Optimal Control Realizations of Lagrangian Systems with Symmetry

TL;DR

This paper establishes a correspondence between optimal control systems with symmetry and Lagrangian systems, using Clebsch variables and Lin constraints, providing new insights into their equivalence and applications.

Contribution

It introduces a novel explicit realization of the relationship between symmetric optimal control systems and Lagrangian systems using Clebsch variables and Lin constraints.

Findings

Demonstrates the correspondence between control solutions and Lagrangian systems with symmetry.

Uses Clebsch variables and Lin constraints for explicit realization.

Provides applications illustrating the theoretical results.

Abstract

A new relation among a class of optimal control systems and Lagrangian systems with symmetry is discussed. It will be shown that a family of solutions of optimal control systems whose control equation are obtained by means of a group action are in correspondence with the solutions of a mechanical Lagrangian system with symmetry. This result also explains the equivalence of the class of Lagrangian systems with symmetry and optimal control problems discussed in \cite{Bl98}, \cite{Bl00}. The explicit realization of this correspondence is obtained by a judicious use of Clebsch variables and Lin constraints, a technique originally developed to provide simple realizations of Lagrangian systems with symmetry. It is noteworthy to point out that this correspondence exchanges the role of state and control variables for control systems with the configuration and Clebsch variables for the corresponding Lagrangian system. These results are illustrated with various simple applications.