H 1 (S), Q x = (q ℓ,x ) N ℓ=0 .
Then, setting U(t) = (V ( (0.4.25) where U 0 = (V 0 , I 0 , W 0 , ν 0 , η 0 ) ⊤ . First, we prove that the operator A is maximal dissipative. Then, using Lumer-Phillips Theorem (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] we conclude that A generates a C 0 -semigroup of contractions on H. Hence, semigroup theory leads us to an existence and uniqueness result stated in Chapter 2, Theorem 2.2.2.

H U t = AU, t ∈ R * + , U(0) = U 0 ,
Second, we study the strong stability of the system in the absence of the compactness of the resolvent and without any additional conditions on the coefficients of the system. Within this chapter, the reader will realize that analyzing the stability of our system is not immediate. The result is given in the following theorem.

R(iλ -A) = H, ∀λ ∈ R.
The first result is proved by using the dissipativeness of A and our boundary and transmission conditions, while the second one is based on a compact perturbation argument in addition to a new uniqueness result and Lemma 0.4.2. These two Lemmas guarantee that the resolvent set ρ(A) of A contains the whole imaginary axis. Hence, by a general criteria of Arendt-Batty (see [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF], we deduce that the C 0 -semigroup of contractions (e tA ) t≥0 is strongly stable, i.e., the energy of system (0.4.20)-(0.4.23) converges to zero as time t tends to infinity.

Later on, under the condition that a ℓ , b ℓ ∈ W 1,∞ (0, L), for all ℓ ∈ {0, .., N} we prove that the semigroup (e tA ) t≥0 is exponentially stable, i.e. there exist two positive constants M and ϵ > 0 such that

||e tA U 0 || H ≤ Me -ϵt ||U 0 || H , ∀t ≥ 0, ∀U 0 ∈ H.
Here contrary to Nicaise, 2015, we do not require that r ℓ + g ℓ is uniformly bounded from below for each ℓ (see the assumption (25) in [START_REF] Nicaise | Stabilization and asymptotic behavior of a generalized telegraph equation[END_REF] to get exponential decay of the energy but rather exploit the dissipative boundary conditions (0.4.21) at the exterior nodes. To prove our result we combine the frequency domain approach with an ad-hoc multiplier method given in the following lemma.

Lemma 0.4.4. Let ℓ ∈ {0, .., N} be fixed and assume that the functions a ℓ , b ℓ ∈ W 1,∞ (0, L). Then, there exists h ℓ ∈ W 1,∞ (0, L) and a positive real number ρ ℓ such that h ℓ (0) = 0, (0.4.26a) h ℓ,x (x) ≥ ρ ℓ > 0, for a.a. x ∈ (0, L), (0.4.26b)

h ℓ,x (x) a ℓ (x) b ℓ (x) + h ℓ (x) a ℓ (x) b ℓ (x) x ≥ 0, for a.a. x ∈ (0, L). (0.4.26c)
Finally, we discuss the stability of the system (0.4.20) with some extensions, either by considering a general dynamic boundary condition or by considering general networks.

In the last chapter, we consider a coupled system involving a thermo-elastic/elastic transmission problem on networks (see "General notations on graphs" page xiii).

Chapter 3: Let N be a network of elastic and thermo-elastic materials. The set of edges I(N ) is split up into I(N ) = I te ∪ I e , with I e ∩ I te = ∅, in other words, I te (resp. I e ) is the set of thermo-elastic (resp. elastic) edges. Assume that N contains at least one thermoelastic edge, that V ext ̸ = ∅, that every maximal subgraph of elastic edges is a tree whose all of its exterior vertices except one are attached to thermo-elastic edges and that every subgraph of thermo-elastic edges is not a circuit.

In this chapter, we investigate the stabilization of the following thermoelastic/elastic transmission problem on N .

• On every thermo-elastic edge of N (j ∈ I te ) the following equations hold: (0.4.27) where α j , ρ j , κ j and β j are positive constants.

     u j tt -u j xx + α j θ j x -β j y j t = 0 in (0, ℓ j ) × (0, ∞), y j tt -ρ j y j xx + β j u j t = 0 in (0, ℓ j ) × (0, ∞), θ j t -κ j θ j xx + α j u j tx = 0 in (0, ℓ j ) × (0, ∞),
• On every elastic edge of N (j ∈ I e ) one has: (0.4.28) where β j and ρ j are positive constants. We assume that the initial data on the network N are

u j tt -u j xx -β j y j t = 0 in (0, ℓ j ) × (0, ∞), y j tt -ρ j y j xx + β j u j t = 0 in (0, ℓ j ) × (0, ∞),
u j (x, 0) = u j 0 (x), u j t (x, 0) = u j 1 (x), y j (x, 0) = y j 0 (x), y j t (x, 0) = y j 1 (x), θ j (x, 0) = θ j 0 (x), ∀j ∈ I(N ). (0.4.29)
Previously, several authors considered the linear one-dimensional thermo-elastic system on a thermo-elastic rod (0, L) given by:

u tt -u xx + αθ x = 0, in (0, L) × (0, ∞), θ t -θ xx + αu tx = 0, in (0, L) × (0, ∞), (0.4.30)
with the initial conditions

u(x, 0) = u 0 , u t (x, 0) = u 1 , θ(x, 0) = θ 0 , x ∈ (0, L), (0.4.31)
where u is the displacement, θ is the temperature deviation from the reference temperature and α is a positive constant.

Both [START_REF] Hansen | Exponential energy decay in a linear thermoelastic rod[END_REF][START_REF] Liu | Locally Distributed Control and Damping for the Conservative Systems[END_REF]Zheng, 1993 considered the stabilization of system (0.4.30)-(0.4.31) on a thermo-elastic rod (see Figure 14). In Shel, 2012, the author studied the stability problem of a thermoelastic system on particular cases of networks of elastic and thermoelastic materials. In the first case, they suppose that two elastic edges cannot be adjacent. In the second one, they consider a tree of elastic materials, the leaves of which thermoelastic materials are added as follows: the thermoelastic body is related to only one leaf by an end, and the second is free or connects two leaves, with the condition that each leaf is connected to only one thermoelastic body, see Figure 16. Under the continuity condition of the displacement, the Neumann condition for the temperature at the internal nodes, and the balance condition, an exponential stability was proved. Later on Han and Zuazua, 2017 discussed the asymptotic behaviour of a transmission problem of the thermo-elastic system on star shaped networks of elastic and thermo-elastic rods, see Figure 17. When only one purely elastic edge is present the uniform exponential decay rate was proved by a frequency domain analysis. Otherwise, a polynomial decay rate was deduced under a suitable irrationality condition on the lengths of the rods when more than one purely elastic edge is involved.

In this chapter, our main interest is to generalize the previous results by considering more general networks and by studying a more complicated thermo-elastic system such that on each thermo-elastic edge of the considered networks, we have a system of two wave equations coupled by velocity, such that one wave equation is coupled to a heat equation with a thermal effect while, on each purely undamped elastic edges, we have only a system of two conservative wave equations coupled by velocity, namely we consider system (0.4.27)-(0.4.29).

We prove that the dissipation due to the thermal effect is also strong enough to prove the exponential stability of the energy of the whole system when the two wave equation propagate with the same speed at each thermo-elastic edge of N . Otherwise, a polynomial decay is established.

To complete formulating our system, we need to indicate the boundary conditions considered with system (0.4.27)-(0.4.29). Denote by I te (a k ) (resp. I e (a k )) the set of indices of thermo-elastic (resp. elastic) edges adjacent to a k and denote by V ′ ext ( resp. V ′ int ) the set of exterior (resp. interior) nodes of maximal subgraph of thermo-elastic edges. Then the boundary condition on N are described as follows:

The displacement and temperature satisfies the Dirichlet boundary condition,

     u j (a k , t) = 0, j ∈ I(a k ), a k ∈ V ext , y j (a k , t) = 0, j ∈ I(a k ), a k ∈ V ext , θ j (a k , t) = 0, j ∈ I te (a k ), a k ∈ V ′ ext .
(0.4.32)

The displacement and temperature are continuous,

     u j (a k , t) = u ℓ (a k , t), j, ℓ ∈ I(a k ), a k ∈ V int , y j (a k , t) = y ℓ (a k , t), j, ℓ ∈ I(a k ), a k ∈ V int , θ j (a k , t) = θ ℓ (a k , t), j, ℓ ∈ I te (a k ), a k ∈ V ′ int .
(0.4.33)

The system satisfies the balance condition on y at every interior node,

∑ j∈I (a k ) d kj ρ j y j x (a k ) = 0, a k ∈ V int . (0.4.34)
The system satisfies the following balance conditions on u and θ,

       ∑ j∈I te (a k ) d kj κ j θ j x (a k , t) = 0, a k ∈ V ′ int , ∑ j∈I te (a k ) d kj (u j x (a k ) -α j θ j (a k )) + ∑ j∈I e (a k ) d kj u j x (a k ) = 0, a k ∈ V int .
(0.4.35)

First, we prove that system (0.4.27)-(0.4.29) with the boundary conditions (0.4.32)-(0.4.35) admits a unique solution in an appropriate Hilbert space H using semigroup theory. Then, we give sufficient conditions that guarantee the strong stability of the system. In fact, we prove that the energy of the system converges to zero as t tends to infinity, if one of the following conditions holds, 1) Each maximal subgraph of thermo-elastic edges has an exterior vertex that belongs to V ext .

2) There exists a maximal subgraph of thermo-elastic edges with no exterior vertices that belong to V ext and β j = β, for all j ∈ I(N ).

We rely on the Theorem of [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF] combined with a unique continuation result derived in [START_REF] Hayek | A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface[END_REF] and an iteration method to prove that our system is strongly stable. Later on, we distinguish between two different energy decay rates. More precisely, our main results are given in the following theorems.

Theorem 0.4.5. Let N be an arbitrary network for which system {(0. :

θ j (a k ) = 0 : θ j x (a k ) = 0 (P ) (G 1 ) (G 2 )
Thermo-elastic edge

Elastic edge

The boundary conditions of system (0.4.27)-(0.4.29) on the considered networks P, G 1 and G 2 will be as follows.

The system satisfies the Dirichlet boundary condition for the displacement and temperature at the exterior nodes,

     u j (a k , t) = 0, j ∈ I(a k ), a k ∈ V ext , y j (a k , t) = 0, j ∈ I(a k ), a k ∈ V ext , θ j (a k , t) = 0, j ∈ I te (a k ), a k ∈ V ext .
(0.4.37)

The displacement is continuous at every interior node,

u j (a k , t) = u ℓ (a k , t), j, ℓ ∈ I(a k ), a k ∈ V int , y j (a k , t) = y ℓ (a k , t), j, ℓ ∈ I(a k ), a k ∈ V int . (0.4.38)
The temperature satisfies the Neumann condition at the interior nodes,

θ j x (a k , t) = 0, j ∈ I te (a k ), a k ∈ V int . (0.4.39)
The system satisfies the balance condition at every interior node,

             ∑ j∈I te (a k ) d kj (u j x (a k ) -α j θ j (a k )) + ∑ j∈I e (a k ) d kj u j x (a k ) = 0, a k ∈ V int , ∑ j∈I (a k ) d kj ρ j y j x (a k ) = 0, a k ∈ V int .
(0.4.40)

Using semigroup theory, we easily prove the well-posedness of the system. Then, we discuss its strong stability and we show that under some sufficient conditions on the lengths of the purely elastic edges attached to the thermo-elastic ones, the system (0.4.27)-(0.4.29) with the boundary conditions (0.4.37)-(0.4.40) is strongly stable and then exponentially stable on the above described networks. More precisely, we show that the energy of the solution of the system (0.4.27)-(0.4.29) with the boundary conditions (0.4.37)-(0.4.40) converges to zero as t tends to infinity if one of the following conditions hold.

1) N is the graph G 1 ,
2) N is the graph P, β j = β, ρ j = 1, ∀j ∈ I(P ), and there exists j ∈ {1, 3} such that

ℓ j ̸ = mπ √ 2β j , ∀m ∈ N * , (0.4.41) 3) N is the graph G 2 , β j = β, ρ j = 1, ∀j ∈ I(G 2 )
, and in every circuit C, for the unique j, k ∈ Ω such that e j and e k are edges of C, we have sin(

√ 2β j ℓ j ) + sin( √ 2β k ℓ k ) ̸ = 0. (0.4.42)
Finally, we again guarantee that the system (0.4.27)-(0.4.29) with the boundary conditions (0.4.37)-(0.4.40) is also exponentially stable whenever it is strongly stable.

Part 2

♢A viscoelastic/elastic transmission problem in a bounded domain of R N ♢

Introduction

Let Ω ⊆ R N , N ≥ 1 be a bounded open set with a Lipschitz boundary Γ. We consider the following system of coupled wave equations with viscoelastic dampings: 1.1.1) where

           Z tt -div(∇Z + D∇Z t ) + BZ = 0, in Ω × R * + , Z = 0, on Γ × R * + , Z(0) = Z 0 , Z t (0) = Z 1 in Ω, (
• Z = (u, y) with u = u(x, t) and y = y(x, t), Z t = (∂ t u, ∂ t y) denotes the time derivative of Z.

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface

• D(x) = b(x) 0 0 c(x) with b, c : Ω → R are the visco-elastic damping functions satisfying b, c ∈ L ∞ (Ω), (1.1.2)
and b(x) ≥ 0, c(x) ≥ 0 for almost all x ∈ Ω.

(1.1.3) .1.4) where C 0 denotes the Poincaré constant. More precisely, C 0 is the smallest positive constant such that

• B(x) = 0 α(x) α(x) 0 with α : Ω → R is the coupling function satisfying α ∈ L ∞ (Ω), ||α|| ∞ < 1 C 0 , ( 1
Ω |θ| 2 dx ≤ C 0 Ω |∇θ| 2 dx, ∀ θ ∈ H 1 0 (Ω).
System (1.1.1) models the vibration of bodies which have one part made of a Kelvin-Voigt type viscoelastic material and the other one is made of an elastic material.

Naturally, system (1.1.1) is dissipative. Indeed, the energy of the system defined by

E(t) = 1 2 Ω (|u t | 2 + | ∇u | 2 +|y t | 2 + | ∇y | 2 ) dx + 2Re Ω αuy dx (1.1.5)
is decreasing with respect to t ∈ (0, ∞) since

∂E(t) ∂t = E ′ (t) = - Ω (b(x) | ∇u t | 2 +c(x) | ∇y t | 2 ) dx ≤ 0.
Besides, when no damping is applied to the system (b = c = 0 on Ω), the energy is conserved, i.e. E(t) = E(0), for all t > 0.

The stabilization of systems (simple or coupled) with Kelvin-Voigt damping has attracted the attention of many authors. In particular, in the one dimensional case, it was proved that the smoothness of the damping coefficient at the interface plays a critical role in the stability and regularity of the solution of the system (see [START_REF] Liu | Locally Distributed Control and Damping for the Conservative Systems[END_REF][START_REF] Liu | Spectrum and Stability for Elastic Systems with Global or Local Kelvin-Voigt Damping[END_REF]Liu and Liu, 1998;[START_REF] Liu | Exponential decay of energy of vibrating strings with local viscoelasticity[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF][START_REF] Zhang | Exponential stability of an elastic string with local Kelvin Voigt damping[END_REF]. However, there are only a few number of publications in the multi-dimensional setting. Let us start by recalling some previous studies in the case of simple wave equation system given by:

           u tt -div(a∇u + b∇u t ) = 0, in Ω × R * + , u(x, t) = 0, on Γ × R * + , u(x, 0) = u 0 , u t (x, 0) = u 1 , in Ω.
(1.1.6)

In [START_REF] Huang | On the mathematical model for linear elastic systems with analytic damping[END_REF], the author proved that when the Kelvin-Voigt damping div(b(x)∇u t ) is globally distributed i.e. b(x) ≥ b 0 > 0 for almost all x ∈ Ω, the wave equation (1.1.6) generates an analytic semi-group. In [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF], the authors considered the wave equation with local visco-elastic damping distributed around the boundary of Ω. They proved that the energy of the system Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface decays exponentially to zero as t goes to infinity for all usual initial data under the assumption that the damping coefficient satisfies: b ∈ C Further, in Nasser, Noun, and Wehbe, 2019, the authors investigate the stabilization of the wave equation with Kelvin-Voigt damping localized via non smooth coefficient in a suitable sub-domain of the whole bounded domain. They proved a polynomial stability result in any space dimension, provided that the damping region satisfies some geometric conditions. Finally in Hassine, 2015 this system is considered in the interval (0, L), with L > 0, when b = d1 (α,β) , with d > 0, and 0 ≤ α < β < L, where an energy decay rate in t -4 is obtained. On the other hand, to the best of our knowledge, the stabilization of a system of weakly coupled wave equations with one locally Kelvin-Voigt damping was studied in [START_REF] Najdi | Study of the exponential and polynomial stability of some systems of coupled equations with indirect bounded or unbounded control[END_REF][START_REF] Oquendo | Optimal decay for coupled waves with Kelvin-Voigt damping[END_REF][START_REF] Hassine | Stability of elastic transmission systems with a local Kelvin-Voigt damping[END_REF]Souayeh, 2019. In Najdi, 2016, the author proved a polynomial energy decay rate of type t -1/4 provided that the coupling coefficient is a constant real number, the damping coefficient b ∈ C 1,1 (Ω), and the damping region is covering the whole boundary (that represent a quite restrictive assumption). In addition, it was proved that the polynomial energy decay rate of order t -1/4 is optimal in some sense. A decay rate of the energy in t -1/2 and an optimality result is proved in Oquendo and Pacheco, 2017 provided that the coupling function is a constant, c = 0 and the visco-elastic coefficient b is a positive constant (that means that the damping acts on one equation but on the whole domain). A strongly coupled system (i.e. the term BZ is replaced by BZ t ) in the interval (-1, 1) is considered in Hassine and Souayeh, 2019 when c = 0 and b = d1 (0,1) , with d > 0, where a decay rate of the energy in t -1 6 is obtained. Let us finally mention the paper Hassine, 2016, where a transmission wave/beam system with a local Kelvin-Voigt damping is studied. So, since the stabilization of the system (1.1.1) with more significant geometric situations and more flexible conditions on the damping coefficient is not covered yet and since the problem In this chapter, we consider (1.1.1) with non smooth globally or locally distributed Kelvin-Voigt dampings in richer geometric situations than all previous results. First, we study the strong stability in several geometric situations. More precisely, we assumed that there exist three constants b 0 , c 0 and α 0 and three open sets

ω b , ω α and ω c contained in Ω such that b(x) ≥ b 0 > 0, ∀x ∈ ω b , (1.1.7) α(x) ≥ α 0 > 0, ∀x ∈ ω α , (1.1.8) and c(x) ≥ c 0 > 0, ∀x ∈ ω c . (1.1.9)
Then, we proved that the energy of the system (1.1.1) decays to zero as t tends to infinity if one of the following assumptions hold:

(H1) If α = α 0 is a positive constant, then suppose that either ω b or ω c is a non-empty open subset of Ω (see Figure 1.1). (H2) If α is a non-zero function, then suppose that ω b or ω c is is a non-empty open subset of Ω. Further, assume that either ω 1 = ω b ∩ ω α satisfies meas(ω 1 ∩ Γ) > 0 or ω 2 = ω c ∩ ω α satisfies meas(ω 2 ∩ Γ) > 0 (see Figure 1.1). (H3) ω 3 = ω b ∩ ω c is a non-empty open subset of Ω. FIGURE 1.1: Examples of domains Ω α = 1 ω b α = 1 Ω ω 1 Ω Ω verifies (H2) Ω verifies (H1)
Next, we study the energy decay rate in distinguished several cases. In the first case, we proved that the corresponding semi-group is analytic when the Kelvin-Voigt dampings are globally distributed (i.e. ω b = ω c = Ω). Then, for the system (1.1.1) with one non-smooth local Kelvin-Voigt damping and under some conditions Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface specified below satisfied by ω b or ω c and ω α , we showed that the energy of the system decays polynomially with different decay rates according to these conditions. In particular, we prove that the energy decay rate obtained in Najdi, 2016 is not optimal. We even improve this result by proving a polynomial energy decay rate of type t -1/2 when the system (1.1.1) is partially damped by one non-smooth locally Kelvin-Voigt damping (b ≥ 0, c = 0 or b = 0, c ≥ 0 on Ω) distributed in a flexible geometry that covers several situations (see Remark 5.6). Moreover, in the case where c = 0, b = 1 and α is constant, we prove that our energy decay rate of type t -1/2 is the optimal decay rate (better than the one obtained in [START_REF] Najdi | Study of the exponential and polynomial stability of some systems of coupled equations with indirect bounded or unbounded control[END_REF].

Let us briefly outline the content of this chapter. First, in Section 1.2, we show that the system (1.1.1) is well-posed using semi-group theory. Then, using a unique continuation result based on a Carleman estimate and a general criteria of Arendt-Batty in [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF], we proved the strong stability of the system in the absence of the compactness of the resolvent. In Section 1.3, we prove that the associated semi-group is analytic in the case of the global Kelvin-Voigt damping (ω b = ω c = Ω). In Section 1.4, using a spectral analysis, we prove the non uniform stability of the system in the case where c = 0, b = 1 and α is constant. In Section 1.5, when the system (1.1.1) is partially damped by one locally distributed Kelvin-Voigt damping ( b ≥ 0, c = 0 or b = 0, c ≥ 0 on Ω), we established in different cases polynomial energy decay rates for smooth solutions using a frequency domain approach combined with a multiplier method. In Section 1.6, we prove a better polynomial decay rate in the case of two active damping. Finally, in Section 1.7, we prove the optimality of the polynomial decay rate t -1/2 in the case where c = 0, b = 1 and α is constant.

Well-posedness and strong stability

In this section, we will study the existence, uniqueness and regularity of the solution of system (1.1.1) using a semi-group approach. Later, we will show the strong stability of this system in the absence of the compactness of the resolvent.

Let us start by proving the wellposedness of system (1.1.1).

Well-posedness

First, we define the energy space H by

H = (H 1 0 (Ω) × L 2 (Ω)) 2 (1.2.1)
equipped with the following inner product:

(U, V) H = Ω (∇u • ∇ u + ∇y • ∇ y) dx + Ω (v v + z z) dx +Re Ω α(x)(u y + y u) dx, (1.2.2) for all U = (u, v, y, z) , V = ( u, v, y, z) ∈ H. Note that (1.2.
2) is an inner product due to condition (1.1.4).

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface Next, we define the unbounded linear operator A by:

D(A) = (u, v, y, z) ∈ H : div(∇u + b∇v) ∈ L 2 (Ω), div(∇y + c∇z) ∈ L 2 (Ω) and v, z ∈ H 1 0 (Ω) , AU = (v, div(∇u + b(x)∇v) -αy, z, div(∇y + c(x)∇z) -αu), ∀ U = (u, v, y, z) ∈ D(A).
By putting v = u t , z = y t and if U = (u, u t , y, y t ) is a regular solution of system (1.1.1), we can rewrite this system as the following evolution equation:

U t = AU, U(0) = U 0 , (1.2.3)
where U 0 = (u 0 , u 1 , y 0 , y 1 ).

We recall that the energy of the system is given by

E(t) = 1 2 Ω (|u t | 2 + | ∇u | 2 +|y t | 2 + | ∇y | 2 ) dx + 2Re Ω αuy dx , (1.2.4)
and it is non-negative for any solution U. In fact,

E(t) ≥ 1 2 ||v|| 2 L 2 (Ω) + ||z|| 2 L 2 (Ω) + 1 2 (1 -||α|| ∞ C 0 ) ||u|| 2 H 1 0 (Ω) + ||y|| 2 H 1 0 (Ω) .
Then, under assumption (1.1.4) we deduce that the energy E(t) is non-negative.

Proposition 1.2.1. The unbounded linear operator A generates a C 0 semi-group of contractions on H.

Proof. Using Lumer-Phillips Theorem (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], it is sufficient to prove that A is a maximal dissipative operator so that A generates a C 0 semi-group of contractions on H. First, let U = (u, v, y, z) ∈ D(A). Then, by Green's formula we have

Re(AU, U) H = - Ω (b | ∇v | 2 +c | ∇z | 2 ) dx ≤ 0. (1.2.5)
This implies that A is dissipative. Now, let us go on with maximality.

Let F = ( f 1 , f 2 , f 3 , f 4 ) ∈ H, we look for U = (u, v, y, z) ∈ D(A) solution of -AU = F. (1.2.6)
Equivalently, we have the following system (Ω) respectively, integrating over Ω, and using Green's formula we get .2.11) and .2.12) Replacing (1.2.7) in (1.2.11) and (1.2.9) in (1.2.12), then adding the resulting equations we obtain the following variational problem:

-v = f 1 , (1.2.7) -div(∇u + b(x)∇v) + αy = f 2 , (1.2.8) -z = f 3 , (1.2.9) -div(∇y + c(x)∇z) + αu = f 4 . ( 1
Ω ∇u • ∇φ 1 dx + Ω b∇v • ∇φ 1 dx + Ω αyφ 1 dx = Ω f 2 φ 1 dx, ( 1
Ω ∇y • ∇φ 2 dx + Ω c∇z • ∇φ 2 dx + Ω αuφ 2 dx = Ω f 4 φ 2 dx. ( 1
a((u, y)(φ 1 , φ 2 )) = L(φ 1 , φ 2 ), ∀(φ 1 , φ 2 ) ∈ H 1 0 (Ω) × H 1 0 (Ω), (1.2.13)
where

a((u, y)(φ 1 , φ 2 )) = Ω (∇u • ∇φ 1 + ∇y • ∇φ 2 + αyφ 1 + αuφ 2 ) dx and L(φ 1 , φ 2 ) = Ω ( f 2 φ 1 + b∇ f 1 • ∇φ 1 + f 4 φ 2 + c∇ f 3 • ∇φ 2 ) dx.
First, using (1.1.4), a is a continuous and coercive form on

H 1 0 (Ω) × H 1 0 (Ω) then, by Lax-Milgram theorem, the variational problem (1.2.13) admits a unique solution (u, y) ∈ H 1 0 (Ω) × H 1 0 (Ω). Next, taking φ 1 ∈ C ∞ c (Ω), φ 2 ≡ 0 in (1.2.13
) and applying Green's formula again, we deduce that .2.14) This implies that div(∇u

Ω div(∇u -b∇ f 1 ) + αy φ 1 dx = Ω f 2 φ 1 dx, ∀φ 1 ∈ C ∞ c (Ω). ( 1
-b∇ f 1 ) = f 2 -αy ∈ L 2 (Ω). Similarly, by taking φ 1 = 0 and φ 2 ∈ C ∞ c (Ω) in (1.2.13) we deduce that div(∇y -c∇ f 3 ) = f 4 -αu ∈ L 2 (Ω). Finally, by setting v = -f 1 and z = -f 3 , we conclude that U = (u, v, y, z) ∈ D(A) is solution of equation (1.2.6).
To conclude, we need to show the uniqueness of such a solution. So, let U = (u, v, y, z) ∈ D(A) be a solution of equation ( 1

.2.6) with f 1 = f 2 = f 3 = f 4 = 0, then we directly deduce that v = z = 0 and therefore (u, y) ∈ H 1 0 (Ω) × H 1 0 (Ω) satisfies (1.2.13) with L(φ 1 , φ 2 ) = 0. As a is a sesquilinear, continuous coercive form on H 1 0 (Ω) × H 1 0 (Ω), we deduce that u = y = 0, in other words, ker A = {0} and 0 belongs to the resolvent set ρ(A) of A. The proof is thus complete since ρ(A) is open.
As A generates a C 0 -semigroup of contraction (e tA ) t≥0 (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], we have the following result: Theorem 1.2.2. (Existence and uniqueness of the solution)

(1) If U 0 = (u 0 , u 1 , y 0 , y 1 ) ∈ D(A), then problem (1.2.3) admits a strong unique solution U = (u, v, y, z) satisfying: U ∈ C 1 (R + , H) ∩ C 0 (R + , D(A)).
(2) If U 0 = (u 0 , u 1 , y 0 , y 1 ) ∈ H, then problem (1.2.3) admits a unique weak solution U = (u, v, y, z) satisfying:

U ∈ C 0 (R + , H). with Kelvin-Voigt dampings and non-smooth coefficient at the interface Now, we are able to study the strong stability of system (1.1.1).

Strong Stability with non compact resolvent

In this section, we will prove the strong stability of the system using a unique continuation result based on a Carleman estimate and a general criteria of Arendt-Batty (see [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF]. But, before stating the main result of this section, our goal is to prove a local unique continuation result for a coupled system of wave equation.

We define the following elliptic operator P defined on a product space by

P : H 2 (V) × H 2 (V) -→ L 2 (V) × L 2 (V) (u, y) -→ (∆u, ∆y) (1.2.15)
and the following function g defined by

g : L 2 (V) × L 2 (V) -→ L 2 (V) × L 2 (V) (u, y) -→ (-λ 2 u + αy, -λ 2 y + αu) (1.2.16)
In order to prove our result, we need a Carleman estimate represented in the following Lemma (see [START_REF] Hörmander | Linear Partial Differential Operators[END_REF], Hörmander, 2009and Theorem 3.5 in Lebeau, 1996):

Lemma 1.2.3. Let V be a bounded open set in R N and let φ = e ρψ with ψ ∈ C ∞ (R N , R); |∇ x ψ| > 0 and ρ > 0 large enough. Then, there exist τ 0 > 0 large enough and C > 0 such that τ 3 ||e τ φ u|| 2 L 2 (V) + τ||e τ φ ∇ x u|| 2 L 2 (V) ≤ C||e τ φ ∆u|| 2 L 2 (V) (1.2.17)
for all u ∈ H 2 0 (V) and τ > τ 0 . Now, we are ready to state our first result in this section (see Section 4 in [START_REF] Lebeau | Équation des ondes amorties[END_REF].

Proposition 1.2.4. Let Ω be a bounded open set in R N and x 0 be a point in Ω. In a neighborhood V of x 0 ∈ Ω, take a function f such that ∇ f ̸ = 0 in V. Moreover, let (u, y) ∈ H 2 (V) × H 2 (V) be a solution of P(u, y) = g(u, y). If u = y = 0 in {x ∈ V; f (x) ≥ f (x 0 )} then, u = y = 0 in a neighborhood of x 0 . Proof. Set W := {x ∈ V; f (x) ≥ f (x 0 )}. Choose V ′ and V ′′ neighborhoods of x 0 such that V ′′ ⊆ V ′ ⊆ V (see Figure 1.2). Let χ ∈ C ∞ c (V ′ ) such that χ = 1 in V ′′ . Set ũ = χu and ỹ = χy. Then, ( ũ, ỹ) ∈ H 2 0 (V) × H 2 0 (V). Let ψ = f (x) -c|x -x 0 | 2 and set φ = e ρψ .
Then, apply the Carleman estimate of Lemma 1.2.3 to ũ and ỹ respectively then add both inequalities to obtain

τ 3 V ′ e 2τ φ (| ũ| 2 + | ỹ| 2 )dx + τ V ′ e 2τ φ (|∇ ũ| 2 + |∇ ỹ| 2 )dx ⩽ C V ′ e 2τ φ (|∆ ũ| 2 + |∆ ỹ| 2 )dx. (1.2.18) As V ′′ ⊆ V ′ and χ ∈ C ∞ c (V ′ ) such that χ = 1 in V ′′ , we get τ 3 V ′′ e 2τ φ (|u| 2 + |y| 2 )dx + τ V ′′ e 2τ φ (|∇u| 2 + |∇y| 2 )dx ⩽ C V ′′ e 2τ φ (|∆u| 2 + |∆y| 2 )dx +C V ′ \V ′′ e 2τ φ (|∆ ũ| 2 + |∆ ỹ| 2 )dx.
with Kelvin-Voigt dampings and non-smooth coefficient at the interface

x 0 S f (x) = f (x 0 ) W φ(x) = φ(x 0 ) φ(x) = φ(x 0 ) -ϵ/2 φ(x) = φ(x 0 ) -ϵ B 0 V ′′ V ′ V FIGURE 1.2: W is the region above the curve f (x) = f (x 0 ) and S is
the region shaded with blue.

