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The internal stabilization by noiseof the linearized Navier-Stokes equation*

Published online by Cambridge University Press:  30 October 2009

Viorel Barbu
Viorel Barbu*
Affiliation:
Al.I. Cuza University and Octav Mayer Institute of Mathematics of Romanian Academy, Iaşi, Romania. vb41@uaic.ro
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Abstract

One shows that the linearized Navier-Stokes equation in ${\mathcal{O}}{\subset} R^d,\;d \ge 2$ , around an unstable equilibriumsolution is exponentially stabilizable in probability by aninternal 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 $\{\beta_i\}^N_{i=1}$ areindependent Brownian motions in a probability space and $\{\psi_i\}^N_{i=1}$ is a system of functions on ${\mathcal{O}}$ withsupport in an arbitrary open subset ${\mathcal{O}}_0\subset {\mathcal{O}}$ . Thestochastic control input $\{V_i\}^N_{i=1}$ is found in feedbackform. One constructs also a tangential boundary noise controllerwhich exponentially stabilizes in probability the equilibriumsolution.

Keywords

Navier-Stokes equationfeedback controllerstochastic processStokes-Oseen operator
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 1 , January 2011 , pp. 117 - 130
Copyright
© EDP Sciences, SMAI, 2009

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