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Stabilization of a transmission wave plate equation

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18 pages
Stabilization of a transmission wave/plate equation Kaıs Ammari ? and Serge Nicaise † Abstract. We consider a stabilization problem, for a model arising in the control of noise, coupling the damped wave equation with a damped Kirchoff plate equation. We prove an expo- nential stability result under some geometric condition. Our method is based on an identity with multipliers that allows to show an appropriate energy estimate. Keywords: Wave/plate equation, transmission, boundary stabilization, multiplier. AMS 2000 subject classification: 35B37, 35B40, 93B07, 93D15. 1 Introduction and main results In this paper we consider the stabilization of a system coupling the wave equation with a Kirchhoff system (see [3] for the unidimensional model) damped through a dissipation law on the Kirchhoff system and on the wave system. More precisely we consider a bounded domain ? of IR2 with a Lipschitz boundary such that ?¯ = ?¯1 ? ?¯2, where ?i, i = 1, 2 are bounded domains with a Lipschitz boundary such that ?1 ? ?2 = ?. We then denote by I the interior of ?¯1 ? ?¯2, that is called the interface between ?1 and ?2. For i = 1 or 2, we also set ?i = ∂?i \ I¯, the “exterior” boundary of ?i. We consider the wave equation in ?1 coupled with the Kirchhoff system in ?2, more precisely we consider the following system: ∂2t u1(x, t)?∆u1(x, t) = 0, in ?1 ? (0,+

  • ?u1 ?

  • lumer-phillips theorem

  • then there

  • ?v1 ?∆u1

  • †universite de valenciennes et du hainaut cambresis

  • exists ?

  • plate equation


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Stabilization of a transmission wave/plate equation Kaı¨sAmmari and Serge Nicaise
Abstract. We consider a stabilization problem, for a model arising in the control of noise, coupling the damped wave equation with a damped Kirchoff plate equation. We prove an expo-nential stability result under some geometric condition. Our method is based on an identity with multipliers that allows to show an appropriate energy estimate. Keywords: Wave/plate equation, transmission, boundary stabilization, multiplier. AMS 2000 subject classification: 35B37, 35B40, 93B07, 93D15. 1 Introduction and main results In this paper we consider the stabilization of a system coupling the wave equation with a Kirchhoff system (see [3] for the unidimensional model) damped through a dissipation law on the Kirchhoff system and on the wave system. More precisely we consider a bounded 2 ¯ ¯ ¯ domain Ω of IR with a Lipschitz boundary such that Ω = Ω 1 Ω 2 , where Ω i , i = 1 , 2 are bounded domains with a Lipschitz boundary such that Ω 1 Ω 2 = . ¯ ¯ We then denote by I the interior of Ω 1 Ω 2 , that is called the interface between Ω 1 and Ω 2 . ¯ For i = 1 or 2, we also set Γ i = Ω i \ I , the “exterior” boundary of Ω i . We consider the wave equation in Ω 1 coupled with the Kirchhoff system in Ω 2 , more precisely we consider the following system: t 2 u 1 ( x, t ) Δ u 1 ( x, t ) = 0 , in Ω 1 × (0 , + ) , (1.1) t 2 u 2 ( x, t ) + Δ 2 u 2 ( x, t ) = 0 , in Ω 2 × (0 , + ) , (1.2) u i ( x, 0) = u i 0 ( x ) , ∂ t u i ( x, 0) = u i 1 ( x ) , in Ω i , i = 1 , 2 , (1.3) u 1 = u 2 , B 1 u 2 = 0 , B 2 u 2 = ν 1 u 1 on I × (0 , + ) , (1.4) ν 1 u 1 = α 1 u 1 t u 1 on Γ 1 × (0 , + ) , (1.5) B 1 u 2 = β∂ ν 2 u 2 ν 2 t u 2 on Γ 2 × (0 , + ) , (1.6) B 2 u 2 = α 2 u 2 + t u 2 on Γ 2 × (0 , + ) , (1.7) where ν i = ( ν i 1 , ν i 2 ) is the unit normal vector of Ω i pointing towards the exterior of Ω i , i = 1 , 2 , and τ i = ( ν i 2 , ν i 1 ) is the unit tangent vector along Ω i . We further denote by ν i (resp. τ i , t ) the normal (resp. tangent, time) derivative. The constant µ (0 , 12 ) is the Poisson coefficient and the boundary operator B j , j = 1 , 2 are defined on Ω 2 as follows: B 1 y = Δ y + (1 µ ) 2 ν 21 ν 22 x12 yx 2 ν 221 2 xy 22 ν 222 2 xy 12 , De´partementdeMath´ematiques,Facult´edesSciencesdeMonastir,5019Monastir,Tunisie,email: kais.ammari@fsm.rnu.tn Universit´edeValenciennesetduHainautCambr´esis,LAMAV,FRCNRS2956,LeMontHouy,59313 Valenciennes Cedex 9, France, email : snicaise@univ-valenciennes.fr 1