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。     

一类不连续系统的推广Matrosov稳定性定理

在一般Filippov解意义下给出利用一族满足一定条件的Lyapunov函数的更一般的Matrosov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.     

一类具有Beddington-DeAngelis响应函数的阶段结构捕食模型的稳定性

该文讨论一类具有Beddington-DeAngelis响应函数的阶段结构捕食模型.利用RouthHurwitz判别定理讨论了其正常数平衡解的局部渐近稳定性,通过构造Lyapunov函数方法得到了全局渐近稳定性.     

时标动力学方程的稳定性

本文主要研究非自治时标动力学方程x△=f(t,x),(t,x)∈T×Rn的平凡解稳定、一致稳定、渐近稳定,与不稳定的充分必要条件.在定理的证明中充分性主要利用K类函数严格单调递增和Lyapunov函数的连续性;必要性的证明中,分别构造Lyapunov函数,满足定理的条件,使得定理的结论成立.     

含有离散时滞及分布时滞分数阶神经网络的渐近稳定性分析

研究了含有离散时滞及分布时滞的分数阶神经网络在Caputo导数意义下的渐近稳定性问题.通过构造Lyapunov函数和利用分数阶Razumikhin定理给出了含有离散时滞和分布时滞的分数阶神经网络渐近稳定性的充分条件,并给出4个例子验证了定理条件的有效性.     

非线性有限时滞脉冲泛函微分系统的稳定性

研究了一类非线性有限时滞脉冲泛函微分系统,利用比较原理和Lyapunov函数,得到了系统零解一致最终稳定性及一致最终渐近稳定性的充分条件.     

中立型随机泛函微分方程的稳定性

该文研究了一般中立型随机微分方程解的渐近性质,利用Lyapunov函数和上鞅收敛定理,得到 了该方程解的一些渐近稳定性、多项式渐近稳定性及指数稳定性等渐近性质,其结果涵盖并 推广了已有文献的结论。     

常微分方程解的渐近性质(Ⅰ)

前言在工程设计中,人们不仅对方程的Lyapunov意义稳定性感兴趣,而且对方程的各种渐近性态如方程的界感兴趣,甚至有时Lyapunov意义稳定性对工程设计没有必要或完全没有用。如导弹的发射,仅要求初始条件选在某一范围而轨线保持在另一范围内足矣,而并不要求Lyapunov意义稳定。文[1]—[6]给出集合稳定性,不稳定性,一致稳定性的定义,并且给出其判定定理,并且给出特殊意义下的渐近稳定及收缩稳定的定义及其判定定理。本文引进一般意义下的集合渐近稳定,全局渐近稳定,(拟)收缩稳定,全局(拟)收缩稳定的概念,给出其判定定理。本文给出的集合稳定,一致稳定及不     

关于随机Lyapunov函数的存在性

本文对于一类非时齐的Ito型随机微分系统及可分离变量的常微辅助系统,建立了[1]的随机稳定性比较准则中的纯量Lyapunov函数及条件随机稳定性比较准则中的向量Lyapunov函数的存在性定理(这些Lyapunov函数我们就称其为随机Lyapunov函数).作为纯量随机Lyapunov函数存在性定理的一个推论,即为[2]中定理2的推广,并且在推论中所构造的随机Lyapunov函数,即为[4]中的Lyapunov函数.这些存在性定理也是[5]中常微分方程稳定性及条件稳定性比较准则的逆定理,对于随机微分系统的推广.     

Stability analysis of time-varying switched systems via indefinite difference Lyapunov functions

Asymptotic stability of time-varying switched systems is investigated in this paper. The less conservative sufficient criteria for asymptotic stability of time-varying discrete-time switched systems are proposed via common indefinite difference Lyapunov functions and multiple indefinite difference Lyapunov functions introduced in this note, respectively. Common indefinite difference Lyapunov functions can be used to analyze stability of a switched system with asymptotic stable subsystems and arbitrary switching signal. Multiple indefinite difference Lyapunov functions can be used to investigate stability of a switched system with unstable subsystems and a given switching signal. The difference of the proposed Lyapunov function may be positive at some instants for an asymptotically stable subsystem. We compare these main results and illustrate the effectiveness of the obtained theorems by three numerical examples.     

