Variational integrators for mechanical control systems with symmetries

Leonardo Colombo; Fernando Jiménez; David Martín de Diego

doi:10.3934/jcd.2015003

Article Contents

2015, Volume 2, Issue 2: 193-225. Doi: 10.3934/jcd.2015003

This issue Previous Article Compressed sensing and dynamic mode decomposition Next Article Computing continuous and piecewise affine lyapunov functions for nonlinear systems

Variational integrators for mechanical control systems with symmetries

1.
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109
2.
Zentrum Mathematik der Technische Universität München, D-85747 Garching bei Munchen
3.
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Campus de Cantoblanco, UAM C/ Nicolas Cabrera, 15 - 28049 Madrid

Received: June 2013

Revised: May 2015

Published: June 2015

Abstract / Introduction Related Papers Cited by

Abstract

Optimal control problems for underactuated mechanical systems can be seen as a higher-order variational problem subject to higher-order constraints (that is, when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces). In this paper we discuss the variational formalism for the class of underactuated mechanical control systems when the configuration space is a trivial principal bundle and the construction of variational integrators for such mechanical control systems.
An interesting family of geometric integrators can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators, being one of their main properties the preservation of geometric features as the symplecticity, momentum preservation and good behavior of the energy. We construct variational integrators for higher-order mechanical systems on trivial principal bundles and their extension for higher-order constrained systems, paying particular attention to the case of underactuated mechanical systems.

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Mathematics Subject Classification: Primary: 70G45; Secondary: 70Hxx, 17B66, 22A22.

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