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 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 -

References

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  1. [1] V.M. Alekseev, V.M. Tikhomirov and S.V. Fomin, Optimal control. Consultants Bureau, New York (1987). Zbl0689.49001MR924574
  2. [2] D. Chae, O.Yu. Imanuvilov and S.M. Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions. J. Dynam. Control Systems 2 (1996) 449–483. Zbl0946.93007
  3. [3] J.-M. Coron, On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier-Slip boundary conditions. ESAIM: COCV 1 (1996) 35–75. Zbl0872.93040
  4. [4] J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996) 155–188. Zbl0848.76013
  5. [5] J.-M. Coron, Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles bidimensionnels. C. R. Acad. Sci. Paris Sér. I Math. 317 (1993) 271–276. Zbl0781.76013
  6. [6] J.-M. Coron and A.V. Fursikov, Global exact controllability of the 2-D Navier-Stokes equations on manifold without boundary. Russian J. Math. Phys. 4 (1996) 1–20. Zbl0938.93030
  7. [7] C. Fabre, Résultats d’unicité pour les équations de Stokes et applications au contrôle. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 1191–1196. Zbl0861.35073
  8. [8] C. Fabre and G. Lebeau, Prolongement unique des solutions de l’équation de Stokes. Comm. Partial Differential Equations 21 (1996) 573–596. Zbl0849.35098
  9. [9] A.V. Fursikov and O.Yu. Imanuvilov, Local exact controllability of two dimensional Navier-Stokes system with control on the part of the boundary. Sb. Math. 187 (1996) 1355–1390. Zbl0869.35074
  10. [10] A.V. Fursikov and O.Yu. Imanuvilov, Local exact boundary controllability of the Boussinesq equation. SIAM J. Control Optim. 36 (1988) 391–421. Zbl0907.76020
  11. [11] A.V. Fursikov and O.Yu. Imanuvilov, Local exact controllability of the Navier-Stokes Equations. C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 275–280. Zbl0873.76020
  12. [12] A.V. Fursikov and O.Yu. Imanuvilov, Controllability of evolution equations, Lecture notes series (1996), no. 34 SNU, Seoul. Zbl0862.49004MR1406566
  13. [13] A.V. Fursikov and O.Yu. Imanuvilov, On approximate controllability of the Stokes system. Ann. Fac. Sci. Toulouse 11 (1993) 205–232. Zbl0925.93416
  14. [14] A.V. Fursikov and O.Yu. Imanuvilov, Exact controllability of the Navier-Stokes equations and the Boussinesq system. Russian Math. Surveys 54 (1999) 565–618. Zbl0970.35116
  15. [15] O. Glass, Contrôlabilité de l’équation d’Euler tridimensionnelle pour les fluides parfaits incompressibles, Séminaire sur les Équations aux Dérivées Partielles, 1997-1998, Exp No XV. École Polytechnique, Palaiseau (1998) 11. Zbl1175.93030
  16. [16] O. Glass, Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles en dimension 3. C. R. Acad. Sci. Paris Sér. I Math. (1997) 987–992. Zbl0897.76014
  17. [17] L. Hörmander, Linear partial differential operators. Springer-Verlag, Berlin (1963). Zbl0108.09301MR404822
  18. [18] T. Horsin, On the controllability of the Burgers equations. ESAIM: COCV 3 (1998) 83–95. Zbl0897.93034
  19. [19] O.Yu. Imanuvilov, On exact controllability for the Navier-Stokes equations. ESAIM: COCV 3 (1998) 97–131. Zbl1052.93502
  20. [20] O.Yu. Imanuvilov, Boundary controllability of parabolic equations. Sb. Math. 186 (1995) 879–900. Zbl0845.35040
  21. [21] O.Yu. Imanuvilov, Local exact controllability for the 2-D Navier-Stokes equations with the Navier slip boundary conditions. Lecture Notes in Phys. 491 (1977) 148–168. Zbl0897.35063
  22. [22] O.Yu. Imanuvilov and M. Yamamoto, On Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, UTMS 98-46. Zbl1065.35079
  23. [23] A.N. Kolmogorov and S.V. Fomin, Introductory real analysis. Dover Publications, INC, New York (1996). Zbl0213.07305MR377445
  24. [24] O.A. Ladyzenskaja and N.N. Ural’ceva, Linear and quasilinear equations of elliptic type. Academic Press, New York (1968). 
  25. [25] J.L. Lions, Contrôle des systèmes distribués singuliers. Gauthier-Villars, Paris (1983). Zbl0514.93001MR712486
  26. [26] J.L. Lions, Optimal control of systems governed by partial differential equations. Springer-Verlag (1971). Zbl0203.09001MR271512
  27. [27] J.-L. Lions, Are there connections between turbulence and controllability?, in 9 e Conférence internationale de l’INRIA. Antibes (1990). 
  28. [28] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems. Springer-Verlag, Berlin (1971). 
  29. [29] M. Taylor, Pseudodifferential operators. Princeton Univ. Press (1981). Zbl0453.47026MR618463
  30. [30] M. Taylor, Pseudodifferential operators and Nonlinear PDE. Birkhäuser (1991). Zbl0746.35062MR1121019
  31. [31] R. Temam, Navier-Stokes equations. North-Holland Publishing Company, Amsterdam (1979). Zbl0426.35003MR603444

Citations in EuDML Documents

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  1. S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system
  2. Jean-Pierre Puel, Inégalités de Carleman globales pour les problèmes elliptiques non homogènes
  3. Sergio Guerrero, Controllability of systems of Stokes equations with one control force : existence of insensitizing controls
  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. Muriel Boulakia, Axel Osses, Local null controllability of a two-dimensional fluid-structure interaction problem
  10. Sergio Guerrero, Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions

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