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Stochastic suppression and stabilization of functional differential equations. (English) Zbl 1217.93177
Summary: 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 $\dot x (t) = f(x_t,t)$. 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 $dx(t) = f(x_t,t)dt + qx(t)dw_1 (t) + \sigma |x(t)|^\beta x(t)dw_2 (t)$ 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.

##### MSC:
 93E15 Stochastic stability 93C23 Systems governed by functional-differential equations 60H10 Stochastic ordinary differential equations 93D21 Adaptive or robust stabilization
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##### References:
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