THE DISCRETE CLASSICAL OPTIMAL CONTROL PROBLEM of A SEMILINEAR PARABOLIC EQUATION (COCP)

Abstract

In this paper, the continuous classical optimal control for systems of a semilinear parabolic partial differential equations is described with a equality and inequality state constraints. First, the considered continuous classical optimal control problem is discretized into a discrete classical optimal control problem by using the Galerkin finite element method in space and the theta finite difference scheme (             -method)in time. The classical continuous controls are approximated by picewise constants. Second the existence of a unique solution of the discrete state equations for fixed discrete classical control is studied. Third, the existence theory for optimality of the discrete classical problem is proved, and the discrete adjoint equations are developed corresponding to the discrete state equations. Finally the necessary conditions and a picewise minimum principle are derived for optimality of the discrete classical problem so as the sufficient conditions.