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

Umberto Biccari; Mahamadi Warma; Enrique Zuazua

doi:10.3934/cpaa.2020086

Article Contents

2020, Volume 19, Issue 4: 1949-1978. Doi: 10.3934/cpaa.2020086

This issue Previous Article Longtime behavior for 3D Navier-Stokes equations with constant delays Next Article Stability and forward attractors for non-autonomous impulsive semidynamical systems

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

Abstract / Introduction Full Text(HTML) Figure(13) Related Papers Cited by

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 $ (-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.

Keywords:

Mathematics Subject Classification: 35K05, 35R11, 35S05, 93B05, 93C20.

Citation:

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Figure 1. Graphic of the function $ q(x) $

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Figure 2. 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

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Figure 8. 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

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Figure 3. 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

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Figure 4. 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)

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Figure 10. 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)

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Figure 5. 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

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Figure 6. 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

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Figure 7. 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

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Figure 9. 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

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Figure 11. 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

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Figure 12. 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

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Figure 13. 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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