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Local Controllability to Trajectories of the Magnetohydrodynamic Equations

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Abstract

In this paper we prove the local controllability to trajectories of the three dimensional magnetohydrodynamic equations by means of two internal controls, one in the velocity equations and the other in the magnetic field equations and both localized in an arbitrary small subset with not empty interior of the domain. This paper improves the previous results (Barbu et al. in Comm Pure Appl Math 56:732–783, 2003; Barbu et al. in Adv Differ Equ 10:481–504, 2005; Havârneanu et al. in Adv Differ Equ 11:893–929, 2006; Havârneanu, in SIAM J Control Optim 46:1802–1830, 2007) where the second control is not localized and it allows to deduce the local controllability to trajectories with boundary controls. The proof relies on the Carleman inequality for the Stokes system of Imanuvilov et al. (Carleman estimates for second order nonhomogeneous parabolic equations, preprint) to deal with the velocity equations and on a new Carleman inequality for a Dynamo-type equation to deal with the magnetic field equations.

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References

  1. Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21(9), 823–864 (1998)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  2. Amrouche, C., Girault, V.: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslovak Math. J. 44(119)(1), 109–140 (1994)

  3. Badra, M.: Global Carleman inequalities for Stokes and penalized Stokes equations. Math. Control Relat. Fields 1(2), 149–175 (2011)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  4. Badra, M., Ervedoza, S., Guerrero, S.: Local controllability to trajectories for non-homogeneous 2-d incompressible Navier-Stokes equations (preprint)

  5. Badra, M., Takahashi, T.: On Fattorini criterion for approximate controllability and stabilizability of parabolic equations ESAIM Control Optim. Calc. Var. 20(3), 924–956 (2014)

  6. Barbu, V., Havârneanu, T., Popa, C., Sritharan, S.S.: Exact controllability for the magnetohydrodynamic equations. Comm. Pure Appl. Math. 56(6), 732–783 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barbu, V., Havârneanu, T., Popa, C., Sritharan, S.S.: Local exact controllability for the magnetohydrodynamic equations revisited. Adv. Differ. Equ. 10(5), 481–504 (2005)

    MATH  Google Scholar 

  8. Barbu, V., Rodrigues, S.S., Shirikyan, A.: Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations. SIAM J. Control Optim. 49(4), 1454–1478 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bensoussan, A., Da Prato, G., Delfour, M.C., Mitter, S.K.: Representation and control of infinite dimensional systems. Systems & Control: Foundations & Applications Birkhäuser Boston Inc., Boston, MA, second edition (2007)

  10. Coron, J.-M., Fursikov, A.V.: Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4(4), 429–448 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Coron, J.-M., Guerrero, S.: Null controllability of the N-dimensional Stokes system with N−1 scalar controls. J. Differ. Equ. 246(7), 2908–2921 (2009)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  12. Dautray, R., Lions, J.-L.: Analyse mathématique et calcul numérique pour les sciences et les techniques, vol. 5. INSTN: Collection Enseignement. [INSTN: Teaching Collection]. Masson, Paris, 1988. Spectre des opérateurs. [The operator spectrum], With the collaboration of Michel Artola, Michel Cessenat, Jean Michel Combes and Bruno Scheurer, Reprinted from the 1984 edition

  13. Davidson, P.A.: An introduction to magnetohydrodynamics. In: Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2001)

  14. Fernández-Cara, E., González-Burgos, M., Guerrero, S., Puel, J.-P.: Null controllability of the heat equation with boundary Fourier conditions: the linear case. ESAIM Control Optim. Calc. Var. 12(3), 442–465 (electronic) (2006)

  15. Fernández-Cara, E., Guerrero, S.: Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45(4), 1399–1446 (electronic) (2006)

  16. Fernández-Cara, E., Guerrero, S.: Local exact controllability of micropolar fluids. J. Math. Fluid Mech. 9(3), 419–453 (2007)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. Fernández-Cara, E., Guerrero, S., Imanuvilov, O.Yu., Puel, J.-P.: Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. (9) 83(12), 1501–1542 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fernández-Cara, E., Guerrero, S., Imanuvilov, O.Yu., Puel, J.-P.: Some controllability results for the N-dimensional Navier-Stokes and Boussinesq systems with N−1 scalar controls. SIAM J. Control Optim. 45(1), 146–173 (electronic) (2006)

