arXiv2006
For $α\in(0,π/2)$, let $(Σ)_α$ 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 $\cos(α)$ 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 $α$ 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 $U_α$ of the south pole $(0,0,-1)^T$ of diameter at most $Ø(α)$. We next show that, inside $U_α$, 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 $(α_k)_{k\geq 0}$ satisfying $(a)$$r(α_k)=\bar{r}$; $(b)$ $r(α_k)=Cα_k$ and $(c)$ $r(α_k)=0$, 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 $α$ tends to zero, of the corresponding TOS inside $U_α$.