An n degree-of-freedom (DOF) non-integrable Hamiltonian system subject to light damping and weak stochastic excitation is called quasi-non-integrable Hamiltonian system. In the present paper, the stochastic averaging of quasi-non-integrable Hamiltonian systems is brie5y reviewed. A new norm in terms of the square root of Hamiltonian is introduced in the de6nitions of stochastic stability and Lyapunov exponent and the formulas for the Lyapunov exponent are derived from the averaged Itˆo equations of the Hamiltonian and of the square root of Hamiltonian. It is inferred that the Lyapunov exponent so obtained is the 6rst approximation of the largest Lyapunov exponent of the original quasi-non-integrable Hamiltonian systems and the necessary and su8cient condition for the asymptotic stability with probability one of the trivial solution of the original systems can be obtained approximately by letting the Lyapunov exponent to be negative. This inference is con6rmed by comparing the stability conditions obtained from negative Lyapunov exponent and by examining the sample behaviors of averaged Hamiltonian or the square root of averaged Hamiltonian at trivial boundary for two examples. It is also veri6ed by the largest Lyapunov exponent obtained using small noise expansion for the second example.

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) independent 0rst integrals which are in involution is calledpartially integrable Hamiltonian system. A partially integrable Hamiltonian system subject to light dampings andweak stochastic excitations is called quasi-partially integrable Hamiltonian system. In the present paper, the procedures for studying the 0rst-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems are proposed. First, the stochastic averaging methodfor quasi-partially integrable Hamiltonian systems is brie4y reviewed. Then, basedon the averagedIto equations, a backwardKolmogorov equation governing the conditional reliability function, a set of generalized Pontryagin equations governing the conditional moments of 0rst-passage time and their boundary and initial conditions are established. After that, the dynamical programming equations and their associated boundary and 0nal time conditions for the control problems of maximization of reliability andof maximization of mean 0rst-passage time are formulated. The relationship between the backwardKolmogorov equation andthe dynamical programming equation for reliability maximization, andthat between the Pontryagin equation andthe dynamical programming equation for maximization of mean 0rst-passage time are discussed. Finally, an example is worked out to illustrate the proposed procedures and the e9ectiveness of feedback control in reducing 0rst-passage failure.

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.

A strategy for optimal nonlinear feedback control of randomly excited structural systems is proposed based on the stochastic averaging method for quasi-Hamiltonian systems and the stochastic dynamic programming principle. A randomly excited structural system is formulated as a quasi-Hamiltonian system and the control forces are divided into conservative and dissipative parts. The conservative parts are designed to change the integrability and resonance of the associated Hamiltonian system and the energy distribution among the controlled system. After the conservative parts are determined, the system response is reduced to a controlled diffusion process by using the stochastic averaging method. The dissipative parts of control forces are then obtained from solving the stochastic dynamic programming equation. Both the responses of uncontrolled and controlled structural systems can be predicted analytically. Numerical results for a controlled and stochastically excited Duffing oscillator and a two-degree-of-freedom system with linear springs and linear and nonlinear dampings, show that the proposed control strategy is very effective and efficient.

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

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.

It is shown that the structure and property of the exact stationary solution of a stochastically excited and dissipated n-degree-of-freedom Hamiltonian system depend upon the integrability and resonant property of the Hamiltonian system modified by the Wong-Zakai correct terms. For a stochastically excited and dissipated nonintegra-ble Hamiltonian system, the exact stationary solution is a functional of the Hamilto-nian and has the protYerty of equipartition of energy. For a stochastically excited and dissipated integrable Hamiltonian system, the exact stationary solution is a functional of n independent integrals of motion or n action variables of the modified Hamiltonian system in nonresonant case, or a functional of both n action variables and binations of phase angles in resonant case with a (1≤α≤n-1) resonant relations, and has the property that the partition of the energy among n degrees-of-freedom can be adjusted by the magnitudes and distributions of dampings and stochastic excitations. All the exact stationary solutions obtained to date for nonlinear stochastic systems are those for stochastically excited and dissipated nonin-tegrable Hamiltonian systems, which are further generalized to account for the modi-fication of the Hamiltonian by Wong-Zakai correct terms. Procedures to obtain the exact stationary solutions of stochastically excited and dissipated integrable Hamilto-nian systems in both resonant and nonresonant cases are proposed and the conditions for such solutions to exist are deduced. The above procedures and results are further extended to a more general class of systems, which include the stochastically excited and dissipated Hamiltonian systems as special cases. Examples are given to illustrate the applications of the procedures.