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Null controllability from the exterior of a one-dimensional nonlocal heat equation

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 $\frac 12<s<1$.