Exponential stabilization without geometric control

TL;DR

This paper demonstrates exponential stabilization of the damped wave equation on compact manifolds even when the geometric control condition fails, using a dynamical approach involving topological pressure.

Contribution

It introduces a novel dynamical method based on topological pressure to achieve stabilization without geometric control conditions.

Findings

Exponential stabilization achieved without geometric control.

Dynamical argument involving topological pressure is effective.

Applicable to damped wave equations on compact manifolds.

Abstract

We present examples of exponential stabilization for the damped wave equation on a compact manifold in situations where the geometric control condition is not satisfied. This follows from a dynamical argument involving a topological pressure on a suitable uncontrolled set.