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The Hamilton-Jacobi equation, integrability, and nonholonomic systems
Higher-order variational problems on lie groups and optimal control applications
1. | Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, United States |
2. | Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Campus de Cantoblanco, UAM C/ Nicolas Cabrera, 15 - 28049 Madrid, Spain |
References:
[1] |
R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition,, Springer-Verlag, (1988).
doi: 10.1007/978-1-4612-1029-0. |
[2] |
L. Abrunheiro, M. Camarinha, J. F. Cariñena, J. Clemente-Gallardo, E. Martínez and P. Santos, Some applications of quasi-velocities in optimal control,, Int. J. Geom. Methods Mod. Phys., 08 (2011), 835.
doi: 10.1142/S0219887811005427. |
[3] |
M. Barbero-Liñán, A.Echeverría Enríquez, D. Martín de Diego, M.C Muñoz-Lecanda and N. Román-Roy, Skinner-Rusk unified formalism for optimal control systems and applications,, J. Phys. A: Math Theor., 40 (2007), 12071.
doi: 10.1088/1751-8113/40/40/005. |
[4] |
L. Bates and R. Cushman, Global Aspect of Classical Integrable Systems,, Birkhäuser Verlag, (1997).
doi: 10.1007/978-3-0348-8891-2. |
[5] |
R. Benito, Roberto and D. Martín de Diego, Hidden symplecticity in Hamilton's principle algorithms,, Differential geometry and its applications, (2005), 411.
|
[6] |
A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, (2003).
doi: 10.1007/b97376. |
[7] |
A. M. Bloch, I. I. Hussein, M. Leok and A. K. Sanyal, Geometric Structure-Preserving Optimal Control of the Rigid Body,, Journal of Dynamical and Control Systems, 15 (2009), 307.
doi: 10.1007/s10883-009-9071-2. |
[8] |
C. Burnett, D. D. Holm and D. Meier, Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework,, Proc. R. Soc. A., 469 (2013).
doi: 10.1098/rspa.2013.0249. |
[9] |
F. Bullo and A. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems,, Texts in Applied Mathematics, (2005).
doi: 10.1007/978-1-4899-7276-7. |
[10] |
M. Crampin, W. Sarlet and F. Cantrijn, Higher order differential equations and higher order Lagrangian Mechanics,, Math. Proc. Camb. Phil. Soc., 99 (1986), 565.
doi: 10.1017/S0305004100064501. |
[11] |
L. Colombo, F. Jimenez and D. Martín de Diego, Discrete Second-Order Euler-Poincaré Equations. An application to optimal control,, International Journal of Geometric Methods in Modern Physics, 9 (2012).
doi: 10.1142/S0219887812500375. |
[12] |
L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometrical approach,, Journal Mathematical Physics, 51 (2010).
doi: 10.1063/1.3456158. |
[13] |
L. Colombo and D. Martín de Diego, Quasivelocities and Optimal Control of Underactuated Mechanical Systems,, Geometry and Physics: XVIII Fall Workshop on Geometry and Physics. AIP Conference Proceedings, 1260 (2011), 133.
|
[14] |
J. Cortés, M. de León, D. Martín de Diego and S. Martínez, Geometric description of vakonomic and nonholonomic dynamics,, SIAM J. Control Optim., 41 (2002), 1389.
doi: 10.1137/S036301290036817X. |
[15] |
P. Crouch and F. Silva-Leite, Geometry and the dynamic interpolation problem,, American Control Conference, (1991), 1131. Google Scholar |
[16] |
M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of nonlinear systems,Introductory theory and examples,, International Journal of Control, 61 (1995), 1327.
doi: 10.1080/00207179508921959. |
[17] |
F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems,, Communications in Mathematical Physics, 309 (2012), 413.
doi: 10.1007/s00220-011-1313-y. |
[18] |
F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X Vialard, Invariant higher-order variational problems II,, Journal of Nonlinear Science, 22 (2012), 553.
doi: 10.1007/s00332-012-9137-2. |
[19] |
F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Higher-Order Lagrange-Poincaré and Hamilton-Poincaré Reductions,, Bulletin of the Brazilian Math. Soc., 42 (2011), 579.
