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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, (2000), 1273.
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.
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.
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.
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.
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.
doi: 10.1023/A:1007654605901. |
[7] |
V. G. Boltyanskii, "Optimal Control of Discrete Systems,", John Wiley, (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.
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.
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.
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.
doi: 10.1007/BF02429858. |
[12] |
P. E. Crouch, N. Nordkvist and A. K. Sanyal, Optimal control and geodesics on matrix Lie groups,, in, (2010). Google Scholar |
[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.
doi: 10.1007/s10444-004-4093-5. |
[14] |
Y. N. Federov and V. V. Kozlov, Various aspects of n-dimensional rigid body dynamics,, in, 168 (1995), 141.
|
[15] |
F. Gay-Balmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics,, Journal of Geometric Mechanics, 3 (2011), 41.
doi: 10.3934/jgm.2011.3.41. |
[16] |
E. Hairer, C. Lubich and G. Wanner., "Geometric Numerical Integration,", Springer Verlag, (2002).
|
[17] |
D. D. Holm, Riemannian optimal control formulation of incompressible ideal fluid flow,, preprint., (). Google Scholar |
[18] |
V. Jurdjevic, "Geometric Control Theory,", Cambridge University Press, (1997).
|
[19] |
M. Kobilarov and J. E. Marsden, Discrete geometric optimal control on Lie groups,, IEEE Transactions on Robotics, 27 (2011), 641.
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, (2005), 962. Google Scholar |
[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.
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.
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.
|
[24] |
J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincare and Lie-Poisson equations,, Nonlinearity, 12 (1999), 1647.
doi: 10.1088/0951-7715/12/6/314. |
[25] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", $2^{nd}$ edition, 17 (1999).
|
[26] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 357.
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. Google Scholar |
[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.
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, (2010), 5414.
doi: 10.1109/CDC.2010.5717622. |
[30] |
J. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Birkhäuser Verlag, (2004).
|
[31] |
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] |
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.
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.
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.
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.
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.
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.
doi: 10.1023/A:1007654605901. |
[7] |
V. G. Boltyanskii, "Optimal Control of Discrete Systems,", John Wiley, (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.
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.
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.
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.
doi: 10.1007/BF02429858. |
[12] |
P. E. Crouch, N. Nordkvist and A. K. Sanyal, Optimal control and geodesics on matrix Lie groups,, in, (2010). Google Scholar |
[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.
doi: 10.1007/s10444-004-4093-5. |
[14] |
Y. N. Federov and V. V. Kozlov, Various aspects of n-dimensional rigid body dynamics,, in, 168 (1995), 141.
|
[15] |
F. Gay-Balmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics,, Journal of Geometric Mechanics, 3 (2011), 41.
doi: 10.3934/jgm.2011.3.41. |
[16] |
E. Hairer, C. Lubich and G. Wanner., "Geometric Numerical Integration,", Springer Verlag, (2002).
|
[17] |
D. D. Holm, Riemannian optimal control formulation of incompressible ideal fluid flow,, preprint., (). Google Scholar |
[18] |
V. Jurdjevic, "Geometric Control Theory,", Cambridge University Press, (1997).
|
[19] |
M. Kobilarov and J. E. Marsden, Discrete geometric optimal control on Lie groups,, IEEE Transactions on Robotics, 27 (2011), 641.
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, (2005), 962. Google Scholar |
[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.
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.
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.
|
[24] |
J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincare and Lie-Poisson equations,, Nonlinearity, 12 (1999), 1647.
doi: 10.1088/0951-7715/12/6/314. |
[25] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", $2^{nd}$ edition, 17 (1999).
|
[26] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 357.
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. Google Scholar |
[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.
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, (2010), 5414.
doi: 10.1109/CDC.2010.5717622. |
[30] |
J. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Birkhäuser Verlag, (2004).
|
[31] |
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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