Abstract

We investigate the null controllability of the wave equation with a Kelvin-Voigt damping on the two-dimensional torus \BbbT 2. We consider a distributed control supported in a moving domain \omega (t) with a uniform motion at a constant velocity c = (1, \zeta ). The results we obtain depend strongly on the topological features of the geodesics of \BbbT 2 with constant velocity c. When \zeta \in \BbbQ, writing \zeta = p/q with p, q relatively prime, we prove that the null controllability holds if roughly the diameter of \omega (0) is larger than 1/p and if the control time is larger than q. We also prove that for almost every \zeta \in \BbbR + \setminus \BbbQ, and also for some particular values including, e.g., \zeta = e, the null controllability holds for any choice of \omega (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 topological assumption on the control region often encountered in the literature. Diophantine approximations are also needed when \zeta is irrational.