Stabilization for the semilinear wave equation with geometric control condition

TL;DR

This paper proves exponential stabilization of a semilinear wave equation with damping under geometric control, using unique continuation and analyticity techniques, and explores implications for controllability and attractors.

Contribution

It introduces a novel proof of large-time unique continuation for undamped equations using asymptotic smoothing and analyticity, advancing control theory for nonlinear wave equations.

Findings

Exponential stabilization under geometric control condition.

Unique continuation result for large time in undamped equations.

Implications for controllability and attractor existence.

Abstract

In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results, is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity due to Robbiano, Zuily, Tataru and H\"ormander. Some other consequences are also given for the controllability and the existence of a compact attractor.