Abstract
The stochastic stabilization of Navier–Stokes equations is an alternative approach to stabilization techniques described in Chap. 3, which have two important advantages: the simplicity of the stabilizable feedback law and its robustness to (deterministic and stochastic) perturbations. A long time ago, it was observed that the noise might stabilize the finite and infinite-dimensional dynamical systems and several empirical observations in fluid dynamics suggested that noise might have a dissipation effect comparable with increasing the viscosity of fluid. This is exactly what will be rigorously proven here by designing stabilizing noise feedback controller with internal or boundary support.
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References
Apleby JAD, Mao X, Rodkina A (2008) Stabilization and destabilization of nonlinear differential equations by noise. IEEE Trans. Autom. Control 53:683–691
Arnold L, Craul H, Wihstutz V (1983) Stabilization of linear systems by noise. SIAM J. Control. Optim. 21:451–461
Barbu V (2009) The internal stabilization by noise of the linearized Navier–Stokes equation. ESAIM COCV (online)
Barbu V (2010) Stabilization of a plane periodic channel flow by noise wall normal controllers. Syst. Control Lett. 50(10):608–618
Barbu V (2010) Exponential stabilization of the linearized Navier–Stokes equation by pointwise feedback controllers. Automatica. doi:10.1016/j.automatica.2010.08.013
Barbu V, Da Prato G (2010) Internal stabilization by noise of the Navier–Stokes equation. SIAM J. Control Optim. (to appear)
Caraballo T, Robinson JC (2004) Stabilization of linear PDEs by Stratonovich noise. Syst. Control Lett. 53:41–50
Caraballo T, Liu K, Mao X (2001) On stabilization of partial differential equations by noise. Nagoya Math. J. 101:155–170
Cerrai S (2000) Stabilization by noise for a class of stochastic reaction-diffusion equations. Probab. Theory Relat. Fields 133:190–214
Da Prato G, Zabczyk J (1991) Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge UK
Da Prato G, Zabczyk J (1996) Ergodicity for Infinite Dimensional Systems. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge
Deng H, Krstic M, Williams RJ (2001) Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Trans. Autom. Control 46:1237–1253
Duan J, Fursikov AV (2005) Feedback stabilization for Oseen Fluid Equations. A stochastic approach. J. Math. Fluids Mech. 7:574–610
Kuksin S, Shirikyan A (2001) Ergodicity for the randomly forced 2D Navier–Stokes equations. Math. Phys. Anal. Geom. 4:147–195
Kato T (1966) Perturbation Theory of Linear Operators. Springer, Berlin, Heidelberg, New York
Lipster R, Shiraev AN (1989) Theory of Martingals. Kluwer, Dordrecht
Mao XR (2003) Stochastic stabilization and destabilization. Syst. Control Lett. 23:279–290
Shirikyan A (2004) Exponential mixing 2D Navier–Stokes equations perturbed by an unbounded noise. J. Math. Fluids Mech. 6:169–193
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Barbu, V. (2011). Stabilization by Noise of Navier–Stokes Equations. In: Stabilization of Navier–Stokes Flows. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-043-4_4
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DOI: https://doi.org/10.1007/978-0-85729-043-4_4
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