Local null controllability of the penalized Boussinesq system with a reduced number of controls

Jon Asier Bárcena-Petisco; Kévin Le Balc'h

doi:10.3934/mcrf.2021038

Article Contents

2022, Volume 12, Issue 3: 641-666. Doi: 10.3934/mcrf.2021038

This issue Previous Article Numerical analysis and simulations of a frictional contact problem with damage and memory Next Article On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

Local null controllability of the penalized Boussinesq system with a reduced number of controls

Jon Asier Bárcena-Petisco^1,2, , and
Kévin Le Balc'h^3,

1.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
2.
Chair of Computational Mathematics, Fundación Deusto, Avenida de las Universidades 24, 48007 Bilbao, Basque Country, Spain
3.
Institut de Mathématiques de Bordeaux, 351 Cours de la Libération, 33400 Bordeaux, France

^* Corresponding author: Jon Asier Bárcena-Petisco
^* Corresponding author: Jon Asier Bárcena-Petisco

Received: July 31, 2020

Revised: May 31, 2021

Early access: July 2021

Published: July 2022

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Abstract

In this paper we consider the Boussinesq system with homogeneous Dirichlet boundary conditions, defined in a regular domain $ \Omega\subset\mathbb R^N $ for $ N = 2 $ and $ N = 3 $. The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter $ \varepsilon > 0 $. We prove that our system is locally null controllable using a control with a restricted number of components, localized in an open set $ \omega $ contained in $ \Omega $. We also show that the control cost is bounded uniformly with respect to $ \varepsilon \rightarrow 0 $. The proof is based on a linearization argument. The null controllability of the linearized system is obtained by proving a new Carleman estimate for the adjoint system. This inequality is derived by exploiting the coercivity of some second order differential operator involving crossed derivatives.

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

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