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

The transient responses of an optimally controlled multi-delayed quasi-linear system driven by colored noise excitations are studied through Fokker-Planck-Kolmogorov equation. The time-delayed system is firstly transformed to an equivalent delay-freee system. The standard stochastic averaging method is then applied to obtain the partially averaged It? stochastic differential equation for the amplitude process of the original system. Afterwards, based on the dynamical programming principle and the bounded value condition of control, the dynamical programming equation is built and the optimal bounded control algorithm for minimizing the system response is obtained, thereby leading to the complete averaged FPK equation for the amplitude process. 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 complete averaged FPK equation in the orthogonal basis space. Finally, the analysis procedure is applied to study a time-delayed Duffing-Van Der Pol oscillator with optimal bounded control and a colored noise excitation, wherein the effects of the colored noise, the time delay, the control force and the resonance on the nonstationary response of system are discussed. All the results are verified through conducting Monte Carlo Simulation.