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Stabilisation frontière de l’equation des ondes: Une méthode directe. (Boundary stabilization of the wave equation: A direct method). (French) Zbl 0661.93054
We consider the evolution equation: $$y''-\Delta y=0$$ in $$\Omega \times (0,+\infty)$$; $$\partial y/\partial v=F(y,y')$$ on $$\Gamma_ 0\times (0,+\infty)$$; $$y=0$$ on $$\Gamma_ 1\times (0,+\infty)$$, where $$\{\Gamma_ 0,\Gamma_ 1\}$$ denotes a partition of the boundary $$\partial \Omega$$ and $$y'=\partial y/\partial t$$. Its stabilizability is studied, i.e. the existence of a feedback F(y,y’) such that, for given initial data, the solution $$y=y(x,t)$$ decays exponentially in the energy space when $$t\to +\infty$$. If $$\Omega$$ satisfies very strong geometrical hypotheses several authors have proved that the system is stabilized by the feedback $$F(y')=-by'$$ with $$b\in L^{\infty}(\Gamma_ 0)$$, $$b\geq b_ 0>0$$ if the partition $$\{\Gamma_ 0,\Gamma_ 1\}$$ is suitably chosen. We prove the stabilizability of the system without geometrical hypotheses for the case of dimension $$n=2$$, i.e. $$\Omega \subset {\mathbb{R}}^ 2$$. The proof is based on the use of the feedback $$F(y')=-by'$$ with $$b\in L^{\infty}(\Gamma_ 0)$$, $$b\geq 0$$ and $$b(x)=0$$ on the interface-points $$x\in {\bar \Gamma}_ 0\cap {\bar \Gamma}_ 1$$ and on the construction of energy functionals well adapted to the system.

MSC:
 93D15 Stabilization of systems by feedback 35L05 Wave equation 93C20 Control/observation systems governed by partial differential equations 93C05 Linear systems in control theory