哈密顿系统正则变换在时变最优控制中的应用

利用哈密顿系统正则变换和生成函数理论求解线性时变最优控制问题,构造了新的最优控制律形式并提出了控制增益计算的保结构算法. 利用生成函数求解最优控制导出的哈密顿系统两端边值问题,并构造线性时变系统的最优控制律,由第2类生成函数所构造的最优控制律避免了末端时刻出现无穷大反馈增益. 控制系统设计中需求解生成函数满足的时变矩阵微分方程组. 根据生成函数与哈密顿系统状态转移矩阵之间的关系,从正则变换的辛矩阵描述出发,导出了求解这组微分方程组的保结构递推算法.为了保持递推计算中的辛矩阵结构,哈密顿系统状态转移矩阵的计算中利用了Magnus级数.      

Symplectic solution system for reissner plate bending

Based on the Hellinger-Reissner variatonal principle for Reissner plate bendingand introducing dual variables, Hamiltonian dual equations for Reissner plate bending werepresented. Therefore Hamiltonian solution system can also be applied to Reissner platebending problem, and the transformation from Euclidian space to symplectic space and fromLagrangian system to Hamiltonian system was realized. So in the symplectic space whichconsists of the original variables and their dual variables, the problem can be solved viaeffective mathematical physics methods such as the method of separation of variables andeigenfunction-vector expansion. All the eigensolutions and Jordan canonical formeigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail, and their physical meanings are showed clearly. The adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue are formed. It is showed that the alleigensolutions for zero eigenvalue are basic solutions of the Saint-Venant problem and theyform a perfect symplectic subspace for zero eigenvalue. And the eigensolutions for nonzeroeigenvalue are covered by the Saint-Venant theorem. The symplectic solution method is notthe same as the classical semi-inverse method and breaks through the limit of the traditional semi-inverse solution. The symplectic solution method will have vast application.     

High order symplectic conservative perturbation method for time-varying Hamiltonian system

This paper presents a high order symplectic conservative perturbation method for linear time-varying Hamiltonian system.Firstly,the dynamic equation of Hamiltonian system is gradually changed into a high order perturbation equation,which is solved approximately by resolving the Hamiltonian coefficient matrix into a "major component" and a "high order small quantity" and using perturbation transformation technique,then the solution to the original equation of Hamiltonian system is determined through a series of inverse transform.Because the transfer matrix determined by the method in this paper is the product of a series of exponential matrixes,the transfer matrix is a symplectic matrix;furthermore,the exponential matrices can be calculated accurately by the precise time integration method,so the method presented in this paper has fine accuracy,efficiency and stability.The examples show that the proposed method can also give good results even though a large time step is selected,and with the increase of the perturbation order,the perturbation solutions tend to exact solutions rapidly.     

圆柱型正交各向异性弹性楔的佯谬解

圆柱型正交各向异性弹性楔体顶端受有集中力偶的经典解,当顶角满足一定关系时,其应力成为无穷大,这是个佯谬．该文在哈密顿体系下将该问题进行重新求解,即利用极坐标各向异性弹性力学哈密顿体系．在原变量和其对偶变量组成的辛几何空间求解特殊本征值的约当型本征解,从而直接给出该佯谬问题的解析解．结果再次表明经典力学中的弹性楔佯谬解对应的是哈密顿体系下辛几何的约当型解．     

粘弹性厚壁筒问题的辛本征解方法

将哈密顿体系引进到粘弹性力学厚壁筒问题中,在辛体系下重新描述了基本问题,即建立了正则方程组。借助于积分变换,得到了拉伸、扭转和弯曲等问题的解以及有边界局部效应的解。将原问题归结为辛几何空间中的零本征值本征解和非零本征值本征解问题,从而建立了一种有效的分析问题方法和数值方法。为解决同类问题提供了一条可行的路径。     

HAMILTONIAN SYSTEM AND SIMPLETIC GEOMETRY IN MECHANICS OF COMPOSITE MATERIALS(Ⅰ)——FUNDAMENTAL THEORY

For the first time, Hamiltonian system used in dynamics is introduced to formulate statics and Hamiltonian equation is derived corresponding to the original governing equation, which enables separation of variables to work and eigen function to be obtained for the boundary problem. Consequently, analytical and semi-analytical solutions can be got. The method is especially suitable to solve rectangular plane problem and spatial prism in elastic mechanics. The paper presents a new idea to solve partially differential equation in solid mechanics. The flexural problem and plane stress problem of laminated plate are studied in detail. This project is supported by National Natural Science Foundation of China for Post-Doctorate Research.     

HAMILTONIAN SYSTEM AND SIMPLETIC GEOMETRY IN MECHANICS OF COMPOSITE MATERIALS (Ⅰ)——FUNDAMENTAL THEORY

For the first time, Hamiltonian system used in dynamics is introduced to formulate statics and Hamdtonian equation is derived corresponding to the original governing equation, which enables separation of variables to work and eigen function to be obtained for the boundary problem. Consequently, analytical and semi-analytical solutions can be got. The method is especially suitable to solve rectangular plane problem and spatial prism in elastic mechanics. The paper presents a new idea to solve partially differential equation in solid mechanics. The flexural problem and phane stress problem of laminated plate are studied in detail.     

