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

## TL;DR

This paper introduces a new technique for deriving guaranteed lower bounds for cost functionals in time-periodic parabolic optimal control problems, enhancing the ability to estimate and control solutions accurately.

## Contribution

The paper presents a novel method for obtaining computable lower bounds for two cost functionals in time-periodic parabolic control problems, complementing existing upper bounds and enabling two-sided estimates.

## Key findings

- Derived guaranteed lower bounds for cost functionals.
- Validated bounds through numerical experiments.
- Established a basis for adaptive time-periodic optimization schemes.

## Abstract

In this paper, a new technique is shown for 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. Both upper and lower bounds are derived for the second new cost functional subject to the same parabolic PDE-constraints, but where the target is a desired gradient. 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 optimization problem is discussed in more detail. Finally, several numerical experiments for both optimal control problems are presented confirming the theoretical results obtained. This work provides the basis for an adaptive scheme for time-periodic optimization problems.