November  2016, 36(11): 6023-6064. doi: 10.3934/dcds.2016064

Second-order variational problems on Lie groupoids and optimal control applications

1. 

Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109

2. 

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Calle Nicolás Cabrera 15, Campus UAM, Cantoblanco, Madrid, 28049, Spain

Received  June 2015 Revised  May 2016 Published  August 2016

In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational techniques for second-order variational problems on Lie groupoids and their applications to the construction of variational integrators for optimal control problems of mechanical systems. Next, we show how Lagrangian submanifolds of a symplectic groupoid gives intrinsically the discrete dynamics for second-order systems, both unconstrained and constrained, and we study the geometric properties of the implicit flow which defines the dynamics in a Lagrangian submanifold. We also study the theory of reduction by symmetries and the corresponding Noether theorem.
Citation: Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064
References:
[1]

L. Abrunheiro and M. Camarinha, Optimal control of affine connection control systems, from the point of view of lie algebroids, Int. J. Geom. Methods Mod. Phys., (2014). doi: 10.1142/S0219887814500388.

[2]

R. Benito, de León M and D. Martín de Diego, Higher-order discrete lagrangian mechanics, Int. Journal of Geometric Methods in Modern Physics, 3 (2006), 421-436. doi: 10.1142/S0219887806001235.

[3]

A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics Series, 24, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[4]

A. M. Bloch, L. Colombo, R. Gupta and D. Martín de Diego, A geometric approach to the optimal control of nonholonomic mechanical systems, Analysis and geometry in control theory and its applications, Springer, 2015. doi: 10.1007/978-3-319-06917-3_2.

[5]

A. M. Bloch and P. E. Crouch, On the equivalence of higher order variational problems and optimal control problems, Proceedings of 35rd IEEE Conference on Decision and Control, (1996), 1648-1653. doi: 10.1109/CDC.1996.572780.

[6]

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), 307-330. doi: 10.1007/s10883-009-9071-2.

[7]

AI. Bobenko and YB. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations and semidirect products, Lett. Math. Phys., 49, (1999). doi: 10.1023/A:1007654605901.

[8]

N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups, Foundations of Computational Mathematics, 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4.

[9]

A. J. Bruce, K. Grabowska and J. Grabowski, Higher order mechanics on graded bundles, J. Phys. A, 48 (2015), 205203. doi: 10.1088/1751-8113/48/20/205203.

[10]

A. J. Bruce, K. Grabowska, J. Grabowski and P. Urbanski, New Developments In Geometric Mechanics, To appear in Banach Center Publications. Preprint available at arXiv:1510.00296 [math-ph], 2015.

[11]

F. Bullo and A. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, Springer Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.

[12]

C. Burnett, D. Holm and D. Meier, Geometric integrators for higher-order mechanics on Lie groups, Proc. R. Soc. A., 469 (2013), 20130249. doi: 10.1098/rspa.2013.0249.

[13]

J. A. Cadzow, Discrete Calculus of Variations, Int. J. Control, 11 (1970), 393-407. doi: 10.1080/00207177008905922.

[14]

M. Camarinha, P. Crouch and F. Silva-Leite, Splines of class $C^k$ on non-Euclidean spaces, IMA J. Math. Control Info., 12 (1995), 399-410. doi: 10.1093/imamci/12.4.399.

[15]

D. Chang and J. Marsden, Asymptotic stabilization of the heavy top using controlled Lagrangians, Proceedings of the 39th IEEE Conference on Decision and Control, (2000), 269-273. doi: 10.1109/CDC.2000.912771.

[16]

L. Colombo, Geometric and Numerical Methods For Optimal Control of Mechanical Systems, Ph.D Thesis, Instituto de Ciencias Matemáticas, ICMAT (CSIC-UAM-UCM-UC3M), 2014.

[17]

L. Colombo, S. Ferraro and D. Martín de Diego, Geometric integrators for higher-order variational systems and their application to optimal control, Preprint, available at arXiv:1410.5766, (2014). doi: 10.1007/s00332-016-9314-9.

[18]

L. Colombo and D. Martín de Diego, Second-order constrained variational problems on Lie algebroids, Preprint, 2016, 47p. Available for private distribution.

[19]

L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometrical approach, Journal Mathematical Physics, 51 (2010), 083519. doi: 10.1063/1.3456158.

[20]

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, 133-140, 2010.

[21]

L. Colombo and D. Martín de Diego, Higher-order variational problems on Lie groups and optimal control applications, J. Geom. Mech., 6 (2014), 451-478. doi: 10.3934/jgm.2014.6.451.

[22]

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.

[23]

J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213.

[24]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids, SIAM J. Control Optim., 41 (2002), 1389-1412.

[25]

A. Coste, P. Dazord and A. Weinstein, Grupoïdes symplectiques, Pub. Dep. Math. Lyon, 2/A (1987), 1-62.

[26]

P. Crouch and P. Silva Leite, The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces, J. Dynam. Control Systems, 1 (1995), 177-202. doi: 10.1007/BF02254638.

[27]

F. Gay-Balmaz, D. D Holm, D. Meier, T. Ratiu and F. Vialard, Invariant higher-order variational problems, Communications in Mathematical Physics, 309 (2012), 413-458. doi: 10.1007/s00220-011-1313-y.

[28]

F. Gay-Balmaz, D. D. Holm, D. Meier, T. Ratiu, F. Vialard, Invariant higher-order variational problems II, Journal of Nonlinear Science, 22 (2012), 553-597. doi: 10.1007/s00332-012-9137-2.

[29]

F. Gay-Balmaz, D. D. Holm and T. Ratiu, Higher-order Lagrange-Poincaré and Hamilton-Poincaré reductions, Bulletin of the Brazilian Math. Soc., 42 (2011), 579-606. doi: 10.1007/s00574-011-0030-7.

[30]

Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A, 133 (1988), 134-139. doi: 10.1016/0375-9601(88)90773-6.

[31]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-05018-7.

[32]

P. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra, 129 (1990), 194-230. doi: 10.1016/0021-8693(90)90246-K.

[33]

D. D. Holm, T. Schamah and C. Stoica, Geometry Mechanics and Symmetry, Oxford Text in Applied Mathematics, 2009.

[34]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martinez, Discrete nonholonomic Lagrangian systems on Lie groupoids, Journal of Nonlinear Science, 18 (2008), 221-276. doi: 10.1007/s00332-007-9012-8.

[35]

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), 351-397. doi: 10.1080/14689360802294220.

[36]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Padrón, Discrete dynamics in implicit form, Discrete and Continuous Dynamical Systems - Series A, 33 (2013), 1117-1135. doi: 10.3934/dcds.2013.33.1117.

[37]

A. Iserles, H. Munthe-Kaas, S. Norsett and A. Zanna, Lie-group methods, Acta Numerica, (2005). doi: 10.1017/S0962492900002154.

[38]

M. Jóźwikowski and M. Rotkiewicz, Bundle-theoretic methods for higher-order variational calculus, J. Geom. Mech. 6 (2014), 99-120. doi: 10.3934/jgm.2014.6.99.

[39]

M. Kobilarov and J. Marsden, Discrete geometric optimal control on Lie groups, to appear in IEEE Transactions on Robotics, 2010. doi: 10.1109/TRO.2011.2139130.

[40]

M. Kobilarov, Discrete Geometric Motion Control of Autonomous Vehicles, PhD thesis, University of Southern California, 2008.

[41]

T. Lee, M. Leok and N. McClamroch, Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on $SO(3)$, Journal of Dynamical and Control Systems, 14 (2008), 465-487. doi: 10.1007/s10883-008-9047-7.

[42]

T. Lee, N. H. McClamroch and M. Leok, Attitude maneuvers of a rigid spacecraft in a circular orbit, In American Control Conference, Minneapolis, Minnesota, USA, 1742-1747, 2006.

