A procedure for designing optimal bounded control to minimize the response of quasi-integrable Hamiltonian systems is proposed based on the stochastic averaging method for quasi-integrable Hamiltonian systems and the stochastic dynamical programming principle. The equations of motion of a controlled quasi-integrable Hamiltonian system are 5rst reduced to a set of partially completed averaged Itˆo stochastic di7erential equations by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, the dynamical programming equation for the control problems of minimizing the response of the averaged system is formulated based on the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraints without solving the dynamical programming equation. The response of optimally controlled systems is predicted through solving the Fokker-Planck-Kolmogrov equation associated with fully completed averaged Ito equations. Finally, two examples are worked out in detail to illustrate the application and e7ectiveness of the proposed control strategy.

An n-degree-of-freedom Hamiltonian system with r (1＜r＜n) integrals of motion which are in involution is called partially integrable Hamiltonian system. In the present paper, the exact stationary solutions of stochastically excited and dissipated partially integrable Hamiltonian systems are first reviewed. Then an equivalent non-linear system method for this class of systems in both nonresonant and resonant cases is developed. Three criteria are proposed to obtain the equivalent non-linear systems. The application and e!ectiveness of the method are illustrated by an example.

A new procedure for analyzing the stochastic Hopf bifurcation of quasi-non-integrable-Hamiltonian systems is proposed. A quasi-non-integrable-Hamiltonian system is

A survey of stochastic averaging methods in random vibration is given. After a brief introduction to the basic ideas, the advantages and the history of the methods, three kinds of stochastic averaging methods are formulated, and their applicability and recent developments are stated. In the second part, the applications of the methods in response prediction, stability decision, and reliability estimation of randomly excited nonlinear and parametric systems are reviewed. The possibility of further developments and applications of the methods is also pointed out.