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

Viorel Barbu

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

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

Abstract

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One shows that the linearized Navier-Stokes equation in 𝒪 ⊂ R d , d ≥ 2 , around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller V ( t , ξ ) = ∑ i = 1 N V i ( t ) ψ i ( ξ ) β ˙ i ( t ) , ξ ∈ 𝒪 , where { β i } i = 1 N are independent Brownian motions in a probability space and { ψ i } i = 1 N is a system of functions on 𝒪 with support in an arbitrary open subset 𝒪 0 ⊂ 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.

How to cite

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Barbu, 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/272758>.

@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 $\lbrace \beta _i\rbrace ^N_\{i=1\}$ are independent Brownian motions in a probability space and $\lbrace \psi _i\rbrace ^N_\{i=1\}$ 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 $\lbrace V_i\rbrace ^N_\{i=1\}$ 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},
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/272758},
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
PY - 2011
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 $\lbrace \beta _i\rbrace ^N_{i=1}$ are independent Brownian motions in a probability space and $\lbrace \psi _i\rbrace ^N_{i=1}$ 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 $\lbrace V_i\rbrace ^N_{i=1}$ 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/272758
ER -

References

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