Optimal Bang-Bang Trajectories in Sub-Finsler Problems on the Engel 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.