Switching in time-optimal problem. The 3-D case with 2-D control

TL;DR

This paper investigates the local structure of time-optimal controls for a 3D control-affine system with 2D control inputs, providing conditions for smoothness or isolated jumps based on Lie bracket relations.

Contribution

It offers new sufficient conditions, expressed via Lie brackets, that determine when optimal controls are smooth or exhibit isolated discontinuities in the 3D case.

Findings

Conditions for smooth optimal controls derived from Lie brackets.

Criteria for isolated jump discontinuities in controls.

Enhanced understanding of control structure in 3D systems.

Abstract

We study local structure of time-optimal controls and trajectories for a 3-dimensional control-affine system with a 2-dimensional control parameter with values in the disk. In particular, we give sufficient conditions, in terms of Lie bracket relations, for optimal controls to be smooth or to have only isolated jump discontinuities.