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The internal stabilization by noise of the linearized Navier-Stokes equation. (English) Zbl 1210.35302

Summary: One shows that the linearized Navier-Stokes equation in \({\mathcal O}\subset\mathbb R^d\), \(d\geq 2\), around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller \(V(t,\xi)=\sum_{i=1}^N V_i(t)\psi_i(\xi)\dot\beta_i(t)\), \(\xi\in{\mathcal O}\), where \(\{\beta_i\}_{i=1}^N\) are independent Brownian motions in a probability space and \(\{\psi_i\}_{i=1}^N\) 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 \(\{V_i\}_{i=1}^N\) is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium solution.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35Q30 Navier-Stokes equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
76M35 Stochastic analysis applied to problems in fluid mechanics
93B52 Feedback control
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References:

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