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 $𝒪\subset {R}^{d},\phantom{\rule{0.277778em}{0ex}}d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller $V\left(t,\xi \right)=\sum _{i=1}^{N}{V}_{i}\left(t\right){\psi }_{i}\left(\xi \right){\stackrel{˙}{\beta }}_{i}\left(t\right)$, $\xi \in 𝒪$, where ${\left\{{\beta }_{i}\right\}}_{i=1}^{N}$ are independent Brownian motions in a probability space and ${\left\{{\psi }_{i}\right\}}_{i=1}^{N}$ is a system of functions on $𝒪$ with support in an arbitrary open subset ${𝒪}_{0}\subset 𝒪$. The stochastic control input ${\left\{{V}_{i}\right\}}_{i=1}^{N}$ 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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