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

Abstract

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Communications on Pure and Applied Analysis following peer review. The definitive publisher-authenticated version Biccari, U., Warma, M., & Zuazua, E. (2020). Controllability of the one-dimensional fractional heat equation under positivity constraints. Communications on Pure & Applied Analysis, 19(4), 1949-1978. is available online at https://www.aimsciences.org/article/doi/10.3934/cpaa.2020086In 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 resultsThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement NO. 694126-DyCon). The work of the three authors is partially supported by the Air Force Office of Scientific Research under Award NO: FA9550-18-1-0242. The work of the first and of the third author was partially supported by the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain) and by the ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government. The work of the third author was partially supported by the Alexander von Humboldt-Professorship program, the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement NO. 765579-ConFlex, and by the Grant ICON-ANR-16-ACHN-0014 of the French AN