随机激励的耗散的Hamilton系统理论的研究进展

ADVANCES IN THEORY OF STOCHASTICALLY EXCITED AND DISSIPATED HAMILTONIAN SYSTEMS

近几年中，利用Ｈａｍｉｌｔｏｎ系统的可积性与共振性概念及Ｐｏｉｓｓｏｎ括号性质等，提出了高斯白噪声激励下多自由度非线性随机系统的精确平稳解的泛函构造与求解方法，并在此基础上提出了等效非线性系统法，提出了拟Ｈａｍｉｌｔｏｎ系统的随机平均法，并在该法基础上研究了拟Ｈａｍｉｌｔｏｎ系统随机稳定性、随机分岔、可靠性及最优非线性随机控制，从而基本上形成了一个非线性随机动力学与控制的Ｈａｍｉｌｔｏｎ理论框架．本文简要介绍了这方面的进展．

In recent years, the functional form and solution method of the exact stationary solution for multi-degree-of-freedom nonlinear stochastic oscillatory systems under Gaussian white noise excitation were proposed by the present authors using the concepts of integrability and resonance and the property of Poisson bracket in Hamiltonian dynamics. Based on the exact stationary solution, an equivalent nonlinear system method was developed for similar systems. The stochastic averaging method for quasi-Hamiltonian systems was also proposed. The stochastic stability, stochastic bifurcation, reliability and optimal stochastic control of quasi-Hamiltonian systems were studied based on the stochastic averaging method. Thus, a Hamiltonian framework of nonlinear stochastic dynamics and control has been basically formulated. In the present paper, the advances in the theory of stochastically excited and dissipated Hamiltonian systems is reviewed.