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

TL;DR

This paper investigates the approximate controllability of a class of parabolic integro-differential equations, establishing conditions under which control solutions exist and can be approximated numerically.

Contribution

It demonstrates that approximate controllability for the unperturbed system implies controllability for the integro-differential case and develops a penalty-based approximation method.

Findings

The set of approximate controls is nonempty under certain conditions.

Optimal controls can be approximated via unconstrained penalized problems.

Numerical schemes converge to the optimal control solutions.

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 $0 \leq s \leq t \leq T$ 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.