关于稳定性的几个基本定理

本文给出了非自治常微分方程稳定性，一致稳定性，渐近稳定性与一致渐近稳定性的几个定理。这些定理允许Lyapunov函数的导数为变号函数，放宽了Marachkov定理对系统状态变量变化率有界的要求，推广了文[4-7]的相应结果。     

基于曲线导数的二元函数微分中值定理

给定二元函数,文献[1]定义了其在光滑曲线上的方向导数(简称为曲线导数).本文主要利用曲线导数建立二元函数的微分中值定理,比如罗尔定理,拉格朗日中值定理,柯西中值定理.这些中值定理可视作一元函数微分中值定理在二维情形的推广.     

具有可变时滞的Hopfield型随机神经网络的指数稳定性

研究了具有可变时滞的Ｈｏｐｆｉｅｌｄ型随机种经网络的指数稳定性,应用Ｒａｚｕｍｉｋｈｉｎ定理与 Ｌｙａｐｕｎｏｖ函数,建立了这种神经网络的均方指数稳定与几乎必然指数稳定的两类判据,一类是时 滞相关而另一类是时滞无关．     

Fuzzy数测度与积分

本文利用文[2]所给出的Fuzzy数测度的概念,定义了(—)fuzzy值函数关于(—)fuzzy数测度的积分,并且研究了这种积分的性质,得到了各种收敛定理,其中包括广义Lebesgue单调收敛定理、Fatou引理及Lebesgue控制收敛定理。在最后,讨论了(—)fuzzy数测度的R—N导数的存在性,并且给出了Fubini定理。     

Marachkov-Barbashin-Krasovskii型渐近稳定性定理

本文研究非自治系统的渐近稳定性.且得到了不要求Liapunov函数正定,也不要求其沿系统的解的导数负定的渐近稳定性定理.一致渐近稳定性定理及全局渐近稳定性定理.     

离散广义系统的平稳振荡

为了研究离散广义系统的平稳振荡，本文利用广义Ｌｙａｐｕｎｏｖ函数方法，给出了一个ｍ周期解存在的充要条件，得出了离散广义系统的周期解的存在性、唯一性、稳定性的有关定理，进而研究具有某种分解的复杂离散广义系统的平稳振荡问题，方法简单易行．     

两类稳定性定理的改进

本文对两类稳定性定理进行了改进.通过K函数性质,将原来马尔金稳定性定理中V函数导数在有限时间内可变号的性质,推广到长时间内,得到了新的非自治微分方程解的稳定性判定方法.对积分收敛判定定理中的不足进行了补充说明,并且对定理进行了改进和推广,得到了新的一致稳定性判定方法.     

稳定性理论中几个基本定理的推广

本文推广了经典的稳定性,有界性定理及关于渐近稳定的充分条件。去掉了原始定理中,函数的导数在其定义域内常负或定负的要求,并应用于集合稳定性判定.     

一类连续半鞅型随机微分方程解的随机稳定性

本文利用Ｌｙａｐｕｎｏｖ函数方法，讨论了时齐Ｄｏｌｅａｎｓ－Ｄａｄｅｒ－Ｐｒｏｔｔｅｒ方程ｄＸ＿ｔ＝σ（Ｘ＿ｔ）ｄＭ＿ｔ＝ｂ（Ｘ＿ｔ）ｄＡ＿ｔ＋（（Ｍ＿ｔ）为连续局部平方可积鞅；（Ａ＿ｔ）为连续有限变差过程）平凡解的随机稳定性。本文建立了随机稳定性的判定定理并给出了相应的Ｌｙａｐｕｎｏｖ函数的一种具体形式。     

无RNP Banach空间中集值测度的Radon-Nikodym定理

自七十年代初以来关于集值测度的Radon-Nikodym导数取得了不少成果,所有这些成果中均要求所考虑的Banach空间具有RNP,本文在无RNP的Banach空间中建立了集值测度的Radon-Nikodym定理.     

Stability and boundedness of nonlinear impulsive systems in terms of two measures via perturbing Lyapunov functions

This paper develops the concepts of stability, practical stability and boundedness in terms of two measures for nonlinear impulsive differential systems using the method of perturbing Lyapunov functions. The notion of perturbing Lyapunov functions enables us to discuss stability properties of solutions of nonlinear impulsive differential systems in terms of two measures under much weaker assumptions. The novel results offer a way to unify a variety of stability results found in the relative literature.     

