The internal stabilization by noise of the linearized Navier-Stokes equation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 117-130.

One shows that the linearized Navier-Stokes equation in 𝒪R d ,d2, 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), ξ𝒪, 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.

DOI : 10.1051/cocv/2009042
Classification : 35Q30, 60H15, 35B40
Mots clés : Navier-Stokes equation, feedback controller, stochastic process, Stokes-Oseen operator
@article{COCV_2011__17_1_117_0,
     author = {Barbu, Viorel},
     title = {The internal stabilization by noise of the linearized {Navier-Stokes} equation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {117--130},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {1},
     year = {2011},
     doi = {10.1051/cocv/2009042},
     mrnumber = {2775189},
     zbl = {1210.35302},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2009042/}
}
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
PY  - 2011
SP  - 117
EP  - 130
VL  - 17
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2009042/
DO  - 10.1051/cocv/2009042
LA  - en
ID  - COCV_2011__17_1_117_0
ER  - 
%0 Journal Article
%A Barbu, Viorel
%T The internal stabilization by noise of the linearized Navier-Stokes equation
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2011
%P 117-130
%V 17
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2009042/
%R 10.1051/cocv/2009042
%G en
%F COCV_2011__17_1_117_0
Barbu, Viorel. The internal stabilization by noise of the linearized Navier-Stokes equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 117-130. doi : 10.1051/cocv/2009042. http://www.numdam.org/articles/10.1051/cocv/2009042/

[1] J.A.D. Apleby, X. Mao and A. Rodkina, Stochastic stabilization of functional differential equations. Syst. Control Lett. 54 (2005) 1069-1081. | Zbl

[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. | MR

[3] L. Arnold, H. Craul and V. Wihstutz, Stabilization of linear systems by noise. SIAM J. Contr. Opt. 21 (1983) 451-461. | MR | Zbl

[4] V. Barbu, Feedback stabilization of Navier-Stokes equations. ESAIM: COCV 9 (2003) 197-205. | Numdam | MR | Zbl

[5] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite dimensional controllers. Indiana Univ. Math. J. 53 (2004) 1443-1494. | MR | Zbl

[6] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Memoires Amer. Math. Soc. AMS, USA (2006). | MR | Zbl

[7] T. Caraballo, K. Liu and X. Mao, On stabilization of partial differential equations by noise. Nagoya Math. J. 101 (2001) 155-170. | MR | Zbl

[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. | MR | Zbl

[9] S. Cerrai, Stabilization by noise for a class of stochastic reaction-diffusion equations. Prob. Th. Rel. Fields 133 (2000) 190-214. | MR | Zbl

[10] G. Da Prato, An Introduction to Infinite Dimensional Analysis. Springer-Verlag, Berlin, Germany (2006). | MR | Zbl

[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. | MR | Zbl

[12] J. Duan and A. Fursikov, Feedback stabilization for Oseen Fluid Equations. A stochastic approach. J. Math. Fluids Mech. 7 (2005) 574-610. | MR | Zbl

[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 Seminar 206, M.S. Agranovic and M.A. Shubin Eds. (2002) 95-123. | Zbl

[14] A. Fursikov, Stabilization for the 3-D Navier-Stokes systems by feedback boundary control. Discrete Contin. Dyn. Syst. 10 (2004) 289-314. | MR | Zbl

[15] T. Kato, Perturbation Theory of Linear Operators. Springer-Verlag, New York, Berlin (1966). | MR | Zbl

[16] S. Kuksin and A. Shirikyan, Ergodicity for the randomly forced 2D Navier-Stokes equations. Math. Phys. Anal. Geom. 4 (2001) 147-195. | MR | Zbl

[17] T. Kurtz, Lectures on Stochastic Analysis. Lecture Notes Online, Wisconsin (2007), available at http://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. | MR | Zbl

[20] J.P. Raymond, Feedback boundary stabilization of the two dimensional Navier-Stokes equations. SIAM J. Contr. Opt. 45 (2006) 790-828. | MR | Zbl

[21] J.P. Raymond, Feedback boundary stabilization of the three dimensional incompressible Navier-Stokes equations. J. Math. Pures Appl. 87 (2007) 627-669. | MR | Zbl

[22] A. Shirikyan, Exponential mixing 2D Navier-Stokes equations perturbed by an unbounded noise. J. Math. Fluids Mech. 6 (2004) 169-193. | MR | Zbl

Cité par Sources :