Asymptotic Behavior of a Bi-Dimensional Hybrid System

We study the asymptotic behavior of the solutions of a Hybrid System wrapping an elliptic operator.


Introduction
In this paper, we address some issues related to the asymptotic behavior a hybrid system with two types of vibrations of different nature.The model under consideration is inspired in and introduced in [1].However, there are some important differences between these two models.In [1] the flexible part of the boundary 0 Γ is occu- pied by a flexible damped beam instead of a flexible.Most of the relevant properties see [2].In [3] the authors are interested on the existence of periodic solutions of this system.Due to the localization of the damping term in a relatively small part of the boundary and to the effect of the hybrid structure of the system, the existence of periodic solutions holds for a restricted class of non homogeneous terms.Some resonance-type phenomena are also exhibited.Cindea, Sorin and Pazoto [4] consider the motion of a stretched string coupled with a rigid body at one end and we study the existence of periodic solution when a periodic force facts on the body.The main difficulty of the study is related to the weak dissipation that characterizes this hybrid system, which does not ensure a uniform decay rate of the energy.For more examples of hybrid systems see [5] [6].We refer to [7] for a discussion on the model and references therein.In [8] the authors to discern exact controllability properties of two coupled wave equations, one of which holds on the interior of a bounded open domain Ω , and the other on a segment 0 Γ of the boundary ∂Ω .Moreover, the coupling is accomplished through terms on the boundary.Because of the particular physical application involved the attenuation of acoustic waves within a chamber by means of active controllers on the chamber walls control is to be implemented on the boundary only.
We consider the bi-dimensional cavity Ω an open class C 2 with limited boundary contained in Ω 1 , filled with an elastic, in viscid, compressible fluid, in which the acoustic vibrations are coupled with the mechanical vibration of a string located in the subset The subset 1 Γ is assumed to be rigid and we impose zero normal velocity of the fluids on it.The subset 0 Γ is supposed to be flexible and occupied by a flexible string that vibrates under the pressure of the fluid on the plane where Ω lies.The displacement of 0 Γ , described by the scalar function ( ) , w w x t = , obeys the one-dimensional dissipative wave equation.As Ω is compressible fluid where the velocity field v is given by the potential All deformations are supposed to be small enough so that linear theory applies.The linear motion of this system is described by means of the coupled wave equations where ν denote the unit outward normal to Ω .
We define the energy associated with this system.Proceeding formally, multiply the first equation by t ϕ and then integrate over Ω .
However, the integral Multiplying by w in the second equation of the system (1) and then integrate over 0 Γ Integrating by parts Replacing the above equation over ( 5) we obtain which leads us to assert that, the energy of the system is given by ( ) ( ) for each 0 t ≥ .Remark 1 The first two terms represents the energy of acoustic wave and the other terms is the energy of bungee wave.
The system has a natural dissipation.Indeed, to observe this fact multiply the first equation of (1) by t ϕ and then the second equation of (1) by t w , as was done in calculations ( ) in his doctoral thesis [7] shows non-exponential decay of the energy of the hybrid system (1).

Mathematical Formulation
Define the face space ( ) ( ) ( ) ( ) endowed with the Hilbertian scalar product given by ( ) ( ) for all ( ) ( ) ∈  We can show that the pair ( ) , ,  is a Hilbert space.Since the first and second equation of the system (1), we obtain ( )  in this sense for all ( ) Note that U ∈   if and only if Now, we consider the problem with Neumann boundary conditions ( ) ( ) where we can say that ( ) [9].Similarly, consider the problem We can say that ( ) In this sense we can define the domain of the operator  which we denote ( )   , as the set of ( ) Remark 2 By previous observations we can say that the hybrid system (1) is equivalent to the Cauchy problem where

Solution Existence
We want to show that  is a dissipative operator and ( ) In particular, ( ) where ( ) By previous observations that there have ( ).
U ∈   Using the application of Lummer Phillips Theorem [10] [11], we have the following result.
Theorem 1 The operator  set to (10) is the infinitesimal generator of a contraction semigroup 0 C .Theorem 2 The  is the infinitesimal generator of a semigroup 0 C and verifies ( ) then the solution of (13) satisfies

Asymptotic Behavior
We now show that the energy associated with the system decays exponentially.Multiplying by ϕ the first eq- uation in (1) and integrating over Ω yields Observe that From the second equation in (1), we obtain ( ) On the other hand, From ( 17)-( 19), we obtain Replacing (20) into (16) Now, since Poincaré inequality we have where 1 λ is the Poincaré constant.In a similar way, From ( 22), ( 23) and ( 24) we have We define the operator ( ) ( ) Differentiating (26) and using (8) we obtain ( ) ( ) Considering n large enough, we can obtain a constant C such that ( ) ( ) On the other hand, using Poincaré, we can obtain .The result Remark 4 In the case of 1 0 γ = can be also said that a power decays exponentially.
The above results support the conclusion.