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>0T>0 there exist square integrable initial data which cannot be steered to hit zero at time TT, using square integrable controls.We show that the counterexample we present complies with the restrictions imposed by the second principle of thermodynamics.