Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 635823, 21 pages
Research Article

Optimal Bounded Control for Stationary Response of Strongly Nonlinear Oscillators under Combined Harmonic and Wide-Band Noise Excitations

1Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China
2Institute of Mechatronic Engineering, Hangzhou Dianzi University, Hangzhou 310018, China
3Department of Mechanics, Zhejiang University, Hangzhou 310027, China

Received 29 March 2011; Revised 8 July 2011; Accepted 25 July 2011

Academic Editor: Angelo Luongo

Copyright © 2011 Yongjun Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the stochastic optimal bounded control for minimizing the stationary response of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. The stochastic averaging method and the dynamical programming principle are combined to obtain the fully averaged Itô stochastic differential equations which describe the original controlled strongly nonlinear system approximately. The stationary joint probability density of the amplitude and phase difference of the optimally controlled systems is obtained from solving the corresponding reduced Fokker-Planck-Kolmogorov (FPK) equation. An example is given to illustrate the proposed procedure, and the theoretical results are verified by Monte Carlo simulation.