A note on functional a posteriori estimates for elliptic optimal control problems

Abstract

Abstract. In this work, new theoretical results on functional type a posteriori estimates for elliptic optimal control problems with control constraints are presented. More precisely, we derive new, sharp, guaranteed and fully computable lower bounds for the cost functional in addition to the already existing upper bounds. Using both, the lower and the upper bounds, we arrive at two-sided estimates for the cost functional. We prove that these bounds finally lead to sharp, guaranteed and fully computable upper estimates for the discretization error in the state and the control of the optimal control problem. First numerical tests are presented confirming the efficiency of the a posteriori estimates derived.

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

“…These lower bounds are guaranteed and computable. Together with using the results from [44] as well as [16] one can apply the method also to time-periodic optimal control problems, where box constraints are being imposed on the Fourier coefficients of the control.…”

Functional a posteriori estimates for elliptic optimal control problems

“…The aim of this work was to derive first results on minorants for cost functionals of distributed elliptic optimal control problems with control constraints and to present a fully computable upper bound for the discretization error in the state and the control. Details on the derivation of the new minorants and majorants including the proofs can be found in [5].…”

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

“…In this work, we complement the guaranteed upper bounds for the discretization error in state and control of minimizing cost functional J defined in (9) subject to (1)-( 3). This is done by obtaining fully computable lower bounds for J following the technique from [44] (derived for elliptic problems) leading to two-sided bounds for the cost functional (9).…”

“…In [26], majorants for one cost functional of a time-periodic parabolic optimal control and for the corresponding optimality system were presented. This work presents the corresponding minorants for this cost functional using the new technique presented in [44], which makes use of ideas derived by Mikhlin [34] but generalized for the class of optimal control problems. We mention here that [30] presents a different approach for the derivation of a lower bound for a class of elliptic optimal control problems.…”