This implies that,

τ 3 V ′′ e 2τ φ (|u| 2 + |y| 2 )dx ⩽ C V ′′ e 2τ φ (|∆u| 2 + |∆y| 2 )dx + C V ′ \V ′′ e 2τ φ (|∆ ũ| 2 + |∆ ỹ| 2 )dx.
But ∆u = -λ 2 u + αy and ∆y = -λ 2 y + αu. Hence, there exists C λα > 0 such that

(τ 3 -C λα ) V ′′ e 2τ φ (|u| 2 + |y| 2 )dx ⩽ C V ′ \V ′′ e 2τ φ (|∆ ũ| 2 + |∆ ỹ| 2 )dx.
Then, there exist τ > 0 large enough and C > 0 such that .2.20) where

τ 3 V ′′ e 2τ φ (|u| 2 + |y| 2 )dx ⩽ C V ′ \V ′′ e 2τ φ (|∆ ũ| 2 + |∆ ỹ| 2 )dx. (1.2.19) As u = y = 0 in W we obtain, τ 3 V ′′ e 2τ φ (|u| 2 + |y| 2 )dx ⩽ C S e 2τ φ (|∆ ũ| 2 + |∆ ỹ| 2 )dx, ( 1
S = V ′ \ V ′′ ∪ W. Set J ϵ = {x ∈ V; φ(x) ≤ φ(x 0 ) -ϵ} and J ′ ϵ = {x ∈ V; φ(x) ≥ φ(x 0 ) -ϵ 2 }. There exists ϵ > 0 such that S ⊂ J ϵ . Then choose a ball B 0 with center x 0 such that B 0 ⊂ V ′′ ∩ J ′ ϵ .
Then, using (1.2.20), we have

B 0 (|u| 2 + |y| 2 )dx ⩽ Ce -ϵτ τ 3 S (|∆ ũ| 2 + |∆ ỹ| 2 )dx. (1.2.21) Chapter 1.
A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface Taking τ tends to infinity, we obtain that u = y = 0 in B 0 . So the desired goal is achieved.

Theorem 1.2.5. (Calderón theorem). Let Ω be a connected open set in R N and let ω ⊂ Ω, u, y) in Ω and u = y = 0 in ω, then u and y vanishes in Ω.

with ω ̸ = ∅. If (u, y) in H 2 (Ω) × H 2 (Ω) satisfies P(u, y) = g(
Proof. By setting F = supp u ∪ supp y and using Proposition 1.2.4 instead of Proposition 4.1 in the proof of Theorem 4.2 in Rousseau and Lebeau, 2011 the result holds (see Figure 1.3).

Ω F A = Ω \ F B 1 B 0 FIGURE 1.3: A dimonstrated figure
Now, we are in position to state our stability result.

Theorem 1.2.6. Let Ω be a connected open set in R N and assume that either (H1), (H2) or (H3) holds. The C 0 -semi group of contractions (e tA ) t≥0 is strongly stable in the energy space H in the sense that lim 2.6 shows that for the strong stability of the semi-group, we only need one of the damping coefficients b or c to be non-zero function.

t→∞ ∥e tA U 0 ∥ = 0, ∀U 0 ∈ H. Remark 1.2.7. Theorem 1.
The resolvent of A is not compact. Then classical methods such as Lasalle's invariance principle in [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with ana priori bounded control[END_REF] or the spectrum decomposition theory of [START_REF] Benchimol | A Note on Weak Stabilizability of Contraction Semigroups[END_REF] are not applicable in this case. We prove the strong stability with a more general criteria of [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF] that states that a C 0 -semigroup of contractions in a reflexive Banach space is strongly stable, if there is no eigenvalues on the imaginary axis and the set σ(A) ∩ iR is countable. In order to prove Theorem 1.2.6 we need the following two Lemmas.

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface Lemma 1.2.8. Assume that either (H1), (H2) or (H3) holds. Then, we have

ker(iλ -A) = {0} , ∀ λ ∈ R.
Proof. From Proposition 1.2.1 we know that 0 ∈ ρ(A), then we may assume that λ ̸ = 0 and let U = (u, v, y, z) ∈ D(A) such that AU = iλU.

(1.2.22)

Using the identity (1.2.5) and the above assumption we get .2.23) This implies that,

0 = Re(iλ∥U∥ 2 ) = Re(AU, U) = - Ω (b(x)|∇v| 2 + c(x)|∇z| 2 )dx. ( 1
√ b∇v = 0 in Ω and √ c∇z = 0 in Ω. (1.2.24)
Inserting (1.2.24) into (1.2.22), we get

       v = iλu, in Ω, ∆u -αy = iλv, in Ω, z = iλy, in Ω, ∆y -αu = iλz, in Ω. (1.2.25)
Here, we distinguish between the following three cases :

Case 1. If (H3) holds i.e. ω b ̸ = ∅, ω c ̸ = ∅ and ω 3 = ω b ∩ ω c satisfies meas(ω 3 ∩ Γ) > 0. Then, by Poincaré inequality we have v = z = 0 in ω 3 .
So, the first and the third equation in (1.2.25) imply that u = 0 and y = 0 respectively in ω 3 . Consequently, we have the following system

   λ 2 u + ∆u -αy = 0, in Ω, λ 2 y + ∆y -αu = 0, in Ω, u = y = 0, in ω 3 .
(1.2.26)

Then, by applying Theorem 1.2.5 we get u = y = 0 in Ω. Consequently, by the first and third equation of (1.2.25) we deduce that v = z = 0 in Ω and the desired goal U = 0 holds.

Case 2. For simplicity, assume that

ω b ̸ = ∅ and ω c = ∅ ( The same proof holds if ω c ̸ = ∅ and ω b = ∅). If (H2) holds then, α is a non-zero function and ω 1 = ω b ∩ ω α satisfies meas(ω 1 ∩ Γ) > 0. Then, Poincaré inequality imply that v = 0 in ω 1 .
So, the first equation in (1.2.25) gives that u = 0 in ω 1 . From the second and third equations in (1.2.25) and as ω 1 ⊆ ω α we deduce respectively that y ≡ 0 and z ≡ 0 in Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface ω 1 . Then, we get the following system

   λ 2 u + ∆u -αy = 0, in Ω, λ 2 y + ∆y -αu = 0, in Ω, u = y = 0, in ω 1 .
(1.2.27)

Similarly, by applying Theorem 1.2.5 we get u = y = 0 in Ω. Consequently, by the first and third equation of (1.2.25) we deduce that v = z = 0 in Ω. By differentiating the second equation of (1.2.25) we obtain

∂ j (αy) = 0 in ω b , ∀j = 1, .., N,
but since α is constant, we deduce that

∂ j y = 0 in ω b , ∀j = 1, .., N.
Then, for all j = 1, .., N, we have the following system

   λ 2 ∂ j u + ∆∂ j u -α∂ j y = 0, in Ω, λ 2 ∂ j y + ∆∂ j y -α∂ j u = 0, in Ω, ∂ j u = ∂ j y = 0, in ω b .
(1.2.28)

Repeating the same argument of the first case and applying Theorem 1.2.5 we obtain

∂ j u = ∂ j y = 0 in Ω, ∀j = 1, .., N.
Then,

u = y = c in Ω,
for some constant c. But as u |Γ = y |Γ = 0, we get u = y = 0 in Ω. Consequently, U = 0 in Ω.

Lemma 1.2.9. Under the assumptions of Lemma 1.2.8, we have .2.29) This is equivalent to v = iλuf 1 , z = iλyf 3 and to the following system

R(iλI -A) = H, ∀λ ∈ R. Proof. Let F = ( f 1 , f 2 , f 3 , f 4 ) ∈ H, we look for U ∈ D(A) such that iλU -AU = F. ( 1
λ 2 u + div(∇u + iλb(x)∇u) -αy = f , in Ω, λ 2 y + div(∇y + iλc(x)∇z) -αu = g, in Ω, (1.2.30) where f = -f 2 -iλ f 1 + div(b∇ f 1 ) ∈ H -1 (Ω) and g = -f 4 -iλ f 3 + div(c∇ f 3 ) ∈ H -1 (Ω). Now, define the linear operator L : H 1 0 (Ω) × H 1 0 (Ω) -→ H -1 (Ω) × Chapter 1.
A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface

H -1 (Ω) by L u y :=   -div(∇u + iλb∇u) + αy -div(∇y + iλc∇y) + αu   . (1.2.31) Let U = (u, y) T and F = ( f , g) T , then (1.2.30) is equivalent to (λ 2 I -L)U = F. (1.2.32) Since the operator L is an isomorphism from H 1 0 (Ω) × H 1 0 (Ω) onto H -1 (Ω) × H -1 (Ω) and I is a compact operator from H 1 0 (Ω) × H 1 0 (Ω) onto H -1 (Ω) × H -1 (Ω). Then, using Fredholm's Alternative theorem, problem (1.2.32) admits a unique so- lution in H 1 0 (Ω) × H 1 0 (Ω) if and only if λ 2 I -L is injective. For that purpose, let U = (u, y) ∈ ker(λ 2 I -L). Then, if we set v = iλu and z = iλy, we deduce that U = (u, v, y, z) ∈ D(A) is a solution of (iλ -A)U = 0. Using Lemma 1.2.8, we deduce that u = v = y = z = 0. This implies that equation (1.2.32) admits a unique solution U = (u, y) ∈ H 1 0 (Ω) × H 1 0 (Ω) and div(∇y + iλc∇y -c∇ f 3 ), div(∇u + iλb∇u -b∇ f 1 ) ∈ L 2 (Ω). By setting v = iλu -f 1 and z = iλy -f 3 , we deduce that U = (u, v, y, z) ∈ D(A)
is the unique solution of equation (1.2.29) and the proof is thus complete.

Proof of Theorem 1.2.6. Using Lemma 1.2.8, the operator A has no pure imaginary eigenvalues and by Lemma 1.2.9, we have R(iλI -A) = H, for all λ in R. Therefore, the closed graph theorem implies that σ(A) ∩ iR = ∅. Following Arendt-Batty (see [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF], the C 0 -semi group of contractions (e tA ) t≥0 is strongly stable and the proof is complete.

Analytic Stability

In this section, under the condition that ω b = ω c = Ω, we will prove that the associated semigroup of the system (1.1.1) is analytic. For this aim, we will use the following result (see Corollary 3.7.18 page 157 in [START_REF] Arendt | Vector-valued Laplace transforms and Cauchy problems[END_REF]. Theorem 1.3.1. Let (S(t) = e tA ) t≥0 be a C 0 -semi group of contractions in a Hilbert space. Assume that iR ⊂ ρ(A) (A1).

Then, (e tA ) t≥0 is analytic if and only if Proof. We know that condition (A1) is verified. Our goal now is to prove that condition (A2) is satisfied. To this aim, we proceed by a contradiction argument. Suppose that (A2) does not hold, then there exist a sequence (λ n ) ⊂ R and a sequence

lim sup λ∈R,|λ|→∞ 1 |λ| -1 ∥(iλ -A) -1 ∥ L(H) < ∞ (A2).
(U n ) ⊂ D(A) such that |λ n | -→ +∞, ∥U n ∥ H = ∥(u n , v n , y n , z n )∥ H = 1 (1.3.1) and λ -1 n (iλ n -A)U n = ( f n 1 , g n 1 , f n 2 , g n 2 ) -→ 0 in H (1.3.2)
are satisfied.

Step 1. (The dissipation).

Multiply in H equation (1.3.2) by the uniformly bounded sequence

U n = (u n , v n , y n , z n ), we get λ -1 n Ω (b|∇v n | 2 + c|∇z n | 2 ) dx = -λ -1 n Re((iλ n I -A)U n , U n ) H = o(1).
Then, using (1.1.7), (1.1.9) and as

ω b = ω c = Ω, it follows that λ -1/2 n ∥∇v n ∥ L 2 (Ω) = o(1), (1.3.3) and λ -1/2 n ∥∇z n ∥ L 2 (Ω) = o(1). (1.3.4)
Using Poincaré inequality, we get

λ -1/2 n ∥v n ∥ L 2 (Ω) = o(1), λ -1/2 n ∥z n ∥ L 2 (Ω) = o(1). (1.3.5)
In what follows, we drop the index n for simplicity.

Step 2. (Information on ∇u and ∇y).

By detailing equation (1.3.2), we get the following system

λ -1 (iλu -v) = f 1 -→ 0 in H 1 0 (Ω), (1.3.6) λ -1 (iλv -div(∇u + b∇v) + αy) = g 1 -→ 0 in L 2 (Ω), (1.3.7) λ -1 (iλy -z) = f 2 -→ 0 in H 1 0 (Ω), (1.3.8) λ -1 (iλz -div(∇y + c∇z) + αu) = g 2 -→ 0 in L 2 (Ω). (1.3.9)
From equations (1.3.3), (1.3.6), (1.3.8), (1.3.4) and (1.3.5), we get 1).

∥∇u∥ L 2 (Ω) = o(1) and ∥∇y∥ L 2 (Ω) = o(1), ∥u∥ L 2 (Ω) = o(1) and ∥y∥ L 2 (Ω) = o(
(1.3.10)

Step 3. (Information on v and z).

Multiplying equation (1.3.7) by v, then integrating over Ω gives

i Ω |v| 2 dx + λ -1 Ω ∇u • ∇v dx + λ -1 Ω b|∇v| 2 dx +λ -1 Ω αyv dx = Ω g 1 v dx.
( 1.3.11) Then, taking the imaginary part of (1.3.11) and using (1.3.3), (1.3.10) and the fact that 1).

∥g 1 ∥ H 1 0 (Ω) = o(1), we get ∥v∥ L 2 (Ω) = o(
(1.3.12)

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface Similarly, multiplying equation (1.3.9) by z, then integrating over Ω gives

i Ω |z| 2 dx + λ -1 Ω ∇y • ∇z dx + λ -1 Ω c|∇z| 2 dx +λ -1 Ω αuz dx = Ω g 2 z dx.
( 1.3.13) Then, taking the imaginary part of (1.3.13) and using (1.3.4), (1.3.10) and the fact that .3.14) Consequently, by (1.3.10), (1.3.12) and (1.3.14) we deduce the desired contradiction.

∥g 2 ∥ H 1 0 (Ω) = o(1), we get ∥z∥ L 2 (Ω) = o(1). ( 1

Non-Uniform stability of the system

In this section, we will prove the non-uniform stability of system (1.1.1). Our main result is represented in the following theorem.

Theorem 1.4.1. Let c = 0, b = 1 and α = α 0 be a positive constant. Then the energy of the system (1.1.1) does not decay uniformly to zero as t tends to infinity.

To prove Theorem 1.4.1 we study the asymptotic behavior of the eigenvalues of the operator A near the imaginary axis. So, let λ be an eigenvalue of A and Φ = (u, v, y, z) be an associated eigenvector, i.e,

AΦ = λΦ, (1.4.1) equivalently,                            v = λu, in Ω, ∆u + λ∆u -αy = λv, in Ω, z = λy, in Ω, ∆y -αu = λz, in Ω, u = y = 0, on Γ. (1.4.2)
Eliminating v and z from (1.4.2), we get

           λ 2 u -∆u -λ∆u + αy = 0, , in Ω, λ 2 y -∆y + αu = 0, in Ω, u = y = 0, on Γ. (1.4.3)
From the second equation of (1.4.3), we have .4.4) with Kelvin-Voigt dampings and non-smooth coefficient at the interface Substituting (1.4.4) in the first equation of (1.4.3), we obtain

u = 1 α ∆y -λ 2 y . ( 1
   ∆ 2 y -λ 2 2 + λ 1 + λ ∆y + λ 4 -α 4 1 + λ y = 0, in Ω, y = ∆y = 0, on Γ.
(1.4.5) Now, let {µ 2 k } ∞ k=1 be the sequence of eigenvalues of the Laplace operator with Dirichlet boundary conditions in Ω and eigenvectors φ k , i.e,

-∆φ k = µ 2 k φ k , in Ω, φ k = 0, on Γ. (1.4.6)
Then by taking y = φ k in (1.4.5), we deduce that λ will be an eigenvalue of

A if λ is a root of P(λ) = λ 4 + µ 2 k λ 3 + 2µ 2 k λ 2 + µ 4 k λ + µ 4 k -α 2 = 0. (1.4.7)
We have the following result,

λ + k = iµ k - α 2 2µ 4 k + O 1 µ 9 k , ∀|k| ≥ k 0 (1.4.8)
and

λ - k = -iµ k - α 2 2µ 4 k + O 1 µ 9 k , ∀|k| ≥ k 0 .
(1.4.9)

Proof. First, set ξ = λ µ k and ζ k = 1 µ k . Then, multiply equation (1.4.7) by 1 µ 5 k , we get h(ξ) = ξ 3 + ξ + ζ k + 2ξ 2 ζ k + ξ 4 ζ k -α 2 ζ 5 k = 0. (1.4.10)
Then, in order to find the roots of P, i.e. the eigenvalues of A we need before to calculate the roots of h. We will continue the proof with three steps:

Step 1. Let f (ξ) = ξ 3 + ξ and g(ξ) = ξ 4 ζ k + ζ k + 2ξ 2 ζ k -α 2 ζ 5 k . We look for r k sufficiently small such that | f | > |h -f | = |g|, on ∂D, where, D = {ξ ∈ C; |ξ -i| ≤ r k }. Let ξ ∈ ∂D(i, r k ), then ξ = i + r k e iθ with 0 ≤ θ ≤ 2π. We have f (ξ) = ξ 3 + ξ = ξr k (2ie iθ + r k e 2iθ ). But, if r k ≤ 1 2 then, |ξ| ≥ |1 -r k | ≥ 1 2 50 Chapter 1.
A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface

and |2ie iθ + r k e 2iθ | ≥ |2ie iθ | -r k ≥ 2 -r k ≥ 3/2.
This gives that

| f | = |ξ 3 + ξ| ≥ 3r k 4 , if r k ≤ 1/2.
On the other hand, since ξ is bounded in D and ζ k → 0 we have,

|g(ξ)| ≤ c 1 ζ k , for some constant c 1 > 0.
So, it is enough to choose r k = 4c 1 3 ζ k . Similarly, we can find r k sufficiently small such that

| f | > |h -f | = |g|, on ∂D ′ = ∂{ξ ∈ C; |ξ + i| ≤ r k }.
Step 2. In this step, we prove that P admits an infinity of simple roots denoted by .4.11) Using

λ + k and λ - k for |k| ≥ k 0 with k 0 large enough such that λ ± k = ±µ k + µ k ϵ ± k , with lim |k|→∞ ϵ ± k = 0. ( 1
Step 1 and thanks to Rouché's theorem, there exists k 0 large enough such that for all |k| ≥ k 0 the roots of the polynomial h are close to the roots of the polynomial f (ξ) = ξ 3 + ξ. Then,

ξ + k = i + ϵ + k and ξ - k = -i + ϵ - k , with lim |k|→∞ ϵ ± k = 0. (1.4.12)
Using the fact that λ ± k = µ k ξ ± k we deduce (1.4.11).

Step 3. Asymptotic behavior of ϵ ± k . Inserting equation (1.4.12) into equation (1.4.10), we get .4.13) From equation (1.4.13), we can predict that ϵ ± k is in the form

ϵ ± k -2 + ± 3i - 4 µ k ϵ ± k + 1 ± 4i µ k (ϵ ± k ) 2 + 1 µ k (ϵ ± k ) 3 = α 2 µ 5 k . ( 1
ϵ ± k = -α 2 2µ 5 k + η ± k , (1.4.14)
with η ± k tends to zero as k tends to infinity. Now, inserting expression (1.4.14) in (1.4.13), we find .4.15) with Kelvin-Voigt dampings and non-smooth coefficient at the interface where,

a ± 0,k + a ± 1,k η ± k + a ± 2,k (η ± k ) 2 + a ± 3,k (η ± k ) 3 + a ± 4,k (η ± k ) 4 = 0, ( 1
                                       a ± 0,k = α 8 16µ 21 k -α 6 8µ 15 k -α 4 µ 11 k ± (-α 6 2µ 16 k + 3α 4 4µ 10 k )i, a ± 1,k = -α 6 2µ 16 k + 3α 4 4µ 10 k + 4α 2 µ 6 k -2 ± (-3α 2 µ 5 k + 3α 4 µ 11 k )i, a ± 2,k = 3α 4 2µ 11 k -3α 2 2µ 5 -4 µ k ± (3 -6α 2 µ 6 k )i, a ± 3,k = -2α 2 µ 6 k + 1 ± 4i µ k , a ± 4,k = 1 µ k .
( 1.4.16) We can notice that a ± 1,k ̸ = 0 for k large enough, a ± i,k is bounded uniformly for k large enough, i = 0, .., 4 and |a ± 1,k | ≥ 1, ∀k ≥ k 0 . Therefore, using (1.4.15) there exists C > 0 (independent of k) such that

a ± 0,k a ± 1,k + η ± k ≤ C|η ± k | 2 .
(1.4.17)

Hence, for k large enough, we deduce that

|η ± k | ≤ 2 a ± 0,k a ± 1,k . Simple calculations yields a ± 0,k a ± 1,k = O 1 µ 10 k .
This implies that

ϵ ± k = -α 2 2µ 5 k + O 1 µ 10 k . Consequently ξ ± k = ±i + -α 2 2µ 5 k + O 1 µ 10 k .
Finally, using the fact that λ ± k = µ k ξ ± k we deduce that (1.4.8) and (1.4.9) hold.

Polynomial Stability with one damping

From now on, we assume that α(x) ≥ 0 for all x ∈ Ω. In this section, we will study the polynomial decay of our system with only one damping term (b ≥ 0, c = 0 or b = 0, c ≥ 0 on Ω). One of the main ingredients is to use the exponential or polynomial decay of the wave equation with frictional damping in ω 1 = ω b ∩ ω α , namely Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface

     φ tt -∆φ + 1 ω 1 φ t = 0 in Ω × (0, ∞), φ = 0 on Γ × (0, ∞), φ(t = 0) = φ 0 , φ t (t = 0) = φ 1 .
(

ω 1 = ω b ∩ ω α satisfies meas(ω 1 ∩ Γ) > 0.
Further, assume that the energy of the system (1.5.1) is exponentially stable. Then, there exists a constant C > 0 such that for every initial data U 0 = (u 0 , u 1 , y 0 , y 1 ) ∈ D(A), the energy of system (1.1.1) verifies the following estimate:

E(t) ≤ C 1 √ t ∥U 0 ∥ 2 D(A) , ∀t > 0. (1.5.2)
Proof. Following [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] (see also [START_REF] Liu | Locally Distributed Control and Damping for the Conservative Systems[END_REF]Rao, 2005, Batty and[START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF], a C 0 semigroup of contractions (e tA ) t≥0 on a Hilbert space H verifies

(1.5.2) if iR ⊂ ρ(A) (S1) and lim sup |λ|→∞ 1 λ 4 ∥(iλ -A) -1 ∥ L(H) < ∞ (S2).
We know that condition (S1) is verified. Our goal now is to prove that condition (S2) is satisfied. To this aim, we proceed by a contradiction argument. Suppose that (S2) does not hold, then there exist a sequence (λ n ) ⊂ R and a sequence

(U n ) ⊂ D(A) such that |λ n | -→ +∞, ∥U n ∥ H = ∥(u n , v n , y n , z n )∥ H = 1 (1.5.3)
and

λ 4 n (iλ n -A)U n = ( f n 1 , g n 1 , f n 2 , g n 2 ) -→ 0 in H (1.5.4) are satisfied.
In what follows, we drop the index n for simplicity.

Lemma 1.5.2. Under all the above assumptions, we have

∥∇v∥ L 2 (ω 1 ) = o(1) λ 2 and ∥v∥ L 2 (ω 1 ) = o(1) λ 2 .
(1.5.5)

Proof. Multiply in H equation (1.5.4) by the uniformly bounded sequence U = (u, v, y, z), we get

Ω b(x)|∇v| 2 dx = -Re((iλI -A)U, U) H = o(1) λ 4 . It follows that ∥ √ b∇v∥ L 2 (Ω) = o(1) λ 2 .
(1.5.6) with Kelvin-Voigt dampings and non-smooth coefficient at the interface Using (1.1.7), (1.5.6) and Poincaré inequality (since meas(ω 1 ∩ Γ) > 0) we get (1.5.5).

By detailing equation (1.5.4), we get the following system

λ 4 (iλu -v) = f 1 -→ 0 in H 1 0 (Ω), (1.5.7) λ 4 (iλv -div(∇u + b∇v) + αy) = g 1 -→ 0 in L 2 (Ω), (1.5.8) λ 4 (iλy -z) = f 2 -→ 0 in H 1 0 (Ω), (1.5.9) λ 4 (iλz -∆y + αu) = g 2 -→ 0 in L 2 (Ω).
(1.5.10)

From equations (1.5.3), (1.5.7) and (1.5.9) we get .5.11) and using (1.5.3), (1.5.5) and ( 1.5.7) we also have

∥u∥ L 2 (Ω) = O(1) λ , ∥y∥ L 2 (Ω) = O(1) λ , ( 1
∥∇u∥ L 2 (ω 1 ) = o(1)
λ 3 , and ∥u∥ L 2 (ω 1 ) = o(1)

λ 3 .
(1.5.12)

Lemma 1.5.3. Under all the above assumptions, we have

Ω |λu| 2 dx = o(1). (1.5.13) Proof. For all n ∈ N, let φ n , ψ n ∈ H 2 (Ω) ∩ H 1 0 (Ω)
be the solution of the following system .5.16) where (u n , v n , y n , z n ) is the solution of (1.5.7)- (1.5.10). If (1.5.1) with the local viscous damping 1 ω 1 φ t ( or 1 ω 1 ψ t ) is exponentially stable then, following [START_REF] Huang | Characteristics conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF], there exists M > 0 independent of n such that

λ 2 n φ n + ∆φ n -iλ n 1 ω 1 φ n = u n , in Ω, (1.5.14) λ 2 n ψ n + ∆ψ n -iλ n 1 ω 1 ψ n = y n , in Ω, (1.5.15) φ n = ψ n = 0, in Γ, ( 1
||λφ n || 2 L 2 (Ω) + ||∇φ n || 2 L 2 (Ω) + ||λψ n || 2 L 2 (Ω) + ||∇ψ n || 2 L 2 (Ω) ≤ M(||u n || 2 L 2 (Ω) + ||y n || 2 L 2 (Ω) )
.

(1.5.17)

For simplicity, we drop the index n. Now, multiplying (1.5.7) and (1.5.8) by iλ 3 φ and λ 2 φ respectively and applying Green's formula, we get

- Ω λ 2 u(λ 2 φ + ∆φ) dx + Ω λ 2 b∇v • ∇φ dx + Ω λ 2 αyφ dx = Ω ( g 1 φ λ 2 + i f 1 φ λ ) dx.
(1.5.18)

Using (1.5.14) we get

- Ω |λu| 2 dx + Ω iλ 3 1 ω 1 uφ dx + Ω λ 2 b∇v • ∇φ dx + Ω λ 2 αyφ dx = Ω ( g 1 φ λ 2 + i f 1 φ λ ) dx.
(1.5.19)

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface But, using Cauchy-Schwarz inequality, (1.5.12), (1.5.17) and (1.5.3) we have,

Ω iλ 3 1 ω 1 uφ dx ≤ ||λ 3 u|| L 2 (ω 1 ) ||φ|| L 2 (Ω) = o(1)
( 1.5.20) and by (1.5.6), (1.5.17) and (1.5.3) 1).

Ω λ 2 b∇v • ∇φ dx ≤ ||λ 2 b∇v|| L 2 (Ω) ||∇φ|| L 2 (Ω) = o(
(1.5.21)

Again, using Cauchy-Schwarz inequality, (1.5.11), (1.5.17) and the fact ||u||

L 2 (Ω) = o(1) we obtain, Ω λ 2 αyφ dx ≤ ||α|| ∞ ||λy|| L 2 (Ω) ||λφ|| L 2 (Ω) = o(1). (1.5.22)
Consequently, by (1.5.19) and the fact that φ, f 1 and g 1 are bounded in L 2 (Ω) and using (1.5.20)-( 1

-λ 2 u -div(∇u + b∇v) + αy = g 1 λ 4 + i f 1 λ 3 .
(1.5.24)

Similarly, inserting equation (1.5.9) in equation ( 1.5.10) we obtain

-λ 2 y -∆y + αu = g 2 λ 4 + i f 2 λ 3 .
(1.5.25)

Now multiplying (1.5.24) and (1.5.25) by y and u respectively then integrating over Ω and applying Green's formula we obtain

- Ω λ 2 uy dx + Ω ∇u • ∇y dx + Ω b∇v • ∇y dx + Ω α|y| 2 dx = Ω ( g 1 y λ 4 + i f 1 y λ 3 ) dx, (1.5.26) and - Ω λ 2 yu dx + Ω ∇y • ∇u dx + Ω α|u| 2 dx = Ω ( g 2 u λ 4 + i f 2 u λ 3 ) dx.
(1.5.27)

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface Taking the real part of (1.5.26) and (1.5.27) then subtracting both equations, using (1.5.11) and the fact that f 1 , g 1 , f 2 and g 2 are bounded in L 2 (Ω) we get

Ω α|u| 2 dx + Re Ω b∇v • ∇y dx - Ω α|y| 2 dx = o(1) λ 4 .
(1.5.28)

Using Cauchy-Schwarz inequality, (1.5.6) and the fact that ∇y is bounded in L 2 (Ω) we have

Re

Ω b∇v • ∇y dx ≤ Ω b∇v • ∇y dx ≤ ||b∇v|| L 2 (Ω) ||∇y|| L 2 (Ω) = o(1) λ 2 . (1.5.29)
This yields,

Ω α|u| 2 dx - Ω α|y| 2 dx = o(1) λ 2 .
(1.5.30)

Using (1.5.13) and the fact that α ∈ L ∞ (Ω) we obtain

Ω α|λy| 2 dx = o(1).
But, using condition (1.1.8) we have

α 0 ω 1 |λy| 2 dx ≤ Ω α|λy| 2 dx = o(1).
Consequently,

ω 1 |λy| 2 dx = o(1)
.

(1.5.31)

Now, multiplying (1.5.9) and (1.5.10) by iλ 3 ψ and λ 2 ψ respectively and applying Green's formula, we get .5.32) Using (1.5.15) we get

- Ω λ 2 y(λ 2 ψ + ∆ψ) dx + Ω λ 2 αuψ dx = Ω ( g 2 ψ λ 2 + i f 2 ψ λ ) dx. ( 1
- Ω |λy| 2 dx + Ω iλ 3 1 ω 1 yψ dx + Ω λ 2 αuψ dx = Ω ( g 2 ψ λ 2 + i f 2 ψ λ ) dx.
(1.5.33)

But, using Cauchy-Schwarz inequality, (1.5.31), (1.5.17) and (1.5.3) we have

Ω iλ 3 1 ω 1 yψ dx ≤ ||λ 2 y|| L 2 (ω 1 ) ||λψ|| L 2 (Ω) = o(1)
(1.5.34)

and

Ω λ 2 αuψ dx ≤ ||α|| ∞ ||λu|| L 2 (Ω) ||λψ|| L 2 (Ω) = o(1). (1.5.35) Chapter 1.
A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface Then, by (1.5.33), the fact that ψ is bounded in L 2 (Ω) by (1.5.11) and (1.5.17), the fact that f 2 , g 2 are bounded in L 2 (Ω) and using (1.5.34)- (1.5.35), we obtain

Ω |λy| 2 dx = o(1).
Lemma 1.5.5. Under all the above assumptions, we have

Ω |∇u| 2 dx = o(1)
and 1).

Ω |∇y| 2 dx = o(
(1.5.36)

Proof. Multiply (1.5.24) and ( 1.5.25) by u and y respectively, then integrate over Ω and apply Green's formula, we obtain

- Ω |λu| 2 dx + Ω |∇u| 2 dx + Ω b∇v • ∇u dx + Ω αyu dx = Ω ( g 1 u λ 4 + i f 1 u λ 3 ) dx, (1.5.37) and - Ω |λy| 2 dx + Ω |∇y| 2 dx + Ω αuy dx = Ω ( g 2 y λ 4 + i f 1 y λ 3 ) dx. (1.5.38)
But, using Cauchy-Schwarz inequality, (1.5.6) and (1.5.3) we have

Ω b∇v • ∇u dx ≤ ||b∇v|| L 2 (Ω) ||∇u|| L 2 (Ω) = o(1)
and

Ω αyu dx ≤ ||α|| ∞ ||u|| L 2 (Ω) ||y|| L 2 (Ω) = o(1)
.

Consequently, we get

Ω |∇u| 2 dx = Ω |λu| 2 dx + o(1)
and

Ω |∇y| 2 dx = Ω |λy| 2 dx + o(1)
.

By (1.5.13) and (1.5.23) we deduce that

Ω |∇u| 2 dx = o(1)
and

Ω |∇y| 2 dx = o(1).
In conclusion, by (1.5.13), (1.5.23) and (1.5.36) we deduce that ∥U n ∥ H = o(1) which leads to the desired contradiction.

Remark 1.5.6.

1) Theorem 1.5.1 shows that we can have b ̸ = 0, c ≥ 0 or b ≥ 0, c ̸ = 0. In other words, to prove the polynomial energy decay rate of type t -1/2 we only need one of the dampings to be effective.

