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)over dot(t) = f(x(t), t). Since f may defy the linear growth condition or the onesided 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 dx(t) = f (x(t), t)dt + qx(t)dw(1)(t)+sigma vertical bar x(t)vertical bar(beta)x(t)dw(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 beta. Moreover, for sufficiently large q, this stochastic functional system is exponentially stable. These results can be used to examine stochastic stabilization.