Optimization Methods for Optimal Control of Nonlinear Elliptic Systems

We consider an optimal control problem for systems governed by a highly nonlinear second order elliptic partial differential equation, with control and state constraints. The problem is formulated in the classical and in the relaxed form, and various necessary conditions for optimality are given. For the numerical solution of these problems, we propose a penalized gradient projection method generating classical controls and a penalized conditional descent method generating relaxed controls. We study the behavior in the limit of sequences constructed by these methods, using relaxation theory. Finally, numerical examples are given. Key-Words: Optimal control, nonlinear elliptic systems, state constraints, classical penalized gradient projection method, relaxed penalized conditional descent method, relaxed controls.