First passage failure of quasi-partial integrable generalized Hamiltonian systems

The first passage failure of quasi-partial integrable generalized Hamiltonian systems is studied by using the stochastic averaging method. First, the stochastic averaging method for quasi-partial integrable generalized Hamiltonian systems is introduced briefly. Then, the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of first passage time are derived from the averaged Itô equations. The conditional reliability function, the conditional probability density and mean of the first passage time are obtained from solving these equations together with suitable initial condition and boundary conditions, respectively. Finally, one example is given to illustrate the proposed procedure in detail and the solutions are confirmed by using the results from Monte Carlo simulation of the original system.     

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.     

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.     

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.     

Stochastic optimal control of quasi non-integrable Hamiltonian systems with stochastic maximum principle

A new procedure for designing optimal control of quasi non-integrable Hamiltonian systems under stochastic excitations is proposed based on the stochastic averaging method for quasi non-integrable Hamiltonian systems and the stochastic maximum principle. First, the control problem consisting of 2n-dimensional equations governing the controlled quasi non-integrable system and performance index is converted into a partially averaged one consisting of one-dimensional equation of the controlled system and performance index by using the stochastic averaging method. Then, the adjoint equation and the maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The optimal control forces are determined from the maximum condition and solving the forward?Cbackward stochastic differential equations (FBSDE). For infinite time-interval ergodic control, the adjoint variable is a stationary process and the FBSDE is reduced to a partial differential equation. Finally, the response statistics of optimally controlled system is predicted by solving the Fokker?CPlank equation (FPE) associated with the fully averaged It? equation of the controlled system. An example of two degree-of-freedom (DOF) quasi non-integrable Hamiltonian system is worked out to illustrate the proposed procedure and its effectiveness.     

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.     

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.     

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。     

Optimal Bounded Control for Minimizing the Response of Quasi Non-Integrable Hamiltonian Systems

A strategy for designing optimal bounded control to minimize theresponse of quasi non-integrable Hamiltonian systems is proposed basedon the stochastic averaging method for quasi non-integrable Hamiltoniansystems and the stochastic dynamical programming principle. Theequations of motion of a controlled quasi non-integrable Hamiltoniansystem are first reduced to an one-dimensional averaged Itô stochasticdifferential equation for the Hamiltonian by using the stochasticaveraging method for quasi non-integrable Hamiltonian systems. Then, thedynamical programming equation for the control problem of minimizing theresponse of the averaged system is formulated based on the dynamicalprogramming principle. The optimal control law is derived from thedynamical programming equation and control constraints without solvingthe equation. The response of optimally controlled systems is predictedthrough solving the Fokker–Planck–Kolmogrov (FPK) equation associatedwith completely averaged Itô equation. Finally, two examples are workedout in detail to illustrate the application and effectiveness of theproposed control strategy.     

Stochastic stabilization of quasi non-integrable Hamiltonian systems

A procedure for designing a feedback control to asymptotically stabilize in probability a quasi non-integrable Hamiltonion system is proposed. First, an one-dimensional averaged Itô stochastic differential equation for controlled Hamiltonian is derived from given equations of motion of the system by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. Second, a dynamical programming equation for an ergodic control problem with undetermined cost function is established based on the stochastic dynamical programming principle and solved to yield the optimal control law. Third, the asymptotic stability in probability of the system is analysed by examining the sample behaviors of the completely averaged Itô differential equation at its two boundaries. Finally, the cost function and the optimal control forces are determined by the requirement of stabilizing the system. Two examples are given to illustrate the application of the proposed procedure and the effect of control on the stability of the system.     

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.     

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.     

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.     

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.     

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.     

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.     

Reliability of a class of nonlinear systems under switching random excitations

Systems subjected to switching random excitations are practically significant because they include many safety-critical systems such as power plants and communication networks. In this paper, the reliability of multi-degree, nonlinear, non-integrable Hamiltonian systems subjected to switching random excitations is investigated. Such a system is formulated as a continuous–discrete hybrid based upon the Markov jump theory. Stochastic averaging is applied to suppress the rapidly varying parameters of the Markov jump process in order to generate a probability-weighted diffusion equation. The associated backward Kolmogorov equation is then set up, from which the approximate reliability function and probability density of first passage time are obtained. The utility and accuracy of this approximate procedure are demonstrated by two examples.     

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.     

Feedback Stabilization of Quasi Nonintegrable Hamiltonian Systems by Using Lyapunov Exponent

A procedure for designing a feedback control to asymptotically stabilize, with probability one, a quasi nonintegrable Hamiltonian system is proposed. First, the motion equations of a system are reduced to a one-dimensional averaged Itô stochastic differential equation for controlled Hamiltonian by using the stochastic averaging method for quasi nonintegrable Hamiltonian systems. Second, a dynamical programming equation for the ergodic control problem of the averaged system with undetermined cost function is established based on the dynamical programming principle. This equation is then solved to yield the optimal control law. Third, a formula for the Lyapunov exponent of the completely averaged Itô equation is derived by introducing a new norm for the definitions of stochastic stability and Lyapunov exponent in terms of the square root of Hamiltonian. The asymptotic stability with probability one of the originally controlled system is analysed approximately by using the Lyapunov exponent. Finally, the cost function is determined by the requirement of stabilizing the system. Two examples are given to illustrate the application of the proposed procedure and the effectiveness of control on stabilizing the system.