Stabilisation de l’équation des ondes au moyen d’un feedback portant sur la condition aux limites de Dirichlet. (Stabilization of the wave equation by Dirichlet type boundary feedback). (French) Zbl 0764.35055

Let \(\Omega\) be a bounded \(C^ \infty\) domain in \(\mathbb{R}^ n\) whose boundary is composed of two disjoint components \(\Gamma_ 0\) and \(\Gamma_ 1\). Consider the wave equation \(y''-\Delta y=0\) with boundary conditions \(y=\partial/\partial\nu(Gy')\) on \(\Gamma_ 0\times(0,\infty)\) and \(y=0\) on \(\Gamma_ 1\times(0,\infty)\), where \(G=(-\Delta)^{-1}: H^{-1}(\Omega)\to H_ 0^ 1(\Omega)\). The initial data are in \(L^ 2(\Omega)\times H^{-1}(\Omega)\). The authors show that under suitable conditions on the partition \(\{\Gamma_ 0,\Gamma_ 1\}\), the solution \((y(t),y'(t))\) decays exponentially to zero in \(L^ 2(\Omega)\times H^{-1}(\Omega)\) as \(t\to+\infty\). The proof uses microlocal analysis techniques. A similar result is established for a class of boundary conditions containing the classical \(y'+\partial y/\partial\nu=0\) and, as a limit case, the above boundary conditions. A stabilization result for the case of absorbing boundary conditions is also given.
Reviewer: C.Popa (Iaşi)


35L20 Initial-boundary value problems for second-order hyperbolic equations
35B35 Stability in context of PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs