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

• Published in: SIAM Journal on Control and Optimization
• Main Authors: Mason, Paolo, Salmoni, Rebecca, Boscain, Ugo, Chitour, Yacine
• Other Authors: Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), SISSA MathLab Trieste
• Format: Article in Journal/Newspaper
• Language: English
• Published: HAL CCSD 2008
• Subjects: Asymptotics; Control of quantum systems; Control-affine systems; Minimum time; Optimal synthesis; [MATH]Mathematics [math]; North Pole; South Pole; South pole
• Online Access: https://hal.science/hal-02320804; https://doi.org/10.1137/060675988

International audience 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.