Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control

Chowdhury, Shirshendu; Mitra, Debanjana; Renardy, Michael

doi:10.3934/eect.2018022

Cited by 4 publications

(5 citation statements)

References 18 publications

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“…As we said in Section 2, all the Fourier modes except the Fourier mode 0 satisfy the unique continuation property. The article [16], which basically extends Lemma 4.1 to any non-zero Fourier mode, shows that the linearized system (1.4) can be controlled to zero provided the initial datum contains no Fourier mode 0.…”

Section: Null Controllability Issuesmentioning

confidence: 86%

“…This regularity is needed to obtain the regularity of the controlled trajectory. Let us also note that, as pointed out in the recent work [16], similar arguments as the one used for establishing Lemma 4.1 can be developed to show that for all k ∈ N \ {0}, the k-mode of the equation (1.4) is null-controllable with controls in L 2 (0, T ). The work [16] also shows that this family of equations is null-controllable uniformly with respect to the Fourier parameter k ∈ N\{0} through a deeper spectral analysis as the one we propose here.…”

Section: Null Controllability Of the 1-modes Of (39)mentioning

confidence: 87%

“…Let us also note that, as pointed out in the recent work [16], similar arguments as the one used for establishing Lemma 4.1 can be developed to show that for all k ∈ N \ {0}, the k-mode of the equation (1.4) is null-controllable with controls in L 2 (0, T ). The work [16] also shows that this family of equations is null-controllable uniformly with respect to the Fourier parameter k ∈ N\{0} through a deeper spectral analysis as the one we propose here. Still, we have chosen to present a detailed proof of Lemma 4.1 to underline how controls in H 1 0 (0, T ) can be constructed and to introduce several spectral computations that will be useful afterwards.…”

Section: Null Controllability Of the 1-modes Of (39)mentioning

confidence: 87%

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Open loop stabilization of incompressible Navier–Stokes equations in a 2d channel using power series expansion

Chowdhury

Ervedoza

2019

Journal de Mathématiques Pures et Appliquées

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In this article, we discuss the stabilization of incompressible Navier-Stokes equations in a 2d channel around a fluid at rest when the control acts only on the normal component of the upper boundary. In this case, the linearized equations are not controllable nor stabilizable at an exponential rate higher than νπ 2 /L 2 , when the channel is of width L and of length 2π and ν denotes the viscosity parameter. Our main result allows to go above this threshold and reach any exponential decay rate by using the non-linear term to control the directions which are not controllable for the linearized equations. Our approach therefore relies on writing the controlled trajectory as an expansion of order two taking the form εα + ε 2 β for some ε > 0 small enough. This method is inspired by the previous work [18] by J.-M. Coron and E. Crépeau on the controllability of the Korteweg de Vries equations.In this section, we recall some basic facts on the linearized equation (1.4) and give a modal description adapted to our setting.

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Section: Null Controllability Issuesmentioning

confidence: 86%

Section: Null Controllability Of the 1-modes Of (39)mentioning

confidence: 87%

Section: Null Controllability Of the 1-modes Of (39)mentioning

confidence: 87%

See 1 more Smart Citation

Open loop stabilization of incompressible Navier–Stokes equations in a 2d channel using power series expansion

Chowdhury

Ervedoza

2019

Journal de Mathématiques Pures et Appliquées

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“…To the best of our knowledge, there are almost no results regarding the controllability of Stokes system with controls acting on only normal or tangential components. We are only aware of [17] for the case of tangential controls on the whole boundary and of the results in [8] for the Stokes equation in a channel when the control is localized on the whole boundary of one side of the channel.…”

Section: Boundary Control Of 3d Stokes Equations Again One Can Considermentioning

confidence: 99%

Switching Controls for Analytic Semigroups and Applications to Parabolic Systems

Chaves-Silva¹,

Ervedoza²,

Souza³

2021

SIAM J. Control Optim.

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In this work, we extend the analysis of the problem of switching controls proposed in [E. Zuazua, J. Eur. Math. Soc. (JEMS), 13 (2011), pp. 85--117]. The problem asks the following question: Assuming that one can control a system using two or more actuators, does there exist a control strategy such that at all times, only one actuator is active? We answer positively when the controlled system corresponds to an analytic semigroup spanned by a positive self-adjoint operator which is null-controllable in arbitrary small times. Similarly to [E. Zuazua, J. Eur. Math. Soc. (JEMS), 13 (2011), pp. 85--117], our proof relies on analyticity arguments and will also work in finite dimensional settings and under some further spectral assumptions when the operator spans an analytic semigroup but is not necessarily self-adjoint.

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“…4.1 Null-controllability of the Boussinesq system A natural perspective that could be addressed in the future is the null-controllability of the linearized Boussinesq system (2). In this context, the techniques employed in the article [CMR18], which establishes the null-controllability of the incompressible Stokes equation, would be useful. The difficulty comes from the lengthy computations, due to the eigenvalue problem (24) and (25) associated to the ordinary differential equation of order six, instead of an ordinary differential equation of order four that appears for the Stokes problem, see [CMR18, Equation (2.4)].…”

Section: Perspectives and Related Referencesmentioning

confidence: 99%

Approximate controllability of the linearized Boussinesq system in a two dimensional channel

Balc’h¹

2019

Preprint

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In this paper, we prove an approximate controllability result for the linearized Boussinesq system around a fluid at rest, in a two dimensional channel, when the control acts only on the temperature, through the upper boundary. This result can be seen as a first step to obtain an open loop stabilization result of the nonlinear Boussinesq system, in the spirit of the article [CE19] by Chowdhury, Ervedoza concerning the Navier Stokes equations. The proof relies on the well-known Fattorini criterion, i.e. we show an unique continuation property for the adjoint system, by expanding the solution in Fourier series and using ordinary differential equations arguments. More precisely, we prove that the spectrum of the adjoint operator splits into two parts corresponding respectively to the Stokes eigenvalues and the Dirichlet Laplacian eigenvalues. Whereas the second part can be treated easily thanks to the wellknown form of the eigenfunctions, the first part requires to show that a matrix of size three, depending analytically of the parameters of the problem, is "generically" invertible.

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Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control

Cited by 4 publications

References 18 publications

Open loop stabilization of incompressible Navier–Stokes equations in a 2d channel using power series expansion

Open loop stabilization of incompressible Navier–Stokes equations in a 2d channel using power series expansion

Switching Controls for Analytic Semigroups and Applications to Parabolic Systems

Approximate controllability of the linearized Boussinesq system in a two dimensional channel

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