| The stochastic response,reliability and asymptotic Lyapunov stability with probability one of strongly nonlinear systems with time-delayed feedback control under several kinds of narrow-band random excitations are investigated.The time-delayed feedback control forces are approximated with the control forces without time delay by using the general harmonic functions.Then the motion equations are reduced to a set of averaged It(?)stochastic differential equations by using the stochastic averaging method.The stationary solution of the Fokker-Plank-Kolmogorov(FPK)equation associated with the averaged It(?)equations is obtained.The effects of time delay on the stochastic response of the controlled systems are discussed.By solving the backward Kolmogorov equation and the generalized Pontryagin equation associated with the averaged It(?)equations,the effects of the time delay on the conditional reliability function,the conditional probability density of the first-passage time and the mean first-passage time are analyzed.Based on the averaged It(?)equations and by introducing a norm in terms of response amplitude,the Lyapunov exponent which determines the asymptotic Lyapunov stability with probability one of the controlled system is obtained.How the asymptotic Lyapunov stability with probability one of the controlled system affected by the time delay is analyzed.Based on the proposed method,in chapter 2,the effects of time delay on stochastic response,reliability and stability of strongly nonlinear systems under combined harmonic and Gaussian white-noise excitations are studied. In chapter 3,the effects of time delay on stochastic response,reliability and stability of strongly nonlinear systems under combined harmonic and wide-band noise excitations are studied.In chapter 4,the effects of time delay on stochastic response and stability of strongly nonlinear systems under bounded noise excitation are studied. All the theoretical results are verified by Monte Carlo digital simulations.
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