Higher-order variational problems on lie groups and optimal control applications

Leonardo Colombo; David Martín de Diego

doi:10.3934/jgm.2014.6.451

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December 2014, 6(4): 451-478. doi: 10.3934/jgm.2014.6.451

Higher-order variational problems on lie groups 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, United States
2.	Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Campus de Cantoblanco, UAM C/ Nicolas Cabrera, 15 - 28049 Madrid, Spain

Received May 2014 Revised August 2014 Published December 2014

Abstract
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In this paper, we describe a geometric setting for higher-order La- grangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, we deduce an intrinsic framework for this type of dynamical systems. Interesting applications as, for instance, a geometric derivation of the higher-order Euler-Poincaré equations, optimal control of underactuated control systems whose configuration space is a Lie group are shown, among others, along the paper.

Keywords: higher-order tangent bundles, Lie groups, optimal control of mechanical systems, Pontryagin's bundle., Skinner-Rusk formalism.

Mathematics Subject Classification: Primary: 70H50, 70H45, 79J15; Secondary: 70G65, 70Hx.

Citation: 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

References:

[1]	R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1029-0. Google Scholar
[2]	L. Abrunheiro, M. Camarinha, J. F. Cariñena, J. Clemente-Gallardo, E. Martínez and P. Santos, Some applications of quasi-velocities in optimal control,, Int. J. Geom. Methods Mod. Phys., 08 (2011), 835. doi: 10.1142/S0219887811005427. Google Scholar
[3]	M. Barbero-Liñán, A.Echeverría Enríquez, D. Martín de Diego, M.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. doi: 10.1088/1751-8113/40/40/005. Google Scholar
[4]	L. Bates and R. Cushman, Global Aspect of Classical Integrable Systems,, Birkhäuser Verlag, (1997). doi: 10.1007/978-3-0348-8891-2. Google Scholar
[5]	R. Benito, Roberto and D. Martín de Diego, Hidden symplecticity in Hamilton's principle algorithms,, Differential geometry and its applications, (2005), 411. Google Scholar
[6]	A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, (2003). doi: 10.1007/b97376. Google Scholar
[7]	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
[8]	C. Burnett, D. D. Holm and D. Meier, Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework,, Proc. R. Soc. A., 469 (2013). doi: 10.1098/rspa.2013.0249. Google Scholar
[9]	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
[10]	M. Crampin, W. Sarlet and F. Cantrijn, Higher order differential equations and higher order Lagrangian Mechanics,, Math. Proc. Camb. Phil. Soc., 99 (1986), 565. doi: 10.1017/S0305004100064501. Google Scholar
[11]	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
[12]	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
[13]	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* (2011), 133. Google Scholar*
[14]	J. Cortés, M. de León, D. Martín de Diego and S. Martínez, Geometric description of vakonomic and nonholonomic dynamics,, SIAM J. Control Optim., 41 (2002), 1389. doi: 10.1137/S036301290036817X. Google Scholar
[15]	P. Crouch and F. Silva-Leite, Geometry and the dynamic interpolation problem,, American Control Conference, (1991), 1131. Google Scholar
[16]	M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of nonlinear systems,Introductory theory and examples,, International Journal of Control, 61 (1995), 1327. doi: 10.1080/00207179508921959. Google Scholar
[17]	F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems,, Communications in Mathematical Physics, 309 (2012), 413. doi: 10.1007/s00220-011-1313-y. Google Scholar
[18]	F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X Vialard, Invariant higher-order variational problems II,, Journal of Nonlinear Science, 22 (2012), 553. doi: 10.1007/s00332-012-9137-2. Google Scholar
[19]	F. Gay-Balmaz, D. D. Holm and T. S. 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
[20]	M. J. Gotay and J. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence theorem,, Ann. Inst. Henri Poincaré, 30 (1979), 129. Google Scholar
[21]	M. J. Gotay, J. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints,, J. Math. Phys., 19 (1978), 2388. doi: 10.1063/1.523597. Google Scholar
[22]	D. D. Holm, Geometric mechanics. Part I and II,, Imperial College Press, (2008). Google Scholar
[23]	D. D. Holm, J. E. Marsden and T. S. Ratiu, he Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721. Google Scholar
[24]	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
[25]	T. Lee, M. Leok and N. H. 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
[26]	A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment,, Internat. J. Non-Linear Mech., 30 (1995), 793. doi: 10.1016/0020-7462(95)00024-0. Google Scholar
[27]	M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (1985). Google Scholar
[28]	J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry,, Second Edition, 17 (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar
[29]	D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D thesis, (2013). Google Scholar
[30]	T. Mestdag and M. Crampin, Anholonomic frames in constrained dynamics,, Dynamical Systems. An International Journal, 25 (2010), 159. doi: 10.1080/14689360903360888. Google Scholar
[31]	M. van Nieuwstadt, M. Rathinam and R. M. Murray, Differential Flatness and Absolute Equivalence of Nonlinear Control Systems,, SIAM J. Control Optim., 36 (1998), 1225. doi: 10.1137/S0363012995274027. Google Scholar
[32]	H. Poincaré, Sur une forme nouvelle des équations de la mécanique,, C. R. Acad. Sci., 132* (1901), 369. Google Scholar*
[33]	R. Skinner and R. Rusk, Generalized Hamiltonian dynamics I. Formulation on $T^{}Q\oplus TQ$,, Journal of Mathematical Pyhsics, 24 (1983), 2589. Google Scholar*
[34]	K. Spindler, Optimal attitude control of a rigid body,, Applied Mathematics& Optimization, 34 (1996), 79. doi: 10.1007/BF01182474. Google Scholar

show all references

References:

