0
2013
Impact Factor

    Symmetries and Parameterization of Abnormal Extremals in the Sub-Riemannian Problem with the Growth Vector (2, 3, 5, 8)

    2019, Vol. 15, no. 4, pp.  577-585

    Author(s): Sachkov Y. L., Sachkova E. F.

    The left-invariant sub-Riemannian problem with the growth vector (2, 3, 5, 8) is considered. A two-parameter group of infinitesimal symmetries consisting of rotations and dilations is described. The abnormal geodesic flow is factorized modulo the group of symmetries. A parameterization of the vertical part of abnormal geodesic flow is obtained.
    Keywords: sub-Riemannian geometry, abnormal extremals, symmetries
    Citation: Sachkov Y. L., Sachkova E. F., Symmetries and Parameterization of Abnormal Extremals in the Sub-Riemannian Problem with the Growth Vector (2, 3, 5, 8), Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp.  577-585
    DOI:10.20537/nd190417


    Download File
    PDF, 248.32 Kb

    References

    [1] Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesics and Applications, Math. Surveys Monogr., 91, AMS, Providence, R.I., 2002, xx+259 pp.  mathscinet  zmath
    [2] Agrachev, A. A. and Sachkov, Yu. L., Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci., 87, Springer, Berlin, 2004  crossref  mathscinet  zmath
    [3] Sachkov, Yu. L. and Sachkova, E. F., “Degenerate Abnormal Trajectories in a Sub-Riemannian Problem with Growth Vector $(2,3,5,8)$”, Differ. Equ., 53:3 (2017), 352–365  mathnet  crossref  mathscinet  zmath  elib; Differ. Uravn., 2017, no. 3, 362–374 (Russian)  zmath
    [4] Gauthier, J.-P. and Sachkov, Yu. L., “On the Free Carnot $(2, 3, 5, 8)$ Group”, Program Systems: Theory and Applications, 6:2(25) (2015), 45–61  mathnet  crossref
    [5] Sachkov, Yu. L. and Sachkova, E. F., “Structure of Abnormal Extremals in Sub-Riemannian Problem with the Growth Vector $(2,3,5,8)$” (to appear)
    [6] Lokutsievsky, L. V. and Sachkov, Yu. L., “On Liouville Integrability of Sub-Riemannian Problems on Carnot Groups of Step $4$ and More”, Sb. Math., 209:5 (2018), 672–713  mathnet  crossref  mathscinet; Mat. Sb., 209:5 (2018), 74–119 (Russian)  crossref  mathscinet
    [7] Sachkov, Yu. L., “Symmetries of Flat Rank Two Distributions and Sub-Riemannian Structures”, Trans. Amer. Math. Soc., 356:2 (2004), 457–494  crossref  mathscinet  zmath



    Creative Commons License
    This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License