## 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)

### MSC:

 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