Time-Optimal Synthesis for Left-Invariant Control Systems on SO (3)

Consider the control system (Σ) given by x=x(f+ug), where x∈ SO(3), |u|≤ 1, and f,g∈ so(3) define two perpendicular left-invariant vector fields normalized so that \|f\|=cos() and \|g\|=sin(), ∈ ]0,π/4[. In this paper, we provide an upper bound and a lower bound for N(α), the maximum number of switchings for time-optimal trajectories of (Σ). More precisely, we show that NS()≤ N()≤ NS()+4, where NS() is a suitable integer function of such that NS()2pt~→ 0π/(4α). The result is obtained by studying the time-optimal synthesis of a projected control problem on P², where the projection is defined by an appropriate Hopf fibration. Finally, we study the projected control problem on the unit sphere S². It exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations.