Controllability of the one-dimensional fractional heat equation under positivity constraints

Abstract

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 (−dx2)s (0 < s < 1) on the interval (−1, 1). We prove the existence of 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 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.