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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 
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), 12731279, 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), 13091341. doi: 10.1088/09517715/15/4/316. 
[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), 469500. doi: 10.1007/s1020800890251. 
[7] 
A. M. Bloch, P. E. Crouch and A. K. Sanyal, A variational problem on Stiefel manifolds, Nonlinearity, 19 (2006), 22472276. doi: 10.1088/09517715/19/10/002. 
[8] 
Y. N. Federov and V. V. Kozlov, Various aspects of ndimensional rigid body dynamics, in "Dynamical Systems in Classical Mechanics," American Mathematical Society Translations, 168, Amer. Math. Soc., Providence, RI, (1995), 141171. 
[9] 
F. GayBalmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics,, preprint., (). 
[10] 
I. M. Gelfand and S. V. Fomin, "Calculus of Variations," PrenticeHall, 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 ndimensional rigid body, Functional Analysis and Its Applications, 10 (1976), 328329. 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), 15911593. 
[16] 
T. S. Ratiu, The motion of the free ndimensional rigid body, Indiana University Mathematics Journal, 29 (1980), 609629. 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), 12731279, 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), 13091341. doi: 10.1088/09517715/15/4/316. 
[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), 469500. doi: 10.1007/s1020800890251. 
[7] 
A. M. Bloch, P. E. Crouch and A. K. Sanyal, A variational problem on Stiefel manifolds, Nonlinearity, 19 (2006), 22472276. doi: 10.1088/09517715/19/10/002. 
[8] 
Y. N. Federov and V. V. Kozlov, Various aspects of ndimensional rigid body dynamics, in "Dynamical Systems in Classical Mechanics," American Mathematical Society Translations, 168, Amer. Math. Soc., Providence, RI, (1995), 141171. 
[9] 
F. GayBalmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics,, preprint., (). 
[10] 
I. M. Gelfand and S. V. Fomin, "Calculus of Variations," PrenticeHall, 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 ndimensional rigid body, Functional Analysis and Its Applications, 10 (1976), 328329. 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), 15911593. 
[16] 
T. S. Ratiu, The motion of the free ndimensional rigid body, Indiana University Mathematics Journal, 29 (1980), 609629. doi: 10.1512/iumj.1980.29.29046. 
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