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.     

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.     

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.     

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 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.     

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.     

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.     

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 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.     

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.     

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.     

加性非平稳激励下结构速度响应的FPK方程降维

结构在随机激励下的非线性响应分析是具有高度挑战性的困难问题. 对于白噪声或过滤白噪声激励,求解FPK方程将获得结构响应 的精确解. 遗憾的是,对于非线性多自由度系统,FPK方程难以直接求解. 事实上,其数值解法严重受限于方程维度,而解析求解 则仅适用于少数特定的系统,且多是稳态解. 因此,将FPK方程进行降维,是求解高维随机动力响应分析问题的重要途径. 本文针 对幅值调制的加性白噪声激励下多自由度非线性结构的非平稳随机响应分析问题,将联合概率密度函数满足的高维FPK方程进行降 维. 针对结构速度响应概率密度函数求解,通过引入等价漂移系数,原FPK方程可转化为一维FPK型方程. 建议了构造等价漂移系数 的条件均值函数方法. 进而,采用路径积分方法求解降维FPK型方程,得到速度概率密度函数的数值解答. 结合单自由度Rayleigh 振子、十层线性剪切型框架和非线性剪切型框架结构在幅值调制的加性白噪声激励下的非平稳速度响应求解,讨论了本文方法的精 度和效率,验证了其有效性.      

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.     

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.     

Response statistics of van der Pol oscillators excited by white noise

The joint probability density function of the state space vector of a white noise exoited van der Pol oscillator satisfics a Fokker-Planck-Kolmogorov (FPK) equation. The paper describes a numerical procedure for solving the transient FPK equation based on the path integral solution (PIS) technique. It is shown that by combining the PIS with a cubic B-spline interpolation method, numerical solution algorithms can be implemented giving solutions of the FPK equation that can be made accurate down to very low probability levels. The method is illustrated by application to two specific examples of a van der Pol oscillator.     

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.     

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.     

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 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.