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface 2) From Zuazua, 1990, Theorem 1.1 and Remark 1.2 we deduce that if the boundary of Ω is C 2 and ω 1 is a neighbourhood of the boundary, then the solution of (1.5.1) is exponentially decaying. In fact, we know that given any point x 0 ∈ R N , (1.5.17) is satisfied when ω 1 is a neighborhood of Γ(x 0 ) where

Γ(x 0 ) = {x ∈ Γ : (x -x 0 ).ν(x) > 0},
where ν(x) is the unit outward normal at x ∈ Γ.

3) Also, from [START_REF] Zuazua | Exponential decay for the semilinear wave equation with locally distributed damping[END_REF] and by [START_REF] Bardos | Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary[END_REF] (see also [START_REF] Lebeau | Équation des ondes amorties[END_REF]Lebeau, -1994) ) for domains with boundaries, we know that when Ω is of class C ∞ , (1.5.17) is satisfied when ω 1 satisfies the Geometric Control Condition (GCC).

Recall that the GCC can be formulated as follows: For a subset ω of Ω, we shall say that ω satisfies the Geometric Control Condition 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. β with β > 0, then the energy of the system (1.1.1) decays polynomially as t Let ω α satisfies meas(ω α ∩ Γ) > 0 and let ω α = supp α ⊂ ω b . Also, assume that the energy of the system (1.5.1) is exponentially stable (β = 0) or polynomially stable as t -4 β , with β > 0. Then, there exists a constant C > 0 such that for every initial data U 0 = (u 0 , u 1 , y 0 , y 1 ) ∈ D(A), the energy of system (1.1.1) verifies the following estimate:

4) In

E(t) ≤ C t 2 ℓ ∥U 0 ∥ 2 D(A) , ℓ = 2β + 4, β ≥ 0, ∀t > 0.
(1.5.39)

Proof. Following [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], (see also [START_REF] Liu | Locally Distributed Control and Damping for the Conservative Systems[END_REF]Rao, 2005, Batty and[START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF], a C 0 -semigroup of contractions (e tA ) t≥0 on a Hilbert space H verifies (1.5.39) if (S1) holds and lim sup

|λ|→∞ 1 λ ℓ ∥(iλ -A) -1 ∥ L(H) < ∞ (S3).
We know that condition (S1) is verified. Our goal now is to prove that condition (S3) is satisfied. To this aim, we proceed by a contradiction argument. Suppose that (S3) does not hold, then there exist a sequence (λ n ) ⊂ R and a sequence (U n ) ⊂ D(A) such that (1.5.3) and In what follows, we drop the index n for simplicity.

λ ℓ n (iλ n -A)U n = ( f n 1 , g n 1 , f n 2 , g n 2 ) -→ 0 in H (
Lemma 1.5.8. Under all the above assumptions, we have

∥∇v∥ L 2 (ω b ) = o(1)
λ ℓ/2 and ∥v∥ L 2 (ω b ) = o(1) λ ℓ/2 . (1.5.41)
The proof is the same as the one of Lemma 1.5.2.

By detailing equation ( 1.5.40), we get the following system .5.45) From equations (1.5.42) and (1.5.44) and using (1.5.3) we get (1.5.11), and using (1.5.41), (1.5.42), (1.5.3), the fact that ω α = supp α ⊂ ω b , and Poincaré inequality (since meas(ω α ∩ Γ) > 0), we also have

λ ℓ (iλu -v) = f 1 -→ 0 in H 1 0 (Ω), (1.5.42) λ ℓ (iλv -div(∇u + b∇v) + αy) = g 1 -→ 0 in L 2 (Ω), (1.5.43) λ ℓ (iλy -z) = f 2 -→ 0 in H 1 0 (Ω), (1.5.44) λ ℓ (iλz -∆y + αu) = g 2 -→ 0 in L 2 (Ω). ( 1
∥∇u∥ L 2 (ω α ) = o(1) λ ℓ 2 +1
and ∥u∥

L 2 (ω α ) = o(1) λ ℓ 2 +1
.

(1.5.46) Lemma 1.5.9. Under all above assumptions, we have .5.48) Similarly, inserting equation (1.5.44) in equation ( 1.5.45) we obtain

-λ 2 u -div(∇u + b∇v) + αy = g 1 λ ℓ + i f 1 λ ℓ-1 . ( 1
-λ 2 y -∆y + αu = g 2 λ ℓ + i f 2 λ ℓ-1 .
(1.5.49)

Now multiplying (1.5.48) and (1.5.49) by y and u respectively, then integrating over Ω and applying Green's formula we obtain

- Ω λ 2 uy dx + Ω ∇u • ∇y dx + Ω b∇v • ∇y dx + Ω α|y| 2 dx = Ω ( g 1 y λ ℓ + i f 1 y λ ℓ-1 ) dx, (1.5.50) and - Ω λ 2 yu dx + Ω ∇y • ∇u dx + Ω α|u| 2 dx = Ω ( g 2 u λ ℓ + i f 2 u λ ℓ-1 ) dx.
(1.5.51)

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface Taking the real part of (1.5.50) and (1.5.51) then subtracting both equations, using (1.5.11) and the fact that f 1 , g 1 , f 2 and g 2 converge to zero in L 2 (Ω) we get

Ω α|u| 2 dx + Re Ω b∇v • ∇y dx - Ω α|y| 2 dx = o(1) λ ℓ .
(1.5.52)

Using Cauchy-Schwarz inequality, (1.5.41) and the fact that ∇y is bounded in L 2 (Ω) we have

Re .5.53) This yields,

Ω b∇v • ∇y dx ≤ Ω b∇v • ∇y dx ≤ ||b∇v|| L 2 (Ω) ||∇y|| L 2 (Ω) = o(1) λ ℓ/2 . ( 1
Ω α|u| 2 dx - Ω α|y| 2 dx = o(1) λ ℓ/2 .
(1.5.54)

But, using (1.5.46) and the fact that ω α = suppα, we get

Ω α|u| 2 dx ≤ ||α|| ∞ ω α |u| 2 dx = o(1) λ ℓ 2 +1
.

Then, (1.5.54) becomes

Ω α|y| 2 dx = o(1)
λ ℓ/2 . But, using condition (1.1.8) and as ω α = suppα, we deduce that (1.5.56)

Proof. For all n ∈ N, let φ n , ψ n ∈ H 2 (Ω) ∩ H 1 0 (Ω) be the solution of (1.5.14)- (1.5.16). If the system (1.5.1) with ω 1 = ω α decays exponentially (β = 0) or polynomially as t -4

β , with β > 0. Then, following [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], there exists M > 0 independent of n such that

||λφ n || 2 L 2 (Ω) + ||∇φ n || 2 L 2 (Ω) + ||λψ n || 2 L 2 (Ω) + ||∇ψ n || 2 L 2 (Ω) ≤ M|λ| β (||u n || 2 L 2 (Ω) + ||y n || 2 L 2 (Ω) )
.

(1.5.57)

Hence, using (1.5.11) we have,

         ||λφ|| L 2 (Ω) = O(1) λ 1-β 2 , ||∇φ|| L 2 (Ω) = O(1) λ 1-β 2 , ||λψ|| L 2 (Ω) = O(1) λ 1-β 2 , ||∇ψ|| L 2 (Ω) = O(1) λ 1-β 2 .
(1.5.58)

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface Now, multiplying (1.5.42) and (1.5.43) by iλ 3 φ and λ 2 φ respectively and applying Green's formula, we get

- Ω λ 2 u(λ 2 φ + ∆φ) dx + Ω λ 2 b∇v • ∇φ dx + Ω λ 2 αyφ dx = Ω ( g 1 φ λ ℓ-2 + i f 1 φ λ ℓ-3 ) dx.
(1.5.59)

Using (1.5.14) we get

- Ω |λu| 2 dx + Ω iλ 3 1 ω α uφ dx + Ω λ 2 b∇v • ∇φ dx + Ω λ 2 αyφ dx = Ω ( g 1 φ λ ℓ-2 + i f 1 φ λ ℓ-3 ) dx.
(1.5.60)

But, using Cauchy-Schwarz inequality, (1.5.46) and ( 1.5.58) we have, .5.61) This means that we need

Ω iλ 3 1 ω α uφ dx ≤ |λ| 2 ||u|| L 2 (ω α ) ||λφ|| L 2 (Ω) = o(1) λ ℓ-β 2 . ( 1
ℓ-β 2 ≥ 0, i.e, ℓ ≥ β so that Ω λ 3 1 ω α uφ dx = o(1).
Also, by (1.5.41) and ( 1.5.58) we have .5.62) This means that we should have

Ω λ 2 b∇v • ∇φ dx ≤ |λ| 2 ||b∇v|| L 2 (Ω) ||∇φ|| L 2 (Ω) = o(1) λ ℓ-β 2 -1 . ( 1
ℓ-β 2 -1 ≥ 0, i.e, ℓ ≥ β + 2 so that Ω λ 2 b∇v • ∇φ dx = o(1).
Again, using Cauchy-Schwarz inequality, (1.5.47), (1.5.58) and as ω α = suppα, we obtain .5.63) This means that we should have

Ω λ 2 αyφ dx ≤ ||α|| ∞ |λ|||y|| L 2 (ω α ) ||λφ|| L 2 (Ω) = o(1) λ ℓ-2β 4 . ( 1
ℓ-2β 4 ≥ 0, i.e, ℓ ≥ 2β so that Ω λ 2 αyφ dx = o(1).
On the other hand, using Cauchy-Schwarz inequality, (1.5.58) and the fact that f 1 and g 1 converge to zero in L 2 (Ω) we have

Ω ( g 1 φ λ ℓ-2 + i f 1 φ λ ℓ-3 ) dx ≤ 1 |λ| ℓ-2 ||g 1 || L 2 (Ω) + 1 |λ| ℓ-3 || f 1 || L 2 (Ω) ||φ|| L 2 (Ω) = o(1)
λ ℓ-β 2 + o(1) λ ℓ-1-β 2 .
(1.5.64)

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface This means that we need ℓ -

1 -β 2 ≥ 0, i.e, ℓ ≥ β 2 + 1 so that Ω ( g 1 φ λ ℓ-2 + i f 1 φ λ ℓ-3 ) dx = o(1
- Ω λ 2 y(λ 2 ψ + ∆ψ) dx + Ω λ 2 αuψ dx = Ω ( g 2 ψ λ ℓ-2 + i f 2 ψ λ ℓ-3 ) dx.
(1.5.66)

Using (1.5.15) we get

- Ω |λy| 2 dx + Ω iλ 3 1 ω α yψ dx + Ω λ 2 αuψ dx = Ω ( g 2 ψ λ ℓ-2 + i f 2 ψ λ ℓ-3 ) dx.
(1.5.67)

But, using Cauchy-Schwarz inequality, (1.5.55) and (1.5.58) we have .5.68) This means that we need ℓ ≥ 2β + 4 so that

Ω iλ 3 1 ω α yψ dx ≤ |λ| 2 ||y|| L 2 (ω α ) ||λψ|| L 2 (Ω) = o(1) λ ℓ-2β 4 -1 . ( 1
Ω λ 3 1 ω α yψ dx = o(1).
Again using Cauchy-Schwarz inequality, (1.5.58), (1.5.46) and the fact that ω α = supp α, we have .5.69) This means that we need ℓ ≥ β -2 so that Ω λ 2 αuψ dx = o(1). with Kelvin-Voigt dampings and non-smooth coefficient at the interface On the other hand, by Cauchy-Schwarz inequality, (1.5.58) and the fact that f 2 , g 2 converge to zero in L 2 (Ω) we have

Ω λ 2 αuψ dx ≤ ||α|| ∞ ||λu|| L 2 (ω α ) ||λψ|| L 2 (Ω) = o(1) λ 1+ ℓ-β 2 . ( 1
Ω ( g 2 ψ λ ℓ-2 + i f 2 ψ λ ℓ-3 ) dx ≤ 1 |λ| ℓ-1 ||g 2 || L 2 (Ω) + 1 |λ| ℓ-2 || f 2 || L 2 (Ω) ||λψ|| L 2 (Ω) = o(1)
λ ℓ-β 2 + o(1) λ ℓ-1-β 2 .
( .5.70) This means that we need ℓ -

1 -β 2 ≥ 0, i.e, ℓ ≥ β 2 + 1 so that Ω ( g 2 ψ λ ℓ-2 + i f 2 ψ λ ℓ-3 ) dx = o(1).
As a conclusion, if ℓ ≥ 2β + 4 we obtain

Ω |λy| 2 dx = o(1)
. Thus, we conclude that ∥U n ∥ H = ∥(u n , v n , y n , z n )∥ H = o(1) which leads to the desired contradiction.

Consequently, if ℓ = 2β + 4 we deduce that Ω |λu| 2 dx = o(1), Ω |λy| 2 dx = o(
Remark 1.5.12.

1) In Stahn, 2017, the damped wave equation (1.5.1) with Dirichlet boundary conditions was considered on the unit square (0, 1) 2 . In fact, it is proved that if ω 1 = {(x, y) ∈ (0, 1) 2 ; x < σ}, where σ is some fixed number from the interval (0, 1). Then, the energy of (1.5.1) decays polynomially as t -4/3 , which corresponds to β = 3 in Theorem 1.5.7. The author noticed that the same decay remains true in higher dimensional spaces, namely in a hyper cube (0, 1) N and ω 1 = {(x 1 , .., x N ) ∈ (0, 1) N ; x 1 < σ}.

2) In example 3 of Liu and Rao, 2005, the authors considered the wave equation with local viscous damping in the square (0, π) 2 . Based on their results, if we impose that the damping region ω 1 contains a vertical strip of the square domain, i.e,

ω 1 ⊃ Ω s = {(x 1 , x 2 )/a < x 1 < b, 0 < x 2 < π}.
Then, the energy of (1.5.1) decays polynomially as t -1 , i.e, β = 4 in Theorem 1.5.7.

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface

Polynomial stability with two dampings

In this section, we will improve the polynomial decay obtained in section 5, by using two damping terms. One of the main ingredients is to use the exponential or polynomial decay of the wave equation with frictional damping in

ω 3 = ω b ∩ ω c , namely      φ tt -∆φ + 1 ω 3 φ t = 0 in Ω × (0, ∞), φ = 0 on Γ × (0, ∞), φ(t = 0) = φ 0 , φ t (t = 0) = φ 1 .
(1.6. β , with β > 0. Then, there exists a constant C > 0 such that for every initial data U 0 = (u 0 , u 1 , y 0 , y 1 ) ∈ D(A), the energy of system (1.1.1) verifies the following estimate:

E(t) ≤ C t 2 β+2 ∥U 0 ∥ 2 D(A) , ∀t > 0. (1.6.2)
Proof. Following [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], (see also [START_REF] Liu | Locally Distributed Control and Damping for the Conservative Systems[END_REF]Rao, 2005, Batty and[START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF], a C 0 -semigroup of contractions (e tA ) t≥0 on a Hilbert space H verifies (1.6.2) if (S1) holds and lim sup

|λ|→∞ 1 λ ℓ ∥(iλ -A) -1 ∥ L(H) < ∞, ℓ = β + 2 (S4).
As (S1) already holds, our goal is to prove that condition (S4) is satisfied. To this aim, we proceed by a contradiction argument. Suppose that (S4) does not hold, then there exist a sequence (λ n ) ⊂ R and a sequence (U n ) ⊂ D(A) such that (1.5. In what follows, we drop the index n for simplicity.

Lemma 1.6.2. Under all the above assumptions, we have

     ∥∇v∥ L 2 (ω 3 ) = o(1) λ ℓ/2 , ∥v∥ L 2 (ω 3 ) = o(1) λ ℓ/2 , ∥∇z∥ L 2 (ω 3 ) = o(1) λ ℓ/2 , ∥z∥ L 2 (ω 3 ) = o(1) λ ℓ/2 .
(1.6.3)

Proof. Multiply in H equation (1.5.40) by the uniformly bounded sequence U = (u, v, y, z), we get

Ω (b(x)|∇v| 2 + c(x)|∇z| 2 ) dx = -Re((iλI -A)U, U) H = o(1) λ ℓ . It follows that      ∥b 1/2 ∇v∥ L 2 (Ω) = o(1) λ ℓ/2 , ∥c 1/2 ∇z∥ L 2 (Ω) = o(1) λ ℓ/2 .
(1.6.4)

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface Using (1.1.7), (1.6.4) and Poincaré inequality (since meas(ω 3 ∩ Γ) > 0) we get (1.6.3).

By detailing equation (1.5.40), we get (1.5.42)- (1.5.44) and

λ ℓ (iλz -div(∇y + c∇z) + αu) = g 2 -→ 0 in L 2 (Ω).
(1.6.5)

From equations (1.5.42) and (1.5.44) and using (1.5.3) we get (1.5.11).

Using (1.5.3), (1.6.3), (1.5.42) and (1.5.44) we also have

       ∥∇u∥ L 2 (ω 3 ) = o(1) λ ℓ 2 +1
and ∥u∥ L 2 (ω 3 ) = o( 1)

λ ℓ 2 +1 , ∥∇y∥ L 2 (ω 3 ) = o(1) λ ℓ 2 +1
and ∥y∥

L 2 (ω 3 ) = o(1) λ ℓ 2 +1
.

(1.6.6)

Lemma 1.6.3. Under all the above assumptions, if ℓ ≥ β + 2 then, we have

Ω |λu| 2 dx = o(1)
and

Ω |λy| 2 dx = o(1)
.

(1.6.7)

Proof. For all n ∈ N, let φ n , ψ n ∈ H 2 (Ω) ∩ H 1 0 (Ω) be the solution of (1.5. 14)- (1.5.16). If (1.6.1) decays exponentially (β = 0) or polynomially as t -4

β , β > 0. Then, following Borichev and Tomilov, 2009 and using (1.5.11), we know that (1.5.57) and (1.5.58) hold.

As before we deduce that (1.5.59) holds, i.e,

- Ω |λu| 2 dx + Ω iλ 3 1 ω 3 uφ dx + Ω λ 2 b∇v • ∇φ dx + Ω λ 2 αyφ dx = Ω ( g 1 φ λ ℓ-2 + i f 1 φ λ ℓ-3 ) dx.
(1.6.8)

But, using Cauchy-Schwarz inequality, (1.6.6) and (1.5.58) we have, .6.9) This means that we need

Ω iλ 3 1 ω 3 uφ dx ≤ |λ| 2 ||u|| L 2 (ω 3 ) ||λφ|| L 2 (Ω) = o(1) λ ℓ-β 2 . ( 1
ℓ-β 2 ≥ 0 i.e. ℓ ≥ β so that Ω λ 3 1 ω 3 uφ dx = o(1).
Also, by (1.6.4) and (1.5.58) we have .6.10) This means that we should have

Ω λ 2 b∇v • ∇φ dx ≤ |λ| 2 ||b∇v|| L 2 (Ω) ||∇φ|| L 2 (Ω) = o(1) λ ℓ-β 2 -1 . ( 1
ℓ-β 2 -1 ≥ 0 i.e. ℓ ≥ β + 2 so that Ω λ 2 b∇v • ∇φ dx = o(1)
. with Kelvin-Voigt dampings and non-smooth coefficient at the interface Again, using Cauchy-Schwarz inequality, (1.6.6) and (1.5.58) we obtain,

Ω λ 2 αyφ dx ≤ ||α|| ∞ |λ|||y|| L 2 (ω 3 ) ||λφ|| L 2 (Ω) = o(1) λ ℓ-β 2 +1
. (1.6.11) This means that we should have

ℓ-β 2 + 1 > 0 i.e. ℓ ≥ β -2 so that Ω λ 2 αyφ dx = o(1).
On the other hand, (1.5.64) holds, so we need ℓ -

1 -β 2 ≥ 0 i.e ℓ ≥ β 2 + 1 so that Ω ( g 1 φ λ ℓ-2 + i f 1 φ λ ℓ-3 ) dx = o(1).
As a conclusion, if ℓ ≥ β + 2 we get

Ω |λu| 2 dx = o(1).
Similarly, multiplying (1.5.44) and (1.6.5) by iλ 3 ψ and λ 2 ψ respectively, applying Green's formula, and Cauchy-Schwarz inequality, we prove that Proof. By inserting equation (1.5.42) in equation ( 1.5.43) we get

Ω |λy| 2 dx = o(1), if ℓ ≥ β + 2.
-λ 2 u -div(∇u + b∇v) + αy = g 1 λ ℓ + i f 1 λ ℓ-1 .
(1.6.12)

Similarly, inserting equation (1.5.44) in equation (1.6.5) we obtain -λ 2 y --div(∇y

+ c∇z) + αu = g 2 λ ℓ + i f 2 λ ℓ-1 .
(1.6.13)

Now, multiply (1.6.12) and (1.6.13) by u and y respectively then integrate over Ω and apply Green's formula, we obtain

- Ω |λu| 2 dx + Ω |∇u| 2 dx + Ω b∇v • ∇u dx + Ω αyu dx = Ω ( g 1 u λ ℓ + i f 1 u λ ℓ-1 ) dx (1.6.14)
Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface and

- Ω |λy| 2 dx + Ω |∇y| 2 dx + Ω c∇z • ∇y dx + Ω αuy dx = Ω ( g 2 y λ ℓ + i f 1 y λ ℓ-1 ) dx.
(1.6.15)

But, using Cauchy-Schwarz inequality, (1.6.4) and (1.5.3) we have

Ω b∇v • ∇u dx ≤ ||b∇v|| L 2 (Ω) ||∇u|| L 2 (Ω) = o(1)
λ ℓ/2 , Ω αyu dx ≤ ||α|| ∞ ||u|| L 2 (Ω) ||y|| L 2 (Ω) = o(1),
and

Ω c∇z • ∇y dx ≤ ||c∇z|| L 2 (Ω) ||∇y|| L 2 (Ω) = o(1) λ ℓ/2 .
Consequently, we get

Ω |∇u| 2 dx = Ω |λu| 2 dx + o(1),
and

Ω |∇y| 2 dx = Ω |λy| 2 dx + o(1).
Consequently, if ℓ = β + 2 we deduce that Thus, we conclude that 1) which leads to the desired contradiction.

∥U n ∥ H = ∥(u n , v n , y n , z n )∥ H = o(
Remark 1.6.5. In Theorem 1.6.1, if supp α ⊈ ω 3 then, we can easily prove that the energy of system (1.1.1) decays polynomially as t -2

β+2 , with β < 2. Indeed, in order to prove that (compare with (1.6.11))

Ω αλ 2 yψ dx = o(1),
we will need β < 2.

Optimality of the polynomial decay rate

We study here the optimality of the polynomial decay rate obtained for the Ndimensional coupled wave system in Theorem 1.5.1.

Theorem 1.7.1. Assume that c = 0, b = 1 and α = α 0 > 0. Then the energy decay rate (1.5.2) is optimal in the sense that for any ϵ > 0 we cannot expect the decay rate 1 t 1/2+ϵ for all initial data U 0 ∈ D(A).

Proof. Assume that there exist ϵ > 0 and C such that

||e tA || H ≤ C t 1 4 +ϵ ||U|| D(A) , ∀U ∈ D(A). (1.7.1)
Then, by Theorem 2.4 of [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], we deduce that there exists

C ′ > 0 such that ||(is -A) -1 || L(H) ≤ C ′ |s| 4 1+4ϵ , ∀s ∈ R, (1.7.2)
or equivalently,

||U|| H ≤ C ′ |s| 4 1+4ϵ ||(is -A)U|| H , ∀U ∈ D(A). (1.7.3)
Now, let λ + k , with k ≥ k 0 , be the sequence of eigenvalues of A described in Proposition 1.4.2 and let U n ∈ D(A) be the normalized eigenfunction. Moreover, set

β k = ℑ(λ + k ), ∀k ≥ k 0 .
(1.7.4)

Then, using (1.4.8) we have

||(iβ k I -A)U k || H ∼ 1 µ 4 k .
(1.7.5)

Consequently, by (1.7.3), we will have

1 = ||U k || H ≲ C ′ |β k | 4 1+4ϵ µ 4 k ≲ C ′ µ 4 1+4ϵ -4 k ≲ C ′ µ -16ϵ 1+4ϵ k .
This leads to a contradiction.

Conclusion

The following table summarizes our main results.

Chapter 1. A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface

Kelvin-Voigt Damping

Sufficient conditions

Obtained decay rate

Covered geometrical situations One nonsmooth (H2), (1.1.7) and (1.1.8), (1.5.1) exponentially decaying

pol. t -1/2 1) Ω C ∞ , ω 1 satisfies GCC.
2) Ω C 2 , ω 1 neighborhood of the boundary.

3) See Remark 1.5.6.

One non smooth

(1.1.7) and ( 1

.1.8), meas(ω α ∩ Γ) > 0, ω α = supp α ⊂ ω b , (1.5.1) exponentially sta- ble or polynomially stable as t -4 β , with β > 0. pol. t -1 β+2 1) Ω C ∞ , ω α satisfies GCC. 2) Ω C 2 , ω α is a neigh- borhood of the bound- ary.
3) ω α contains a vertical strip of the square (0, π) 2 . 4) See Remark 1.5.6, 1.5.12.

Two non smooth

(1.1.7), (1.1.8) and (1.1.9),

ω 3 = ω b ∩ ω c satis- fies meas(ω 3 ∩ Γ) > 0, supp α ⊆ ω 3 , (1.6.1) is exponentially stable or polynomially stable as t -4 β , with β > 0. pol. t -1 β/2+1 1) Ω C ∞ , ω 3 satisfies GCC. 2) Ω C 2 , ω 3 is a neigh- borhood of the bound- ary.
3) ω 3 contains a vertical strip of the square (0, π) 2 . 4) See Remark 1.5.6, 1.5.12.

Two non smooth

(1.1.7), (1.1.9),

ω b = ω c = Ω.

Analytic

Lipschitz boundary

Introduction

Over decades, telegraph equations have gained attention and interest among scientists due to their different applications in the transmission of electrical signals along transmission lines of all frequencies, in addition to many other physical, biological and engineering applications (see [START_REF] Bohme | Non-Newtonian Fluid Mechanics[END_REF][START_REF] Evans | The numerical solution of the telegraph equation by the alternating group explicit (AGE) method[END_REF][START_REF] Jordan | Causal implications of viscous damping in compressible fluid flows[END_REF][START_REF] Mohanty | New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations[END_REF][START_REF] Pascal | Pressure wave propagation in a fluid flowing through a porous medium and problems related to interpretation of Stoneley's wave attenuation in acoustical well logging[END_REF]. As a consequence, many mathematical models were set up, for instance in Imperiale and Joly, 2014, a general and realistic situation was considered and a mathematical model of electromagnetic wave propagation in heterogeneous lossy coaxial cables was derived. Recently, referring to Nicaise, 2015, S. Nicaise has considered the stabilization of the generalized telegraph equation set in a real interval (model on a cable from Imperiale and Joly, 2014):

     V t + gV + aI x + kW = 0, in (0, L) × (0, ∞), I t + rI + bV x = 0, in (0, L) × (0, ∞), W t + cW = V, in (0, L) × (0, ∞), (2.1.1)
Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks with the following boundary conditions

V(0, t) = V(L, t) = 0, t ∈ R * + ,
and the following initial conditions

V(x, 0) = V 0 (x), I(x, 0) = I 0 (x), W(x, 0) = W 0 (x), x ∈ (0, L),
where, a, b, c, r, k and g are all non-negative functions in L ∞ (0, L) that verify some assumptions mentioned in Nicaise, 2015, see (2.1.2) below for the exact conditions. The generalized telegraph equation is a coupling between the usual telegraph equation where the electric unknowns are V and I representing the electric potential and the electric current respectively with a first order differential equation of parabolic type involving an auxiliary variable W representing the non-local effects. In Nicaise, 2015, the author was interested in studying the energy decay rate of system (2.1.1).

In fact, under the additional condition that r + g > 0 in (0, L), an exponential energy decay rate was established. Otherwise, he proved a polynomial energy decay rate of type t -1 for smooth initial data. Moreover, the obtained polynomial decay rate is optimal in the case r = g = 0. On the other hand, first order hyperbolic systems set on graphs, also called networks, appear in recent applications, like electrical circuits, arterial networks, networks of open channels, traffic flows on networks (see [START_REF] Bastin | Stability and Boundary Stabilization of 1-D Hyperbolic Systems[END_REF][START_REF] Gugat | Gas Flow in Fan-Shaped Networks: Classical Solutions and Feedback Stabilization[END_REF][START_REF] Leugering | Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[END_REF][START_REF] Valein | Stabilization of the wave equation on 1-d networks[END_REF][START_REF] Zhang | Exponential and Super Stability of a Wave Network[END_REF]Kramar, Mugnolo, and Nicaise, 2020b;Kramar, Mugnolo, and Nicaise, 2020a;[START_REF] Kramar | Spectral properties and asymptotic periodicity of flows in networks[END_REF][START_REF] Dorn | The semigroup approach to transport processes in networks[END_REF] for heat, wave or beam equations on graphs, see [START_REF] Mehmeti | Nonlinear wave in networks[END_REF][START_REF] Below | A qualitative theory for parabolic problems under dynamical boundary conditions[END_REF][START_REF] Gugat | Stars of vibrating strings: switching boundary feedback stabilization[END_REF][START_REF] Mugnolo | Dynamic and generalized Wentzell node conditions for network equations[END_REF][START_REF] Mugnolo | Vector-valued heat equations and networks with coupled dynamic boundary conditions[END_REF][START_REF] Zhang | Stability of a complex network of Euler-Bernoulli beams[END_REF][START_REF] Mugnolo | Semigroup Methods for Evolution Equations on Networks[END_REF] and the references there. In particular, referring to Nicaise, 2017, the usual telegraph equation was considered on a network with a general class of transmission conditions, including reasonable ones (like the Kirchhoff law) so that with the dissipative boundary conditions at the exterior vertices, an exponential stability was established. Note that some exact controllability results or stability properties have been obtained in [START_REF] Diagne | Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws[END_REF][START_REF] Xu | Abstract Second Order Hyperbolic System and Applications to Controlled Network of Strings[END_REF][START_REF] Lagnese | On the analysis and control of hyperbolic systems associated with vibrating networks[END_REF][START_REF] Maffucci | A Unified Approach for the Analysis of Networks Composed of Transmission Lines and Lumped Circuits[END_REF][START_REF] Perrollaz | Finite-Time Stabilization of 2 × 2 Hyperbolic Systems on Tree-Shaped Networks[END_REF][START_REF] Zhou | A Simple Derivation of Microstrip Transmission Line Equations[END_REF][START_REF] Gugat | Stars of vibrating strings: switching boundary feedback stabilization[END_REF][START_REF] Zhang | Stability of a complex network of Euler-Bernoulli beams[END_REF] for instance. But to the best of our knowledge, the stability of the generalized telegraph equation on a network is an open problem. This motivates us to study this equation on star shaped graphs.

In the present work, we consider system (2.1.1) on a star shaped network S made of N + 1 cables of equal length L > 0, with a positive integer N (see Figure 2.1). For that purpose, we fix different real valued and non-negative functions

a = (a ℓ ) N ℓ=0 , b = (b ℓ ) N ℓ=0 , c = (c ℓ ) N ℓ=0 , k = (k ℓ ) N ℓ=0 , r = (r ℓ ) N ℓ=0
, and g = (g ℓ ) N ℓ=0 in (L ∞ (0, L)) N+1 satisfying the following assumption

a ℓ ≥ µ, b ℓ ≥ µ, c ℓ ≥ µ, k ℓ + g ℓ ≥ µ a.e in (0, L), ∀ℓ = 0, . . . , N, (2.1.2)
where µ > 0 is a positive real number. These assumptions are in agreement with the physical setting from Imperiale and Joly, 2012; Imperiale and Joly, 2014.

On each edge e ℓ with index ℓ ∈ {0, ...., N}, we consider the generalized telegraph

     V ℓ,t + g ℓ V ℓ + a ℓ I ℓ,x + k ℓ W ℓ = 0, (x, t) ∈ (0, L) × R * + , I ℓ,t + r ℓ I ℓ + b ℓ V ℓ,x = 0, (x, t) ∈ (0, L) × R * + , W ℓ,t + c ℓ W ℓ = V ℓ , (x, t) ∈ (0, L) × R * + , (2.1.3)
where V ℓ represents the electric potential, I ℓ represents the electric current and W ℓ represents the non-local effects. Moreover, we denote by V = (V ℓ ) N ℓ=0 , I = (I ℓ ) N ℓ=0 , and W = (W ℓ ) N ℓ=0 .

