Remarks on exact controllability for the Navier-Stokes equations

Oleg Yu. Imanuvilov

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • Volume: 6, page 39-72
  • ISSN: 1292-8119

Abstract

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We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain Ω with control distributed in a subdomain ω ⊂ Ω ⊂ ℝ n , n ∈ { 2 , 3 } . The result that we obtained in this paper is as follows. Suppose that v ^ ( t , x ) is a given solution of the Navier-Stokes equations. Let v 0 ( x ) be a given initial condition and ∥ v ^ ( 0 , · ) - v 0 ∥ < ε where ε is small enough. Then there exists a locally distributed control u , supp u ⊂ ( 0 , T ) × ω such that the solution v ( t , x ) of the Navier-Stokes equations: ∂ t v - Δ v + ( v , ∇ ) v = ∇ p + u + f , div v = 0 , v | ∂ Ω = 0 , v | t = 0 = v 0 coincides with v ^ ( t , x ) at the instant T : v ( T , x ) ≡ v ^ ( T , x ) .

How to cite

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Imanuvilov, Oleg Yu.. "Remarks on exact controllability for the Navier-Stokes equations." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 39-72. <http://eudml.org/doc/90600>.

@article{Imanuvilov2001,
abstract = {We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain $\Omega $ with control distributed in a subdomain $\omega \subset \Omega \subset \mathbb \{R\}^n, n\in \lbrace 2,3\rbrace $. The result that we obtained in this paper is as follows. Suppose that $\hat\{v\}(t,x)$ is a given solution of the Navier-Stokes equations. Let $ v_0(x)$ be a given initial condition and $\Vert \hat\{v\}(0,\cdot ) - v_0 \Vert &lt; \varepsilon $ where $\varepsilon $ is small enough. Then there exists a locally distributed control $u, \text\{supp\}\, u\subset (0,T)\times \omega $ such that the solution $v(t,x)$ of the Navier-Stokes equations:\[ \partial \_tv-\Delta v+(v,\nabla )v=\nabla p+u+f, \,\, \text\{div\}\, v=0,\,\, v\vert \_\{\partial \Omega \}=0, \,\, v \vert \_\{t=0\} = v\_0 \]coincides with $\hat\{v\}(t,x)$ at the instant $T$ : $v(T,x) \equiv \hat\{v\}(T,x)$.},
author = {Imanuvilov, Oleg Yu.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {locally distributed control; Navier-Stokes system; Carleman estimate},
language = {eng},
pages = {39-72},
publisher = {EDP-Sciences},
title = {Remarks on exact controllability for the Navier-Stokes equations},
url = {http://eudml.org/doc/90600},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Imanuvilov, Oleg Yu.
TI - Remarks on exact controllability for the Navier-Stokes equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 39
EP - 72
AB - We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain $\Omega $ with control distributed in a subdomain $\omega \subset \Omega \subset \mathbb {R}^n, n\in \lbrace 2,3\rbrace $. The result that we obtained in this paper is as follows. Suppose that $\hat{v}(t,x)$ is a given solution of the Navier-Stokes equations. Let $ v_0(x)$ be a given initial condition and $\Vert \hat{v}(0,\cdot ) - v_0 \Vert &lt; \varepsilon $ where $\varepsilon $ is small enough. Then there exists a locally distributed control $u, \text{supp}\, u\subset (0,T)\times \omega $ such that the solution $v(t,x)$ of the Navier-Stokes equations:\[ \partial _tv-\Delta v+(v,\nabla )v=\nabla p+u+f, \,\, \text{div}\, v=0,\,\, v\vert _{\partial \Omega }=0, \,\, v \vert _{t=0} = v_0 \]coincides with $\hat{v}(t,x)$ at the instant $T$ : $v(T,x) \equiv \hat{v}(T,x)$.
LA - eng
KW - locally distributed control; Navier-Stokes system; Carleman estimate
UR - http://eudml.org/doc/90600
ER -

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Citations in EuDML Documents

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  1. S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system
  2. Sergio Guerrero, Controllability of systems of Stokes equations with one control force : existence of insensitizing controls
  3. Jean-Pierre Puel, Inégalités de Carleman globales pour les problèmes elliptiques non homogènes
  4. Yuning Liu, Takéo Takahashi, Marius Tucsnak, Single input controllability of a simplified fluid-structure interaction model
  5. Muriel Boulakia, Axel Osses, Local null controllability of a two-dimensional fluid-structure interaction problem
  6. Fágner D. Araruna, Enrique Fernández-Cara, Diego A. Souza, Uniform local null control of the Leray-α model
  7. Mehdi Badra, Takéo Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems
  8. Sylvain Ervedoza, Local exact controllability for the 1 -d compressible Navier-Stokes equations
  9. Sergio Guerrero, Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions
  10. Muriel Boulakia, Axel Osses, Local null controllability of a two-dimensional fluid-structure interaction problem

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