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A Family of Stabilization Problems for the Oseen Equations

  • Jean-Pierre Raymond
Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 155)

Abstract

The feedback stabilization of the Navier-Stokes equations around an unstable stationary solution is related to the feedback stabilization of the Oseen equations (the linearized Navier-Stokes equations about the unstable stationary solution). In this paper we investigate the regularizing properties of feedback operators corresponding to a family of optimal control problems for the Oseen equations.

Keywords

Weak Solution Optimal Control Problem Regularity Result Unbounded Operator Analytic Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    V. Barbu, Feedback stabilization of the Navier-Stokes equations, ESAIM COCV, 9 (2003), 197–206.MATHMathSciNetGoogle Scholar
  2. [2]
    V. Barbu, S.S. Sritharan, H∞-control theory of fluid dynamics, Proc. R. Soc. Lond. A 454 (1998), 3009–3033.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    V. Barbu, R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana University Journal, 52(5) (2004), 1443–1494.CrossRefMathSciNetGoogle Scholar
  4. [4]
    V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier-Stokes equations, Memoirs of the A.M.S., 2005, to appear.Google Scholar
  5. [5]
    A. Bensoussan, G. Da Prato, M.C. Delfour, S.K. Mitter, Representation and Control of Infinite-Dimensional Systems, Vol. 1, Birkhäuser, 1992.Google Scholar
  6. [6]
    A. Bensoussan, G. Da Prato, M.C. Delfour, S.K. Mitter, Representation and Control of Infinite-Dimensional Systems, Vol. 2, Birkhäuser, 1993.Google Scholar
  7. [7]
    E. Fernandez-Cara, S. Guerrero, O. Yu.Imanuvilov, J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl., Vol. 83 (2004), 1501–1542.MATHMathSciNetGoogle Scholar
  8. [8]
    A.V. Fursikov, M.D. Gunzburger, L.S. Hou, Inhomogeneous boundary value problems for the three-dimensional evolutionary Navier-Stokes equations, J. Math. Fluid Mech., 4 (2002), 45–75.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    A.V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control, J. Math. Fluid Mech., 3 (2001), 259–301.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    A.V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete and Cont. Dyn. Systems, 10 (2004), 289–314.MATHMathSciNetGoogle Scholar
  11. [11]
    I. Lasiecka, R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control, part 1, Appl. Math. Optim., Vol. 16 (1987), 147–168.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    I. Lasiecka, R. Triggiani, Control Theory for Partial Differential Equations, Vol. 1, Cambridge University Press, 2000.Google Scholar
  13. [13]
    I. Lasiecka, R. Triggiani, Control Theory for Partial Differential Equations, Vol. 2, Cambridge University Press, 2000.Google Scholar
  14. [14]
    J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, 2005, submitted.Google Scholar
  15. [15]
    J.-P. Raymond, Boundary feedback stabilization of the two-dimensional Navier-Stokes equations, 2005, to appear in SIAM J. Control and Optim..Google Scholar
  16. [16]
    J.-P. Raymond, Boundary feedback stabilization of the three-dimensional Navier-Stokes equations, in preparation.Google Scholar
  17. [17]
    J.-P. Raymond, Feedback boundary stabilization of the Navier-Stokes equations, in Control Systems: Theory, Numerics and Applications, Proceedings of Science, PoS(CSTNA2005)003, htpp://pos.sissa.it.Google Scholar
  18. [18]
    R. Temam, Navier-Stokes equations, North-Holland, 1984.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Jean-Pierre Raymond
    • 1
  1. 1.Laboratoire MIP, UMR CNRS 5640Université Paul SabatierToulouse Cedex 9France

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