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

A strategy is proposed based on the stochastic averaging method for quasi non- integrable Hamiltonian systems and the stochastic dynamical programming principle.The pro- posed 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 av- eraged It stochastic differential equation.By using the stochastic dynamical programming princi- ple 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 It stochastic differential equation.An example is worked out in detail to illustrate the application of the control strategy proposed.

Some applications of stochastic averaging method for quasi Hamiltonian systems in physics

Many physical systems can be modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems can be applied to yield reasonable approximate response sta-tistics.In the present paper,the basic idea and procedure of the stochastic averaging method for quasi Hamiltonian systems are briefly introduced.The applications of the stochastic averaging method in studying the dynamics of active Brownian particles,the reaction rate theory,the dynamics of breathing and denaturation of DNA,and the Fermi resonance and its effect on the mean transition time are reviewed.

STOCHASTIC HOPF BIFURCATION IN QUASIINTEGRABLE-HAMILTONIAN SYSTEMS

A new procedure is developed to study the stochastic Hopf bifurcation in quasi- integrable-Hamiltonian systems under the Gaussian white noise excitation.Firstly,the singular bound- aries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system's energy levels with respect to the stochastic aver- aging method.Secondly,the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones.Lastly,a quasi-integrable- Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure. Moreover,simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure.It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system's parameters.Therefore,one can see that the numerical results are consistent with the theoretical predictions.

Science Letters:A minimax optimal control strategy for uncertain quasi-Hamiltonian systems

A minimax optimal control strategy for quasi-Hamiltonian systems with bounded parametric and/or external disturbances is proposed based on the stochastic averaging method and stochastic differential game. To conduct the system energy control,the partially averaged It stochastic differential equations for the energy processes are first derived by using the stochastic averaging method for quasi-Hamiltonian systems. Combining the above equations with an appropriate performance index,the proposed strategy is searching for an optimal worst-case controller by solving a stochastic differential game problem. The worst-case disturbances and the optimal controls are obtained by solving a Hamilton-Jacobi-Isaacs(HJI) equation. Numerical results for a controlled and stochastically excited Duffing oscillator with uncertain disturbances exhibit the efficacy of the proposed control strategy.

Establishment of infinite dimensional Hamiltonian system of multilayer quasi-geostrophic flow & study on its linear stability

A multilayer flow is a stratified fluid composed of a finite number of layers with densities homogeneous within one layer but different from each other. It is an intermediate system between the single-layer barotropic model and the continuously stratified baroclinic model. Since this system can simulate the baroclinic effect simply, it is widely used to study the large-scale dynamic process in atmosphere and ocean. The present paper is concerned with the linear stability of the multilayer quasi-geostrophic flow, and the associated linear stability criteria are established. Firstly, the nonlinear model is turned into the form of a Hamiltonian system, and a basic flow is defined. But it cannot be an extreme point of the Hamiltonian function since the system is an infinite-dimensional one. Therefore, it is necessary to reconstruct a new Hamiltonian function so that the basic flow becomes an extreme point of it. Secondly, the linearized equations of disturbances in the multilayer quasi-geostrophic flow are derived by introducing infinitesimal disturbances superposed on the basic flows. Finally, the properties of the linearized system are discussed, and the linear stability criteria in the sense of Liapunov are derived under two different conditions with respect to certain norms.

Energy diffusion controlled reaction rate in dissipative Hamiltonian systems

In this paper the energy diffusion controlled reaction rate in dissipative Hamiltonian systems is investigated by using the stochastic averaging method for quasi Hamiltonian systems. The boundary value problem of mean first- passage time （MFPT） of averaged system is formulated and the energy diffusion controlled reaction rate is obtained as the inverse of MFPT. The energy diffusion controlled reaction rate in the classical Kramers bistable potential and in a two-dimensional bistable potential with a heat bath are obtained by using the proposed approach respectively. The obtained results are then compared with those from Monte Carlo simulation of original systems and from the classical Kraraers theory. It is shown that the reaction rate obtained by using the proposed approach agrees well with that from Monte Carlo simulation and is more accurate than the classical Kramers rate.

