Periodic controls in step 2 sub-Finsler problems

TL;DR

This paper studies time-optimal control problems on step 2 Carnot groups, revealing that extremal controls are either constant or periodic, with detailed analysis of Casimirs and symplectic structures.

Contribution

It characterizes extremal controls in step 2 Carnot groups, especially for rank 3, and describes the symplectic foliation and Casimirs, advancing understanding of sub-Finsler problems.

Findings

Extremal controls are either constant or periodic.

All linear-in-momenta Casimirs are described.

Symplectic foliation is characterized for rank 3 Lie groups.

Abstract

We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with strictly convex set of control parameters (in particular, sub-Finsler problems). We describe all linear-in-momenta Casimirs on the dual of the Lie algebra. In the case of rank 3 Lie groups we describe the symplectic foliation on the dual of the Lie algebra. On this basis we show that extremal controls are either constant or periodic. Some related results for other Carnot groups are presented.