Sufficient conditions for time optimality of systems with control on the disk

TL;DR

This paper investigates time optimal control problems for affine systems with control constrained to a disk, providing verifiable conditions for optimality and analyzing singularities and flow regularity.

Contribution

It introduces new verifiable sufficient conditions for local optimality in systems with control on a disk, focusing on singularities and flow regularity analysis.

Findings

Established conditions for local optimality involving limits of Jacobi fields

Analyzed the regularity of extremal flows with control on a disk

Provided a framework for handling singularities in time optimal control

Abstract

The case of time minimization for affine control systems with control on the disk is studied. After recalling the standard sufficient conditions for local optimality in the smooth case, the analysis focusses on the specific type of singularities encountered when the control is prescribed to the disk. Using a suitable stratification, the regularity of the flow is analyzed, which helps to devise verifiable sufficient conditions in terms of left and right limits of Jacobi fields at a switching point. Under the appropriate assumptions, piecewise regularity of the field of extremals is obtained.