A sub-Finsler problem on the Cartan group

TL;DR

This paper investigates a sub-Finsler geometric problem on the Cartan group, a specific Carnot group, using control theory to characterize extremals, abnormal and singular arcs, and bang-bang flows.

Contribution

It provides a detailed analysis of extremal curves and control structures on the Cartan group with an $ ext{l}_ ext{infinity}$ norm, advancing understanding of sub-Finsler geometry in this context.

Findings

Characterization of extremal curves via Pontryagin maximum principle

Description of abnormal and singular arcs

Construction of bang-bang flow patterns

Abstract

In this paper we study a sub-Finsler geometric problem on the free-nilpotent group of rank 2 and step 3. Such a group is also called Cartan group and has a natural structure of Carnot group, which we metrize considering the $\ell_\infty$ norm on its first layer. We adopt the point of view of time-optimal control theory. We characterize extremal curves via Pontryagin maximum principle. We describe abnormal and singular arcs, and construct the bang-bang flow.