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Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems
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 |
3. | Mathematics and Sciences Division, Leeward Community College, Pearl City, HI 96782, United States |
References:
[1] |
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, (Dec. 2000), 1273-1279.
doi: 10.1109/CDC.2000.912030. |
[2] |
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. |
[3] |
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. |
[4] |
A. M. Bloch, P. E. Crouch, N. Nordkvist and A. K. Sanyal, Embedded geodesic problems and optimal control for matrix Lie groups, Journal of Geometric Mechanics, 3 (2011), 197-223.
doi: 10.3934/jgm.2011.3.197. |
[5] |
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. |
[6] |
A. I Bobenko and Y. B. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations, and semidirect products, Letters in Mathematical Physics, 49 (1999), 79-93.
doi: 10.1023/A:1007654605901. |
[7] |
V. G. Boltyanskii, "Optimal Control of Discrete Systems," John Wiley, New York, 1978. |
[8] |
N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups part I: Introduction and structure-preserving properties, Foundations of Computational Mathematics, 9 (2008), 197-219.
doi: 10.1007/s10208-008-9030-4. |
[9] |
J. R. Cardoso and F. Silva Leite, The Moser-Veselov equation, Linear Algebra and its Applications, 360 (2003), 237-248.
doi: 10.1016/S0024-3795(02)00450-0. |
[10] |
C. J. Cotter and D. D. Holm, Continuous and discrete Clebsch variational principles, Foundations of Computational Mathematics, 9 (2009), 221-242.
doi: 10.1007/s10208-007-9022-9. |
[11] |
P. E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on manifolds, Journal of Nonlinear Science, 3 (1993), 1-33.
doi: 10.1007/BF02429858. |
[12] |
P. E. Crouch, N. Nordkvist and A. K. Sanyal, Optimal control and geodesics on matrix Lie groups, in "Proc. 9th Portuguese Conference on Automatic Control - CONTROLO'2010,'' Coimbra, Portugal, (Sep. 2010). |
[13] |
M. de León, D. Martín de Diego and A. Santamaría-Merino, Discrete variational integrators and optimal control theory, Advances in Computational Mathematics, 26 (2007), 251-268.
doi: 10.1007/s10444-004-4093-5. |
[14] |
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. |
[15] |
F. Gay-Balmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics, Journal of Geometric Mechanics, 3 (2011), 41-79.
doi: 10.3934/jgm.2011.3.41. |
[16] |
E. Hairer, C. Lubich and G. Wanner., "Geometric Numerical Integration," Springer Verlag, New York, 2002. |
[17] |
D. D. Holm, Riemannian optimal control formulation of incompressible ideal fluid flow, preprint. |
[18] |
V. Jurdjevic, "Geometric Control Theory," Cambridge University Press, Cambridge, UK, 1997. |
[19] |
M. Kobilarov and J. E. Marsden, Discrete geometric optimal control on Lie groups, IEEE Transactions on Robotics, 27 (2011), 641-655.
doi: 10.1007/s10208-011-9089-1. |
[20] |
T. Lee, M. Leok and N. H. McClamroch, A Lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum, in "Proc. IEEE Conf. on Control Applications," Toronto, Canada, (Aug. 2005), 962-967. |
[21] |
T. Lee, M. Leok and N. H. McClamroch, Lie group variational integrators for the full body problem, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2907-2924.
doi: 10.1016/j.cma.2007.01.017. |
[22] |
T. Lee, M. Leok and N. H. McClamroch, Lagrangian mechanics and variational integrators on two-spheres, International Journal for Numerical Methods in Engineering, 79 (2009), 1147-1174.
doi: 10.1002/nme.2603. |
[23] |
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. |
[24] |
J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincare and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.
doi: 10.1088/0951-7715/12/6/314. |
[25] |
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. |
[26] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[27] |
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. |
[28] |
J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Communications in Mathematical Physics, 139 (1991), 217-243.
doi: 10.1007/BF02352494. |
[29] |
N. Nordkvist and A. K. Sanyal, A Lie group variational integrator for rigid body motion in $SE(3)$ with applications to underwater vehicles, in "IEEE Conf. on Decision and Control," Atlanta, GA, (Dec. 2010), 5414-5419.
