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

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

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  • 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.

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

    Citation:

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

    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

    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

    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

    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)

    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)

    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

    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

    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

    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

    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

    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

    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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