Symmetry Reduction of Optimal Control Systems and Principal Connections

TL;DR

This paper develops a geometric framework for symmetry reduction in nonlinear and optimal control systems, providing explicit methods to derive reduced systems using principal connections for various symmetry groups.

Contribution

It introduces a general approach to symmetry reduction in optimal control systems, explicitly constructing principal connections without assuming prior knowledge, and applies this to affine and kinematic systems.

Findings

Explicit expressions for reduced systems using principal connections.

Application to affine and kinematic optimal control systems.

Examples demonstrating symmetry reduction techniques.

Abstract

This paper explores the role of symmetries and reduction in nonlinear control and optimal control systems. The focus of the paper is to give a geometric framework of symmetry reduction of optimal control systems as well as to show how to obtain explicit expressions of the reduced system by exploiting the geometry. In particular, we show how to obtain a principal connection to be used in the reduction for various choices of symmetry groups, as opposed to assuming such a principal connection is given or choosing a particular symmetry group to simplify the setting. Our result synthesizes some previous works on symmetry reduction of nonlinear control and optimal control systems. Affine and kinematic optimal control systems are of particular interest: We explicitly work out the details for such systems and also show a few examples of symmetry reduction of kinematic optimal control problems.