An Optimization Method for Semilinear Parabolic Relaxed Constrained Optimal Control Problems

This paper addresses optimal control problems governed by semilinear parabolic partial differential equations, subject to control constraints and state constraints of integral type. Since such problems may not have classical solutions, a relaxed optimal control problem is considered. The relaxed control problem is discretized by using a finite element method and the behavior in the limit of discrete optimality, admissibility and extremality properties is studied. A conditional descent method with penalties applied to the discrete problems is proposed. It is shown that the accumulation points of sequences produced by this method are admissible and extremal for the discrete problem. Finally, numerical examples are given.