Remarks on exact controllability for the Navier-Stokes equations

Imanuvilov, Oleg Yu.

Imanuvilov, Oleg Yu.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 39-72.

Résumé
Abstract

On étudie le problème de contrôlabilité locale exacte pour les équations de Navier-Stokes incompressibles dans un domaine $Ω$ borné avec un contrôle réparti dans un sous-domaine $ω \subset Ω$ $\subset ℝ^{n}, n \in {2, 3} .$ On obtient le résultat suivant. Supposons que $\hat{v} (t, x)$ soit une solution des équations de Navier-Stokes et $v_{0} (x)$ une condition initiale telle que $∥ \hat{v} (0, \cdot) - v_{0} ∥ < ε$ pour $ε$ assez petit. On montre alors qu’il existe un contrôle localement réparti $u, supp u \subset [0, T] \times ω,$ tel que la solution $v (t, x)$ des équations de Navier-Stokes :

\partial_{t} v - Δ v + (v, \nabla) v = \nabla p + u + f, div v = 0, v {|_{\partial Ω} = 0, v |}_{t = 0} = v_{0}

coïncide avec

\hat{v} (t, x)

au temps

T

v (T, x) \equiv \hat{v} (T, x) .

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 $ω \subset Ω \subset ℝ^{n}, n \in {2, 3}$ . 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 $∥ \hat{v} (0, \cdot) - v_{0} ∥ < ε$ where $ε$ is small enough. Then there exists a locally distributed control $u, supp u \subset (0, T) \times ω$ such that the solution $v (t, x)$ of the Navier-Stokes equations:

\partial_{t} v - Δ v + (v, \nabla) v = \nabla p + u + f, div v = 0, v {|_{\partial Ω} = 0, v |}_{t = 0} = v_{0}

coincides with

\hat{v} (t, x)

at the instant

T

v (T, x) \equiv \hat{v} (T, x)

MR Zbl | 23 citations dans Numdam

Classification : 35K10, 35K55, 35K60, 93B05, 93C10, 93C20
Mots-clés : locally distributed control, Navier-Stokes system

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     author = {Imanuvilov, Oleg Yu.},
     title = {Remarks on exact controllability for the {Navier-Stokes} equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {39--72},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2001},
     mrnumber = {1804497},
     zbl = {0961.35104},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2001__6__39_0/}
}

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Imanuvilov, Oleg Yu. Remarks on exact controllability for the Navier-Stokes equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 39-72. http://www.numdam.org/item/COCV_2001__6__39_0/

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