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

Abstract

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 $3\times 3$ 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 $Ø(\alpha )$. 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 $({\alpha }_k{)}_{k\ge 0}$ 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 $U_{\alpha }$.

28 pages, 10 figures