High order variational integrators in the optimal control of mechanical systems

Cédric M. Campos; Sina Ober-Blöbaum; Emmanuel Trélat

doi:10.3934/dcds.2015.35.4193

Article Contents

2015, Volume 35, Issue 9: 4193-4223. Doi: 10.3934/dcds.2015.35.4193

This issue Previous Article A model problem for Mean Field Games on networks Next Article Global propagation of singularities for time dependent Hamilton-Jacobi equations

High order variational integrators in the optimal control of mechanical systems

1.
IMUVA, Universidad de Valladolid, 47011 Valladolid
2.
Department of Mathematics, University of Paderborn, 33098 Paderborn
3.
Sorbonne Universités, UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris

Received: May 2014

Revised: October 2014

Published: May 2017

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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.

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Mathematics Subject Classification: Primary: 65P10; Secondary: 65L06, 65K10, 49Mxx.

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