  19. Foiaş, C., Temam, R.: Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(1), 28–63 (1978)

    MathSciNet  Google Scholar 

  20. Fursikov, A.V.: Exact boundary zero controllability of three-dimensional Navier-Stokes equations. J. Dynam. Control Syst. 1(3), 325–350 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fursikov, A.V., Èmanuilov, O.Yu.: Exact controllability of the Navier-Stokes and Boussinesq equations. Uspekhi Mat. Nauk 54(3(327)), 93–146 (1999)

    Article  Google Scholar 

  22. Fursikov, A.V., Imanuvilov, O.Yu.: On exact boundary zero-controllability of two-dimensional Navier-Stokes equations. Acta Appl. Math. 37(1–2), 67–76, 1994 Mathematical problems for Navier-Stokes equations (Centro, 1993)

  23. Fursikov, A.V., Imanuvilov, O.Yu.: On controllability of certain systems simulating a fluid flow. In: Flow control (Minneapolis, MN 1992) volume 68 of IMA Vol. Math. Appl. pages 149–184. Springer, New York, 1995

  24. Fursikov, A.V., Imanuvilov, O.Yu.: Controllability of evolution equations. In: Lecture Notes Series, vol. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1996)

  25. Galdi, G.P. An introduction to the mathematical theory of the Navier-Stokes equations, vol. I. In: Linearized Steady Problems, vol. 38. Springer Tracts in Natural Philosophy. Springer, New York

  26. González-Burgos, M., Guerrero, S., Puel, J.-P.: Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Commun. Pure Appl. Anal. 8(1), 311–333 (2009)

    MathSciNet  MATH  Google Scholar 

  27. Grisvard, P.: Caractérisation de quelques espaces d’interpolation. Arch. Rational Mech. Anal. 25, 40–63 (1967)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  28. Guerrero, S.: Local exact controllability to the trajectories of the Boussinesq system. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(1), 29–61 (2006)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  29. Guerrero, S.: Controllability of systems of Stokes equations with one control force: existence of insensitizing controls. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(6), 1029–1054 (2007)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  30. Guerrero, S., Guillén-González, F.: On the controllability of the hydrostatic Stokes equations. J. Math. Fluid Mech. 10(3), 402–422 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Havârneanu, T., Popa, C., Sritharan, S.S.: Exact internal controllability for the magnetohydrodynamic equations in multi-connected domains. Adv. Differ. Equ. 11(8), 893–929 (2006)

    MATH  Google Scholar 

  32. Havârneanu, T., Popa, C., Sritharan, S.S.: Exact internal controllability for the two-dimensional magnetohydrodynamic equations. SIAM J. Control Optim. 46(5), 1802–1830 (electronic) (2007)

  33. Imanuvilov, O.Y., Yamamoto, M.: Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. Res. Inst. Math. Sci. 39(2), 227–274 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Imanuvilov, O.Yu.: Remarks on exact controllability for the Navier-Stokes equations. ESAIM Control Optim. Calc. Var. 6, 39–72 (electronic) (2001)

  35. Imanuvilov, O.Yu., Puel, J.-P., Yamamoto, M.: Carleman estimates for parabolic equations with nonhomogeneous boundary conditions. Chin. Ann. Math. Ser. B 30(4), 333–378 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Imanuvilov, O.Yu., Puel, J.-P., Yamamoto, M.: Carleman estimates for second order nonhomogeneous parabolic equations. Preprint

  37. Imanuvilov, O.Yu., Takahashi, T.: Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. (9) 87(4), 408–437 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kim, T., Cao, D.: Local exact controllability of the Navier-Stokes equations with the condition on the pressure on parts of the boundary. SIAM J. Control Optim. 48(6), 3805–3837 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lefter, C.-G.: On a unique continuation property related to the boundary stabilization of magnetohydrodynamic equations. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 56(1), 1–15 (2010)

    MathSciNet  MATH  Google Scholar 

  40. Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 1. Travaux et Recherches Mathématiques, No. 17. Dunod, Paris (1968)

  41. Raymond, J.-P., Vanninathan, M.: Null controllability in a fluid-solid structure model. J. Differ. Equ. 248(7), 1826–1865 (2010)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  42. Tucsnak, M., Weiss, G.: Observation and control for operator semigroups. In: Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel

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  1. Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, 64013, Pau Cedex, France

    Mehdi Badra

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Communicated by A. V. Fursikov

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Badra, M. Local Controllability to Trajectories of the Magnetohydrodynamic Equations. J. Math. Fluid Mech. 16, 631–660 (2014). https://doi.org/10.1007/s00021-014-0186-1

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