Optimal Bang-Bang Trajectories in Sub-Finsler Problem on the Cartan Group

    Received 24 October 2018

    2018, Vol. 14, no. 4, pp.  583-593

    Author(s): Sachkov Y. L.

    The Cartan group is the free nilpotent Lie group of step 3, with 2 generators. This paper studies the Cartan group endowed with the left-invariant sub-Finsler $\ell_\infty$ norm. We adopt the viewpoint of time-optimal control theory. By Pontryagin maximum principle, all sub-Finsler length minimizers belong to one of the following types: abnormal, bang-bang, singular, and mixed. Bang-bang controls are piecewise controls with values in the vertices of the set of control parameter.
    In a previous work, it was shown that bang-bang trajectories have a finite number of patterns determined by values of the Casimir functions on the dual of the Cartan algebra. In this paper we consider, case by case, all patterns of bang-bang trajectories, and obtain detailed upper bounds on the number of switchings of optimal control.
    For bang-bang trajectories with low values of the energy integral, we show optimality for arbitrarily large times.
    The bang-bang trajectories with high values of the energy integral are studied via a second order necessary optimality condition due to A. Agrachev and R.Gamkrelidze. This optimality condition provides a quadratic form, whose sign-definiteness is related to optimality of bangbang trajectories. For each pattern of these trajectories, we compute the maximum number of switchings of optimal control. We show that optimal bang-bang controls may have not more than 11 switchings. For particular patterns of bang-bang controls, we obtain better bounds. In such a way we improve the bounds obtained in previous works.
    On the basis of results of this work we can start to study the cut time along bang-bang trajectories, i.e., the time when these trajectories lose their optimality. This question will be considered in subsequent works.
    Keywords: sub-Finsler geometry, optimal control, switchings, bang-bang trajectories
    Citation: Sachkov Y. L., Optimal Bang-Bang Trajectories in Sub-Finsler Problem on the Cartan Group, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp.  583-593