Let us mention that the case when the edges have different lengths

L ℓ , ℓ = 0, • • • , N, namely system      V ℓ,t + g ℓ V ℓ + a ℓ I ℓ,x + k ℓ W ℓ = 0, (x, t) ∈ (0, L ℓ ) × R * + , I ℓ,t + r ℓ I ℓ + b ℓ V ℓ,x = 0, (x, t) ∈ (0, L ℓ ) × R * + , W ℓ,t + c ℓ W ℓ = V ℓ , (x, t) ∈ (0, L ℓ ) × R * + ,
can be treated similarly since it can be reduced to (2.1.3) as follows. Indeed by performing on edge ℓ, the change of variable x ℓ = L L ℓ x, with a fixed L > 0, and the change of unknowns

Îℓ (x ℓ , t) = I ℓ ( L ℓ L x ℓ , t), Vℓ (x ℓ , t) = V ℓ ( L ℓ L x ℓ , t), Ŵℓ (x ℓ , t) = W ℓ ( L ℓ L x ℓ , t),
we transform this system into (2.1.3) where the coefficients a ℓ and b ℓ are changed into L L ℓ a ℓ and L L ℓ b ℓ respectively (and that continue to satisfy (2.1.2)).

System (2.1.3) is considered with the following dissipative boundary condition at the exterior vertices

V ℓ (L, t) -α ℓ I ℓ (L, t) = 0, in R * + , α ℓ > 0, ∀ℓ ∈ {0, ....,
         N ∑ k=1 Z ℓk I k,t (0, t) = 1 δ (V 0 (0, t) -V ℓ (0, t)), in R * + , ∀ℓ ∈ {1, ...., N}, V 0,t (0, t) = -1 δY N ∑ ℓ=0 I ℓ (0, t) in R * + , (2.1.5)
where, Y and δ are two positive constants and Z = (Z ℓk ) N×N is a symmetric, positive definite matrix. These boundary conditions are derived in Beck, 2016, §8.2 or Joly and Semin, 2008 by using the method of matched asymptotics starting from thin coaxial cables of thickness δ. The coefficient Y and the matrix Z are defined from the material properties of the medium and from 3D potentials defined in the reference domain.

Remark 2.1.1. If r ℓ = 0, then differentiating (2.1.4) with respect to t and using the second equation of (2.1.3), we exactly find the Neumann velocity feedback law

V ℓ,t (L, t) = -α ℓ b ℓ V ℓ,x (L, t), in R * + , ∀ℓ ∈ {0, ...., N},
from Cox and Zuazua, 1995, (4) or Gugat, 2014, (4).

By setting ν(t) = V 0 (0, t) and η(t) = (η ℓ (t)) N ℓ=1 with η ℓ (t) = N ∑ k=1 Z ℓk I k (0, t), System (2.1.3)-(2.1.5) is completed with the following initial conditions      V (x, 0) = V 0 (x), I(x, 0) = I 0 (x), W (x, 0) = W 0 (x), x ∈ (0, L), ν(0) = ν 0 , η(0) = η 0 .
(2.1.6)

As suggested before, our goal is then to find sufficient conditions on the functions a ℓ , b ℓ , c ℓ , k ℓ , r ℓ , and g ℓ that guarantee that the above system is strongly stable and then exponentially stable.

Let us briefly outline the content of this chapter. First, in Section 2.2, we show that System (2.1.3)-(2.1.6) admits a unique solution in an appropriate Hilbert space using semi-group theory. Next, using a general criteria of Arendt-Batty (see [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF], we prove the strong stability of the system in the absence of the compactness of the resolvent. Then, in Section 2.3, based on a frequency domain approach combined with an ad-hoc multiplier technique, we establish an exponential energy decay rate. Finally, in Section 2.4, we discuss the stability of our system with some extensions, either by considering general dynamic boundary conditions or by considering general networks.

Let us finish this introduction with some notations used in the remainder of the pa-

per: for all p ∈ [1, ∞], L p (S) = L p (0, L) N+1 , 1 ≤ p ≤ ∞ and H 1 (S) = H 1 (0, L) N+1 .
We denote by (•, •) C N the inner product in C N .

Well posedness and strong stability

In this section, we will study the existence, uniqueness, regularity and asymptotic behavior of the solution of System (2.1.3)-(2.1.6).

Well posedness of the problem

First, we define the energy space

H = (L 2 (S)) 3 × C N+1 ,
that is a Hilbert space equipped with the following inner product:

⟨(V, I, W, ν, η) ⊤ , (V * , I * , W * , ν * , η * ) ⊤ ⟩ H = N ∑ ℓ=0 L 0 (θ ℓ V ℓ V * ℓ + β ℓ I ℓ I * ℓ + γ ℓ W ℓ W * ℓ ) dx + δY νν * + δ (η, Z -1 η * ) C N where θ = (θ ℓ ) N ℓ=0 , β = (β ℓ ) N ℓ=0 , γ = (γ ℓ ) N
ℓ=0 ∈ L ∞ (S) will be fixed later but are such that

θ ℓ ≥ µ 0 , β ℓ ≥ µ 0 , γ ℓ ≥ µ 0 a.e in Ω, ℓ = 0, . . . , N, (2.2.1)
for some µ 0 > 0.

Next, we define the unbounded linear operator A : D(A) → H by

D(A) = (V, I, W, ν, η) ∈ H : V ∈ H 1 (S), I ∈ H 1 (S), V ℓ (L) = α ℓ I ℓ (L), ∀ℓ ∈ {0, . . . , N}, ν = V 0 (0), and η = (η ℓ ) N ℓ=1 with η ℓ = N ∑ k=1 Z ℓk I k (0) , and A       V I W ν η       = -g ⊗ V + a ⊗ I x + k ⊗ W, r ⊗ I + b ⊗ V x , c ⊗ W -V , 1 δY N ∑ ℓ=0 I ℓ (0), 1 δ (V ℓ (0) -ν) N ℓ=1 ⊤ ,
where for two vector functions

P = (p ℓ ) N ℓ=0 ∈ L ∞ (S), and Q = (q ℓ ) N ℓ=0 ∈ L 2 (S), we set P ⊗ Q = (p ℓ q ℓ ) N ℓ=0 , while for Q = (q ℓ ) N ℓ=0 ∈ H 1 (S), Q x = (q ℓ,x ) N ℓ=0 .
Then, setting U(t) = (V ( where U 0 = (V 0 , I 0 , W 0 , ν 0 , η 0 ) ⊤ . For this system, we have the following existence result.

Theorem 2.2.1. Assume that condition (2.1.2) holds. Then, there exist θ, β, γ ∈ L ∞ (S) satisfying (2.2.1) such that the operator A generates a C 0 -semigroup of contractions (e tA ) t≥0 on H.

Proof. Using Lumer-Phillips Theorem (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF][START_REF] Liu | Semigroups associated with dissipative systems[END_REF], it is sufficient to prove that A is a maximal dissipative operator so that A generates a C 0 -semigroup of contractions on H.

Step 1. Let us start by the dissipativity. Let U = (V, I, W, ν, η) ⊤ ∈ D(A) then, we have

ℜ⟨AU, U⟩ H =ℜ - N ∑ ℓ=0 L 0 θ ℓ (g ℓ V ℓ + a ℓ I ℓ,x + k ℓ W ℓ )V ℓ dx - N ∑ ℓ=0 L 0 β ℓ (r ℓ I ℓ + b ℓ V ℓ,x )I ℓ dx + N ∑ ℓ=0 L 0 γ ℓ (V ℓ -c ℓ W ℓ )W ℓ dx -V 0 (0) N ∑ ℓ=0 I ℓ (0) - N ∑ ℓ=1 (V ℓ (0) -V 0 (0))I ℓ (0) = - N ∑ ℓ=0 L 0 (θ ℓ g ℓ |V ℓ | 2 + β ℓ r ℓ |I ℓ | 2 + γ ℓ c ℓ |W ℓ | 2 ) dx -ℜ N ∑ ℓ=0 L 0 θ ℓ a ℓ I ℓ,x V ℓ dx -ℜ N ∑ ℓ=0 L 0 β ℓ b ℓ V ℓ,x I ℓ dx + ℜ N ∑ ℓ=0 L 0 (γ ℓ -θ ℓ k ℓ )V ℓ W ℓ dx -ℜV 0 (0) N ∑ ℓ=0 I ℓ (0) -ℜ N ∑ ℓ=1 (V ℓ (0) -V 0 (0))I ℓ (0).
Now, as in Nicaise, 2015, we chose θ ℓ = a -1 ℓ , β ℓ = b -1 ℓ for all ℓ, and apply Green's formula on the third term of the right-hand side, we obtain

ℜ⟨AU, U⟩ = - N ∑ ℓ=0 L 0 (a -1 ℓ g ℓ |V ℓ | 2 + b -1 ℓ r ℓ |I ℓ | 2 + γ ℓ c ℓ |W ℓ | 2 ) dx -ℜ N ∑ ℓ=0 L 0 I ℓ,x V ℓ dx + ℜ N ∑ ℓ=0 L 0 I ℓ,x V ℓ dx -ℜ N ∑ ℓ=0 V ℓ (L)I ℓ (L) + ℜ N ∑ ℓ=0 V ℓ (0)I ℓ (0) + ℜ N ∑ ℓ=0 L 0 (γ ℓ -θ ℓ k ℓ )V ℓ W ℓ dx -ℜV 0 (0) N ∑ ℓ=0 I ℓ (0) + ℜV 0 (0) N ∑ ℓ=1 I ℓ (0) -ℜ N ∑ ℓ=1 V ℓ (0)I ℓ (0) = - N ∑ ℓ=0 L 0 (a -1 ℓ g ℓ |V ℓ | 2 + b -1 ℓ r ℓ |I ℓ | 2 + γ ℓ c ℓ |W ℓ | 2 ) dx + ℜ N ∑ ℓ=0 L 0 (γ ℓ -θ ℓ k ℓ )V ℓ W ℓ dx - N ∑ ℓ=0 α ℓ |I ℓ (L)| 2 .
Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks Finally, by estimating the second term of this right-hand side by Young's inequality we get, ℜ⟨AU, U⟩ ≤ -

N ∑ ℓ=0 L 0 (a -1 ℓ g ℓ |V ℓ | 2 + b -1 ℓ r ℓ |I ℓ | 2 + γ ℓ c ℓ |W ℓ | 2 ) dx - N ∑ ℓ=0 α ℓ |I ℓ (L)| 2 + N ∑ ℓ=0 L 0 |γ ℓ -k ℓ θ ℓ | 2 (ϵ ℓ |V ℓ | 2 + 1 ϵ ℓ |W ℓ | 2 ) dx,
for all ϵ(x) = (ϵ ℓ (x)) N ℓ=0 positive vector ( i.e. ϵ ℓ (x) > 0, ∀ℓ ∈ {0, .., N}, x ∈ (0, L)). Now, we need to find ϵ(x) = (ϵ ℓ (x)) N ℓ=0 and γ = (γ ℓ ) N ℓ=0 to satisfy for all ℓ ∈ {0, .., N} :

|γ ℓ -k ℓ θ ℓ |ϵ ℓ 2 -θ ℓ g ℓ ≤ 0 and |γ ℓ -k ℓ θ ℓ | 2ϵ ℓ -γ ℓ c ℓ ≤ 0.
But according to the proof of Theorem 2.1 of Nicaise, 2015 (see p. 3224-3225 in Nicaise, 2015) such a choice is always possible (we do not give the details and refer to that paper for the details). With such a choice, we find

ℜ⟨AU, U⟩ ≤ - 1 2 N ∑ ℓ=0 L 0 (θ ℓ g ℓ |V ℓ | 2 + 2β ℓ r ℓ |I ℓ | 2 + γ ℓ c ℓ |W ℓ | 2 )dx - N ∑ ℓ=0 α ℓ |I ℓ (L)| 2 ≤ 0, (2.2.3)
and consequently, A is dissipative.

Step 2. Now, let us go on with the maximality. Let λ > 0 be fixed. Given

F = ( f 1 , f 2 , f 3 , f 4 , f 5 ) ⊤ ∈ H, we look for U = (V, I, W, ν, η) ⊤ ∈ D(A) solution of (λI -A)U = F, or equivalently,                      λV ℓ + a ℓ I ℓ,x + g ℓ V ℓ + k ℓ W ℓ = f ℓ,1 , ∀ℓ ∈ {0, .., N}, λI ℓ + b ℓ V ℓ,x + r ℓ I ℓ = f ℓ,2 , ∀ℓ ∈ {0, .., N}, λW ℓ + c ℓ W ℓ -V ℓ = f ℓ,3 , ∀ℓ ∈ {0, .., N}, λν + 1 δY N ∑ ℓ=0 I ℓ (0) = f 4 , λη ℓ + 1 δ (V ℓ (0) -ν) = f ℓ,5 , ∀ℓ ∈ {1, .., N}.
(2.2.4)

Assume for the moment that a solution U ∈ D(A) of (2.2.4) exists. Then, the third equation is equivalent to

W ℓ = V ℓ + f ℓ,3 λ + c ℓ ,
= f 4 -1 δY N ∑ ℓ=0 I ℓ (0) λ , (2.2.6)
and

η ℓ = f ℓ,5 -1 δ (V ℓ (0) -ν) λ
, for all ℓ ∈ {1, .., N}.

(2.2.7)

Replacing (2.2.5) in the first equation of (2.2.4), we get

V ℓ + a ℓ,1 (λ)I ℓ,x = f λ,ℓ , V ℓ,x + a ℓ,2 (λ)I ℓ = g λ,ℓ , (2.2.8) where                        a ℓ,1 (λ) = a ℓ λ + g ℓ + k ℓ λ + c ℓ , a ℓ,2 (λ) = λ + r ℓ b ℓ , f λ,ℓ = f ℓ,1 - k ℓ f ℓ,3 λ + c ℓ λ + g ℓ + k ℓ λ + c ℓ , g λ,ℓ = f ℓ,2 b ℓ , (2.2.9)
for all ℓ ∈ {0, .., N}.

Let ψ = (ψ ℓ ) N ℓ=0 ∈ H 1 (S). Multiplying the second equation of (2.2.8) by ψ ℓ , integrating over (0, L) and summing on ℓ ∈ {0, .., N}, yields

N ∑ ℓ=0 L 0 V ℓ,x ψ ℓ dx + N ∑ ℓ=0 L 0 a ℓ,2 (λ)I ℓ ψ ℓ dx = N ∑ ℓ=0 L 0 g λ,ℓ ψ ℓ dx. (2.2.10)
Similarly, multiplying the first equation of (2.2.8) by ψ ℓ,x , integrating over (0, L) and summing on ℓ ∈ {0, .., N}, gives .2.11) By integrating by parts the first term of the left hand side of (2.2.11), we obtain

N ∑ ℓ=0 L 0 V ℓ ψ ℓ,x dx + N ∑ ℓ=0 L 0 a ℓ,1 (λ)I ℓ,x ψ ℓ,x dx = N ∑ ℓ=0 L 0 f λ,ℓ ψ ℓ,x dx. ( 2
- N ∑ ℓ=0 L 0 V ℓ,x ψ ℓ dx + N ∑ ℓ=0 V ℓ (L)ψ ℓ (L) - N ∑ ℓ=0 V ℓ (0)ψ ℓ (0) + N ∑ ℓ=0 L 0 a ℓ,1 (λ)I ℓ,x ψ ℓ,x dx = N ∑ ℓ=0 L 0 f λ,ℓ ψ ℓ,x dx.
(2.2.12)

Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks Then, adding (2.2.10) and (2.2.12) leads to

N ∑ ℓ=0 L 0 a ℓ,1 (λ)I ℓ,x ψ ℓ,x + a ℓ,2 (λ)I ℓ ψ ℓ dx + N ∑ ℓ=0 V ℓ (L)ψ ℓ (L) - N ∑ ℓ=0 V ℓ (0)ψ ℓ (0) = N ∑ ℓ=0 L 0 f λ,ℓ ψ ℓ,x + g λ,ℓ ψ ℓ dx.
(2.2.13)

By taking into account the boundary conditions in D(A), in addition to (2.2.6) and (2.2.7), we find

N ∑ ℓ=0 V ℓ (0)ψ ℓ (0) = V 0 (0)ψ 0 (0) + N ∑ ℓ=1 V ℓ (0)ψ ℓ (0) = V 0 (0)ψ 0 (0) + δ N ∑ ℓ=1 f ℓ,5 ψ ℓ (0) + V 0 (0) N ∑ ℓ=1 ψ ℓ (0) -δλ N ∑ ℓ=1 N ∑ k=1 Z ℓk I k (0) ψ ℓ (0). Then, N ∑ ℓ=0 V ℓ (0)ψ ℓ (0) = V 0 (0) N ∑ ℓ=0 ψ ℓ (0) + δ N ∑ ℓ=1 f ℓ,5 ψ ℓ (0) -δλ N ∑ ℓ=1 N ∑ k=1 Z ℓk I k (0) ψ ℓ (0) = f 4 λ N ∑ ℓ=0 ψ ℓ (0) - 1 δYλ N ∑ ℓ=0 I ℓ (0) N ∑ ℓ=0 ψ ℓ (0) + δ N ∑ ℓ=1 f ℓ,5 ψ ℓ (0) -δλ N ∑ ℓ=1 N ∑ k=1 Z ℓk I k (0) ψ ℓ (0) and N ∑ ℓ=0 V ℓ (L)ψ ℓ (L) = N ∑ ℓ=0 α ℓ I ℓ (L)ψ ℓ (L).
By replacing these two identities in (2.2.13), we arrive at

N ∑ ℓ=0 L 0 a ℓ,1 (λ)I ℓ,x ψ ℓ,x + a ℓ,2 (λ)I ℓ ψ ℓ dx + N ∑ ℓ=0 α ℓ I ℓ (L)ψ ℓ (L) + 1 δYλ N ∑ ℓ=0 I ℓ (0) N ∑ ℓ=0 ψ ℓ (0) + δλ N ∑ ℓ=1 N ∑ k=1 Z ℓk I k (0) ψ ℓ (0) = N ∑ ℓ=0 L 0 ( f λ,ℓ ψ ℓ,x + g λ,ℓ ψ ℓ ) dx + f 4 λ N ∑ ℓ=0 ψ ℓ (0) + δ N ∑ ℓ=1 f ℓ,5 ψ ℓ (0), ( 2
∑ ℓ=0 L 0 (a ℓ,1 (λ)|I ℓ,x | 2 + a ℓ,2 (λ)|I ℓ | 2 ) dx + N ∑ ℓ=0 α ℓ |I ℓ (L)| 2 + 1 δYλ N ∑ ℓ=0 I ℓ (0) 2 +δλ(ZI * (0), I * (0)) C N ≥ C(λ)||I|| 2 H 1 (S) ,
with I * (0) = (I ℓ (0)) N ℓ=1 . Secondly, the right-hand side of (2.2. 14) is a continuous form on H 1 (S).

Now, according to the first equation of (2.2.8) we set

V ℓ = f λ,ℓ -a ℓ,1 (λ)I ℓ,x , (2.2.15)
that clearly belongs to L 2 (0, L), for all ℓ. Then, taking in (2.2.14) a test function ψ such that ψ ℓ ∈ D(0, L) for some ℓ = 0, .., N and ψ j = 0, for all j ̸ = ℓ, we find

L 0 a ℓ,1 (λ)I ℓ,x ψ ℓ,x dx + L 0 a ℓ,2 (λ)I ℓ ψ ℓ dx = L 0 f λ,ℓ ψ ℓ,x dx + L 0 g λ,ℓ ψ ℓ dx. ( 2
- L 0 V ℓ ψ ℓ,x dx + L 0 a ℓ,2 (λ)I ℓ ψ ℓ dx = L 0 g λ,ℓ ψ ℓ dx.
(2.2.17) Applying Green's formula on the first integral of the left-hand side of (2.2.17), we obtain V ℓ,x + a ℓ,2 (λ)I ℓ = g λ,ℓ , in D ′ (0, L).

(2.2.18)

As -a ℓ,2 (λ)I ℓ + g λ,ℓ ∈ L 2 (0, L), we deduce that V ℓ ∈ H 1 (0, L) and that (2.2.8) holds.

Then, after defining W by (2.2.5), ν by (2.2.6) and η by (2.2.7), it remains to check the boundary conditions appearing in D(A). For this purpose, we introduce two Hilbert spaces

H 1 R = {u ∈ H 1 (0, L); u(0) = 0}, and H 1 L = {u ∈ H 1 (0, L); u(L) = 0}
. Now, we first take different test functions ψ in (2.2.14). By Green's formula and taking into account (2.2.8), we have

N ∑ ℓ=0 V ℓ (L)ψ ℓ (L) - N ∑ ℓ=0 V ℓ (0)ψ ℓ (0) - N ∑ ℓ=0 α ℓ I ℓ (L)ψ ℓ (L) - 1 δYλ N ∑ ℓ=0 I ℓ (0) N ∑ ℓ=0 ψ ℓ (0) -δλ N ∑ ℓ=1 N ∑ k=1 Z ℓk I k (0) ψ ℓ (0) +δ N ∑ ℓ=1 f ℓ,5 ψ ℓ (0) + 1 λ N ∑ ℓ=0 f 4 ψ ℓ (0) = 0, ∀ψ ∈ H 1 (S). ( 2
∏ ℓ=0 H 1 R such that ψ ℓ (L) = z arbitrary in C and ψ j (L) = 0, ∀j ̸ = ℓ, we get (V ℓ (L) -α ℓ I ℓ (L)) z = 0, ∀z ∈ C.
Consequently, V ℓ (L) = α ℓ I ℓ (L). Second, we fix ℓ and take ψ ∈

N ∏ ℓ=0 H 1 L such that ψ ℓ (0) = z arbitrary in C and ψ j (0) = 0, ∀j ̸ = ℓ, we obtain -V ℓ (0) - 1 δYλ N ∑ ℓ=0 I ℓ (0) -δλ N ∑ k=1 Z ℓk I k (0) + δ f ℓ,5 + 1 λ f 4 z = 0, ∀z ∈ C.
Using (2.2.6) and (2.2.7) we deduce that η ℓ = N ∑ k=1 Z ℓk I k (0). Third, we fix ℓ and take

ψ ∈ N ∏ ℓ=0
H 1 L such that ψ ℓ = 0, ∀ℓ ̸ = 0, and ψ 0 (0) = z, arbitrary in C we obtain

f 4 λ - 1 δYλ N ∑ ℓ=0 I ℓ (0) = V 0 (0).
Using (2.2.6), we deduce that ν = V 0 (0). The proof of Theorem 2.2.1 is thus complete.

As A generates a C 0 -semigroup on H, problem (2.2.2) admits a unique solution given by U(t) = e tA U 0 , t ≥ 0.

Hence, the semi-group theory allows us to deduce the following existence and uniqueness results.

∈ C 0 ([0, ∞); H). Moreover if U 0 ∈ D(A), problem (2.2.2) admits a unique strong solution U ∈ C 1 ([0, ∞); H) ∩ C 0 ([0, ∞); D(A)).
As usual the energy associated with (2.2.2) is defined by

E(t) = 1 2 ||(V, I, W, ν, η)|| 2 H and for U = (V, I, W, ν, η) ∈ C 1 ([0, ∞); H) ∩ C 0 ([0, ∞); D(A)), we have E ′ (t) ≤ - 1 2 N ∑ ℓ=0 L 0 (θ ℓ g ℓ |V ℓ | 2 + 2β ℓ r ℓ |I ℓ | 2 + γ ℓ c ℓ |W ℓ | 2 )dx - N ∑ ℓ=0 α ℓ |I ℓ (L)| 2 ≤ 0.
For such a solution, we then have

E(t 2 ) ≤ E(t 1 ), ∀0 ≤ t 1 ≤ t 2 ,
in other words, our system is dissipative in the sense that its energy is nonincreasing with respect to t. By the density of D(A) into H, this last property remains valid for weak solution U ∈ C 0 ([0, ∞); H).

Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks Now, we are able to study the strong stability of our system.

Strong stability

To show the strong stability of the C 0 -semigroup of contractions (e tA ) t≥0 we will rely on the following result due to [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF].

Theorem 2.2.3. [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF]. Let A : D(A) ⊂ H → H generates a C 0semigroup of contractions on the Hilbert space H. If 1) A has no pure imaginary eigenvalues, 2) σ(A) ∩ iR is countable, where σ(A) denotes the spectrum of A. Then, the C 0 -semigroup (e tA ) t≥0 is strongly stable, namely lim

t→∞ ||e tA U 0 || H = 0, ∀U 0 ∈ H.
Let us now state the main result of this subsection.

Theorem 2.2.4. Assume that assumption (2.1.2) holds. Then, the C 0 -semigroup (e tA ) t≥0 is strongly stable on the energy space H.

Its proof is based on the use of Theorem 2.2.3. Since in our situation the resolvent of A is not compact, we have to analyze its full spectrum on the imaginary axis. More precisely with the help of the following Lemmas, we will deduce that σ(A) ∩ iR is empty, hence the result. Lemma 2.2.5. Under the same condition of Theorem 2.2.4, we have iλ -A is injective for all λ ∈ R, i.e.

ker(iλI -A) = {0}, ∀λ ∈ R.

Proof. Let λ ∈ R and U = (V, I, W, ν, η) ⊤ ∈ D(A) be such that

(iλI -A)U = 0, equivalently,                    iλV ℓ + a ℓ I ℓ,x + g ℓ V ℓ + k ℓ W ℓ = 0, ∀ℓ ∈ {0, .., N}, iλI ℓ + b ℓ V ℓ,x + r ℓ I ℓ = 0, ∀ℓ ∈ {0, .., N}, iλW ℓ + c ℓ W ℓ -V ℓ = 0, ∀ℓ ∈ {0, .., N}, iλν + 1 δY N ∑ ℓ=0 I ℓ (0) = 0, iλη ℓ + 1 δ (V ℓ (0) -ν) = 0, ∀ℓ ∈ {1, .., N}.
(2.2.20)

Hence, as ℜ⟨AU, U⟩ H = 0, using inequality (2.2.3) and the fact that c ℓ > 0 and α ℓ > 0, we get

g ℓ V ℓ = r ℓ I ℓ = W ℓ = I ℓ (L) = 0, ∀ ℓ ∈ {0, .., N}. (2.2.21)
Using the third identity of (2.2.20) we deduce that V ℓ = 0, for all ℓ ∈ {0, .., N}. On the other hand, the first equation yields that I ℓ,x = 0, so I ℓ is constant on (0, L) for all ℓ ∈ {0, .., N}. As I ℓ (L) = 0, we deduce that I ℓ = 0, for all ℓ ∈ {0, .., N}. This implies

that ν = V 0 (0) = 0 and η ℓ = N ∑ k=1 Z ℓk I k (0) = 0,
= ( f 1 , f 2 , f 3 , f 4 , f 5 ) ⊤ ∈ H, we look for a unique solution U = (V, I, W, ν, η) ⊤ ∈ D(A) of (iλI -A)U = F, or equivalently,                    iλV ℓ + a ℓ I ℓ,x + g ℓ V ℓ + k ℓ W ℓ = f ℓ,1 , ∀ℓ ∈ {0, .., N}, iλI ℓ + b ℓ V ℓ,x + r ℓ I ℓ = f ℓ,2 , ∀ℓ ∈ {0, .., N}, iλW ℓ + c ℓ W ℓ -V ℓ = f ℓ,3 , ∀ℓ ∈ {0, .., N}, iλν + 1 δY N ∑ ℓ=0 I ℓ (0) = f 4 , iλη ℓ + 1 δ (V ℓ (0) -ν) = f ℓ,5 , ∀ℓ ∈ {1, .., N}.
(2.2.22)

Suppose such a U = (V, I, W, ν, η) ⊤ ∈ D(A) exists, then .2.25) Replace W ℓ in the first and second identities of (2.2.22) to get

W ℓ = V ℓ + f ℓ,3 (iλ + c ℓ ) , ∀ℓ ∈ {0, .., N}, (2.2.23) ν = f 4 -1 δY N ∑ ℓ=0 I ℓ (0) iλ , (2.2.24) η ℓ = f ℓ,5 -1 δ (V ℓ (0) -ν) iλ , ∀ℓ ∈ {1, .., N}. ( 2
                   V ℓ + a ℓ (iλ + g ℓ + k ℓ iλ + c ℓ ) I ℓ,x = ( f ℓ,1 - k ℓ f ℓ,3 iλ + c ℓ ) (iλ + g ℓ + k ℓ iλ + c ℓ ) , iλ + r ℓ b ℓ I ℓ + V ℓ,x = f ℓ,2 b ℓ , ℓ ∈ {0, .., N}.
Then, to find a solution of (2.2.22) it is enough to find a solution of (2.2.26) where a ℓ,1 (λ), a ℓ,2 (λ), f λ,ℓ and g λ,ℓ are defined in (2.2.9). Now following the arguments of the proof of Theorem 2.2.1, we deduce that I ∈ Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks 2.2.27) for all ψ ∈ H 1 (S). Here we note that Lax-Milgram Lemma cannot be applied because coercivity is not available. Therefore, we use a perturbation argument. For that purpose, let us introduce the sesquilinear form

V ℓ + a ℓ,1 (iλ)I ℓ,x = f iλ,ℓ , V ℓ,x + a ℓ,2 (iλ)I ℓ = g iλ,ℓ ,
H 1 (S) is a solution of N ∑ ℓ=0 L 0 a ℓ,1 (iλ)I ℓ,x ψ ℓ,x + a ℓ,2 (iλ)I ℓ ψ ℓ dx + N ∑ ℓ=0 α ℓ I ℓ (L)ψ ℓ (L) + 1 iδYλ N ∑ ℓ=0 I ℓ (0) N ∑ ℓ=0 ψ ℓ (0) + iδλ N ∑ ℓ=1 N ∑ k=1 Z ℓk I k (0) ψ ℓ (0) = N ∑ ℓ=0 L 0 f iλ,ℓ ψ ℓ,x + g iλ,ℓ ψ ℓ dx + f 4 iλ N ∑ ℓ=0 ψ ℓ (0) + δ N ∑ ℓ=1 f ℓ,5 ψ ℓ (0), (
b λ (I, ψ) = N ∑ ℓ=0 L 0 a ℓ,1 (iλ)I ℓ,x ψ ℓ,x dx + N ∑ ℓ=0 L 0 1 b ℓ I ℓ ψ ℓ dx + N ∑ ℓ=0 α ℓ I ℓ (L)ψ ℓ (L) + 1 iδYλ N ∑ ℓ=0 I ℓ (0) N ∑ ℓ=0 ψ ℓ (0) + iδλ N ∑ ℓ=1 N ∑ k=1 Z ℓk I k (0) ψ ℓ (0), ∀I, ψ ∈ H 1 (S).
This sesquilinear form b λ is continuous and coercive on H 1 (S), since for all ℓ

ℜ(a ℓ,1 (iλ)) = a ℓ λ 2 g ℓ + c ℓ (c ℓ g ℓ + k ℓ ) (c ℓ g ℓ + k ℓ -λ 2 ) 2 + λ 2 (c ℓ + g ℓ ) 2 ≥ C(λ),
where C(λ) is a positive constant that depends only on λ. Then, by Lax-Milgram Lemma, the operator

B λ : H 1 (S) → (H 1 (S)) * : I → B λ I, with B λ I(ψ) = b λ (I, ψ), is an isomorphism. Now, let us set R λ : H 1 (S) → (H 1 (S)) * : I → R λ I, with R λ I(ψ) = N ∑ ℓ=0 L 0 (iλ + r ℓ -1) b ℓ I ℓ ψ ℓ dx.
As R λ is a compact operator, we deduce that

B λ + R λ is a Fredholm operator of index zero from H 1 (S) to (H 1 (S)) * .
Now by setting

L λ (ψ) = N ∑ ℓ=0 L 0 (g iλ,ℓ ψ ℓ + f iλ,ℓ ψ ℓ,x ) dx + δ N ∑ ℓ=1 f ℓ,5 ψ ℓ (0) + 1 iλ N ∑ ℓ=0 f 4 ψ ℓ (0),
we notice that (2.2.27) is equivalent to Let us now fix I ∈ ker(B λ + R λ ), then it satisfies

(B λ + R λ )I = L λ in (H 1 (S)) * . ( 2
N ∑ ℓ=0 L 0 a ℓ,1 (iλ)I ℓ,x ψ ℓ,x + a ℓ,2 (iλ)I ℓ ψ ℓ dx + N ∑ ℓ=0 α ℓ I ℓ (L)ψ ℓ (L) + 1 iδYλ N ∑ ℓ=0 I ℓ (0) N ∑ ℓ=0 ψ ℓ (0) + iδλ N ∑ ℓ=1 N ∑ k=1 Z ℓk I k (0) ψ ℓ (0) = 0, ∀ψ ∈ H 1 (S).
Thus, if we set

V ℓ = - I ℓ,x a ℓ,1 (iλ) , W ℓ = V ℓ iλ + c ℓ and ν = - N ∑ ℓ=0 I ℓ (0) iλδY and η ℓ = -V ℓ (0) + ν iλδ , we conclude that (V, I, W, ν, η) ∈ D(A) is a solution of (iλ -A)(V, I, W, ν, η) ⊤ = 0.
Using Lemma 2.2.5, we deduce that

V ℓ = I ℓ = W ℓ = ν = η ℓ = 0 for all ℓ ∈ {0, .., N}.
This shows that B λ + R λ is invertible and therefore a unique solution (I ℓ ) ℓ ∈ H 1 (S) of (2.2.28) exists. At this stage, by setting V ℓ = f iλ,ℓa ℓ,1 (iλ)I ℓ,x , we conclude as in the proof of Theorem 2.2.1 that (V, I, W, ν, η) ∈ D(A) is a solution of (2.2.22) and the proof is thus complete.

