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

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

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

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topBarbu, 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 -

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