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Zbl 1175.93234
Wu, Fuke; Hu, Shigeng
Suppression and stabilisation of noise. (English)
[J] Int. J. Control 82, No. 11, 2150-2157 (2009). ISSN 0020-7179; ISSN 1366-5820/e

Summary: We investigate the stochastic suppression and stabilisation of nonlinear systems. Given an unstable differential equation $\dot x (t) = f(x(t),t)$, in which $f$ satisfies the one-sided polynomial growth condition, we introduce two Brownian noise feedbacks and therefore stochastically perturb this system and transform it into the nonlinear stochastic differential equation $dx(t)= f(x(t),t)\,dt + qx(t)\,dw_1(t)+\sigma |x(t)|^\beta x(t)\,w_2(t)$. This article shows that appropriate values of $\beta $ guarantee that this stochastic system has a unique global solution although the corresponding deterministic may explode in a finite time. Then sufficiently large $q$ may ensure that this stochastic system is almost surely exponentially stable.

MSC 2000:: *93E15 Stochastic stability
60H10 Stochastic ordinary differential equations
93C10 Nonlinear control systems
93D21 Adaptive and robust stabilization

Keywords: Brownian motion; suppression; stabilisation; Itô formula

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Zentralblatt MATH Berlin [Germany]