Abstract
The stability and response for delayed dynamical systems with delayed feedback control under additive or multiplicative Gaussian white noise excitations are studied by using the stochastic averaging method. The stochastic differential equation with time delay is transformed into Ito stochastic differential equation without time delay first. Then, the averaged Ito stochastic differential equations for the system are established and the stationary solutions of the averaged Fokker-Planck equations are derived. Finally, the analytical expressions of the response and stability conditions are derived for both cases through two examples. The boundedness conditions of the mean square of the amplitude for additive Gaussian white noise are obtained. Meanwhile, the moment stability condition for the case of multiplicative Gaussian white noise depends on the noise intensity, the time delays and the feedback gains. The numerical simulation results demonstrate the effectiveness of the proposed method
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