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

Barbu, Viorel

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 117-130

One shows that the linearized Navier-Stokes equation in $𝒪\subset R^d,d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller ${\displaystyle V(t,\xi )=\sum _{i=1}^N{V}_i\left(t\right){\psi }_i\left(\xi \right){\dot{\beta }}_i\left(t\right)}$, $\xi \in 𝒪$, where ${\left\{{\beta }_i\right\}}_{i=1}^N$ are independent Brownian motions in a probability space and ${\left\{{\psi }_i\right\}}_{i=1}^N$ is a system of functions on $𝒪$ with support in an arbitrary open subset ${𝒪}_0\subset 𝒪$. The stochastic control input ${\left\{V_i\right\}}_{i=1}^N$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium solution.

Classification : 35Q30, 60H15, 35B40
Keywords: Navier-Stokes equation, feedback controller, stochastic process, Stokes-Oseen operator

[1] J.A.D. Apleby, X. Mao and A. Rodkina, Stochastic stabilization of functional differential equations. Syst. Control Lett. 54 (2005) 1069-1081. | Zbl

[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. | MR

[3] L. Arnold, H. Craul and V. Wihstutz, Stabilization of linear systems by noise. SIAM J. Contr. Opt. 21 (1983) 451-461. | Zbl | MR

[4] V. Barbu, Feedback stabilization of Navier-Stokes equations. ESAIM: COCV 9 (2003) 197-205. | Zbl | MR | Numdam

[5] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite dimensional controllers. Indiana Univ. Math. J. 53 (2004) 1443-1494. | Zbl | MR

[6] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Memoires Amer. Math. Soc. AMS, USA (2006). | Zbl | MR

[7] T. Caraballo, K. Liu and X. Mao, On stabilization of partial differential equations by noise. Nagoya Math. J. 101 (2001) 155-170. | Zbl | MR

[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. | Zbl | MR

[9] S. Cerrai, Stabilization by noise for a class of stochastic reaction-diffusion equations. Prob. Th. Rel. Fields 133 (2000) 190-214. | Zbl | MR

[10] G. Da Prato, An Introduction to Infinite Dimensional Analysis. Springer-Verlag, Berlin, Germany (2006). | Zbl | MR

[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. | Zbl | MR

[12] J. Duan and A. Fursikov, Feedback stabilization for Oseen Fluid Equations. A stochastic approach. J. Math. Fluids Mech. 7 (2005) 574-610. | Zbl | MR

[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. | Zbl

[14] A. Fursikov, Stabilization for the 3-D Navier-Stokes systems by feedback boundary control. Discrete Contin. Dyn. Syst. 10 (2004) 289-314. | Zbl | MR

[15] T. Kato, Perturbation Theory of Linear Operators. Springer-Verlag, New York, Berlin (1966). | Zbl | MR

[16] S. Kuksin and A. Shirikyan, Ergodicity for the randomly forced 2D Navier-Stokes equations. Math. Phys. Anal. Geom. 4 (2001) 147-195. | Zbl | MR

[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. | Zbl | MR

[20] J.P. Raymond, Feedback boundary stabilization of the two dimensional Navier-Stokes equations. SIAM J. Contr. Opt. 45 (2006) 790-828. | Zbl | MR

[21] J.P. Raymond, Feedback boundary stabilization of the three dimensional incompressible Navier-Stokes equations. J. Math. Pures Appl. 87 (2007) 627-669. | Zbl | MR

[22] A. Shirikyan, Exponential mixing 2D Navier-Stokes equations perturbed by an unbounded noise. J. Math. Fluids Mech. 6 (2004) 169-193. | Zbl | MR