Error estimates for the approximation of multibang control problems

TL;DR

This paper derives error estimates for approximating multibang control problems, combining regularization and discretization techniques, with numerical results validating the theoretical analysis.

Contribution

It provides new error estimates for the Moreau--Yosida approximation and discretization of multibang control problems, enabling efficient numerical solutions.

Findings

Error bounds for the Moreau--Yosida approximation

Convergence rates for discretized solutions

Numerical validation of theoretical error estimates

Abstract

This work is concerned with optimal control problems where the objective functional consists of a tracking-type functional and an additional "multibang" regularization functional that promotes optimal control taking values from a given discrete set pointwise almost everywhere. Under a regularity condition on the set where these discrete values are attained, error estimates for the Moreau--Yosida approximation (which allows its solution by a semismooth Newton method) and the discretization of the problem are derived. Numerical results support the theoretical findings.