Local null controllability of the N-dimensional Navier–Stokes system with nonlinear Navier-slip boundary conditions and N − 1 scalar controls

Guerrero, Sergio; Montoya, Cristhian

doi:10.1016/j.matpur.2018.03.004

Cited by 13 publications

(24 citation statements)

References 11 publications

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“…Our theorems extend the results of [2] and [12]. Taking into account the relation between the observability and controllability property, it will be appropriate to consider the following adjoint system related to (2):…”

Section: Introductionsupporting

confidence: 62%

“…For instance, N. Carreño and S. Guerrero in [2] have proven the local null controllability for the Navier-Stokes system with Dirichlet conditions and N − 1 scalar controls. The recent work made by S. Guerrero and C. Montoya shows that the local null controllability property is achieved for the N -dimensional Navier-Stokes system with Navier-slip conditions and N − 1 scalar controls [12]. The methodology in the previous articles are Carleman estimates.…”

Section: Introductionmentioning

confidence: 99%

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Remarks on local controllability for the Boussinesq system with Navier boundary condition

Montoya¹

2020

Comptes Rendus. Mathématique

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This note deals with the local exact controllability to a particular class of trajectories for the Boussinesq system with nonlinear Navier-slip boundary conditions and internal controls having vanishing components. Briefly speaking, in two dimensions, the local exact controllability property is obtained using only one control in the heat equation, whereas two scalar controls are required in three dimensions.Résumé. Cette note concerne la contrôlabilité locale d'une classe particulière de trajectoires, ceci pour le système de Boussinesq avec la condition de Navier non linéaire et certains contrôles internes. En bref, la propriété de contrôlabilité exacte locale s'obtient en dimension deux en n'utilisant que le contrôle associé à l'équation de la chaleur. En revanche, deux contrôles scalaires sont nécessaires pour obtenir notre résultat dans le cas de dimension trois Funding. This work has been supported by FONDECYT grant 3180100.

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Section: Introductionsupporting

confidence: 62%

Section: Introductionmentioning

confidence: 99%

Remarks on local controllability for the Boussinesq system with Navier boundary condition

Montoya¹

2020

Comptes Rendus. Mathématique

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“…The study of control problems associated to Stokes systems with Navier-slip boundary condition (see (1.1)) is not new in the literature: the existence of d dimensional controls leading to null controllability was proved in [18], while the existence of d − 1 dimensional controls has recently been proved in [21]. The need of working in a specific domain to simplify the problem is not new either.…”

Section: Historical Backgroundmentioning

confidence: 99%

Uniform Controllability of a Stokes Problem with a Transport Term in the Zero-Diffusion Limit

Bárcena-Petisco¹

2020

SIAM J. Control Optim.

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In this paper we consider a Stokes system with Navier-slip boundary conditions. The main results concern the behaviour of the cost of null controllability with respect to the diffusion coefficient when the control acts in the interior. In particular, we prove in (0, π) 2 that for a sufficiently large time the cost decays exponentially as the diffusion coefficient vanishes, whereas in (0, π) 3 we prove that for most of the control domains and for any time T > 0 the cost explodes exponentially as the diffusion coefficient vanishes.

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“…Local estimates. In this step, we turn back to our original function and use bootstrap arguments as in [7] and [21] to estimate the local term associated to |w xx |.…”

Section: Carleman Inequalitymentioning

confidence: 99%

“…Proof. The proof follows some ideas [21] and therefore we only give a sketch of the proof. Let us now set…”

Section: Null Controllability Of the Linearized Systemmentioning

confidence: 99%

Local exact controllability to the trajectories of the Korteweg–de Vries–Burgers equation on a bounded domain with mixed boundary conditions

Cerpa

Montoya

Zhang

2020

Journal of Differential Equations

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This paper studies the internal control of the Korteweg-de Vries-Burgers (KdVB) equation on a bounded domain. The diffusion coefficient is time-dependent and the boundary conditions are mixed in the sense that homogeneous Dirichlet and periodic Neumann boundary conditions are considered. The exact controllability to the trajectories is proven for a linearized system by using duality and getting a new Carleman estimate. Then, using an inversion theorem we deduce the local exact controllability to the trajectories for the original KdVB equation, which is nonlinear.

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Local null controllability of the N-dimensional Navier–Stokes system with nonlinear Navier-slip boundary conditions and N − 1 scalar controls

Cited by 13 publications

References 11 publications

Remarks on local controllability for the Boussinesq system with Navier boundary condition

Remarks on local controllability for the Boussinesq system with Navier boundary condition

Uniform Controllability of a Stokes Problem with a Transport Term in the Zero-Diffusion Limit

Local exact controllability to the trajectories of the Korteweg–de Vries–Burgers equation on a bounded domain with mixed boundary conditions

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Local null controllability of the N-dimensional Navier–Stokes system with nonlinear Navier-slip boundary conditions and N − 1 scalar controls

Cited by 13 publications

References 11 publications

Remarks on local controllability for the Boussinesq system with Navier boundary condition

Remarks on local controllability for the Boussinesq system with Navier boundary condition

Uniform Controllability of a Stokes Problem with a Transport Term in the Zero-Diffusion Limit

Local exact controllability to the trajectories of the Korteweg–de Vries–Burgers equation on a bounded domain with mixed boundary conditions

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Local null controllability of the N-dimensional Navier–Stokes system with nonlinear Navier-slip boundary conditions and N − 1 scalar controls