Necessary Optimality Conditions for the Control of Partial Integro-Differential Equations

Abstract

In this paper we derive necessary optimality conditions for optimization problems with partial integro-differential equations. We use the concept of mild solutions coming from semi-group theory for evolution equations. The application considered is a model for a cell adhesion process which leads to a two-dimensional system of nonlinear partial integro-differential equations. The objective function is of tracking type with the coefficients in the integral operators as unknown control or design variables. We derive necessary optimality conditions in the form of an adjoint system of partial integro-differential equations.