Switching in time-optimal problem with control in a ball

TL;DR

This paper investigates the regularity of time-optimal controls in an n-dimensional affine control system with controls constrained in a k-dimensional ball, providing conditions for smoothness or isolated jumps when k equals n-1.

Contribution

It offers new sufficient conditions based on Lie brackets for the smoothness or isolated discontinuities of optimal controls in such systems.

Findings

Conditions for control smoothness when k=n-1

Criteria for isolated jump discontinuities

Analysis of local regularity of optimal trajectories

Abstract

In this paper we analyse local regularity of time-optimal controls and trajectories for an n-dimensional affine control system with a control parameter, taking values in a k-dimensional closed ball. In the case of k equal to n-1, we give sufficient conditions in terms of Lie bracket relations for all optimal controls to be smooth or to have only isolated jump discontinuities.