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

10/31/2019
by   Harbir Antil, et al.
0

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 (-∂_x^2)^s (with 0<s<1) on the interval (-1,1). Our control function is localized in an open set O in the exterior of (-1,1), that is, O⊂ (R∖ (-1,1)). We show that there exists a minimal (strictly positive) time T_ 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 an exterior positive control, if and only if 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.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset