Lack of controllability of the heat equation with memory

We consider a model for the heat equation with memory, which has infinite propagation speed, like the standard heat equation. We prove that, in spite of this, for every T > 0 there exist square integrable initial data which cannot be steered to hit zero at time T , using square integrable controls. w that the counterexample we present complies with the restrictions imposed by the second principle of thermodynamics.