March 2017, 9(1): 1-45. doi: 10.3934/jgm.2017001

Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control

Department of Mathematics, University of Michigan, 530 Church Street, 3828 East Hall, Ann Arbor, Michigan, 48109, USA

Received  June 2016 Revised  January 2017 Published  March 2017

The aim of this work is to study, from an intrinsic and geometric point of view, second-order constrained variational problems on Lie algebroids, that is, optimization problems defined by a cost function which depends on higher-order derivatives of admissible curves on a Lie algebroid. Extending the classical Skinner and Rusk formalism for the mechanics in the context of Lie algebroids, for second-order constrained mechanical systems, we derive the corresponding dynamical equations. We find a symplectic Lie subalgebroid where, under some mild regularity conditions, the second-order constrained variational problem, seen as a presymplectic Hamiltonian system, has a unique solution. We study the relationship of this formalism with the second-order constrained Euler-Poincaré and Lagrange-Poincaré equations, among others. Our study is applied to the optimal control of mechanical systems.

Citation: 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
References:
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M. Barbero-LiñánA. Echeverría EnríquezD. Martín de DiegoM. C. Muñoz-Lecanda and N. Román-Roy, Skinner-Rusk unified formalism for optimal control systems and applications, J. Phys. A: Math Theor., 40 (2007), 12071-12093. doi: 10.1088/1751-8113/40/40/005.

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A. J. Bruce, Higher contact-like structures and supersymmetry, J. Phys. A: Math. Theor., 45 (2012), 265205, 12PP. doi: 10.1088/1751-8113/45/26/265205.

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A. J. BruceK. GrabowskaJ. Grabowski and P. Urbanski, New developments in geometric mechanics, Conference proceedings "Geometry of Jets and Fields" (Bedlewo, 10-16 May, 2015), Banach Center Publ., 110 (2016), 57-72, Polish Acad. Sci., Warsaw, 2016.

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

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J. CariñenaJ. Nunes da Costa and P. Santos, Quasi-coordinates from the point of view of Lie algebroid, J. Phys. A: Math. Theor., 40 (2007), 10031-10048. doi: 10.1088/1751-8113/40/33/008.

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J. Cariñena and M. Rodríguez-Olmos, Gauge equivalence and conserved quantities for Lagrangian systems on Lie algebroids, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 265209.

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

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show all references

References:
[1]

L. AbrunheiroM. CamarinhaJ. CarinenaJ. Clemente-GallardoE. Martínez and P. Santos, Some applications of quasi-velocities in optimal control, International Journal of Geometric Methods in Modern Physics, 8 (2011), 835-851. doi: 10.1142/S0219887811005427.

[2]

M. Barbero LiñánA. Echeverría-EnríquezD. Martín de DiegoM.C. Muñoz-Lecanda and N. Román-Roy, Unified formalism for non-autonomous mechanical systems, J. Math. Phys., 49 (2008), 062902, 14pp. doi: 10.1063/1.2929668.

[3]

M. Barbero LiñanM. de LeónD. Martín de Diego and M. Muñoz Lecanda, Kinematic reduction and the Hamilton-Jacobi equation, J. Geometric Mechanics, 4 (2012), 207-237. doi: 10.3934/jgm.2012.4.207.

[4]

M. Barbero-LiñánA. Echeverría EnríquezD. Martín de DiegoM. C. Muñoz-Lecanda and N. Román-Roy, Skinner-Rusk unified formalism for optimal control systems and applications, J. Phys. A: Math Theor., 40 (2007), 12071-12093. doi: 10.1088/1751-8113/40/40/005.

[5]

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

[6]

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.

[7]

A. BlochJ. Marsden and D. Zenkov, Quasivelocities and symmetries in non-holonomic systems, Dynamical Systems, 24 (2009), 187-222. doi: 10.1080/14689360802609344.

[8]

A. J. BruceK. Grabowska and J. Grabowski, Higher order mechanics on graded bundles, J. Phys. A, 48 (2015), 205203, 32PP. doi: 10.1088/1751-8113/48/20/205203.

[9]

A. J. Bruce, Higher contact-like structures and supersymmetry, J. Phys. A: Math. Theor., 45 (2012), 265205, 12PP. doi: 10.1088/1751-8113/45/26/265205.

[10]

A. J. BruceK. GrabowskaJ. Grabowski and P. Urbanski, New developments in geometric mechanics, Conference proceedings "Geometry of Jets and Fields" (Bedlewo, 10-16 May, 2015), Banach Center Publ., 110 (2016), 57-72, Polish Acad. Sci., Warsaw, 2016.

