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.     

Stochastic Stabilization of Quasi-Partially Integrable Hamiltonian Systems by Using Lyapunov Exponent

A procedure for designing a feedback control to asymptoticallystabilize, with probability one, a quasi-partially integrableHamiltonian system is proposed. First, the averaged stochasticdifferential equations for controlled r first integrals are derived fromthe equations of motion of a given system by using the stochasticaveraging method for quasi-partially integrable Hamiltonian systems.Second, a dynamical programming equation for the ergodic control problemof the averaged system with undetermined cost function is establishedbased on the dynamical programming principle. The optimal control law isderived from minimizing the dynamical programming equation with respectto control. Third, the asymptotic stability with probability one of theoptimally controlled system is analyzed by evaluating the maximalLyapunov exponent of the completely averaged Itô equations for the rfirst integrals. Finally, the cost function and optimal control forces aredetermined by the requirements of stabilizing the system. An example isworked out in detail to illustrate the application of the proposedprocedure and the effect of optimal control on the stability of thesystem.     

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.     

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 stability of quasi-partially integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

The asymptotic Lyapunov stability with probability one of multi-degree-of freedom quasi-partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises is studied. First, the averaged stochastic differential equations for quasi partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are derived by means of the stochastic averaging method and the stochastic jump-diffusion chain rule. Then, the expression of the largest Lyapunov exponent of the averaged system is obtained by using a procedure similar to that due to Khasminskii and the properties of stochastic integro-differential equations. Finally, the stochastic stability of the original quasi-partially integrable and non-resonant Hamiltonian systems is determined approximately by using the largest Lyapunov exponent. An example is worked out in detail to illustrate the application of the proposed method. The good agreement between the analytical results and those from digital simulation show that the proposed method is effective.     

Nonstationary probability densities of strongly nonlinear single-degree-of-freedom oscillators with time delay

The approximate nonstationary probability density of a nonlinear single-degree-of-freedom (SDOF) oscillator with time delay subject to Gaussian white noises is studied. First, the time-delayed terms are approximated by those without time delay and the original system can be rewritten as a nonlinear stochastic system without time delay. Then, the stochastic averaging method based on generalized harmonic functions is used to obtain the averaged Itô equation for amplitude of the system response and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the nonstationary probability density of amplitude is deduced. Finally, the approximate solution of the nonstationary probability density of amplitude is obtained by applying the Galerkin method. The approximate solution is expressed as a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. The proposed method is applied to predict the responses of a Van der Pol oscillator and a Duffing oscillator with time delay subject to Gaussian white noise. It is shown that the results obtained by the proposed procedure agree well with those obtained from Monte Carlo simulation of the original systems.     

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.     

Optimal bounded control of stochastically excited MDOF nonlinear viscoelastic systems

A new procedure for designing optimal bounded control of stochastically excited multi-degree-of-freedom (MDOF) nonlinear viscoelastic systems is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the system is formulated as a quasi-integrable Hamiltonian system with viscoelastic terms and each viscoelastic term is replaced approximately by an elastically restoring force and a visco-damping force based on the randomly periodic behavior of the motion of quasi-integrable Hamiltonian system. Thus, a stochastically excited MDOF nonlinear viscoelastic system is converted to an equivalent quasi-integrable Hamiltonian system without viscoelastic terms. Then, by applying stochastic averaging, the system is further reduced to a partially averaged system of less dimension. The adjoint equation and maximum condition for the optimal control problem of the partially averaged system are derived by using the stochastic maximum principle, and the optimal bounded control force is determined from the maximum condition. Finally, the probability and statistics of the stationary response of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation (FPK) associated with the fully averaged Itô equation of the controlled system. An example is worked out to illustrate the proposed procedure and its effectiveness.     

To obtain approximate probability density functions for a class of axially moving viscoelastic plates under external and parametric white noise excitation

The probability density function plays an essential role to investigate the behaviors of stochastic linear or nonlinear systems. This function can be evaluated by several approaches but due to its analytical theme, the Fokker–Planck–Kolmlgorov (FPK) approach is preferable. FPK equation is a nonlinear PDE gives the probability density function for a stochastic linear or nonlinear system. Many researches have been done in literature tried to specify the conditions, in which the FPK equation gives an exact solution. Although, the exact probability density function can be achieved by solving the FPK equation even for some nonlinear systems, many types of systems cannot satisfy the conditions for exact solution. In this article, the axially moving viscoelastic plates under both external and parametric white noise excitation as one of the newest and applicable research areas are studied. Due to strong nonlinearities recognized in the governing equation of the system, the exact probability density function cannot be obtained, however, via an approximate method; some precise approximate solutions for different but comprehensive case studies are evaluated, validated, and discussed.     

