Optimal control and geodesics on quadratic matrix Lie groups

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

Research output: Contribution to journalArticle

21 Citations (Scopus)

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

Fingerprint

Lie groups
Matrix Groups
Rigid Body
Geodesic
Optimal Control
Unit matrix
Optimal Control Problem
Lie Algebra
Algebra
Invariant Metric
General Linear Group
Matrix Algebra
Semisimple
Numerical Algorithms
Discretization
Integrate
Subgroup
Symmetry
Generalise

Keywords

  • Generalized rigid body equations
  • Geodesics
  • Optimal control

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Mathematics(all)
  • Applied Mathematics
  • Computational Mathematics

Cite this

Optimal control and geodesics on quadratic matrix Lie groups. / Bloch, Anthony M.; Crouch, Peter E.; Marsden, Jerrold E.; Sanyal, Amit.

In: Foundations of Computational Mathematics, Vol. 8, No. 4, 01.08.2008, p. 469-500.

Research output: Contribution to journalArticle

Bloch, Anthony M. ; Crouch, Peter E. ; Marsden, Jerrold E. ; Sanyal, Amit. / Optimal control and geodesics on quadratic matrix Lie groups. In: Foundations of Computational Mathematics. 2008 ; Vol. 8, No. 4. pp. 469-500.
@article{550ca233837540b2a065f7c3a7a713b9,
title = "Optimal control and geodesics on quadratic matrix Lie groups",
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.",
keywords = "Generalized rigid body equations, Geodesics, Optimal control",
author = "Bloch, {Anthony M.} and Crouch, {Peter E.} and Marsden, {Jerrold E.} and Amit Sanyal",
year = "2008",
month = "8",
day = "1",
doi = "10.1007/s10208-008-9025-1",
language = "English (US)",
volume = "8",
pages = "469--500",
journal = "Foundations of Computational Mathematics",
issn = "1615-3375",
publisher = "Springer New York",
number = "4",

}

TY - JOUR

T1 - Optimal control and geodesics on quadratic matrix Lie groups

AU - Bloch, Anthony M.

AU - Crouch, Peter E.

AU - Marsden, Jerrold E.

AU - Sanyal, Amit

PY - 2008/8/1

Y1 - 2008/8/1

N2 - 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.

AB - 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.

KW - Generalized rigid body equations

KW - Geodesics

KW - Optimal control

UR - http://www.scopus.com/inward/record.url?scp=47249123754&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=47249123754&partnerID=8YFLogxK

U2 - 10.1007/s10208-008-9025-1

DO - 10.1007/s10208-008-9025-1

M3 - Article

AN - SCOPUS:47249123754

VL - 8

SP - 469

EP - 500

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

IS - 4

ER -