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The internal stabilization by noise of the linearized Navier-Stokes equation*
Published online by Cambridge University Press: 30 October 2009
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.
- 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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