doi: 10.1007/s00574-011-0030-7. |
[20] |
M. J. Gotay and J. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence theorem,, Ann. Inst. Henri Poincaré, 30 (1979), 129.
|
[21] |
M. J. Gotay, J. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints,, J. Math. Phys., 19 (1978), 2388.
doi: 10.1063/1.523597. |
[22] |
D. D. Holm, Geometric mechanics. Part I and II,, Imperial College Press, (2008).
|
[23] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, he Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1.
doi: 10.1006/aima.1998.1721. |
[24] |
D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids,, Dyn. Syst., 23 (2008), 351.
doi: 10.1080/14689360802294220. |
[25] |
T. Lee, M. Leok and N. H. McClamroch, Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on $SO(3)$,, Journal of Dynamical and Control Systems, 14 (2008), 465.
doi: 10.1007/s10883-008-9047-7. |
[26] |
A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment,, Internat. J. Non-Linear Mech., 30 (1995), 793.
doi: 10.1016/0020-7462(95)00024-0. |
[27] |
M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (1985).
|
[28] |
J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry,, Second Edition, 17 (1999).
doi: 10.1007/978-0-387-21792-5. |
[29] |
D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D thesis, (2013). Google Scholar |
[30] |
T. Mestdag and M. Crampin, Anholonomic frames in constrained dynamics,, Dynamical Systems. An International Journal, 25 (2010), 159.
doi: 10.1080/14689360903360888. |
[31] |
M. van Nieuwstadt, M. Rathinam and R. M. Murray, Differential Flatness and Absolute Equivalence of Nonlinear Control Systems,, SIAM J. Control Optim., 36 (1998), 1225.
doi: 10.1137/S0363012995274027. |
[32] |
H. Poincaré, Sur une forme nouvelle des équations de la mécanique,, C. R. Acad. Sci., 132 (1901), 369. Google Scholar |
[33] |
R. Skinner and R. Rusk, Generalized Hamiltonian dynamics I. Formulation on $T^{*}Q\oplus TQ$,, Journal of Mathematical Pyhsics, 24 (1983), 2589. Google Scholar |
[34] |
K. Spindler, Optimal attitude control of a rigid body,, Applied Mathematics& Optimization, 34 (1996), 79.
doi: 10.1007/BF01182474. |
show all references
References:
[1] |
R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition,, Springer-Verlag, (1988).
doi: 10.1007/978-1-4612-1029-0. |
[2] |
L. Abrunheiro, M. Camarinha, J. F. Cariñena, J. Clemente-Gallardo, E. Martínez and P. Santos, Some applications of quasi-velocities in optimal control,, Int. J. Geom. Methods Mod. Phys., 08 (2011), 835.
doi: 10.1142/S0219887811005427. |
[3] |
M. Barbero-Liñán, A.Echeverría Enríquez, D. Martín de Diego, M.C Muñoz-Lecanda and N. Román-Roy, Skinner-Rusk unified formalism for optimal control systems and applications,, J. Phys. A: Math Theor., 40 (2007), 12071.
doi: 10.1088/1751-8113/40/40/005. |
[4] |
L. Bates and R. Cushman, Global Aspect of Classical Integrable Systems,, Birkhäuser Verlag, (1997).
doi: 10.1007/978-3-0348-8891-2. |
[5] |
R. Benito, Roberto and D. Martín de Diego, Hidden symplecticity in Hamilton's principle algorithms,, Differential geometry and its applications, (2005), 411.
|
[6] |
A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, (2003).
doi: 10.1007/b97376. |
[7] |
A. M. Bloch, I. I. Hussein, M. Leok and A. K. Sanyal, Geometric Structure-Preserving Optimal Control of the Rigid Body,, Journal of Dynamical and Control Systems, 15 (2009), 307.
doi: 10.1007/s10883-009-9071-2. |
[8] |
C. Burnett, D. D. Holm and D. Meier, Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework,, Proc. R. Soc. A., 469 (2013).
doi: 10.1098/rspa.2013.0249. |
[9] |
F. Bullo and A. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems,, Texts in Applied Mathematics, (2005).
doi: 10.1007/978-1-4899-7276-7. |
[10] |
M. Crampin, W. Sarlet and F. Cantrijn, Higher order differential equations and higher order Lagrangian Mechanics,, Math. Proc. Camb. Phil. Soc., 99 (1986), 565.