Stochastic averaging of quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems to combined harmonic and white noise excitations. According to the proposed method, an n+α+β-dimensional averaged Fokker-Planck-Kolmogorov (FPK) equation governing the transition probability density of n action variables or independent integrals of motion, α combinations of angle variables and β combinations of angle variables and excitation phase angles can be constructed when the associated Hamiltonian system has α internal resonant relations and the system and harmonic excitations have β external resonant relations. The averaged FPK equation is solved by using the combination of the finite difference method and the successive over relaxation method. Two coupled Duffing-van der Pol oscillators under combined harmonic and white noise excitations is taken as an example to illustrate the application of the proposed procedure and the stochastic jump and its bifurcation as the system parameters change are examined.     

Time-delayed stochastic optimal control of strongly non-linear systems with actuator saturation by using stochastic maximum principle

A time-delayed stochastic optimal bounded control strategy for strongly non-linear systems under wide-band random excitations with actuator saturation is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the partially averaged Itô equation for the system amplitude is derived by using the stochastic averaging method for strongly non-linear systems. The time-delayed feedback control force is approximated by a control force without time delay based on the periodically random behavior of the displacement and velocity of the system. The partially averaged Itô equation for the system energy is derived from that for the system amplitude by using Itô formula and the relation between system amplitude and system energy. Then, the adjoint equation and maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The saturated optimal control force is determined from maximum condition and solving the forward–backward stochastic differential equations (FBSDEs). For infinite time-interval ergodic control, the adjoint variable is stationary process and the FBSDE is reduced to a ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained from solving the Fokker–Plank–Kolmogorov (FPK) equation associated with the fully averaged Itô equation of the controlled system. For comparison, the optimal control forces obtained from the time-delayed bang–bang control and the control without considering time delay are also presented. An example is worked out to illustrate the proposed procedure and its advantages.     

Stochastic averaging of quasi-generalized Hamiltonian systems

A stochastic averaging method for generalized Hamiltonian systems (GHS) subject to light dampings and weak stochastic excitations is proposed. First, the GHS are briefly reviewed and classified into five classes, i.e., non-integrable GHS, completely integrable and non-resonant GHS, completely integrable and resonant GHS, partially integrable and non-resonant GHS and partially integrable and resonant GHS. Then, the averaged and FPK equations and the drift and diffusion coefficients for the five classes of quasi-GHS are derived. Finally, the stochastic averaging for a nine-dimensional quasi-partially integrable GHS is given to illustrate the application of the proposed procedure, and the results are confirmed by using those from Monte Carlo simulation.     

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.     

Approximate stationary solution and stochastic stability for a class of differential equations with parametric colored noise

This paper aims to study a class of differential equations with parametric Gaussian colored noise. We present the general framework to get the solvability conditions of the approximate stationary probability density function, which is determined by the Fokker-Planck-Kolmogorov (FPK) equations. These equations are derived using the stochastic averaging method and the operator theory with the perturbation technique. An illustrative example is proposed to demonstrate the procedure of our proposed method. The analytical expression of approximate stationary probability density function is obtained. Numerical simulation is carried out to verify the analytical results and excellent agreement can be easily found. The FPK equation for the probability density function of order ε ⁰ is used to examine the almost-sure stability for the amplitude process. Finally, the stability in probability of the amplitude process is investigated by Lin and Cai’s method.     

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.     

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.     

FPK方程的近似闭合解

讨论了FPK方程的近似闭合解问题。假设FPK方程的解具有指数多项式的形式，利用比较系数的方法确定其中待定的常数。计算表明，本方法适用于强非线性系统，在特殊情形下还能求出原方程的精确解。     

Optimal bounded control of quasi-nonintegrable Hamiltonian systems using stochastic maximum principle

A new procedure for designing optimal bounded control of quasi-nonintegrable Hamiltonian systems with actuator saturation is proposed based on the stochastic averaging method for quasi-nonintegrable Hamiltonian systems and the stochastic maximum principle. First, the stochastic averaging method for controlled quasi-nonintegrable Hamiltonian systems is introduced. The original control problem is converted into one for a partially averaged equation of system energy together with a partially averaged performance index. Then, the adjoint equation and the maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The bounded optimal control forces are obtained from the maximum condition and solving the forward–backward stochastic differential equations (FBSDE). For infinite time-interval ergodic control, the adjoint variable is stationary process, and the FBSDE is reduced to an ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation associated with the fully averaged Itô equation of the controlled system. For comparison, the bang–bang control is also presented. An example of two degree-of-freedom quasi-nonintegrable Hamiltonian system is worked out to illustrate the proposed procedure and its effectiveness. Numerical results show that the proposed control strategy has higher control efficiency and less discontinuous control force than the corresponding bang–bang control at the price of slightly less control effectiveness.     

The stationary solution of a random dynamical model

This paper studies the stationary probability density function (PDF) solution of a nonlinear business cycle model subjected to random shocks of Gaussian white-noise type. The PDF solution is controlled by a Fokker–Planck–Kolmogorov (FPK) equation, and we use exponential polynomial closure (EPC) method to derive an approximate solution for the FPK equation. Numerical results obtained from EPC method, better than those from Gaussian closure method, show good agreement with the probability distribution obtained with Monte Carlo simulation including the tail regions.