The internal stabilization by noise of the linearized Navier-Stokes equation*

Viorel Barbu

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 1, page 117-130
  • ISSN: 1292-8119

Abstract

top
One shows that the linearized Navier-Stokes equation in 𝒪 R d , d 2 , around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller V ( t , ξ ) = i = 1 N V i ( t ) ψ i ( ξ ) β ˙ i ( t ) , ξ 𝒪 , where { β i } i = 1 N are independent Brownian motions in a probability space and { ψ i } i = 1 N is a system of functions on 𝒪 with support in an arbitrary open subset 𝒪 0 𝒪 . The stochastic control input { V i } i = 1 N is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium solution.

How to cite

top

Barbu, Viorel. "The internal stabilization by noise of the linearized Navier-Stokes equation*." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 117-130. <http://eudml.org/doc/197364>.

@article{Barbu2011,
abstract = { One shows that the linearized Navier-Stokes equation in $\{\mathcal\{O\}\}\{\subset\} R^d,\;d \ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller $V(t,\xi)=\displaystyle\sum\limits_\{i=1\}^\{N\} V_i(t)\psi_i(\xi) \dot\beta_i(t)$, $\xi\in\{\mathcal\{O\}\}$, where $\\{\beta_i\\}^N_\{i=1\}$ are independent Brownian motions in a probability space and $\\{\psi_i\\}^N_\{i=1\}$ is a system of functions on $\{\mathcal\{O\}\}$ with support in an arbitrary open subset $\{\mathcal\{O\}\}_0\subset \{\mathcal\{O\}\}$. The stochastic control input $\\{V_i\\}^N_\{i=1\}$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium solution. },
author = {Barbu, Viorel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Navier-Stokes equation; feedback controller; stochastic process; Stokes-Oseen operator},
language = {eng},
month = {2},
number = {1},
pages = {117-130},
publisher = {EDP Sciences},
title = {The internal stabilization by noise of the linearized Navier-Stokes equation*},
url = {http://eudml.org/doc/197364},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Barbu, Viorel
TI - The internal stabilization by noise of the linearized Navier-Stokes equation*
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/2//
PB - EDP Sciences
VL - 17
IS - 1
SP - 117
EP - 130
AB - One shows that the linearized Navier-Stokes equation in ${\mathcal{O}}{\subset} R^d,\;d \ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller $V(t,\xi)=\displaystyle\sum\limits_{i=1}^{N} V_i(t)\psi_i(\xi) \dot\beta_i(t)$, $\xi\in{\mathcal{O}}$, where $\{\beta_i\}^N_{i=1}$ are independent Brownian motions in a probability space and $\{\psi_i\}^N_{i=1}$ is a system of functions on ${\mathcal{O}}$ with support in an arbitrary open subset ${\mathcal{O}}_0\subset {\mathcal{O}}$. The stochastic control input $\{V_i\}^N_{i=1}$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium solution.
LA - eng
KW - Navier-Stokes equation; feedback controller; stochastic process; Stokes-Oseen operator
UR - http://eudml.org/doc/197364
ER -

References

top
  1. J.A.D. Apleby, X. Mao and A. Rodkina, Stochastic stabilization of functional differential equations. Syst. Control Lett.54 (2005) 1069–1081.  
  2. J.A.D. Apleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise. IEEE Trans. Automat. Contr.53 (2008) 683–691.  
  3. L. Arnold, H. Craul and V. Wihstutz, Stabilization of linear systems by noise. SIAM J. Contr. Opt.21 (1983) 451–461.  Zbl0514.93069
  4. V. Barbu, Feedback stabilization of Navier-Stokes equations. ESAIM: COCV9 (2003) 197–205.  Zbl1076.93037
  5. V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite dimensional controllers. Indiana Univ. Math. J.53 (2004) 1443–1494.  Zbl1073.76017
  6. V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Memoires Amer. Math. Soc. AMS, USA (2006).  Zbl1098.35026
  7. T. Caraballo, K. Liu and X. Mao, On stabilization of partial differential equations by noise. Nagoya Math. J.101 (2001) 155–170.  Zbl0986.60058
  8. T. Caraballo, H. Craul, J.A. Langa and J.C. Robinson, Stabilization of linear PDEs by Stratonovich noise. Syst. Control Lett.53 (2004) 41–50.  Zbl1157.60332
  9. S. Cerrai, Stabilization by noise for a class of stochastic reaction-diffusion equations. Prob. Th. Rel. Fields133 (2000) 190–214.  Zbl1077.60046
  10. G. Da Prato, An Introduction to Infinite Dimensional Analysis. Springer-Verlag, Berlin, Germany (2006).  Zbl1109.46001
  11. H. Ding, M. Krstic and R.J. Williams, Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Trans. Automat. Contr.46 (2001) 1237–1253.  Zbl1008.93068
  12. J. Duan and A. Fursikov, Feedback stabilization for Oseen Fluid Equations. A stochastic approach. J. Math. Fluids Mech.7 (2005) 574–610.  Zbl1085.93024
  13. A. Fursikov, Real processes of the 3-D Navier-Stokes systems and its feedback stabilization from the boundary, in AMS Translations, Partial Differential Equations, M. Vîshnik Seminar206, M.S. Agranovic and M.A. Shubin Eds. (2002) 95–123.  Zbl1036.76010
  14. A. Fursikov, Stabilization for the 3-D Navier-Stokes systems by feedback boundary control. Discrete Contin. Dyn. Syst.10 (2004) 289-314.  Zbl1174.93675
  15. T. Kato, Perturbation Theory of Linear Operators. Springer-Verlag, New York, Berlin (1966).  Zbl0148.12601
  16. S. Kuksin and A. Shirikyan, Ergodicity for the randomly forced 2D Navier-Stokes equations. Math. Phys. Anal. Geom.4 (2001) 147–195.  Zbl1013.37046
  17. T. Kurtz, Lectures on Stochastic Analysis. Lecture Notes Online, Wisconsin (2007), available at .  URIhttp://www.math.wisc.edu/~kurtz/735/main735.pdf
  18. R. Lipster and A.N. Shiraev, Theory of Martingals. Dordrecht, Kluwer (1989).  
  19. X.R. Mao, Stochastic stabilization and destabilization. Syst. Control Lett.23 (2003) 279–290.  Zbl0820.93071
  20. J.P. Raymond, Feedback boundary stabilization of the two dimensional Navier-Stokes equations. SIAM J. Contr. Opt.45 (2006) 790–828.  Zbl1121.93064
  21. J.P. Raymond, Feedback boundary stabilization of the three dimensional incompressible Navier-Stokes equations. J. Math. Pures Appl.87 (2007) 627–669.  Zbl1114.93040
  22. A. Shirikyan, Exponential mixing 2D Navier-Stokes equations perturbed by an unbounded noise. J. Math. Fluids Mech.6 (2004) 169–193.  Zbl1095.35032

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.