Stabilization of a transmission wave plate equation

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