First passage failure of quasi non-integrable generalized Hamiltonian systems

The first passage failure of quasi non-integrable generalized Hamiltonian systems is studied. First, the generalized Hamiltonian systems are reviewed briefly. Then, the stochastic averaging method for quasi non-integrable generalized Hamiltonian systems is applied to obtain averaged Itô stochastic differential equations, from which the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of the first passage time are established. The conditional reliability function and the conditional mean of first passage time are obtained by solving these equations together with suitable initial condition and boundary conditions. Finally, an example of power system under Gaussian white noise excitation is worked out in detail and the analytical results are confirmed by using Monte Carlo simulation of original system.     

First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems

An n degree-of-freedom Hamiltonian system with r(1<r<n) independent first integrals which are in involution is called partially integrable Hamiltonian system. A partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi-partially integrable Hamiltonian system. In the present paper, the procedures for studying the first-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems are proposed. First, the stochastic averaging method for quasi-partially integrable Hamiltonian systems is briefly reviewed. Then, based on the averaged Itô equations, a backward Kolmogorov equation governing the conditional reliability function, a set of generalized Pontryagin equations governing the conditional moments of first-passage time and their boundary and initial conditions are established. After that, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and of maximization of mean first-passage time are formulated. The relationship between the backward Kolmogorov equation and the dynamical programming equation for reliability maximization, and that between the Pontryagin equation and the dynamical programming equation for maximization of mean first-passage time are discussed. Finally, an example is worked out to illustrate the proposed procedures and the effectiveness of feedback control in reducing first-passage failure.     

Feedback minimization of first-passage failure of quasi integrable Hamiltonian systems

A nonlinear stochastic optimal control strategy for minimizing the first-passage failure of quasi integrable Hamiltonian systems (multi-degree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is proposed. The equations of motion for a controlled quasi integrable Hamiltonian system are reduced to a set of averaged Itô stochastic differential equations by using the stochastic averaging method. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The final dynamical programming equations for these control problems are determined and their relationships to the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are separately established. The conditional reliability function and the mean first-passage time of the controlled system are obtained by solving the final dynamical programming equations or their equivalent Kolmogorov and Pontryagin equations. An example is presented to illustrate the application and effectiveness of the proposed control strategy.     

Optimal Bounded Control of First-Passage Failure of Quasi-Integrable Hamiltonian Systems with Wide-Band Random Excitation

The optimal bounded control of quasi-integrable Hamiltonian systems with wide-band random excitation for minimizing their first-passage failure is investigated. First, a stochastic averaging method for multi-degrees-of-freedom (MDOF) strongly nonlinear quasi-integrable Hamiltonian systems with wide-band stationary random excitations using generalized harmonic functions is proposed. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximizinig reliability and maximizing mean first-passage time are formulated based on the averaged Itô equations by applying the dynamical programming principle. The optimal control law is derived from the dynamical programming equations and control constraints. The relationship between the dynamical programming equations and the backward Kolmogorov equation for the conditional reliability function and the Pontryagin equation for the conditional mean first-passage time of optimally controlled system is discussed. Finally, the conditional reliability function, the conditional probability density and mean of first-passage time of an optimally controlled system are obtained by solving the backward Kolmogorov equation and Pontryagin equation. The application of the proposed procedure and effectiveness of control strategy are illustrated with an example.     

First-passage problem of a class of internally resonant quasi-integrable Hamiltonian system under wide-band stochastic excitations

In this paper, first-passage problem of a class of internally resonant quasi-integrable Hamiltonian system under wide-band stochastic excitations is studied theoretically. By using stochastic averaging method, the equations of motion of the original internally resonant Hamiltonian system are reduced to a set of averaged Itô stochastic differential equations. The backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are established under appropriate boundary and (or) initial conditions. An example is given to show the accuracy of the theoretical method. Numerical solutions of high-dimensional backward Kolmogorov and Pontryagin equation are obtained by finite difference. All theoretical results are verified by Monte Carlo simulation.     

First-passage failure of strongly non-linear oscillators with time-delayed feedback control under combined harmonic and wide-band noise excitations

A procedure for studying the first-passage failure of strongly non-linear oscillators with time-delayed feedback control under combined harmonic and wide-band noise excitations is proposed. First, the time-delayed feedback control forces are expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method. A backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. An example is worked out in detail to illustrate the proposed procedure. The effects of time delay in feedback control forces on the conditional reliability function, conditional probability density and moments of first-passage time are analyzed. The validity of the proposed method is confirmed by digital simulation.     

First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations

First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations is studied. The motion equation of the system is reduced to a set of averaged Itô stochastic differential equations by stochastic averaging in the case of resonance. Then, the backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function and the conditional probability density and mean first-passage time are obtained by solving the backward Kolmogorov equation and Pontryagin equation with suitable initial and boundary conditions. The procedure is applied to Duffing–van der Pol system in resonant case and the analytical results are verified by Monte Carlo simulation.     

