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On the Stabilization Problem by Feedback Control for Some Hydrodynamic Type Systems

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Fluids Under Control

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

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Abstract

The main hydrodynamic system that we study here is three-dimensional (3D) Navier–Stokes system. This chapter is divided into two parts: investigation of the cases of local and non-local stabilization.

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Notes

  1. 1.

    That is, linearized Navier–Stokes system.

  2. 2.

    This is one of the millennium problems.

  3. 3.

    Because we assume that right side \(f(x)=0\).

  4. 4.

    This is a very previous definition of the feedback notion. This notion will be discussed below in a detailed way.

  5. 5.

    Note that the analog of this extension operator \(\mbox{Ext}_\sigma v_0\) from \(\Omega \) to G is the extension operator (1.14) from the stabilization problem described in Sect. 1.2.1.

  6. 6.

    The definition and properties of sets \(M_-, M_+, M_g\) are given below in Sect. 1.8.3 in the case of differentiated Burgers equation.

  7. 7.

    That is, \(\| y(t,\cdot ;y_0)\|{ }_0\to \infty \) as \(0<t<t_0\) and \(t\to t_0.\)

  8. 8.

    The most complete results on geometrical structure of the sets \(M_-, M_+, M_g\) had been obtained in [15] in the case NPE, generated with 3D Helmholtz system. The same results for NPE corresponding differentiated Burgers equation can be obtained absolutely similarly.

  9. 9.

    Relation \([-\pi ,\pi ]:=\mathbb {T}\) means that the segment \([-\pi ,\pi ]\) is identified with the unit circumference of length \(2\pi \).

References

  1. A.V. Babin, M.I. Vishik, Attractors of Evolution Equations (North-Holland, Amsterdam, London, 1992)

    Google Scholar 

  2. V. Barbu, S.S. Sritharan, \(H^\infty \)-control theory of fluid dynamics. Proc. R. Soc. Lond. A 454, 3009–3033 (1998)

    Google Scholar 

  3. V. Barbu, I. Lasiecka, R. Triggiani, Abstract setting of tangential boundary stabilization strategies of Navier-Stokes equations by high-and low-gain feedback controllers. Nonlinear Analy. 64(12), 2704–2746 (2006)

    Article  Google Scholar 

  4. V. Barbu, I. Lasiecka, R. Triggiani, Local exponential stabilization strategies of Navier-Stokes equations, d=2,3 via feedback stabilization of its linearization, International Series of Numerical Mathematics, vol. 155 (Birkhäuser, Verlag, 2007), pp. 13–46

    Google Scholar 

  5. J.M. Coron, On null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domains. SIAM J. Control Optim. 37(6), 1874–1896 (1999)

    Article  MathSciNet  Google Scholar 

  6. J.M. Coron, Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136 (AMS, Providence, 2007)

    Google Scholar 

  7. J.M. Coron, A.V. Fursikov, Global exact controllability of the 2D Navier-Stokes equations on manifold without boundary. J. Russian Math. Phys. 4(3), 1–20 (1996)

    Google Scholar 

  8. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (AMS, Providence, 2002)

    Google Scholar 

  9. C. Foias, R. Temam, Remarques sur les équations de Navier-Stokes et les phénomènes successifs de bifurcation. Annali Scuola Norm. Sup. di Pisa Series IV V, 29–63 (1978)

    Google Scholar 

  10. A.V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications. Translations of Mathematical Monographs, vol. 187 (American Mathematical Society, Providence, 2000)

    Google Scholar 

  11. A.V. Fursikov, Stabilizability of quasi linear parabolic equation by feedback boundary control. Sbornik Mathematics 192(4), 593–639 (2001)

    Article  MathSciNet  Google Scholar 

  12. A.V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of boundary feedback control. J. Math. Fluid Mech. 3, 259–301 (2001)

    Article  MathSciNet  Google Scholar 

  13. A.V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete Cont. Dyn. Syst. 10, 289–314 (2004)

    Article  MathSciNet  Google Scholar 

  14. A.V. Fursikov, The simplest semilinear parabolic equation of normal type. Math. Control Rel. Fields 2(N2), 141–170 (2012)

    Article  MathSciNet  Google Scholar 

  15. A.V. Fursikov, On parabolic system of normal type corresponding to 3D Helmholtz system, advances in mathematical analysis of PDEs. AMS Transl. Ser. 2 232, 99–118 (2014)

    Google Scholar 

  16. A.V. Fursikov, Stabilization of the simplest normal parabolic equation by starting control. Commun. Pure Appl. Analy. 13(5), 1815–1854 (2014)

