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

Abstract In this paper we consider the heat equation with memory in a bounded region Ω ⊂ Rd, d ≥ 1, in the case that the propagation speed of the signal is infinite (i.e. the Colemann-Gurtin model). The memory kernel is of class C1. We examine its controllability properties both under the action of boundary controls or when the controls are distributed in a subregion of Ω. We prove approximate controllability of the system and, in contrast with 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 heat equation.