Stabilisation of wave equations on the torus with rough dampings

TL;DR

This paper establishes a simple necessary and sufficient geometric condition for the uniform stabilization of wave equations with polygonal damping on the 2D torus, extending the classical control condition to rough, $L^ olinebreak^ olinebreak ext{infty}$ dampings.

Contribution

It introduces a new geometric condition for stabilization with rough dampings on the torus and proves its necessity generally, with sufficiency shown in a specific case.

Findings

A simple necessary and sufficient condition for stabilization on the torus.

The generalized geometric control condition is necessary for any $L^ olinebreak^ olinebreak ext{infty}$ damping.

Sufficiency of this condition is proven for polygonal dampings on the 2D torus.

Abstract

For the damped wave equation on a compact manifold with {\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two dimensional torus, in the special case where $a(x) = \sum\_{j=1}^N a\_j 1\_{x\in R\_j}$ ($R\_j$ are polygons), we give a very simple necessary and sufficient geometric condition for uniform stabilisation. We also propose a natural generalization of the geometric control condition which makes sense for $L^\infty$ dampings. We show that this condition is always necessary for uniform stabilisation (for any compact (smooth) manifold and any $L^\infty$ damping), and we prove that it is sufficient in our particular case on $\mathbb{T}^2$ (and for our particular dampings).