[1]	R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1029-0. Google Scholar
[2]	L. Abrunheiro, M. Camarinha, J. F. Cariñena, J. Clemente-Gallardo, E. Martínez and P. Santos, Some applications of quasi-velocities in optimal control,, Int. J. Geom. Methods Mod. Phys., 08 (2011), 835. doi: 10.1142/S0219887811005427. Google Scholar
[3]	M. Barbero-Liñán, A.Echeverría Enríquez, D. Martín de Diego, M.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. doi: 10.1088/1751-8113/40/40/005. Google Scholar
[4]	L. Bates and R. Cushman, Global Aspect of Classical Integrable Systems,, Birkhäuser Verlag, (1997). doi: 10.1007/978-3-0348-8891-2. Google Scholar
[5]	R. Benito, Roberto and D. Martín de Diego, Hidden symplecticity in Hamilton's principle algorithms,, Differential geometry and its applications, (2005), 411. Google Scholar
[6]	A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, (2003). doi: 10.1007/b97376. Google Scholar
[7]	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
[8]	C. Burnett, D. D. Holm and D. Meier, Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework,, Proc. R. Soc. A., 469 (2013). doi: 10.1098/rspa.2013.0249. Google Scholar
[9]	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
[10]	M. Crampin, W. Sarlet and F. Cantrijn, Higher order differential equations and higher order Lagrangian Mechanics,, Math. Proc. Camb. Phil. Soc., 99 (1986), 565. doi: 10.1017/S0305004100064501. Google Scholar
[11]	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
[12]	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
[13]	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* (2011), 133. Google Scholar*
[14]	J. Cortés, M. de León, D. Martín de Diego and S. Martínez, Geometric description of vakonomic and nonholonomic dynamics,, SIAM J. Control Optim., 41 (2002), 1389. doi: 10.1137/S036301290036817X. Google Scholar
[15]	P. Crouch and F. Silva-Leite, Geometry and the dynamic interpolation problem,, American Control Conference, (1991), 1131. Google Scholar
[16]	M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of nonlinear systems,Introductory theory and examples,, International Journal of Control, 61 (1995), 1327. doi: 10.1080/00207179508921959. Google Scholar
[17]	F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems,, Communications in Mathematical Physics, 309 (2012), 413. doi: 10.1007/s00220-011-1313-y. Google Scholar
[18]	F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X Vialard, Invariant higher-order variational problems II,, Journal of Nonlinear Science, 22 (2012), 553. doi: 10.1007/s00332-012-9137-2. Google Scholar
[19]	F. Gay-Balmaz, D. D. Holm and T. S. 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
[20]	M. J. Gotay and J. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence theorem,, Ann. Inst. Henri Poincaré, 30 (1979), 129. Google Scholar
[21]	M. J. Gotay, J. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints,, J. Math. Phys., 19 (1978), 2388. doi: 10.1063/1.523597. Google Scholar
[22]	D. D. Holm, Geometric mechanics. Part I and II,, Imperial College Press, (2008). Google Scholar
[23]	D. D. Holm, J. E. Marsden and T. S. Ratiu, he Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721. Google Scholar
[24]	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
[25]	T. Lee, M. Leok and N. H. 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
[26]	A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment,, Internat. J. Non-Linear Mech., 30 (1995), 793. doi: 10.1016/0020-7462(95)00024-0. Google Scholar
[27]	M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (1985). Google Scholar
[28]	J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry,, Second Edition, 17 (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar
[29]	D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D thesis, (2013). Google Scholar
[30]	T. Mestdag and M. Crampin, Anholonomic frames in constrained dynamics,, Dynamical Systems. An International Journal, 25 (2010), 159. doi: 10.1080/14689360903360888. Google Scholar
[31]	M. van Nieuwstadt, M. Rathinam and R. M. Murray, Differential Flatness and Absolute Equivalence of Nonlinear Control Systems,, SIAM J. Control Optim., 36 (1998), 1225. doi: 10.1137/S0363012995274027. Google Scholar
[32]	H. Poincaré, Sur une forme nouvelle des équations de la mécanique,, C. R. Acad. Sci., 132* (1901), 369. Google Scholar*
[33]	R. Skinner and R. Rusk, Generalized Hamiltonian dynamics I. Formulation on $T^{}Q\oplus TQ$,, Journal of Mathematical Pyhsics, 24 (1983), 2589. Google Scholar*
[34]	K. Spindler, Optimal attitude control of a rigid body,, Applied Mathematics& Optimization, 34 (1996), 79. doi: 10.1007/BF01182474. Google Scholar

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