[11]

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

[12]

C. M. CamposM. de LeónD. Martín de Diego and K. Vankerschaver, Vankerschaver, Unambigous formalism for higher-order lagrangian field theories, J. Phys A: Math Theor., 42 (2009), 475207.

[13]

A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Amer. Math. Soc., Providence, RI, 1999

[14]

J. CariñenaJ. Nunes da Costa and P. Santos, Quasi-coordinates from the point of view of Lie algebroid, J. Phys. A: Math. Theor., 40 (2007), 10031-10048. doi: 10.1088/1751-8113/40/33/008.

[15]

J. Cariñena and E. Martínez, Lie Algebroid Generalization of Geometric Mechanics, In Lie Algebroids and Related Topics in Differential Geometry Banach Center Publications, 54 (2011), P201.

[16]

J. Cariñena and M. Rodríguez-Olmos, Gauge equivalence and conserved quantities for Lagrangian systems on Lie algebroids, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 265209.

[17]

H. CendraJ. Marsden and T. Ratiu, Lagrangian reduction by stages, Memoirs of the American Mathematical Society, 152 (2001), x+108 pp. doi: 10.1090/memo/0722.

[18]

S. A. Chaplygin, On the Theory of Motion of Nonholonomic Systems. The Theorem on the Reducing Multiplier, Math. Sbornik XXVIII, 303314, (in Russian) 1911.

[19]

L. ColomboD. Martín de Diego D and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometrical approach, Journal Mathematical Physics, 51 (2010), 083519, 24pp. doi: 10.1063/1.3456158.

[20]

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.

[21]

L. ColomboM. de LeónP. D. Prieto-Martínez and N. Román-Roy, Unified formalism for the generalized kth-order Hamilton-Jacobi problem, Int. J. Geom. Methods Mod. Phys., 11 (2014), 1460037, 9pp. doi: 10.1088/1751-8113/47/23/235203.

[22]

L. ColomboM. de LeónP. D. Prieto-Martínez and N. Román-Roy, Unified formalism for the generalized kth-order Hamilton-Jacobi problem, Int. J. Geom. Methods Mod. Phys., 11 (2014), 1460037, 9pp. doi: 10.1142/S0219887814600378.

[23]

L. Colombo and P. D. Prieto-Martínez, Unified formalism for higher-order variational problems and its applications in optimal control, Int. J. Geom. Methods Mod. Phys., 11 (2014), 1450034, 31pp. doi: 10.1142/S0219887814500340.

[24]

L. A. Cordero, C. T. J. Dodson and M. de León, Differential Geometry of Frame Bundles, Mathematics and Its Applications, Kluwer, Dordrecht, 1989. doi: 10.1007/978-94-009-1265-6.

[25]

J. CortesS. Martínez and F. Cantrijn, Skinner-Rusk approach to time-dependent mechanics, Phys. Lett. A, 300 (2002), 250-258. doi: 10.1016/S0375-9601(02)00777-6.

[26]

J. CortésM. de LeónJ. 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.

[27]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids, SIAM J. Control Optim., 41 (2002), 1389-1412. doi: 10.1137/S036301290036817X.

[28]

M. CrampinW. Sarlet and F. Cantrijn, Higher order differential equations and higher order Lagrangian Mechanics, Math. Proc. Camb. Phil. Soc., 99 (1986), 565-587. doi: 10.1017/S0305004100064501.

[29]

A. Echeverría-EnríquezC. LópezJ. Marín-SolanoM. C. Muñoz-Lecanda and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for field theory, J. Math. Phys., 45 (2004), 360-380.

[30]

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

[31]

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

[32]

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

[33]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.

[34]

K. GrabowskaP. Urbański and J. Grabowski, Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids, Ann. Global Anal. Geom., 15 (1997), 447-486. doi: 10.1023/A:1006519730920.

[35]

K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory, J. Geometric Mechanics, 7 (2015), 1-33. doi: 10.3934/jgm.2015.7.1.

[36]

J. GrabowskiM. de LeónJ. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint, Journal of Mathematical Physics, 50 (2009), 013520, 17pp. doi: 10.1063/1.3049752.

[37]

J. Grabowski and M. Jozwikowski, Pontryagin maximum principle on almost Lie algebroids, SIAM J. on Control and Optimization, 49 (2011), 1306-1357. doi: 10.1137/090760246.

[38]

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Figure 1.  Second order Skinner and Rusk formalism on Lie algebroids
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