Nonstationary response of nonlinear oscillators with optimal bounded control and broad-band noises

Nonstationary response of nonlinear oscillators with optimal bounded control and broad-band noise excitations is investigated. First, the stochastic averaging method is applied to obtain an averaged Itô stochastic differential equation for the amplitude process. Then, the dynamical programming equation is employed to minimize the system response and establish an optimal control law with a control constraint. The nonstationary probability density of the amplitude process can be solved from the corresponding Fokker-Planck-Kolmogorov equation by using the Galerkin method if only external excitations exist. In the case of parametric excitations are present, Monte Carlo simulations can be carried out for the simplified averaged system of the amplitude process with much less computational efforts. Two examples are given to illustrate the feasibility of the proposed procedure and the effectiveness of the optimal control strategy. The accuracy and efficiency of the proposed procedure are substantiated by those obtained from Monte Carlo simulation of the original system.