We study the controllability of trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator (−∂x2)s (with 0 < s < 1) on the interval (−1, 1). Our control function is localized in a bounded open set O in the exterior of (−1, 1), that is, O ⊂ R \ (−1, 1). We show that there exists a minimal (strictly positive) time Tmin such that the fractional heat dynamics can be controlled from any initial datum in L2 (−1, 1) to a positive trajectory through the action of an exterior positive control, if and only if 1/2 < s < 1. In addition, we prove that at this minimal controllability time, the constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. Finally, we provide several numerical illustrations that confirm our theoretical results.
