Microsoft Word - Math040228 final version-f.doc SQU Journal For Science, 9 (2004) 33-40 © 2004 Sultan Qaboos University 33 A New Approach to the Stabilization of the Wave Equation with Boundary Damping Control Boumediène Chentouf and Mohamed S. Boudellioua Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Al Khod 123, Muscat, Sultanate of Oman, Email: chentouf@squ.edu.om.. إقرار معادلة موجة بنظام تحكم حدي مخمد بومدين شنتوف ومحمد صالح بودليوة يهتم هذا البحث بدراسة مسألة إقرار نظام مكون من معادلة موجة في مجال محدود مع شروط حدية :خالصة كانه أن حتى تتم عملية االقرار نقترح نظام تحكم مخمد ونبرهن أن هذا النوع من التحكم بإم .من نوع نومان وباإلضافة إلى ذلك ، نبرهن أن موضع االستقرار . يجعل حلول معادلة الموجة تتقارب نحو موضع إستقرار . يتعلق بالشروط األساسية فقط ABSTRACT: 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). KEYWORDS: Wave equation, Neumann condition, boundary damping control, energy function, asymptotic behavior. 1. Introduction et Ω be a bounded open connected set in n having a smooth boundary Γ = ∂Ω of class 2C . Given a partition ( )0 1,Γ Γ of Γ , consider the following wave equation: ( ) ( ) ( ), , 0, in 0,tty x t y x t− ∆ = Ω× ∞ (1.1) with Neumann boundary conditions and initial conditions L BOUMEDIENE CHENTOUF and MOHAMED S. BOUDELLIOUA 34 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 0 0 , 0, 0, , , ( ), 0, , , 0 , , 0 ,t y x t on y x t U t on y x y x y x z x υ υ ∂ = Γ × ∞∂  ∂ = Γ × ∞ ∂  = =  (1.2) 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 1Γ is supposed to be nonempty whereas 0Γ may be empty. As in Chentouf and Boudellioua (2004), we propose a feedback depending only on a damping term, that is, ( ) ( ) ( ) ( ) ( )1, , , 0,tU t a x y x t x t= − ∈Γ × ∞ (1.3) where ( )1a L∞∈ Γ satisfies ( ) 0 0a x a≥ > for any 1x ∈Γ . Then, it is proved that the solutions of the closed loop system asymptotically tend towards a constant depending on the initial data 0 0andy z . There is a rich literature concerning the stabilization problem of the wave equation (see for instance Bardos et al. (1992), Chen (1979a, 1979b, 1981a, 1981b), Komornik (1994), Lax et al. (1963), Lagnese (1988), Lions (1988a, 1988b), Morawetz (1975), Quinn and Russell (1977), Triggiani (1975) and the references therein). In all references cited above, at least one of the following conditions is assumed to be satisfied: • The equation (1.1) involves also the displacement term y. • The stabilizing feedback law ( )U t contains not only a boundary dissipation term ty but also the boundary displacement y . • The first boundary condition in (1.2) involves the displacement term y (the boundary condition (1.2) is replaced for instance by 0 or 0 y y y υ ∂ = + = ∂ on ( )0 0,Γ × ∞ . 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). 2. 