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In this paper, we examine the controllability problem of 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 root, then the system cannot be driven to equilibrium in finite time.

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