[43]

M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Mathematical Studies 112, North-Holland, Amsterdam, 1985.

[44]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A, 2005.

[45]

L. Machado, F. Silva Leite and K. Krakowski, Higher-order smoothing splines versus least squares problems on Riemannian manifolds, J. Dyn. Control Syst., 16 (2010), 121-148. doi: 10.1007/s10883-010-9080-1.

[46]

K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical society Lecture Notes, 213, Cambridge University Press, 2005. doi: 10.1017/CBO9781107325883.

[47]

J. C. Marrero, E. Martínez and D. Martín de Diego, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348. doi: 10.1088/0951-7715/19/6/006.

[48]

J. C. Marrero, E. Martínez and D Martín de Diego, The local description of discrete mechanics, Geometry, Mechanics and Dynamics. Fields Institute Communications, (2015), 285-317. doi: 10.1007/978-1-4939-2441-7_13.

[49]

J. C. Marrero, D. Martín de Diego and A. Stern, Symplectic groupoids and discrete constrained Lagrangian mechanics, Discrete and Continuous Mechanical Systems, Serie A, 35 (2015), 367-397.

[50]

E. Martínez, Variational calculus on Lie algebroids, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.

[51]

E. Martínez, Geometric formulation of mechanics on Lie algebroids, In Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, (1999), Publicaciones de la RSME, 2, 209-222, (2001).

[52]

E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.

[53]

E. Martínez, Higher-order variational calculus on Lie algebroids, J. Geometric Mechanics, 7 (2015), 81-108. doi: 10.3934/jgm.2015.7.81.

[54]

E. Martínez and J. Cortés, Lie algebroids in classical mechanics and optimal control, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), 050. doi: 10.3842/SIGMA.2007.050.

[55]

J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincaré and Lie- Poisson equations, Nonlinearity, 12 (1999), 1647-1662. doi: 10.1088/0951-7715/12/6/314.

[56]

J. E. Marsden, S. Pekarsky and S. Shkoller, Symmetry reduction of discrete Lagrangian mechanics on Lie groups, J. Geom. Phys., 36 (1999). doi: 10.1016/S0393-0440(00)00018-8.

[57]

M. Marsden and M. West, Discrete Mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[58]

D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications, Ph.D Thesis, Imperial College London, 2013.

[59]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243.

[60]

L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA J. Math. Control Inform., 6 (1989), 465-473. doi: 10.1093/imamci/6.4.465.

[61]

P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems, J. Phys. A, 44 (2011), 385203. doi: 10.1088/1751-8113/44/38/385203.

[62]

D. Saunders, Prolongations of Lie groupoids and Lie algebroids, Houston J. Math., 30 (2004), 637-655.

[63]

J. Sniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds, Indiana Univ. Math. J., 22 (1972). doi: 10.1512/iumj.1973.22.22021.

[64]

WM. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sc. Paris, Série A, 283 (1976), 15-18

[65]

WM. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sc. Paris, Série A, 283 (1976), 675-678.

[66]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494.

[67]

A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math Soc. Japan, 40 (1988), 705-727. doi: 10.2969/jmsj/04040705.

[68]

A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231.

show all references

References:
[1]

L. Abrunheiro and M. Camarinha, Optimal control of affine connection control systems, from the point of view of lie algebroids, Int. J. Geom. Methods Mod. Phys., (2014). doi: 10.1142/S0219887814500388.

[2]

R. Benito, de León M and D. Martín de Diego, Higher-order discrete lagrangian mechanics, Int. Journal of Geometric Methods in Modern Physics, 3 (2006), 421-436. doi: 10.1142/S0219887806001235.

[3]

A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics Series, 24, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[4]

A. M. Bloch, L. Colombo, R. Gupta and D. Martín de Diego, A geometric approach to the optimal control of nonholonomic mechanical systems, Analysis and geometry in control theory and its applications, Springer, 2015. doi: 10.1007/978-3-319-06917-3_2.

[5]

A. M. Bloch and P. E. Crouch, On the equivalence of higher order variational problems and optimal control problems, Proceedings of 35rd IEEE Conference on Decision and Control, (1996), 1648-1653. doi: 10.1109/CDC.1996.572780.

[6]

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), 307-330. doi: 10.1007/s10883-009-9071-2.

[7]

AI. Bobenko and YB. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations and semidirect products, Lett. Math. Phys., 49, (1999). doi: 10.1023/A:1007654605901.

[8]

N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups, Foundations of Computational Mathematics, 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4.

[9]

A. J. Bruce, K. Grabowska and J. Grabowski, Higher order mechanics on graded bundles, J. Phys. A, 48 (2015), 205203. doi: 10.1088/1751-8113/48/20/205203.

[10]

A. J. Bruce, K. Grabowska, J. Grabowski and P. Urbanski, New Developments In Geometric Mechanics, To appear in Banach Center Publications. Preprint available at arXiv:1510.00296 [math-ph], 2015.

[11]

F. Bullo and A. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, Springer Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.

[12]

C. Burnett, D. Holm and D. Meier, Geometric integrators for higher-order mechanics on Lie groups, Proc. R. Soc. A., 469 (2013), 20130249. doi: 10.1098/rspa.2013.0249.

[13]

J. A. Cadzow, Discrete Calculus of Variations, Int. J. Control, 11 (1970), 393-407. doi: 10.1080/00207177008905922.

[14]

M. Camarinha, P. Crouch and F. Silva-Leite, Splines of class $C^k$ on non-Euclidean spaces, IMA J. Math. Control Info., 12 (1995), 399-410. doi: 10.1093/imamci/12.4.399.

[15]

D. Chang and J. Marsden, Asymptotic stabilization of the heavy top using controlled Lagrangians, Proceedings of the 39th IEEE Conference on Decision and Control, (2000), 269-273. doi: 10.1109/CDC.2000.912771.

[16]

L. Colombo, Geometric and Numerical Methods For Optimal Control of Mechanical Systems, Ph.D Thesis, Instituto de Ciencias Matemáticas, ICMAT (CSIC-UAM-UCM-UC3M), 2014.

[17]

L. Colombo, S. Ferraro and D. Martín de Diego, Geometric integrators for higher-order variational systems and their application to optimal control, Preprint, available at arXiv:1410.5766, (2014). doi: 10.1007/s00332-016-9314-9.

[18]

L. Colombo and D. Martín de Diego, Second-order constrained variational problems on Lie algebroids, Preprint, 2016, 47p. Available for private distribution.

[19]

L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometrical approach, Journal Mathematical Physics, 51 (2010), 083519. doi: 10.1063/1.3456158.

[20]

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, 133-140, 2010.

[21]

L. Colombo and D. Martín de Diego, Higher-order variational problems on Lie groups and optimal control applications, J. Geom. Mech., 6 (2014), 451-478. doi: 10.3934/jgm.2014.6.451.

[22]

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.

[23]

J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213.

[24]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids, SIAM J. Control Optim., 41 (2002), 1389-1412.

[25]

A. Coste, P. Dazord and A. Weinstein, Grupoïdes symplectiques, Pub. Dep. Math. Lyon, 2/A (1987), 1-62.

[26]

P. Crouch and P. Silva Leite, The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces, J. Dynam. Control Systems, 1 (1995), 177-202. doi: 10.1007/BF02254638.

[27]

F. Gay-Balmaz, D. D Holm, D. Meier, T. Ratiu and F. Vialard, Invariant higher-order variational problems, Communications in Mathematical Physics, 309 (2012), 413-458. doi: 10.1007/s00220-011-1313-y.

[28]

F. Gay-Balmaz, D. D. Holm, D. Meier, T. Ratiu, F. Vialard, Invariant higher-order variational problems II, Journal of Nonlinear Science, 22 (2012), 553-597. doi: 10.1007/s00332-012-9137-2.

