\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 70H50, 70H45, 79J15; Secondary: 70G65, 70Hxx.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(353) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return