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In this paper, we discuss the distributed control problem governed by thefollowing parabolic integro-differential equation (PIDE) in the abstract form\begin{eqnarray*} \frac{\partial y}{\partial t} + A y &=& \int_0^t B(t, s) y(s)ds + Gu, \;\, t \in [0, T], \;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \hfill{(\ast)}\\y(0) &=& y_0 \, \in X, \nonumber \end{eqnarray*} where, $y$ denotes the statespace variable, $u$ is the control variable, $A$ is a self adjoint, positivedefinite linear (not necessarily bounded) operator in a Hilbert space $X$ withdense domain $D(A) \subset X,$ $B(t,s)$ is an unbounded operator, smooth withrespect to $t$ and $s$ with $D(A) \subset D(B(t,s)) \subset X$ for $0 \leq s\leq t \leq T$ and $G$ is a bounded linear operator from the control space to$X.$ Assuming that the corresponding evolution equation ($B \equiv 0$ in($\ast$)) is approximately controllable, it is shown that the set ofapproximate controls of the distributed control problem ($\ast$) is nonempty.The problem is first viewed as constrained optimal control problem and then itis approximated by unconstrained problem with a suitable penalty function. Theoptimal pair of the constrained problem is obtained as the limit of optimalpair sequence of the unconstrained problem. The approximation theorems, whichguarantee the convergence of the numerical scheme to the optimal pair sequence,are also proved.

Approximate Controllability of a Class of Partial Integro-Differential Equations of Parabolic Type