On stability in terms of two measures for impulsive systems of functional differential equations

This paper considers the stability problem for a class of impulsive systems of functional differential equations. By using Lyapunov functions and the Razumikhin technique, several stability criteria in terms of two measures are established. Some examples are worked out to illustrate the theorems.     

Stability of impulsive stochastic differential equations in terms of two measures via perturbing Lyapunov functions

In this paper, we study the stability of nonlinear impulsive stochastic differential equations in terms of two measures. The concept of perturbing Lyapunov functions is introduced to discuss stability properties of solutions of nonlinear impulsive stochastic differential equations in terms of two measures. By using perturbing Lyapunov functions and comparison method, some sufficient conditions for the above stability are given.     

时滞微分方程依照两种测度的稳定性

运用变异Liapunov方法,讨论了时滞微分方程依照两种测度的稳定性。借助于中间测度h*(t,x),在未扰动系统为常微分方程的情形下,得到了关于时滞微分方程非一致和一致稳定性的判定定理。     

PERTURBING FAMILIES OF GENERALIZED LYAPUNOV FUNCTIONS AND RELATIVE STABILITY IN TERMS OF TWO MEASURES

ＰＥＲＴＵＲＢＩＮＧＦＡＭＩＬＩＥＳＯＦＧＥＮＥＲＡＬＩＺＥＤＬＹＡＰＵＮＯＶ　ＦＵＮＣＴＩＯＮＳＡＮＤＲＥＬＡＴＩＶＥＳＴＡＢＩＬＩＴＹＩＮＴＥＲＭＳＯＦＴＷＯ　ＭＥＡＳＵＲＥＳＳｈａｎｇＬｉｊｕｎ（商立军）（ＴｈｅＦｏｕｒｔｈＭｉｌｉｔａｒｙＭｅ...     

On the stability of abstract monotone impulsive differential equations in terms of two measures

We consider differential equations in a Banach space subjected to an impulsive influence at fixed times. It is assumed that a partial ordering is introduced in the Banach space by using a normal cone and that the differential equations are monotone with respect to the initial data. We propose a new approach to the construction of comparison systems in finite-dimensional spaces without using auxiliary Lyapunov-type functions. On the basis of this approach, we establish sufficient conditions for the stability of this class of differential equations in terms of two measures. In this case, a Birkhoff measure is chosen as the measure of initial displacements, and the norm in the given Banach space is used as the measure of current displacements. We present some examples of investigations of the impulsive systems of differential equations in the critical cases and linear impulsive systems of partial differential equations.     

Stability criteria in terms of two measures for functional differential equation with variable‐time impulses

This paper focuses on the stability in terms of two measures for functional differential equation with variable‐time impulses. Being different from most of existing literatures, the impulses of functional differential equation are assumed to be closely associated to the current state. We propose a new comparison principle for the considered systems and establish several stability criteria in terms of two measures. Also, the theoretical results are applied in a class of delayed neural network systems with variable‐tine impulses, and numerical simulations are introduced to illustrate the effectiveness of our results. Copyright © 2014 John Wiley & Sons, Ltd.     

A nonlinear inequality and application to global asymptotic stability of perturbed systems

The goal of this work is to present a new nonlinear inequality which is used in a study of the Lyapunov uniform stability and uniform asymptotic stability of solutions to time‐varying perturbed differential equations. New sufficient conditions for global uniform asymptotic stability and/or practical stability in terms of Lyapunov‐like functions for nonlinear time‐varying systems is obtained. Our conditions are expressed as relation between the Lyapunov function and the existence of specific function which appear in our analysis through the solution of a scalar differential equation. Moreover, an example in dimensional two is given to illustrate the applicability of the main result. Copyright © 2014 John Wiley & Sons, Ltd.     

有限时滞微分系统依照双测度的实际稳定性

本文对具有有限时滞的泛函微分方程建立了关于依照两种测度的实际稳定性的Razumikhin型判定定理，其中未采用通常的辅助函数，且可运用多个含有状态变量x的部分变元的Lyapunov函数，得出部分变远实际稳定性的判定定理，从而改进了已有的结果。