# American Institute of Mathematical Sciences

June  2011, 3(2): 197-223. doi: 10.3934/jgm.2011.3.197

## Embedded geodesic problems and optimal control for matrix Lie groups

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

Received  July 2009 Revised  June 2011 Published  July 2011

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.
Citation: Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist, Amit K. Sanyal. Embedded geodesic problems and optimal control for matrix Lie groups. Journal of Geometric Mechanics, 2011, 3 (2) : 197-223. doi: 10.3934/jgm.2011.3.197
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