Dirac structures and reduction of optimal control problems with symmetries

TL;DR

This paper explores how Dirac structures can be used to understand and reduce optimal control problems with symmetries, linking geometric reduction methods with the Pontryagin Maximum Principle.

Contribution

It extends Dirac structure reduction techniques to optimal control problems with Lie group symmetries and connects them with existing reduction approaches.

Findings

Reduction of variational principles and Dirac structures coincide.

The approach reproduces Martinez's Lie algebroid reduction.

Illustrated with a subriemannian geodesic problem on the Heisenberg group.

Abstract

We discuss the use of Dirac structures to obtain a better understanding of the geometry of a class of optimal control problems and their reduction by symmetries. In particular we will show how to extend the reduction of Dirac structures recently proposed by Yoshimura and Marsden [Yo09] to describe the reduction of a class of optimal control problems with a Lie group of symmetry. We will prove that, as in the case of reduction of implicit Hamiltonian or Lagrangian systems, the reduction of the variational principle and the reduction of the Dirac structure describing the Pontryagin Maximum Principle first order differential conditions coincide. Moreover they will also reproduce E. Mart\'inez Lie algebroids reduction approach [Mr04] to optimal control systems with symmetry. The geodesic subriemannian problem considered as an optimal control problem on the Heisenberg group, will be discussed as a simple example illustrating these results.