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

Abstract

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 $0\le s\le t\le T$ 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.