Extremals on Lie groups with asymmetric polyhedral Finsler structures

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 $\mathfrak a(t)$.