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

We consider an optimal control problem described by a semilinear parabolic partial differential equation, with control and state constraints, where the state constraints and cost involve also the state gradient. Since this problem may have no classical solutions, it is reformulated in the relaxed form. The relaxed control problem is discretized by using a finite element method in space involving numerical integration and an implicit theta-scheme in time for space approximation, while the controls are approximated by blockwise constant relaxed controls. Under appropriate assumptions, we prove that relaxed accumulation points of sequences of optimal (resp. admissible and extremal) discrete relaxed controls are optimal (resp. admissible and extremal) for the continuous relaxed problem. We then apply a penalized conditional descent method to each discrete problem, and also a progressively refining version of this method to the continuous relaxed problem. We prove that accumulation points of sequences generated by the first method are extremal for the discrete problem, and that relaxed accumulation points of sequences of discrete controls generated by the second method are admissible and extremal for the continuous relaxed problem. Finally, numerical examples are given