Conjugate time in sub-Riemannian problem on Cartan group

TL;DR

This paper investigates the local optimality of geodesics in a sub-Riemannian problem on the Cartan group, establishing bounds on conjugate times and their relation to Maxwell times, with implications for geodesic optimality.

Contribution

It proves that the first conjugate time is not less than the Maxwell time and characterizes cases where they are equal in the Cartan group sub-Riemannian problem.

Findings

First conjugate time is bounded below by the Maxwell time.

Characterization of geodesics with equal conjugate and Maxwell times.

Continuity of the first conjugate time near infinity.

Abstract

The Cartan group is the free nilpotent Lie group of rank 2 and step 3. We consider the left-invariant sub-Riemannian problem on the Cartan group defined by an inner product in the first layer of its Lie algebra. This problem gives a nilpotent approximation of an arbitrary sub-Riemannian problem with the growth vector $(2,3,5)$. In previous works we described a group of symmetries of the sub-Riemannian problem on the Cartan group, and the corresponding Maxwell time -- the first time when symmetric geodesics intersect one another. It is known that geodesics are not globally optimal after the Maxwell time. In this work we study local optimality of geodesics on the Cartan group. We prove that the first conjugate time along a geodesic is not less than the Maxwell time corresponding to the group of symmetries. We characterize geodesics for which the first conjugate time is equal to the first Maxwell time. Moreover, we describe continuity of the first conjugate time near infinite values.