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Zbl 0769.93017
Zuazua, E.
Exact controllability for semilinear wave equations in one space dimension. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 10, No.1, 109-129 (1993). ISSN 0294-1449

Summary: The exact controllability of the semilinear wave equation $y''- y\sb{xx}+f(y)=h$ in one space dimension with Dirichlet boundary conditions is studied. We prove that if $\vert f(s)\vert/\vert s\vert\log\sp 2\vert s\vert\to 0$ as $\vert s\vert\to\infty$, then the exact controllability holds in $H\sp 1\sb 0(\Omega)\times L\sp 2(\Omega)$ with controls $h\in L\sp 2(\Omega\times(0,T))$ 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 --- $s \log\sp p(1+\vert s\vert)$ with $p>2$ as $\vert s\vert\to\infty$, 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 2000:: *93B05 Controllability
35L05 Wave equation
93C20 Control systems governed by PDE

Keywords: Hilbert uniqueness method; exact controllability; semilinear wave equation

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