Remarks on exact controllability for the Navier-Stokes equations

Oleg Yu. Imanuvilov

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 (2010): 39-72. <http://eudml.org/doc/197343>.

@article{Imanuvilov2010,
abstract = { 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 $\omega\subset\Omega\subset \mathbb\{R\}^n, n\in\\{2,3\\}$. 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 $\Vert \hat v(0,\cdot) - v_0 \Vert < \varepsilon$ where ε 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: $$ \partial\_tv-\Delta v+(v,\nabla)v=\nabla p+u+f, \,\, \text\{\rm div\}\, v=0,\,\, v\vert\_\{\partial\Omega\}=0, \,\, v \vert\_\{t=0\} = v\_0 $$ 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.; locally distributed control; Navier-Stokes system; Carleman estimate},
language = {eng},
month = {3},
pages = {39-72},
publisher = {EDP Sciences},
title = {Remarks on exact controllability for the Navier-Stokes equations},
url = {http://eudml.org/doc/197343},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Imanuvilov, Oleg Yu.
TI - Remarks on exact controllability for the Navier-Stokes equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
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 Ω with control distributed in a subdomain $\omega\subset\Omega\subset \mathbb{R}^n, n\in\{2,3\}$. 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 $\Vert \hat v(0,\cdot) - v_0 \Vert < \varepsilon$ where ε 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: $$ \partial_tv-\Delta v+(v,\nabla)v=\nabla p+u+f, \,\, \text{\rm div}\, v=0,\,\, v\vert_{\partial\Omega}=0, \,\, v \vert_{t=0} = v_0 $$ 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.; locally distributed control; Navier-Stokes system; Carleman estimate
UR - http://eudml.org/doc/197343
ER -

References

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