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For $\alpha \in ]0,\pi/2[$, let $(\Sigma)_\alpha$ be the control system $\dot{x}=(F+uG)x$, where $x$ belongs to the two-dimensional unit sphere $S^2$, $u\in [-1,1]$, and $F,G$ are $3\times3$ skew-symmetric matrices generating rotations with perpendicular axes and of respective norms $\cos(\alpha)$ and $\sin(\alpha)$. In this paper, we study the time optimal synthesis (TOS) from the north pole $(0,0,1)^T$ associated to $(\Sigma)_\alpha$, as the parameter $\alpha$ 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 $S^2$, except for a neighborhood $U_\alpha$ of the south pole $(0,0,-1)^T$ of diameter at most ${\cal O}(\alpha)$. We next show that, inside $U_\alpha$, the TOS depends on the relationship between $r(\alpha):=\pi/2\alpha-[\pi/2\alpha]$ and $\alpha$. More precisely, we characterize three main relationships by considering sequences $(\alpha_k)_{k\geq 0}$ satisfying (a) $r(\alpha_k)=\bar{r}$, (b) $r(\alpha_k)=C\alpha_k$, and (c) $r(\alpha_k)=0$, where $\bar{r}\in (0,1)$ and $C0$. In each case, we describe the TOS and provide, after a suitable rescaling, the limiting behavior, as $\alpha$ tends to zero, of the corresponding TOS inside $U_\alpha$.

Limit Time Optimal Synthesis for a Control-Affine System on $S^2$