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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 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)+σ|x(t)| β 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 β. Moreover, for sufficiently large q, this stochastic functional system is exponentially stable. These results can be used to examine stochastic stabilization.
MSC:
93E15Stochastic stability
93C23Systems governed by functional-differential equations
60H10Stochastic ordinary differential equations
93D21Adaptive or robust stabilization
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