Second-order variational problems on Lie groupoids and optimal control applications

TL;DR

This paper develops a geometric variational framework for second-order problems on Lie groupoids, enabling the construction of variational integrators for optimal control of mechanical systems with symmetry considerations.

Contribution

It introduces new variational techniques on Lie groupoids and links symplectic geometry to discrete dynamics, advancing the geometric understanding of optimal control systems.

Findings

Variational integrators for second-order systems are constructed using Lie groupoid structures.

Lagrangian submanifolds of symplectic groupoids encode the discrete dynamics.

The framework incorporates symmetry reduction and Noether's theorem.

Abstract

In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational techniques for second-order variational problems on Lie groupoids and their applications to the construction of variational integrators for optimal control problems of mechanical systems. Next, we show how Lagrangian submanifolds of a symplectic groupoid gives intrinsically the discrete dynamics for second-order systems, both unconstrained and constrained, and we study the geometric properties of the implicit flow which defines the dynamics in the Lagrangian submanifold. We also study the theory of reduction by symmetries and the corresponding Noether theorem.