Embedded geodesic problems and optimal control for matrix Lie groups

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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

Abstract
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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

References:

[1]	R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2^nd edition, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1978.
[2]	V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," 2^nd edition, Graduate Texts in Mathematics, 60, Springer Verlag, New York, 1989.
[3]	A. M. Bloch, J. Ballieul, P. E. Crouch and J. E. Marsden, "Nonholonomic Mechanics and Control," number 24 in "Interdisciplinary Texts in Mathematics," Springer Verlag, New York, 2003.
[4]	A. M. Bloch, P. E. Crouch, D. D. Holm and J. E. Marsden, An optimal control formulation for inviscid incompressible ideal fluid flow, In "Proc. IEEE Conf. on Decision and Control," Sydney, Australia, (December 2000), 1273-1279, arXiv:nlin/0103042.
[5]	A. M. Bloch, P. E. Crouch, J. E. Marsden and T. S. Ratiu, The symmetric representation of the rigid body equations and their discretization, Nonlinearity, 15 (2002), 1309-1341. doi: 10.1088/0951-7715/15/4/316.
[6]	A. M. Bloch, P. E. Crouch, J. E. Marsden and A. K. Sanyal, Optimal control and geodesics on quadratic matrix Lie groups, Foundations of Computational Mathematics, 8 (2008), 469-500. doi: 10.1007/s10208-008-9025-1.
[7]	A. M. Bloch, P. E. Crouch and A. K. Sanyal, A variational problem on Stiefel manifolds, Nonlinearity, 19 (2006), 2247-2276. doi: 10.1088/0951-7715/19/10/002.
[8]	Y. N. Federov and V. V. Kozlov, Various aspects of n-dimensional rigid body dynamics, in "Dynamical Systems in Classical Mechanics," American Mathematical Society Translations, 168, Amer. Math. Soc., Providence, RI, (1995), 141-171.
[9]	F. Gay-Balmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics,, preprint., ().
[10]	I. M. Gelfand and S. V. Fomin, "Calculus of Variations," Prentice-Hall, Inc., Englewood Cliffs, NJ, (reprinted by Dover, 2000), 1963.
[11]	D. D. Holm, Riemannian optimal control formulation of incompressible ideal fluid flow,, preprint., ().
[12]	D. E. Kirk, "Optimal Control Theory: An Introduction," Dover Publications, New York, 2004.
[13]	S. V. Manakov, Note on the integration of Euler's equations of the dynamics of an n-dimensional rigid body, Functional Analysis and Its Applications, 10 (1976), 328-329. doi: 10.1007/BF01076037.
[14]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," 2^nd edition, Texts in Applied Mathematics, 17, Springer Verlag, New York, 1999.
[15]	A. S. Mischenko and A. T. Fomenko, On the integration of the Euler equations on semisimple Lie algebras, Sov. Math. Dokl., 17 (1976), 1591-1593.
[16]	T. S. Ratiu, The motion of the free n-dimensional rigid body, Indiana University Mathematics Journal, 29 (1980), 609-629. doi: 10.1512/iumj.1980.29.29046.

show all references

References:

[1]	R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2^nd edition, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1978.
[2]	V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," 2^nd edition, Graduate Texts in Mathematics, 60, Springer Verlag, New York, 1989.
[3]	A. M. Bloch, J. Ballieul, P. E. Crouch and J. E. Marsden, "Nonholonomic Mechanics and Control," number 24 in "Interdisciplinary Texts in Mathematics," Springer Verlag, New York, 2003.
[4]	A. M. Bloch, P. E. Crouch, D. D. Holm and J. E. Marsden, An optimal control formulation for inviscid incompressible ideal fluid flow, In "Proc. IEEE Conf. on Decision and Control," Sydney, Australia, (December 2000), 1273-1279, arXiv:nlin/0103042.
[5]	A. M. Bloch, P. E. Crouch, J. E. Marsden and T. S. Ratiu, The symmetric representation of the rigid body equations and their discretization, Nonlinearity, 15 (2002), 1309-1341. doi: 10.1088/0951-7715/15/4/316.
[6]	A. M. Bloch, P. E. Crouch, J. E. Marsden and A. K. Sanyal, Optimal control and geodesics on quadratic matrix Lie groups, Foundations of Computational Mathematics, 8 (2008), 469-500. doi: 10.1007/s10208-008-9025-1.
[7]	A. M. Bloch, P. E. Crouch and A. K. Sanyal, A variational problem on Stiefel manifolds, Nonlinearity, 19 (2006), 2247-2276. doi: 10.1088/0951-7715/19/10/002.
[8]	Y. N. Federov and V. V. Kozlov, Various aspects of n-dimensional rigid body dynamics, in "Dynamical Systems in Classical Mechanics," American Mathematical Society Translations, 168, Amer. Math. Soc., Providence, RI, (1995), 141-171.
[9]	F. Gay-Balmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics,, preprint., ().
[10]	I. M. Gelfand and S. V. Fomin, "Calculus of Variations," Prentice-Hall, Inc., Englewood Cliffs, NJ, (reprinted by Dover, 2000), 1963.
[11]	D. D. Holm, Riemannian optimal control formulation of incompressible ideal fluid flow,, preprint., ().
[12]	D. E. Kirk, "Optimal Control Theory: An Introduction," Dover Publications, New York, 2004.
[13]	S. V. Manakov, Note on the integration of Euler's equations of the dynamics of an n-dimensional rigid body, Functional Analysis and Its Applications, 10 (1976), 328-329. doi: 10.1007/BF01076037.
[14]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," 2^nd edition, Texts in Applied Mathematics, 17, Springer Verlag, New York, 1999.
[15]	A. S. Mischenko and A. T. Fomenko, On the integration of the Euler equations on semisimple Lie algebras, Sov. Math. Dokl., 17 (1976), 1591-1593.
[16]	T. S. Ratiu, The motion of the free n-dimensional rigid body, Indiana University Mathematics Journal, 29 (1980), 609-629. doi: 10.1512/iumj.1980.29.29046.

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