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Optimal Control and Geodesics on Quadratic Matrix Lie Groups

Bloch, Anthony M. and Crouch, Peter E. and Marsden, Jerrold E. and Sanyal, Amit K. (2008) Optimal Control and Geodesics on Quadratic Matrix Lie Groups. Foundations of Computational Mathematics, 8 (4). pp. 469-500. ISSN 1615-3375.

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The purpose of this paper is to extend the symmetric representation of the rigid body equations from the group SO(n) to other groups. These groups are matrix subgroups of the general linear group that are defined by a quadratic matrix identity. Their corresponding Lie algebras include several classical semisimple matrix Lie algebras. The approach is to start with an optimal control problem on these groups that generates geodesics for a left-invariant metric. Earlier work by Bloch, Crouch, Marsden, and Ratiu defines the symmetric representation of the rigid body equations, which is obtained by solving the same optimal control problem in the particular case of the Lie group SO(n). This paper generalizes this symmetric representation to a wider class of matrix groups satisfying a certain quadratic matrix identity. We consider the relationship between this symmetric representation of the generalized rigid body equations and the generalized rigid body equations themselves. A discretization of this symmetric representation is constructed making use of the symmetry, which in turn give rise to numerical algorithms to integrate the generalized rigid body equations for the given class of matrix Lie groups.

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Additional Information:© SFoCM 2008. Received: 17 January 2007. Revised: 5 December 2007. Accepted: 28 January 2008. Published online: 28 February 2008. Published online: 28 February 2008. The research reported in this paper was partially supported by the National Science Foundation. We would also like thank the referee whose suggestions greatly improved the exposition of this paper. Communicated by Hans Munthe-Kaas
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Subject Keywords:Geodesics · Optimal control · Generalized rigid body equations
Classification Code:MSC: (2000) 34H05 · 70E40 · 49K15
Record Number:CaltechAUTHORS:20100715-090459941
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19063
Deposited By: Ruth Sustaita
Deposited On:15 Jul 2010 19:04
Last Modified:26 Dec 2012 12:14

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