本文研究了一类随机激励的多自由度非线性内共振拟可积哈密顿系统的首次穿越可靠性的最优控制问题。基于随机平均法与动态规划原理,得到了最优控制系统的Itô随机微分方程,建立了最优控制系统条件可靠性函数满足的后向Kolmogorov方程及平均首次穿越时间满足的Pontryagin方程。通过具体的算例,结合Monte Carlo数值模拟验证了理论方法的有效性。
Abstract
Here, the optimal control problem for first-passage reliability of a class of randomly excited multi-DOF nonlinear quasi-integrable Hamiltonian systems with internal resonance was investigated.Based on the stochastic averaging method and dynamic programming principle, the It stochastic differential equations of the optimally controlled system were obtained.Then the backward Kolmogorov equation governing conditional reliability function and the Pontryagin equation governing the mean first-passage time were established.An illustrative numerical example was given.The validity of the theoretical method was verified by Monte Carlo digital simulation.
关键词
拟可积哈密顿系统 /
内共振 /
随机平均 /
动态规划 /
首次穿越
{{custom_keyword}} /
Key words
Quasi-integrable Hamiltonian system /
internal resonance /
stochastic averaging /
dynamic programming principle /
first-passage
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] Cox D R, Miller H D. The Theory of Stochastic Processes [M]. Chapman and Hall, New York, 1965.
[2] ZHU Wei-qiu. Nonlinear stochastic dynamics and control in Hamiltonian formulation [J]. Applied Mechanics Reviews, 2006, 59: 230-248.
[3] 朱位秋. 非线性随机动力学与控制-Hamiltonian理论体系框架[M],科学出版社,北京,2003.
ZHU Wei-qiu. Nonlinear Stochastic Dynamics and Control-Hamiltonian Theoretical Framework [M]. Science Press, Beijing, 2003.
[4] Bellman R. Dynamic Programming [M]. Princeton University Press, Princeton, 1957.
[5] Huang C S, Wang S, Teo K L. Solving Hamilton-Jacobi-Bellman equations by a modified method of characteristics [J]. Nonlinear Analysis, Theory, Methods and Applications, 2000, 40(1): 279–293.
[6] Besselink B, Tabak U, Lutowska A, et al. A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control [J]. Journal of Sound and Vibration, 2013, 332: 4403-4422.
[7] Roberts J B, Spanos P D. Stochastic averaging: an approximate method of solving random vibration problems [J]. International Journal of Non-Linear Mechanics, 1986, 21: 111-134.
[8] Deng M L, Zhu W Q. Feedback minimization of first-passage failure of quasi integrable Hamiltonian systems [J]. Acta Mechanica Sinica, 2007, 23(4): 437-444.
[9] Zhu W Q, Huang Z L, Deng M L. First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems [J]. International Journal of Non-Linear Mechanics, 2003, 38(8): 1133-1148.
[10] Wu Y J, Huan R H. First-passage failure minimization of stochastic Duffing-Rayleigh-Mathieu system [J], Mechanics Research Communications, 2008, 35(7): 447-453.
[11] Cheung Y K, Xu Z. Internal resonance of strongly non-linear autonomous vibrating systems with many degrees of freedom [J]. Journal of Sound and Vibration, 1995, 180(2): 229-238.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}