Mixed Discretization-Optimization Methods for Nonlinear Elliptic Optimal Control Problems

Abstract

An optimal control problem is considered, for systems governed by a nonlinear elliptic partial differential equation, with control and state constraints. 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, where the controls are approximated by elementwise constant, linear, or multilinear, controls. Our first result is that strong accumulation points in L2 of sequences of admissible and extremal discrete controls are admissible and weakly extremal classical for the continuous classical problem, and that relaxed accumulation points of sequences of admissible and extremal discrete controls are admissible and weakly extremal relaxed for the continuous relaxed problem. We then propose a penalized gradient projection method, applied to the discrete problem, and a corresponding discretization-optimization method, applied to the continuous classical problem, that progressively refines the discretization during the iterations, thus reducing computing time and memory. We prove that accumulation points of sequences generated by the first method are admissible and 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.