Feedback stabilization of Navier–Stokes equations

Viorel Barbu

ESAIM: Control, Optimisation and Calculus of Variations (2003)

  • Volume: 9, page 197-205
  • ISSN: 1292-8119

Abstract

top
One proves that the steady-state solutions to Navier–Stokes equations with internal controllers are locally exponentially stabilizable by linear feedback controllers provided by a L Q control problem associated with the linearized equation.

How to cite

top

Barbu, Viorel. "Feedback stabilization of Navier–Stokes equations." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 197-205. <http://eudml.org/doc/245418>.

@article{Barbu2003,
abstract = {One proves that the steady-state solutions to Navier–Stokes equations with internal controllers are locally exponentially stabilizable by linear feedback controllers provided by a $LQ$ control problem associated with the linearized equation.},
author = {Barbu, Viorel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Navier–Stokes system; Riccati equation; linearized system; steady-state solution; weak solution; Navier-Stokes system},
language = {eng},
pages = {197-205},
publisher = {EDP-Sciences},
title = {Feedback stabilization of Navier–Stokes equations},
url = {http://eudml.org/doc/245418},
volume = {9},
year = {2003},
}

TY - JOUR
AU - Barbu, Viorel
TI - Feedback stabilization of Navier–Stokes equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2003
PB - EDP-Sciences
VL - 9
SP - 197
EP - 205
AB - One proves that the steady-state solutions to Navier–Stokes equations with internal controllers are locally exponentially stabilizable by linear feedback controllers provided by a $LQ$ control problem associated with the linearized equation.
LA - eng
KW - Navier–Stokes system; Riccati equation; linearized system; steady-state solution; weak solution; Navier-Stokes system
UR - http://eudml.org/doc/245418
ER -

References

top
  1. [1] F. Abergel and R. Temam, On some control problems in fluid mechanics. Theoret. Comput. Fluid Dynam. 1 (1990) 303-325. Zbl0708.76106
  2. [2] V. Barbu, Mathematical Methods in Optimization of Differential Systems. Kluwer, Dordrecht (1995). Zbl0819.49002MR1325922
  3. [3] V. Barbu, Local controllability of Navier–Stokes equations. Adv. Differential Equations 6 (2001) 1443-1462. Zbl1034.35012
  4. [4] V. Barbu, The time optimal control of Navier–Stokes equations. Systems & Control Lett. 30 (1997) 93-100. Zbl0898.49011
  5. [5] V. Barbu and S. Sritharan, H ∞ -control theory of fluid dynamics. Proc. Roy. Soc. London 454 (1998) 3009-3033. Zbl0919.93026MR1658234
  6. [6] V. Barbu and S. Sritharan, Flow invariance preserving feedback controller for Navier–Stokes equations. J. Math. Anal. Appl. 255 (2001) 281-307. Zbl1073.93030
  7. [7] Th.R. Bewley and S. Liu, Optimal and robust control and estimation of linear path to transition. J. Fluid Mech. 365 (1998) 305-349. Zbl0924.76028MR1631954
  8. [8] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems. Birkhäuser, Boston, Bassel, Berlin (1992). Zbl0781.93002MR2273323
  9. [9] C. Cao, I.G. Kevrekidis and E.S. Titi, Numerical criterion for the stabilization of steady states of the Navier–Stokes equations. Indiana Univ. Math. J. 50 (2001) 37-96. Zbl0997.35048
  10. [10] P. Constantin and C. Foias, Navier–Stokes Equations. University of Chicago Press, Chicago, London (1989). Zbl0687.35071
  11. [11] J.M. Coron, On the controllability for the 2-D incompresssible Navier–Stokes equations with the Navier slip boundary conditions. ESAIM: COCV 1 (1996) 33-75. Zbl0872.93040
  12. [12] J.M. Coron, On the null asymptotic stabilization of the 2-D incompressible Euler equations in a simple connected domain. SIAM J. Control Optim. 37 (1999) 1874-1896. Zbl0954.76010MR1720143
  13. [13] J.M. Coron and A. Fursikov, Global exact controllability of the 2-D Navier–Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4 (1996) 429-448. Zbl0938.93030
  14. [14] O.A. Imanuvilov, Local controllability of Navier–Stokes equations. ESAIM: COCV 3 (1998) 97-131. Zbl1052.93502
  15. [15] O.A. Imanuvilov, On local controllability of Navier–Stokes equations. ESAIM: COCV 6 (2001) 49-97. 
  16. [16] I. Lasiecka and R. Triggianni, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Encyclopedia of Mathematics and its Applications. Cambridge University Press (2000). Zbl0961.93003
  17. [17] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis. SIAM Philadelphia (1983). Zbl0833.35110

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.