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

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

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

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.
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, New York, 1988. doi: 10.1007/978-1-4612-1029-0.

[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), 835-851. doi: 10.1142/S0219887811005427.

[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), 12071-12093. doi: 10.1088/1751-8113/40/40/005.

[4]

L. Bates and R. Cushman, Global Aspect of Classical Integrable Systems, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8891-2.

[5]

R. Benito, Roberto and D. Martín de Diego, Hidden symplecticity in Hamilton's principle algorithms, Differential geometry and its applications, Matfyzpress, Prague, (2005), 411-419.

[6]

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

[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), 307-330. doi: 10.1007/s10883-009-9071-2.

[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), 20130249, 24pp. doi: 10.1098/rspa.2013.0249.

[9]

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.

[10]

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

[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), 1250037, 20pp. doi: 10.1142/S0219887812500375.

[12]

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

[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), 133-140.

[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), 1389-1412. doi: 10.1137/S036301290036817X.

[15]

P. Crouch and F. Silva-Leite, Geometry and the dynamic interpolation problem, American Control Conference, (1991), 1131-1136.

[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), 1327-1361. doi: 10.1080/00207179508921959.

[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), 413-458. doi: 10.1007/s00220-011-1313-y.

[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), 553-597. doi: 10.1007/s00332-012-9137-2.

[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), 579-606. doi: 10.1007/s00574-011-0030-7.

[20]

M. J. Gotay and J. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence theorem, Ann. Inst. Henri Poincaré, 30 (1979), 129-142.

[21]

M. J. Gotay, J. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19 (1978), 2388-2399. doi: 10.1063/1.523597.

[22]

D. D. Holm, Geometric mechanics. Part I and II, Imperial College Press, London; distributed by World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.

[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), 1-81. doi: 10.1006/aima.1998.1721.

[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), 351-397. doi: 10.1080/14689360802294220.

[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), 465-487. doi: 10.1007/s10883-008-9047-7.

[26]

A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment, Internat. J. Non-Linear Mech., 30 (1995), 793-815. doi: 10.1016/0020-7462(95)00024-0.

[27]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Mathematical Studies 112, North-Holland, Amsterdam, 1985.

[28]

J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Second Edition, Springer-Verlag, Text in Applied Mathematics, 17, 1999. doi: 10.1007/978-0-387-21792-5.

[29]

D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications, Ph.D thesis, Imperial College London, 2013.

[30]

T. Mestdag and M. Crampin, Anholonomic frames in constrained dynamics, Dynamical Systems. An International Journal, 25 (2010), 159-187. doi: 10.1080/14689360903360888.

[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), 1225-1239. doi: 10.1137/S0363012995274027.

[32]

H. Poincaré, Sur une forme nouvelle des équations de la mécanique, C. R. Acad. Sci., 132 (1901), 369-371.

[33]

R. Skinner and R. Rusk, Generalized Hamiltonian dynamics I. Formulation on $T^{*}Q\oplus TQ$, Journal of Mathematical Pyhsics, 24 (1983), 2589-2594 and 2595-2601.

[34]

K. Spindler, Optimal attitude control of a rigid body, Applied Mathematics& Optimization, 34 (1996), 79-90. doi: 10.1007/BF01182474.

show all references

References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.

[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), 835-851. doi: 10.1142/S0219887811005427.

[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), 12071-12093. doi: 10.1088/1751-8113/40/40/005.

[4]

L. Bates and R. Cushman, Global Aspect of Classical Integrable Systems, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8891-2.

[5]

R. Benito, Roberto and D. Martín de Diego, Hidden symplecticity in Hamilton's principle algorithms, Differential geometry and its applications, Matfyzpress, Prague, (2005), 411-419.

[6]

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

[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), 307-330. doi: 10.1007/s10883-009-9071-2.

[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), 20130249, 24pp. doi: 10.1098/rspa.2013.0249.

[9]

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.

[10]

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

[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), 1250037, 20pp. doi: 10.1142/S0219887812500375.

[12]

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

[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), 133-140.

[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), 1389-1412. doi: 10.1137/S036301290036817X.

[15]

P. Crouch and F. Silva-Leite, Geometry and the dynamic interpolation problem, American Control Conference, (1991), 1131-1136.

[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), 1327-1361. doi: 10.1080/00207179508921959.

[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), 413-458. doi: 10.1007/s00220-011-1313-y.

[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), 553-597. doi: 10.1007/s00332-012-9137-2.

[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), 579-606. doi: 10.1007/s00574-011-0030-7.

[20]

M. J. Gotay and J. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence theorem, Ann. Inst. Henri Poincaré, 30 (1979), 129-142.

[21]

M. J. Gotay, J. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19 (1978), 2388-2399. doi: 10.1063/1.523597.

[22]

D. D. Holm, Geometric mechanics. Part I and II, Imperial College Press, London; distributed by World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.

[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), 1-81. doi: 10.1006/aima.1998.1721.

[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), 351-397. doi: 10.1080/14689360802294220.

[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), 465-487. doi: 10.1007/s10883-008-9047-7.

[26]

A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment, Internat. J. Non-Linear Mech., 30 (1995), 793-815. doi: 10.1016/0020-7462(95)00024-0.

[27]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Mathematical Studies 112, North-Holland, Amsterdam, 1985.

[28]

J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Second Edition, Springer-Verlag, Text in Applied Mathematics, 17, 1999. doi: 10.1007/978-0-387-21792-5.

[29]

D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications, Ph.D thesis, Imperial College London, 2013.

[30]

T. Mestdag and M. Crampin, Anholonomic frames in constrained dynamics, Dynamical Systems. An International Journal, 25 (2010), 159-187. doi: 10.1080/14689360903360888.

[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), 1225-1239. doi: 10.1137/S0363012995274027.

[32]

H. Poincaré, Sur une forme nouvelle des équations de la mécanique, C. R. Acad. Sci., 132 (1901), 369-371.

[33]

R. Skinner and R. Rusk, Generalized Hamiltonian dynamics I. Formulation on $T^{*}Q\oplus TQ$, Journal of Mathematical Pyhsics, 24 (1983), 2589-2594 and 2595-2601.

[34]

K. Spindler, Optimal attitude control of a rigid body, Applied Mathematics& Optimization, 34 (1996), 79-90. doi: 10.1007/BF01182474.

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