H. Antil, U. Biccari, R. Ponce, M. Warma, S. Zamorano (2024) Controllability properties from the exterior under positivity constraints for a 1-D fractional heat equation, Vol. 13, pp. 893-924, https://doi.org/10.3934/eect.2024010
Abstract. 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 ( with ) on the interval . Our control function is localized in an open set in the exterior of , that is, . We show that there exists a minimal (strictly positive) time min such that the fractional heat dynamics can be controlled from any initial datum in to a positive trajectory through the action of an exterior positive control, if and only if . 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.