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 ( )1,x a x a∈Γ = , where a is a positive constant. Then, consider the state space ( ) ( )1 2 ,H H L= Ω × Ω (2) equipped with the inner product A NEW APPROACH TO STABILIZATION OF THE WAVE EQUATION 35 ( )( ) ( ) ( )( ) 1 1 , , σ σ , H y z y z y y zz dx z dx ay d zdx ay dε Ω Ω Γ Ω Γ = ∇ ∇ + + + + ∫ ∫ ∫ ∫ ∫ (2.1) where 0ε > is a constant to be determined. The first result is stated in the following proposition: Proposition 1. The state space ( ) ( )1 2H H L= Ω × 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 ( ) ( )1 2H LΩ × Ω ⋅ , that is, the existence of two positive constants K and K such that ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 1 2 , , , H L H H L K y z y z K y z Ω × Ω Ω × Ω ≤ ≤ . (2.2) Applying Holder's inequality and using trace Theorem Adams (1976) (see also Mikhaïlov (1980)) yield: ( ) ( )( ) ( ) ( )( )( ) 2 2 2 2 22 1 1 , 1 2 2 1 2 , H y z y mes z dx a C mes y y dx ε ε ε Ω Ω ≤ ∇ + + Ω   + Γ ∇ + + ∫ ∫ where 1C is a positive constant depending on Ω (see Adams (1976) or Mikhaïlov (1980)). Therefore (2.2) holds for a positive constant K depending on ( ) ( )1 and, ,a mes mesε Γ Ω . For the reverse inequality, we proceed as follows: ( ) ( ) ( ) ( ) ( )( ) 1 1 222 2 2 , σ 2 σ . H y z y z dx z dx ay d ay d z dx ε ε ε Ω Ω Γ Γ Ω = ∇ + + + + ∫ ∫ ∫ ∫ ∫ (2.3) Obviously for any 0,δ > , Young's inequality yields ( )( ) ( ) ( ) 1 1 22 12 σ σ .ay d z dx a z dx a y dδ δ− Γ Ω Ω Γ ≥ − −∫ ∫ ∫ ∫ (2.4) Combining (2.4) and (2.3), we get ( ) ( ) ( )( ) ( )( ) 1 222 2 2 1, 1 σ . H y z y z dx a z dx a a y dε δ ε δ− Ω Ω Γ ≥ ∇ + + − + −∫ ∫ ∫ (2.5) Furthermore, using a classical compactness argument, one can show the following generalized Poincaré inequality: ( ) 1 2 2 2 2 σ ,y dx C y dx y dΩ Ω Γ   ≤ ∇ +    ∫ ∫ ∫ where 2 0C > depends on Ω . This, together with (2.5), implies that BOUMEDIENE CHENTOUF and MOHAMED S. BOUDELLIOUA 36 ( ) ( ) ( ){ } ( ) ( ) 2 2 2 21 2 2 1 , 1 , H y z a a C y a a y z dx a z dx ε δ ε δ ε δ δ − Ω − Ω  ≥ − + − − ∇ +    + − ∫ ∫ (2.6) for any aδ < . Finally, applying again Holder's inequality to the last term in (2.6), one can show the existence of a positive constant K depending on ,a δ and ( )mes Ω such that the second inequality in (2.2) holds, provided that aδ < and ε satisfies the following condition ( ) ( ) ( ) 1 min , . a a a mes δ ε δ δ   <   − − Ω  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 .H Let ( ) ( ) ( ) ( ) ( )( ), , and , , , .tz t y t t y t z t⋅ = ⋅ Φ = ⋅ ⋅ Then, the closed loop system can be written as follows ( ) ( ) ( ) ( )0 0 0 , 0 , , t t t y z ΑΦ = Φ  Φ = Φ = (2.7) where A is an unbounded linear operator defined by ( ) ( ) ( ) ( ) ( )1 1 2 0 1 , ; ; , 0 on ; 0 on y z H H y L D A y y az υ υ  ∈ Ω × Ω ∆ ∈ Ω   =  ∂ ∂ = Γ + = Γ  ∂ ∂  (2.8) and for any ( ) ( ), ,y z D A∈ ( ) ( ), , .A y z z y= ∆ (2.9) 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 0C semigroup of contractions ( )S t on ( ).H D A= (ii) For any initial data ( ) ( )0 0 0, ,y z D AΦ = ∈ the system (2.7) has a unique strong solution ( ) ( ) ( )( ) ( ) ( )0, , ,tt y t y t S t D AΦ = ⋅ ⋅ = Φ ∈ for all 0t ≥ such that ( ) ( ), ty yΦ ⋅ = ∈ ( ) ( )( )1 ; ;C H C D A+ +∩ . Moreover, the function ( ) H t A tΦ is decreasing. (iii) For any initial data ( )0 0 0,y z HΦ = ∈ , the system (2.7) has a unique weak solution ( ) ( ) ( )( ) ( ) 0, , ,tt y t y t S t HΦ = ⋅ ⋅ = Φ ∈ such that ( ) ( ) ( )0, ;ty y C H+Φ ⋅ = ∈ . A NEW APPROACH TO STABILIZATION OF THE WAVE EQUATION 37 3. Asymptotic Behavior In this section, we will show an asymptotic behavior result for the unique solution of (2.7) in the state space ( ) ( )1 2H H L= Ω × Ω . To do so, we shall first show the following lemma: Lemma 2. The resolvent operator ( ) 1 :I A H Hλ −− → is compact for any 0λ > and hence the canonical embedding ( ):i D A H→ is compact, where is equipped with the graph norm. Proof of Lemma 2. Let ( ), ,f g H∈ . We seek ( ) ( ),y z D A∈ such that ( )( ) ( ), ,I A y z f g− = . Equivalently, ( ) ( ) ( ) 0 1 1 2 , , 0, on , , on , ; . z y f y y f g y y ay af y H y L υ υ   = −  ∆ − = − + ∂ = Γ ∂ ∂ + = Γ∂ ∈ Ω ∆ ∈ Ω Using Lax-Milgram Theorem (see Brezis (1992)), one can readily show that the above system has a unique solution and thus the operator ( ) 1I A −− exists and maps ( )intoH D A . Finally, by using Sobolev embedding, we deduce that ( ) 1I A −− is compact. The proof of Lemma 2 follows then from the well- known result of Kato (1976). ■ The main result of this paper is: Theorem 1 . For any initial data, ( )0 0 0, ,y z HΦ = ∈ the solution ( ) ( ) ( )( ), , ,tt y t y tΦ = ⋅ ⋅ of (2.7) tends in H to ( ), 0 ,C as t → +∞ where ( ){ } ( ) 1 1 1 0 0 σC a mes z dx ay d − Ω Γ = Γ +∫ ∫ . Proof of Theorem 1. By a standard argument of density of ( )2D A in H and the contraction of the semigroup ( ),S t it suffices to prove Theorem 1 for smooth initial data ( ) ( )20 0 0,y z D AΦ = ∈ . Let ( ) ( ) ( )( ) ( ) 0, , ,tt y t y t S tΦ = ⋅ ⋅ = Φ be the solution of (2.7). It follows from (ii) of Lemma 1 that the trajectory of solution ( ){ } 0t t ≥ Φ is a bounded set for the graph norm and thus precompact in H by virtue BOUMEDIENE CHENTOUF and MOHAMED S. BOUDELLIOUA 38 of Lemma 2. Applying LaSalle's principle, we deduce that ( )0ω Φ is non-empty, compact, invariant under the semigroup ( )S t and ( ) ( ) [ ]0 0 as 10S t tωΦ → Φ → ∞ . In order to prove the strong stability, it suffices to show that ( )0ω Φ reduces to ( ), 0C . To this end, let ( ) ( ) ( )0 0 0 0,y z D AωΦ = ∈ Φ ⊂ and ( ) ( ) ( )( ) ( ) ( )0, , ,tt y t y t S t D AΦ = ⋅ ⋅ = Φ ∈ the unique strong solution of (2.7). Recall that it is well-known that ( ) H tΦ is constant Haraux (1991) and thus ( )( )2 0,Hd tdt Φ = i.e., 1 2 , σ 0.HA a z dΓ< Φ Φ > = − =∫ (3.1) This implies that 0tz y= = on 1Γ and therefore y is solution of the system System: ( ) ( ) ( ) 0 1 0 0 1 2 0, 0, on , 0, on , 0 ; , ; , tt t t y y y y y y y y z y H y L υ υ − ∆ = ∂ = Γ ∂  ∂ = = Γ ∂  = =  ∈ Ω ∆ ∈ Ω  (3.2) whereas tz y= is solution of the system ( ) ( ) 0 1 0 0 0, 0, on , 0, on , 0 ; 0 . tt t z z z z z z z z y υ υ − ∆ = ∂ = Γ ∂  ∂ = = Γ  ∂  = = ∆ (3.3) Obviously, to deduce the desired result, it suffices to show that y = constant is the only solution of (3.2). To do so, we first use the standard Holmgren's uniqueness theorem (see John (1982)) for the system (3.3) to conclude that 0z ≡ . Thus the system (3.2) is reduced to the elliptic problem: 0, 0, on , y y υ ∆ =  ∂ = Γ∂ which clearly yields that y ≡ constant. 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