Controllability of the one-dimensional fractional heat equation under positivity constraints
EntityUAM. Departamento de Matemáticas
PublisherAmerican Institute of Mathematical Sciences (A I M S Press)
10.3934/cpaa.2020086Communications on Pure and Applied Analysis 19.4 (2020):1949-1978
ISSN1553-5258 (online); 1534-0392 (print)
Funded byThis 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 ANR
ProjectGobierno de España. MTM2017-92996-C2-1-R; info:eu-repo/grantAgreement/EC/H2020/694126/EU//DYCON; info:eu-repo/grantAgreement/EC/H2020/765579/EU//ConFlex
SubjectsConstrained controllability; Fractional heat equation; Waiting time; Matemáticas
NoteThis 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.2020086
Rights© 2020 American Institute of Mathematical Sciences. All rights reserved.
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
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