Extremals on Lie groups with asymmetric polyhedral Finsler structures

TL;DR

This paper investigates extremal curves on Lie groups with asymmetric polyhedral Finsler structures using Pontryagin's Maximal Principle, representing the problem as a control system and analyzing the uniqueness of solutions via asymptotic curvature.

Contribution

It introduces a control system framework for extremals on Lie groups with polyhedral Finsler metrics and links solution uniqueness to asymptotic curvature analysis.

Findings

Control system representation of extremals on Lie groups.

Uniqueness of extremal controls related to asymptotic curvature.

Application of Pontryagin's Maximal Principle to polyhedral Finsler structures.

Abstract

In this work we study extremals on Lie groups $G$ endowed with a left invariant polyhedral Finsler structure. We use the Pontryagin's Maximal Principle (PMP) to find curves on the cotangent bundle of the group, such that its projections on $G$ are extremals. Let $\mathfrak g$ and $\mathfrak g^\ast$ be the Lie algebra of $G$ and its dual space respectively. We represent this problem as a control system $\mathfrak a^\prime (t)= -\mathrm{ad}^\ast(u(t))(\mathfrak a(t))$ of Euler-Arnold type equation, where $u(t)$ is a measurable control in the unit sphere of $\mathfrak g$ and $\mathfrak a(t)$ is an absolutely continuous curve in $\mathfrak g^\ast$. A solution $(u(t), \mathfrak a(t))$ of this control system is a Pontryagin extremal and $\mathfrak a(t)$ is its vertical part. In this work we show that for a fixed vertical part of the Pontryagin extremal $\mathfrak a(t)$, the uniqueness of $u(t)$ such that $(u(t),\mathfrak a(t))$ is a Pontryagin extremal can be studied through an asymptotic curvature of $\mathfrak a(t)$.