# Stabilization of the wave equation with moving boundary

## TL;DR

This paper presents a feedback control method to exponentially stabilize the wave equation with a periodically moving boundary, ensuring energy decay, based on a reduction theorem of Yoccoz.

## Contribution

It introduces a novel feedback control strategy for wave equations with moving boundaries, leveraging Yoccoz's reduction theorem for stability analysis.

## Key findings

- Exponential decay of energy achieved under the proposed feedback.
- Stability holds for boundaries moving slower than the speed of light.
- Provides insights into moving-pointwise stabilization problems.

## Abstract

We deal with the wave equation with assigned moving boundary ($0<x<a(t)$) upon which Dirichlet-Neuman boundary conditions are satisfied, here $a(t)$ is assumed to move slower than the light and periodically. We give a feedback which guarantees the exponential decay of the energy. The proof relies on a reduction theorem of Yoccoz. At the end we give a remark on the moving-pointwise stabilization problem.