Abstract
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.
Highlights
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 well-known tool to solve the problem of stochastic optimal control is the dynamical programming principle, which was proposed by Bellman 1
The procedure consists of applying the stochastic averaging method for weakly controlled strongly nonlinear systems under combined harmonic and wide-band noise excitations, establishing the dynamical programming equation for the control problem of minimizing the response based on the partially averaged Itostochastic differential equations and the dynamical programming principle, determining the optimal control from the dynamical programming equation and the control constraint without solving the dynamical programming equation
Summary
The well-known tool to solve the problem of stochastic optimal control is the dynamical programming principle, which was proposed by Bellman 1. By using stochastic averaging method, many researchers have studied the linear or strongly nonlinear systems under combined harmonic and wide-band random excitations 11–16. Optimal bounded control of a linear or nonlinear oscillator subject to combined harmonic and Gaussian white noise excitations has been studied 17–19. Little work has been done on the optimal bounded control of a strongly nonlinear oscillator subject to combined harmonic and wide-band noise excitations 20. A procedure for designing optimal bounded control to minimize the stationary response of a strongly nonlinear oscillator under combined harmonic and wide-band colored noise excitations is proposed.
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