Time Optimal Synthesis for a so(3)-left-invariant Control System on a Sphere
Résumé
Consider the control system (σ) given by ẋ = 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 NS(α) ≤ N(α) ≤ NS(α) + 4, where NS(α) is a suitable integer function of a such that NS(α) α → 0 ̃ π/4α. 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.