The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag

This paper is concerned with the exponential stability of singularly perturbed delay differential equations with a bounded (state-independent) lag. A generalized Halanay inequality is derived in Section 2, and in Section 3 a sufficient condition will be provided to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is exponentially stable uniformly for sufficiently small ε>0. This type of exponential asymptotic stability can obviously be applied to general delay differential equations with a bounded lag.     

变时滞的退化滞后型微分系统的稳定性

主要研究了具有变时滞的退化时滞微分系统的稳定性.利用退化时滞微分系统的变易公式和Gronwall-Bellman积分不等式给出了该系统的指数估计以及稳定和指数渐近稳定的充分条件.     

变时滞的退化中立型微分系统的稳定性

研究了具有变时滞的退化中立型微分系统的稳定性．利用退化时滞微分系统的变易公式和Gronwall-Bellman积分不等式给出了该系统的指数估计以及稳定和指数渐近稳定的充分条件．     

中立型驱动反应BAM 神经网络的全局渐近同步性

针对一类具有时滞的中立型驱动反应BAM神经网络,提出全局渐近同步性问题.不使用现有文献中传统的李雅普诺夫泛函、矩阵测度和线性矩阵不等式(LMI)等已被广泛应用于研究神经网络全局渐近同步性的方法,而是通过构造2个微分不等式U 1(t)和U 1(t),利用微分不等式方程和不等式技巧解出2个不等式,得到能够确保中立型驱动反应BAM神经网络全局渐近同步的两个新的充分条件.     

Stability of nonlinear variable-time impulsive differential systems with delayed impulses

This paper investigates the stability of nonlinear variable-time impulsive differential systems with delayed impulses. By using the comparison principle, the Lyapunov method and inequality techniques, some sufficient conditions for quasi-uniformly asymptotic stability and quasi-exponential stability of a given solution of variable-time impulsive differential systems with delayed impulses are established.     

Dynamics of a hysteretic neuron model

A mathematical model of a single isolated artificial neuron with hysterisis is formulated by means of a neutral delay differential equation. The asymptotic and exponential stability of such a model are investigated. Sufficient conditions for the exponential stability of a linear integral difference inequality are obtained. In the absence of hysterisis effect, our model reduces to a known model of a single neuron. Usually asymptotic stability of neutral delay differential equations is studied by means of degenerate Lyapunov–Kravsovskii functionals. In this article, perhaps for the first time exponential stability of a class of neutral differential equations are studied by means of the exponential stability of an affiliated difference inequality. While generalization to Hopfield type hysteretic neural networks is possible, such a generalization is not considered in this article.     

一类无穷时滞微分系统的周期解和全局渐近稳定性

利用重合度理论中的延拓定理和微分不等式讨论一类无穷时滞微分系统的周期解的存在性和全局渐近稳定性,获得了简便的判别条件.     

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.     

On the stability of time-varying differential equations *

This paper deals with stability of time-varying differential equations. New asymptotic stability conditions for time-varying continuous systems with more general assumptions are given. The Gronwall's inequality and its discrete-time versions play a crucial role in our investigation.     

Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations

This paper is concerned with the dissipativity and asymptotic stability of the theoretical solutions of a class of nonlinear neutral delay integro-differential equations (NDIDEs). We first give a generalization of the Halanay inequality which plays an important role in the study of dissipativity and stability of differential equations. Then, we apply the generalization of the Halanay inequality to NDIDEs and the dissipativity and the asymptotic stability results of the theoretical solution of NDIDEs are obtained. From a numerical point of view, it is important to study the potential of numerical methods in preserving the qualitative behavior of the analytical solutions. Therefore, the results, presented in this paper, provide the theoretical foundation for analyzing the dissipativity and the asymptotic stability of the numerical methods when they are applied to these systems.     

Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices

In this paper a generalization of the delayed exponential defined by Khusainov and Shuklin (2003) [1] for autonomous linear delay systems with one delay defined by permutable matrices is given for delay systems with multiple delays and pairwise permutable matrices. Using this multidelay-exponential a solution of a Cauchy initial value problem is represented. By an application of this representation and using Pinto’s integral inequality an asymptotic stability results for some classes of nonlinear multidelay differential equations are proved.     

