Mixed Discretization-Optimization Methods for Relaxed Optimal Control of Nonlinear Parabolic Systems

A nonconvex optimal control problem is considered, for systems governed by a parabolic partial differential equation, nonlinear in the state and control variables, with control and state constraints. Since this problem may have no classical solutions, it is reformulated in the relaxed form. The relaxed problem is discretized by using a finite element method in space and an implicit theta-scheme in time, while the controls are approximated by blockwise constant relaxed controls. Results are obtained on the behavior in the limit of discrete optimality, and of discrete admissibility and extremality. We then propose a penalized conditional descent method, applied to the discrete relaxed problem, and a progressively refining version of this method, applied to the continuous relaxed problem, that reduces computing time and memory. The behavior in the limit of sequences constructed by these methods is examined. Finally, numerical examples are given. Key-Words: Optimal control, nonlinear parabolic systems, state constraints, relaxed controls, discretization, finite elements, theta-scheme, discrete penalized conditional descent method, progressive refining.