A New Approach to the Stabilization of the Wave Equation with Boundary Damping Control

This paper deals with boundary feedback stabilization of a system, which consists of a wave equation in a bounded domain of n , with Neumann boundary conditions. To stabilize the system, we propose a boundary feedback law involving only a damping term. Then using a new energy function, we show that the solutions of the system asymptotically converge to a stationary position, which depends on the initial data. Similar results were announced without proof in (Chentouf and Boudellioua, 2004).


Introduction
et Ω be a bounded open connected set in n having a smooth boundary Γ = ∂Ω of class 2  C .
Given a partition ( ) where υ is the unit normal of Γ pointing towards the exterior of Ω and U is a feedback law to be determined.Note that 1 Γ is supposed to be nonempty whereas 0 Γ may be empty.
As in Chentouf and Boudellioua (2004), we propose a feedback depending only on a damping term, that is, Then, it is proved that the solutions of the closed loop system asymptotically tend towards a constant depending on the initial data 0 0 and y z .
The main contribution of this paper is to provide an alternative proof of Lagnese's result (1983) by means of a simple and direct method.The key idea of the proof is to introduce a new energy norm (see Chentouf and Boudellioua (2004) for similar systems and Conrad et al. (2002) for one-dimensional wave equation).

Preliminaries and Well-posedness of the Problem
In this section, we study the existence and uniqueness of the solutions of the closed-loop system (1.1)-(1.3).Assume, without loss of generality, that for any = , where a is a positive constant.
Then, consider the state space ( ) ( ) equipped with the inner product where 0 ε > is a constant to be determined.
The first result is stated in the following proposition: Proposition 1.The state space ( ) ( ) Ω endowed with the inner product (2.1) is a Hilbert space provided that ε is small enough.
Proof of Proposition 1.It suffices to show that the norm H ⋅ induced by the inner product (2.1) is equivalent to the usual one , that is, the existence of two positive constants K and K such that Applying Holder's inequality and using trace Theorem Adams (1976) (see also Mikhaïlov (1980)) yield: C is a positive constant depending on Ω (see Adams (1976) or Mikhaïlov (1980)).Therefore (2.2) holds for a positive constant K depending on , , a mes mes For the reverse inequality, we proceed as follows: Obviously for any 0, δ > , Young's inequality yields Combining (2.4) and (2.3), we get Furthermore, using a classical compactness argument, one can show the following generalized Poincaré inequality: ( ) where 2 0 C > depends on Ω .This, together with (2.5), implies that This concludes the proof of Proposition 1. ■ We turn now to the formulation of the closed-loop system (1.1)-( 1.3) in an abstract form on .
Then, the closed loop system can be written as follows where A is an unbounded linear operator defined by and for any ( ) ( ) It is a simple task to check that the operator , A − defined by (2.8)-(2.9), is maximal monotone.Therefore, it follows from semigroups theory Pazy (1983) (see also Brezis (1992)) that: Lemma 1. (i) The linear operator A generates a 0 C semigroup of contractions ( )

H D A =
(ii) For any initial data ( ) ( ) (iii) For any initial data ( )

Asymptotic Behavior
In this section, we will show an asymptotic behavior result for the unique solution of (2.7) in the state space ( ) ( ) To do so, we shall first show the following lemma: Lemma 2. The resolvent operator ( ) is compact for any 0 λ > and hence the canonical embedding ( ) , where is equipped with the graph norm.
Proof of Lemma 2. Let ( ) . Brezis (1992)), one can readily show that the above system has a unique solution and thus the operator ( )  (1976).■ The main result of this paper is: Theorem 1 .For any initial data, ( ) The proof of Lemma 2 follows then from the wellknown result of Kato consider the following wave equation: set for the graph norm and thus precompact in H by virtue Remark 1. Integrating with respect to x and t and using Green formula for the closed loop system (1.1)- (1.3), we obtain the following identity: