Stochastic optimal control of MDOF nonlinear systems under combined harmonic and wide-band noise excitations

A strategy for stochastic optimal control of multi-degrees-of-freedom (MDOF) strongly nonlinear systems under combined harmonic and wide-band noise excitations is proposed. First, a stochastic averaging procedure is developed to obtain the partially averaged Itô stochastic differential equations for weakly controlled strongly nonlinear systems under combined harmonic and wide-band noise excitations. Then, the dynamical programming equation for stochastic optimal control problem is derived and solved to yield the optimal control law. Finally, the responses of optimally controlled MDOF nonlinear systems are predicted by solving the Fokker–Planck–Kolmogorov equation associated with the fully averaged Itô stochastic differential equations. As an example, the stochastic optimal control of two coupled Duffing oscillators under combined harmonic and wide-band noise excitations is worked out in detail to illustrate the strategy. The effectiveness of the proposed control strategy is verified using the results from Monte Carlo simulation.