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The internal stabilization by noiseof the linearized Navier-Stokes equation*
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- Journal:
- ESAIM: Control, Optimisation and Calculus of Variations / Volume 17 / Issue 1 / January 2011
- Published online by Cambridge University Press:
- 30 October 2009, pp. 117-130
- Print publication:
- January 2011
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- Article
- Export citation
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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.
