Vector-valued multibang control of differential equations

TL;DR

This paper introduces a convex relaxation approach for vector-valued multibang control problems in differential equations, ensuring well-posedness and stability, and demonstrates numerical solutions for specific physical systems.

Contribution

It develops a convex relaxation framework for vector-valued multibang control problems, providing well-posedness, stability analysis, and a semismooth Newton method for numerical solutions.

Findings

Convex relaxation ensures well-posedness of multibang control problems.

Stability analysis confirms robustness of the approach.

Numerical experiments demonstrate effectiveness on physical models.

Abstract

We consider a class of (ill-posed) optimal control problems in which a distributed vector-valued control is enforced to pointwise take values in a finite set $\mathcal{M}\subset\mathbb{R}^m$. After convex relaxation, one obtains a well-posed optimization problem, which still promotes control values in $\mathcal{M}$. We state the corresponding well-posedness and stability analysis and exemplify the results for two specific cases of quite general interest, optimal control of the Bloch equation and optimal control of an elastic deformation. We finally formulate a semismooth Newton method to numerically solve a regularized version of the optimal control problem and illustrate the behavior of the approach for our example cases.