On the Controllability of a Class of Degenerate Parabolic Equations with Memory

Abstract

In this paper, we study the null controllability and approximate controllability for a class of weakly degenerate parabolic equations with memory by means of boundary controls. Unlike the known result for the degenerate parabolic equation, the degenerate parabolic equation with memory in general is not null controllable. This is based on the observability inequality for the adjoint system, which does not hold in the corresponding space. On the other hand, we prove the approximate controllability property of it in a suitable state space with a boundary control, which acts on the degenerate boundary or the nondegenerate boundary.