# American Institute of Mathematical Sciences

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.

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##### 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. 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