On the minimum time problem for driftless left-invariant control systems on SO(3)

In this paper, we investigate the structure of time-optimal trajectories for a driftless control system on $SO(3)$ of the type $\dot x=x(u_1f_1+u_2f_2), \quad |u_1|, \quad |u_2|\leq 1$, where $f_1,\quad f_2\in so(3)$ define two linearly independent left-invariant vector fields on $SO(3)$. We show that every time-optimal trajectory is a finite concatenation of at most five (bang or singular) arcs. More precisely, a time-optimal trajectory is, on the one hand, bang-bang with at most either two consecutive switchings relative to the same input or three switchings alternating between two inputs, or, on the other hand, a concatenation of at most two bangs followed by a singular arc and then two other bangs. We end up finding a finite number of three-parameters trajectory types that are sufficient for time-optimality.