    Article  Google Scholar 

  17. A.V. Fursikov, A.V. Gorshkov, Certain questions of feedback stabilization for Navier-Stokes equations. Evolut. Eq. Control Theory 1(1), 109–140 (2012)

    Article  MathSciNet  Google Scholar 

  18. A.V. Fursikov, O.Y. Imanuvilov, Controllability of Evolution Equations. Lecture Notes Series (Global Anan. Res. Ctr.), vol. 34, (Seoul National University, Seoul, 1996), pp. 1–163

    Google Scholar 

  19. A.V. Fursikov, O.Y. Immanuvilov, Exact controllability of Navier-Stokes and Boussinesq equations. Russian Math. Surv. 54(3), 565–618 (1999)

    Article  MathSciNet  Google Scholar 

  20. A.V. Fursikov, A.A. Kornev, Feedback stabilization for Navier-Stokes equations: Theory and calculations, in Mathematical Aspects of Fluid Mechanics. LMS Lecture Notes Series, vol. 402 (Cambridge University Press, Cambridge, 2012), pp. 130–172

    Google Scholar 

  21. A.V. Fursikov, L.S. Osipova, One method for the nonlocal stabilization of a burgers-type equation by an impulsive control. Diff. Eq. 55(5), 1–15 (2019). Pleiades Publishing, Ltd. (2019)

    Google Scholar 

  22. A.V. Fursikov, L.S. Shatina, Nonlocal stabilization of the normal equation connected with Helmholtz system by starting control. Discrete Cont. Dyn. Syst. 38, 1187–1242 (2018)

    Article  Google Scholar 

  23. A.V. Fursikov, M.D. Gunzburger, L.S. Hou, Trace theorems for three- dimensional time-dependent solenoidal vector fields and their applications. Trans. Amer. Math. Soc. 354, 1079–1116 (2002)

    Article  MathSciNet  Google Scholar 

  24. M.V. Keldysh, On completeness of eigenfunctions for certain classes of not self-adjoint linear operators. Russian Math. Surveys 26(4) , 15–41 (1971) (in Russian)

    Article  Google Scholar 

  25. A.N. Kolmogorov, S.V. Fomin, Introductory Real Analysis (Dover Publications, New York, 1975)

    Google Scholar 

  26. M. Krstic, On global stabilization of Burgers’ equation by boundary control. Syst. Control Lett. 37, 123–141 (1999)

    Article  MathSciNet  Google Scholar 

  27. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids (Gordon and Breach, New York, 1963). (SIAM, Philadelphia, 1989)

    Google Scholar 

  28. J.-L. Lions, R. Magenes, Problems Aux Limites Non Homogenes et Applications, vol. 1 (Dunod, Paris, 1968)

    Google Scholar 

  29. J.-P. Raymond, Feedback boundary stabilization of the two-dimensional incompressible Navier-Stokes equations. SIAM J. Control Optim. 45(3), 790–828 (2006)

    Article  MathSciNet  Google Scholar 

  30. J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations. J. Math. Pures Appl. 87(6), 627–669 (2007)

    Article  MathSciNet  Google Scholar 

  31. J.-P. Raymond, A family of stabilization problems for Oseen equations, in Control of Coupled Partial Differential Equations. International Series of Numerical Mathematics, vol. 155,, 269–291 (2007)

    Google Scholar 

  32. S.L. Sobolev, Méthode nouvelle á résoudre le probléme de Cauchy pour les équations linéaries hyperboliques normales. Matematicheskii Sbornik 1(43), 39–72 (1936)

    Google Scholar 

  33. R. Temam, Navier-Stokes Equations-Theory and Numerical Analysis, 3rd edn. (revised) (Elsevier Science Publishers B.V., Amsterdam, 1984)

    Google Scholar 

  34. M.I. Vishik, A.V. Fursikov, Mathematical Problems of Statistical Hydromechanics (Kluwer Academic Publishers, Dordrecht, 1988)

    Book  Google Scholar 

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Author information

Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    A. V. Fursikov

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  1. A. V. Fursikov
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Editors and Affiliations

  1. Department of Technical Mathematics, Czech Technical University, Prague, Czech Republic

    Tomáš Bodnár

  2. University of Pittsburgh, Pittsburgh, PA, USA

    Giovanni P. Galdi

  3. Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

    Šárka Nečasová

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Fursikov, A.V. (2024). On the Stabilization Problem by Feedback Control for Some Hydrodynamic Type Systems. In: Bodnár, T., Galdi, G.P., Nečasová, Š. (eds) Fluids Under Control. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-47355-5_1

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