Approximate controllability of linear parabolic equation with memory

In this paper, we consider an optimal control problem governed by linear parabolic differential equations with memory. Under the assumption that the corresponding linear parabolic differential equation without memory term is approximately controllable, it is shown that the set of approximate controls is nonempty. The problem is first viewed as a constrained optimal control problem, and then it is approximated by an unconstrained problem with a suitable penalty function. The optimal pair of the constrained problem is obtained as the limit of the optimal pair sequence of the unconstrained problem. The result is proved by using the theory of strongly continuous semigroups and the Banach fixed point theorem. The approximation theorems, which guarantee the convergence of the numerical scheme to the optimal pair sequence, are also proved. Finally, we present an algorithm to compute an optimal control with a numerical example.