Stochastic suppression and stabilization of functional differential equations

Without the linear growth condition or the one-sided linear growth condition, this paper discusses whether or not stochastic noise feedback can stabilize a given unstable nonlinear functional system x ̇ ( t ) = f ( x t , t ) . Since f may defy the linear growth condition or the one-sided linear growth condition, this system may explode in a finite time. To stabilize this system, this paper stochastically perturbs this system into the stochastic functional differential system d x ( t ) = f ( x t , t ) d t + q x ( t ) d w 1 ( t ) + σ | x ( t ) | β x ( t ) d w 2 ( t ) by two independent Brownian motions w 1 ( t ) and w 2 ( t ) . This paper shows that the Brownian motion w 2 ( t ) may suppress the potential explosion of the solution for appropriate β . Moreover, for sufficiently large q , this stochastic functional system is exponentially stable. These results can be used to examine stochastic stabilization.