Yuri Sachkov

We study the left-invariant sub-Riemannian 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.

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.

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.

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.