# American Institute of Mathematical Sciences

April  2020, 19(4): 1949-1978. doi: 10.3934/cpaa.2020086

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

 1 Chair of Computational Mathematics, Fundación Deusto, Av. de las Universidades 24, 48007 Bilbao, Basque Country, Spain 2 Facultad de Ingeniería, Universidad de Deusto, Av. de las Universidades 24, 48007 Bilbao, Basque Country, Spain 3 George Mason University, Department of Mathematical Sciences, Fairfax, VA 22030, USA 4 Chair in Applied Analysis, Alexander von Humboldt-Professorship, Department of Mathematics, Friedrich-Alexander-Universität, Erlangen-Nürnberg, 91058 Erlangen, Germany 5 Departamento de Matemáticas, Universidad Autonóma de Madrid, 24049, Madrid, Spain

Dedicated to professor Tomás Caraballo on the occasion of his 60th birthday

Received  May 2019 Revised  October 2019 Published  January 2020

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 $(-d_x^{\,2})^{s}{}$ ($0<s<1$) on the interval $(-1,1)$. We prove the 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.

Citation: Umberto Biccari, Mahamadi Warma, Enrique Zuazua. Controllability of the one-dimensional fractional heat equation under positivity constraints. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1949-1978. doi: 10.3934/cpaa.2020086
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Graphic of the function $q(x)$
Evolution in the time interval $(0,T_{\rm min})$ of the solution of (2.1) with $s = 0.8$. The blue curve is the target we want to reach while the green bullets indicate the target we computed numerically
Evolution in the time interval $(0,T_{\rm min})$ of the solution of (2.1) with $s = 0.8$. The blue curve is the target we want to reach while the green bullets indicate the target we computed numerically
Minimal-time control: space-time distribution of the impulses. The white lines delimit the control region $\omega = (-0.3,0.8)$. The regions in which the control is active are marked in yellow
Minimal-time control: intensity of the impulses in logarithmic scale. In the $(t,x)$ plane in blue the time $t$ varies from $t = 0$ (left) to $t = T_{\rm min}$ (right)
Minimal-time control: intensity of the impulses in logarithmic scale. In the $(t,x)$ plane in blue the time $t$ varies from $t = 0$ (left) to $t = T_{\rm min}$ (right)
Evolution in the time interval $(0,0.9)$ of the solution of (2.1) with $s = 0.8$. The blue curve is the target we want to reach while the green bullets indicate the target we computed numerically
Behavior of the control in time $T = 0.9$. The white lines delimit the control region $\omega = (-0.3,0.8)$. The regions in which the control is active are marked in yellow. The atomic nature is lost
Evolution in the time interval $(0,0.7)$ of the solution of (2.1) with $s = 0.8$ (left) and of the control $u$ (right), under the constraint $u\geq 0$. The bold characters highlight the control region $\omega = (-0.3,0.8)$. The control remains inactive during the entire time interval, and the equation is not controllable
Minimal-time control: space-time distribution of the impulses. The white lines delimit the control region $\omega = (-0.3,0.8)$. The regions in which the control is active are marked in yellow
Evolution in the time interval $(0,0.4)$ of the solution of (2.1) with $s = 0.8$. The blue curve is the target we want to reach while the green bullets indicate the target we computed numerically
Behavior of the control in time $T = 0.4$. The white lines delimit the control region $\omega = (-0.3,0.8)$. The regions in which the control is active are marked in yellow. The atomic nature is lost
Evolution in the time interval $(0,0.15)$ of the solution of (2.1) with $s = 0.8$ (left) and of the control $u$ (right), under the constraint $u\geq 0$. The bold characters highlight the control region $\omega = (-0.3,0.8)$. The equation is not controllable
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