  In this paper, we discuss the distributed control problem governed by the
following 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 state
space variable, $u$ is the control variable, $A$ is a self adjoint, positive
definite linear (not necessarily bounded) operator in a Hilbert space $X$ with
dense domain $D(A) \subset X,$ $B(t,s)$ is an unbounded operator, smooth with
respect 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 of
approximate controls of the distributed control problem ($\ast$) is nonempty.
The problem is first viewed as constrained optimal control problem and then it
is approximated by unconstrained problem with a suitable penalty function. The
optimal pair of the constrained problem is obtained as the limit of optimal
pair sequence of the unconstrained problem. The approximation theorems, which
guarantee the convergence of the numerical scheme to the optimal pair sequence,
are also proved.
