Presymplectic integrators for optimal control problems via retraction maps

TL;DR

This paper introduces presymplectic integrators for optimal control problems using retraction maps to discretize tangent and cotangent bundles, enabling symplectic integration of Hamiltonian systems derived from Pontryagin's principle.

Contribution

The paper develops a novel framework combining retraction maps and integrability algorithms to construct presymplectic integrators specifically for optimal control problems.

Findings

Presymplectic integrators effectively preserve geometric properties of optimal control systems.

The approach leverages discretization maps to lift dynamics to the cotangent bundle.

Results demonstrate improved numerical stability and accuracy in control simulations.

Abstract

Retractions maps are used to define a discretization of the tangent bundle of the configuration manifold as two copies of the configuration manifold where the dynamics take place. Such discretization maps can be conveniently lifted to the cotangent bundle so that symplectic integrators are constructed for Hamilton's equations. Optimal control problems are provided with a Hamiltonian framework by Pontryagin's Maximum Principle. That is why we use discretization maps and the integrability algorithm to obtain presymplectic integrators for optimal control problems.