Non Controllability to Rest of the Two-Dimensional Distributed System Governed by the Integrodifferential Equation

The paper deals with controllability problem for a distributed system governed by the two-dimensional Gurtin-Pipkin equation. We consider a system with compactly supported distributed control and show that if the memory kernel is a twice continuously differentiable function, such that its Laplace transformation has at least one non-zero root, then the system cannot be driven to the equilibrium in a finite time.