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A direct method for the boundary stabilization of the wave equation. (English) Zbl 0636.93064
We consider the wave equation \(y''-\Delta y=0\) in a bounded domain \(\Omega\subset {\mathbb{R}}^ n \)with smooth boundary \(\Gamma\), subject to mixed boundary conditions \(y=0\) on \(\Gamma_ 1\) and \(\partial y/\partial v=F(x,y')\) on \(\Gamma_ 0\), \((\Gamma_ 0,\Gamma_ 1)\) being a partition of \(\Gamma\). We study the boundary stabilizability of the solutions i.e. the existence of a partition \((\Gamma_ 0,\Gamma_ 1)\) and of a boundary feedback F(\(\cdot,\cdot)\) such that every solution decays exponentially in the energy space as \(t\to \infty\). We prove in this paper the stabilizability of the system without geometrical hypotheses. The proof is based on the use of a feedback \(F(x,y')=-b(x)y'\) with b(x)\(\geq 0\) and \(b(x)=0\) on the interface points \(x\in cl(\Gamma_ 0)\cap cl(\Gamma_ 1)\), and on the construction of energy functionals, well adapted to the system. This method is rather general and can be adapted to other evolution systems (e.g. models of plates, elasticity systems) as well. The semilinear wave equation \(y''-\Delta y+f(y)=0\) is also treated.
Reviewer: E.Zuazua

93D15 Stabilization of systems by feedback
35L05 Wave equation
93C20 Control/observation systems governed by partial differential equations
93C05 Linear systems in control theory
93C10 Nonlinear systems in control theory