  We consider the null controllability problem from the exterior for the one
dimensional heat equation on the interval $(0,1)$ associated with the
fractional Laplace operator $(-\partial_x^2)^s$, where $0<s<1$. We show that
there is a control function which is localized in a non-empty open set
$\mathcal{O}\subset \left(\mathbb{R}\setminus(0,1)\right)$, that is, at the
exterior of the interval $(0,1)$, such that the system is null controllable at
any time $T>0$ if and only if $\frac 12<s<1$.
