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 & Continuous Dynamical Systems, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064
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show all references

References:
[1]

Int. J. Geom. Methods Mod. Phys., (2014). doi: 10.1142/S0219887814500388.  Google Scholar

[2]

Int. Journal of Geometric Methods in Modern Physics, 3 (2006), 421-436. doi: 10.1142/S0219887806001235.  Google Scholar

[3]

Interdisciplinary Applied Mathematics Series, 24, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar

[4]

Analysis and geometry in control theory and its applications, Springer, 2015. doi: 10.1007/978-3-319-06917-3_2.  Google Scholar

[5]

Proceedings of 35rd IEEE Conference on Decision and Control, (1996), 1648-1653. doi: 10.1109/CDC.1996.572780.  Google Scholar

[6]

Journal of Dynamical and Control Systems, 15 (2009), 307-330. doi: 10.1007/s10883-009-9071-2.  Google Scholar

[7]

Lett. Math. Phys., 49, (1999). doi: 10.1023/A:1007654605901.  Google Scholar

[8]

Foundations of Computational Mathematics, 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4.  Google Scholar

[9]

J. Phys. A, 48 (2015), 205203. doi: 10.1088/1751-8113/48/20/205203.  Google Scholar

[10]

To appear in Banach Center Publications. Preprint available at arXiv:1510.00296 [math-ph], 2015. Google Scholar

[11]

Texts in Applied Mathematics, Springer Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[12]

Proc. R. Soc. A., 469 (2013), 20130249. doi: 10.1098/rspa.2013.0249.  Google Scholar

[13]

Int. J. Control, 11 (1970), 393-407. doi: 10.1080/00207177008905922.  Google Scholar

[14]

IMA J. Math. Control Info., 12 (1995), 399-410. doi: 10.1093/imamci/12.4.399.  Google Scholar

[15]

Proceedings of the 39th IEEE Conference on Decision and Control, (2000), 269-273. doi: 10.1109/CDC.2000.912771.  Google Scholar

[16]

Ph.D Thesis, Instituto de Ciencias Matemáticas, ICMAT (CSIC-UAM-UCM-UC3M), 2014. Google Scholar

[17]

Preprint, available at arXiv:1410.5766, (2014). doi: 10.1007/s00332-016-9314-9.  Google Scholar

[18]

Preprint, 2016, 47p. Available for private distribution. Google Scholar

[19]

Journal Mathematical Physics, 51 (2010), 083519. doi: 10.1063/1.3456158.  Google Scholar

[20]

AIP Conference Proceedings, 1260, 133-140, 2010.  Google Scholar

[21]

J. Geom. Mech., 6 (2014), 451-478. doi: 10.3934/jgm.2014.6.451.  Google Scholar

[22]

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[23]

Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213.  Google Scholar

[24]

SIAM J. Control Optim., 41 (2002), 1389-1412. Google Scholar

[25]

Pub. Dep. Math. Lyon, 2/A (1987), 1-62.  Google Scholar

[26]

J. Dynam. Control Systems, 1 (1995), 177-202. doi: 10.1007/BF02254638.  Google Scholar

[27]

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[28]

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[29]

Bulletin of the Brazilian Math. Soc., 42 (2011), 579-606. doi: 10.1007/s00574-011-0030-7.  Google Scholar

[30]

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[31]

Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-05018-7.  Google Scholar

[32]

J. Algebra, 129 (1990), 194-230. doi: 10.1016/0021-8693(90)90246-K.  Google Scholar

[33]

Oxford Text in Applied Mathematics, 2009. Google Scholar

[34]

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[35]

Dyn. Syst., 23 (2008), 351-397. doi: 10.1080/14689360802294220.  Google Scholar

[36]

Discrete and Continuous Dynamical Systems - Series A, 33 (2013), 1117-1135. doi: 10.3934/dcds.2013.33.1117.  Google Scholar

[37]

Acta Numerica, (2005). doi: 10.1017/S0962492900002154.  Google Scholar

[38]

J. Geom. Mech. 6 (2014), 99-120. doi: 10.3934/jgm.2014.6.99.  Google Scholar

[39]

to appear in IEEE Transactions on Robotics, 2010. doi: 10.1109/TRO.2011.2139130.  Google Scholar

[40]

PhD thesis, University of Southern California, 2008. Google Scholar

[41]

Journal of Dynamical and Control Systems, 14 (2008), 465-487. doi: 10.1007/s10883-008-9047-7.  Google Scholar

[42]

In American Control Conference, Minneapolis, Minnesota, USA, 1742-1747, 2006. Google Scholar

[43]

North-Holland Mathematical Studies 112, North-Holland, Amsterdam, 1985.  Google Scholar

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J. Phys. A, 2005. Google Scholar

[45]

J. Dyn. Control Syst., 16 (2010), 121-148. doi: 10.1007/s10883-010-9080-1.  Google Scholar

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London Mathematical society Lecture Notes, 213, Cambridge University Press, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

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Nonlinearity, 19 (2006), 1313-1348. doi: 10.1088/0951-7715/19/6/006.  Google Scholar

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Geometry, Mechanics and Dynamics. Fields Institute Communications, (2015), 285-317. doi: 10.1007/978-1-4939-2441-7_13.  Google Scholar

[49]

Discrete and Continuous Mechanical Systems, Serie A, 35 (2015), 367-397.  Google Scholar

[50]

ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.  Google Scholar

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In Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, (1999), Publicaciones de la RSME, 2, 209-222, (2001).  Google Scholar

[52]

Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.  Google Scholar

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J. Geometric Mechanics, 7 (2015), 81-108. doi: 10.3934/jgm.2015.7.81.  Google Scholar

[54]

SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), 050. doi: 10.3842/SIGMA.2007.050.  Google Scholar

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Nonlinearity, 12 (1999), 1647-1662. doi: 10.1088/0951-7715/12/6/314.  Google Scholar

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J. Geom. Phys., 36 (1999). doi: 10.1016/S0393-0440(00)00018-8.  Google Scholar

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Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.  Google Scholar

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Ph.D Thesis, Imperial College London, 2013. Google Scholar

[59]

Comm. Math. Phys., 139 (1991), 217-243.  Google Scholar

[60]

IMA J. Math. Control Inform., 6 (1989), 465-473. doi: 10.1093/imamci/6.4.465.  Google Scholar

[61]

J. Phys. A, 44 (2011), 385203. doi: 10.1088/1751-8113/44/38/385203.  Google Scholar

[62]

Houston J. Math., 30 (2004), 637-655.  Google Scholar

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Indiana Univ. Math. J., 22 (1972). doi: 10.1512/iumj.1973.22.22021.  Google Scholar

[64]

C. R. Acad. Sc. Paris, Série A, 283 (1976), 15-18  Google Scholar

[65]

C. R. Acad. Sc. Paris, Série A, 283 (1976), 675-678.  Google Scholar

[66]

Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494.  Google Scholar

[67]

J. Math Soc. Japan, 40 (1988), 705-727. doi: 10.2969/jmsj/04040705.  Google Scholar

[68]

Fields Inst. Comm., 7 (1996), 207-231.  Google Scholar

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