Reliability of quasi integrable and non-resonant Hamiltonian systems under fractional Gaussian noise excitation

The reliability of quasi integrable and non-resonant Hamiltonian system under fractional Gaussian noise(fGn)excitation is studied.Noting rather flat fGn power spectral density(PSD)in most part of frequency band,the fGn is innovatively regarded as a wide-band process.Then,the stochastic averaging method for quasi integrable Hamiltonian systems under wide-band noise excitation is applied to reduce 2n-dimensional original system into n-dimensional averaged ltd stochastic differential equations(SDEs).Reliability function and mean first passage time are obtained by solving the associated backward Kolmogorov equation and Pontryagin equation.The validity of the proposed procedure is tested by applying it to an example and comparing the numerical results with those from Monte Carlo simulation.

Compensation for time-delayed feedback bang-bang control of quasi-integrable Hamiltonian systems

The stochastic averaging method for quasi-integrable Hamiltonian systems with time-delayed feedback bang-bang control is first introduced. Then, two time delay compensation methods, namely the method of changing control force amplitude (CFA) and the method of changing control delay time (CDT), are proposed. The conditions applicable to each compensation method are discussed. Finally, an example is worked out in detail to illustrate the application and effectiveness of the proposed methods and the two compensation methods in combination.

Stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems

The stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems is investigated. First, the stochastic optimal control problem of a partially observable nonlinear quasi-integrable Hamiltonian system is converted into that of a completely observable linear system based on a theorem due to Charalambous and Elliot. Then, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. The response of the controlled quasi Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation and the Riccati equation for the estimated error of system states. As an example to illustrate the procedure and effectiveness of the proposed method, the stochastic optimal control problem of a partially observable two-degree-of-freedom quasi-integrable Hamiltonian system is worked out in detail.

The Stability of the Elliptic Equilibrium of Planar Quasi-periodic Hamiltonian Systems

In this paper, we study the planar Hamiltonian systemwhere where f is real analytic in x and θ,A（θ） is a 2×2real analytic symmetric matrix,J=（1^-1） and w is a Diophantine vector. Under the assumption that the unperturbed systemreducible and stable, we obtain a series of criteria for the stability and instability of the equilibrium of the perturbed system.

Stochastic averaging of quasi partially integrable Hamiltonian systems under fractional Gaussian noise

A stochastic averaging method for predicting the response of quasi partially integrable and non-resonant Hamiltoniansystems to fractional Gaussian noise （fGla） with the Hurst index 1/2〈H〈l is proposed. The averaged stochastic differential equa-tions （SDEs） for the first integrals of the associated Hamiltonian system are derived. The dimension of averaged SDEs is less thanthat of the original system. The stationary probability density and statistics of the original system are obtained approximately fromsolving the averaged SDEs numerically. Two systems are worked out to illustrate the proposed stochastic averaging method. It isshown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of originalsystem agree well, and the computational time for the former results is less than that for the latter ones.

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

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 fGn and related fractional Brownian motion （iBm） 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.

Feedback maximization of reliability of MDOF quasi integrable-Hamiltonian systems under combined harmonic and white noise excitations

We studied the feedback maximization of reliability of multi-degree-of-freedom (MDOF) quasi integrable-Hamiltonian systems under combined harmonic and white noise excitations. First, the partially averaged It equations are derived by using the stochastic averaging method for quasi integrable-Hamiltonian systems under combined harmonic and white noise excitations. Then, the dynamical programming equation and its boundary and final time conditions for the control problems of maximizing the reliability is established from the partially averaged equations by using the dynamical programming principle. The nonlinear stochastic optimal control for maximizing the reliability is determined from the dynamical programming equation and control constrains. The reliability function of optimally controlled systems is obtained by solving the final dynamical programming equation. Finally, the application of the proposed procedure and effectiveness of the control strategy are illustrated by using an example.

