最优有界控制下色噪声驱动多时滞拟线性系统瞬态响应

基于Fokker-Planck-Kolmogorov方程瞬态求解研究了受最优有界控制的色噪声驱动的多时滞拟线性系统的瞬态响应。利用等价变换将时滞系统转化为非时滞系统。在弱扰动假设下应用标准随机平均法得到振幅过程的部分平均It?随机微分方程。由动态规划原理和控制力界值条件得到最优有界控制率从而得到完全平均的Fokker-Planck-Kolmogorov方程。通过原系统的退化线性系统导出一组正交基并在该基空间内进行Galerkin变分得到近似瞬态响应。最后将该方法应用到受最优有界控制率和色噪声共同作用的时滞Duffing-Van Der Pol振子进行理论求解并综合讨论了色噪声、时滞、控制力和共振对系统瞬态响应的影响，采用Monte-Carlo模拟验证了所有理论和计算结果的正确性。

The transient response of the multi-delayed quasi-linear system with colored noise excitations and optimal bounded control

The transient response of the optimally controlled multi-delayed quasi-linear system driven by colored noise excitations are studied based on solving the Fokker-Planck-Kolmogorov equation. The system with time delay is firstly transformed to an equivalent system without time delay. The standard stochastic averaging method is adopted to obtain the partially averaged It? stochastic differential equation for the amplitude process of the original system. Based on the dynamical programming principle, the dynamical programming equation is built and the optimal control algorithm for minimizing the system response is obtained combined with the control bounded value. The completed averaged FPK equation for the system amplitude process is then deduced. A set of orthogonal basis functions are obtained by applying the eigenfunction method to the degenerated linear FPK equation. The approximate probability densities are obtained by applying the Galerkin method to the completed averaged FPK equation in the orthogonal basis space. The time-delayed Duffing-Van Der Pol oscillator with optimal bounded control and a colored noise excitation is given as an example to show the proposed procedure. The results obtained by the analytical method are verified by those obtained from Monte Carlo Simulation. The effects of the colored noise, the time delay, the control force and the resonance on the system nonstationary response are discussed. Results obtained show that the proposed procedure can be used to obtain the nonstationary response of the multi-delayed quasi-linear system with optimal bounded control and colored noises effectively.