Time optimal synthesis for a so(3)-left-invariant control system on a sphere

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(α) and ||g|| = sin(α), α ∈ (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 N_S(α) ≤ N(α) ≤ N_S(α) ≤+4, where N_S(a) is a suitable integer function of α such that N_S(α) _α→0^∼. The result is obtained by studying the time optimal synthesis of a projected control problem on RP², where the projection is defined by an appropriate Hopf fibration.