Remarks on exact controllability for the Navier-Stokes equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001), p. 39-72

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 ωΩ 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,suppu(0,T)×ω such that the solution v(t,x) of the Navier-Stokes equations: t v-Δv+(v,)v=p+u+f,divv=0,v| Ω =0,v| t=0 =v 0 coincides with v ^(t,x) at the instant T : v(T,x)v ^(T,x).

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 ωΩ n ,n{2,3}. On obtient le résultat suivant. Supposons que v ^(t,x) soit une solution des équations de Navier-Stokes et v 0 (x) une condition initiale telle que v ^(0,·)-v 0 <ε pour ε assez petit. On montre alors qu’il existe un contrôle localement réparti u,suppu[0,T]×ω, tel que la solution v(t,x) des équations de Navier-Stokes : t v-Δv+(v,)v=p+u+f,divv=0,v| Ω =0,v| t=0 =v 0 coïncide avec v ^(t,x) au temps T : v(T,x)v ^(T,x).

Classification:  35K10,  35K55,  35K60,  93B05,  93C10,  93C20
Keywords: locally distributed control, Navier-Stokes system
@article{COCV_2001__6__39_0,
     author = {Imanuvilov, Oleg Yu.},
     title = {Remarks on exact controllability for the Navier-Stokes equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2001},
     pages = {39-72},
     zbl = {0961.35104},
     mrnumber = {1804497},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2001__6__39_0}
}
Imanuvilov, Oleg Yu. Remarks on exact controllability for the Navier-Stokes equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001) pp. 39-72. http://www.numdam.org/item/COCV_2001__6__39_0/

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