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In this paper, an optimal time-delay control strategy is designed for multi-degree-of-freedom (multi-DOF) strongly nonlinear systems excited by harmonic and wide-band noises. First, by using the generalized harmonic functions, a stochastic averaging method (SAM) is employed for the time-delay controlled strongly nonlinear system under combined harmonic and wide-band noise excitations, by which a set of partially averaged Itô equations are obtained. Then, by solving the dynamical programming equation associated with the partially averaged Itô equations, the optimal control law can be obtained. Finally, by solving the Fokker–Planck–Kolmogorov (FPK) equation, the responses of the optimally time-delay controlled system are predicted. The analytical results are compared with the Monte Carlo simulation to verify the effectiveness and efficiency of the proposed control strategy.

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The stationary response of multidegree-of-freedom (MDOF) strongly nonlinear system to fractional Gaussian noise (FGN) with Hurst index 1/2 < H < 1 is studied. First, the system is modeled as FGN-excited and -dissipated Hamiltonian system. Based on the integrability and resonance of the associated Hamiltonian system, the system is divided into five classes: partially integrable and resonant, partially integrable and nonresonant, completely integrable and resonant, completely integrable and nonresonant, and nonintegrable. Then, the averaged fractional stochastic differential equations (SDEs) for five classes of quasi-Hamiltonian systems with lower dimension and involving only slowly varying processes are derived. Finally, the approximate stationary probability densities and other statistics of two example systems are obtained by numerical simulation of the averaged fractional SDEs to illustrate the application and compared with those from original systems to show the advantages of the proposed procedure.

An optimal control strategy for the random vibration reduction of nonlinear structures using piezoelectric stack inertial actuator is proposed. First, the dynamic model of the nonlinear structure considering the dynamics of a piezoelectric stack inertial actuator is established, and the motion equation of the coupled system is described by a quasi-non-integrable-Hamiltonian system. Then, using the stochastic averaging method, this quasi-non-integrable-Hamiltonian system is reduced to a one-dimensional averaged system for total energy. The optimal control law is determined by establishing and solving the dynamic programming equation. The proposed control law is analytical and can be fully executed by a piezoelectric stack inertial actuator. The responses of optimally controlled and uncontrolled systems are obtained by solving the Fokker–Planck–Kolmogorov (FPK) equation to evaluate the control effectiveness of the proposed strategy. Numerical results show that our proposed control strategy is effective for random vibration reduction of the nonlinear structures using piezoelectric stack inertial actuator, and the theoretical method is verified by comparing with the simulation results.