Optimal Bang-Bang Trajectories in Sub-Finsler Problem on the Cartan Group

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.