doi: 10.1109/CDC.2010.5717622. |
[30] |
J. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Birkhäuser Verlag, Boston, 2004. |
[31] |
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] |
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, (Dec. 2000), 1273-1279.
doi: 10.1109/CDC.2000.912030. |
[2] |
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. |
[3] |
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. |
[4] |
A. M. Bloch, P. E. Crouch, N. Nordkvist and A. K. Sanyal, Embedded geodesic problems and optimal control for matrix Lie groups, Journal of Geometric Mechanics, 3 (2011), 197-223.
doi: 10.3934/jgm.2011.3.197. |
[5] |
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. |
[6] |
A. I Bobenko and Y. B. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations, and semidirect products, Letters in Mathematical Physics, 49 (1999), 79-93.
doi: 10.1023/A:1007654605901. |
[7] |
V. G. Boltyanskii, "Optimal Control of Discrete Systems," John Wiley, New York, 1978. |
[8] |
N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups part I: Introduction and structure-preserving properties, Foundations of Computational Mathematics, 9 (2008), 197-219.
doi: 10.1007/s10208-008-9030-4. |
[9] |
J. R. Cardoso and F. Silva Leite, The Moser-Veselov equation, Linear Algebra and its Applications, 360 (2003), 237-248.
doi: 10.1016/S0024-3795(02)00450-0. |
[10] |
C. J. Cotter and D. D. Holm, Continuous and discrete Clebsch variational principles, Foundations of Computational Mathematics, 9 (2009), 221-242.
doi: 10.1007/s10208-007-9022-9. |
[11] |
P. E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on manifolds, Journal of Nonlinear Science, 3 (1993), 1-33.
doi: 10.1007/BF02429858. |
[12] |
P. E. Crouch, N. Nordkvist and A. K. Sanyal, Optimal control and geodesics on matrix Lie groups, in "Proc. 9th Portuguese Conference on Automatic Control - CONTROLO'2010,'' Coimbra, Portugal, (Sep. 2010). |
[13] |
M. de León, D. Martín de Diego and A. Santamaría-Merino, Discrete variational integrators and optimal control theory, Advances in Computational Mathematics, 26 (2007), 251-268.
doi: 10.1007/s10444-004-4093-5. |
[14] |
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. |
[15] |
F. Gay-Balmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics, Journal of Geometric Mechanics, 3 (2011), 41-79.
doi: 10.3934/jgm.2011.3.41. |
[16] |
E. Hairer, C. Lubich and G. Wanner., "Geometric Numerical Integration," Springer Verlag, New York, 2002. |
[17] |
D. D. Holm, Riemannian optimal control formulation of incompressible ideal fluid flow, preprint. |
[18] |
V. Jurdjevic, "Geometric Control Theory," Cambridge University Press, Cambridge, UK, 1997. |
[19] |
M. Kobilarov and J. E. Marsden, Discrete geometric optimal control on Lie groups, IEEE Transactions on Robotics, 27 (2011), 641-655.
doi: 10.1007/s10208-011-9089-1. |
[20] |
T. Lee, M. Leok and N. H. McClamroch, A Lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum, in "Proc. IEEE Conf. on Control Applications," Toronto, Canada, (Aug. 2005), 962-967. |
[21] |
T. Lee, M. Leok and N. H. McClamroch, Lie group variational integrators for the full body problem, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2907-2924.
doi: 10.1016/j.cma.2007.01.017. |
[22] |
T. Lee, M. Leok and N. H. McClamroch, Lagrangian mechanics and variational integrators on two-spheres, International Journal for Numerical Methods in Engineering, 79 (2009), 1147-1174.
doi: 10.1002/nme.2603. |
[23] |
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. |
[24] |
J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincare and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.
doi: 10.1088/0951-7715/12/6/314. |
[25] |
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. |
[26] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[27] |
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. |
[28] |
J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Communications in Mathematical Physics, 139 (1991), 217-243.
doi: 10.1007/BF02352494. |
[29] |
N. Nordkvist and A. K. Sanyal, A Lie group variational integrator for rigid body motion in $SE(3)$ with applications to underwater vehicles, in "IEEE Conf. on Decision and Control," Atlanta, GA, (Dec. 2010), 5414-5419.
doi: 10.1109/CDC.2010.5717622. |
[30] |
J. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Birkhäuser Verlag, Boston, 2004. |
[31] |
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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