Approximate controllability and lack of controllability to zero of the heat equation with memory

In this paper we consider the heat equation with memory in a bounded region Ω⊂RdΩ⊂Rd, d≥1d≥1, in the case that the propagation speed of the signal is infinite (i.e. the Coleman–Gurtin model). The memory kernel is of class C1C1. We examine its controllability properties both under the action of boundary controls and when the controls are distributed in a subregion of Ω. We prove approximate controllability of the system and, in contrast to this, we prove the existence of initial conditions which cannot be steered to hit the target 0 in a certain time T, of course when the memory kernel is not identically zero. In both the cases we derive our results from well known properties of the (memoryless) heat equation.