Stabilization of stochastic cycles and control of noise-induced chaos

We consider a nonlinear control system forced by stochastic disturbances. The problem addressed is a design of the feedback regulator which stabilizes a limit cycle of the closed-loop deterministic system and synthesizes a required dispersion of random states of the forced cycle for the corresponding stochastic system. To solve this problem, we develop a method based on the stochastic sensitivity function technique. The problem of a synthesis of the required stochastic sensitivity for cycles by feedback regulators is reduced to the solution of the linear algebraic equation for the gain matrix of the regulator. For this matrix, in the general n-dimensional case, a full parametric representation is found. An attractive case of nonlinear 3D systems which exhibit both regular and chaotic regimes is studied in detail. To construct a regulator, we use a new technique based on a singular decomposition of the assigned stochastic sensitivity matrix. Explicit formulas for parameters of this regulator synthesizing the required stochastic sensitivity for 3D-cycle are obtained. The constructiveness of the developed theory is shown on the example of the stabilization of the cycle for stochastic Lorenz model which exhibits a noise-induced transition to chaos. Using our technique for this model we provide a required small sensitivity for stochastically forced cycle and suppress chaos successfully.