Controllability properties from the exterior under positivity constraints for a 1-D fractional heat equation

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 (x2)s (−∂^2_x)^s ( with 0<s<1 0<s<1 ) on the interval (1,1) (−1,1) . Our control function is localized in an open set O O in the exterior of (1,1) (−1,1) , that is, O(R(1,1)) O⊂(R∖(−1,1)) . We show that there exists a minimal (strictly positive) time T T min such that the fractional heat dynamics can be controlled from any initial datum in L2(1,1) L^2(−1,1) to a positive trajectory through the action of an exterior positive control, if and only if 1/2<s<1 1/2<s<1 . 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.

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