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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
References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1978.  Google Scholar

[2]

V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," 2nd edition, Graduate Texts in Mathematics, 60, Springer Verlag, New York, 1989.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

I. M. Gelfand and S. V. Fomin, "Calculus of Variations," 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]

D. E. Kirk, "Optimal Control Theory: An Introduction," Dover Publications, New York, 2004. Google Scholar

[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.  Google Scholar

[14]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," 2nd edition, Texts in Applied Mathematics, 17, Springer Verlag, New York, 1999.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1978.  Google Scholar

[2]

V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," 2nd edition, Graduate Texts in Mathematics, 60, Springer Verlag, New York, 1989.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

I. M. Gelfand and S. V. Fomin, "Calculus of Variations," 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]

D. E. Kirk, "Optimal Control Theory: An Introduction," Dover Publications, New York, 2004. Google Scholar

[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.  Google Scholar

[14]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," 2nd edition, Texts in Applied Mathematics, 17, Springer Verlag, New York, 1999.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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