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

Embedded geodesic problems and optimal control for matrix Lie groups

Anthony M. Bloch 1, , Peter E. Crouch 2, , Nikolaj Nordkvist 3, and  Amit K. Sanyal 4,

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.
Keywords: generalized rigid body mechanics., Optimal control, geodesics.
Mathematics Subject Classification: Primary: 37J05, 49K15, 53C22, 70H30; Secondary: 34G2.
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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show all references

References:
[1]

2nd edition, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1978.  Google Scholar

[2]

2nd edition, Graduate Texts in Mathematics, 60, Springer Verlag, New York, 1989.  Google Scholar

[3]

number 24 in "Interdisciplinary Texts in Mathematics," Springer Verlag, New York, 2003.  Google Scholar

[4]

In "Proc. IEEE Conf. on Decision and Control," Sydney, Australia, (December 2000), 1273-1279, arXiv:nlin/0103042. Google Scholar

[5]

Nonlinearity, 15 (2002), 1309-1341. doi: 10.1088/0951-7715/15/4/316.  Google Scholar

[6]

Foundations of Computational Mathematics, 8 (2008), 469-500. doi: 10.1007/s10208-008-9025-1.  Google Scholar

[7]

Nonlinearity, 19 (2006), 2247-2276. doi: 10.1088/0951-7715/19/10/002.  Google Scholar

[8]

in "Dynamical Systems in Classical Mechanics," American Mathematical Society Translations, 168, Amer. Math. Soc., Providence, RI, (1995), 141-171.  Google Scholar

[9]

F. Gay-Balmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics,, preprint., ().   Google Scholar

[10]

Prentice-Hall, Inc., Englewood Cliffs, NJ, (reprinted by Dover, 2000), 1963.  Google Scholar

[11]

D. D. Holm, Riemannian optimal control formulation of incompressible ideal fluid flow,, preprint., ().   Google Scholar

[12]

Dover Publications, New York, 2004. Google Scholar

[13]

Functional Analysis and Its Applications, 10 (1976), 328-329. doi: 10.1007/BF01076037.  Google Scholar

[14]

2nd edition, Texts in Applied Mathematics, 17, Springer Verlag, New York, 1999.  Google Scholar

[15]

Sov. Math. Dokl., 17 (1976), 1591-1593. Google Scholar

[16]

Indiana University Mathematics Journal, 29 (1980), 609-629. doi: 10.1512/iumj.1980.29.29046.  Google Scholar

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