Controllability properties from the exterior under positivity constraints for a 1-d fractional heat equation

We study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator (-partial derivative(2)(x))s (with 0 < s < 1) on the interval (-1,1). Our control function is localized in an open set O in the exterior of (-1,1), that is, O subset of R\(-1,1)). We show that there exists a minimal (strictly positive) time T-min such that the fractional heat dynamics can be controlled from any initial datum in L-2(-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.