Higher-order variational problems on lie groups and optimal control applications

Leonardo Colombo; David Martín de Diego

doi:10.3934/jgm.2014.6.451

Article Contents

2014, Volume 6, Issue 4: 451-478. Doi: 10.3934/jgm.2014.6.451

This issue Previous Article The Hamilton-Jacobi equation, integrability, and nonholonomic systems Next Article Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems

Higher-order variational problems on lie groups and optimal control applications

Leonardo Colombo^1, and
David Martín de Diego^2,

1.
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109
2.
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Campus de Cantoblanco, UAM C/ Nicolas Cabrera, 15 - 28049 Madrid

Received: May 2014

Revised: August 2014

Published: December 2014

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Abstract

In this paper, we describe a geometric setting for higher-order La- grangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, we deduce an intrinsic framework for this type of dynamical systems. Interesting applications as, for instance, a geometric derivation of the higher-order Euler-Poincaré equations, optimal control of underactuated control systems whose configuration space is a Lie group are shown, among others, along the paper.

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Mathematics Subject Classification: Primary: 70H50, 70H45, 79J15; Secondary: 70G65, 70Hxx.

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