## 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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### References:

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