The variational discretization of the constrained higher-order Lagrange-Poincaré equations

Anthony Bloch; Leonardo Colombo; Fernando Jiménez

doi:10.3934/dcds.2019013

Article Contents

2019, Volume 39, Issue 1: 309-344. Doi: 10.3934/dcds.2019013

This issue Previous Article Cauchy problem for the Kuznetsov equation Next Article Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

1.
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan, 48109, USA
2.
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Calle Nicolás Cabrera 13-15, Campus UAM, Madrid, 28049, Spain
3.
Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United Kindom

Received: December 31, 2017

Revised: June 30, 2018

Published: December 2018

A. Bloch was supported by NSF grant DMS-1613819 and AFOSR grant FA 9550-18-0028. L. Colombo was supported by MINECO (Spain) grant MTM2016-76072-P. F. Jiménez was supported by the EPSRC project: "Fractional Variational Integration and Optimal Control"; ref: EP/P020402/1. We thank the reviewers for their valuable comments, that have helped to improve this work

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Abstract

In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincaré equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincaré equations.

Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.

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Mathematics Subject Classification: Primary: 37K05; Secondary: 37J15, 37N05, 49M25, 49S05, 49J15.

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