外共振耦合Duffing-van der Pol系统的首次穿越

研究了谐和力与宽带噪声激励下二自由度强非线性Duffing-van derPol系统的首次穿越问题. 在外共振情形, 应用随机平均法将系统动力学方程化为关于振幅与角变量的Itô随机微分方程. 然后建立了系统的可靠性函数满足的后向Kolmogorov方程以及平均首次穿越时间满足的Pontryagin方程. 在一定的边界条件和初始条件下, 用有限差分法求解了这两个高维偏微分方程, 得到系统的条件可靠性函数、平均首次穿越时间以及平均首次穿越时间的条件概率密度. 讨论了不同参数对系统可靠性以及平均首次穿越时间的影响. 用Monte Carlo数值模拟验证了理论方法的有效性.     

Gauss白噪声外激下Rayleigh振子的平稳响应与首次穿越

研究了Rayleigh振子在Gauss白噪声外激下的平稳响应和首次穿越。首先利用随机平均法给出了系统随机平均It^o微分方程,对平均方程的稳态概率密度做了数值分析;然后建立了条件可靠性函数的后向Kolmogorov方程及首次穿越时间条件矩的Pontragin方程;最后对三组不同的参数值分析了首次穿越的概率统计特性。     

First-Passage Time of Duffing Oscillator under Combined Harmonic and White-Noise Excitations

The first-passage time of Duffing oscillator under combined harmonic andwhite-noise excitations is studied. The equation of motion of the system is firstreduced to a set of averaged Itô stochastic differential equations by using thestochastic averaging method. Then, a backward Kolmogorov equation governing theconditional reliability function and a set of generalized Pontryagin equationsgoverning the conditional moments of first-passage time are established. Finally, theconditional reliability function, and the conditional probability density and momentsof first-passage time are obtained by solving the backward Kolmogorov equation andgeneralized Pontryagin equations with suitable initial and boundary conditions.Numerical results for two resonant cases with several sets of parameter values areobtained and the analytical results are verified by using those from digital simulation.     

Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems

The non-linear stochastic optimal control of quasi non-integrable Hamiltonian systems for minimizing their first-passage failure is investigated. A controlled quasi non-integrable Hamiltonian system is reduced to an one-dimensional controlled diffusion process of averaged Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. The dynamical programming equations and their associated boundary and final time conditions for the problems of maximization of reliability and of maximization of mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The dynamical programming equations for maximum reliability problem and for maximum mean first-passage time problem are finalized and their relationships to the backward Kolmogorov equation for the reliability function and the Pontryagin equation for mean first-passage time, respectively, are pointed out. The boundary condition at zero Hamiltonian is discussed. Two examples are worked out to illustrate the application and effectiveness of the proposed procedure.     

Transient stochastic response of quasi-partially integrable Hamiltonian systems

The approximate transient response of multi-degree-of-freedom (MDOF) quasi-partially integrable Hamiltonian systems under Gaussian white noise excitation is investigated. First, the averaged Itô equations for first integrals and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the transient probability density of first integrals of the system are derived by applying the stochastic averaging method for quasi-partially integrable Hamiltonian systems. Then, the approximate solution of the transient probability density of first integrals of the system is obtained from solving the FPK equation by applying the Galerkin method. The approximate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coefficients. The transient probability densities of displacements and velocities can be derived from that of first integrals. One example is given to illustrate the application of the proposed procedure. It is shown that the results for the example obtained by using the proposed procedure agree well with those from Monte Carlo simulation of the original system.     

Optimal control strategies for stochastically excited quasi partially integrable Hamiltonian systems

In this paper two different control strategies designed to alleviate the response of quasi partially integrable Hamiltonian systems subjected to stochastic excitation are proposed. First, by using the stochastic averaging method for quasi partially integrable Hamiltonian systems, an n-DOF controlled quasi partially integrable Hamiltonian system with stochastic excitation is converted into a set of partially averaged Itô stochastic differential equations. Then, the dynamical programming equation associated with the partially averaged Itô equations is formulated by applying the stochastic dynamical programming principle. In the first control strategy, the optimal control law is derived from the dynamical programming equation and the control constraints without solving the dynamical programming equation. In the second control strategy, the optimal control law is obtained by solving the dynamical programming equation. Finally, both the responses of controlled and uncontrolled systems are predicted through solving the Fokker-Plank-Kolmogorov equation associated with fully averaged Itô equations. An example is worked out to illustrate the application and effectiveness of the two proposed control strategies.     

Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems

An n degree-of-freedom Hamiltonian system with r(1<r<n) independent first integrals which are in involution is called partially integrable Hamiltonian system and a partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi partially integrable Hamiltonian system. In the present paper, the averaged Itô and Fokker-Planck-Kolmogorov (FPK) equations for quasi partially integrable Hamiltonian systems in both cases of non-resonance and resonance are derived. It is shown that the number of averaged Itô equations and the dimension of the averaged FPK equation of a quasi partially integrable Hamiltonian system is equal to the number of independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. The technique to obtain the exact stationary solution of the averaged FPK equation is presented. The largest Lyapunov exponent of the averaged system is formulated, based on which the stochastic stability and bifurcation of original quasi partially integrable Hamiltonian systems can be determined. Examples are given to illustrate the applications of the proposed stochastic averaging method for quasi partially integrable Hamiltonian systems in response prediction and stability decision and the results are verified by using digital simulation.     

耦合Duffing-van der Pol系统的首次穿越问题

利用拟不可积Hamilton系统随机平均法，研究了高斯白噪声激励下耦 合Duffing-van der Pol系统的首次穿越问题. 首先给出了条件可靠性函数满足的后向 Kolmogorov 方程以及首次穿越时间条件矩满足的广义Pontryagin方程. 然后根据 这两类偏微分方程的边界条件和初始条件，详细分析了在外激与参激共 同作用以及纯外激作用等情况下系统的可靠性与首次穿越时间的各阶矩. 最后以图表形式给 出了可靠性函数、首次穿越时间的概率密度以及平均首次穿越时间的数值结果.     

STOCHASTIC OPTIMAL CONTROL FOR THE RESPONSE OF QUASI NON-INTEGRABLE HAMILTONIAN SYSTEMS~

A strategy is proposed based on the stochastic averaging method for quasi nonintegrable Hamiltonian systems and the stochastic dynamical programming principle. The proposed strategy can be used to design nonlinear stochastic optimal control to minimize the response of quasi non-integrable Hamiltonian systems subject to Gaussian white noise excitation. By using the stochastic averaging method for quasi non-integrable Hamiltonian systems the equations of motion of a controlled quasi non-integrable Hamiltonian system is reduced to a one-dimensional averaged Ito stochastic differential equation. By using the stochastic dynamical programming principle the dynamical programming equation for minimizing the response of the system is formulated.The optimal control law is derived from the dynamical programming equation and the bounded control constraints. The response of optimally controlled systems is predicted through solving the FPK equation associated with It5 stochastic differential equation. An example is worked out in detail to illustrate the application of the control strategy proposed.     

Transient stochastic response of quasi integerable Hamiltonian systems

The approximate transient response of quasi integrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged Ito equations for independent motion integrals and the associated Fokker-Planck-Kolmogorov (FPK) equation governing the transient probability density of independent motion integrals of the system are derived by applying the stochastic averaging method for quasi integrable Hamiltonian systems. Then, approximate solution of the transient probability density of independent motion integrals is obtained by applying the Galerkin method to solve the FPK equation. The approximate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coefficients. The transient probability densities of displacements and velocities can be derived from that of independent motion integrals. Three examples are given to illustrate the application of the proposed procedure. It is shown that the results for the three examples obtained by using the proposed procedure agree well with those from Monte Carlo simulation of the original systems.     

Optimal bounded control for minimizing the response of quasi-integrable Hamiltonian systems

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 first reduced to a set of partially completed averaged Itô stochastic differential 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 Itô equations. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed control strategy.     

Stochastic averaging of quasi integrable and resonant Hamiltonian systems excited by fractional Gaussian noise with Hurst index 1/2<H<1

A stochastic averaging method of quasi integrable and resonant Hamiltonian systems under excitation of fractional Gaussian noise(fGn) with the Hurst index 1/2 H 1 is proposed. First, the definition and the basic property of f Gn and related fractional Brownian motion(fBm) are briefly introduced. Then, the averaged fractional stochastic differential equations(SDEs) for the first integrals and combinations of angle variables of the associated Hamiltonian systems are derived. The stationary probability density and statistics of the original systems are then obtained approximately by simulating the averaged SDEs numerically. An example is worked out to illustrate the proposed stochastic averaging method. It is shown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of original system agree well.     

First passage of stochastic fractional derivative systems with power-form restoring force

In this paper, the first-passage failure of stochastic dynamical systems with fractional derivative and power-form restoring force subjected to Gaussian white-noise excitation is investigated. With application of the stochastic averaging method of quasi-Hamiltonian system, the system energy process will converge weakly to an Itô differential equation. After that, Backward Kolmogorov (BK) equation associated with conditional reliability function and Generalized Pontryagin (GP) equation associated with statistical moments of first-passage time are constructed and solved. Particularly, the influence on reliability caused by the order of fractional derivative and the power of restoring force are also examined, respectively. Numerical results show that reliability function is decreased with respect to the time. Lower power of restoring force may lead the system to more unstable evolution and lead first passage easy to happen. Besides, more viscous material described by fractional derivative may have higher reliability. Moreover, the analytical results are all in good agreement with Monte-Carlo data.