On Symmetries in Time Optimal Control, sub-Riemannian Geometries and the K-P Problem

TL;DR

This paper introduces a method to solve time optimal control problems involving sub-Riemannian geometries with symmetry groups, focusing on the K-P problem and applying it to SO(3) for explicit geodesic computation.

Contribution

It develops a reduction technique to analyze optimal control problems on stratified orbit spaces with symmetry, providing explicit solutions for the K-P problem on Lie groups.

Findings

Reduction to orbit space simplifies the problem.

Explicit geodesic formulas for the K-P problem.

Complete optimal synthesis on SO(3).

Abstract

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 space M/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).