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    [1] Boscain, U., Chambrion, Th., and Charlot, G., “Nonisotropic $3$-Level Quantum Systems: Complete Solutions for Minimum Time and Minimum Energy”, Discrete Contin. Dyn. Syst. Ser. B, 5:4 (2005), 957–990  crossref  mathscinet  zmath
    [2] Berestovskii, V. N., “Homogeneous Manifolds with an Intrinsic Metric: 2”, Siberian Math. J., 30:2 (1989), 180–191  crossref  mathscinet; Sibirsk. Mat. Zh., 30:2 (1989), 14–28, 225 (Russian)  mathscinet
    [3] Berestovskii, V. N., “The Structure of Locally Compact Homogeneous Spaces with an Intrinsic Metric”, Siberian Math. J., 30:1 (1989), 16–25  crossref  mathscinet; Sibirsk. Mat. Zh., 30:1 (1989), 23–34 (Russian)  mathscinet
    [4] Breuillard, E. and Le Donne, E., “On the Rate of Convergence to the Asymptotic Cone for Nilpotent Groups and Sub-Finsler Geometry”, Proc. Natl. Acad. Sci. USA, 110:48 (2013), 19220–19226  crossref  mathscinet  zmath  adsnasa
    [5] Clelland, J. N. and Moseley, Ch. G., “Sub-Finsler Geometry in Dimension Three”, Differential Geom. Appl., 24:6 (2006), 628–651  crossref  mathscinet  zmath
    [6] Cowling, M. G. and Martini, A., “Sub-Finsler Geometry and Finite Propagation Speed”, Trends in Harmonic Analysis, Springer INdAM Ser., 3, ed. M. A. Picardello, Springer, Milan, 2013, 147–205, xii+447 pp.  crossref  mathscinet  zmath
    [7] Clelland, J. N., Moseley, Ch. G., and Wilkens, G. R., “Geometry of Sub-Finsler Engel Manifolds”, Asian J. Math., 11:4 (2007), 699–726  crossref  mathscinet  zmath
    [8] Hakavuori, E. and Le Donne, E., Blowups and Blowdowns of Geodesics in Carnot Groups, 2018, arXiv: 1806.09375 [math.MG]
    [9] Le Donne, E., “A Metric Characterization of Carnot Groups”, Proc. Amer. Math. Soc., 143:2 (2015), 845–849  crossref  mathscinet  zmath
    [10] Pansu, P., “Métriques de Carnot – Carathéodory et quasiisométries des espaces symétriques de rang un”, Ann. of Math. (2), 129:1 (1989), 1–60  crossref  mathscinet  zmath
    [11] López, C. and Martínez, E., “Sub-Finslerian Metric Associated to an Optimal Control System”, SIAM J. Control Optim., 39 (2000), 798–811  crossref  mathscinet  zmath
    [12] Sachkov, Yu. L., “Exponential Mapping in Generalized Dido's Problem”, Sb. Math., 194:9 (2003), 1331–1359  mathnet  crossref  mathscinet  zmath; Mat. Sb., 194:9 (2003), 63–90 (Russian)  crossref  mathscinet  zmath
    [13] Agrachev, A. A. and Sachkov, Yu. L., Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci., 87, Springer, Berlin, 2004  crossref  mathscinet  zmath
    [14] Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., and Mishchenko, E. F., The Mathematical Theory of Optimal Processes, Wiley, New York, 1962, 360 pp.  mathscinet  zmath
    [15] Agrachev, A. A. and Gamkrelidze, R. V., “Symplectic Geometry for Optimal Control”, Nonlinear Controllability and Optimal Control, Monogr. Textbooks Pure Appl. Math., 133, ed. H. J. Sussmann, Dekker, New York, 1990, 263–277  mathscinet  zmath
    [16] Barilari, D., Boscain, U., Le Donne, E., and Sigalotti, M., “Sub-Finsler Structures from the Time-Optimal Control Viewpoint for Some Nilpotent Distributions”, J. Dyn. Control Syst., 23:3 (2017), 547–575  crossref  mathscinet  zmath
    [17] Gantmacher, F. R., The Theory of Matrices: In 2 Vols., Chelsea, New York, 1959  mathscinet
    [18] Ardentov, A., Le Donne, E., and Sachkov, Yu., “A Sub-Finsler Problem on the Cartan Group”, Tr. Mat. Inst. Steklova, 2019 (to appear)  mathnet  mathscinet
    [19] Ardentov, A., Le Donne, E., and Sachkov, Yu., “Sub-Finsler Geodesics on the Cartan Group”, Regul. Chaotic Dyn., 2019 (to appear)  mathscinet
    [20] Sachkov, Yu., “Conjugate and Cut Time in the Sub-Riemannian Problem on the Group of Motions of a Plane”, ESAIM Control Optim. Calc. Var., 16:4 (2010), 1018–1039  crossref  mathscinet  zmath
    [21] Ardentov, A. A. and Sachkov, Yu. L., “Cut Time in Sub-Riemannian Problem on Engel Group”, ESAIM Control Optim. Calc. Var., 21:4 (2015), 958–988  crossref  mathscinet  zmath
    [22] Podobryaev, A. V. and Sachkov, Yu. L., “Cut Locus of a Left Invariant Riemannian Metric on $SO(3)$ in the Axisymmetric Case”, J. Geom. Phys., 110 (2016), 436–453  crossref  mathscinet  zmath  adsnasa
    [23] Butt, Y. A., Sachkov, Yu. L., and Bhatti, A. I., “Cut Locus and Optimal Synthesis in Sub-Riemannian Problem on the Lie Group $\rm SH(2)$”, J. Dyn. Control Syst., 23:1 (2017), 155–195  crossref  mathscinet  zmath
    [24] Podobryaev, A. V. and Sachkov, Yu. L., “Symmetric Riemannian Problem on the Group of Proper Isometries of Hyperbolic Plane”, J. Dyn. Control Syst., 24:3 (2018), 391–423  crossref  mathscinet  zmath

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