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Suppression and stabilisation of noise. (English) Zbl 1175.93234
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.

93E15Stochastic stability
60H10Stochastic ordinary differential equations
93C10Nonlinear control systems
93D21Adaptive or robust stabilization
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