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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

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, New York, 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), 835-851. 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), 12071-12093. doi: 10.1088/1751-8113/40/40/005.  Google Scholar

[4]

L. Bates and R. Cushman, Global Aspect of Classical Integrable Systems, Birkhäuser Verlag, Basel, 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, Matfyzpress, Prague, (2005), 411-419.  Google Scholar

[6]

A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics Series, 24, Springer-Verlag, New York, 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), 307-330. 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), 20130249, 24pp. 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, Springer Verlag, New York, 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), 565-587. 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), 1250037, 20pp. 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), 083519, 24pp. 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), 133-140.  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), 1389-1412. doi: 10.1137/S036301290036817X.  Google Scholar

[15]

P. Crouch and F. Silva-Leite, Geometry and the dynamic interpolation problem, American Control Conference, (1991), 1131-1136. 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), 1327-1361. 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), 413-458. 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), 553-597. 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), 579-606. 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), 129-142.  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), 2388-2399. doi: 10.1063/1.523597.  Google Scholar

[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.  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), 1-81. 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), 351-397. 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), 465-487. 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), 793-815. 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, North-Holland, Amsterdam, 1985.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

T. Mestdag and M. Crampin, Anholonomic frames in constrained dynamics, Dynamical Systems. An International Journal, 25 (2010), 159-187. 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), 1225-1239. doi: 10.1137/S0363012995274027.  Google Scholar

[32]

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

[33]

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

[34]

K. Spindler, Optimal attitude control of a rigid body, Applied Mathematics& Optimization, 34 (1996), 79-90. 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, New York, 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), 835-851. 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), 12071-12093. doi: 10.1088/1751-8113/40/40/005.  Google Scholar

[4]

L. Bates and R. Cushman, Global Aspect of Classical Integrable Systems, Birkhäuser Verlag, Basel, 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, Matfyzpress, Prague, (2005), 411-419.  Google Scholar

[6]

A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics Series, 24, Springer-Verlag, New York, 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), 307-330. 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), 20130249, 24pp. 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, Springer Verlag, New York, 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), 565-587. 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), 1250037, 20pp. 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), 083519, 24pp. 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), 133-140.  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), 1389-1412. doi: 10.1137/S036301290036817X.  Google Scholar

[15]

P. Crouch and F. Silva-Leite, Geometry and the dynamic interpolation problem, American Control Conference, (1991), 1131-1136. 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), 1327-1361. 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), 413-458. 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), 553-597. 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), 579-606. 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), 129-142.  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), 2388-2399. doi: 10.1063/1.523597.  Google Scholar

[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.  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), 1-81. 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), 351-397. 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), 465-487. 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), 793-815. 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, North-Holland, Amsterdam, 1985.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

T. Mestdag and M. Crampin, Anholonomic frames in constrained dynamics, Dynamical Systems. An International Journal, 25 (2010), 159-187. 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), 1225-1239. doi: 10.1137/S0363012995274027.  Google Scholar

[32]

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

[33]

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

[34]

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

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