It remains to show the surjectivity of A, but first let us introduce the following Hilbert space

V * = (V, I) ∈ H 1 (S) × H 1 (S) such that V ℓ (L) = α ℓ I ℓ (L), ∀ℓ ∈ {0, .., N}, N ∑ ℓ=0 I ℓ (0) = 0, V ℓ (0) -V 0 (0) = 0, ∀ℓ ∈ {1, .., N} . Lemma 2.2.7. Let (θ, Λ) ∈ N ∏ ℓ=0 D(0, L) × N ∏ ℓ=0 D(0, L), then there exists (ψ, χ) ∈ V * such that ψ ℓ,x = θ ℓ , χ ℓ,x = Λ ℓ , ∀ℓ ∈ {0, .., N}.
(2.2.29)

Proof. As (2.2.29) means that ψ ℓ (resp. χ ℓ ) is a primitive of θ ℓ (resp. Λ ℓ ), we have

ψ ℓ (x) = x 0 θ ℓ (y) dy + d ℓ , χ ℓ (x) = x 0 Λ ℓ (y) dy + p ℓ , ( 2
        N ∑ ℓ=0 χ ℓ (0) = 0, ψ ℓ (0) = ψ 0 (0), ∀ℓ ∈ {1, .., N}, ψ ℓ (L) = α ℓ χ ℓ (L), ∀ℓ ∈ {0, .., N}.
(2.2.31)

Then, knowing that (2.2.32) we see that (2.2.31) is equivalent to the following system with 2N + 2 equations

ψ ℓ (0) = d ℓ , ψ ℓ (L) = L 0 θ ℓ (y) dy + d ℓ , χ ℓ (0) = p ℓ , χ ℓ (L) = L 0 Λ ℓ (y) dy + p ℓ ,
             N ∑ ℓ=0 p ℓ = 0, d ℓ = d 0 , ∀ℓ ∈ {1, .., N}, L 0 θ ℓ (y) dy + d ℓ = α ℓ L 0 Λ ℓ dy + p ℓ , ∀ℓ ∈ {0, .., N}.
(2.2.33)

By the second and third equations of (2.2.33) we find

p ℓ = 1 α ℓ L 0 θ ℓ (y) dy + d 0 - L 0 Λ ℓ (y) dy, ∀ℓ ∈ {0, .., N}. (2.2.34)
Then, replacing (2.2.34) in the first equation of (2.2.33)

leads to N ∑ ℓ=0 1 α ℓ L 0 θ ℓ (y) dy - L 0 Λ ℓ (y) dy + N ∑ ℓ=0 1 α ℓ d 0 = 0. ( 2
Proof. Let F = ( f 1 , f 2 , f 3 , f 4 , f 5 ) ⊤ ∈ H, we look for a unique solution U = (V, I, W, ν, η) ⊤ ∈ D(A) of -AU = F, or equivalently,                      a ℓ I ℓ,x + g ℓ V ℓ + k ℓ W ℓ = f ℓ,1 , b ℓ V ℓ,x + r ℓ I ℓ = f ℓ,2 , c ℓ W ℓ -V ℓ = f ℓ,3 , 1 δY N ∑ ℓ=0 I ℓ (0) = f 4 , 1 δ (V ℓ (0) -ν) = f ℓ,5 .
(2.2.36)

Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks Suppose such a U = (V, I, W, ν, η) ⊤ ∈ D(A) exists, then

W ℓ = V ℓ + f ℓ,3 c ℓ , ∀ℓ ∈ {0, .., N}.
Replace W ℓ in the first and second identities of (2.2.36), we get (2.2.37) where a ℓ,1 (0), a ℓ,2 (0), f 0,ℓ and g 0,ℓ are defined in (2.2.9).

V ℓ + a ℓ,1 (0)I ℓ,x = f 0,ℓ , V ℓ,x + a ℓ,2 (0)I ℓ = g 0,ℓ ,
We will proceed by dividing the proof into three steps.

Step 1. First, fix (ϕ,φ) ∈ H 1 (S) × H 1 (S), such that

ϕ ℓ (0) = δ f ℓ,5 ∀ ℓ ∈ {1, .., N}, ϕ ℓ = 0 on [ϵ 1 , L] for some ϵ 1 ∈ (0, L), ϕ 0 = 0 and φ 0 (0) = δY f 4 , φ 0 = 0 on [ϵ 2 , L] for some ϵ 2 ∈ (0, L), φ ℓ = 0 ∀ℓ ∈ {1, .., N}.
Then, setting V ℓ = V ℓϕ ℓ and I ℓ = I ℓφ ℓ , by (2.2.37) we deduce that they satisfy (2.2.38) where,

V ℓ + a ℓ,1 (0) I ℓ,x = f 0,ℓ , V ℓ,x + a ℓ,2 (0) I ℓ = g 0,ℓ ,
f 0,ℓ = f 0,ℓ -(ϕ ℓ + a ℓ,1 (0)φ ℓ,x ) and g 0,ℓ = g 0,ℓ -(ϕ ℓ,x + a ℓ,2 (0)φ ℓ ),
for all ℓ ∈ {0, .., N}. We further notice that

         N ∑ ℓ=0 I ℓ (0) = 0, V ℓ (0) -V 0 (0) = 0, ∀ℓ ∈ {1, .., N}, V ℓ (L) = α ℓ I ℓ (L), ∀ℓ ∈ {0, .., N}.
(2.2.39)

Step 2. Let (ψ, χ) ∈ V * . Multiplying the first and the second equation of (2.2.38) by χ ℓ,x and ψ ℓ,x respectively, then integrating in (0, L) gives

N ∑ ℓ=0 L 0 ( V ℓ χ ℓ,x + a ℓ,1 (0) I ℓ,x χ ℓ,x ) dx + N ∑ ℓ=0 L 0 ( V ℓ,x ψ ℓ,x + a ℓ,2 (0) I ℓ ψ ℓ,x ) dx = N ∑ ℓ=0 L 0 ( f 0,ℓ χ ℓ,x + g 0,ℓ ψ ℓ,x ) dx.
(2.2.40)

Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks

Again the sesquilinear form from this left-hand side is not necessarily coercive, so again we introduce the sesquilinear form

q(( V , I), (ψ, χ)) = N ∑ ℓ=0 L 0 a ℓ,1 (0) I ℓ,x χ ℓ,x dx + N ∑ ℓ=0 L 0 V ℓ,x ψ ℓ,x dx + N ∑ ℓ=0 L 0 (I ℓ χ ℓ + V ℓ ψ ℓ ) dx, ∀( V , I), (ψ, χ) ∈ V * .
This sesquilinear form q is continuous and coercive on V * since

a ℓ,1 (0) = a ℓ c ℓ (c ℓ g ℓ + k ℓ ) > 0, ∀ℓ ∈ {0, .., N}.
Then, by Lax-Milgram Lemma, the operator

Q : V * → (V * ) * : ( V , I) → Q( V , I), with Q( V , I)(ψ, χ) = q(( V , I), (ψ, χ)),
is an isomorphism. Now, the remainder

S : V * → (V * ) * : ( V , I) → S( V , I)(ψ, χ), with S( V , I)(ψ, χ) = N ∑ ℓ=0 L 0 ( V ℓ χ ℓ,x + a ℓ,2 (0) I ℓ ψ ℓ,x ) dx - N ∑ ℓ=0 L 0 (I ℓ χ ℓ + V ℓ ψ ℓ ) dx,
is a compact operator. Hence, we deduce that Q + S is a Fredholm operator of index zero from V * to (V * ) * .

By setting

L(ψ, χ) = N ∑ ℓ=0 L 0 ( f 0,ℓ χ ℓ,x + g 0,ℓ ψ ℓ,x ) dx,
we again see that (2.2.40) is equivalent to

(Q + S)( V , I) = L in (V * ) * . ( 2

.2.41)

Step 3. Now, let us prove that the system (2.2.38)-(2.2.39) is equivalent to (2.2.41).

First, let (θ, Λ) be an arbitrary test function, such that θ ℓ ′ = 0 ∀ℓ ′ ∈ {0, .., N}, and Λ ℓ ∈ D(0, L), for a fixed ℓ and Λ j = 0, ∀j ̸ = ℓ. Then, by Lemma 2.2.7, we deduce that there exists (ψ, χ) ∈ V * such that

ψ ℓ,x = 0, χ ℓ,x = Λ ℓ , ∀ℓ ∈ {0, .., N}.
Replacing this (ψ, χ) in (2.2.41) we get, L 0

( V ℓ Λ ℓ + a ℓ,1 (0) I ℓ,x Λ ℓ ) dx = L 0 f 0,ℓ Λ ℓ dx.
Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks

Then, V ℓ + a ℓ,1 (0) I ℓ,x = f 0,ℓ in D ′ (0, L),
which means that the first equation of (2.2.38) holds. Second, let (θ, Λ) be an arbitrary test function, such that Λ ℓ = 0 ∀ℓ ∈ {0, .., N}, θ ℓ ∈ D(0, L), for a fixed ℓ and θ j = 0, ∀j ̸ = ℓ. Using Lemma 2.2.7 and following the same arguments as above we deduce that the second equation of ( 2

+ S) = {0}. Let ( V , I) ∈ ker(Q + S) then N ∑ ℓ=0 L 0 ( V ℓ χ ℓ,x + a ℓ,1 (0) I ℓ,x χ ℓ,x ) dx + N ∑ ℓ=0 L 0 ( V ℓ,x ψ ℓ,x + a ℓ,2 (0) I ℓ ψ ℓ,x ) dx = 0, ∀(ψ, χ) ∈ V * .
(2.2.42) Thus, if we set

W ℓ = V ℓ c ℓ , ν = V 0 (0) and η ℓ = N ∑ ℓ=0 Z ℓ,k I k (0). We conclude that ( V , I, W, ν, η) ∈ D(A) is a solution of -A( V , I, W, ν, η) ⊤ = 0.
Using Lemma 2.2.5, we deduce that

V ℓ = I ℓ = W ℓ = ν = η ℓ = 0 for all ℓ ∈ {0, .., N}.
So, Q + S is invertible and a unique solution ( V , I) ∈ V * of (2.2.41) exists. Then, by the previous arguments we deduce that (2.2.38) and (2.2.39) hold.

As we set V ℓ = V ℓϕ ℓ and I ℓ = I ℓφ ℓ , we deduce that a unique solution (V, I) ∈ H 1 (S) × H 1 (S) of (2.2.37) exists, and satisfies

         N ∑ ℓ=0 I ℓ (0) = δY f 4 , V ℓ (0) -V 0 (0) = δ f ℓ,5 , ∀ℓ ∈ {1, .., N}, V ℓ (L) = α ℓ I ℓ (L), ∀ℓ ∈ {0, .., N}.
Finally, by defining

W ℓ = V ℓ + f ℓ,3 c ℓ ∀ℓ ∈ {0, .., N}, ν = V 0 (0), η ℓ = N ∑ ℓ=1 Z ℓk I k (0) ∀ℓ ∈ {1, .., N},
we deduce that a solution (V, I, W, ν, η) ∈ D(A) of (2.2.36) exists. The proof is complete.

Exponential energy decay rate

In this section, we will prove the exponential stability of System (2.1.3)-( 2 [START_REF] Nicaise | Stabilization and asymptotic behavior of a generalized telegraph equation[END_REF] to get exponential decay of the energy but rather exploit the dissipative boundary conditions (2.1.4) at the exterior nodes. Hence our proof fully differs from the proof of Theorem 4.5.4. from [START_REF] Nicaise | Stabilization and asymptotic behavior of a generalized telegraph equation[END_REF]. We here need to combine the frequency domain approach with an ad-hoc multiplier method (see Lemma 2.3.5 above).

Our main result is the following one.

Theorem 2.3.1. Assume that condition (2.1.2) holds. Also, assume that a ℓ , b ℓ ∈ W 1,∞ (0, L), for all ℓ ∈ {0, .., N}. Then, the semigroup (e tA ) t≥0 is exponentially stable, i.e. there exist two positive constants M and ϵ > 0 such that

||e tA U 0 || H ≤ Me -ϵt ||U 0 || H , ∀t ≥ 0, ∀U 0 ∈ H.
Proof. Following [START_REF] Huang | Characteristics conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF] .3.4) Our aim is to show that ||(V n , I n , W n , ν n , η n )|| H → 0. This condition permits to conclude a contradiction with (2.3.3). From now on, for simplicity we drop the index n. Now, writing (2.3.4) in a detailed form we get

U n = (V n , I n , W n , ν n , η n ) in D(A), such that |λ n | → ∞, ||(V n , I n , W n , ν n , η n )|| H = 1, (2.3.3) and (iλ n -A)U n → 0 in H as n → ∞. ( 2
(iλ + g ℓ )V ℓ + a ℓ I ℓ,x + k ℓ W ℓ = f ℓ,1 → 0 in L 2 (0, L), ∀ℓ ∈ {0, . . . , N}, (2.3.5) (iλ + r ℓ )I ℓ + b ℓ V ℓ,x = f ℓ,2 → 0 in L 2 (0, L), ∀ℓ ∈ {0, . . . , N}, (2.3.6) (iλ + c ℓ )W ℓ -V ℓ = f ℓ,3 → 0 in L 2 (0, L), ∀ℓ ∈ {0, . . . , N}, (2.3.7) iλV 0 (0) + 1 δY N ∑ ℓ=0 I ℓ (0) = f 4 → 0 in C, (2.3.8) iλη ℓ + 1 δ (V ℓ (0) -V 0 (0)) = f ℓ,5 → 0 in C, ∀ℓ ∈ {1, . . . , N}.
(2.3.9) Lemma 2.3.2. Under the above assumptions, we have the following estimations for all ℓ ∈ {0, .., N} :

√ g ℓ V ℓ = o(1) in L 2 (0, L), (2.3.10) I ℓ (L) = o(1) in C, (2.3.11) √ r ℓ I ℓ = o(1) in L 2 (0, L),
(2.3.12) .3.13) Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks

W ℓ = o(1) in L 2 (0, L). ( 2
Proof. Taking the inner product in H of (iλ -A)U with U and using (2.2.3) and Cauchy-Schwarz's inequality, we get

1 2 N ∑ ℓ=0 L 0 (θ ℓ g ℓ |V ℓ | 2 + 2β ℓ r ℓ |I ℓ | 2 + γ ℓ c ℓ |W ℓ | 2 ) dx + α ℓ |I ℓ (L)| 2 ≤ ℜ⟨(iλ -A)U, U⟩ H ≤ ||(iλ -A)U|| H ||U|| H .
As ||(iλ -A)U|| H → 0 and ||U|| H = 1, we obtain

1 2 L 0 (θ ℓ g ℓ |V ℓ | 2 + 2β ℓ r ℓ |I ℓ | 2 + γ ℓ c ℓ |W ℓ | 2 ) dx + α ℓ |I ℓ (L)| 2 = o(1), ∀ ℓ ∈ {0, .., N},
and the desired results hold.

From (2.3.3), (2.3.6) and ( 2.3.7) we have

||V ℓ,x || L 2 λ = O(1), ||I ℓ,x || L 2 λ = O(1
λV ℓ I ℓ dx + 2 L 0 g ℓ V ℓ I ℓ dx + 2 L 0 a ℓ I ℓ,x I ℓ dx + 2 L 0 k ℓ W ℓ I ℓ dx = 2 L 0 f ℓ,1 I ℓ dx.
(2.3.18)

The second and fourth integral on the left-hand side of (2.3.18) converge to zero by (2.3.3), (2.3.10) and (2.3.13), similarly the right-hand side converge to zero, using (2.3.3) and the fact that f ℓ,1 converge to zero in L 2 (0, L). This yields 2i L 0 .3.19) Taking the real part of (2.3.19) and then applying Green's formula on the second integral of the left side leads to,

λV ℓ I ℓ dx + 2 L 0 a ℓ I ℓ,x I ℓ dx = o(1). ( 2
-2 ℑ L 0 λV ℓ I ℓ dx - L 0 a ℓ,x |I ℓ | 2 dx + a ℓ (L)|I ℓ (L)| 2 -a ℓ (0)|I ℓ (0)| 2 = o(
λI ℓ V ℓ dx + 2 L 0 r ℓ I ℓ V ℓ dx + 2 L 0 b ℓ V ℓ,x V ℓ dx = 2 L 0 f ℓ,2 V ℓ dx. (2.3.21)
The right hand-side of (2.3.21) converge to zero by (2.3.6) and (2.3.3), similarly the second integral of the left hand-side converge to zero by (2.3.3) and (2.3.12). This yields 2i .3.22) Again, taking the real part of (2.3.22) and then, applying Green's formula in the second integral of the left hand side we get, 1).

L 0 λI ℓ V ℓ dx + 2 L 0 b ℓ V ℓ,x V ℓ dx = o(1). ( 2
-2 ℑ L 0 λI ℓ V ℓ dx - L 0 b ℓ,x |V ℓ | 2 dx + b ℓ (L)|V ℓ (L)| 2 -b ℓ (0)|V ℓ (0)| 2 = o(
(2.3.23) Adding equations (2.3.20) and (2.3.23) (2.3.24) Under the assumption that a ℓ , b ℓ ∈ W 1,∞ (0, L), using (2.1.2), (2.3.3), (2.3.11), (2.3.17) and the fact that V ℓ (L) = α ℓ I ℓ (L), we deduce from (2.3.24) that (2.3.25) and as |V 0 (0)| = O(1) using (2.3.3), we conclude that (2.3.15) holds. On the other hand, for ℓ = 0 and under the assumption that a 0 , b 0 ∈ W 1,∞ (0, L) and using (2.1.2), (2.3.3), (2.3.11) and as V 0 (L) = α 0 I 0 (L), we deduce from (2.3.24) that |I 0 (0)| 2 = O(1). Consequently, with (2.3.17), we conclude that (2.3.16) holds.

leads to - L 0 a ℓ,x |I ℓ | 2 dx - L 0 b ℓ,x |V ℓ | 2 dx + a ℓ (L)|I ℓ (L)| 2 -a ℓ (0)|I ℓ (0)| 2 +b ℓ (L)|V ℓ (L)| 2 -b ℓ (0)|V ℓ (0)| 2 = o(1), ∀ ℓ ∈ {0, .., N}.
|V ℓ (0)| = O(1), ∀ ℓ ∈ {1, .., N},
Lemma 2.3.4. Under the above assumptions, we have 1), (2.3.26) .3.27) Proof. Using (2.3.8), we have by (2.3.16) and the fact that | f 4 | = O(1), we deduce that V 0 (0) = o(1). Similarly, using (2.3.9), we have

ν = V 0 (0) = o(
and η ℓ = o(1) ∀ ℓ ∈ {1, .., N}. ( 2
V 0 (0) = - 1 iλδY N ∑ ℓ=0 I ℓ (0) + f 4 iλ ,
η ℓ = - 1 iλδ V ℓ (0) + 1 iλδ V 0 (0) + 1 iλ f ℓ,5
, by (2.3.15) and the fact that | f ℓ,5 | = O(1), we deduce that (2.3.27) holds.

Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks Lemma 2.3.5. Let ℓ ∈ {0, .., N} be fixed and assume that the functions a ℓ , b ℓ ∈ W 1,∞ (0, L). Then, there exists h ℓ ∈ W 1,∞ (0, L) and a positive real number ρ ℓ such that h ℓ (0) = 0, (2.3.28a)

h ℓ,x (x) ≥ ρ ℓ > 0, for a.a. x ∈ (0, L), (2.3.28b) h ℓ,x (x) a ℓ (x) b ℓ (x) + h ℓ (x) a ℓ (x) b ℓ (x) x ≥ 0, for a.a. x ∈ (0, L). (2.3.28c)
Proof. We first notice that (2.3.28c) is equivalent to

h ℓ,x (x) h ℓ (x) + a ℓ (x) b ℓ (x) x a ℓ (x) b ℓ (x)
≥ 0, for a.a. x ∈ (0, L), (2.3.29) or equivalently

h ℓ,x (x) h ℓ (x) ≥ ω ℓ (x) := - a ℓ (x) b ℓ (x) x a ℓ (x) b ℓ (x)
, for a.a. x ∈ (0, L).

(2.3.30)

Since ω ℓ (x) is bounded on (0, L), then if h ℓ (x) = x near zero, there exists a positive real number ϵ (sufficiently small) such that

1 x ≥ ω ℓ (x), ∀x ∈ (0, ϵ]. (2.3.31) So, we define M ℓ = max x∈(0,L) ω ℓ (x),
and we distinguish the following two cases: i) If M ℓ ≤ 1 L , then (2.3.31) holds with ϵ = L, and we can choose h ℓ (x) = x, ∀x ∈ (0, L). In this case, (2.3.28a), (2.3.28b) and (2.3.28c)

hold. ii) If M ℓ > 1
L , then we choose 0 < ϵ ≤ 1 M ℓ , and define h ℓ (x) = x, ∀x ≤ ϵ, so that (2.3.31) holds. Now, for x ≥ ϵ, we impose that

h ℓ (x) = ϵ e x ϵ m ℓ (y) dy , ∀x ≥ ϵ, (2.3.32) with m ℓ (x) := max{ω ℓ (x), κ}, (2.3.33)
for some κ > 0. Notice that lim x→ϵ h ℓ (x) = ϵ, i.e., h ℓ is continuous on (0, L). Now from the expression (2.3.32), we have .3.34) As m ℓ (x) ≥ ω ℓ (x), we deduce that (2.3.30) (2.3.34) and (2.3.33) implies that

h ℓ,x (x) = ϵ m ℓ (x)e x ϵ m ℓ (y) dy = h ℓ (x)m ℓ (x), ∀x > ϵ. ( 2
h ℓ,x (x) ≥ ϵκ > 0, ∀x > ϵ.
In other words, (2.3.28b) holds for almost all x ∈ (0, L).

Lemma 2.3.6. Under the above assumptions, we have for all ℓ ∈ {0, .., N}, .3.36) Proof. Fix ℓ ∈ {0, .., N} and let h ℓ ∈ W 1,∞ (0, L) be the function defined in Lemma 2.3.5. First, by multiplying (2.3.5) by (2h ℓ V ℓ,x ) and then integrating over (0,L), we get

V ℓ = o(1) in L 2 (0, L). ( 2
2 L 0 h ℓ V ℓ V ℓ,x dx -2i L 0 h ℓ g ℓ V ℓ V ℓ,x λ dx -2i L 0 a ℓ h ℓ V ℓ,x I ℓ,x λ dx -2i L 0 h ℓ k ℓ W ℓ V ℓ,x λ dx = -2i L 0 h ℓ f ℓ,1 V ℓ,x λ dx.
(2.3.37)

The second and fourth integral on the left hand-side of (2.3.37) converge to zero by (2.3.10), (2.3.13) and (2.3.14). Similarly, the right hand-side converge to zero by (2.3.14) and the fact that f ℓ,1 converge to zero in L 2 (0, L), for all ℓ. This yields

2 L 0 h ℓ V ℓ V ℓ,x dx -2i L 0 a ℓ h ℓ V ℓ,x I ℓ,x λ dx = o(1). (2.3.38)
By eliminating V ℓ,x from (2.3.6) in the second term of (2.3.38), we obtain

2 L 0 h ℓ V ℓ V ℓ,x dx -2i L 0 a ℓ b ℓ h ℓ f ℓ,2 I ℓ,x λ dx + 2 L 0 a ℓ b ℓ h ℓ I ℓ I ℓ,x dx +2i L 0 a ℓ b ℓ h ℓ r ℓ I ℓ I ℓ,x λ dx = o(1).
(2.3.39)

The second and fourth integral converge to zero by (2.3.14), (2.3.12) and the fact that f ℓ,2 converge to zero in L 2 (0, L), ∀ℓ. Now, by taking the real part of (2.3.39) and applying Green's formula on the remaining two integrals, we deduce 1).

- L 0 h ℓ,x |V ℓ | 2 dx + h ℓ (L)|V ℓ (L)| 2 -h ℓ (0)|V ℓ (0)| 2 - L 0 a ℓ b ℓ x h ℓ |I ℓ | 2 dx - L 0 a ℓ b ℓ h ℓ,x |I ℓ | 2 dx + a ℓ (L) b ℓ (L) h ℓ (L)|I ℓ (L)| 2 - a ℓ (0) b ℓ (0) h ℓ (0)|I ℓ (0)| 2 = o(
(2.3.40)

Then, due to the fact that V ℓ (L) = α ℓ I ℓ (L) and using (2.3.11) and (2.3.28a), we obtain .3.41) By using (2.3.28b) and (2.3.28c), the result holds.

L 0 h ℓ,x |V ℓ | 2 dx + L 0 a ℓ b ℓ x h ℓ + a ℓ b ℓ h ℓ,x |I ℓ | 2 dx = o(1). ( 2
Lemma 2.3.7. Under the above assumptions, we have for all ℓ ∈ {0, .., N}, .3.42) Proof. First, multiplying (2.3.6) by 1 b ℓ I ℓ and then integrating over (0, L), we find

I ℓ = o(1) in L 2 (0, L). ( 2
L 0 iλ b ℓ |I ℓ | 2 dx + L 0 r ℓ b ℓ |I ℓ | 2 dx + L 0 V ℓ,x I ℓ dx = L 0 f ℓ,2 b ℓ I ℓ dx. (2.3.43)
The second integral of the left hand-side of (2.3.43) converge to zero by (2.3.3) and (2.3.13). Similarly, the integral of the right hand side converge to zero by (2.3.3) and the fact that f ℓ,2 converge to zero in L 2 (0, L), for all ℓ. This yields .3.44) Next, by applying Green's formula on the second integral of the left-hand side of (2.3.44), eliminating I ℓ,x from (2.3.5) and then dividing by iλ, we obtain

L 0 iλ b ℓ |I ℓ | 2 dx + L 0 V ℓ,x I ℓ dx = o(1). ( 2
L 0 |I ℓ | 2 b ℓ dx - L 0 |V ℓ | 2 a ℓ dx + L 0 g ℓ iλa ℓ |V ℓ | 2 dx + L 0 k ℓ iλa ℓ V ℓ W ℓ dx + V ℓ (L)I ℓ (L) iλ - V ℓ (0)I ℓ (0) iλ = L 0 V ℓ f ℓ,1 iλa ℓ dx.
(2.3.45) Bastin and Coron, 2016, §5.6 (see also [START_REF] Gugat | On the limits of stabilizability for networks of strings[END_REF] for second order hyperbolic systems), our boundary conditions are chosen in such a way that there is no limit of stabilization, namely under our previous assumptions, exponential decay occurs for all lengths L but the decay rate ϵ in Theorem 2.3.1 may depend on L. If r ℓ = g ℓ = 0, we conjecture that this decay rate deteriorates as L goes to infinity.

As V ℓ (L) = α ℓ I ℓ (L)

Some extensions and open problems

Other dynamical boundary conditions

The standard Kirchhoff conditions at the interior common vertex are

     V 0 (0, t) -V ℓ (0, t) = 0, in R * + , ∀ℓ ∈ {1, ...., N}, N ∑ ℓ=0 I ℓ (0, t) = 0 in R * + , (2.4.1)
that corresponds to the limit case of (2.1.5) as δ goes to zero. These boundary conditions are stationary ones but can be also used instead of (2.1.5). More generally in the spirit of Kramar, Mugnolo, and Nicaise, 2020a, we can mix up stationary and dynamical boundary conditions at the common vertex v in the following way. First

if u(•, t) = ((V ℓ (•, t), I ℓ (•, t), W ℓ (•, t)) N ℓ=0 ) ⊤ represents the unknowns in (2.1.3), we denote by γ v (u(•, t)) = ((V ℓ (0, t)) N ℓ=0 , (I ℓ (0, t)) N ℓ=0
) ⊤ its trace at v that is a vector in C 2(N+1) . Obviously we only keep the trace of V ℓ and I ℓ since W ℓ is not regular enough in space (see the definition of D(A)). Then we fix Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks

a subspace Y v of C 2(N+1) , another subspace Y (d) v of Y v and finally a linear operator B v from Y v to Y (d) v and set x v (t) = P v γ v (u(•, t)),
where

P v is a projection from C 2(N+1) into Y (d)
v (for an appropriate inner product of C 2(N+1) ).

Hence at v, we can consider the boundary condition

γ v (u(•, t)) ∈ Y v , in R * + , x v,t (t) = B v γ v (u(•, t)), in R * + . (2.4.2)
The boundary conditions (2.1.5) and (2.4.1) enter in this setting. Namely in the first case, we need to take N+1) and

Y v = C 2(
Y (d) v = {((v ℓ ) N ℓ=0 , (i ℓ ) N ℓ=0 ) ⊤ : i 0 = v ℓ = 0, ∀ℓ = 1, • • • , N}, P v is the orthogonal projection into Y (d)
v with respect to the Euclidean inner product, and

B v ((v ℓ ) N ℓ=0 , (i ℓ ) N ℓ=0 ) ⊤ = 1 δ - 1 Y N ∑ ℓ=0 i ℓ , 0, • • • , 0 ⊤ , 0, Z -1 (v 0 1 N -v ⊤ ⊤
, where 1 N is the vector of C N with all entries equal to 1, while v = (v ℓ ) N ℓ=1 . On the contrary in the case (2.4.1), we simply need to take Y

(d) v = {0} and Y v = {((v ℓ ) N ℓ=0 , (i ℓ ) N ℓ=0 ) ⊤ : N ∑ ℓ=0 i ℓ = 0 and v 0 = v ℓ , ∀ℓ = 1, • • • , N}.
First, under some assumptions on Y v and B v , one can show that (2.2.3) remains valid and therefore the associated operator A is dissipative. For the maximality, in some particular situations, using the arguments of subsection 2.2.1 (eliminting the variables W ℓ and one of the other variables to obtain a variational formulation), we can prove that the associated operator is maximal dissipative under additional conditions on Y v and B v and therefore system (2.1.3) with the boundary conditions (2.1.4) and (2.4.2) is well-posed and is governed by a semigroup of contractions. In the general case, using a bounded perturbation argument as in Kramar, Mugnolo, and Nicaise, 2020a, Theorem 3.3 we can prove that a bounded perturbation of A is maximal dissipative under additional conditions on Y v and B v and therefore system (2.1.3) with the boundary conditions (2.1.4) and (2.4.2) is well-posed and is governed by a C 0 semigroup. But using the dissipativeness of A, one can deduce that A generates a semigroup of contractions.

Once the well-posedness of our system holds, one can analyze its stability. First, we notice that Lemma 2.2.5 remains valid because the proof presented before yields first V ℓ = I ℓ = W ℓ = 0, for all ℓ and hence x v = P v γ v (u) will be zero as well. On the contrary, Lemma 2.2.6 is problematic because a compact perturbation argument cannot be invoked due to the third identity in (2.2.22). But if the surjectivity of iλ -A holds for all λ ∈ R, then the system will be exponentially stable. Indeed, in Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks this case the proof of Theorem 2.3.1 remains valid because the use of the multipliers h ℓ from Lemma 2.3.5 allows to skip the use of the boundary condition (2.4.2).

In summary the exponential stability results for system (2.1.3) with the boundary conditions (2.1.4) and (2.4.2) are reduced to an existence result and the surjectivty of the resolvent on the imaginary axis.

General networks

Obviously, we can consider system (2.1.3) on an arbitrary network G with dissipative boundary conditions similar to (2.1.4) at exterior vertices (it is (2.1.4) if the edge having the exterior vertex v as extremity is outgoing, otherwise it is V ℓ (v, t) + α ℓ I ℓ (v, t) = 0) and boundary conditions like (2.4.2) at the interior vertices. As in the previous subsection, existence results can then be obtained under appropriate conditions on Y v and B v . Obviously the surjectivity of iλ -A for all λ ∈ R will remain problematic, but here even its injectivity is not immediate.

The sole case for which injectivity of iλ -A holds for all λ ∈ R * is the case of trees with the boundary conditions (2.1.5) or (2.4.1) at the interior vertices. Indeed, in this case the arguments of Lemma 2.2.5 allow to show that V ℓ = I ℓ = W ℓ = 0, for all edges ℓ of the last generation of the tree. But then using (2.1.5) or (2.4.1), we will get that V ℓ ′ (v ′ ) = I ℓ ′ (v ′ ) = 0 for all edges ℓ ′ of the penultimate generation and the vertex v ′ in common with the last generation. This means that we can use again the arguments of Lemma 2.2.5 to these edges ℓ ′ and by iteration we will find the injectivity result. Finally, the exponential stability results for system (2.1.3) with the boundary conditions (2.1.4) at exterior vertices and boundary conditions (2.1.5) or (2.4.1) at interior vertices is not immediate even for trees.

