We study the memory-type null controllability property for wave equations
involving memory terms. The goal is not only to drive the displacement and the
velocity (of the considered wave) to rest at some time-instant but also to
require the memory term to vanish at the same time, ensuring that the whole
process reaches the equilibrium. This memory-type null controllability problem
can be reduced to the classical null controllability property for a coupled
PDE-ODE system. The later is viewed as a degenerate system of wave equations,
the velocity of propagation for the ODE component vanishing. This fact requires
the support of the control to move to ensure the memory-type null
controllability to hold, under the so-called Moving Geometric Control
Condition. The control result is proved by duality by means of an observability
inequality which employs measurements that are done on a moving observation
open subset of the domain where the waves propagate