Extremals on Lie groups with asymmetric polyhedral Finsler structures
Jéssica B. Prudencio, Ryuichi Fukuoka
TL;DR
This work develops a Pontryagin-maximum-principle framework for extremals on Lie groups equipped with left invariant polyhedral $C^0$-Finsler structures, formulating the Euler-Arnold type equation $\frak{a}'(t)=-\mathrm{ad}^*(u(t))(\frak{a}(t))$ with $u(t)$ constrained to $S_{F_e}$. It introduces an asymptotic (limit) flag curvature $\mathcal{K}_{\mathcal{B}}(\frak{a})$ and a reduced curvature $\mathcal{K}(\frak{a},v_2)$ to relate the vertical part of Pontryagin extremals to the (non)uniqueness of the horizontal control $u(t)$, showing in 3D that this curvature can be independent of the choice of $v_2$ on certain subspaces. The main results connect the measure of the set where $\mathcal{K}(\frak{a},v_2)=0$ to the existence of infinitely many admissible controls achieving the same extremal, providing criteria for when $u(t)$ is unique or not. These findings pave the way for classifying left invariant $p$-Finsler structures in low dimensions and linking curvature-like quantities to geodesic optimality on Lie groups.
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 $\mathfrak a(t)$.