Extremals of a left-invariant sub-Finsler quasimetric on the Cartan group

TL;DR

This paper characterizes extremal trajectories for a class of sub-Finsler quasimetric problems on the Cartan group using Pontryagin's Maximum Principle, providing insights into optimal control in geometric structures.

Contribution

It derives extremals for arbitrary left-invariant sub-Finsler quasimetric on the Cartan group, expanding understanding of optimal control in sub-Finsler geometry.

Findings

Extremals are explicitly characterized using Pontryagin's Maximum Principle.

The analysis applies to any left-invariant sub-Finsler quasimetric on the Cartan group.

Provides a framework for solving time-optimal control problems in this geometric setting.

Abstract

Using the Pontryagin Maximum Principle for the time-optimal problem in coordinates of the first kind, we find extremals of abitrary left-invariant sub-Finsler quasimetric on the Cartan group defined by a distribution of rank two.