Remarks on the controllability of parabolic systems with non-diagonalizable diffusion matrix

Michel Duprez; Manuel González-Burgos; Diego A. Souza

doi:10.3934/dcdss.2023086

Article Contents

2023, Volume 16, Issue 6: 1346-1382. Doi: 10.3934/dcdss.2023086

This issue Previous Article Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation Next Article Blow-up and lifespan estimate for wave equations with critical damping term of space-dependent type related to Glassey conjecture

Remarks on the controllability of parabolic systems with non-diagonalizable diffusion matrix

1.
MIMESIS team, Inria Nancy - Grand Est and, MLMS team, Université de Strasbourg, France
2.
Dpto. Ecuaciones Diferenciales y Análisis Numérico and, Instituto de Matemáticas de la Universidad de Sevilla, Universidad de Sevilla, Spain

^*Corresponding author: Diego A. Souza
^*Corresponding author: Diego A. Souza

Received: July 2022

Revised: February 2023

Early access: April 24, 2023

Published online: April 24, 2023

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Abstract

The distributed null controllability for coupled parabolic systems with non-diagonalizable diffusion matrices with a reduced number of controls has been studied in the case of constant matrices. On the other hand, boundary controllability issues and distributed controllability with non-constant coefficients for this kind of systems is not completely understood. In this paper, we analyze the boundary controllability properties of a class of coupled parabolic systems with non-diagonalizable diffusion matrices in the constant case and the distributed controllability of a $ 2\times 2 $ non-diagonalizable parabolic system with space-dependent coefficients. For the boundary controllability problem, our strategy relies on the moment method. For the distributed controllability problem, our findings provide positive and negative control results by using the Fattorini-Hautus test and a fictitious control strategy.

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Mathematics Subject Classification: Primary: 93B05, 93C20; Secondary: 35K40.

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Figure 1. $ \sigma(L^*) $ for $ d = 2 $, $ n = 4 $ and $ \alpha = 2 $

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Figure 2. $ \sigma(L^*) $ for $ d = 2 $, $ n = 5$ and $ \alpha = 2 $

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Figure 3. Example of a function $ \psi $ in $ [0, \pi] $

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