Tikhonov regularization of control-constrained optimal control problems

TL;DR

This paper studies Tikhonov regularization for control-constrained optimal control problems, providing new error estimates under various conditions, including bang-bang solutions, and confirms findings with numerical experiments.

Contribution

It introduces novel a-priori error estimates for Tikhonov regularization in control problems, especially for bang-bang solutions, and analyzes the necessity of assumptions for convergence.

Findings

New a-priori error estimates under measure conditions

Error analysis for bang-bang solutions

Numerical validation of theoretical results

Abstract

We consider Tikhonov regularization of control-constrained optimal control problems. We present new a-priori estimates for the regularization error assuming measure and source-measure conditions. In the special case of bang-bang solutions, we introduce another assumption to obtain the same convergence rates. This new condition turns out to be useful in the derivation of error estimates for the discretized problem. The necessity of the just mentioned assumptions to obtain certain convergence rates is analyzed. Finally, a numerical example confirms the analytical findings.