Note that for general networks, as in Nicaise, 2015, once iλ -A is bijective for all λ ∈ R, exponential stability will be valid if we assume that r ℓ + g ℓ is uniformly bounded from below for each ℓ (see the assumption (25) in [START_REF] Nicaise | Stabilization and asymptotic behavior of a generalized telegraph equation[END_REF]; without this condition, polynomial stability can be achieved. Nevertheless the type of the decay rate of such systems on general networks remains largely an open problem. Does it remain exponential or polynomial? How many boundary conditions can we remove from the exterior nodes to preserve stability? What will happen if we add some damping terms localized at the interior nodes? Our proof of Theorem 2.3.1 is non constructive (since it uses a contradiction argument), hence it does not give the dependency of the decay rate with respect to the involved parameters, hence its degeneracy as one parameter tends to zero or to infinity (see Remark 2.3.8) cannot be detected. Therefore, it would be interesting to characterize such dependencies by performing some spectral analysis for instance. In power grids, failure of some parts can occur from time to time, this corresponds to switch off one of the boundary controllers (α ℓ = 0) from time to time. As in Gugat and Sigalotti, 2010 for the wave equation, is it possible to find sufficient conditions on the controllers to achieve exponential decay? All of these problems are challenging and merit to be investigated.

The degree of a vertex is the number of incident edges at the vertex. A vertex with degree 1 is called an exterior vertex. On the other hand, a vertex with degree greater than 1 is called an interior vertex. We assume that the network is made of thermo-elastic edges and elastic ones, this means that I(N ) is split up into I(N ) = I te ∪ I e , with I e ∩ I te = ∅, in other words, I te (resp. I e ) is the set of thermo-elastic (resp. elastic) edges.

We further denote by: V ext := set of exterior vertices of N .

d kj =      1 if π j (ℓ j ) = a k , -1 if π j (0) = a k , 0 otherwise, (3.1.6)
and for a function f : N → C, we set f j = f • π j its restriction to the edge e j . For simplicity, we will write f = ( f 1 , .., f N ) and we will denote f j (x) = f j (π j (x)) for any x in (0, ℓ j ). We consider a network of elastic and thermo-elastic materials that coincides with the graph N . We assume that N contains at least one thermo-elastic edge, that V ext ̸ = ∅, that every maximal subgraph of elastic edges is a tree whose all of its exterior vertices except one are attached to thermo-elastic edges and that every subgraph of thermo-elastic edges is not a circuit.

Let u j = u j (x, t) and y j = y j (x, t) be the functions describing the displacement at time t of the edge e j , j ∈ I(N ) and θ j = θ j (x, t) be the temperature difference to a fixed reference temperature of e j , j ∈ I te at time t.

Our system is described as follows:

• On every thermo-elastic edge (j ∈ I te ) the following equations hold:

     u j tt -u j xx + α j θ j x -β j y j t = 0 in (0, ℓ j ) × (0, ∞), y j tt -ρ j y j xx + β j u j t = 0 in (0, ℓ j ) × (0, ∞), θ j t -κ j θ j xx + α j u j tx = 0 in (0, ℓ j ) × (0, ∞), (3.1.7)
where α j , ρ j , κ j and β j are positive constants.

• On every elastic edge (j ∈ I e ) one has: Chapter 3. A transmission problem of a thermo-elastic system on Networks (3.1.8) where β j and ρ j are positive constants.

u j tt -u j xx -β j y j t = 0 in (0, ℓ j ) × (0, ∞), y j tt -ρ j y j xx + β j u j t = 0 in (0, ℓ j ) × (0, ∞),
We assume that the initial data on the network N are

     u j (x, 0) = u j 0 (x), u j t (x, 0) = u j 1 (x), ∀j ∈ I(N ), y j (x, 0) = y j 0 (x), y j t (x, 0) = y j 1 (x), ∀j ∈ I(N ), θ j (x, 0) = θ j 0 (x), ∀j ∈ I te (N ). (3.1.9)
We denote by V ′ ext ( resp. V ′ int ) the set of exterior (resp. interior) nodes of maximal subgraph of thermo-elastic edges. Then, the boundary condition on N are described as follows:

The displacement and temperature satisfies the Dirichlet boundary condition,

     u j (a k , t) = 0, j ∈ I(a k ), a k ∈ V ext , y j (a k , t) = 0, j ∈ I(a k ), a k ∈ V ext , θ j (a k , t) = 0, j ∈ I te (a k ), a k ∈ V ′ ext .
(3.1.10)

The displacement and temperature are continuous,

     u j (a k , t) = u ℓ (a k , t), j, ℓ ∈ I(a k ), a k ∈ V int , y j (a k , t) = y ℓ (a k , t), j, ℓ ∈ I(a k ), a k ∈ V int , θ j (a k , t) = θ ℓ (a k , t), j, ℓ ∈ I te (a k ), a k ∈ V ′ int . (3.1.11)
The system satisfies the balance condition on y at every interior node,

∑ j∈I (a k ) d kj ρ j y j x (a k , t) = 0, a k ∈ V int .
(3.1.12)

The system satisfies the following balance conditions on u and θ, .1.13) Remark that α j > 0 and κ j > 0, for all j ∈ I te while, on each elastic edge only two conservative wave equations hold, i.e, the two wave equations on each elastic edge are neither coupled to a heat equation nor affected by a thermal damping. Hence for j ∈ I e , we may set α j = κ j = 0.

       ∑ j∈I te (a k ) d kj κ j θ j x (a k , t) = 0, a k ∈ V ′ int , ∑ j∈I te (a k ) d kj (u j x (a k , t) -α j θ j (a k ), t) + ∑ j∈I e (a k ) d kj u j x (a k , t) = 0, a k ∈ V int . ( 3
This chapter is organized as follows. In Section 3.2, we prove that system (3.1.7)- (3.1.13) admits a unique solution in an appropriate Hilbert space using semigroup theory. Next, in Section 3.3, using a general criteria of [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF], we discuss the strong stability of the system. In Section 3.4, under the condition that the two waves propagate with the same speed on each thermo-elastic edge of the network, we prove the exponential stability of the system using a frequency domain approach combined with a multiplier technique. Otherwise, we establish a polynomial decay. Finally, in Section 3.5, we present the Neumann boundary condition at the interior nodes of some particular networks, some of which being considered in [START_REF] Shel | Exponential stability of a network of elastic and thermoelastic materials[END_REF]. We show that under some sufficient conditions, the same results as the ones from Section 3.4 hold.

Well-posedness

In this section, we will study the existence, uniqueness and regularity of the solution of system (3.1.7)- (3.1.13), using a semigroup approach.

First, denote by

L 2 = N ∏ j=1 L 2 (0, ℓ j ), H m = N ∏ j=1 H m (0, ℓ j ), m = 1, 2,
and

V = ∏ j∈I te L 2 (0, ℓ j ), V m = ∏ j∈I te H m (0, ℓ j ), m = 1, 2.
Set

H 1 0 = u = (u j ) j∈I (N ) ∈ H 1 / u j (a k ) = 0, ∀j ∈ I(a k ), a k ∈ V ext and u j (a k ) = u ℓ (a k ), ∀j, ℓ ∈ I(a k ), a k ∈ V int . (3.2.1)
We define the energy space H associated with system (3.1.7)- (3.1.13), by

H = H 1 0 × L 2 × H 1 0 × L 2 × V (3.2.2)
equipped with the following inner product:

(U, Ũ) H = N ∑ j=1 ℓ j 0 (u j x ũj x + v j ṽj + ρ j y j x ỹj x + z j zj ) dx + ∑ j∈I te ℓ j 0 θ j θj dx, (3.2.3)
for all U = (u, v, y, z, θ), Ũ = ( ũ, ṽ, ỹ, z, θ) ∈ H. Next, we define the unbounded linear operator A associated to system (3.1.7)- (3.1.13) by

A       u v y z θ       =               v j u j xx -α j θ j x + β j z j z j ρ j y j xx -β j v j κ j θ j xx -α j v j x               j∈I (N ) (3.2.4) whose domain D(A) is given by D(A) = (u, v, y, z, θ) ∈ H ∩ [H 2 × H 1 0 × H 2 × H 1 0 × V 2 ] satisfying (3.2.5) below ,                                    θ j (a k ) = 0, j ∈ I te (a k ), a k ∈ V ′ ext , θ j (a k ) = θ ℓ (a k ), j, ℓ ∈ I te (a k ), a k ∈ V ′ int , ∑ j∈I te (a k ) d kj κ j θ j x (a k ) = 0, a k ∈ V ′ int , ∑ j∈I (a k ) d kj (u j x (a k ) -α j θ j (a k )) = 0, a k ∈ V int , ∑ j∈I (a k ) d kj ρ j y j x (a k ) = 0, a k ∈ V int .
(3.2.5)

If (u, y, θ) is a regular solution of (3.1.7)-(3.1.9), then by setting U = (u, u t , y, y t , θ), we can rewrite this system as the following evolution equation:

U t = AU, U(0) = U 0 , (3.2.6)
where U 0 = (u 0 , u 1 , y 0 , y 1 , θ 0 ).

We recall that the energy associated with system (3.1.7)- (3.1.13) is given by

E(t) = 1 2 N ∑ j=1 ℓ j 0 (|u j x | 2 + |u j t | 2 + ρ j |y j x | 2 + |y j t | 2 ) dx + 1 2 ∑ j∈I te ℓ j 0 |θ j | 2 dx (3.2.7)
and we will see that

E ′ (t) = -∑ j∈I te ℓ j 0 κ j |θ j x | 2 dx, (3.2.8)
for regular solutions. Hence, the system is dissipative in the sense that its energy is non-increasing. Proof. Using Lumer-Phillips Theorem (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], it is sufficient to prove that A is a maximal dissipative operator so that A generates a C 0 -semigroup of contractions on H. First, let U = (u, v, y, z, θ) ∈ D(A). We have, Thus, the operator A is dissipative. Now, in order to prove that A is maximal it is sufficient to show that R(I -A) = H. So, for F = ( f , f , g, g, h) ∈ H, we look for U ∈ D(A) such that (I -A)U = F. (3.2.11) Chapter 3. A transmission problem of a thermo-elastic system on Networks

Re(AU, U) H = Re N ∑ j=1 ℓ j 0 v j x u j x dx + N ∑ j=1 ℓ j 0 (u j xx -α j θ j x + β j z j )v j dx + N ∑ j=1 ℓ j 0 ρ j z j x y j x dx + N ∑ j=1 ℓ j 0 (ρ j y j xx -β j v j )z j dx + ∑ j∈I te ℓ j 0 (κ j θ j xx -α j v j x )θ j dx .
Equivalently, for all j ∈ I(N ), (3.2.13) y jz j = g j , (3.2.14) z jρ j y j xx + β j v j = gj , (3.2.15) .2.16) Assume that U ∈ D(A) exists, then by using equation (3.2.12) and (3.2.14) we obtain for all j ∈ I(N ),

u j -v j = f j , (3.2.12) v j -u j xx + α j θ j x -β j z j = f j ,
θ j -κ j θ j xx + α j v j x = h j . ( 3
v j = u j -f j , ( 3
.2.17)

z j = y j -g j . ( 3
y j -ρ j y j xx + β j u j = F j 2 , (3.2
.20)

θ j -κ j θ j xx + α j u j x = F j 3 (3.2.21)
where, As a is a continuous, coercive form on X × X and L is a continuous form on X , then using Lax-Milgram Theorem there exists a unique solution (u, y, θ) ∈ X of (3.2.29). Now, take in (3.2.29) the test function (φ 1 , 0, 0) such that φ j 1 ∈ C ∞ c (0, ℓ j ), for some Chapter 3. A transmission problem of a thermo-elastic system on Networks fixed j ∈ I(N ) and φ k 1 = 0 for all k ̸ = j, we obtain

F j 1 = f j + f j -β j g j , F j 2 = gj + g j + β j f j , F j 3 = h j + α j f j x . Set X = (φ 1 , φ 2 , φ 3 ) ∈ H 1 0 × H 1 0 × V 1 / φ j 3 (a k ) = 0, j ∈ I te (a k ), a k ∈ V ′ ext and φ j 3 (a k , t) = φ ℓ 3 (a k , t), j, ℓ ∈ I te (a k ), a k ∈ V ′ int . ( 3
ℓ j 0 u j φ j 1 dx - ℓ j 0 u j xx φ j 1 dx + ℓ j 0 α j θ j x φ j 1 dx - ℓ j 0 β j y j φ j 1 dx = ℓ j 0 F j 1 φ j 1 dx, (3.2.23) ℓ j 0 y j φ j 2 dx - ℓ j 0 ρ j y j xx φ j 2 dx + ℓ j 0 β j u j φ j 2 dx = ℓ j 0 F j 2 φ j 2 dx, ( 3
ℓ j 0 u j φ j 1 dx + ℓ j 0 u j x φ j 1,x dx - ℓ j 0 α j θ j φ j 1,x dx - ℓ j 0 β j y j φ j 1 dx = ℓ j 0 F j 1 φ j 1 dx, ∀φ j 1 ∈ C ∞ c (0, ℓ j )
for a fixed j.

(3.2.30) Applying Green's formula on the second and third term of (3.2.31) we get .2.31) This implies that u ju j xx + α j θ j xβ j y j = F j 1 , in D ′ (0, ℓ j ) where, D ′ (0, ℓ j ) is the associated space of distributions. As u j + α j θ j xβ j y j -F j 1 ∈ L 2 (0, ℓ j ), we deduce that u j ∈ H 2 (0, ℓ j ). Similarly, we can prove that (3.2.32) and y j , θ j ∈ H 2 (0, ℓ j ). Now, it remains to prove the transmission conditions in (3.1.12)-(3.1.13). For that aim, fix a k ∈ V int . Let, (3.1.12). Finally, we fix a k ∈ V ′ int , take (0, 0, φ 1 ) in (3.2.29), apply Green's formula and take into account (3.2.19)-(3.2.21), we get (3.1.13). By defining v j by (3.2.17) and z j by (3.2.18), for all j ∈ I(N ), we deduce that (u, v, y, z, θ) ∈ D(A) is a solution of (3.2.12)- (3.2.16) exists and the desired goal is attained.

ℓ j 0 u j φ j 1 dx - ℓ j 0 u j xx φ j 1 dx + ℓ j 0 α j θ j x φ j 1 dx - ℓ j 0 β j y j φ j 1 dx = ℓ j 0 F j 1 φ j 1 dx, ∀φ j 1 ∈ C ∞ c (0, ℓ j ) for a fixed j. ( 3
y j -ρ j y j xx + β j u j = F j 2 , θ j -κ j θ j xx + α j u j x = F j 3
φ j 1 =                  x ℓ j , if j ∈ I(a k ) and π j (ℓ j ) = a k , ℓ j -x ℓ j , if j ∈ I(a k ) and π j (0) = a k , 0, if j / ∈ I(a k ). ( 3
d kj u j x (a k ) -∑ j∈I (a k ) d kj α j θ j (a k ) = 0. ( 3
As A generates a C 0 -semigroup of contractions (e tA ) t≥0 (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], we have the following result: Eliminating v j (resp. z j ) using (3.3.1) (resp. (3.3.3)) and inserting them in (3.3.2), (3.3.4) and (3.3.5) we get the following system for all j ∈ I(N ), λ 2 u j + u j xxα j θ j x + iλβ j y j = 0, (3.3.6) Thus, θ j is constant for all j ∈ I te . But, using the fact that every maximal subgraph of thermo-elastic edges is not a circuit and using (3.1.10) and (3.1.11), we deduce that θ j = 0, ∀j ∈ I te . (3.3.10) Suppose that λ = 0. Then, (3.3.6), (3.3.7) and (3.3.9) implies that u j xx = 0, ∀j ∈ I(N ), ρ j y j xx = 0, ∀j ∈ I(N ). (3.3.11) Multiplying the first equation and second equation of (3.3.11) by u j and y j , respectively. Then, integrating over (0, ℓ j ), summing over j ∈ I(N ) and applying Green's formula, we get .3.12) But using (3.3.10) and the boundary conditions (3.1.10)-(3.1.13), the boundary terms are zero, hence (3.3.12) becomes .3.13) By the fact that ρ j > 0, for all j ∈ I(N ), we obtain that u j x = y j x = 0, for all j ∈ I(N ). Again, by (3.1.10), (3.1.11) and using the fact that V ext ̸ = ∅, we deduce that u j = y j = 0, for all j ∈ I(N ). Consequently, using (3.3.1) and (3.3.3), we conclude that v j = z j = 0, for all j ∈ I(N ) and therefore, U = 0. Now, suppose that λ ̸ = 0. We will distinguish between two cases. Chapter 3. A transmission problem of a thermo-elastic system on Networks Case 1. Assume that each maximal subgraph of thermo-elastic edges has an exterior vertex that belongs to V ext . Using (3.3.10) and (3.3.8), we have u j x = 0, ∀j ∈ I te . (3.3.14) This means that u j is constant for all j ∈ I te . But, using (3.1.10), (3.1.11) and the fact that every maximal subgraph of thermo-elastic edges has an exterior vertex that belongs to V ext , we deduce that u j = 0, for all j ∈ I te . Thus, by (3.3.1), we have v j = 0 and by (3.3.6), (3.3.14) and (3.3.9), we obtain that y j = 0, for all j ∈ I te . Consequently, by (3.3.3), we get z j = 0, for all j ∈ I te . Hence, u j = v j = y j = z j = θ j = 0 on both ends of e j , for all j ∈ I te . Now, let e j be an elastic edge attached only to thermo-elastic edge. As e j is identified by [0, ℓ j ], assume that ℓ j is the extremity in common with the thermo-elastic edge. Then, using (3.1.11), (3.3.10), (3.1.12) and (3.1.13), we have the following system

λ 2 y j + ρ j y j xx -iλβ j u j = 0, ( 3
                   N ∑ j=1 ℓ j 0 |u j x | 2 dx - N ∑ j=1 u j x u j ℓ j 0 = 0, N ∑ j=1 ℓ j 0 ρ j |y j x | 2 dx - N ∑ j=1 ρ j y j x y j ℓ j 0 = 0. ( 3
               N ∑ j=1 ℓ j 0 |u j x | 2 dx = 0, N ∑ j=1 ℓ j 0 ρ j |y j x | 2 dx = 0. ( 3
           λ 2 u j + u j xx + iλβ j y j = 0, λ 2 y j + ρ j y j xx -iλβ j u j = 0, u j (ℓ j ) = y j (ℓ j ) = 0, u j x (ℓ j ) = y j x (ℓ j ) = 0. (3.3.15) Let ũj =
u j , on (0, ℓ j ), 0, on (ℓ j , ℓ j + 1), and ỹj = y j , on (0, ℓ j ), 0, on (ℓ j , ℓ j + 1).

Then, using the boundary conditions of (3.3.15), we deduce that ( ũj , ỹj ) belongs to H 2 (0, ℓ j + 1) × H 2 (0, ℓ j + 1) and satisfies the first two equations of (3.3.15). Consequently, using Theorem 2.5 of [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], we deduce that ũj = ỹj = 0 on (0, ℓ j + 1) and hence, u j = y j = 0 on (0, ℓ j ). Then, v j = z j = 0 by equation (3.3.1) and (3.3.3) respectively. We repeat this technique to every elastic edge connected only to thermo-elastic edges and we proceed by iteration the same method on each maximal subgraph of purely elastic edges (from the leaves to the root), so that u j = v j = y j = z j = 0, for all j ∈ I(N ).

Case 2. Assume that there exists a maximal subgraph of thermo-elastic edges with no exterior vertices that belong to V ext and β j = β, for all j ∈ I(N ). First, notice that (3.3.14) holds and thus, u j xx = 0, ∀j ∈ I te . (3.3.16) Then, using (3.3.16) , (3.3.9) and the fact that λ ̸ = 0, β j = β, equation (3.3.6) becomes λu j + iβy j = 0, ∀j ∈ I te . (3.3.17) Differentiating (3.3.17) twice with respect to x and using (3.3.16), we deduce that y j xx = 0, ∀j ∈ I te . (3.3.18) Chapter 3. A transmission problem of a thermo-elastic system on Networks Then, using (3.3.18) and as λ ̸ = 0, β j = β, (3.3.7) becomes λy jiβu j = 0, ∀j ∈ I te . (3.3.19) Eliminating u j from (3.3.17) and replacing it in (3.3.19) we obtain (λ 2β 2 )y j = 0, ∀j ∈ I te . (3.3.20) Then, for λ ̸ = ±β we deduce that y j = 0, for all j ∈ I te and thus by equation (3.3.19) we get u j = 0, for all j ∈ I te . Again, we proceed using unique continuation Theorem from [START_REF] Hayek | A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface[END_REF] and iteration technique used in Case 1 to conclude that u j = v j = y j = z j = 0, for all j ∈ I(N ).

On the other hand, if λ = ±β. Without loss of generality, assume that λ = β. First, using (3.3.19), we have y ℓ = iu ℓ , ∀ℓ ∈ I te (3.3.21) and thus using (3.3.14), (3.3.21) implies that .3.22) Our aim is to prove that u j xx = 0, ∀j ∈ I(N ), y j xx = 0, ∀j ∈ I(N ). (3.3.23) This would end the proof as in the case λ = 0. As (3.3.16) and (3.3.18) hold, it is enough to prove that (3.3.23) holds for each elastic edge. Let e j be an elastic edge attached to a thermo-elastic edge at the vertex a k . As λ = β, then (3.3.6) and (3.3.7) lead to β 2 u j + u j xx + iβ 2 y j = 0, (3.3.24) .3.25) By eliminating u j from (3.3.24) and inserting it in (3.3.25), we obtain the following equation .3.26) Moreover, using (3.3.25), (3.1.11) and (3.3.21), we have

y ℓ x = iu ℓ x = 0, ∀ℓ ∈ I te . ( 3
β 2 y j + ρ j y j xx -iβ 2 u j = 0. ( 3
y j xxxx + β 2 (ρ j + 1) ρ j y j xx = 0. ( 3
y j xx (a k ) = 0 (3.3.27)
and using (3.3.25), (3.1.12), (3.1.13), (3.3.10) and (3.3.22), we get .3.28) Consequently, by setting Z j = y j xx and using (3.3.24)-(3.3.25), (3.3.27) and (3.3.28) we have the following system

y j xxx (a k ) = 0. ( 3
         Z j xx + β 2 (ρ j + 1) ρ j Z j = 0, Z j (a k ) = 0, Z j x (a k ) = 0.
(3.3.29) Chapter 3. A transmission problem of a thermo-elastic system on Networks Therefore, Z j = 0 and then y j xx = 0. This means that y j x is constant. But using (3.1.12) and (3.3.22), we deduce that y j x (a k ) = 0. Hence, y j x = 0. Therefore, using (3.3.25), we obtain that y j = iu j and then y j x = iu j x = 0. Again, by iteration on each maximal subgraph of purely elastic edges (from the leaves to the root), we repeat the same procedure to prove that (3.3.23) holds. Whenever (3.3.23) is attained, we can proceed as the case λ = 0 which finishes the proof. The same procedure can be used in the case λ = -β.

Let us finish this section by introducing some notations that will be used in the next section. let I ′ ext denotes the set of indices of edges adjacent to a vertex in V ′ ext and G ′ int denotes the set of indices of edges adjacent to two vertices in V ′ int .

Energy decay estimates

Take an arbitrary network N for which the System (3.1.7)- (3.1.13) is stable. In this section, we will prove that under the condition that the two coupled wave equations propagate with the same speed on each thermo-elastic edge, i.e., ρ j = 1, for all j ∈ I te , and using a frequency domain approach combined with a multiplier method, the energy of the system decays exponentially to zero. Otherwise, if there exist j ∈ I te ∩ I ′ ext such that ρ j ̸ = 1, we prove a polynomial decay rate of type t -1/3 , (see [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]. The main results are presented in Theorem 3.4.1 and Theorem 3.4.13. Theorem 3.4.1. Let N be an arbitrary network for which the operator A associated with System (3.1.7)- (3.1.13) satisfies (S1). If ρ j = 1, for all j ∈ I te , then the energy of the system decays exponentially in H. In other words, there exist two positive constants M and ϵ such that ∥e tA x 0 ∥ H ≤ Me -ϵt ∥x 0 ∥ H , ∀ t > 0, ∀ x 0 ∈ H.

Proof. Following [START_REF] Huang | Characteristics conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF], a C 0 -semigroup of contractions (e tA ) t≥0 on a Hilbert space H is exponentially stable if and only if (S1) and and

lim sup |λ|→∞ ∥(iλ -A) -1 ∥ L(H) < ∞ (S2)
hold. As we have assumed that (S1) is satisfied, it remains to prove that condition (S2) is satisfied. To this aim, we proceed by a contradiction argument. Suppose that (S2) does not hold, then there exist a sequence of real numbers λ n ∈ R and a sequence of vectors

U n = (u n , v n , y n , z n , θ n ) ∈ D(A) such that |λ n | -→ +∞, ∥U n ∥ H = ∥(u n , v n , y n , z n , θ n )∥ H = 1, (3.4.1)
and

(iλ n -A)U n = ( f n , fn , g n , gn , h n ) -→ 0 in H, (3.4.2) are satisfied.
In what follows, we drop the index n for simplicity.

Chapter 3. A transmission problem of a thermo-elastic system on Networks

Now by detailing (3.4.2), we get for all j ∈ I(N ) iλu jv j = f j → 0 in H 1 (0, ℓ j ), (3.4.3) iλv ju j xx + α j θ j xβ j z j = f j → 0 in L 2 (0, ℓ j ), (3.4.4) iλy jz j = g j → 0 in H 1 (0, ℓ j ), (3.4.5) iλz jρ j y j xx + β j v j = gj → 0 in L 2 (0, ℓ j ), (3.4.6)

iλθ j -κ j θ j xx + α j v j x = h j → 0 in L 2 (0, ℓ j ). (3.4.7)
Then, by eliminating v j and z j from equations (3.4.3) and (3.4.5) respectively, (3.4.3)-(3.4.7) imply (3.4.8) λ 2 y j + ρ j y j xxiλβ j u j = -β j f jgjiλg j , (3.4.9) (3.4.10) where, ρ j = 1, for all j ∈ I te . Now, we will proceed by dividing the proof into different Lemmas. As κ j > 0, ∀j ∈ I te , it follows that

λ 2 u j + u j xx -α j θ j x + iλβ j y j = β j g j -f j -iλ f j ,
iλθ j -κ j θ j xx + iλα j u j x = h j + α j f j x ,
||θ j x || 2 L 2 (0,ℓ j ) = o(1), ∀j ∈ I te .
Using (3.4.3), (3.4.5) and (3.4.1), we have for all j ∈ I(N ) 1), (3.4.12) .4.13) Also, using (3.4.8), (3.4.9) and (3.4.10), we have for all j ∈ I(N ) Proof. For all j ∈ I te , let Φ j be a function in W 1,∞ (0, ℓ j ), then multiply (3.4.8) by 2Φ j u j

||λu j || L 2 (0,ℓ j ) = O(1), ||u j x || L 2 (0,ℓ j ) = O(
||λy j || L 2 (0,ℓ j ) = O(1), ||y j x || L 2 (0,ℓ j ) = O(1). ( 3
u j xx λ L 2 (0,ℓ j ) = O(1), y j xx λ L 2 (0,ℓ j ) = O(1), θ j xx λ L 2 (0,ℓ j ) = O(1). ( 3
x , integrate over (0, ℓ j ), take the real part and apply Green's formula, we get -

ℓ j 0 Φ j x |λu j | 2 dx + Φ j |λu j | 2 x=ℓ j x=0 - ℓ j 0 Φ j x |u j x | 2 dx + Φ j |u j x | 2 x=ℓ j x=0 -2Re α j ℓ j 0 θ j x Φ j u j x dx + Re 2i ℓ j 0 β j Φ j λy j u j x dx = 2Re ℓ j 0 β j Φ j g j u j x dx -2Re ℓ j 0 f j Φ j u j x dx -2Re ℓ j 0 iλ f j Φ j u j x dx.
( 3.4.19) Using (3.4.12), (3.4.14) and (3.4.11) we obtain, Using (3.4.12) and the fact that f j converges to zero in H 1 (0, ℓ j ), (3.4.21) becomes

Φ j |λu j | 2 x=ℓ j x=0 + Φ j |u j x | 2 x=ℓ j x=0 = O(1) -2Re ℓ j 0 iλ f j Φ j u j x dx. ( 3
-2Re ℓ j 0 iλ f j Φ j u j x dx = -2Re(iλ f j Φ j u j ) x=ℓ j x=0
+ o(1). (3.4.22) Let Φ j = x, for all j ∈ I te . Using Young's inequality, we get 2Re .4.23) Recalling the Gagliardo-Nirenberg inequality in [START_REF] Liu | Semigroups associated with dissipative systems[END_REF]: For all ℓ > 0, there are two positive constants C 1 and C 2 depending on ℓ such that for any Ψ in .4.24) Applying (3.4.24) to Ψ = f j and using the fact that f j converge to zero in H 1 (0, ℓ j ), we deduce that f j (ℓ j ) = o(1). Thus, (3.4.23) yields that 2Re 1). (3.4.25) By inserting (3.4.25) in (3.4.20) and as Φ j = x, we obtain

ℓ j 0 iλ f j Φ j u j x dx ≤ ℓ 2 j ϵ j |λu j (ℓ j )| 2 + | f j (ℓ j )| 2 ϵ j + o(1). ( 3
H 1 (0, ℓ) ⊂ C([0, ℓ]), ||Ψ|| L ∞ (0,ℓ) ≤ C 1 ||Ψ x || 1/2 L 2 (0,ℓ) ||Ψ|| 1/2 L 2 (0,ℓ) + C 2 ||Ψ|| L 2 (0,ℓ) . ( 3
ℓ j 0 iλ f j Φ j u j x dx ≤ ℓ 2 j ϵ j |λu j (ℓ j )| 2 + o(
(ℓ j -ℓ 2 j ϵ j )|λu j (ℓ j )| 2 + ℓ j |u j x (ℓ j )| 2 = O(1)
.

By taking ϵ j = 1 2ℓ j , we deduce that (3.4.15) holds. Similarly, by taking Φ j = (x -ℓ j ), for all j ∈ I te , we conclude that (3.4.16) holds. Also, multiplying (3.4.9) by 2xy j x and 2(x -ℓ j )y j x respectively, we deduce that (3.4.17) and (3.4.18) hold. Taking the imaginary part and using the fact that α j is a positive constant for all j ∈ I te , we deduce that and integrate over (0, ℓ j ), we get

i ℓ j 0 |θ j + α j u j x | 2 dx - ℓ j 0 κ j θ j xx λ (θ j + α j u j x ) dx = ℓ j 0 h j λ (θ j + α j u j x ) dx + ℓ j 0 α j f j x λ (θ j + α j u j x ) dx.
(3.4.30)

By applying Green's formula on the second term of (3.4.30), using Cauchy-Schwarz inequality on the integrals of the right hand side, (3.4.1) and the fact that h j and f j x converge to zero in L 2 (0, ℓ j ), we obtain .4.31) Again, by using Gagliardo-Nirenberg inequality for

i ℓ j 0 |θ j + α j u j x | 2 dx + ℓ j 0 κ j |θ j x | 2 λ dx + ℓ j 0 α j κ j θ j x u j xx λ dx - κ j θ j x √ λ   θ j √ λ + α j u j x √ λ   x=ℓ j x=0 = o(1). ( 3
Ψ = θ j √ λ , Ψ = θ j x √ λ and Ψ = u j x √ λ
, we deduce that the boundary term in (3.4.31) converges to zero. Moreover, using (3.4.11) and (3.4.14), the second and third terms of (3.4.31) converge to zero. Consequently, using (3.4.29), we conclude (3.4.26). Proof. Let j ∈ I te . Multiply (3.4.8) by u j , integrate over (0, ℓ j ) and apply Green's formula, we get

ℓ j 0 |λu j | 2 dx - ℓ j 0 |u j x | 2 dx + u j x u j x=ℓ j x=0 - ℓ j 0 α j θ j x u j dx + ℓ j 0 iβ j λy j u j dx = ℓ j 0 β j g j u j dx - ℓ j 0 f j u j dx -iλ ℓ j 0 f j u j dx.
Using Cauchy-Schwarz inequality, (3.4.26), (3.4.15), (3.4.16), (3.4.11), (3.4.12) and (3.4.13), we deduce that

ℓ j 0 |λu j | 2 dx = o(1).
Using (3.4.3) we conclude that (3.4.32) holds.

