Stabilization of Navier–Stokes Flows

Part of the Communications and Control Engineering book series (CCE)


In this chapter we discuss the feedback stabilization of stationary (equilibrium) solutions to Navier–Stokes equations. The design of a robust stabilizing feedback control is the principal way to suppress instabilities and turbulence occurring in the dynamics of the fluid and we treat this problem in the case of internal and boundary controllers. The first case, already presented in an abstract setting in Chap. 2, is that in which the controller is distributed in a spatial domain \(O\) and has compact support taken arbitrarily small. The second case is that where the controller is concentrated on the boundary \(\partial O\). In both cases, we design a stabilizable feedback linear controller which is robust and has a finite-dimensional structure, that is, it is a linear combination of eigenfunctions for the corresponding linearized systems. From the control theory point of view, this means that the actuation, though infinite-dimensional, is confined to an arbitrary subdomain \(O_{\rm 0} \) or to the boundary.


Stokes Equation Riccati Equation Boundary Stabilization Feedback Controller Algebraic Riccati Equation 
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.


  1. 1.
    Aamo OM, Fosseen TI (2002) Tutorial on feedback control of flows, part I: Stabilization of fluid flows in channels and pipes. Model. Identif. Control 23:161–226 CrossRefGoogle Scholar
  2. 2.
    Aamo OM, Krstic M, Bewley TR (2003) Control of mixing by boundary feedback in 2D-channel. Automatica 39:1597–1606 MathSciNetMATHCrossRefGoogle Scholar
  3. 7.
    Balogh A, Liu W-L, Krstic M (2001) Stability enhancement by boundary control in 2D channel flow. IEEE Trans. Autom. Control 11:1696–1711 MathSciNetCrossRefGoogle Scholar
  4. 12.
    Barbu V (2003) Feedback stabilization of Navier–Stokes equations. ESAIM COCV 9:197–206 MathSciNetMATHCrossRefGoogle Scholar
  5. 13.
    Barbu V (2007) Stabilization of a plane channel flow by wall normal controllers. Nonlinear Anal. Theory-Methods Appl. 56:145–168 MathSciNetMATHGoogle Scholar
  6. 16.
    Barbu V (2010) Optimal stabilizable feedback controller for Navier–Stokes equations. In: Leizarowitz et al. (eds) Nonlinear Analysis and Optimization: Nonlinear Analysis. Contemporary Math. 513. Am. Math. Soc., Providence Google Scholar
  7. 19.
    Barbu V, Coca D, Yan Y (2008) Internal optimal controller synthesis for Navier–Stokes equations. Numer. Funct. Anal. 29:225–242 MathSciNetMATHCrossRefGoogle Scholar
  8. 21.
    Barbu V, Lasiecka I, Triggiani R (2006) Abstract setting for tangential boundary stabilization of Navier–Stokes equations by high and low-gain feedback controllers. Nonlinear Anal. 64:2704–2746 MathSciNetMATHCrossRefGoogle Scholar
  9. 22.
    Barbu V, Lasiecka I, Triggiani R (2006) Tangential boundary stabilization of Navier–Stokes equations. Mem. Am. Math. Soc. 852:1–145 MathSciNetGoogle Scholar
  10. 23.
    Barbu V, Lefter C (2003) Internal stabilizability of the Navier–Stokes equations. Syst. Control Lett. 48:161–167 MathSciNetMATHCrossRefGoogle Scholar
  11. 24.
    Barbu V, Rodriguez S, Shirikyan A (2010) Internal stabilization for Navier–Stokes equations by means of finite dimensional controllers. SIAM J. Control Optim. (to appear) Google Scholar
  12. 26.
    Barbu V, Triggiani R (2004) Internal stabilization of Navier–Stokes equations with finite dimensional controllers. Indiana Univ. Math. J. 53:1443–1494 MathSciNetMATHCrossRefGoogle Scholar
  13. 29.
    Barbu V, Wang G (2005) Feedback stabilization of periodic solutions to nonlinear parabolic-like evolution systems. Indiana Univ. Math. J. 54:1521–1546 MathSciNetMATHCrossRefGoogle Scholar
  14. 30.
    Bedra (2009) Feedback stabilization of the 2-D and 3-D Navier–Stokes equations based on an extended system. ESAIM COCV 15:934–968 CrossRefGoogle Scholar
  15. 31.
    Bedra M (2009) Lyapunov functions and local feedback boundary stabilization of the Navier–Stokes equations. SIAM J. Control Optim. 48:1797–1830 MathSciNetCrossRefGoogle Scholar
  16. 37.
    Burns JA, Singler JR (2006) New scenarios, system sensitivity and feedback control. In: God-il-Hak (ed) Transition and Turbulence Control. Lecture Notes Series, NUS 18:1–35. World Scientific, Singapore Google Scholar
  17. 41.
    Constantin P, Foias C (1989) Navier–Stokes Equations. University of Chicago Press, Chicago, London Google Scholar
  18. 42.
    Coron JM (2007) Control and Nonlinearity. AMS, Providence RI MATHGoogle Scholar
  19. 50.
    Fursikov AV (2000) Optimal Control of Systems Theory and Applications. AMS, Providence RI Google Scholar
  20. 51.
    Fursikov AV (2002) Real processes of the 3-D Navier–Stokes systems and its feedback stabilization from the boundary. In: Agranovic MS, Shubin MA (eds) AMS Translations. Partial Differential Equations. M. Vishnik Seminar 95–123 Google Scholar
  21. 52.
    Fursikov AV (2004) Stabilization for the 3-D Navier–Stokes systems by feedback boundary control. Discrete Contin. Dyn. Syst. 10:289–314 MathSciNetMATHCrossRefGoogle Scholar
  22. 53.
    Fursikov AV, Imanuvilov OY (1998) Local exact controllability of the Boussinesque equation. SIAM J. Control Optim. 36:391–421 MathSciNetMATHCrossRefGoogle Scholar
  23. 55.
    Hormander L (1976) Linear Partial Differential Operators. Springer, Berlin, Heidelberg, New York Google Scholar
  24. 56.
    Imanuvilov OY (1998) On exact controllability for Navier–Stokes equations. ESAIM COCV 3:97–131 MathSciNetMATHCrossRefGoogle Scholar
  25. 57.
    Joseph DD (1976) Stability of Fluid Motions. Springer, Berlin, Heidelberg, New York Google Scholar
  26. 61.
    Lefter C (2009) Feedback stabilization of 2-D Navier–Stokes equations with Navier slip boundary conditions. Nonlinear Anal. 70:553–562 MathSciNetMATHCrossRefGoogle Scholar
  27. 65.
    Munteanu I (2010) Normal feedback stabilization of periodic flows in a 2-D channel (to appear) Google Scholar
  28. 68.
    Ravindran SS (2000) Reduced-order adaptive controllers for fluid flows using POD. J. Sci. Comput. 15(4):457–478 MathSciNetMATHCrossRefGoogle Scholar
  29. 69.
    Smale S (1965) An infinite dimensional version of Sard’s theorem. Am. J. Math. 18:158–174 MathSciNetGoogle Scholar
  30. 70.
    Raymond JP (2006) Feedback boundary stabilization of the two dimensional Navier–Stokes equations. SIAM J. Control Optim. 45:790–828 MathSciNetMATHCrossRefGoogle Scholar
  31. 71.
    Raymond JP (2007) Feedback boundary stabilization of the three dimensional incompressible Navier–Stokes equations. J. Math. Pures Appl. 87:627–669 MathSciNetMATHGoogle Scholar
  32. 73.
    Temam R (1979) Navier–Stokes Equations. North-Holland, Amsterdam MATHGoogle Scholar
  33. 74.
    Temam R (1985) Navier–Stokes Equations and Nonlinear Functional Analysis. SIAM, Philadelphia Google Scholar
  34. 75.
    Triggiani R (2006) Stability enhancement of a 2-D linear Navier–Stokes channel flow by a 2-D wall normal boundary controller. Discrete Contin. Dyn. Syst. SB 8:279–314 MathSciNetCrossRefGoogle Scholar
  35. 76.
    Uhlenbeck K (1974) Eigenfunctions of Laplace operators. Bull. AMS 78:1073–1076 MathSciNetCrossRefGoogle Scholar
  36. 80.
    Vazquez R, Krstic M (2004) A closed form feedback controller for stabilization of linearized Navier–Stokes equations: The 2D Poisseuille system. IEEE Trans. Autom. Control 52:2298–2300 MathSciNetCrossRefGoogle Scholar
  37. 81.
    Vazquez R, Tvelat E, Coron JM (2008) Control for fast and stable Laminar-to-High-Reynolds-Number transfer in a 2D channel flow. Discrete Contin. Dyn. Syst. SB 10:925–956 MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Fac. MathematicsAl. I. Cuza UniversityIaşiRomania

Personalised recommendations