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 also present a numerical example to validate our main
theoretical results.