Skip to main content Accessibility help
×
Home
Hostname: page-component-78dcdb465f-xl52z Total loading time: 1.992 Render date: 2021-04-16T12:22:15.668Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

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

Published online by Cambridge University Press:  30 October 2009

Viorel Barbu
Affiliation:
Al.I. Cuza University and Octav Mayer Institute of Mathematics of Romanian Academy, Iaşi, Romania. vb41@uaic.ro
Get access

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
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below.

References

Apleby, J.A.D., Mao, X. and Rodkina, A., Stochastic stabilization of functional differential equations. Syst. Control Lett. 54 (2005) 10691081. CrossRef
Apleby, J.A.D., Mao, X. and Rodkina, A., Stabilization and destabilization of nonlinear differential equations by noise. IEEE Trans. Automat. Contr. 53 (2008) 683691. CrossRef
Arnold, L., Craul, H. and Wihstutz, V., Stabilization of linear systems by noise. SIAM J. Contr. Opt. 21 (1983) 451461. CrossRef
Barbu, V., Feedback stabilization of Navier-Stokes equations. ESAIM: COCV 9 (2003) 197205. CrossRef
Barbu, V. and Triggiani, R., Internal stabilization of Navier-Stokes equations with finite dimensional controllers. Indiana Univ. Math. J. 53 (2004) 14431494. CrossRef
V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Memoires Amer. Math. Soc. AMS, USA (2006).
Caraballo, T., Liu, K. and Mao, X., On stabilization of partial differential equations by noise. Nagoya Math. J. 101 (2001) 155170. CrossRef
Caraballo, T., Craul, H., Langa, J.A. and Robinson, J.C., Stabilization of linear PDEs by Stratonovich noise. Syst. Control Lett. 53 (2004) 4150. CrossRef
Cerrai, S., Stabilization by noise for a class of stochastic reaction-diffusion equations. Prob. Th. Rel. Fields 133 (2000) 190214. CrossRef
G. Da Prato, An Introduction to Infinite Dimensional Analysis. Springer-Verlag, Berlin, Germany (2006).
Ding, H., Krstic, M. and Williams, R.J., Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Trans. Automat. Contr. 46 (2001) 12371253. CrossRef
Duan, J. and Fursikov, A., Feedback stabilization for Oseen Fluid Equations. A stochastic approach. J. Math. Fluids Mech. 7 (2005) 574610. CrossRef
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.
Fursikov, A., Stabilization for the 3-D Navier-Stokes systems by feedback boundary control. Discrete Contin. Dyn. Syst. 10 (2004) 289-314. CrossRef
T. Kato, Perturbation Theory of Linear Operators. Springer-Verlag, New York, Berlin (1966).
Kuksin, S. and Shirikyan, A., Ergodicity for the randomly forced 2D Navier-Stokes equations. Math. Phys. Anal. Geom. 4 (2001) 147195. CrossRef
T. Kurtz, Lectures on Stochastic Analysis. Lecture Notes Online, Wisconsin (2007), available at http://www.math.wisc.edu/~kurtz/735/main735.pdf.
R. Lipster and A.N. Shiraev, Theory of Martingals. Dordrecht, Kluwer (1989).
Mao, X.R., Stochastic stabilization and destabilization. Syst. Control Lett. 23 (2003) 279290. CrossRef
Raymond, J.P., Feedback boundary stabilization of the two dimensional Navier-Stokes equations. SIAM J. Contr. Opt. 45 (2006) 790828. CrossRef
Raymond, J.P., Feedback boundary stabilization of the three dimensional incompressible Navier-Stokes equations. J. Math. Pures Appl. 87 (2007) 627669. CrossRef
Shirikyan, A., Exponential mixing 2D Navier-Stokes equations perturbed by an unbounded noise. J. Math. Fluids Mech. 6 (2004) 169193. CrossRef

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 52 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 16th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

The internal stabilization by noise of the linearized Navier-Stokes equation*
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

The internal stabilization by noise of the linearized Navier-Stokes equation*
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

The internal stabilization by noise of the linearized Navier-Stokes equation*
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *