Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 869–888 | Cite as

Turnpike Property for Two-Dimensional Navier–Stokes Equations

  • Sebastián ZamoranoEmail author


In this paper we study the turnpike phenomenon arising in the optimal distributed control tracking-type problem for the Navier–Stokes equations. We obtain a positive answer to this property in the case when the control is time-dependent function and also when it is independent of time. In both cases we prove an exponential turnpike property assuming that the stationary optimal state satisfies certain properties of smallness.


Navier–Stokes equations Optimal control problems Turnpike property Asymptotic stability Oseen equation 

Mathematics Subject Classification

Primary 35Q30 Secondary 49J20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abergel, F., Casas, E.: Some optimal control problems of multistate equations appearing in fluid mechanics. RAIRO Modél Math Anal Numér 27(2), 223–247 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abergel, F., Temam, R.: On some control problems in fluid mechanics. Theoret. Comput. Fluid Dyn. 1(6), 303–325 (1990)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Allaire, G.: Shape Optimization by the Homogenization Method, vol. 146. Springer, New York (2012)zbMATHGoogle Scholar
  4. 4.
    Allaire, G., Münch, A., Periago, F.: Long time behavior of a two-phase optimal design for the heat equation. SIAM J. Control Optim. 48(8), 5333–5356 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barbu, V.: Feedback stabilization of Navier–Stokes equations. ESAIM Control Optim. Calc. Var. 9, 197–205 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barbu, V.: Stabilization of Navier–Stokes Flows. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Boyer, F., Fabrie, P.: Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models, vol. 183. Springer, New York (2012)zbMATHGoogle Scholar
  8. 8.
    Casas, E.: An optimal control problem governed by the evolution Navier–Stokes equations. Optim. Control Viscous Flow 59, 79–95 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Casas, E., Mateos, M., Raymond, J.-P.: Error estimates for the numerical approximation of a distributed control problem for the steady-state Navier–Stokes equations. SIAM J. Control Optim. 46(3), 952–982 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    De los Reyes, J.: A primal-dual active set method for bilaterally control constrained optimal control of the Navier–Stokes equations. Numer. Funct. Anal. Optim. 25(7), 657–684 (2004)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fursikov, A.: Stabilizability of two-dimensional Navier–Stokes equations with help of a boundary feedback control. J. Math. Fluid Mech. 3(3), 259–301 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fursikov, A.V., Kornev, A.A.: Feedback stabilization for Navier–Stokes equations: theory and calculations. Math. Asp. Fluid Mech. 402, 130–172 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Volume I: Linearised Steady Problems, vol. 38. Springer, New York (2013)Google Scholar
  14. 14.
    Hinze, M.: Optimal and instantaneous control of the instationary Navier–Stokes equations. PhD thesis, Habilitation thesis, Technische Universität Berlin (2000)Google Scholar
  15. 15.
    Huan, J., Modi, V.: Optimum design of minimum drag bodies in incompressible laminar flow using a control theory approach. Inverse Probl. Eng. 1(1), 1–25 (1994)CrossRefGoogle Scholar
  16. 16.
    Jameson, A., Martinelli, L., Pierce, N.A.: Optimum aerodynamic design using the Navier–Stokes equations. Theor. Comput. Fluid Dyn. 10(1–4), 213–237 (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    Jameson, A., Ou, K.: Optimization methods in computational fluid dynamics. In: Blockley, R., Shyy, W. (eds.) Encyclopedia of Aerospace Engineering. Wiley, New York (2010). Google Scholar
  18. 18.
    Marcus, M., Zaslavski, A.J.: The structure of extremals of a class of second order variational problems. Ann. Inst. Henri Poincare (C) Non Linear Anal. 16(5), 593–629 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Porretta, A., Zuazua, E.: Long time versus steady state optimal control. SIAM J. Control Optim. 51(6), 4242–4273 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Porretta, A., Zuazua, E.: Remarks on long time versus steady state optimal control. In: Ancona, F., Cannarsa, P., Jones, C., Portaluri, A. (eds.) Mathematical Paradigms of Climate Science. Springer INdAM Series, vol. 15, pp. 67–89. Springer, Cham (2016)CrossRefGoogle Scholar
  21. 21.
    Raymond, J.-P.: Feedback boundary stabilization of the two-dimensional Navier–Stokes equations. SIAM J. Control Optim. 45(3), 790–828 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Samuelson, P.A.: A catenary turnpike theorem involving consumption and the golden rule. Am. Econ. Rev. 55(3), 486–496 (1965)Google Scholar
  23. 23.
    Samuelson, P.A.: The Collected Scientific Papers of Paul A. Samuelson. MIT press, Cambridge (1966)Google Scholar
  24. 24.
    Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis, vol. 343. American Mathematical Soc., Philadelphia (2001)zbMATHGoogle Scholar
  25. 25.
    Trélat, E., Zuazua, E.: The turnpike property in finite-dimensional nonlinear optimal control. J. Differ. Equ. 258(1), 81–114 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wachsmuth, D.: Optimal control of the unsteady Navier–Stokes equations. PhD thesis, Technische Universität Berlin (2006)Google Scholar
  27. 27.
    Zaslavski, A.: Turnpike Properties in the Calculus of Variations and Optimal Control, vol. 80. Springer, New York (2006)zbMATHGoogle Scholar
  28. 28.
    Zaslavski, A.J.: Turnpike Phenomenon and Infinite Horizon Optimal Control, vol. 99. Springer, New York (2014)zbMATHGoogle Scholar
  29. 29.
    Zaslavski, A.J.: Turnpike Theory of Continuous-Time Linear Optimal Control Problems, vol. 104. Springer, New York (2015)zbMATHGoogle Scholar
  30. 30.
    Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5(1), 49–62 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemática y Ciencia de la ComputaciónUniversidad de Santiago de ChileSantiagoChile

Personalised recommendations