# Second-order constrained variational problems on Lie algebroids: applications to optimal control

## TL;DR

This paper develops a geometric framework for second-order constrained variational problems on Lie algebroids, deriving dynamical equations and applying the formalism to optimal control of mechanical systems.

## Contribution

It extends the Skinner-Rusk formalism to second-order problems on Lie algebroids and connects it with classical equations, providing a new approach for optimal control.

## Key findings

- Derived dynamical equations for second-order constrained systems on Lie algebroids
- Identified a symplectic Lie subalgebroid where solutions are unique
- Applied the formalism to optimal control problems in mechanics

## Abstract

The aim of this work is to study, from an intrinsic and geometric point of view, second-order constrained variational problems on Lie algebroids, that is, optimization problems defined by a cost functional which depends on higher-order derivatives of admissible curves on a Lie algebroid. Extending the classical Skinner and Rusk formalism for the mechanics in the context of Lie algebroids, for second-order constrained mechanical systems, we derive the corresponding dynamical equations. We find a symplectic Lie subalgebroid where, under some mild regularity conditions, the second-order constrained variational problem, seen as a presymplectic Hamiltonian system, has a unique solution. We study the relationship of this formalism with the second-order constrained Euler-Poincar\'e and Lagrange-Poincar\'e equations, among others. Our study is applied to the optimal control of mechanical systems.