Integrability by quadratures in optimal control of a unicycle on hyperbolic plane

Yasir Awais Butt; Aamer Iqbal Bhatti; Yuri L. Sachkov

doi:10.1109/ACC.2015.7171997

Integrability by quadratures in optimal control of a unicycle on hyperbolic plane

Yasir Awais Butt
, Aamer Iqbal Bhatti
, Yuri L. Sachkov

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

2 Scopus citations

Abstract

We consider the problem of integrability by quadratures of normal Hamiltonian system in sub-Riemannian problem on the groups of motions of hyperbolic plane or pseudo Euclidean plane which form the Lie group SH(2). The first step towards proof of integrability is to calculate the local representation of the Lie group SH(2) in canonical coordinates of second kind. Wei-Norman transformation is applied to obtain this local representation. The Wei-Norman representation shows that the left invariant control system defined on the Lie group SH(2) is equivalent to the motion of a unicycle on hyperbolic plane. Three integrals of motion satisfying the Liouville's integrability conditions are then calculated to prove that the normal Hamiltonian system is integrable by quadratures.

Original language	English
Title of host publication	ACC 2015 - 2015 American Control Conference
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	4251-4256
Number of pages	6
ISBN (Electronic)	9781479986842
DOIs	https://doi.org/10.1109/ACC.2015.7171997
State	Published - 28 Jul 2015

Publication series

Name	Proceedings of the American Control Conference
Volume	2015-July
ISSN (Print)	0743-1619

Bibliographical note

Publisher Copyright:
© 2015 American Automatic Control Council.

Keywords

Hyperbolic Plane
Integrability
Lie Group SH(2)
Optimal Control
Pontryagin Maximum Principle
Sub-Riemannian Geometry

ASJC Scopus subject areas

Electrical and Electronic Engineering

Access to Document

10.1109/ACC.2015.7171997

Cite this

Butt, Y. A., Bhatti, A. I., & Sachkov, Y. L. (2015). Integrability by quadratures in optimal control of a unicycle on hyperbolic plane. In ACC 2015 - 2015 American Control Conference (pp. 4251-4256). Article 7171997 (Proceedings of the American Control Conference; Vol. 2015-July). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ACC.2015.7171997

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title = "Integrability by quadratures in optimal control of a unicycle on hyperbolic plane",

abstract = "We consider the problem of integrability by quadratures of normal Hamiltonian system in sub-Riemannian problem on the groups of motions of hyperbolic plane or pseudo Euclidean plane which form the Lie group SH(2). The first step towards proof of integrability is to calculate the local representation of the Lie group SH(2) in canonical coordinates of second kind. Wei-Norman transformation is applied to obtain this local representation. The Wei-Norman representation shows that the left invariant control system defined on the Lie group SH(2) is equivalent to the motion of a unicycle on hyperbolic plane. Three integrals of motion satisfying the Liouville's integrability conditions are then calculated to prove that the normal Hamiltonian system is integrable by quadratures.",

keywords = "Hyperbolic Plane, Integrability, Lie Group SH(2), Optimal Control, Pontryagin Maximum Principle, Sub-Riemannian Geometry",

author = "Butt, \{Yasir Awais\} and Bhatti, \{Aamer Iqbal\} and Sachkov, \{Yuri L.\}",

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day = "28",

doi = "10.1109/ACC.2015.7171997",

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Integrability by quadratures in optimal control of a unicycle on hyperbolic plane. / Butt, Yasir Awais; Bhatti, Aamer Iqbal; Sachkov, Yuri L.
ACC 2015 - 2015 American Control Conference. Institute of Electrical and Electronics Engineers Inc., 2015. p. 4251-4256 7171997 (Proceedings of the American Control Conference; Vol. 2015-July).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Integrability by quadratures in optimal control of a unicycle on hyperbolic plane

AU - Butt, Yasir Awais

AU - Bhatti, Aamer Iqbal

AU - Sachkov, Yuri L.

PY - 2015/7/28

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N2 - We consider the problem of integrability by quadratures of normal Hamiltonian system in sub-Riemannian problem on the groups of motions of hyperbolic plane or pseudo Euclidean plane which form the Lie group SH(2). The first step towards proof of integrability is to calculate the local representation of the Lie group SH(2) in canonical coordinates of second kind. Wei-Norman transformation is applied to obtain this local representation. The Wei-Norman representation shows that the left invariant control system defined on the Lie group SH(2) is equivalent to the motion of a unicycle on hyperbolic plane. Three integrals of motion satisfying the Liouville's integrability conditions are then calculated to prove that the normal Hamiltonian system is integrable by quadratures.

AB - We consider the problem of integrability by quadratures of normal Hamiltonian system in sub-Riemannian problem on the groups of motions of hyperbolic plane or pseudo Euclidean plane which form the Lie group SH(2). The first step towards proof of integrability is to calculate the local representation of the Lie group SH(2) in canonical coordinates of second kind. Wei-Norman transformation is applied to obtain this local representation. The Wei-Norman representation shows that the left invariant control system defined on the Lie group SH(2) is equivalent to the motion of a unicycle on hyperbolic plane. Three integrals of motion satisfying the Liouville's integrability conditions are then calculated to prove that the normal Hamiltonian system is integrable by quadratures.

KW - Hyperbolic Plane

KW - Integrability

KW - Lie Group SH(2)

KW - Optimal Control

KW - Pontryagin Maximum Principle

KW - Sub-Riemannian Geometry

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T3 - Proceedings of the American Control Conference

SP - 4251

EP - 4256

BT - ACC 2015 - 2015 American Control Conference

PB - Institute of Electrical and Electronics Engineers Inc.

ER -

Integrability by quadratures in optimal control of a unicycle on hyperbolic plane

Abstract

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Bibliographical note

Keywords

ASJC Scopus subject areas

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