关于辛算法稳定性的若干注记

我们讨论辛算法的线性稳定性和非线性稳定性,从动力系统和计算的角度论述了研究辛算法的这两类稳定性问题的重要性,分析总结了相关重要结果.我们给出了解析方法的明确定义,证明了稳定函数是亚纯函数的解析辛方法是绝对线性稳定的.绝对线性稳定的辛方法既有解析方法（如Runge-Kutta辛方法）,也有非解析方法（如基于常数变易公式对线性部分进行指数积分而对非线性部分使用其它数值积分的方法）.我们特别回顾并讨论了R.I.McLachlan,S.K.Gray和S.Blanes,F.Casas,A.Murua等关于分裂算法的线性稳定性结果,如通过选取适当的稳定多项式函数构造具有最优线性稳定性的任意高阶分裂辛算法和高效共轭校正辛算法,这类经优化后的方法应用于诸如高振荡系统和波动方程等线性方程或者线性主导的弱非线性方程具有良好的数值稳定性.我们通过分析辛算法在保持椭圆平衡点的稳定性,能量面的指数长时间慢扩散和KAM不变环面的保持等三个方面阐述了辛算法的非线性稳定性,总结了相关已有结果.最后在向后误差分析基础上,基于一个自由度的非线性振子和同宿轨分析法讨论了辛算法的非线性稳定性,提出了一个新的非线性稳定性概念,目的是为辛算法提供一个实际可用的非线性稳定性判别法.     

$n$-维中立型多时滞微分方程的稳定性分析

This paper is mainly concerned with stability analysis of neutral differential equations with multiple delays. Some criteria on instability, stability, asymptotic stability and exponential stability are obtained. The criterion on asymptotic stability is necessary and sufficient. Two examples are provided to illustrate the applications of our results. Some previous results are extended.     

Exponential stability for set dynamic equations on time scales

A new theory known as set dynamic equations on time scales has been built. The criteria for the equistability, equiasymptotic stability, uniform and uniformly asymptotic stability were developed in Hong (2010) [1]. In this paper, we consider the exponential stability, exponentially asymptotic stability, uniform and uniformly exponentially asymptotic stability for the trivial solution of set dynamic equations on time scales by using Lyapunov-like functions.     

STABILITY OF SOLUTION OF TIME-VARYING SINGULAR SYSTEMS WITH DELAY

1IntroductionInmanydifferentareas)ithasbeenfoundthatmanyproblemsaremodeledbysingularsystems,suchasoptimalcontrolproblemsandconstrainedcontrolproblems,electricalcircuits,somepopulationgrowthmodelsandsingularper-turbations.Sincedelayoftenoccursintheseproble…     

关于插值的稳定性

实际计算表明,高阶多项式插值是不稳定的,分片低阶多项式插值则有良好的稳定性.在[1]中对此有简要的说明,本文将对此进行较详尽的理论分析.     

关于运用Liapunov函数的滞后型系统部分稳定性的注释

Sufficient conditions for the stability with respect to part of the functional differential equation variables are given. These conditions utilize Lyapunov functions to determine the uniform stability and uniform asymptotic stability of functional differential equations. These conditions for the partial stability develop the Razumikhin theorems on uniform stability and uniform asymptotic stability of functional differential equations. An example is presented which demonstrates these results and gives insight into the new stability conditions.     

Vibration and stability of an axially moving Rayleigh beam

In this paper, the vibration and stability of an axially moving beam is investigated. The finite element method with variable-domain elements is used to derive the equations of motion of an axially moving beam based on Rayleigh beam theory. Two kinds of axial motions including constant-speed extension deployment and back-and-forth periodical motion are considered. The vibration and stability of beams with these motions are investigated. For vibration analysis, direct time numerical integration, based on a Runge–Kutta algorithm, is used. For stability analysis of a beam with constant-speed axial extension deployment, eigenvalues of equations of motion are obtained to determine its stability, while Floquet theory is employed to investigate the stability of the beam with back-and-forth periodical axial motion. The effects of oscillation amplitude and frequency of periodical axial movement on the stability of the beam are discussed from the stability chart. Time histories are established to confirm the results from Floquet theory.     

A novel approach on stabilization for linear systems with time-varying input delay

This paper presents new results on delay-dependent stability and stabilization for linear systems with interval time-varying delays. Some less conservative delay-dependent criteria for determining the stability of the time-delay systems are obtained in this paper. Based on the stability conditions, we propose a new state transformation technology to facilitate controller designing efficiently and computationally. The method is also applicable to the existing stability conditions reported by now, while the existing technologies may fail to derive computational control procedures from the stability conditions. Finally, some numerical examples well illustrate the effectiveness of the proposed method.     

Delay-Dependent Criteria for Robust Stabilization of Markovian Switching Networks with Time-Varying Delay

Abstract

A problem of feedback stabilization of hybrid systems with time-varying delay and Markovian switching is considered. Delay-dependent sufficient conditions for stability based on linear matrix inequalities (LMI's) for stochastic asymptotic stability is obtained. The stability result depended on the mode of the system and of delay-dependent. The robustness results of such stability concept against all admissible uncertainties are also investigated. This new delay-dependent stability criteria is less conservative than the existing delay-independent stability conditions. An example is given to demonstrate the obtained results.     

