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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,", 2nd edition, (1978).
|
[2] |
V. I. Arnol'd, "Mathematical Methods of Classical Mechanics,", 2nd edition, 60 (1989).
|
[3] |
A. M. Bloch, J. Ballieul, P. E. Crouch and J. E. Marsden, "Nonholonomic Mechanics and Control,", number 24 in, (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, (2000), 1273. 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.
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.
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.
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, 168 (1995), 141.
|
[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, (2000).
|
[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, (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.
doi: 10.1007/BF01076037. |
[14] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", 2nd edition, 17 (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. Google Scholar |
[16] |
T. S. Ratiu, The motion of the free n-dimensional rigid body,, Indiana University Mathematics Journal, 29 (1980), 609.
doi: 10.1512/iumj.1980.29.29046. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics,", 2nd edition, (1978).
|
[2] |
V. I. Arnol'd, "Mathematical Methods of Classical Mechanics,", 2nd edition, 60 (1989).
|
[3] |
A. M. Bloch, J. Ballieul, P. E. Crouch and J. E. Marsden, "Nonholonomic Mechanics and Control,", number 24 in, (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, (2000), 1273. 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.
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.
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.
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, 168 (1995), 141.
|
[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, (2000).
|
[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, (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.
doi: 10.1007/BF01076037. |
[14] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", 2nd edition, 17 (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. Google Scholar |
[16] |
T. S. Ratiu, The motion of the free n-dimensional rigid body,, Indiana University Mathematics Journal, 29 (1980), 609.
doi: 10.1512/iumj.1980.29.29046. |
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