线性哈密顿系统的本征摄动法

利用哈密顿算子辛自共轭的特点讨论了保守哈密顿体系的摄动问题,给出了哈密顿矩阵的本征值与本征向量的二阶摄动分析方法。即当系统在哈密顿框架下进行较小修改时,不重复求解大型哈密顿矩阵的本征问题,只需在原系统的模态参数基础上进行模态分析即可,这种矩阵摄动法给出了修改后矩阵的二阶本征值和本征向量,为一般线性保守体系的本征摄动求解提出了一个新方法。     

随机激励的耗散的哈密尔顿系统的平稳解

本文首先为一般的随机激励的耗散的哈密尔顿系统得到精确的平稳解,然后在此基础上为类似而更为一般的系统发展了等效非线性系统法     

随机激励的耗散的哈密尔顿系统的平稳解

本文首先为一般的随机激励的耗散的哈密尔顿系统得到精确的平稳解,然后在此基础上为类似而更为一般的系统发展了等效非线性系统法。     

Hamilton体系下环扇形域的Stokes流动问题

基于极坐标下Stokes流的基本方程,将径向坐标模拟为时间坐标,推导了Hamilton体系下Stokes流动问题的对偶方程,采用本征向量展开法对环扇形域Stokes流动问题进行了分析,并给出了相应的实际算例,其结果说明了本文方法的有效性。     

Multi-pulse orbits and chaotic dynamics in a nonlinear forced dynamics of suspended cables

The global bifurcations in mode of a nonlinear forced dynamics of suspended cables are investigated with the case of the 1:1 internal resonance. After determining the equations of motion in a suitable form, the energy phase method proposed by Haller and Wiggins is employed to show the existence of the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the two cases of Hamiltonian and dissipative perturbation. Furthermore, some complex chaos behaviors are revealed for this class of systems.     

一类偏微分方程的Hamilton正则表示

主要给出一系列关于力学中的偏微分方程的无穷维Ｈａｍｉｌｔｏｎ正则表示．其中包括变系数线性偏微分方程,ＫｄＶ方程,ＭＫｄＶ方程,ＫＰ方程,Ｂｏｕｓｉｎｅｓｑ方程等的无穷维Ｈａｍｉｌｔｏｎ正则表示．     

N-pulse homoclinic orbits in perturbations of resonant hamiltonian systems

In this paper we develop an analytical method to detect orbits doubly asymptotic to slow manifolds in perturbations of integrable, two-degree-of-freedom resonant Hamiltonian systems. Our energy-phase method applies to both Hamiltonian and dissipative perturbations and reveals families of multi-pulse solutions which are not amenable to Melnikov-type methods. As an example, we study a two-mode approximation of the nonlinear, nonplanar oscillations of a parametrically forced inextensional beam. In this problem we find unusually complicated mechanisms for chaotic motions and verify their existence numerically.     

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.     

非线性随机动力学与控制的哈密顿理论框架

 近几年来，笔者提出与发展了随机激励的耗散的哈密顿系统理 论，包括精确平稳解、等效非线性系统法、拟哈密顿系统随机平均法、 拟哈密顿系统的随机稳定性与随机分岔、首次穿越损坏分析方法及非 线性随机最优控制策略，从而构成了一个非线性随机动力学与控制的 哈密顿理论框架.本文简要介绍这一理论框架.     

精细辛有限元方法及其相位误差研究

哈密顿系统是一类重要的动力系统,针对哈密顿系统,设计出多类辛方法:SRK、SPRK、辛多步法、生成函数法等.长久以来数值方法在求解哈密顿系统过程中辛特性和保能量特性不能得到同时满足,近年来提出的有限元方法,对于线性系统具有保辛和保能量的优良特性.但是,以上方法都存在相位漂移(轨道偏离)现象,长时间仿真,计算效果会大打折扣.提出精细辛有限元方法(HPD-FEM)求解哈密顿系统,该方法继承时间有限元方法求解哈密顿系统所具有的保哈密顿系统的辛结构和哈密顿函数守恒性的优良特性,同时,通过精细化时间步长极大地减小了时间有限元方法的相位误差.HPD-FEM相较与针对相位误差专门设计的计算格式FSJS、RKN以及SRPK方法具有更好的纠正效果,几乎达到机器精度,误差为O(10-13),同时,HPD-FEM克服了FSJS、RKN和SPRK方法不能保证哈密顿函数守恒的缺点.对于高低混频系统和刚性系统,常规算法很难在较大步长下,同时实现对高低频精确仿真,HPD-FEM通过精细计算时间步长,在大步长情况下,实现高低混频的精确仿真.HPD-FEM方法在计算过程中精细方法没有额外增加计算量,计算效率高.数值结果显示本文提出的方法切实有效.      

Weakly Nonlinear-Dissipative Approximations of Hyperbolic–Parabolic Systems with Entropy

Hyperbolic–parabolic systems have spatially homogenous stationary states. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern perturbations of these constant states. These approximations are quadratically nonlinear. When the original system has an entropy, the approximation is formally dissipative in a natural Hilbert space. We show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that Hilbert space. We also prove a weak-strong uniqueness theorem for it. In addition, we give a Kawashima type criterion for this approximation to be strictly dissipative. We apply the theory to the compressible Navier–Stokes system.     

Dynamic buckling of cylindrical shells subject to an axial impact in a symplectic system

This paper discusses the dynamic pre-buckling of finite cylindrical shells in the propagation and reflection of axial stress waves. By introducing the Hamiltonian system into dynamic buckling of structures, the problem can be described mathematically in a symplectic space. The solutions of Hamiltonian dual equations shown in canonical variables are obtained. The problem is reduced to the determination of eigenvalues and eigensolutions, with the former indicating critical buckling loads and the latter buckling modes. Numerical example presented shows phenomena of axisymmetric and non-axisymmetric dynamic buckling subject to impacts of axial load.     

Form invariance, Noether symmetry and Lie symmetry of Hamiltonian systems in phase space

For the holonomic and non-holonomic Hamiltonian systems in phase space, the definitions and criterions of the form invariance of both Hamilton and generalized Hamilton canonical equations are given. The relations among the form invariance, Noether symmetry and Lie symmetry are studied. Two examples are given to illustrate these results.