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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 thespecial case where $a(x) = \sum\_{j=1}^N a\_j 1\_{x\in R\_j}$ ($R\_j$ arepolygons), we give a very simple necessary and sufficient geometric conditionfor uniform stabilisation. We also propose a natural generalization of thegeometric control condition which makes sense for $L^\infty$ dampings. We showthat this condition is always necessary for uniform stabilisation (for anycompact (smooth) manifold and any $L^\infty$ damping), and we prove that it issufficient in our particular case on $\mathbb{T}^2$ (and for our particulardampings).

Stabilisation of wave equations on the torus with rough dampings