Optimal control and geodesics on quadratic matrix Lie groups

Anthony M. Bloch, Peter E. Crouch, Jerrold E. Marsden, Amit K. Sanyal

Research output: Contribution to journalArticle

22 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)469-500
Number of pages32
JournalFoundations of Computational Mathematics
Volume8
Issue number4
DOIs
StatePublished - Aug 1 2008
Externally publishedYes

Keywords

  • Generalized rigid body equations
  • Geodesics
  • Optimal control

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics

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