Optimality of broken extremals

TL;DR

This paper investigates the conditions under which broken Pontryagin extremals are optimal in affine control systems, providing specific results for systems with different dimensions and control constraints.

Contribution

It establishes new optimality conditions for broken extremals in affine control systems with control parameters in a closed ball, including special cases for contact distributions and abnormal extremals.

Findings

Optimality of broken normal extremals when n=3 and controllable fields form a contact distribution.

Optimality of broken extremals when the Lie algebra is orthogonal to the singular locus.

Any broken extremal is optimal when k=2 and the controllable fields do not form a contact distribution.

Abstract

In this paper we analyse the optimality of broken Pontryagin extremal for an n-dimensional affine control system with a control parameter, taking values in a k- dimensional closed ball. We prove the optimality of broken normal extremals when n = 3 and the controllable vector fields form a contact distribution, and when the Lie algebra of the controllable fields is locally orthogonal to the singular locus and the drift does not belong to it. Moreover, if k = 2, we show the optimality of any broken extremal even abnormal when the controllable fields do not form a contact distribution in the point of singularity.