As a conclusion, we have for every j ∈ I te v j → 0, in L 2 (0, ℓ j ),

u j → 0, in H 1 (0, ℓ j ), θ j → 0, in H 1 (0, ℓ j ).
Lemma 3.4.6. Assume that all above assumptions hold. Then, for every thermo-elastic edge, we have .4.35) Proof. By the proof of Lemma 3.4.3, for all j ∈ I te , and any Φ j in W 1,∞ (0, ℓ j ), (3.4.19) holds. Then, using (3.4.26), (3.4.32), (3.4.11) and (3.4.13) we obtain .4.36) Then, by taking Φ j = x, for all j ∈ I te , and using (3.4.25), we deduce that (3.4.36) becomes, (ℓ j -ℓ

λu j (ℓ j ) = o(1), u j x (ℓ j ) = o(1), Re iλ f j (ℓ j )u j (ℓ j ) = o(1), (3.4.33) λu j (0) = o(1), u j x (0) = o(1), Re iλ f j (0)u j (0) = o(1), (3.4.34) θ j (0) = o(1), θ j (ℓ j ) = o(1). ( 3
Φ j |λu j | 2 x=ℓ j x=0 + Φ j |u j x | 2 x=ℓ j x=0 = o(1) -2Re ℓ j 0 iλ f j Φ j u j x dx. ( 3
2 j ϵ j )|λu j (ℓ j )| 2 + ℓ j |u j x (ℓ j )| 2 = o(1)
.

Taking ϵ j = 1 2ℓ j , we deduce that

λu j (ℓ j ) = o(1) and u j x (ℓ j ) = o(1).
Consequently, by (3.4.36) and (3.4.22), we conclude that (3.4.33) holds. Similarly, by taking Φ j = (x -ℓ j ), for all j ∈ I te , we conclude that (3.4.34) holds. On the other hand, applying Gagliardo-Nirenberg inequality for Ψ = θ j , using (3.4.26) and (3.4.11) we deduce that (3.4.35) holds. Using Cauchy-Schwarz inequality, (3.4.11), (3.4.14) and the fact that g j and f j converge to zero in L 2 (0, ℓ j ), we obtain But, using (3.4.33), (3.4.34), (3.4.17), (3.4.18), (3.4.13) and the fact that f j converge to zero in H 1 (0, ℓ j ), we deduce that - .4.39) Similarly, multiplying (3.4.9) by u j xx λ , integrating over (0, ℓ j ) and using the fact that ρ j = 1 for all j ∈ I te , we get

ℓ j 0 λu j x y j x dx + ℓ j 0 u j xx y j xx λ dx -i ℓ j 0 β j |y j x | 2 dx = o(1). ( 3
ℓ j 0 λy j u j xx dx + ℓ j 0 y j xx u j xx λ dx -i ℓ j 0 β j u j u j xx dx = - ℓ j 0 β j f j u j xx λ dx - ℓ j 0 gj u j xx λ dx -i ℓ j 0 g j u j xx dx.
Using Cauchy-Schwarz inequality, (3.4.14) and the fact that f j and gj converge to zero in L 2 (0, ℓ j ), we obtain But, using (3.4.17), (3.4.18), (3.4.26), (3.4.33), (3.4.34), and the fact that g j coverges to zero in H 1 (0, ℓ j ), we deduce that Proof. Multiply (3.4.9) by y j then, integrate over (0, ℓ j ) and apply Green's formula, we get

ℓ j 0 |λy j | 2 dx - ℓ j 0 |y j x | 2 dx + y j y j x x=ℓ j x=0 -i ℓ j 0 β j λu j y j dx = - ℓ j 0 β j f j y j dx - ℓ j 0 gj y j dx -i ℓ j 0 g j λy j dx.
But, using Cauchy-Schwarz inequality, (3.4.12), (3.4.13), (3.4.17), (3.4.18), (3.4.37) and the fact that f j , g j converge to zero in H 1 (0, ℓ j ) and gj converges to zero in L 2 (0, ℓ j ), we deduce that (3.4.42) holds.

Lemma 3.4.9. Assume that all above assumptions hold. Then, for every thermo-elastic edge, we have Proof. Let e j be an elastic edge attached to a thermoelastic one at an interior vertex a k , , where a k is a leaf of a maximal subgraph of elastic edge. Recall that α j = 0, and let Φ j ∈ W 1,∞ (0, ℓ j ). Multiply (3.4.8) by 2Φ j u j x then integrate over (0, ℓ j ), take the real part and apply Green's formula, we obtain -

λy j (ℓ j ) = o(1), y j x (ℓ j ) = o(1), Re iλg j (ℓ j )y j (ℓ j ) = o(1), (3.4
ℓ j 0 Φ j x |λu j | 2 dx + Φ j |λu j | 2 x=ℓ j x=0 - ℓ j 0 Φ j x |u j x | 2 dx + Φ j |u j x | 2 x=ℓ j x=0 +Re 2i ℓ j 0 λβ j Φ j y j u j x dx = 2Re ℓ j 0 β j Φ j g j u j x dx -2Re ℓ j 0 f j Φ j u j x dx -2Re ℓ j 0 iλ f j Φ j u j x dx.
Again applying Green's formula on the fifth term of the left hand side and on the third term of the right hand side, we get - (3.4.48) Also, using Cauchy-Schwarz inequality, (3.4.12), (3.4.13) and the fact that f j converge to zero in H 1 (0, ℓ j ) and f j , g j converge to zero in L 2 (0, ℓ j ), we have 2Re Similarly, multiply (3.4.9) by 2Φ j y j x , integrate over (0, ℓ j ), take the real part and apply Green's formula, we obtain -

ℓ j 0 Φ j x |λu j | 2 dx + Φ j |λu j | 2 x=ℓ j x=0 - ℓ j 0 Φ j x |u j x | 2 dx + Φ j |u j x | 2 x=ℓ j x=0 -Re 2i ℓ j 0 λu j β j Φ j y j x dx -Re 2i ℓ j 0 λu j β j Φ j x y j dx + Re (2iβ j Φ j λy j u j ) x=ℓ j x=0 = 2Re ℓ j 0 β j Φ j g j u j x dx -2Re ℓ j 0 f j Φ j u j x dx + Re 2i ℓ j 0 λu j f j x Φ j dx +Re 2i ℓ j 0 f j Φ j x λu j dx -2Re(iλ f j Φ j u j )
ℓ j 0 β j Φ j g j u j x dx -2Re ℓ j 0 f j Φ j u j x dx + Re 2i ℓ j 0 λu j f j x Φ j dx +Re 2i ℓ j 0 f j Φ j x λu j dx = o (1).
ℓ j 0 Φ j x |λy j | 2 dx + Φ j |λy j | 2 x=ℓ j x=0 - ℓ j 0 ρ j Φ j x |y j x | 2 dx + ρ j Φ j |y j x | 2 x=ℓ j x=0 -Re 2i ℓ j 0 λβ j Φ j u j y j x dx = -Re 2 ℓ j 0 β j f j Φ j y j x dx -Re 2 ℓ j 0 gj Φ j y j x dx -Re 2i ℓ j 0 λg j Φ j y j x dx.
123 Chapter 3. A transmission problem of a thermo-elastic system on Networks Applying Green's formula on the last integral of the right hand side, we get -

ℓ j 0 Φ j x |λy j | 2 dx + Φ j |λy j | 2 x=ℓ j x=0 - ℓ j 0 ρ j Φ j x |y j x | 2 dx + ρ j Φ j |y j x | 2 x=ℓ j x=0 -Re 2i ℓ j 0 λβ j Φ j u j y j x dx = -Re 2 ℓ j 0 β j f j Φ j y j x dx -Re 2 ℓ j 0 gj Φ j y j x dx + Re 2i ℓ j 0 λg j x Φ j y j dx + Re 2i ℓ j 0 λg j Φ j x y j dx -2 Re(iλg j Φ j y j ) x=ℓ j x=0 .
(3.4.51) But, using (3.4.13) and the fact that g j converge to zero in H 1 (0, ℓ j ) and f j , gj converge to zero in L 2 (0, ℓ j ), we conclude that -Re 2 Without loss of generality, assume that π j (a k ) = 0 and let Φ j = x -ℓ j (otherwise, let Φ j = x). Then, adding the two equations (3.4.50) and (3.4.53), using Lemma 3.4.6, Lemma 3.4.9, the fact that u, y, f and g satisfy the continuity conditions in (3.1.11) and u, θ and y satisfy the balance conditions (3.1.12)-(3.1.13), we deduce that

- ℓ j 0 Φ j x |λy j | 2 dx + Φ j |λy j | 2 x=ℓ j x=0 - ℓ j 0 ρ j Φ j x |y j x | 2 dx + ρ j Φ j |y j x | 2
ℓ j 0 |λu j | 2 dx + ℓ j 0 |u j x | 2 dx + ℓ j 0 |λy j | 2 dx + ℓ j 0 ρ j |y j x | 2 dx = o(1).
Consequently, as ρ j > 0 for all j ∈ I e , (3.4.45) and (3.4.46) hold. Repeating the same technique of Lemma 3.4.6 and Lemma 3.4.9, we conclude that 1), Re iλg j (ℓ j )y j (ℓ j ) = o(1). Chapter 3. A transmission problem of a thermo-elastic system on Networks Then, by iteration on each maximal subgraph of purely elastic edges (from leaves to the root), we prove that 1), (3.4.54) for all j ∈ I e .

λu j (0) = o(1), u j x (0) = o(1), Re iλ f j (0)u j (0) = o(1), λy j (0) = o(1), y j x (0) = o(1), Re iλg j (0)y j (0) = o(1), λu j (ℓ j ) = o(1), u j x (ℓ j ) = o(1), Re iλ f j (ℓ j )u j (ℓ j ) = o(1), λy j (ℓ j ) = o(1), y j x (ℓ j ) = o(
ℓ j 0 |λu j | 2 dx + ℓ j 0 |u j x | 2 dx + ℓ j 0 |λy j | 2 dx + ℓ j 0 |y j x | 2 dx = o(
In conclusion, using Lemmas 3.4.4,3.4.5,3.4.7,3.4.8 and 3.4 Remark 3.4.12. If there exists an elastic edge (j ∈ I e ) such that ρ j ̸ = 1 then, using Lemma 3.4.10, we show that Theorem 3.4.1 holds (i.e., the energy of the system decays exponentially to zero). But, if there exists a thermo-elastic edge (j ∈ I te ) such that ρ j ̸ = 1 then, it seems that the energy of the system does not decay exponentially, but polynomially (see Theorem 3.4.13 below).

Theorem 3.4.13. Let N be an arbitrary network for which the operator associated with System (3.1.7)-(3.1.13) satisfies (S1). Assume that there exists j ∈ I te ∩ I ′ ext such that ρ j ̸ = 1. Then, the energy of the system satisfies .4.55) for some positive constant C > 0.

E(t) ≤ C t 1/3 ||U 0 || 2 D(A) , ∀U 0 ∈ D(A), t > 0, ( 3
Proof. Following Borichev and Tomilov, 2009, a C 0 -semigroup of contractions (e tA ) t≥0 on a Hilbert space H is polynomially stable if and only if (S1) and and lim sup

|λ|→∞ 1 λ 6 ∥(iλ -A) -1 ∥ L(H) < ∞ (S2)
hold. As we have assumed that (S1) is satisfied, it remains to prove that condition (S2) is satisfied. To this aim, we proceed by a contradiction argument. Suppose that (S2) does not hold, then there exist a sequence of real numbers λ n and a sequence of vectors

U n = (u n , v n , y n , z n , θ n ) ∈ D(A) such that (3.4.1) and λ 6 (iλ n -A)U n = ( f n , fn , g n , gn , h n ) -→ 0 in H (3.4.56) are satisfied.
In what follows, we drop the index n for simplicity. Now by detailing (3.4.56), we get for all j ∈ I(N ) 3.4.58) λ 6 (iλy jz j ) = g j → 0 in H 1 (0, ℓ j ), (3.4.59)

λ 6 (iλu j -v j ) = f j → 0 in H 1 (0, ℓ j ), (3.4.57) λ 6 (iλv j -u j xx + α j θ j x -β j z j ) = f j → 0 in L 2 (0, ℓ j ), (
λ 6 (iλz j -ρ j y j xx + β j v j ) = gj → 0 in L 2 (0, ℓ j ), (3.4.60) λ 6 (iλθ j -κ j θ j xx + α j v j x ) = h j → 0 in L 2 (0, ℓ j ). ( 3
λ 2 u j + u j xx -α j θ j x + iλβ j y j = β j g j λ 6 - f j λ 6 -i f j λ 5 , (3.4.62) λ 2 y j + ρ j y j xx -iλβ j u j = - β j f j λ 6 - gj λ 6 -i g j λ 5 , (3.4.63) iλθ j -κ j θ j xx + iλα j u j x = h j λ 6 + α j f j x λ 6 . (3.4.64)
First, our aim is to prove that for each thermo-elastic edge, we have

               v j → 0, in L 2 (0, ℓ j ), u j → 0, in H 1 (0, ℓ j ), θ j → 0, in H 1 (0, ℓ j ), z j → 0, in L 2 (0, ℓ j ), y j → 0, in H 1 (0, ℓ j ).
(3.4.65)

Following the same proof of Lemmas 3.4.4,3.4.5,3.4.7,3.4.8, we can prove that (3.4.65) holds for all j ∈ I te with ρ j = 1. Hence, we only need to prove that (3.4.65) holds for all j ∈ I te ∩ I ′ ext with ρ j ̸ = 1. We will proceed by dividing the proof into different Lemmas. Proof. Let Φ j be a function in W 1,∞ (0, ℓ j ), for all j ∈ I te . Multiplying (3.4.62) by 2Φ j u j x , integrating over (0, ℓ j ) then taking the real part and applying Green's Formula, we obtain -

ℓ j 0 Φ j x |λu j | 2 dx + Φ j |λu j | 2 x=ℓ j x=0 - ℓ j 0 Φ j x |u j x | 2 dx + Φ j |u j x | 2 x=ℓ j x=0 -2Re α j ℓ j 0 θ j x Φ j u j x dx + Re 2i ℓ j 0 β j λy j Φ j u j x dx = 2Re ℓ j 0 β j Φ j g j u j x λ 6 dx -2Re ℓ j 0 f j Φ j u j x λ 6 dx -2Re ℓ j 0 i f j Φ j u j x
λ 5 dx.

(3.4.67)

Using (3.4.12), (3.4.13) and (3.4.66) we get,

Φ j |λu j | 2 x=ℓ j x=0 + Φ j |u j x | 2 x=ℓ j x=0 = O(1). 126
Chapter 3. A transmission problem of a thermo-elastic system on Networks Let Φ j = x, for all j ∈ I te . We deduce that (3.4.15) holds. Similarly, by taking Φ j = (x -ℓ j ), we conclude (3.4.16). Also, multiplying (3.4.9) by 2xy j x and 2(x -ℓ j )y j x respectively, we deduce that (3.4.17) and (3.4.18) hold.

Lemma 3.4.16. Under all above assumptions, we have .4.68) Proof. Let j ∈ I te ∩ I ′ ext . Multiply (3.4.64) by u j x λ , and integrate over (0, ℓ j ), we get

ℓ j 0 |u j x | 2 dx = o(1) λ 2 , ∀ j ∈ I te ∩ I ′ ext . ( 3
i ℓ j 0 θ j u j x dx - ℓ j 0 κ j θ j xx u j x λ dx + i ℓ j 0 α j |u j x | 2 dx = ℓ j 0 h j u j x λ 7 dx + ℓ j 0 α j f j x u j x
λ 7 dx.

(3.4.69)

Applying Green's formula on the first and second term of (3.4.69), using Cauchy-Schwarz inequality, (3.4.12) and the fact that f j converge to zero in H 1 (0, ℓ j ) and h j converge to zero in L 2 (0, ℓ j ), we obtain -i On the other hand, by applying Gagliardo-Nirenberg inequality to Ψ = θ j and Ψ = θ j x λ , using Poincaré inequality, (3.4.66) and (3.4.14) we deduce that Proof. Let j ∈ I te ∩ I ′ ext . Multiply (3.4.62) by u j , integrate over (0, ℓ j ) and apply Green's formula, we get

ℓ j 0 θ j x u j dx + iθ j u j x=ℓ j x=0 + ℓ j 0 κ j θ j x u j xx λ dx -κ j θ j x λ u j x x=ℓ j x=0 +i ℓ j 0 α j |u j x | 2 dx = o(1) λ 7 . ( 3
θ j (0) = o(1) λ 3 , θ j (ℓ j ) = o(1) λ 3 , j ∈ I te ∩ I ′ ext , (3.4.73) and θ j x (0) λ = o(1) λ 2 , θ j x (ℓ j ) λ = o(1) λ 2 , j ∈ I te ∩ I ′ ext . ( 3
ℓ j 0 |λu j | 2 dx - ℓ j 0 |u j x | 2 dx + u j x u j x=ℓ j x=0 - ℓ j 0 α j θ j x u j dx + ℓ j 0 iβ j λy j u j dx = ℓ j 0 β j g j u j λ 6 dx - ℓ j 0 f j u j λ 6 dx -i ℓ j 0 f j u j λ 5 dx.
Using Cauchy-Schwarz inequality, (3.4.12), (3.4.13), (3.4.66), (3.4.68), (3.4.15) and (3.4.16) we deduce that

ℓ j 0 |λu j | 2 dx = o(1).
Using (3.4.57), we conclude that (3.4.75) holds.

Lemma 3.4.18. Assume that all above assumptions hold. Then, for j ∈ I te ∩ I ′ ext , we have .4.78) Proof. By the proof of Lemma 3.4.15, for all j ∈ I te ∩ I ′ ext , and any Φ j in W 1,∞ (0, ℓ j ), (3.4.67) holds. Then, using (3.4.68), (3.4.75), (3.4.66) and (3.4.13) we obtain Then, by taking Φ j = x, for all j ∈ I te ∩ I ′ ext , we deduce that (3.4.76) holds. Similarly, by taking Φ j = (x -ℓ j ), for all j ∈ I te ∩ I ′ ext , we conclude that (3.4.77) holds. On the other hand, applying Gagliardo-Nirenberg inequality, using (3.4.66) and Poincaré inequality, we deduce that (3.4.78) 128 Chapter 3. A transmission problem of a thermo-elastic system on Networks But, using (3.4.76), (3.4.77), (3.4.68) (3.4.75), (3.4.17), (3.4.18) and (3.4.13), we deduce that ℓ j 0 λ 3 y j u j dx = o(1). (3.4.81) Similarly, multiply (3.4.62) by λy j then integrate over (0, ℓ j ) and apply Green's formula, we obtain Then, taking the imaginary part of the above equality, using (3.4.81) and the fact that β j is a positive constant for all j ∈ I te ∩ I ′ ext , we deduce that λ 5 dx.

λu j (ℓ j ) = o(1), u j x (ℓ j ) = o(1), (3.4.76) λu j (0) = o(1), u j x (0) = o(1), (3.4.77) θ j (0) = o(1), θ j (ℓ j ) = o(1). ( 3
Then, using (3.4.79), (3.4.17), (3.4.18) and (3.4.13), we deduce that for all j ∈ I te ∩ I ′ Hence, using (??) and Lemma 3.4.18 and by iteration on each maximal subgraph of purely elastic edges (from leaves to root), the results of Lemma 3.4.10 holds for every elastic edge. Finally, using Lemmas 3.4.16,3.4.17,3.4.19,3.4.20 and 3.4.10, we conclude that ||U|| H = o(1), which contradicts (3.4.1). Remark 3.4.21. 1) If every maximal subgraph of thermo-elastic edges is composed of maximum three thermo-elastic edges and there exist j ∈ I te ∩ G ′ int such that Chapter 3. A transmission problem of a thermo-elastic system on Networks ρ j ̸ = 1 then, the same result of Theorem 3.4.13 holds. The key step to prove that result is to show that (3.4.73) still holds. Due to (3.1.11) and as (3.4.73) holds for all j ∈ I te ∩ I ′ ext , we deduce that (3.4.73) is achieved for all j ∈ I te (in particular for j ∈ I te ∩ G ′ int ). Hence, the desired result is attained.

2) If we replace condition (3.1.13) ii) There exists a maximal subgraph of thermo-elastic edges that contains no interior vertices and contains no exterior vertex that belongs to V ext and β j = β, for all j ∈ I(N ). As A * = A | D(A * ) , we obtain, (3.3.1)-(3.3.10). If λ = 0, we proceed exactly as in the proof of Theorem 3.3.1. Otherwise, if λ ̸ = 0 then, using (3.3.10) and (3.3.8) we have (3.3.14), which means that u j is constant for every j ∈ I te but, due to the fact that each maximal subgraph of thermo-elastic edges contains at least one interior vertex (i.e., V ′ int ̸ = ∅) and using (3.3.9), (3.3.1) and (3.1.11), the balance condition (3.4.84) asserts that u j (a k ) = 0 for some j ∈ I te (a k ), a k ∈ V ′ int . Again, using (3.1.11) we deduce that u j = 0, for all j ∈ I te . Finally, we proceed exactly as Case 1 of the proof of Theorem 3.3.1 to reach the desired aim.

Stabilization of Thermo-elastic System with Neumann Boundary condition at the interior nodes of some particular networks

In this section, we investigate the stabilization of a thermo-elastic system with Neumann boundary condition at the interior nodes of some particular networks (composed of elastic and thermoelastic materials) similar to the particular networks considered in [START_REF] Shel | Exponential stability of a network of elastic and thermoelastic materials[END_REF] In the first case, we consider trees (G 1 ) for which all exterior edges (except one) are thermo-elastic. In the second case, we consider the path (P ) composed of two exterior elastic edges and an interior thermoelastic edge.

In the third case, we consider (G 2 ), trees of elastic materials, whose leaves (exterior nodes of the last generation) are connected to thermoelastic materials as follows: the thermoelastic body connects two leaves issued from the same vertex, with the condition that each leaf is connected to only one thermoelastic body.

In fact, the considered networks (G 1 ), (P ) and (G 2 ) are particular graphs of the general networks covered in Section 3.1-Section 3.4. Notice that, if we apply the boundary conditions of Section 3.1 on these particular networks, we can deduce that the stabilization of the thermo-elastic system on (G 1 ), (P ) and (G 2 ) is achieved when θ satisfies Dirichlet condition on each end of every thermo-elastic edge (see Section 3.1-Section 3.4). In this section, we discuss the stabilization of the thermo-elastic system on these particular networks such that θ satisfies Neumann boundary condition at each interior node connected to a thermo-elastic edge (see The system is described as follows:

• On every thermo-elastic edge (j ∈ I te ) the following equations hold:

     u j tt -u j xx + α j θ j
xβ j y j t = 0 in (0, ℓ j ) × (0, ∞), y j ttρ j y j xx + β j u j t = 0 in (0, ℓ j ) × (0, ∞), θ j tκ j θ j xx + α j u j tx = 0 in (0, ℓ j ) × (0, ∞), (3.5.1) where α j , ρ j , κ j and β j are positive constants.

• On every elastic edge (j ∈ I e ) one has: u j ttu j xxβ j y j t = 0 in (0, ℓ j ) × (0, ∞), y j ttρ j y j xx + β j u j t = 0 in (0, ℓ j ) × (0, ∞), (3.5.2) where β j and ρ j are positive constants. We assume that the initial data on the network N are The boundary conditions of system (3.5.1)- (3.5.3) on the considered networks will be as follows.

The system satisfies the Dirichlet boundary condition for the displacement and temperature at the exterior nodes,      u j (a k , t) = 0, j ∈ I(a k ), a k ∈ V ext , y j (a k , t) = 0, j ∈ I(a k ), a k ∈ V ext , θ j (a k , t) = 0, j ∈ I te (a k ), a k ∈ V ext . (3.5.4) The displacement is continuous at every interior node, u j (a k , t) = u ℓ (a k , t), j, ℓ ∈ I(a k ), a k ∈ V int , y j (a k , t) = y ℓ (a k , t), j, ℓ ∈ I(a k ), a k ∈ V int .

(3.5.5)

The temperature satisfies the Neumann condition at the interior nodes, θ j x (a k , t) = 0, j ∈ I te (a k ), a k ∈ V int .

(3.5.6)

The system satisfies the balance condition at every interior node,

             ∑ j∈I te (a k )
d kj (u j x (a k , t)α j θ j (a k ), t) + ∑ j∈I e (a k )

d kj u j x (a k , t) = 0, a k ∈ V int , ∑ j∈I (a k )
d kj ρ j y j x (a k ) = 0, a k ∈ V int .

(3.5.7)

Mainly, we find sufficient conditions on the lengths of the purely elastic edges attached to the thermo-elastic ones so that the system is strongly stable and then exponentially stable on the above described networks.

Here the energy space H 1 is given by H 1 = (u, u t , y, y t , θ) ∈ H 1 0 × L 2 × H 1 0 × L 2 × V satisfying (3.5.9) below , (3.5.8) ℓ j 0 (α j u j x + θ j ) dx = 0, ∀ j ∈ G int ∩ I te (3.5.9) where, G int is the set of indices of edges adjacent to two interior vertices. The Hilbert space H 1 is equipped with the inner product given in (3.2.3). (3.5.10) with domain

A 1       u v y z θ       =               v j u j xx -α j θ j x + β j z j z j ρ j y j xx -β j v j κ j θ j xx -α j v j x               j∈I (N )
D(A 1 ) = (u, v, y, z, θ) ∈ H 1 ∩ [H 2 × H 1 0 × H 2 × H 1 0 × V 2 ], satisfying (3.5.11)                  θ j (a k ) = 0, j ∈ I te (a k ), a k ∈ V ext , θ j x (a k ) = 0, j ∈ I te (a k ), a k ∈ V int , ∑ j∈I (a k ) d kj (u j x (a k ) -α j θ j (a k )) = 0, a k ∈ V int , ∑ j∈I (a k )
d kj ρ j y j x (a k ) = 0, a k ∈ V int . (3.5.11) Let Ω denotes the set of indices of purely elastic edges attached to thermo-elastic edges in the network N . The main results of this section are stated in the following Theorems: Proof. The same proof as the one of Theorem 3.2.1 implies that A 1 is a maximal dissipative operator. Then, using Lumer-Phillips Theorem (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], A 1 generates a C 0 -semigroup of contractions (e tA 1 ) t≥0 on H 1 . Theorem 3.5.2. Consider the system (3.5.1)-(3.5.7) on N . Assume that one of the following conditions holds:

1) N is the graph G 1 ,
2) N is the graph P, β j = β, ρ j = 1, ∀j ∈ I(P ), and there exists j ∈ {1, 3} such that ℓ j ̸ = mπ √ 2β j , ∀m ∈ N * , (3.5.12)

3) N is the graph G 2 , β j = β, ρ j = 1, ∀j ∈ I(G 2 ), and in every circuit C, for the unique j, k ∈ Ω such that e j and e k are edges of C, we have sin( .5.13) Then, iR ⊂ ρ(A 1 ) and therefore the C 0 -semigroup of contractions (e tA 1 ) t≥0 is strongly stable.

√ 2β j ℓ j ) + sin( √ 2β k ℓ k ) ̸ = 0. ( 3
Proof. As before (I -A 1 ) -1 being compact, then σ(A 1 ) = σ p (A 1 ). Thus, it is sufficient to prove that σ p (A 1 ) ∩ iR = ∅. Let λ ∈ R and U = (u, v, y, z, θ) ∈ D(A 1 ) be such that A 1 U = iλU, equivalently, for all j ∈ I(N ) we have (3.3.1)-(3.3.5). Using (3.3.1), (3.3.3) to eliminate v j and z j , we obtain (3.3.6)-(3.3.8). Further, we easily check that This implies that (3.3.9) holds. Thus, θ j is constant for all j ∈ I te .

If λ = 0 then, using (3.3.6), (3.3.8) and (3. where α j = 0, ∀j ∈ I e .

Multiplying the second equation and third equation of (3.5.14) by u j and y j , respectively. Then, integrating over (0, ℓ j ), summing over j ∈ I(N ) and applying Green's formula, we get This implies that u j x = y j x = 0, for all j ∈ I(N ) and θ j = 0, for all j ∈ I te . Again, by (3.5.4)-(3.5.7), we deduce that u j = y j = 0, for all j ∈ I(N ). Consequently, using (3.3.1) and (3.3.3), we conclude that U = 0. Now, assume that λ ̸ = 0. We will proceed by distinguishing different cases: Case i. Assume that N is the graph G 1 . Then, the proof in this case is exactly the same as the proof of Case 1 in Theorem 3.3.1. In fact, under the boundary conditions of Section 3.5, on each thermo-elastic edge e j of (G 1 ), θ j satisfies the Dirichlet boundary condition on one end and the Neumann boundary condition on the other end. While, under the boundary conditions of Section 3.1, θ j satisfies the Dirichlet boundary condition on the both ends of each thermo-elastic edge e j of (G 1 ). This shows that on networks like (G 1 ), if θ j satisfies Dirichlet boundary condition on only one end of each thermo-elastic edge e j , then it is enough to prove that the system is strongly stable. Now, before proceeding the other cases (Case ii and Case iii below), remark that for an arbitrary network N with β j = β, ρ j = 1, for all j ∈ I(N ), and using (3.3.8) and (3.3.9), (3.3.16)-(3.3.20) hold for all j ∈ I te . Then, for λ ̸ = ±β we have y j = 0, for all j ∈ I te and thus by equation (3.3.19) we get u j = 0, for all j ∈ I te . Then, by (3.3.8), we obtain that θ j = 0, for all j ∈ I te . Again, by proceeding using unique continuation Theorem in [START_REF] Hayek | A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface[END_REF] and iteration technique used in Case 1 of the proof of Theorem 3.3.1, we conclude that u j = v j = y j = z j = 0, for all j ∈ I(N ). So, it is enough to treat the remaining cases (Case ii and Case iii below) with λ = ±β. Without loss of generality, assume that λ = β.

                  
Case ii. Assume that N is the path P, composed of two exterior elastic edges and an interior thermo-elastic edge, β j = β, ρ j = 1, for all j ∈ I(P ) and there exists j ∈ {1, 3} such that (3.5.12) holds. Without loss of generality, assume that π 1 (0) = a 0 , π 2 (0) = a 1 and π 3 (0) = a 2 . For the thermoelastic edge e 2 , we have u 2 xx = y 2 xx = 0 but, using (3.3.17) and dividing by λ = β, we also have y 2 = iu 2 . Thus, u 2 and y 2 can be written in the following form y 2 = ax + b, u 2 = -iaxib, for some a, b ∈ C. (3.5.15) Moreover, using (3.3.9), (3.5.15) and (3.5.9), we can write .5.16) For the elastic edges {e 1 , e 3 }, (3.3.6)-(3.3.8) becomes, β 2 u j + u j xx + iβ 2 y j = 0, on (0, ℓ j ), j = 1, 3, β 2 y j + y j xxiβ 2 u j = 0, on (0, ℓ j ), j = 1, 3.

θ 2 = iα 2 a. ( 3
(3.5.17) Using (3.5.4)-(3.5.7), (3.5.15) and (3.5.16), we have the following boundary conditions:

    
u 1 (0) = y 1 (0) = 0, u 1 (ℓ 1 ) = u 2 (0) = -ib, y 1 (ℓ 1 ) = y 2 (0) = b, u 1

x (ℓ 1 ) = u 2 x (0)α 2 θ 2 (0) = -i(1 + α 2 2 )a, y 1 x (ℓ 1 ) = y 2 x (0) = a (3.5.18) and

     u 3 (ℓ 3 ) = y 3 (ℓ 3 ) = 0, u 3 (0) = u 2 (ℓ 2 ) = -iaℓ 2 -ib, y 3 (0) = y 2 (ℓ 2 ) = aℓ 2 + b, u 3 x (0) = u 2 x (ℓ 2 ) -α 2 θ 2 (ℓ 2 ) = -i(1 + α 2
2 )a, y 3 x (0) = y 2 x (ℓ 2 ) = a.

(3.5.19)

Consequently, (3.5.17), (3.5.18) and (3.5.19) leads to the following system Φ j x = AΦ j , j = 1, 3 (3.5.20) where, Case iii. Assume that N is the graph G 2 , β j = β, ρ j = 1, for all j ∈ I(G 2 ) and that in every circuit C, for the unique j, k ∈ Ω such that e j and e k are edges of C, (3.5.13) holds. Notice that for all j ∈ I te , we have (3.3.21) holds. As in the proof of Case 2 of Theorem 3.3.1, our aim is to prove (3.5.14). This would end the proof as in the case λ = 0. But since, (3.3.9), (3.3.16) and (3.3.18) hold, then it is enough to prove that (3.5.14) holds for every elastic edge. First, for a fixed circuit C of G 2 without loss of generality, we may use the parametrizations from (3.5.28)

Φ j =     
Our aim is to find {u 1 (0), y 1 (0), u 3 (ℓ 3 ), y 3 (ℓ 3 )}. For that purpose, we use (3.5.24) to find Φ 1 (0) and Φ 3 (ℓ 3 ). Then, using (3.5.28) and technical computations, we have

       u 1 (0) = i √ 2α 2 2 a
4β sin(

√ 2βℓ 1 ) + iaℓ 1 -ib + iα 2 2 ℓ 1 a 2 , y 1 (0) = √ 2α 2 2 a 4β sin( √ 2βℓ 1 ) -aℓ 1 + b - α 2 2 ℓ 1 a 2 .
This implies that y 1 xx (0) = β 2 (y 1 (0)iu 1 (0)) = On the other hand, using (3.5.4)-(3.5.7), we have u 1 (0) = u 3 (ℓ 3 ) and y 1 (0) = y 1 (ℓ 3 ).

