Progressively refining discrete gradient projection method for semilinear parabolic optimal control problems

TL;DR: It is proved that strong accumulation points of sequences generated by the proposed discrete, progressively refining, gradient projection method satisfy the weak optimality conditions for the continuous classical problem, and that relaxed accumulation points satisfy the Weak Optimality Conditions for the Continuous relaxed problem.

Abstract: We consider an optimal control problem defined by semilinear parabolic partial differential equations, with convex control constraints. Since this problem may have no classical solutions, we also formulate it in relaxed form. The classical problem is then discretized by using a finite element method in space and a theta-scheme in time, where the controls are approximated by blockwise constant classical ones. We then propose a discrete, progressively refining, gradient projection method for solving the classical, or the relaxed, problem. We prove that strong accumulation points (if they exist) of sequences generated by this method satisfy the weak optimality conditions for the continuous classical problem, and that relaxed accumulation points (which always exist) satisfy the weak optimality conditions for the continuous relaxed problem. Finally, numerical examples are given.