Optimal control of a parabolic equation with memory

Abstract

An optimal control problem for a semilinear parabolic partial differential equation with memory is considered. The well-posedness as well as the first and the second order differentiability of the state equation is established by means of Schauder fixed point theorem and the implicity function theorem. For the corresponding optimal control problem with the quadratic cost functional, the existence of optimal control is proved. The first and the second order necessary conditions are presented, including the investigation of the adjoint equations which are linear parabolic equations with a measure as a coefficient of the operator. Finally, the sufficiency of the second order optimality condition for the local optimal control is proved.