Exact Controllability of the Two-Dimensional Distributed System Governed by Integrodifferential Equation

We will consider the exact controllability of the distributed system, governed by two-dimensional Gurtin-Pipkin equation. It will be proved that this mechanical system can not be driven to an equilibrium point in a finite time if the distributed control has a compact support properly contained in the considered domain. In this case the memory kernel is a twice continuously differentiable function such that its Laplace transformation has at least one root $\lambda_0\neq 0$.