Bounds on time-optimal concatenations of arcs for two-input driftless 3D systems

TL;DR

This paper establishes that for certain 3D driftless control systems with two inputs, time-optimal trajectories are composed of at most five segments, under specific nondegeneracy conditions, using advanced optimality conditions.

Contribution

It provides a new upper bound on the number of arcs in time-optimal controls for 3D two-input driftless systems, extending understanding of their structure.

Findings

Optimal trajectories have at most five arcs.

The result applies locally around points satisfying nondegeneracy conditions.

First and second-order conditions are used to derive the bounds.

Abstract

We study a driftless system on a three-dimensional manifold driven by two scalar controls. We assume that each scalar control has an independent bound on its modulus and we prove that, locally around every point where the controlled vector fields satisfy some suitable nondegeneracy Lie bracket condition, every time-optimal trajectory has at most five bang or singular arcs. The result is obtained using first-and second-order necessary conditions for optimality.