Internal stabilization for the wave equation

Internal stabilization for the wave equation Talk at Chengdu (China) 2004 1 Motivation The purpose of this talk is to give some appli ations of the quanti ation of unique ontinuation property in the framework of stabilization for the wave equation in bounded domain. Here, we deal with the following unique ontinuation property: if the solution of the wave equation with homogeneous Diri hlet boundary ondition is null in a subdomain then the initial data are identi ally zero. The more strong quanti ation of unique ontinuation property is given by the following observability estimate: Theorem, Observabilty estimate [ BLR?, [ Bu? .- Let be a bounded onne ted C 3 do- main in R n , n 1, and ! be a non-empty open subset of . Let us onsider the following wave equation in R, with initial data and homogeneous Diri hlet boundary ondition : 8 < : 2 t uu = 0 in R u = 0 on R u (; 0) = u 0 , t u (; 0) = u 1 in . Under geometri al hypothesis, there exist a onstant > 0 and a time T > 0 su h that for all initial data (u 0 ; u 1 ) 2 H 1 0 ( ) L 2 ( ), the solution u satises the following estimate : k(u 0 ; u 1

• wave equation


• homogeneous diri


• equation des ondes


• solutions des problemes hyperboliques


• following estimate


• diri hlet


• estimate


• stabilisation de l'equation des ondes par le bord


• boundary ondition