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

TL;DR

This paper demonstrates that a finite-dimensional feedback control can locally stabilize solutions of a damped semi-linear wave equation in a bounded domain, using linearized analysis and geometric conditions.

Contribution

It introduces a method to stabilize non-stationary solutions of a non-linear wave equation via finite-dimensional feedback control supported on a geometric subset.

Findings

Finite-dimensional feedback control stabilizes solutions.

Control based on linearized equation analysis.

Applicable under geometric conditions on the domain.

Abstract

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.