Two-sided bounds for cost functionals of time-periodic parabolic optimal control problems.

In this paper, a new technique is applied on deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Moreover, a different cost functional is discussed with the same PDE-constraints. Both upper and lower bounds are derived. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to large systems of linear equations having a saddle point structure. The derivation of preconditioners for the minimal residual method for the new optimal control problem is discussed in more detail. Finally, several numerical experiments for both optimal control problems are presented confirming the theoretical results obtained.