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In this paper, a new technique is shown for deriving computable, guaranteedlower bounds of functional type (minorants) for two different cost functionalssubject to a parabolic time-periodic boundary value problem. Together withprevious results on upper bounds (majorants) for one of the cost functionals,both minorants and majorants lead to two-sided estimates of functional type forthe optimal control problem. Both upper and lower bounds are derived for thesecond new cost functional subject to the same parabolic PDE-constraints, butwhere the target is a desired gradient. The time-periodic optimal controlproblems are discretized by the multiharmonic finite element method leading tolarge systems of linear equations having a saddle point structure. Thederivation of preconditioners for the minimal residual method for the newoptimization problem is discussed in more detail. Finally, several numericalexperiments for both optimal control problems are presented confirming thetheoretical results obtained. This work provides the basis for an adaptivescheme for time-periodic optimization problems.

Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems