Feedback boundary stabilization of wave equations with interior delay

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Feedback boundary stabilization of wave equations with interior delay Kaıs AMMARI ? , Serge NICAISE † and Cristina PIGNOTTI ‡ Abstract. In this paper we consider a boundary stabilization problem for the wave equa- tion with interior delay. We prove an exponential stability result under some Lions geometric condition. The proof of the main result is based on an identity with multipliers that allows to obtain a uniform decay estimate for a suitable Lyapunov functional. Mathematics Subject Classification (2000): 35B05, 93D15, 93D20 Keywords: boundary stabilization, interior delay, wave equations 1 Introduction We study the boundary stabilization of a wave equation in an open bounded domain ? of Rn, n ≥ 2. We denote by ∂? the boundary of ? and we assume that ∂? = ?0??1, where ?0, ?1 are closed subsets of ∂? with ?0 ? ?1 = ?. Moreover we assume meas?0 > 0. The system is given by : utt(x, t)?∆u(x, t) + aut(x, t? ?) = 0, x ? ?, t > 0, (1.1) u(x, t) = 0, x ? ?0, t > 0 (1.2) ∂u ∂? (x, t) = ?kut(x, t), x ? ?1, t > 0 (1.3) u(x, 0) = u0(x), ut(x, 0) = u1(x), x ? ?, (1.4) ut(x, t) =

tn †universite de valenciennes et du hainaut cambresis

univ-valenciennes

sipative boundary

there exist positive

cc?v ?

wave equations

positive constant

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Feedback boundary stabilization of wave equations with interior delay

∗ † Ka¨ıs AMMARI , Serge NICAISE and

‡ Cristina PIGNOTTI

Abstract.In this paper we consider a boundary stabilization problem for the wave equa-tion with interior delay. We prove an exponential stability result under some Lions geometric condition. The proof of the main result is based on an identity with multipliers that allows to obtain a uniform decay estimate for a suitable Lyapunov functional.

Mathematics Subject Classiﬁcation (2000)93D15, 93D20: 35B05, Keywords: boundary stabilization, interior delay, wave equations

Introduction

We study the boundary stabilization of a wave equation in an open bounded domain Ω of n R, n≥denote by2. We ∂Ω the boundary of Ω and we assume that∂Ω = Γ0∪Γ1,where Γ0,Γ1are closed subsets of∂Ω with Γ0∩Γ1=∅. Moreover we assumemeasΓ0>0. The system is given by :

utt(x, t)−Δu(x, t) +aut(x, t−τ) = 0, u(x, t) = 0, ∂u (x, t) =−kut(x, t), ∂ν u(x,0) =u0(x), ut(x,0) =u1(x), ut(x, t) =g(x, t),

x∈Ω>, t 0, x∈Γ0, t >0

x∈Γ1, t >0 x∈Ω, x∈Ω, t∈(−τ,0),

(1.1) (1.2)

(1.3) (1.4) (1.5)

whereνstands for the unit normal vector of∂Ω pointing towards the exterior of Ω and ∂u is the normal derivative. Moreover, the constantτ >0 is the time delay,aandk ∂ν are two positive numbers and the initial data are taken in suitable spaces. Denoting bymthe standard multiplier, that ism(x) =x−x0,we assume

m(x)∙ν(x)≤0,

x∈Γ0,

and

m(x)∙ν(x)≥δ >0,

x∈Γ1.

(1.6)

∗ D´epartementdeMathe´matiques,Facult´edesSciencesdeMonastir,5019Monastir,Tunisie,e-mail : kais.ammari@fsm.rnu.tn † Universite´deValenciennesetduHainautCambre´sis,InstitutdesSciencesetTechniquesdeValen-ciennes, 59313 Valenciennes Cedex 9, France, e-mail: snicaise@univ-valenciennes.fr ‡ DipartimentodiMatematicaPuraeApplicata,Universit`adiL’Aquila,ViaVetoio,Loc.Coppito, 67010 L’Aquila, Italy, e-mail : pignotti@univaq.it

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