Yu. L. Sachkov, Optimal bang-bang trajectories in sub-Finsler problem on the Cartan group

Title: Optimal bang-bang trajectories in sub-Finsler problem on the Cartan group
Authors: Yu. L. Sachkov
Journal title: Russian Journal of Nonlinear Dynamics
Year: 2018
Issue: 4
Volume: 4
Pages: 583-593

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 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 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.