# American Institute of Mathematical Sciences

March  2016, 6(1): 1-25. doi: 10.3934/mcrf.2016.6.1

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

 1 Département de Mathématiques, Faculté des Sciences de Monastir , Université de Monastir, 5019 Monastir, Tunisia 2 LAGA (UMR 7539), Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse 3 Département de Mathématiques, Université de Cergy-Pontoise, UMR CNRS 8088, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France

Received  January 2015 Revised  October 2015 Published  January 2016

We study a damped semi-linear wave equation in a bounded domain of $\mathbb{R}^3$ 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.
Citation: Kaïs Ammari, Thomas Duyckaerts, Armen Shirikyan. Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation. Mathematical Control & Related Fields, 2016, 6 (1) : 1-25. doi: 10.3934/mcrf.2016.6.1
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