Limit time optimal synthesis for a control-affine system on S2

For α ∈]0,π/2[, let (Σ)α be the control system ẋ = (F+uG)x, where x belongs to the two-dimensional unit sphere S2, u ∈ [-1, 1], and F, G are 3 × 3 skew-symmetric matrices generating rotations with perpendicular axes and of respective norms cos(α) and sin(α). In this paper, we study the time optimal synthesis (TOS) from the north pole (0, 0, 1)T' associated to (Σ)α, as the parameter a tends to zero; this problem is motivated by specific issues in the control of quantum systems. We first prove that the TOS is characterized by a "two-snakes" configuration on the whole S2, except for a neighborhood Uα of the south pole (0, 0, -1)T of diameter at most Ο(α). We next show that, inside Uα, the TOS depends on the relationship between r(α) = π/2α -[π/2α] and α. More precisely, we characterize three main relationships by considering sequences (αk)k≥0 satisfying (a) r(αk) = r̄, (b) r(αk) = Cαk, and (c) r(αk) = 0, where r̄ ∈ (0, 1) and C andgt; 0. In each case, we describe the TOS and provide, after a suitable rescaling, the limiting behavior, as α tends to zero, of the corresponding TOS inside Uα. © 2008 Society for Industrial and Applied Mathematics