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

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

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

Leonardo Colombo ^1, and David Martín de Diego ^2,

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

Abstract
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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.

Keywords: Lie algebroids, geometric integration., higher-order variational problems, optimal control, Discrete Lagrangian mechanics, Lagrangian submanifolds, Lie groupoids.

Mathematics Subject Classification: Primary: 70G45; Secondary: 70Hxx, 49J15, 53D17, 37M1.

Citation: Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1142/S0219887806001235.
[3]	A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, 24 (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, (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. 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. 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. 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). 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 , (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, (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). doi: 10.1098/rspa.2013.0249.
[13]	J. A. Cadzow, Discrete Calculus of Variations,, Int. J. Control, 11 (1970), 393. 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. 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. doi: 10.1109/CDC.2000.912771.
[16]	L. Colombo, Geometric and Numerical Methods For Optimal Control of Mechanical Systems,, Ph.D Thesis, (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, (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).
[19]	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.
[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 (2010), 133.
[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. 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. 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.
[25]	A. Coste, P. Dazord and A. Weinstein, Grupoïdes symplectiques,, Pub. Dep. Math. Lyon, 2/A (1987), 1.
[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. 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. 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. 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. 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. 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 (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. 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. 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. 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. 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), 6 (2014), 99. 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, (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. 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, (2006), 1742.
[43]	M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (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. doi: 10.1007/s10883-010-9080-1.
[46]	K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, London Mathematical society Lecture Notes, 213 (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. 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, (2015), 285. 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, 35 (2015), 367.
[50]	E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 (2008), 356. 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, 2 (1999), 209.
[52]	E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295. doi: 10.1023/A:1011965919259.
[53]	E. Martínez, Higher-order variational calculus on Lie algebroids,, J. Geometric Mechanics, 7 (2015), 81. 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). 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. 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. doi: 10.1017/S096249290100006X.
[58]	D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D Thesis, (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.
[60]	L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces,, IMA J. Math. Control Inform., 6 (1989), 465. 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). doi: 10.1088/1751-8113/44/38/385203.
[62]	D. Saunders, Prolongations of Lie groupoids and Lie algebroids,, Houston J. Math., 30 (2004), 637.
[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, 283* (1976), 15.*
[65]	WM. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, C. R. Acad. Sc. Paris, 283* (1976), 675.*
[66]	J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials,, Comm. Math. Phys., 139 (1991), 217. doi: 10.1007/BF02352494.
[67]	A. Weinstein, Coisotropic calculus and Poisson groupoids,, J. Math Soc. Japan, 40 (1988), 705. doi: 10.2969/jmsj/04040705.
[68]	A. Weinstein, Lagrangian mechanics and groupoids,, Fields Inst. Comm., 7 (1996), 207.

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. doi: 10.1142/S0219887806001235.
[3]	A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, 24 (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, (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. 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. 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. 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). 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 , (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, (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). doi: 10.1098/rspa.2013.0249.
[13]	J. A. Cadzow, Discrete Calculus of Variations,, Int. J. Control, 11 (1970), 393. 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. 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. doi: 10.1109/CDC.2000.912771.
[16]	L. Colombo, Geometric and Numerical Methods For Optimal Control of Mechanical Systems,, Ph.D Thesis, (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, (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).
[19]	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.
[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 (2010), 133.
[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. 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. 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.
[25]	A. Coste, P. Dazord and A. Weinstein, Grupoïdes symplectiques,, Pub. Dep. Math. Lyon, 2/A (1987), 1.
[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. 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. 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. 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. 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. 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 (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. 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. 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. 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. 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), 6 (2014), 99. 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, (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. 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, (2006), 1742.
[43]	M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (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. doi: 10.1007/s10883-010-9080-1.
[46]	K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, London Mathematical society Lecture Notes, 213 (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. 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, (2015), 285. 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, 35 (2015), 367.
[50]	E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 (2008), 356. 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, 2 (1999), 209.
[52]	E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295. doi: 10.1023/A:1011965919259.
[53]	E. Martínez, Higher-order variational calculus on Lie algebroids,, J. Geometric Mechanics, 7 (2015), 81. 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). 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. 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. doi: 10.1017/S096249290100006X.
[58]	D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D Thesis, (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.
[60]	L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces,, IMA J. Math. Control Inform., 6 (1989), 465. 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). doi: 10.1088/1751-8113/44/38/385203.
[62]	D. Saunders, Prolongations of Lie groupoids and Lie algebroids,, Houston J. Math., 30 (2004), 637.
[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, 283* (1976), 15.*
[65]	WM. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, C. R. Acad. Sc. Paris, 283* (1976), 675.*
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