Abstract

The exact controllability of the semilinear wave equation y″ – yxx + f(y) = h in one space dimension with Dirichlet boundary conditions is studied. We prove that if |f(s)|/|s |log2 |s| → 0 as |s| → ∞, then the exact controllability holds in ${\mathrm{H}}_0^1(\mathrm{\Omega })\times {\mathrm{L}}^2(\mathrm{\Omega })$ with controls h ∈ L2(Ω × (0, T)) supported in any open and non empty subset of Ω. 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 logp(1 + |s|) with p > 2 as |s| → ∞, 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| log2 |s| → 0 lorsque |s| → ∞. 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.