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

Neighboring extremal optimal control for mechanical systems on Riemannian manifolds

Abstract Related Papers Cited by
  • In this paper, we extend neighboring extremal optimal control, which is well established for optimal control problems defined on a Euclidean space (see, e.g., [8]) to the setting of Riemannian manifolds. We further specialize the results to the case of Lie groups. An example along with simulation results is presented.
    Mathematics Subject Classification: Primary: 49J15, 49K40; Secondary: 37N35.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    R. Abraham and J. E. Marsden, Foundations of Mechanics, AMS Chelsea Publishing, 1978.doi: 10.1090/chel/364.

    [2]

    A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Springer Science & Business Media, 2004.doi: 10.1007/978-3-662-06404-7.

    [3]

    C. Altafini, Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric, ESAIM: Control, Optimisation and Calculus of Variations, 10 (2004), 526-548.doi: 10.1051/cocv:2004018.

    [4]

    M. Barbero-Liñán, A Geometric Study of Abnormality in Optimal Control Problems for Control and Mechanical Control Systems, PhD thesis, Technical University of Catalonia, 2008.

    [5]

    M. Barbero-Liñán, Characterization of accessibility for affine connection control systems at some points with nonzero velocity, in Proceedings of IEEE Conference on Decision and Control and European Control Conference, 2011, 6528-6533.

    [6]

    A. M. Bloch, Nonholonomic Mechanics and Control, Springer Science & Business Media, 2003.doi: 10.1007/978-1-4939-3017-3.

    [7]

    J. V. Breakwell and H. Yu-Chi, On the conjugate point condition for the control problem, International Journal of Engineering Science, 2 (1965), 565-579.doi: 10.1016/0020-7225(65)90037-6.

    [8]

    A. E. Bryson, Applied Optimal Control: Optimization$,$ Estimation and Control, CRC Press, 1975.

    [9]

    F. Bullo, Invariant Affine Connections and Controllability on Lie Groups, Technical Report Final Project Report for CIT-CDS 141a, Control and Dynamical Systems, California Institute of Technology, 1995.

    [10]

    F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, Springer Science & Business Media, 2005.doi: 10.1007/978-1-4899-7276-7.

    [11]

    F. Bullo and A. D. Lewis, Reduction, linearization, and stability of relative equilibria for mechanical systems on Riemannian manifolds, Acta Applicandae Mathematicae, 99 (2007), 53-95.doi: 10.1007/s10440-007-9155-5.

    [12]

    J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems, Optimization Methods and Software, 27 (2012), 177-196.doi: 10.1080/10556788.2011.593625.

    [13]

    N. Caroff and H. Frankowska, Conjugate points and shocks in nonlinear optimal control, Transactions of the American Mathematical Society, 348 (1996), 3133-3153.doi: 10.1090/S0002-9947-96-01577-2.

    [14]

    P. Crouch and F. Silva Leite, The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces, Journal of Dynamical and Control Systems, 1 (1995), 177-202.doi: 10.1007/BF02254638.

    [15]

    P. Crouch, F. Silva Leite and M. Camarinha, A second order Riemannian variational problem from a Hamiltonian perspective, 1998.

    [16]

    M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992.doi: 10.1007/978-1-4757-2201-7.

    [17]

    A. L. Dontchev and W. W. Hager, Lipschitzian stability in nonlinear control and optimization, SIAM Journal on Control and Optimization, 31 (1993), 569-603.doi: 10.1137/0331026.

    [18]

    A. L. Dontchev, W. W. Hager, A. B. Poore and B. Yang, Optimality, stability, and convergence in nonlinear control, Applied Mathematics and Optimization, 31 (1995), 297-326.doi: 10.1007/BF01215994.

    [19]

    A. L. Dontchev and W. W. Hager, Lipschitzian stability for state constrained nonlinear optimal control, SIAM Journal on Control and Optimization, 36 (1998), 698-718.doi: 10.1137/S0363012996299314.

    [20]

    R. Gupta, A. M. Bloch and I. V. Kolmanovsky, Combined homotopy and neighboring extremal optimal control, Optimal Control Applications and Methods, (2016), to appear.

    [21]

    R. V. Iyer, R. Holsapple and D. Doman, Optimal control problems on parallelizable Riemannian manifolds: Theory and applications, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 1-11.doi: 10.1051/cocv:2005026.

    [22]

    J. M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 2003.doi: 10.1007/978-0-387-21752-9.

    [23]

    P. D. Loewen and H. Zheng, Generalized conjugate points for optimal control problems, Nonlinear Analysis: Theory, Methods & Applications, 22 (1994), 771-791.doi: 10.1016/0362-546X(94)90226-7.

    [24]

    J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer Science & Business Media, 1999.doi: 10.1007/978-0-387-21792-5.

    [25]

    P. M. Mereau and W. F. Powers, Conjugate point properties for linear quadratic problems, Journal of Mathematical Analysis and Applications, 55 (1976), 418-433.

    [26]

    J. W. Milnor, Morse Theory, Princeton University Press, 1963.

    [27]

    L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA Journal of Mathematical Control and Information, 6 (1989), 465-473.doi: 10.1093/imamci/6.4.465.

    [28]

    S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds I, Tohoku Mathematical Journal, Second Series, 10 (1958), 338-354.

    [29]

    S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds II, Tohoku Mathematical Journal, Second Series, 14 (1962), 146-155.

    [30]

    H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Science & Business Media, 2012.doi: 10.1007/978-1-4614-3834-2.

    [31]

    F. Silva Leite, M. Camarinha and P. Crouch, Elastic curves as solutions of Riemannian and sub-Riemannian control problems, Mathematics of Control, Signals, and Systems, 13 (2000), 140-155.doi: 10.1007/PL00009863.

    [32]

    J. L. Speyer and D. H. Jacobson, Primer on Optimal Control Theory, SIAM, 2010.doi: 10.1137/1.9780898718560.

    [33]

    D. R. Tyner and A. D. Lewis, Geometric jacobian linearization and LQR theory, Journal of Geometric Mechanics, 2 (2010), 397-440.doi: 10.3934/jgm.2010.2.397.

    [34]

    V. Zeidan and P. Zezza, The conjugate point condition for smooth control sets, Journal of Mathematical Analysis and Applications, 132 (1988), 572-589.doi: 10.1016/0022-247X(88)90085-6.

    [35]

    V. Zeidan and P. Zezza, Conjugate points and optimal control: Counterexamples, IEEE Transactions on Automatic Control, 34 (1989), 254-256.doi: 10.1109/9.21115.

    [36]

    V. Zeidan, The riccati equation for optimal control problems with mixed state-control constraints: Necessity and sufficiency, SIAM Journal on Control and Optimization, 32 (1994), 1297-1321.doi: 10.1137/S0363012992233640.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(411) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return