Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

High order variational integrators in the optimal control of mechanical systems

Pages: 4193 - 4223, Volume 35, Issue 9, September 2015      doi:10.3934/dcds.2015.35.4193

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Cédric M. Campos - IMUVA, Universidad de Valladolid, 47011 Valladolid, Spain (email)
Sina Ober-Blöbaum - Department of Mathematics, University of Paderborn, 33098 Paderborn, Germany (email)
Emmanuel Trélat - Sorbonne Universités, UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris, France (email)

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.

Keywords:  Optimal control, mechanical systems, geometric integration, variational integrator, high order, Runge-Kutta, direct methods, commutation property.
Mathematics Subject Classification:  Primary: 65P10; Secondary: 65L06, 65K10, 49Mxx.

Received: May 2014;      Revised: October 2014;      Available Online: April 2015.