Error estimates for a certain class of elliptic optimal control problems

TL;DR

This paper develops a posteriori error estimates for elliptic optimal control problems with PDE constraints, providing bounds on the cost functional and introducing an error measure that quantifies the gap between approximate and optimal controls.

Contribution

It applies functional a posteriori error estimates to derive bounds for the cost functional and introduces a new error quantity for control approximation accuracy.

Findings

Error bounds for the cost functional are established.

A new error measure for control approximation is proposed.

Numerical tests confirm the theoretical estimates.

Abstract

In this paper, error estimates are presented for a certain class of optimal control problems with elliptic PDE-constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in late 90's are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.