Adaptive finite element method for an elliptic optimal control problem with integral state constraints

TL;DR

This paper develops an adaptive finite element method with a posteriori error analysis for a linear quadratic elliptic optimal control problem involving integral and pointwise constraints, enhancing solution accuracy.

Contribution

It introduces a novel a posteriori error estimator for a nonconforming finite element method applied to constrained elliptic optimal control problems, with numerical validation.

Findings

The error estimator is reliable and efficient.

Numerical results confirm the estimator's effectiveness.

The method improves solution accuracy for constrained control problems.

Abstract

In this article, we develop a posteriori error analysis of a nonconforming finite element method for a linear quadratic elliptic distributed optimal control problem with two different set of constraints, namely (i) integral state constraint and integral control constraint (ii) integral state constraint and pointwise control constraints. In the analysis, we have taken the approach of reducing the state-control constrained minimization problem into a state minimization problem obtained by eliminating the control variable. The reliability and efficiency of a posteriori error estimator are discussed. Numerical results are reported to illustrate the behavior of the error estimator.