Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation

Kaïs Ammari (1) , Thomas Duyckaerts (2) , Armen Shirikyan (3)

1 FSM - Faculté des Sciences de Monastir
2 LAGA - Laboratoire Analyse, Géométrie et Applications
3 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

We study a damped semi-linear wave equation in a bounded domain with smooth boundary. It is proved that any sufficiently 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.

Mots clés: Non-linear wave equation; distributed control; feedback stabilisation; truncated observability inequality

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Kaïs Ammari, Thomas Duyckaerts, Armen Shirikyan. Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation. 2012. ⟨hal-00759119⟩