Sub-Finsler geodesics on the Cartan group

TL;DR

This paper analyzes sub-Finsler geodesics on the Cartan group, detailing extremal trajectories, bounds on switchings, and the structure of minimizers, advancing understanding of sub-Finsler geometry in this context.

Contribution

It provides a detailed classification of extremals and bounds on the complexity of geodesics in the sub-Finsler Cartan group, extending previous work with new structural insights.

Findings

Upper bounds on switchings in bang-bang extremals

Classification of extremals as bang-bang, singular, or mixed

Existence of piecewise smooth minimizers connecting any two points

Abstract

This paper is a continuation of the work by the same authors on the Cartan group equipped with the sub-Finsler $\ell_\infty$ norm. We start by giving a detailed presentation of the structure of bang-bang extremal trajectories. Then we prove upper bounds on the number of switchings on bang-bang minimizers. We prove that any normal extremal is either bang-bang, or singular, or mixed. Consequently, we study mixed extremals. In particular, we prove that every two points can be connected by a piecewise smooth minimizer, and we give a uniform bound on the number of such pieces.