Symmetries in left-invariant optimal control problems

TL;DR

This paper explores symmetries in left-invariant optimal control problems on Lie groups, introducing a new construction for exponential map symmetries that aid in analyzing trajectory optimality.

Contribution

It introduces a novel construction for symmetries of the exponential map in left-invariant control problems on Lie groups, enhancing the analysis of extremal trajectories.

Findings

Symmetries of the exponential map are characterized for specific Lie groups.

The construction aids in determining optimality of extremal trajectories.

Applicable to problems with connected stabilizers of coadjoint actions.

Abstract

We consider left-invariant optimal control problems on connected Lie groups such that generic stabilizer of the coadjoint action is connected and has dimension not more than 1. We introduce a construction for symmetries of the exponential map. These symmetries play a key role in investigation of optimality of extremal trajectories.