A posteriori error estimates for a bang-bang optimal control problem

TL;DR

This paper develops residual-based a posteriori error estimates for control-constrained optimal control problems with bang-bang solutions, focusing on reliability and efficiency without control discretization or Tikhonov regularization.

Contribution

It introduces a novel residual-type a posteriori error estimator for a variational approach to bang-bang control problems, analyzing its reliability and efficiency.

Findings

The error estimator accurately measures discretization errors in state and adjoint variables.

Numerical examples confirm the theoretical reliability and efficiency of the estimator.

The approach avoids control discretization and Tikhonov regularization, simplifying analysis.

Abstract

We propose and analyze a posteriori error estimates for a control-constrained optimal control problem with bang-bang solutions. We consider a solution strategy based on the variational approach, where the control variable is not discretized; no Tikhonov regularization is made. We design, for the proposed scheme, a residual-type a posteriori error estimator that can be decomposed as the sum of two individual contributions related to the discretization of the state and adjoint equations. We explore reliability and efficiency properties of the aforementioned error estimator. We illustrate the theory with numerical examples.