doi: 10.1017/S0305004100064501. |
[11] |
L. Colombo, F. Jimenez and D. Martín de Diego, Discrete Second-Order Euler-Poincaré Equations. An application to optimal control,, International Journal of Geometric Methods in Modern Physics, 9 (2012).
doi: 10.1142/S0219887812500375. |
[12] |
L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometrical approach,, Journal Mathematical Physics, 51 (2010).
doi: 10.1063/1.3456158. |
[13] |
L. Colombo and D. Martín de Diego, Quasivelocities and Optimal Control of Underactuated Mechanical Systems,, Geometry and Physics: XVIII Fall Workshop on Geometry and Physics. AIP Conference Proceedings, 1260 (2011), 133.
|
[14] |
J. Cortés, M. de León, D. Martín de Diego and S. Martínez, Geometric description of vakonomic and nonholonomic dynamics,, SIAM J. Control Optim., 41 (2002), 1389.
doi: 10.1137/S036301290036817X. |
[15] |
P. Crouch and F. Silva-Leite, Geometry and the dynamic interpolation problem,, American Control Conference, (1991), 1131. Google Scholar |
[16] |
M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of nonlinear systems,Introductory theory and examples,, International Journal of Control, 61 (1995), 1327.
doi: 10.1080/00207179508921959. |
[17] |
F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems,, Communications in Mathematical Physics, 309 (2012), 413.
doi: 10.1007/s00220-011-1313-y. |
[18] |
F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X Vialard, Invariant higher-order variational problems II,, Journal of Nonlinear Science, 22 (2012), 553.
doi: 10.1007/s00332-012-9137-2. |
[19] |
F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Higher-Order Lagrange-Poincaré and Hamilton-Poincaré Reductions,, Bulletin of the Brazilian Math. Soc., 42 (2011), 579.
doi: 10.1007/s00574-011-0030-7. |
[20] |
M. J. Gotay and J. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence theorem,, Ann. Inst. Henri Poincaré, 30 (1979), 129.
|
[21] |
M. J. Gotay, J. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints,, J. Math. Phys., 19 (1978), 2388.
doi: 10.1063/1.523597. |
[22] |
D. D. Holm, Geometric mechanics. Part I and II,, Imperial College Press, (2008).
|
[23] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, he Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1.
doi: 10.1006/aima.1998.1721. |
[24] |
D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids,, Dyn. Syst., 23 (2008), 351.
doi: 10.1080/14689360802294220. |
[25] |
T. Lee, M. Leok and N. H. McClamroch, Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on $SO(3)$,, Journal of Dynamical and Control Systems, 14 (2008), 465.
doi: 10.1007/s10883-008-9047-7. |
[26] |
A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment,, Internat. J. Non-Linear Mech., 30 (1995), 793.
doi: 10.1016/0020-7462(95)00024-0. |
[27] |
M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (1985).
|
[28] |
J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry,, Second Edition, 17 (1999).
doi: 10.1007/978-0-387-21792-5. |
[29] |
D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D thesis, (2013). Google Scholar |
[30] |
T. Mestdag and M. Crampin, Anholonomic frames in constrained dynamics,, Dynamical Systems. An International Journal, 25 (2010), 159.
doi: 10.1080/14689360903360888. |
[31] |
M. van Nieuwstadt, M. Rathinam and R. M. Murray, Differential Flatness and Absolute Equivalence of Nonlinear Control Systems,, SIAM J. Control Optim., 36 (1998), 1225.
doi: 10.1137/S0363012995274027. |
[32] |
H. Poincaré, Sur une forme nouvelle des équations de la mécanique,, C. R. Acad. Sci., 132 (1901), 369. Google Scholar |
[33] |
R. Skinner and R. Rusk, Generalized Hamiltonian dynamics I. Formulation on $T^{*}Q\oplus TQ$,, Journal of Mathematical Pyhsics, 24 (1983), 2589. Google Scholar |
[34] |
K. Spindler, Optimal attitude control of a rigid body,, Applied Mathematics& Optimization, 34 (1996), 79.
doi: 10.1007/BF01182474. |
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