Yuri Sachkov
Publications:
Sachkov Y. L., Sachkova E. F.
Abnormal Extremals in the SubRiemannian Problem with Growth Vector $(2, 3, 5, 8, 14)$
2023, Vol. 19, no. 4, pp. 559573
Abstract
We study the leftinvariant subRiemannian problem on the free nilpotent Lie group of
rank 2 and step 5. We describe some abnormal trajectories and some properties of the set filled
by nice abnormal trajectories starting at the identity of the group.

Sachkov Y. L.
Optimal BangBang Trajectories in SubFinsler Problems on the Engel Group
2020, Vol. 16, no. 2, pp. 355367
Abstract
The Engel group is the fourdimensional nilpotent Lie group of step 3, with 2 generators.
We consider a oneparameter family of leftinvariant rank 2 subFinsler 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 timeoptimal control theory. By Pontryaginâ€™s
maximum principle, all subFinsler length minimizers belong to one of the following types:
abnormal, bangbang, singular, and mixed. Bangbang controls are piecewise controls with
values in the vertices of the set of control parameters. We describe the phase portrait for bangbang extremals. In previous work, it was shown that bangbang trajectories with low values of the energy integral are optimal for arbitrarily large times. For optimal bangbang 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 bangbang trajectories via a secondorder necessary optimality condition due to A. Agrachev and R.Gamkrelidze. This optimality condition provides a quadratic form, whose signdefiniteness 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 bangbang controls may have not more than 9 switchings. For particular patterns of bangbang 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 bangbang trajectories, i.e., the time when these trajectories lose their optimality. This question will be considered in subsequent work. 
Sachkov Y. L., Sachkova E. F.
Symmetries and Parameterization of Abnormal Extremals in the SubRiemannian Problem with the Growth Vector (2, 3, 5, 8)
2019, Vol. 15, no. 4, pp. 577585
Abstract
The leftinvariant subRiemannian problem with the growth vector (2, 3, 5, 8) is considered.
A twoparameter 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.

Sachkov Y. L.
Optimal BangBang Trajectories in SubFinsler Problem on the Cartan Group
2018, Vol. 14, no. 4, pp. 583593
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 leftinvariant subFinsler $\ell_\infty$ norm. We adopt the viewpoint of timeoptimal control theory. By Pontryagin maximum principle, all subFinsler length minimizers belong to one of the following types: abnormal, bangbang, singular, and
mixed. Bangbang controls are piecewise controls with values in the vertices of the set of control parameter. In a previous work, it was shown that bangbang 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 bangbang trajectories, and obtain detailed upper bounds on the number of switchings of optimal control. For bangbang trajectories with low values of the energy integral, we show optimality for arbitrarily large times. The bangbang 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 signdefiniteness 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 bangbang controls may have not more than 11 switchings. For particular patterns of bangbang 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 bangbang trajectories, i.e., the time when these trajectories lose their optimality. This question will be considered in subsequent works. 