Null-controllability properties of the wave equation with a second order memory term

We study the internal controllability of a wave equation with memory in the principal part, defined on the one-dimensional torus $\mathbb{T}=\mathbb{R}\mathrm{/}2\pi \mathbb{Z}$. We assume that the control is acting on an open subset $\omega (t)\subset \mathbb{T}$, which is moving with a constant velocity $c\in \mathbb{R}\setminus \{-1,0,1\}$. The main result of the paper shows that the equation is null controllable in a sufficiently large time $T$ and for initial data belonging to suitable Sobolev spaces. Its proof follows from a careful analysis of the spectrum associated to our problem and from the application of the classical moment method.