U. Biccari, M. Warma, E. Zuazua. Controllability of the one-dimensional fractional heat equation under positivity constraints Commun. Pure Appl. Anal., Vol 19. No. 4 (2020), pp. 1949-1978. DOI: 10.3934/cppaa.2020086
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 (-\Delta)^s (0<s<1) on the interval (-1,1) . We prove 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.