[29]

F. Gay-Balmaz, D. D. Holm and T. Ratiu, Higher-order Lagrange-Poincaré and Hamilton-Poincaré reductions, Bulletin of the Brazilian Math. Soc., 42 (2011), 579-606. doi: 10.1007/s00574-011-0030-7.

[30]

Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A, 133 (1988), 134-139. doi: 10.1016/0375-9601(88)90773-6.

[31]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-05018-7.

[32]

P. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra, 129 (1990), 194-230. doi: 10.1016/0021-8693(90)90246-K.

[33]

D. D. Holm, T. Schamah and C. Stoica, Geometry Mechanics and Symmetry, Oxford Text in Applied Mathematics, 2009.

[34]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martinez, Discrete nonholonomic Lagrangian systems on Lie groupoids, Journal of Nonlinear Science, 18 (2008), 221-276. doi: 10.1007/s00332-007-9012-8.

[35]

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), 351-397. doi: 10.1080/14689360802294220.

[36]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Padrón, Discrete dynamics in implicit form, Discrete and Continuous Dynamical Systems - Series A, 33 (2013), 1117-1135. doi: 10.3934/dcds.2013.33.1117.

[37]

A. Iserles, H. Munthe-Kaas, S. Norsett and A. Zanna, Lie-group methods, Acta Numerica, (2005). doi: 10.1017/S0962492900002154.

[38]

M. Jóźwikowski and M. Rotkiewicz, Bundle-theoretic methods for higher-order variational calculus, J. Geom. Mech. 6 (2014), 99-120. doi: 10.3934/jgm.2014.6.99.

[39]

M. Kobilarov and J. Marsden, Discrete geometric optimal control on Lie groups, to appear in IEEE Transactions on Robotics, 2010. doi: 10.1109/TRO.2011.2139130.

[40]

M. Kobilarov, Discrete Geometric Motion Control of Autonomous Vehicles, PhD thesis, University of Southern California, 2008.

[41]

T. Lee, M. Leok and N. McClamroch, Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on $SO(3)$, Journal of Dynamical and Control Systems, 14 (2008), 465-487. doi: 10.1007/s10883-008-9047-7.

[42]

T. Lee, N. H. McClamroch and M. Leok, Attitude maneuvers of a rigid spacecraft in a circular orbit, In American Control Conference, Minneapolis, Minnesota, USA, 1742-1747, 2006.

[43]

M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Mathematical Studies 112, North-Holland, Amsterdam, 1985.

[44]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A, 2005.

[45]

L. Machado, F. Silva Leite and K. Krakowski, Higher-order smoothing splines versus least squares problems on Riemannian manifolds, J. Dyn. Control Syst., 16 (2010), 121-148. doi: 10.1007/s10883-010-9080-1.

[46]

K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical society Lecture Notes, 213, Cambridge University Press, 2005. doi: 10.1017/CBO9781107325883.

[47]

J. C. Marrero, E. Martínez and D. Martín de Diego, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348. doi: 10.1088/0951-7715/19/6/006.

[48]

J. C. Marrero, E. Martínez and D Martín de Diego, The local description of discrete mechanics, Geometry, Mechanics and Dynamics. Fields Institute Communications, (2015), 285-317. doi: 10.1007/978-1-4939-2441-7_13.

[49]

J. C. Marrero, D. Martín de Diego and A. Stern, Symplectic groupoids and discrete constrained Lagrangian mechanics, Discrete and Continuous Mechanical Systems, Serie A, 35 (2015), 367-397.

[50]

E. Martínez, Variational calculus on Lie algebroids, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.

[51]

E. Martínez, Geometric formulation of mechanics on Lie algebroids, In Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, (1999), Publicaciones de la RSME, 2, 209-222, (2001).

[52]

E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.

[53]

E. Martínez, Higher-order variational calculus on Lie algebroids, J. Geometric Mechanics, 7 (2015), 81-108. doi: 10.3934/jgm.2015.7.81.

