On Symmetries in Time Optimal Control, Sub-Riemannian Geometries, and the K−P Problem
On Symmetries in Time Optimal Control, Sub-Riemannian Geometries, and the K−P Problem
复制标题
关于时间最优控制的对称性、次黎曼几何和 K−P 问题
DOI：
10.1007/s10883-016-9351-6
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发表时间：
2016
影响因子：
0.9
通讯作者：
D. D’Alessandro
中科院分区：
文献类型：
--
作者：
F. Albertini;D. D’Alessandro
本文的目标是描述一种解决一类时间最优控制问题的方法，该问题相当于在流形 M 上寻找亚黎曼最小化测地线。特别地，我们假设流形 M 受到群 G 的作用，群 G 是动力学的对称群。 G对M的作用是适当的，但不一定是自由的。因此，轨道空间 M/G 不一定是流形，但它呈现了分层空间的更一般结构。该方法的主要内容是将问题简化为轨道空间 M/G 并分析该空间上的可达集。我们给出了与轨道空间的分层结构相关的一般结果，以及将其分解为轨道类型，并进行了最佳综合。我们更详细地考虑所谓的 K−P 问题的情况，其中流形 M 本身是李群，群 G 由 M 的嘉当分解确定。在这种情况下，测地线可以显式计算并且是解析的。作为说明，我们将我们的方法和结果应用于 SO(3) 的完整优化合成。
The goal of this paper is to describe a method to solve a class of time optimal control problems which are equivalent to finding the sub-Riemannian minimizing geodesics on a manifold M. In particular, we assume that the manifold M is acted upon by a group G which is a symmetry group for the dynamics. The action of G on M is proper but not necessarily free. As a consequence, the orbit spaceM/G is not necessarily a manifold but it presents the more general structure of a stratified space. The main ingredients of the method are a reduction of the problem to the orbit space M/G and an analysis of the reachable sets on this space. We give general results relating the stratified structure of the orbit space, and its decomposition into orbit types, with the optimal synthesis. We consider in more detail the case of the so-called K−P problem where the manifold M is itself a Lie group and the group G is determined by a Cartan decomposition of M. In this case, the geodesics can be explicitly calculated and are analytic. As an illustration, we apply our method and results to the complete optimal synthesis on SO(3).