具有非对称多面体芬斯勒结构的李群上的极值曲线
Extremals on Lie groups with asymmetric polyhedral Finsler structures
AI总结：
该研究针对赋予左不变多面体芬斯勒结构的李群极值问题，借助庞特里亚金极大值原理转化为欧拉-阿诺德型控制系统，提出可通过极值垂直部分的渐近曲率分析控制量的唯一性。
摘要：
本文研究了赋予左不变多面体芬斯勒结构的李群$G$上的极值曲线。我们利用庞特里亚金极大值原理（Pontryagin's Maximal Principle, PMP）寻找李群余切丛上的曲线，使得其在$G$上的投影为极值曲线。设$\frak g$和$\frak g^\f$分别为$G$的李代数及其对偶空间。我们将该问题表示为欧拉-阿诺德（Euler-Arnold）型方程的控制系统$\frak a^\f(t)= -\frak{ad}^\f(u(t))(\frak a(t))$，其中$u(t)$是$\frak g$单位球面上的可测控制量，$\frak a(t)$是$\frak g^\f$中的绝对连续曲线。该控制系统的解$(u(t), \frak a(t))$为庞特里亚金极值，$\frak a(t)$为其垂直部分。本文证明，对于固定的庞特里亚金极值垂直部分$\frak a(t)$，使得$(u(t),\frak a(t))$成为庞特里亚金极值的$u(t)$的唯一性可通过$\frak a(t)$的渐近曲率进行研究。
英文摘要：
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 $\mathfrak a(t)$.
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