论文标题
具有不对称多面体鳍结构的谎言组的极端物
Extremals on Lie groups with asymmetric polyhedral Finsler structures
论文作者
论文摘要
在这项工作中，我们研究了谎言组的极端$ g $，这些$ g $赋予了左左而不变的多面体鳍结构。我们使用Pontryagin的最大原理（PMP）在小组的Cotangent捆绑包上找到曲线，以使其对$ G $的预测是极端的。令$ \ Mathfrak g $和$ \ Mathfrak g^\ ast $分别为$ g $的Lie代数及其双空间。我们将此问题表示为控制系统$ \ mathfrak a^\ prime（t）= - \ Mathrm {ad}^\ ast（u（t））（\ Mathfrak a（t））$ euler -arnold Type等式的$ $ \ mathfrak g^\ ast $。该控制系统的解决方案$（u（t），\ mathfrak a（t））$是极值的pontryagin，而$ \ mathfrak a（t）$是其垂直部分。在这项工作中，我们表明，对于pontryagin umpory $ \ mathfrak a（t）$的固定垂直部分，$ u（t）$的唯一性使$（u（t），\ mathfrak a（t））$是pontryagin的极值，可以通过$ \ mathfrak a（t）$（t）$（t）$（t）$ \ m m iathervator curvator corvator cornemptotic curvator。
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 $\mathfrak a(t)$.
