Insensitizing controls for a semilinear parabolic equation: A numerical approach

Franck Boyer; Víctor Hernández-Santamaría; Luz De Teresa

doi:10.3934/mcrf.2019007

Article Contents

2019, Volume 9, Issue 1: 117-158. Doi: 10.3934/mcrf.2019007

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Insensitizing controls for a semilinear parabolic equation: A numerical approach

1.
Institut de Mathématiques de Toulouse & Institut universitaire de France, UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France
2.
Depto. de Control Automático, CINVESTAV-IPN, Apartado Postal 14-740, 07000 México, D.F., México
3.
DeustoTech, University of Deusto, 48007 Bilbao, Basque Country, Spain
4.
Facultad de Ingeniería, Universidad de Deusto, Avda Universidades 24, 48007 Bilbao, Basque Country, Spain
5.
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U. 04510 México D.F., México

* Corresponding author: F. Boyer
* Corresponding author: F. Boyer

Received: May 2017

Revised: November 2017

Published: January 2019

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Abstract

In this paper, we study the insensitizing control problem in the discrete setting of finite-differences. We prove the existence of a control that insensitizes the norm of the observed solution of a 1-D semi discrete parabolic equation. We derive a (relaxed) observability estimate that yields a controllability result for the cascade system arising in the insensitizing control formulation. Moreover, we deal with the problem of computing numerical approximations of insensitizing controls for the heat equation by using the Hilbert Uniqueness Method (HUM). We present various numerical illustrations.

Keywords:

Mathematics Subject Classification: 35K15, 65M06, 93C20.

Citation:

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Figure 1. $f(y) = -0.1 \sin(y)$, $y_0 = 0$, $\xi(x, t) = {\bf{1}}_{[0.4, 1]}(t)$. Uncontrolled solution

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Figure 2. $f(y) = -0.1 \sin(y)$, $y_0 = 0$, $\xi(x, t) = {\bf{1}}_{[0.4, 1]}(t)$. Controlled solution

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Figure 3. Convergence properties of the method for insensitizing problem

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Figure 4. Value of $\Psi(y)$ for different parameters $\tau$ and initial perturbations $w_0$

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Figure 5. $y_0(x) = {\bf{1}}_{(0.2, 0.7)}(x)$, $\xi = 0$, $f = 0$, $\omega = (0.3, 0.8)$. Same legend as in Figure 3

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Figure 6. The case where $\mathcal{O} = \Omega$ with $\xi = 0$, $\omega = (0, 0.5)$. Same legend as in Figure 3

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Figure 7. Different values of $\mathcal{M}$ in the source term. Same legend as in Figure 3

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Figure 8. The case where $\omega \cap \mathcal{O} = \emptyset$. Same legend as in Figure 3

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Figure 9. Convergence properties for the quadratic case. Same legend as in Figure 3.

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Figure 10. $f(y) = -y^2$, $y_0 = 0$, $\xi(x, t) = 8\times {\bf{1}}_{[0.2, 1]}(t)$. Time evolution

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Figure 11. $f(y) = -y^2$, $y_0 = 0$, $\xi(x, t) = 8\times {\bf{1}}_{[0.2, 1]}(t)$. Controlled solution

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Figure 12. Simultaneous insensitizing and null-control

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