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Exact controllability for semilinear wave equations in one space dimension. (English) Zbl 0769.93017
Summary: The exact controllability of the semilinear wave equation \(y''- y_{xx}+f(y)=h\) in one space dimension with Dirichlet boundary conditions is studied. We prove that if \(| f(s)|/| s|\log^ 2| s|\to 0\) as \(| s|\to\infty\), then the exact controllability holds in \(H^ 1_ 0(\Omega)\times L^ 2(\Omega)\) with controls \(h\in L^ 2(\Omega\times(0,T))\) supported in any open and non-empty subset of \(\Omega\). The exact controllability time is that of the linear case where \(f=0\). Our method of proof is based on HUM (Hilbert Uniqueness Method) and on a fixed point technique. We also show that this result is almost optimal by proving that if \(f\) behaves like — \(s \log^ p(1+| s|)\) with \(p>2\) as \(| s|\to\infty\), then the system is not exactly controllable. This is due to blow-up phenomena. The method of proofs is rather general and applied also to the wave equation with Neumann type boundary conditions.

MSC:
93B05 Controllability
35L05 Wave equation
93C20 Control/observation systems governed by partial differential equations
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