Unified formalism for higher-order variational problems and its applications in optimal control

TL;DR

This paper develops a unified geometric framework for higher-order variational problems with constraints, extending classical methods to derive equations of motion and optimal control for complex mechanical systems.

Contribution

It introduces an intrinsic Skinner-Rusk based formalism for higher-order constrained variational problems and applies it to optimal control of underactuated systems on principal bundles.

Findings

Established a symplectic framework for higher-order variational problems.

Derived equations of motion for constrained systems.

Applied the formalism to optimal control of underactuated mechanical systems.

Abstract

In this paper we consider an intrinsic point of view to describe the equations of motion for higher-order variational problems with constraints on higher-order trivial principal bundles. Our techniques are an adaptation of the classical Skinner-Rusk approach for the case of Lagrangian dynamics with higher-order constraints. We study a regular case where it is possible to establish a symplectic framework and, as a consequence, to obtain a unique vector field determining the dynamics. As an interesting application we deduce the equations of motion for optimal control of underactuated mechanical systems defined on principal bundles.