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

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

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

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topBarbu, 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

top- [1] J.A.D. Apleby, X. Mao and A. Rodkina, Stochastic stabilization of functional differential equations. Syst. Control Lett.54 (2005) 1069–1081. Zbl1129.34330
- [2] J.A.D. Apleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise. IEEE Trans. Automat. Contr.53 (2008) 683–691. MR2401021
- [3] L. Arnold, H. Craul and V. Wihstutz, Stabilization of linear systems by noise. SIAM J. Contr. Opt.21 (1983) 451–461. Zbl0514.93069MR696907
- [4] V. Barbu, Feedback stabilization of Navier-Stokes equations. ESAIM: COCV 9 (2003) 197–205. Zbl1076.93037MR1957098
- [5] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite dimensional controllers. Indiana Univ. Math. J.53 (2004) 1443–1494. Zbl1073.76017MR2104285
- [6] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Memoires Amer. Math. Soc. AMS, USA (2006). Zbl1098.35026MR2215059
- [7] T. Caraballo, K. Liu and X. Mao, On stabilization of partial differential equations by noise. Nagoya Math. J.101 (2001) 155–170. Zbl0986.60058MR1820216
- [8] T. Caraballo, H. Craul, J.A. Langa and J.C. Robinson, Stabilization of linear PDEs by Stratonovich noise. Syst. Control Lett.53 (2004) 41–50. Zbl1157.60332MR2077187
- [9] S. Cerrai, Stabilization by noise for a class of stochastic reaction-diffusion equations. Prob. Th. Rel. Fields133 (2000) 190–214. Zbl1077.60046MR2198698
- [10] G. Da Prato, An Introduction to Infinite Dimensional Analysis. Springer-Verlag, Berlin, Germany (2006). Zbl1109.46001MR2244975
- [11] H. Ding, M. Krstic and R.J. Williams, Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Trans. Automat. Contr.46 (2001) 1237–1253. Zbl1008.93068MR1847327
- [12] J. Duan and A. Fursikov, Feedback stabilization for Oseen Fluid Equations. A stochastic approach. J. Math. Fluids Mech. 7 (2005) 574–610. Zbl1085.93024MR2189675
- [13] A. Fursikov, Real processes of the 3-D Navier-Stokes systems and its feedback stabilization from the boundary, in AMS Translations, Partial Differential Equations, M. Vîshnik Seminar 206, M.S. Agranovic and M.A. Shubin Eds. (2002) 95–123. Zbl1036.76010
- [14] A. Fursikov, Stabilization for the 3-D Navier-Stokes systems by feedback boundary control. Discrete Contin. Dyn. Syst.10 (2004) 289-314. Zbl1174.93675MR2026196
- [15] T. Kato, Perturbation Theory of Linear Operators. Springer-Verlag, New York, Berlin (1966). Zbl0435.47001MR203473
- [16] S. Kuksin and A. Shirikyan, Ergodicity for the randomly forced 2D Navier-Stokes equations. Math. Phys. Anal. Geom.4 (2001) 147–195. Zbl1013.37046MR1860883
- [17] T. Kurtz, Lectures on Stochastic Analysis. Lecture Notes Online, Wisconsin (2007), available at http://www.math.wisc.edu/~kurtz/735/main735.pdf.
- [18] R. Lipster and A.N. Shiraev, Theory of Martingals. Dordrecht, Kluwer (1989).
- [19] X.R. Mao, Stochastic stabilization and destabilization. Syst. Control Lett.23 (2003) 279–290. Zbl0820.93071MR1298174
- [20] J.P. Raymond, Feedback boundary stabilization of the two dimensional Navier-Stokes equations. SIAM J. Contr. Opt.45 (2006) 790–828. Zbl1121.93064MR2247716
- [21] J.P. Raymond, Feedback boundary stabilization of the three dimensional incompressible Navier-Stokes equations. J. Math. Pures Appl.87 (2007) 627–669. Zbl1114.93040MR2335090
- [22] A. Shirikyan, Exponential mixing 2D Navier-Stokes equations perturbed by an unbounded noise. J. Math. Fluids Mech.6 (2004) 169–193. Zbl1095.35032MR2053582

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