Stabilization by Noise of Navier–Stokes Equations
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.
KeywordsStochastic Differential Equation Feedback Controller Stochastic Stabilization Unique Mild Solution Independent Brownian Motion
- 14.Barbu V (2009) The internal stabilization by noise of the linearized Navier–Stokes equation. ESAIM COCV (online) Google Scholar
- 18.Barbu V (2010) Exponential stabilization of the linearized Navier–Stokes equation by pointwise feedback controllers. Automatica. doi: 10.1016/j.automatica.2010.08.013
- 20.Barbu V, Da Prato G (2010) Internal stabilization by noise of the Navier–Stokes equation. SIAM J. Control Optim. (to appear) Google Scholar
- 44.Da Prato G, Zabczyk J (1991) Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge UK Google Scholar
- 63.Lipster R, Shiraev AN (1989) Theory of Martingals. Kluwer, Dordrecht Google Scholar