Multiharmonic finite element analysis of a time-periodic parabolic optimal control problem

This paper presents the multiharmonic analysis of a distributed parabolic optimal control problem in a time-periodic setting. We prove the existence and uniqueness of the solution of some weak space-time variational formulation for the parabolic time-periodic boundary value problem appearing in the constraints for the optimal control problem. Since the cost functional is quadratic, the optimal control problem is uniquely solvable as well. In order to solve the optimal control problem, we state its optimality system and discretize it by the multiharmonic finite element method leading to a system of linear algebraic equations which decouples into smaller systems. We construct preconditioners for these systems which yield robust convergence rates and optimal complexity for the preconditioned minimal residual method. All systems can be solved totally in parallel. Furthermore, we present a complete analysis for the error introduced by the multiharmonic finite element discretization as well as some numerical results confirming our theoretical findings.