High order variational integrators in the optimal control of mechanical systems

TL;DR

This paper develops high order variational integrators for mechanical systems, demonstrating their effectiveness in optimal control problems by ensuring convergence and the commutation of dualization and discretization.

Contribution

It introduces two novel high order variational integrators with enhanced approximation capabilities for mechanical systems in optimal control.

Findings

Proved convergence of state and control variables.

Showed dualization and discretization commute for these integrators.

Validated effectiveness in optimal control applications.

Abstract

In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimensional optimal control problems posed with mechanical systems. The convergence of state and control variables of the approximated problem is shown. Furthermore, by analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that, for these particular integrators, dualization and discretization commute.