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

Leonardo Colombo; David Martín de  Diego

doi:10.3934/dcds.2016064

Previous Article
Differential geometry of rigid bodies collisions and non-standard billiards
DCDS Home
This Issue
Next Article
Euler-Poincaré-Arnold equations on semi-direct products II

November 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
Related Papers

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. Google Scholar
[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. Google Scholar
[3]	A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, 24 (2003). doi: 10.1007/b97376. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	J. A. Cadzow, Discrete Calculus of Variations,, Int. J. Control, 11 (1970), 393. doi: 10.1080/00207177008905922. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	L. Colombo, Geometric and Numerical Methods For Optimal Control of Mechanical Systems,, Ph.D Thesis, (2014). Google Scholar
[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. Google Scholar
[18]	L. Colombo and D. Martín de Diego, Second-order constrained variational problems on Lie algebroids,, Preprint, (2016). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids,, SIAM J. Control Optim., 41 (2002), 1389. Google Scholar
[25]	A. Coste, P. Dazord and A. Weinstein, Grupoïdes symplectiques,, Pub. Dep. Math. Lyon, 2/A (1987), 1. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	D. D. Holm, T. Schamah and C. Stoica, Geometry Mechanics and Symmetry,, Oxford Text in Applied Mathematics, (2009). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[37]	A. Iserles, H. Munthe-Kaas, S. Norsett and A. Zanna, Lie-group methods,, Acta Numerica, (2005). doi: 10.1017/S0962492900002154. Google Scholar
[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. Google Scholar
[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. Google Scholar
[40]	M. Kobilarov, Discrete Geometric Motion Control of Autonomous Vehicles,, PhD thesis, (2008). Google Scholar
[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. Google Scholar
[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. Google Scholar
[43]	M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (1985). Google Scholar
[44]	M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A, (2005). Google Scholar
[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. Google Scholar
[46]	K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, London Mathematical society Lecture Notes, 213 (2005). doi: 10.1017/CBO9781107325883. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[50]	E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 (2008), 356. doi: 10.1051/cocv:2007056. Google Scholar
[51]	E. Martínez, Geometric formulation of mechanics on Lie algebroids,, In Proceedings of the VIII Fall Workshop on Geometry and Physics, 2 (1999), 209. Google Scholar
[52]	E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295. doi: 10.1023/A:1011965919259. Google Scholar
[53]	E. Martínez, Higher-order variational calculus on Lie algebroids,, J. Geometric Mechanics, 7 (2015), 81. doi: 10.3934/jgm.2015.7.81. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[57]	M. Marsden and M. West, Discrete Mechanics and variational integrators,, Acta Numerica, 10 (2001), 357. doi: 10.1017/S096249290100006X. Google Scholar
[58]	D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D Thesis, (2013). Google Scholar
[59]	J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials,, Comm. Math. Phys., 139 (1991), 217. Google Scholar
[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. Google Scholar
[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. Google Scholar
[62]	D. Saunders, Prolongations of Lie groupoids and Lie algebroids,, Houston J. Math., 30 (2004), 637. Google Scholar
[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. Google Scholar
[64]	WM. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne,, C. R. Acad. Sc. Paris, 283* (1976), 15. Google Scholar*
[65]	WM. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, C. R. Acad. Sc. Paris, 283* (1976), 675. Google Scholar*
[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. Google Scholar
[67]	A. Weinstein, Coisotropic calculus and Poisson groupoids,, J. Math Soc. Japan, 40 (1988), 705. doi: 10.2969/jmsj/04040705. Google Scholar
[68]	A. Weinstein, Lagrangian mechanics and groupoids,, Fields Inst. Comm., 7 (1996), 207. Google Scholar

