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中文摘要：
该文研究了受色噪声激励的Rayleigh—During振子瞬态响应及其最优有界控制问题。在弱扰动假设下应用标准随机平均法得到了原系统振幅过程的部分平均It6随机微分方程。应用Bellman动态规划原理，结合控制力有界条件，得到了最优有界控制率。完成所有平均过程得到了系统振幅过程的完全平均It6随机微分方程和相应的Fokker—Planck．Kolmogorov方程。基于退化线性系统得到一组正交基空间，在此基空间内进行Galerkin变分近似求解Fokker—Planck-Kolmogorov方程得到了受控系统的瞬态响应。采用Monte—Carlo模拟验证了所有理论结果的有效性。计算表明：1）所提方法求解受最优有界控制率作用的随机非线性系统瞬态响应有效；2）最优有界控制率成功降低了系统瞬态响应；3）该方法的求解效率高于Monte—Carlo数值模拟方法。
英文摘要：
The transient response and its optimal bounded control of a Rayleigh-Duffing oscillator driven by colored noises are studied theoretically. The disturbances are assumed to be weak. The standard stochastic averaging method is firstly adopted to obtain the partly averaged It6 stochastic differential equation for a original system amplitude process. The optimal bounded control algorithm is built by Bellman dynamic programming principle combined with control constraints. Finishing all the averaging procedures, the completed averaged It6 stochastic differential equation and corresponding Fokker-Planck-Kolmogorov equation are established. Then, the controlled system transient responses are predicted by applying Galerkin method to the obtained Fokker-Planck-Kolmogorov equation. The base functions used in the Galerkin scheme are obtained from a degenerated linear system. Finally, Monte Carlo simulation is used to verify the theoretical results reliability. Calculations show that： 1） the proposed method is effective at solving the transient response of an optimally bounded controlled stochastic nonlinear system; 2） the control algorithm does successfully reduce the transient response of the system; 3） the calculation efficiency of the theoretical method is higher than that of Monte Carlosimulation.
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