Left-invariant optimal control problems on Lie groups

TL;DR

This paper reviews the theory, methods, and results of left-invariant optimal control problems on Lie groups, highlighting their symmetry, applications, and fundamental properties like extremal trajectories and optimal synthesis.

Contribution

It provides a comprehensive overview of the main concepts, techniques, and findings related to left-invariant optimal control problems on Lie groups, including nilpotent cases.

Findings

Analysis of extremal trajectories and their optimality

Description of cut time and cut locus

Development of optimal synthesis methods

Abstract

Left-invariant optimal control problems on Lie groups form an important class of problems with big symmetry group. They are interesting from the theoretical point of view since they often can be completely studied, and general features can be investigated on these model problems. In particular, problems on nilpotent Lie groups give a fundamental nilpotent approximation of general problems. Left-invariant problems also often arise in applications: in classical and quantum mechanics, geometry, robotics, models of vision and image processing. This work aims to review the main notions, methods and results for left-invariant optimal control problems on Lie groups. The main attention is paid to description of extremal trajectories and their optimality, cut time and cut locus, optimal synthesis. Bibliography: 238 titles.