Mean-square Stabilization of Invariant Manifolds for SDEs

We consider systems of Ito’s stochastic differential equations with smooth compact invariant manifolds. The problem addressed is an exponential mean square (EMS) stabilization of these manifolds. The necessary and sufficient conditions of the stabilizability are derived on the base of the spectral criterion of the EMS-stability of invariant manifolds. We suggest methods for the design of the feedback stabilizing regulator for SDEs. Parametrical criteria of the stochastic stabilizability for limit cycles and tori are given. These criteria reduce the stabilization problem to the minimization of quadratic functionals. An analysis of the minimization problem of the quadratic functional for the case of the cycle of 2D stochastic system is presented in detail. Constructiveness of the elaborated theory is demonstrated for the stabilization of stochastically forced cycles of the Hopf system.