The geometry of optimal control problems on some six dimensional lie groups

James Biggs, William Holderbaum

Research output: Contribution to conferencePaper

Abstract

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E3 , the spheres S3 and the hyperboloids H3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.
LanguageEnglish
Pages1427-1432
Number of pages6
Publication statusPublished - 2006
Event44th IEEE Conference on Decision and Control/European Control Conference - Seville, Spain
Duration: 12 Dec 2005 → …

Conference

Conference44th IEEE Conference on Decision and Control/European Control Conference
CitySeville, Spain
Period12/12/05 → …

Fingerprint

Lie groups
optimal control
geometry
bundles
Geometry
curvature
control system
Euclidean geometry
elliptic functions
Control systems

Keywords

  • geometry
  • optimal control problems
  • six dimensional lie groups
  • control systems

Cite this

Biggs, J., & Holderbaum, W. (2006). The geometry of optimal control problems on some six dimensional lie groups. 1427-1432. Paper presented at 44th IEEE Conference on Decision and Control/European Control Conference, Seville, Spain, .
Biggs, James ; Holderbaum, William. / The geometry of optimal control problems on some six dimensional lie groups. Paper presented at 44th IEEE Conference on Decision and Control/European Control Conference, Seville, Spain, .6 p.
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Biggs, J & Holderbaum, W 2006, 'The geometry of optimal control problems on some six dimensional lie groups' Paper presented at 44th IEEE Conference on Decision and Control/European Control Conference, Seville, Spain, 12/12/05, pp. 1427-1432.

The geometry of optimal control problems on some six dimensional lie groups. / Biggs, James; Holderbaum, William.

2006. 1427-1432 Paper presented at 44th IEEE Conference on Decision and Control/European Control Conference, Seville, Spain, .

Research output: Contribution to conferencePaper

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AU - Holderbaum, William

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N2 - This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E3 , the spheres S3 and the hyperboloids H3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.

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Biggs J, Holderbaum W. The geometry of optimal control problems on some six dimensional lie groups. 2006. Paper presented at 44th IEEE Conference on Decision and Control/European Control Conference, Seville, Spain, .