THE DISCRETE CLASSICAL OPTIMAL CONTROL PROBLEM OF A NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION (DCOCP)

In this paper, we consider a continuous classical optimal control for systems of nonlinear hyperbolic partial differential equations, with several 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 implicit finite difference scheme 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, we develop the existence theory for optimality of the discrete classical problem, and the discrete adjoint equations are developed corresponding to the discrete state equations. Finally the necessary conditions and a picewise minimum principle are developed for optimality of the discrete classical problem.