基于拟哈密顿理论的随机电力系统暂态稳定性分析

蒙特卡洛法分析随机稳定性所需的计算量大,且难以同时考虑随机过程中发生的大量不确定性。基于拟哈密顿系统随机平均方法,提出随机电力系统暂态稳定性分析法。首先,根据扩展等面积法,将受扰多机系统动态映射为两群系统,建立其随机微分方程模型。忽略相关的非哈密顿因素,按哈密顿能量函数确定其安全区域。然后,考虑随机扰动、阻尼等的影响,对等效拟哈密顿系统进行随机平均,求出支配暂态能量转移的平均扩散方程,并基于扩散理论,根据系统条件可靠性分析切除时间、阻尼系数及激励强度等对随机电力系统暂态稳定性的影响。通过4机2区随机系统验证了该方法的有效性。

拟哈密顿系统非线性随机最优控制

主要介绍近十几年来拟哈密顿系统非线性随机最优控制理论方法及其应用的研究成果,包括基于拟哈密顿系统随机平均法与随机动态规划原理的非线性随机最优控制基本策略,即响应极小化控制、随机稳定化、首次穿越损坏最小化控制、以概率密度为目标的控制,为将它们应用于工程实际而作的部分可观测系统最优控制、有界控制、时滞控制、半主动控制、极小极大控制的进一步研究,以及综合考虑这些实际问题的非线性随机最优控制的综合策略,非线性随机最优控制在滞迟系统、分数维系统等中的若干应用,介绍与这些研究有关的背景,并指出今后有待进一步研究的问题.

基于拟可积Hamilton系统的铁路桥梁动力可靠度计算研究

铁路运输提速增载,迫切需要对铁路桥梁动力可靠度进行研究。本文基于振型空间,首先推导出结构体系的广义动能和势能,继而基于拟Hamilton系统理论,推导出铁路混凝土桥梁的拟可积Hamilton系统方程。只考虑横向和扭转位移,得到铁路混凝土桥梁的条件可靠性函数所满足的BK方程及其定量边界、初值条件,可用中心差分法进行求解。以实际桥梁为算例,用上述方程求解其在列车荷载下的动力可靠度,得出或验证了与实际情况相符的若干重要结论,结果表明:动力可靠度和概率密度峰值,随桥梁初始能量的增大而减小,随桥梁边界能量的增大而增大;不同跨度桥梁分析结果与实际情况相符,说明基于拟可积Hamilton系统理论计算铁路桥梁的动力可靠度是可行的。

一类拟 Hamilton 碰振系统的全局分岔及多解共存现象分析

研究了一类拟Hamilton碰振系统的全局动力学特性,参照同宿轨道的Melnikov函数形式,构造了周期轨道次谐Melnikov函数.并用一类拟Hamilton碰振系统详细介绍了其计算方法和运用,数值结果验证了构造的次谐Melnikov函数的有效性.另外用改进的胞映射方法对这类系统的全局分岔和多解共存现象进行了分析,发现随着外激励力的变动吸引子数量发生变化,各个吸引域形态复杂且相互缠绕.

拟可积哈密顿系统中噪声诱发的混沌运动

研究拟可积哈密顿系统在谐和与噪声激励联合作用下的混沌运动。通过对噪声性质的假 定，并利用动力系统理论，给出了高维梅尔尼科夫方法应用于随机拟可积哈密顿系统的推广形 式．根据这种推广的方法，研究了谐和与高斯白噪声激励联合作用下两自由度拟可积哈密顿系 统的同宿分岔，得出了系统发生混沌运动的参数阈值，并由此讨论了噪声对系统的混沌运动的 影响．蒙特－卡罗方法模拟、李雅普诺夫指数等数值结果表明。

随机非线性物价模型的最优控制

将政府对价格系统的宏观调控作为外部控制力,建立受控的随机非线性物价模型:利用拟Hamilton系统随机平均法和随机动态规划原理的非线性随机控制策略对系统实施最优控制,控制目标是实现系统的稳定性变大;并通过对比控制前后的Lyapunov指数值说明了控制的有效性.

fGn激励下非线性系统近似方法适用性的解析分析

由于受分数高斯噪声(fGn)激励的非线性系统响应不再具有马尔科夫性,基于扩散过程的理论方法不能直接用于研究此类问题。作为近似方法,宽带噪声激励的拟哈密顿系统随机平均法已经被用于解决此类问题。虽然,该理论方法在响应预测和可靠性分析方面取得了较好的效果,但是到目前为止还没有做过对近似方法的误差和适用性的解析分析。在本研究中,将近似方法用于分析fGn激励下的单自由度非线性系统,得到了系统响应的近似解析解,再结合已报道的精确解析解,用渐近分析的方法进行了误差分析,从而对近似方法的适用性进行了论证,为将来能够进一步扩展近似方法的应用提供了理论依据。