基于投影寻踪-随机森林的制造企业供应链稳定性评价

基于扎根理论提取制造企业供应链稳定性影响因素,鉴于因素的随机性和模糊性特征,引入云模型,将定性概念转化为定量数据,从众多影响因素中筛选出重要因素,构建制造企业供应链稳定性评价指标体系。采用加速遗传算法对建立的投影寻踪模型进行求解,确定指标权重与稳定性评价值。利用随机森林方法根据评价值划分供应链稳定性类型,检验评价方法科学性,并确定当前制造企业供应链稳定性的整体状态,据此提出优化制造企业供应链稳定性建议。     

我国股市网络稳定性及其宏观影响因素——基于三种网络拓扑结构、鲁棒性的实证研究

基于金融时间序列的多重分形特征及衡量市场风险的VaR模型,建立我国沪深股市的股票关联网络,实证研究三种网络拓扑结构特征,并使用协整检验方法分析网络稳定性和宏观经济变量间的长期均衡关系。结果表明:股票价格网络不具有无标度性,多标度网络和风险网络都具有无标度性;在三种网络中,风险网络具有更强的鲁棒性。此外,股市波动率和网络稳定系数间互为格兰杰因果关系,股市波动的前期变化能有效解释网络稳定性系数的变化;网络稳定性与宏观经济变量间具有长期的均衡关系,GDP增长率、消费者物价水平CPI对网络稳定性具有正向效应,利率对网络稳定性具有负向效应。风险网络的提出有助于分析我国股市的短期风险及稳定性,并为制定系统风险防御策略提供参考。     

非均匀Gabor框架的稳定性

研究了当窗函数变化时非均匀Gabor框架的稳定性.对紧支撑Gabor框架,将均匀情况下关于稳定性的结论推广到了非均匀的情况;对一般的Gabor框架,利用W（L^∞,e^1）范数给出了其稳定的一个充分条件.     

变时滞退化Lurie控制系统的绝对稳定性

研究了变时滞退化Lurie控制系统的绝对稳定性问题.基于Lyapunov稳定性理论,利用线性矩阵不等式方法给出了系统绝对稳定的判别准则.讨论了变时滞退化Lurie直接控制系统和间接控制系统的绝对稳定性,得到绝对稳定性的充分条件仅依赖于时滞导数的大小,且时滞可以是无界函数：最后给出了实例说明本文结果的有效性.     

电力市场实用稳定性的判定条件

首次把实用稳定性的理论用于电力市场稳定性的研究中.结合Alvarado提出的电力市场动态模型,利用微分代数方程与特征值技术,从理论上研究电力市场的实用稳定性,并且给出了判断电力市场实用稳定、一致实用稳定和实用渐近稳定性的充分条件.利用这些实用稳定性条件,对于Alvarado给出的数值算例,可方便地利用初始数据判断电力市场模型的实用稳定性,并通过实例提供了假设模型中某个参数在电力市场变化中起主要作用,如何控制电力市场模型实用稳定性的方法.最后利用Maple软件包给出了参数变化引起模型实用稳定和不稳定的图形演示.     

Stability and positivity of equilibria for subhomogeneous cooperative systems

Building on recent work on homogeneous cooperative systems, we extend results concerning stability of such systems to subhomogeneous systems. We also consider subhomogeneous cooperative systems with constant input, and relate the global asymptotic stability of the unforced system to the existence and stability of positive equilibria for the system with input.     

Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays

This paper is concerned with the study of the stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays. We are interested in the comparison between the analytical and numerical stability regions. First, we focus on scalar equations with real coefficients. It is proved that all Gauss-Pouzet methods can retain the asymptotic stability of the analytical solution. Then, we consider the multidimensional case. A new stability condition for the stability of the analytical solution is given. Under this condition, the asymptotic stability of Gauss-Pouzet methods is investigated.      

非定常线性系统在经常作用干扰下的稳定性定理及其在机器人动态稳定性分析中的应用

本文在文献[1]、[2]的基础上,给出并证明了非定常线性系统在经常作用干扰下稳定的充分和必要条件,并推得该系统在李雅普诺夫意义下一致渐近稳定和在经常作用干扰下稳定等价的定理.此外,作者还应用文中的定理,对机器人动态稳定性进行了比较切合实际的分析.     

Robust stability of uncertain impulsive dynamical systems

This paper studies robust stability of uncertain impulsive dynamical systems. By introducing the concepts of uniformly positive definite matrix functions and Hamilton–Jacobi/Riccati inequalities, several criteria on robust stability, robust asymptotic stability and robust exponential stability are established. An example is also worked through to illustrate our results.     

滞后型分段连续随机微分方程的稳定性

本文研究了滞后型分段连续随机微分方程的解析稳定性和数值稳定性问题.首先,利用伊藤公式等方法获得了解析解均方稳定的条件,其次,对于包括均方稳定和T-稳定在内的Euler-Maruyama方法的数值稳定性问题,运用不等式技术和随机分析方法获得了一些新的结果,证明了在一定条件下,Euler-Maruyama方法既是均方稳定又是T-稳定的,推广了随机延迟微分方程的数值稳定性结论.