  In this paper, we analyze the controllability properties under positivity
constraints on the control or the state of a one-dimensional heat equation
involving the fractional Laplacian $(-\Delta)^s$ ($0<s<1$) on the interval
$(-1,1)$. We prove the existence of a minimal (strictly positive) time $T_{\rm
min}$ 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 a positive
control, when $s>1/2$. Moreover, we show that in this minimal time constrained
controllability is achieved by means of a control that belongs to a certain
space of Radon measures. We also give some numerical simulations that confirm
our theoretical results.
