The internal stabilization by noise of the linearized Navier-Stokes equation
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 1, p. 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 : https://doi.org/10.1051/cocv/2009042
Classification:  35Q30,  60H15,  35B40
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {1},
     year = {2011},
     pages = {117-130},
     doi = {10.1051/cocv/2009042},
     zbl = {1210.35302},
     mrnumber = {2775189},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 17 (2011) no. 1, pp. 117-130. doi : 10.1051/cocv/2009042. http://www.numdam.org/item/COCV_2011__17_1_117_0/

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