Cut Locus and Optimal Synthesis in Sub-Riemannian Problem on the Lie Group SH(2)

TL;DR

This paper analyzes the global optimality and cut locus in a sub-Riemannian problem on the Lie group SH(2), providing explicit descriptions and proving the exponential map's diffeomorphism on certain domains.

Contribution

It introduces a detailed analysis of the cut locus and optimal synthesis in SH(2), including explicit descriptions and the proof of the exponential map's diffeomorphism on key domains.

Findings

Explicit description of the cut locus in SH(2)

Proof of the exponential map being a diffeomorphism on open dense domains

Calculation of the cut time for optimal trajectories

Abstract

Global optimality analysis in sub-Riemannian problem on the Lie group SH(2) is considered. We cutout open dense domains in the preimage and in the image of the exponential mapping based on the description of Maxwell strata. We then prove that the exponential mapping restricted to these domains is a diffeomorphism. Based on the proof of diffeomorphism, the cut time, i.e., time of loss of global optimality is computed on SH(2). We also consider the global structure of the exponential mapping and obtain an explicit description of cut locus and optimal synthesis.