2016/06/29 by Francesca Albertini, Domenico D’Alessandro, Albertini, Francesca +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1606.09280
openalex publication_date 2016/06/29 · openalex created_date 2022/08/07 · openalex updated_date 2026/07/28
The goal of this paper is to describe a method to solve a class of time\noptimal control problems which are equivalent to finding the sub-Riemannian\nminimizing geodesics on a manifold M. In particular, we assume that the\nmanifold M is acted upon by a group G which is a symmetry group for the\ndynamics. The action of G on M is proper but not necessarily free. As a\nconsequence, the orbit space M/G is not necessarily a manifold but it presents\nthe more general structure of a stratified space. The main ingredients of the\nmethod are a reduction of the problem to the orbit space M/G and an analysis of\nthe reachable sets on this space. We give general results relating the\nstratified structure of the orbit space, and its decomposition into orbit\ntypes, with the optimal synthesis. We consider in more detail the case of the\nso-called K-P problem where the manifold M is itself a Lie group and the group\nG is determined by a Cartan decomposition of M. In this case, the geodesics can\nbe explicitly calculated and are analytic. As an illustration, we apply our\nmethod and results to the complete optimal synthesis on SO(3).\n