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 (−1, 1), associated with the fractional Laplace operator (−∂2 x)s, where 0 < s < 1. We show that there is a control function, which is localized in a nonempty open set O ⊂ (R \ (−1, 1)), that is, at the exterior of the interval (−1, 1), such that the system is null controllable at any time T > 0 if and only if 1/2 < s < 1.