Feedback Stabilization for Oseen Fluid Equations: A Stochastic Approach

Abstract.

The authors consider stochastic aspects of the stabilization problem for two and three-dimensional Oseen equations with help of feedback control defined on a part of the fluid boundary. Stochastic issues arise when inevitable unpredictable fluctuations in numerical realization of stabilization procedures are taken into account and they are supposed to be independent identically distributed random variables. Under this assumption the solution to the stabilization problem obtained via boundary feedback control can be described by a Markov chain or a discrete random dynamical system. It is shown that this random dynamical system possesses a unique, exponentially attracting, invariant measure, namely, this random dynamical system is ergodic. This gives adequate statistical description of the stabilization process on the stage when stabilized solution has to be retained near zero (i.e. near unstable state of equilibrium).