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Uniform stabilization of the wave equation by nonlinear boundary feedback. (English) Zbl 0695.93090

Summary: The question of uniformly stabilizing the solution of the wave equation \(y''-\Delta y=0\) in \(\Omega\) \(\times (0,\infty)\) (\(\Omega\) is a bounded domain of \({\mathbb{R}}^ n)\) by means of a nonlinear feedback law of the following form is studied: \[ \partial y/\partial v=-k(x)g(y')\quad on\quad \Gamma_ 0\times (0,\infty),\quad y=0\quad on\quad \Gamma_ 1\times (0,\infty), \] (\(\Gamma_ 0,\Gamma_ 1)\) being a suitable partition of the boundary of \(\Omega\) and g a continuous nondecreasing function such that \(g(0)=0\). We choose \(k(x)\in L^{\infty}(\Gamma_ 0)\), k(x)\(\geq 0\) such that k(x) vanishes linearly at the interface points \(x\in {\bar \Gamma}_ 0\cap {\bar \Gamma}_ 1\). Then, if g(s) behaves like \(| s|^{p-1}s\) as \(| s| \to 0\) with \(p>1\) and linearly as \(| s| \to \infty\), it is proved that the energy of every solution decays like \(t^{-2/(p-1)}\) as \(t\to \infty\). In the case where \(p=1\) the exponential decay rate is proved.

MSC:

93D15 Stabilization of systems by feedback
35B40 Asymptotic behavior of solutions to PDEs
93C20 Control/observation systems governed by partial differential equations
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