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

TL;DR

This paper develops new sharp, guaranteed, and fully computable a posteriori estimates for elliptic optimal control problems, providing two-sided bounds for the cost functional and discretization error, with numerical validation.

Contribution

It introduces novel theoretical bounds for the cost functional and error estimates in elliptic optimal control problems with control constraints.

Findings

Derived new lower bounds for the cost functional.

Established two-sided estimates for discretization errors.

Numerical tests confirm the efficiency of the estimates.

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.