Exact controllability for semilinear wave equations in one space dimension

Abstract

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 ${\mathrm{H}}_0^1(\mathrm{\Omega })\times {\mathrm{L}}^2(\mathrm{\Omega })$ with controls $h\in {\mathrm{L}}^2(\Omega \times (0,T))$ 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 proof is rather general and applied also to the wave equation with Neumann type boundary conditions.

Résumé

On démontre la contrôlabilité exacte de l’équation des ondes semi-linéaire à une dimension d’espace pour des nonlinéarités $f$ que satisfont $|s|/|s|{\log }^2|s|\to 0$ lorsque $|s|\to \infty$. La méthode de démonstration combine HUM et une technique de point fixe. En utilisant des arguments d’explosion on démontre que la condition de croissance imposée à la nonlinéarité est presque optimale.