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

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 of respective length $\cos(\alpha)$ 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. 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 $\O(\alpha)$. 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 $C>0$. 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 $U_\alpha$.