[54]

E. Martínez and J. Cortés, Lie algebroids in classical mechanics and optimal control, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), 050. doi: 10.3842/SIGMA.2007.050.

[55]

J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincaré and Lie- Poisson equations, Nonlinearity, 12 (1999), 1647-1662. doi: 10.1088/0951-7715/12/6/314.

[56]

J. E. Marsden, S. Pekarsky and S. Shkoller, Symmetry reduction of discrete Lagrangian mechanics on Lie groups, J. Geom. Phys., 36 (1999). doi: 10.1016/S0393-0440(00)00018-8.

[57]

M. Marsden and M. West, Discrete Mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[58]

D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications, Ph.D Thesis, Imperial College London, 2013.

[59]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243.

[60]

L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA J. Math. Control Inform., 6 (1989), 465-473. doi: 10.1093/imamci/6.4.465.

[61]

P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems, J. Phys. A, 44 (2011), 385203. doi: 10.1088/1751-8113/44/38/385203.

[62]

D. Saunders, Prolongations of Lie groupoids and Lie algebroids, Houston J. Math., 30 (2004), 637-655.

[63]

J. Sniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds, Indiana Univ. Math. J., 22 (1972). doi: 10.1512/iumj.1973.22.22021.

[64]

WM. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sc. Paris, Série A, 283 (1976), 15-18

[65]

WM. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sc. Paris, Série A, 283 (1976), 675-678.

[66]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494.

[67]

A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math Soc. Japan, 40 (1988), 705-727. doi: 10.2969/jmsj/04040705.

[68]

A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231.

[1]

Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451

[2]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81

[3]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421

[4]

Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213

[5]

Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, 2017, 9 (1) : 1-45. doi: 10.3934/jgm.2017001

[6]

Juan Carlos Marrero, David Martín de Diego, Ari Stern. Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367

[7]

Gennadi Sardanashvily. Lagrangian dynamics of submanifolds. Relativistic mechanics. Journal of Geometric Mechanics, 2012, 4 (1) : 99-110. doi: 10.3934/jgm.2012.4.99

[8]

Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105

[9]

Pedro D. Prieto-Martínez, Narciso Román-Roy. Higher-order mechanics: Variational principles and other topics. Journal of Geometric Mechanics, 2013, 5 (4) : 493-510. doi: 10.3934/jgm.2013.5.493

[10]

Víctor Manuel Jiménez Morales, Manuel De León, Marcelo Epstein. Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies. Journal of Geometric Mechanics, 2019, 11 (3) : 301-324. doi: 10.3934/jgm.2019017

[11]

Theodore Voronov. Book review: General theory of Lie groupoids and Lie algebroids, by Kirill C. H. Mackenzie. Journal of Geometric Mechanics, 2021, 13 (3) : 277-283. doi: 10.3934/jgm.2021026

[12]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990

[13]

Cédric M. Campos, Elisa Guzmán, Juan Carlos Marrero. Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (1) : 1-26. doi: 10.3934/jgm.2012.4.1

[14]

Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist, Amit K. Sanyal. Embedded geodesic problems and optimal control for matrix Lie groups. Journal of Geometric Mechanics, 2011, 3 (2) : 197-223. doi: 10.3934/jgm.2011.3.197

[15]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213

[16]

Monica Motta, Caterina Sartori. Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4527-4552. doi: 10.3934/dcds.2015.35.4527

[17]

Michele Zadra, Elizabeth L. Mansfield. Using Lie group integrators to solve two and higher dimensional variational problems with symmetry. Journal of Computational Dynamics, 2019, 6 (2) : 485-511. doi: 10.3934/jcd.2019025

[18]

Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017

[19]

Dennise García-Beltrán, José A. Vallejo, Yurii Vorobiev. Lie algebroids generated by cohomology operators. Journal of Geometric Mechanics, 2015, 7 (3) : 295-315. doi: 10.3934/jgm.2015.7.295

[20]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether currents for higher-order variational problems of Herglotz type with time delay. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 91-102. doi: 10.3934/dcdss.2018006

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (89)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]