中立型比例延迟微分系统线性θ-方法的渐近估计

本文研究了一类线性非自治中立型比例延迟微分系统线性θ-方法的渐近稳定性,并借助于泛函不等式得到了数值解的渐近估计.此渐近估计不仅比数值渐近稳定性描述得更加精确,而且还能给出非稳定情形数值解的上界估计式.数值算例验证了相关理论结果.     

一类三阶中立型时滞微分方程的渐近稳定性

讨论了一类三阶中立型时滞微分方程的零解的渐近稳定性,借助于构造函数、推广的Halanay一维时滞微分不等式及泰塔格利亚公式,得到了判定其零解是渐近稳定的且与时滞无关的一个充分条件.     

Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory

The asymptotic log-Harnack inequality is established for several kinds of models on stochastic differential systems with infinite memory: non-degenerate SDEs, neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate, uniqueness of the invariant probability measure, asymptotic gradient estimate (hence, asymptotically strong Feller property), as well as asymptotic irreducibility.     

具概率延迟反馈金融系统的脉冲控制

研究了概率时滞脉冲金融系统平衡点的全局渐近稳定性问题。首先,通过定义合适的时滞分段区间上的随机变量,给出了概率时滞的脉冲金融系统的数学模型,根据脉冲微分不等式特点构造了一个简便合适的Lyapunov函数利用脉冲微分不等式引理、控制脉冲间隔与脉冲量以及概率时滞分析技巧,获得了较大时滞允许范畴下的平衡点的全局指数稳定,并通过数值实例验证了方法的可行性以及概率时滞的优势。特别地,稳定性判定准则的时滞允许上限的增大,扩大了准则的实用性.     

A LMI approach to stability analysis and synthesis of impulsive switched systems with time delays

This paper studies the asymptotic stability problem for a class of impulsive switched systems with time invariant delays based on linear matrix inequality (LMI) approach. Some sufficient conditions, which are independent of time delays and impulsive switching intervals, for ensuring asymptotical stability of these systems are derived by using a Lyapunov–Krasovskii technique. Moreover, some appropriate feedback controllers, which can stabilize the closed-loop systems, are constructed. Illustrative examples are presented to show the effectiveness of the results obtained.     

Stability criterion for synchronization of linearly coupled unified chaotic systems

This paper investigates the synchronization of two linearly coupled unified chaotic systems. A new stability criterion for asymptotic synchronization is attained using the Lyapunov stability theory and linear matrix inequality (LMI) approach. A numerical example is given to illuminate the design procedure and advantage of the result derived.     

New stability conditions for neural networks with constant and variable delays

In this paper, by utilizing Lyapunov functional method, we analyze global asymptotic stability of neural networks with constant delays. A new sufficient condition ensuring global asymptotic stability of the unique equilibrium point of delayed neural networks is obtained. Furthermore, based on the method of delay differential inequality, the conditions checking global exponential stability of the equilibrium point of neural networks with variable delays are given. The results extend and improve the earlier publications.     

New stability conditions for neural networks with constant and variable delays

In this paper, by utilizing Lyapunov functional method, we analyze global asymptotic stability of neural networks with constant delays. A new sufficient condition ensuring global asymptotic stability of the unique equilibrium point of delayed neural networks is obtained. Furthermore, based on the method of delay differential inequality, the conditions checking global exponential stability of the equilibrium point of neural networks with variable delays are given. The results extend and improve the earlier publications.     

Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations

Summary. We consider systems of delay differential equations (DDEs) of the form with the initial condition . Recently, Torelli [10] introduced a concept of stability for numerical methods applied to dissipative nonlinear systems of DDEs (in some inner product norm), namely RN-stability, which is the straighforward generalization of the wellknown concept of BN-stability of numerical methods with respect to dissipative systems of ODEs. Dissipativity means that the solutions and corresponding to different initial functions and , respectively, satisfy the inequality , and is guaranteed by suitable conditions on the Lipschitz constants of the right-hand side function . A numerical method is said to be RN-stable if it preserves this contractivity property. After showing that, under slightly more stringent hypotheses on the Lipschitz constants and on the delay function , the solutions and are such that , in this paper we prove that RN-stable continuous Runge-Kutta methods preserve also this asymptotic stability property. Received March 29, 1996 / Revised version received August 12, 1996