  In this paper, we consider the wave equation with both a viscous Kelvin-Voigt
and frictional damping as a model of viscoelasticity in which we incorporate an
internal control with a moving support. We prove the null controllability when
the control region, driven by the flow of an ODE, covers all the domain. The
proof is based upon the interpretation of the system as, roughly, the coupling
of a heat equation with an ordinary differential equation (ODE). The presence
of the ODE for which there is no propagation along the space variable makes the
controllability of the system impossible when the control is confined into a
subset in space that does not move. %Accordingly, we consider the control on a
moving support that, along the time interval, covers the whole domain. The null
controllability of the system with a moving control is established in using the
observability of the adjoint system and some Carleman estimates for a coupled
system of a parabolic equation and an ODE with the same singular weight,
adapted to the geometry of the moving support of the control. This extends to
the multi-dimensional case the results by P. Martin et al. on the
one-dimensional case, employing 1-d Fourier analysis techniques.
