  We study the internal controllability of a wave equation with memory in the
principal part, defined on the one-dimensional torus
$\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$. 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.
