Stability and genericity of bang-bang controls in affine problems

TL;DR

This paper investigates the stability and genericity of bang-bang controls in affine optimal control problems, showing that stability properties often depend on the bang-bang nature of minimizers and that perturbations tend to produce bang-bang solutions.

Contribution

It establishes that essential stability properties are tied to bang-bang solutions and demonstrates that almost any perturbation results in a bang-bang strict global minimizer, covering various control problems.

Findings

Stability properties are satisfied mainly by bang-bang minimizers.

Perturbations typically lead to bang-bang solutions.

Results apply to problems constrained by differential equations.

Abstract

We analyse the role of the bang-bang property in affine optimal control problems. We show that many essential stability properties of affine problems are only satisfied when minimizers are bang-bang. Moreover, we prove that almost any perturbation in an affine optimal control problem leads to a bang-bang strict global minimizer. We work in an abstract framework that allows to cover many problems in the literature of optimal control, this includes problems constrained by partial and ordinary differential equations. We give examples that show the applicability of our results to specific optimal control problems.