Stability results involving time-averaged Lyapunov function derivatives

The stability results which comprise the Direct Method of Lyapunov involve the existence of auxiliary functions (Lyapunov functions) endowed with certain definiteness properties. Although the Direct Method is very general and powerful, it has some limitations: there are dynamical systems with known stability properties for which there do not exist Lyapunov functions which satisfy the hypotheses of a Lyapunov stability theorem.In the present paper we identify a scalar switched dynamical system whose equilibrium (at the origin) has known stability properties (e.g., uniform asymptotic stability) and we prove that there does not exist a Lyapunov function which satisfies any one of the Lyapunov stability theorems (e.g., the Lyapunov theorem for uniform asymptotic stability). Using this example as motivation, we establish stability results which eliminated some of the limitations of the Direct Method alluded to. These results involve time-averaged Lyapunov function derivatives (TALFD’s). We show that these results are amenable to the analysis of the same dynamical systems for which the Direct Method fails. Furthermore, and more importantly, we prove that the stability results involving TALFD’s are less conservative than the results which comprise the Direct Method (which henceforth, we refer to as the classical Lyapunov stability results).While we confine our presentation to continuous finite-dimensional dynamical systems, the results presented herein can readily be extended to arbitrary continuous dynamical systems defined on metric spaces. Furthermore, with appropriate modifications, stability results involving TALFD’s can be generalized to discontinuous dynamical systems (DDS).     

Stability analysis of quasilinear uncertain dynamical systems

In this paper, we establish new conditions of uniform asymptotic stability for uncertain quasilinear systems. Also, a new class of Lyapunov functions: canonical matrix-valued Lyapunov functions is presented and used for stability analysis. Illustrative examples are given.     

Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response

Two models of a density dependent predator-prey system with Beddington-DeAngelis functional response are systematically considered. One includes the time delay in the functional response and the other does not. The explorations involve the permanence, local asymptotic stability and global asymptotic stability of the positive equilibrium for the models by using stability theory of differential equations and Lyapunov functions. For the permanence, the density dependence for predators is shown to give some negative effect for the two models. Further the permanence implies the local asymptotic stability for a positive equilibrium point of the model without delay. Also the global asymptotic stability condition, which can be easily checked for the model is obtained. For the model with time delay, local and global asymptotic stability conditions are obtained.     

Stability of solutions of nonlinear systems with unbounded perturbations

We study systems of differential equations with perturbations that are unbounded functions of time. We suggest a method for constructing Lyapunov functions to determine conditions under which the perturbations do not affect the asymptotic stability of the solutions.     

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.     

Stability Conditions for Solutions of Multidimensional Completely Integrable Differential Equations

New sufficient tests are given for the stability and asymptotic stability of the zero solution of a nonautonomous completely integrable equation on an arbitrary salient convex closed cone and on a finitely generated cone. The class of Lyapunov functions suitable for studying the asymptotic behavior of solutions of nonautonomous completely integrable equations is significantly extended by substantially weakening the sign negativeness condition, traditional in the Lyapunov second method, for the derivative of the Lyapunov function at the interior points of the cone.     

Stability analysis based on nonlinear inhomogeneous approximation

The asymptotic stability of zero solutions for essentially nonlinear systems of differential equations in triangular inhomogeneous approximation is studied. Conditions under which perturbations do not affect the asymptotic stability of the zero solution are determined by using the direct Lyapunov method. Stability criteria are stated in the form of inequalities between perturbation orders and the orders of homogeneity of functions involved in the nonlinear approximation system under consideration.     

The stability of neutral stochastic delay differential equations with Poisson jumps by fixed points

In the paper, the asymptotic mean square stability of the zero solution for neutral stochastic delay differential equations with Poisson jumps is studied by fixed points theory without Lyapunov functions. The coefficient functions have not been asked for a fixed sign, and the sufficient condition for mean square stability has been obtained. Therefore, some well-known results are improved and generalized.