LOCAL FEEDBACK STABILISATION TO A NON-STATIONARY SOLUTION FOR A DAMPED NON-LINEAR WAVE EQUATION

We study a damped semi-linear wave equation in a bounded domain of R-3 with smooth boundary. It is proved that any H-2-smooth solution can be stabilised locally by a finite-dimensional feedback control supported by a given open subset satisfying a geometric condition. The proof is based on an investigation of the linearised equation, for which we construct a stabilising control satisfying the required properties. We next prove that the same control stabilises locally the non-linear problem.