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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.

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