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

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.

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