Embedded geodesic problems and optimal control for matrix Lie groups

Pages: 197 - 223, Volume 3, Issue 2, June 2011

doi:10.3934/jgm.2011.3.197       Abstract        References        Full Text (490.9K)       Related Articles

Anthony M. Bloch - Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States (email)
Peter E. Crouch - Department of Electrical Engineering, University of Hawai‘i at Mānoa, Honolulu, HI 96822, United States (email)
Nikolaj Nordkvist - Department of Mechanical Engineering, University of Hawai‘i at Mānoa, Honolulu, HI 96822, United States (email)
Amit K. Sanyal - Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, United States (email)

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.

Keywords:  Optimal control, geodesics, generalized rigid body mechanics.
Mathematics Subject Classification:  Primary: 37J05, 49K15, 53C22, 70H30; Secondary: 34G20.

Received: July 2009;      Revised: June 2011;      Published: July 2011.

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