To enhance the reliability of the stochastically excited structure, it is significant to study the problem
of stochastic optimal control for minimizing first-passage failure. Combining the stochastic averaging method
with dynamical programming principle, we study the optimal control for minimizing first-passage failure of multidegrees-
of-freedom (MDoF) nonlinear oscillators under Gaussian white noise excitations. The equations of motion
of the controlled system are reduced to time homogenous diffusion processes by stochastic averaging. The optimal
control law is determined by the dynamical programming equations and the control constraint. The backward Kolmogorov
(BK) equation and the Pontryagin equation are established to obtain the conditional reliability function
and mean first-passage time (MFPT) of the optimally controlled system, respectively. An example has shown that
the proposed control strategy can increase the reliability and MFPT of the original system, and the mathematical
treatment is also facilitated.
GAO Yang-yan (高阳艳), WU Yong-jun* (吴勇军)
. Stochastic Optimal Control for First-Passage Failure of Nonlinear Oscillators with Multi-Degrees-of-Freedom[J]. Journal of Shanghai Jiaotong University(Science), 2013
, 18(5)
: 577
-582
.
DOI: 10.1007/s12204-013-1428-8
[1] Chen L, Wu Z. Maximum principle for the stochastic optimal control problem with delay and application [J]. Automatica, 2010, 46: 1074-1080.
[2] Wu Z, Zhang F. Stochastic maximum principle for optimal control problems of forward-backward systems involving impulse controls [J]. IEEE Transactions on Automatic Control, 2011, 56(6): 1401-1406.
[3] Yu Z Y. The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls [J]. Automatica, 2012, 48: 2420-2432.
[4] Gu X D, Zhu W Q, Xu W. Stochastic optimal control of quasi non-integrable Hamiltonian systems with stochastic maximum principle [J]. Nonlinear Dynamics, 2012, 70: 779-787.
[5] Zhu Wei-qiu. Nonlinear stochastic dynamics and control: Hamiltonian theoretical framework [M]. Beijing: Science Press, 2003 (in Chinese).
[6] Zhu W Q. Nonlinear stochastic dynamics and control in Hamiltonian formulation [J]. Applied Mechanics Reviews, 2006, 59(4): 230-248.
[7] Zhu W Q, Huang Z L, Deng M L. Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems [J]. International Journal of Non-Linear Mechanics, 2002, 37(6): 1057-1071.
[8] Deng M L, Zhu W Q. Feedback minimization of firstpassage failure of quasi-integrable Hamiltonian systems [J]. Acta Mechanica Sinica, 2007, 23(4): 437-444.
[9] Zhu W Q, Wu Y J. Optimal bounded control of firstpassage failure of strongly non-linear oscillators under combined harmonic and white-noise excitations [J]. Journal of Sound and Vibration, 2004, 271(1-2): 83-101.
[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.