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. Google Scholar
[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. Google Scholar
[3]	A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, 24 (2003). doi: 10.1007/b97376. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	J. A. Cadzow, Discrete Calculus of Variations,, Int. J. Control, 11 (1970), 393. doi: 10.1080/00207177008905922. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	L. Colombo, Geometric and Numerical Methods For Optimal Control of Mechanical Systems,, Ph.D Thesis, (2014). Google Scholar
[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. Google Scholar
[18]	L. Colombo and D. Martín de Diego, Second-order constrained variational problems on Lie algebroids,, Preprint, (2016). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids,, SIAM J. Control Optim., 41 (2002), 1389. Google Scholar
[25]	A. Coste, P. Dazord and A. Weinstein, Grupoïdes symplectiques,, Pub. Dep. Math. Lyon, 2/A (1987), 1. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	D. D. Holm, T. Schamah and C. Stoica, Geometry Mechanics and Symmetry,, Oxford Text in Applied Mathematics, (2009). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[37]	A. Iserles, H. Munthe-Kaas, S. Norsett and A. Zanna, Lie-group methods,, Acta Numerica, (2005). doi: 10.1017/S0962492900002154. Google Scholar
[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. Google Scholar
[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. Google Scholar
[40]	M. Kobilarov, Discrete Geometric Motion Control of Autonomous Vehicles,, PhD thesis, (2008). Google Scholar
[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. Google Scholar
[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. Google Scholar
[43]	M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (1985). Google Scholar
[44]	M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A, (2005). Google Scholar
[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. Google Scholar
[46]	K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, London Mathematical society Lecture Notes, 213 (2005). doi: 10.1017/CBO9781107325883. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[50]	E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 (2008), 356. doi: 10.1051/cocv:2007056. Google Scholar
[51]	E. Martínez, Geometric formulation of mechanics on Lie algebroids,, In Proceedings of the VIII Fall Workshop on Geometry and Physics, 2 (1999), 209. Google Scholar
[52]	E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295. doi: 10.1023/A:1011965919259. Google Scholar
[53]	E. Martínez, Higher-order variational calculus on Lie algebroids,, J. Geometric Mechanics, 7 (2015), 81. doi: 10.3934/jgm.2015.7.81. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[57]	M. Marsden and M. West, Discrete Mechanics and variational integrators,, Acta Numerica, 10 (2001), 357. doi: 10.1017/S096249290100006X. Google Scholar
[58]	D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D Thesis, (2013). Google Scholar
[59]	J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials,, Comm. Math. Phys., 139 (1991), 217. Google Scholar
[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. Google Scholar
[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. Google Scholar
[62]	D. Saunders, Prolongations of Lie groupoids and Lie algebroids,, Houston J. Math., 30 (2004), 637. Google Scholar
[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. Google Scholar
[64]	WM. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne,, C. R. Acad. Sc. Paris, 283* (1976), 15. Google Scholar*
[65]	WM. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, C. R. Acad. Sc. Paris, 283* (1976), 675. Google Scholar*
[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. Google Scholar
[67]	A. Weinstein, Coisotropic calculus and Poisson groupoids,, J. Math Soc. Japan, 40 (1988), 705. doi: 10.2969/jmsj/04040705. Google Scholar
[68]	A. Weinstein, Lagrangian mechanics and groupoids,, Fields Inst. Comm., 7 (1996), 207. Google Scholar

[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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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]	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
[9]	Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105
[10]	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
[11]	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
[12]	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
[13]	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
[14]	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
[15]	Monica Motta, Caterina Sartori. Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4527-4552. doi: 10.3934/dcds.2015.35.4527
[16]	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
[17]	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
[18]	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
[19]	Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether currents for higher-order variational problems of Herglotz type with time delay. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 91-102. doi: 10.3934/dcdss.2018006
[20]	Andrew James Bruce, Katarzyna Grabowska, Giovanni Moreno. On a geometric framework for Lagrangian supermechanics. Journal of Geometric Mechanics, 2017, 9 (4) : 411-437. doi: 10.3934/jgm.2017016

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors