Cut time and optimal synthesis in sub-Riemannian problem on the group of motions of a plane

TL;DR

This paper provides a comprehensive solution to the sub-Riemannian problem on SE(2), including optimality conditions, conjugate and cut times, and an explicit description of the cut locus, advancing understanding of optimal trajectories in this geometric setting.

Contribution

It offers the first explicit global description of the cut locus and optimal synthesis for the sub-Riemannian problem on the group of motions of a plane.

Findings

Lower and upper bounds on the first conjugate time established.

Cut time equals the first Maxwell time from group symmetries.

Explicit global description of the cut locus achieved.

Abstract

A solution to the left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is obtained. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. An explicit global description of the cut locus is obtained. Optimal synthesis is described.