Embedded geodesic problems and optimal control for matrix Lie groups

Anthony M. Bloch; Peter E. Crouch; Nikolaj Nordkvist; Amit K. Sanyal

doi:10.3934/jgm.2011.3.197

Article Contents

2011, Volume 3, Issue 2: 197-223. Doi: 10.3934/jgm.2011.3.197

This issue Previous Article Lyapunov constraints and global asymptotic stabilization Next Article Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover

Embedded geodesic problems and optimal control for matrix Lie groups

1.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
2.
Department of Electrical Engineering, University of Hawai‘i at Mānoa, Honolulu, HI 96822
3.
Department of Mechanical Engineering, University of Hawai‘i at Mānoa, Honolulu, HI 96822
4.
Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003

Received: July 2009

Revised: June 2011

Published: June 2011

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Abstract

This paper is devoted to a detailed analysis of the geodesic problem on matrix Lie groups, with left invariant metric, by examining representations of embeddings of geodesic flows in suitable vector spaces. We show how these representations generate extremals for optimal control problems. In particular we discuss in detail the symmetric representation of the so-called $n$-dimensional rigid body equation and its relation to the more classical Euler description. We detail invariant manifolds of these flows on which we are able to define a strict notion of equivalence between representations, and identify naturally induced symplectic structures.

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Mathematics Subject Classification: Primary: 37J05, 49K15, 53C22, 70H30; Secondary: 34G20.

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