Null Controllability of the Structurally Damped Wave Equation on the Two-Dimensional Torus

We investigate the null controllability of the wave equation with a Kelvin-Voigt damping on the two-dimensional torus T 2. We consider a distributed control supported in a moving domain ω(t) with a uniform motion at a constant velocity c = (1, ζ). The results we obtain depend strongly on the topological features of the geodesics of T 2 with constant velocity c. When ζ ∈ Q, writing ζ = p/q with p, q relatively prime, we prove that the null controllability holds if roughly the diameter of ω(0) is larger than 1/p and if the control time is larger than q. We prove also that for almost every ζ ∈ R + \ Q, and also for some particular values including e.g. ζ = e, the null controllability holds for any choice of ω(0) and for a sufficiently large control time. The proofs rely on a delicate construction of the weight function in a Carleman estimate which gets rid of a topogical assumption on the control region often encountered in the literature. Diophantine approximations are also needed when ζ is irrationnal. 2010 Mathematics Subject Classification: 35Q74, 93B05, 93B07, 93C20.