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2013
Impact Factor

    Yuri Sachkov

    Program Systems Institute, Russian Academy of Sciences

    Publications:

    Sachkov Y. L.
    Abstract
    The Engel group is the four-dimensional nilpotent Lie group of step 3, with 2 generators. We consider a one-parameter family of left-invariant rank 2 sub-Finsler problems on the Engel group with the set of control parameters given by a square centered at the origin and rotated by an arbitrary angle. We adopt the viewpoint of time-optimal control theory. By Pontryagin’s 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 parameters.
    We describe the phase portrait for bang-bang extremals.
    In previous work, it was shown that bang-bang trajectories with low values of the energy integral are optimal for arbitrarily large times. For optimal bang-bang trajectories with high values of the energy integral, a general upper bound on the number of switchings was obtained.
    In this paper we improve the bounds on the number of switchings on optimal bang-bang trajectories 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 bang-bang 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 9 switchings. For particular patterns of bang-bang controls, we obtain better bounds. In such a way we improve the bounds obtained in previous work.
    On the basis of the 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 work.
    Keywords: sub-Finsler problem, Engel group, bang-bang extremal, optimality condition
    Citation: Sachkov Y. L.,  Optimal Bang-Bang Trajectories in Sub-Finsler Problems on the Engel Group, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp.  355-367
    DOI:10.20537/nd200209
    Sachkov Y. L., Sachkova E. F.
    Abstract
    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
    Sachkov Y. L.
    Abstract
    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
    DOI:10.20537/nd180411

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