Discretization-Optimization Methods for Nonlinear Elliptic Optimal Control Problems with State Constraints

Abstract

We consider an optimal control problem described by a second order elliptic boundary value problem, jointly nonlinear in the state and control, with control and state constraints, where the state constraints and cost functionals involve also the state gradient. Since this problem may have no classical solutions, it is also formulated in the relaxed form. The classical problem is discretized by using a finite element method for state approximation, while the controls are approximated by elementwise constant, or linear, or multilinear, controls. Various necessary conditions for optimality are given for the classical and the relaxed problem, in the continuous and the discrete case. We then study the behavior in the limit of discrete optimality, and of discrete extremality and admissibility. Next, we apply a penalized gradient projection method to each discrete problem, and also a progressively refining version of this method to the continuous classical problem. We prove that accumulation points of sequences generated by the first method are extremal for the discrete problem, and that strong classical (resp. relaxed) accumulation points of sequences of discrete controls generated by the second method are admissible and weakly extremal classical (resp. relaxed) for the continuous classical (resp. relaxed) problem. Finally, numerical examples are given.