√ 2βℓ 3 ) = 0.
Notice that if sin( √ 2βℓ 1 ) + sin( √ 2βℓ 3 ) ̸ = 0, then, a = 0, i.e., θ 2 = 0. Using (3.3.8), we obtain that u 2 x = 0. Again, repeating the same strategy in every circuit of N and using the fact that (3.5.13) holds, we deduce that θ j = 0, ∀j ∈ I te .

(3.5.32) Hence, u j x = 0, for all j ∈ I te and (3.3.22) holds. Consequently, using (3.5.4)-(3.5.7), (3.5.32), (3.3.21)-(3.3.22) and using iteration method from the leaves to the root, we prove that every elastic edge satisfies (3.3.29). This implies that y j xx = 0, for all j ∈ I e . Finally, we can proceed as the case λ = 0. This finishes the proof. The same procedure is used in the case λ = -β so that the desired goal holds.

Theorem 3.5.3. Let N be an arbitrary network for which the operator (A 1 ) associated with System (3.5.1)-(3.5.7) satisfies iR ⊂ ρ(A 1 ). If ρ j = 1, for all j ∈ I(N ), then the energy of the system decays exponentially in H 1 . In other words, there exist two positive constants M and ϵ such that

∥e tA 1 x 0 ∥ H 1 ≤ Me -ϵt ∥x 0 ∥ H 1 , ∀ t > 0, ∀ x 0 ∈ H 1 .
Proof. Same proof as the one of Theorem 3.4.1 holds. Remark 3.5.4. Examples of networks for which iR ⊂ ρ(A 1 ) are given by Theorem 3.5.2.

FIGURE 1 :

 1 FIGURE 1: waves in sciences

FIGURE 2 :

 2 FIGURE 2: Sea waves FIGURE 3: waves on the strings of a guitar

FIGURE

  FIGURE 4: Electromagnetic waves

FIGURE 5 :

 5 FIGURE 5: A coaxial cable

FIGURE

  FIGURE 6: Heat conduction

FIGURE 7 :

 7 FIGURE 7: A launched rocket

FIGURE 8 :

 8 FIGURE 8: The generation of several waves

FIGURE

  FIGURE 9: viscous fluids

FIGURE 10 :

 10 FIGURE 10: A group of visco-elastic materials

FIGURE 11 :

 11 FIGURE 11: Different localizations of the damping. The pink color represents the damping region.

FIGURE 12 :

 12 FIGURE 12: Chemical and computer networks

FIGURE 13 :

 13 FIGURE 13: Electrical and arterial networks

  with b, c : Ω → R are the visco-elastic damping functions satisfying b, c ∈ L ∞ (Ω), (0.4.4) and b(x) ≥ 0, c(x) ≥ 0 for almost all x ∈ Ω. (0.4.5)

  [START_REF] Hansen | Exponential energy decay in a linear thermoelastic rod[END_REF] considered (0.4.30)-(0.4.31) with u and θ satisfying the Dirichlet and Neumann condition respectively (or vice versa) while, Liu and Zheng, 1993 considered (0.4.30)-(0.4.31) with u and θ satisfying both the Dirichlet condition. Both authors succeeded in proving the exponential stability of the system.

FIGURE 14 :

 14 FIGURE 14: A thermoelastic rod

FIGURE 15 :

 15 FIGURE 15: An elastic/thermo-elastic transmission problem

FIGURE 16 :

 16 FIGURE 16: Elastic/therm-elastic networks

FIGURE 17 :

 17 FIGURE 17: A star shaped network of elastic/ thermo-elastic rods

Chapter 1 .

 1 A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface of energy decay rate remains an open problem, our goal is to answer this open question.

Case 3 .

 3 Otherwise, if (H1) holds, i.e. α = α 0 is a positive constant and ω b is any open subset of Ω. Then, by the first equation of (1.2.25) and using(1.2.24) we get ∇u = 0 in ω b .

  Now, we are ready to state our result. Theorem 1.3.2. (Analytic stability) Assume that (1.1.7), (1.1.9) hold and ω b = ω c = Ω. Then the C 0 -semigroup (e tA ) t≥0 is analytic.

Proposition 1 . 4 . 2 .

 142 Assume that c = 0, b = 1 and α = α 0 is a positive constant. Then, there exists k 0 ∈ N * sufficiently large and two sequences λ + k and λ - k simple roots of P (that are also simple eigenvalues of A) satisfying the following asymptotic behavior

  1.5.1) Without loss of generality, assume that b ̸ = 0 and c = 0 (The same proof holds if c ̸ = 0 and b = 0). Theorem 1.5.1. (Polynomial energy decay rate) Let ω b and ω α be non-empty open subsets of Ω such that (1.1.7) and (1.1.8) hold. Also, suppose that

  .5.22) we get Ω |λu| 2 dx = o(1). Lemma 1.5.4. Under all the above assumptions, we have Ω |λy| 2 dx = o(1). (1.5.23) Proof. By inserting equation (1.5.7) in equation (1.5.8) we get

- 2 ℓ

 2 with ℓ = 2β + 4. In fact, under the additional assumption ω α = suppα ⊂ ω b we will find a better estimation of ||y n || L 2 (ω α ) . Theorem 1.5.7. (Polynomial energy decay rate) Let ω α and ω b be non-empty subsets of Ω. Assume that conditions (1.1.7) and (1.1.8) hold.

  1.5.40) with Kelvin-Voigt dampings and non-smooth coefficient at the interface are satisfied.

  By inserting equation(1.5.42) in equation(1.5.43) we get

10 .

 10 Under all the above assumptions, if ℓ ≥ max{β + 2, 2β} we have Ω |λu| 2 dx = o(1).

Lemma 1 . 6 . 4 .

 164 Under all the above assumptions, if ℓ ≥ 0 then,

Ω |λu| 2

 2 dx = o(1), Ω |λy| 2 dx = o(1), and using Lemma 1.6.4, we obtain Ω |∇u| 2 dx = o(1), and Ω |∇y| 2 dx = o(1).

Chapter 2 .FIGURE 2 . 1 :

 221 FIGURE 2.1: A star-shaped network

V

  int := set of interior vertices of N . I(a k ):= set of indices of edges incident to a k . I te (a k ):= set of indices of thermo-elastic edges adjacent to a k . I e (a k ):= set of indices of elastic edges incident to a k . I ext := set of indices of edges adjacent to an exterior vertex of N . The incidence matrix D = (d kj ) n×N of N is defined by

Theorem 3 . 2 . 1 .

 321 The unbounded linear operator A associated with system (3.1.7)-(3.1.13) generates a C 0 -semigroup of contractions on H.

  transmission problem of a thermo-elastic system on Networks Applying Green's formula on the second and third term of(3.2.23) and taking the sum over I(N ), we obtain using(3.1.13), 's formula on the second term of (3.2.24) and taking the sum over I(N ), using (3.1.12), we get Green's formula on the second term of(3.2.25) and taking the sum over I(N ), condition(3.1.13) yields that, u, y, θ), (φ 1 , φ 2 , φ 3 )) = L(φ 1 , φ 2 , φ 3 ), ∀(φ 1 , φ 2 , φ 3 ) u, y, θ), (φ 1 , φ 2 , φ 3 )) =

2

 2 dx = Re(AU, U) H = Re(iλU, U) H = 0

Lemma 3 . 4 . 2 . 2 2

 34222 Under all above assumptions, we have dx = o(1), ∀j ∈ I te .(3.4.11) Proof. Taking the inner product in H of equation(3.4.2) with the uniformly bounded sequence U = (u, v, y, z, θ), we get dx = -Re((iλI -A)U, U) H = o(1).

3 .

 3 Under all above assumptions, we have for all j ∈ I te λu j (ℓ j ) = O(1

Lemma 3 .

 3 4.4. Under all above assumptions, we have for all j ∈ I te ,ℓ j 0 |u j x | 2 dx = o(1), ℓ j 0 |θ j | 2 dx = o(1).(3.4.26)Proof. Let j ∈ I te . Multiply (3.4.10) by u j x λ and integrate over (0, ℓ j ) we get i

2

 2 dx = o(1). (3.4.29) Chapter 3. A transmission problem of a thermo-elastic system on Networks Now, multiply (3.4.10) by θ j +α j u j x λ

Lemma 3 . 4 . 5 .

 345 Under all above assumptions, we have for all j ∈ I te

Lemma 3 . 4 . 7 .

 347 Under all above assumptions, we have for all j ∈ I te ,

  s formula on the first and third term of the left hand side and on the integral of the right hand side, we get -

  s formula on the first and third term of the left hand side and on the integral of the right hand side, we obtain -

  the imaginary part of equations(3.4.39) and (3.4.41) then, adding the two resulting equations, we conclude that ℓ j 0 |y j x | 2 dx = o(1). Chapter 3. A transmission problem of a thermo-elastic system on Networks and the result holds. Lemma 3.4.8. Under all above assumptions, we have ℓ j 0 |λy j | 2 dx = o(1), ∀j ∈ I te .(3.4.42)

Lemma 3 . 4 . 10 .

 3410 The proof is the same as the one ofLemma 3.4.3 or Lemma 3.4.6, using (3.4.37) and(3.4.42), the result holds. Under all above assumptions, for each elastic edge we have

  -Schwarz inequality,(3.4.12) and(3.4.13), we deduce that Re 2i ℓ j 0 λu j β j Φ j x y j dx = o(1).

  j | 2 dx + Φ j |λu j | 2 j Φ j λy j u j ) = -2Re(iλ f j Φ j u j )

  dx = o(1).

( 3 .

 3 4.52)Then, inserting (3.4.52) in equation(3.4.51), we get

  = o(1) -2 Re(iλg j Φ j y j )

  .4.61) Chapter 3. A transmission problem of a thermo-elastic system on Networks Then, by eliminating v j and z j using(3.4.57) and (3.4.59) respectively, (3.4.57)-(3.4.61) becomes

  Same proof as the one of Lemma 3.4.2.Using(3.4.1),(3.4.57) and(3.4.59), we can easily deduce that (3.4.12)-(3.4.13) holds, for all j ∈ I(N ). Also, using (3.4.62),(3.4.63) and(3.4.64) we conclude(3.4.14) for all j ∈ I(N ). Lemma 3.4.15. Under all above assumptions, (3.4.15)-(3.4.18) holds.

17 .

 17 Under all above assumptions, we have for all j ∈ I te ∩ I ′ ext ℓ j 0 |λu j | 2 dx = o(1) and ℓ j 0 |v j | 2 dx = o(1). (3.4.75) 127 Chapter 3. A transmission problem of a thermo-elastic system on Networks

  holds. Lemma 3.4.19. Under all above assumptions, we have for all j ∈ I te ∩ I ′ ext ℓ j 0 |λy j | 2 dx = o(1) and ℓ j 0 |z j | 2 dx = o(1).(3.4.79)Proof. Let j ∈ I te ∩ I ′ ext . Multiply equation(3.4.63) by λu j , then, integrate over (0, ℓ j ),

β

  s formula on the second term of(3.4.80), using Cauchy-Schwarz inequality and (3.4.12), we obtain ℓ j 0 λ 3 y j u j dxρ j j |λu j | 2 dx = o(1).

λ 3

 3 u j y j dx + iβ j ℓ j 0 |λy j | 2 dx = o(1).

  Under all above assumptions, we have for all j ∈ I te ∩ I ′ Multiply (3.4.63) by y j , integrate over (0, ℓ j ) and apply Green's formula, we getℓ j 0 |λy j | 2 dxρ j

  k ) = 0, a k ∈ V int , ∑ j∈I te (a k ) d kj (α j u j t (a k )κ j θ j x (a k )) = 0, a k ∈ V ′ int .(3.4.84)Then, we can also prove that the same results of Theorem 3.4.1 and Theorem 3.4.13 hold for system {(3.1.7)-(3.1.12),(3.4.84)}. Notice that due to condition (3.4.84), a slight gain appears when proving the system {(3.1.7)-(3.1.12),(3.4.84)} is strongly stable. In other words, the energy associated with system {(3.1.7)-(3.1.12),(3.4.84)} converges to zero if one of the following conditions holds, i) Each maximal subgraph of thermo-elastic edges contains at least one interior vertex or contains an exterior vertex that belongs to V ext .

  Idea of the proof. In comparison with Theorem 3.3.1, it is enough to prove that if each maximal subgraph of thermo-elastic edges contains at least one interior vertex, then the energy associated with system {(3.1.7)-(3.1.12),(3.4.84)} converges to zero. Due to the fact that (I -A * ) -1 is compact where, A * = A | D(A * ) is the operator associated with system {(3.1.7)-(3.1.12),(3.4.84)}, we have σ(A * ) = σ p (A * ). Following the same proof as the one of Theorem 3.3.1, let λ ∈ R and U = (u, v, y, z, θ) ∈ D(A * ) be such that A * U = iλU.

Figure 3 FIGURE 3 . 1 :

 331 FIGURE 3.1: Some particular networks

  ), ∀j ∈ I(N ), θ j (x, 0) = θ j 0 (x), ∀j ∈ I te (N ).

Theorem 3 . 5 . 1 .

 351 The unbounded linear operator A 1 generates a C 0 -semigroup of contractions on H 1 .

2

 2 dx = Re(A 1 U, U) H 1 = Re(iλU, U) H 1 = 0.

Figure 3 FIGURE 3 . 2 : 1 (

 3321 FIGURE 3.2: A circuit and its parametrizations:π 1 (0) = a 1 , π 2 (0) =a 2 , and π 3 (0) = a 3 .

3 )

 3 ia(ℓ 2 + ℓ 3 )ib -

Stabilisation de quelques systèmes couplés impliquant problèmes de transmission (thermo-) viscoélastiques/élastiques ou équations télégraphiques dans des domaines bornés ou sur réseaux

  Cette thèse s'intéresse à la stabilisation de certains systèmes couplés impliquant des problèmes de transmission (thermo-) viscoélastique/élastique et des équations télégraphiques dans des domaines bornés ou sur réseaux. Dans un premier temps, nous considérons la stabilité d'un système d'équations d'ondes faiblement couplées avec un ou deux amortissements de type Kelvin-Voigt. L'amortissement est localisé via un coefficient singulier dans une partie du domaine. Sur chaque bord élastique, on considère deux équations d'onde conservatives tandis que, sur chaque bord thermoélastique, on considère deux équations d'onde couplées avec un seul amortissement thermique. Nous montrons que la stabilité du système est influencée par l'effet thermique et la vitesse de propagation des deux ondes. Différents résultats de stabilité exponentielle ou polynomiale sont établis.

	par Alaa HAYEK
	Résumé

D'abord, en adaptant un théorème de continuation unique, nous obtenons un nouveau résultat d'unicité et nous montrons, par conséquence, que notre système est fortement stable. Ensuite, nous montrons que le système n'est pas exponentiellement stable. Cependant, en utilisant une méthode fréquentielle combinée avec une méthode de multiplicateur par morceaux, nous établissons différents type de stabilisation polynomiale en toute dimension d'espace dès que la région d'amortissement satisfait certaines conditions géométriques. Dans un second temps, nous étudions l'existence, l'unicité et la stabilité de la solution d'une équation télégraphique généralisée sur un réseau en forme d'étoile mono-dimensionnel. Sur le noeud commun intérieur, nous considérons une condition aux limites dynamique (appelée condition de Kirchhoff améliorée), tandis que sur les noeuds extérieurs du réseau, une condition aux limites dissipative est appliquée. En utilisant une méthode fréquentielle combinée avec une nouvelle technique de multiplicateur, nous montrons que le système est uniformément (exponentiellement) stable. Finallement, nous étudions la stabilité indirecte d'un problème de transmission élastique/thermo-élastique sur réseaux.

Theorem 0.4.1. Assume

  that assumptions (0.4.19) and (0.4.24) hold. Then, the C 0semigroup (e tA ) t≥0 is strongly stable on the energy space H, i.e., for any U 0 ∈ H, we have lim

	t→∞	||e tA U 0 || H = 0.
	The proof of this theorem is based on the following lemmas.
	Lemma 0.4.2. Under the assumptions of Theorem 0.4.1, we have
	ker(iλ -A) = {0}, ∀λ ∈ R.
	Lemma 0.4.3. Under the assumptions of Theorem 0.4.1, we have

  Me -ϵt ∥x 0 ∥ H , ∀ t > 0, ∀ x 0 ∈ H.

					e 1	e 2	e 3
					a 0	a 1	a 2	a 3
	a 2	e 2	e 1	a 0 a 1	a 3 e 3		a 2	e 2	e 1	a 0 a 1	e 3	a 3
									4.27) -(0.4.29)} with
	the boundary conditions {(0.4.32) -(0.4.35)} is stable. If ρ j = 1, for all j ∈ I te , then the
	energy of the system decays exponentially in H. In other words, there exist two positive
	constants M and ϵ such that
	∥e tA x 0 ∥ H ≤ Let I ′ ext denotes the set of indices of edges adjacent to a vertex in V ′ ext .
	Theorem 0.4.6. Let N be an arbitrary network for which system (0.4.27) -(0.4.29) with
	the boundary conditions (0.4.32) -(0.4.35) is stable. Assume that there exists j ∈ I te ∩ I ′ ext
	such that ρ j ̸ = 1. Then, the energy of the system decays polynomially in H. More precisely,
	there exists C > 0 such that for all t > 0 we have
					E(t) ≤	C t 1/3 ||U 0 || 2 D(A) , ∀U 0 ∈ D(A).				(0.4.36)
	Finally, we present some other boundary conditions (in particular the Neumann
	boundary conditions) and we study the stabilization of system (0.4.27)-(0.4.29) on
	some particular networks similar to the networks considered in Shel, 2012. The
	networks considered in this case are shown in the following figure.

  1,1 (Ω), ∆b ∈ L ∞ (Ω) and |∇b(x)| 2 ≤ M 0 b(x) for almost every x in Ω where M 0 is a positive constant. On the other hand, in Tebou, 2012, the author studied the stabilization of the wave equation with Kelvin-Voigt damping. He established a polynomial energy decay rate of type

	t -1 provided that the damping region is localized in a neighborhood of a part of
	the boundary and verifies the Piecewise Multiplier Geometric Condition (PMGC in
	short) from Liu, 1997.
	Also in Nicaise and Pignotti, 2016, under the same assumption on b, the authors
	established the exponential stability of the wave equation with local Kelvin-Voigt
	damping localized around a part of the boundary and an extra boundary with
	time delay where they added an appropriate geometric condition. Later on, in
	Cavalcanti, Domingos Cavalcanti, and Tebou, 2017, the authors proved an expo-
	nential decay of the energy of a wave equation equation with two types of locally
	distributed mechanisms; a frictional damping and a Kelvin-Voigt damping where
	the location of each damping is such that none of them is able to exponentially
	stabilize the system. Under an appropriate geometric condition (PMGC) on a
	subset ω of Ω where the dissipation is effective, they proved that the energy of
	the system decays polynomially of type t -1 in the absence of regularity of the
	Kelvin-Voigt damping coefficient b. In Ammari, Hassine, and Robbiano, 2018, the
	wave equation with Kelvin-Voigt damping localized in a subdomain ω far away
	from the boundary without any geometric conditions was considered. The authors
	established a logarithmic energy decay rate for smooth initial data.

  .2.10) with Kelvin-Voigt dampings and non-smooth coefficient at the interface Multiplying equation (1.2.8) by φ 1 ∈ H 1 0 (Ω) and equation (1.2.10) by φ 2 ∈ H 1 0

  ). Under all above assumptions, if ℓ ≥ 2β + 4 then, we have Proof. Now, multiplying(1.5.44) and (1.5.45) by iλ 3 ψ and λ 2 ψ respectively and applying Green's formula, we get

	As a conclusion, if ℓ ≥ max{β + 2, 2β} we get
	Lemma 1.5.11. Ω	|λy| 2 dx = o(1).	(1.5.65)

Ω

|λu| 2 dx = o(1).

1 )

 1 Theorem 1.6.1. (Polynomial energy decay rate) Let ω b , ω c and ω α be non-empty open subsets of Ω. Assume that conditions (1.1.7),(1.1.8) and (1.1.9) hold. Let ω 3 = ω b ∩ ω c satisfies meas(ω 3 ∩ Γ) > 0 and let supp α ⊆ ω 3 . Also, assume that the energy of the system (1.6.1) is exponentially stable (β = 0) or polynomially stable as t

	-4

  Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks and the transmission conditions from Beck, 2016, §8.2 or Joly and Semin, 2008 (called by these authors the improved Kirchhoff conditions) at the interior common vertex

N},

(2.1.4)

  Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped NetworksSimilarly, the last two equations of (2.2.4) allow to eliminate ν and η ℓ , namely ν

for all ℓ ∈ {0, .., N}.

(2.2.5)

  Under the same condition of Theorem 2.2.4, we have iλ -A is surjective for all λ ∈ R

	Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized
	telegraph equation on star shaped Networks
	Lemma 2.2.6.

for all ℓ ∈ {1, .., N}. Hence U = 0 and the desired goal holds. * , i.e.

R(iλ -

A) = H, ∀λ ∈ R * .

Proof. For any λ ∈ R * and F

  .2.30) Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks where d ℓ , p ℓ are two constants to be determined later on. Notice that (ψ, χ) ∈ V * if and only if 

  .2.38) holds. In conclusion, the system (2.2.38)-(2.2.39) admits a unique solution ( V , I) if and only if Q + S is invertible. As Q + S is a Fredholm operator, it is enough to prove that it is injective, i.e.,

	ker(Q

  Chapter 2. Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped Networks

.

1.6

) based on a frequency domain approach. Here contrary to Nicaise, 2015, we do not require that r ℓ + g ℓ is uniformly bounded from below for each ℓ (see the assumption (25) in

  our C 0 -semigroup of contractions (e tA ) t≥0 in H is uniformly (exponentially) stable if and only if

	iR ⊂ ρ(A),	(2.3.1)
	and
	sup

λ∈R ||(iλ -A) -1 || L(H) < ∞. (

2

.3.2) Since (2.3.1) was already proved, the proof of Theorem 2.3.1 is reduced to show that condition (2.3.2) holds. This is checked by using a contradiction argument. Indeed, suppose that (2.3.2) is false, then there exist a sequence of real number λ n ∈ R and a sequence of vectors

  and using (2.1.2),(2.3.3), (2.3.11),(2.3.13) andLemmas 2.3.3 and 2.3.6, the result holds.Consequently, using(2.3.13),Lemma 2.3.4, Lemma 2.3.6 and Lemma 2.3.7, we deduce the desired contradiction. This ends the proof.

	Remark 2.3.8. Contrary to

  .10, we conclude that ||U|| H = o(1), which contradicts(3.4.1).

	Remark 3.4.11. Examples of networks for which (S1) holds are given by Theorem
	3.3.1.

& thermoelastic/elastic transmission problem on network♢

Acknowledgements

This PhD thesis is the output of the effort and support of several people to whom I am extremely grateful. First, I would like to express my sincere gratitude and appreciation to my supervisors Prof. Serge NICAISE, Prof. Ali WEHBE and Dr. Zaynab SALLOUM for the continuous support in my PhD studies, for their patience, motivation, and immense knowledge. Their guidance helped me in all the time of research and I hope to be able to work with them again in the future.

List of Symbols

R

The set of real numbers.

R *

The set of non-zero real numbers.

R *

Chapter 3

A transmission problem of a thermo-elastic system on Networks

Abstract

In this chapter, we investigate a network of elastic and thermo-elastic materials.

On each thermo-elastic edge, we consider two coupled wave equations such that one of them is damped via a coupling with a heat equation. On each elastic edge (undamped), we consider two coupled conservative wave equations. Under some conditions, we prove that the thermal damping is enough to stabilize the whole system. If the two waves propagate with the same speed on each thermo-elastic edge, we show that the energy of the system decays exponentially. Otherwise, a polynomial energy decay is attained. Finally, we present some other boundary conditions and we show that under sufficient conditions on the lengths of some elastic edges, the energy of the system decays exponentially on some particular networks similar to the ones considered in Shel, 2012.

Introduction

Thermoelasticity is a principle concerned with predicting the thermo-mechanical behaviour of elastic solids. Understanding such a principle is needed by many engineers to design different materials. Thus, several scientists were motivated to study the thermoelastic system described by the coupling between the mechanical vibration and the heat (thermal) effect of materials. Mathematically, a linear onedimensional thermo-elastic system satisfied by a thermoelastic bar (0, L) is represented by the following two equations:

with the initial conditions

where, u is the displacement, θ is the temperature deviation from the reference temperature and the mechanical-thermal coupling α is a positive constant. The existence and asymptotic behavior of the solution of the linear thermo-elastic system was firstly studied in [START_REF] Dafermos | On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity[END_REF] but, no decay rate was given. In the one Chapter 3. A transmission problem of a thermo-elastic system on Networks dimensional case, the stabilization of the linear thermo-elastic system satisfied by thermo-elastic materials (damped by thermal effect) with various boundary conditions was investigated by several authors. We will recall some of these results. In [START_REF] Hansen | Exponential energy decay in a linear thermoelastic rod[END_REF], the author considered the stabilization of system (3.1.1)-(3.1.2) on a thermo-elastic rod with u and θ satisfying the Dirichlet and Neumann condition respectively (or vice versa). He succeeded in proving the exponential stability of the system. More precisely, the author established the following energy estimate: There exist two positive constants M and ϵ such that

Similarly, when u and θ satisfy both the Dirichlet condition, it was shown that the estimate (3.1.3) still holds in [START_REF] Liu | Exponential stability of the semigroup associated with a thermoelastic system[END_REF]. Then, the method of Liu and Zheng, 1993 was extended in Burns, Liu, and Zheng, 1993 to prove (3.1.3) when u xαθ satisfies Dirichlet condition on both ends and θ x (0

Later on, the importance of damping and controlling the vibrations of materials composed of both elastic (undamped) and thermo-elastic (damped by thermal effect) parts appears in several physical applications and consequently in several mathematical papers. The main questions that received the interest of the researchers is the kind of stability of the thermo-elastic system on such composite materials and how should the thermo-elastic damping be localized to get the best decay rate or what is the energy decay rate in different localizations of the thermal damping? Such questions were answered in several ways. For example, in [START_REF] Marzocchi | Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity[END_REF], it was considered a one dimensional body which is configurated in [0, L 3 ] ⊂ R and for a given

. The authors proved that the whole system is exponentially stable, i.e, (3.1.3) holds. Then, in [START_REF] Fatori | Transmission problem for hyperbolic thermoelastic systems[END_REF], the authors considered the stabilization of a transmission problem for the thermo-elastic system with local thermal effect which is effective only over the interval [0, L 0 ], L 0 ∈ [0, L], this corresponds to the following system:

where u is the displacement in the thermo-elastic part, v is the displacement in the elastic part and θ is the temperature difference from a reference value. The system is completed with the following boundary conditions

and the following transmission conditions

The authors proved that the localized dissipation due to the thermal effect is strong enough to prove the exponential decay to zero of the energy. We also refer to Lebeau Chapter 3. A transmission problem of a thermo-elastic system on Networks and [START_REF] Lebeau | Decay Rates for the Three-Dimensional Linear System of Thermoelasticity[END_REF][START_REF] Oliveira | Stabilization of a locally damped thermoelastic system[END_REF][START_REF] Oliveira | Stabilization of a locally damped thermoelastic system[END_REF] for the study of the stabilization of multi-dimensional linear thermo-elastic systems.

On the other hand, there are only few publications on the stabilization of networks of thermo-elastic materials. Let us recall some of these results. In Abdallah and Shel, 2012, an exponential stability was proved on a network of thermo-elastic materials under both Fourier's law and Cattaneo's law. In Shel, 2012, the author studied the stability problem of a thermo-elastic system on particular cases of networks of elastic and thermo-elastic materials. Under the continuity condition of the displacement, the Neumann condition for the temperature at the internal nodes, and the balance condition, an exponential stability was proved (see also, Shel, 2014 for the network of elastic and thermo-elastic beams). Later on, in Han and Zuazua, 2017 the authors discussed the asymptotic behaviour of a transmission problem of the thermo-elastic system on star shaped networks of elastic and thermo-elastic rods. The uniform exponential decay rate was proved by a frequency domain analysis when only one purely elastic edge was present. Otherwise, a polynomial decay rate was deduced under a suitable irrationality condition on the lengths of the rods when more than one purely elastic edge is involved. After the review of these results that investigated the stabilization of a thermo-elastic system composed of the coupling between one wave equation and a heat equation, a remarkable question can be asked. What happens if we consider a network of elastic and thermo-elastic materials such that:

• On the thermo-elastic edges, we have a system of two wave equations coupled by velocity, such that one wave equation is coupled to a heat equation with a thermal effect.

• On the purely undamped elastic edges, we have only a system of two conservative wave equations coupled by velocity.

Hence our main question is the following one: Will the dissipation due to the thermal effect be also strong enough to prove the exponential stability of the energy of the whole system? To the best of our knowledge, the answer to this question remains an open problem. Therefore, our aim is to solve this open question.

In this work, we investigate the stabilization of the above described transmission problem on networks of elastic and thermo-elastic materials. We prove the exponential stability of the whole system under the condition that the two waves propagate with the same speed on all the thermo-elastic edges of the network.

On the other hand, if there exists an exterior thermo-elastic edge such that the two waves propagate with different speed on this edge, we show the polynomial stability of the whole system. Our main tool is a frequency domain approach, namely to prove the exponential stability we use a result due to [START_REF] Huang | Characteristics conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF] and to show the polynomial stability we use a result due to [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] Now, let us introduce some notations needed to formulate the problem under consideration, refer to [START_REF] Valein | Stabilization of the wave equation on 1-d networks[END_REF][START_REF] Valein | Stabilization of the wave equation on 1-d networks[END_REF][START_REF] Abdallah | Exponential stability of a general network of 1-d thermoelastic rods[END_REF][START_REF] Shel | Exponential stability of a network of elastic and thermoelastic materials[END_REF] for more details. Let N be a network embedded in the Euclidean space R m , m ∈ N * , with n vertices V = {a 0 , a 2 , .., a n-1 } and N edges E = {e 1 , .., e N }, with I(N ) = {1, .., N}, the set of indices of edges. Each edge e j is a curve, parametrized by π j : [0, ℓ j ] → e j : x j → π j (x j ). (2) If U 0 = (u 0 , u 1 , y 0 , y 1 , θ 0 ) ∈ H, then problem (3.2.6) admits a unique weak solution U = (u, v, y, z, θ) satisfying U ∈ C 0 (R + , H).

Strong stability

In this section, we will give sufficient conditions that guarantee the strong stability of the system (3.1.7)- (3.1.13) in the sense that the energy E(t), of the associated system decreases to zero as t tends to infinity. To show the strong stability of the C 0 -semigroup of contractions (e tA ) t≥0 we will rely on the following result obtained by [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF]. Now, we are in position to state the main result of this section.

Theorem 3.3.1. Consider the system (3.1.7)- (3.1.13) on N . Assume additionally that one of the following conditions holds, 1) Each maximal subgraph of thermo-elastic edges has an exterior vertex that belongs to V ext .

2) There exists a maximal subgraph of thermo-elastic edges with no exterior vertices that belong to V ext and β j = β, for all j ∈ I(N ).

Then, iR ⊂ ρ(A), (S1) and therefore lim t→∞ E(t) → 0.

Proof. Using Sobelev embedding Theorem, we deduce that (I -A) -1 is a compact operator. Then, the spectrum σ(A) of A is reduced to its discrete spectrum σ p (A).

Hence, using [START_REF] Arendt | Tauberian theorems and stability of oneparameter semigroups[END_REF], it is sufficient to prove that σ p (A)

equivalently, for all j ∈ I(N ) we have, So, if there exists j ∈ {1, 3} such that ℓ j ̸ = mπ √ 2β j , ∀m ∈ N * , we deduce that a = 0.

Consequently, by (3.5.25) or (3.5.26) we deduce that b = 0 and hence u 2 = y 2 = 0 and θ 2 = 0. By proceeding using unique continuation Theorem in [START_REF] Hayek | A transmission problem of a system of weakly coupled wave equations with Kelvin-Voigt dampings and non-smooth coefficient at the interface[END_REF] and iteration technique used in Case 1 of the proof of Theorem 3.3.1, we deduce that u j = y j = v j = z j = 0, j = 1, 2, 3. The same procedure can be used in the case λ = -β so that the desired goal holds.