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
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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://archive.numdam.org/articles/10.1051/cocv/2009042/

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