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Neighboring extremal optimal control for mechanical systems on Riemannian manifolds

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  • 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.


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  • [1]

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


    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.


    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.


    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.


    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.


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


    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.


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


    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.


    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.


    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.


    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.


    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.


    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.


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


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


    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.


    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.


    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.


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


    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.


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


    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.


    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.


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


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


    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.


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


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


    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.


    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.


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


    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.


    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.


    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.


    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.

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