Stability of Equilibria of Discrete-Time Systems and Localization of Invariant Compact Sets

The stability (or asymptotic stability) of equilibria of an time-invariant discrete-time system can be verified with the use of stability and asymptotic stability criteria stated in terms of invariant sets. An earlier proposed method reduces the verification of these criteria to some set operations. However, the method is analytical and hard to implement. We propose another approach to the verification of these criteria based on the functional method for localizing invariant compact sets.     

基于一类时滞不等式的时滞微分方程的渐近稳定性

利用Lyapunov的方法讨论了时滞微分方程x.(t)=f(t,x(t),x(t-τ(t)))的全局指数渐近稳定性和全局渐近稳定性.     

Global asymptotic stability of a class of nonautonomous integro-differential systems and applications

In this paper, we study the global asymptotic stability of a class of nonautonomous integro-differential systems. By constructing suitable Lyapunov functionals, we establish new and explicit criteria for the global asymptotic stability in the sense of Definition 2.1. In the autonomous case, we discuss the global asymptotic stability of a unique equilibrium of the system, and in the case of periodic system, we establish sufficient criteria for existence, uniqueness and global asymptotic stability of a periodic solution. Also explored are applications of our main results to some biological and neural network models. The examples show that our criteria are more general and easily applicable, and improve and generalize some existing results.     

$n$-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.     

Stability and boundedness of solutions of neutral functional differential equations with finite delay

Sufficient and necessary criteria are established for the uniform stability and uniformly asymptotic stability of solutions of neutral functional differential equations (NFDEs) with finite delay by using the Liapunov functional approach. We also prove that the uniformly asymptotic stability of solutions implies the existence of bounded solution.     

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.     

一类扩张的细胞神经网络稳定性判别法

本文研究一类含有阻尼项带有时滞细胞神经网络的全局渐进稳定性和一致稳定性的性质,通过构造适当的李雅普诺夫函数及利用分析的有关知识,给出了全局渐进稳定性和一致稳定的判别法．     

Criteria for the Asymptotic Stability of Solutions of Dynamical Systems

We present a new proof for criteria for the asymptotic stability of systems of difference and differential equations based on the properties of monotone operators in a semiordered space. We also establish necessary and sufficient conditions for the asymptotic stability of stochastic systems of differential and difference equations in the mean square.     

Stability criteria for exact and discrete solutions of neutral multidelay-integro-differential equations

This paper deals with the asymptotic stability of exact and discrete solutions of neutral multidelay-integro-differential equations. Sufficient conditions are derived that guarantee the asymptotic stability of the exact solutions. Adaptations of classical Runge–Kutta and linear multistep methods are suggested for solving such systems with commensurate delays. Stability criteria are constructed for the asymptotic stability of these numerical methods and compared to the stability criteria derived for the continuous problem. It is found that, under suitable conditions, these two classes of numerical methods retain the stability of the continuous systems. Some numerical examples are given that illustrate the theoretical results. This research is supported by Fellowship F/02/019 of the Research Council of the K.U.Leuven, NSFC (No.10571066) and SRF for ROCS, SEM.     

Global asymptotic stability analysis for neutral stochastic neural networks with time-varying delays

In this paper, the global asymptotic stability is investigated for a class of neutral stochastic neural networks with time-varying delays and norm-bounded uncertainties. Based on Lyapunov stability theory and stochastic analysis approaches, delay-dependent criteria are derived to ensure the global, robust, asymptotic stability of the addressed system in the mean square for all admissible parameter uncertainties. The criteria can be checked easily by the LMI Control Toolbox in Matlab. A numerical example is given to illustrate the feasibility and effectiveness of the results.     

Stability properties for Hopfield neural networks with delays and impulsive perturbations

In this paper, we consider the uniform asymptotic stability, global asymptotic stability and global exponential stability of the equilibrium point of Hopfield neural networks with delays and impulsive perturbation. Some new stability criteria for such system are derived by using the Lyapunov functional method and the linear matrix inequality approach. The results are related to the size of delays and impulses. Our results are less restrictive and conservative than that given in some earlier references. Finally, two numerical examples showing the effectiveness of the present criteria are given.     

Boundedness and Asymptotic Stability of Multistep Methods for Generalized Pantograph Equations

In this paper, we deal with the boundedness and the asymptotic stability of linear and one-leg multistep methods for generalized pantograph equations of neutral type, which arise from some fields of engineering. Some criteria of the boundedness and the asymptotic stability for the methods are obtained.     

Comparison principle and stability criteria for stochastic differential delay equations with Markovian switching

In the present paper we first obtain the comparison principle for the nonlinear stochastic differential delay equations with Markovian switching. Later, using this comparison principle, we obtain some stability criteria, including stability in probability, asymptotic stability in probability, stability in thepth mean, asymptotic stability in the pth mean and the pth moment exponential stability of such equations. Finally, an example is given to illustrate the effectiveness of our results.     

Stability criteria for a class of differential inclusion systems with discrete and distributed time delays

In this paper, the global asymptotic stability for a class of differential inclusion systems with discrete and distributed time delays is investigated. Some delay-dependent criteria are proposed to guarantee the global asymptotic stability of the systems. Finally, a numerical example is provided to illustrate the use of the main results.     

Delay-dependent asymptotic stability for neural networks with distributed delays

In this paper, dynamical behavior of a class of neural networks with distributed delays is studied by employing suitable Lyapunov functionals, delay-dependent criteria to ensure local and global asymptotic stability of the equilibrium of the neural networks. Our results are applied to classical Hopfield neural networks with distributed delays and some novel asymptotic stability criteria are also derived. The obtained conditions are shown to be less conservative and restrictive than those reported in the known literature.     

Global stability for separable nonlinear delay differential systems

In this paper, we consider separable nonlinear delay differential systems and we establish conditions for global asymptotic stability of the zero solution. Applying these, we offer improved 3/2-type criteria for global asymptotic stability of nonautonomous Lotka-Volterra systems with delays.     

Asymptotic Stability and Boundedness of Stochastic Differential Equations with Respect to Semimartingales

In this paper, we shall use multiple Lyapunov functions to establish some sufficient criteria for locating the limit sets of solutions of stochastic differential equations with respect to semimartingales. From them follow many useful results on stochastic asymptotic stability and boundedness, including some classical results as special cases. In particular, our new asymptotic stability criteria do not require the diffusion operator associated with the underlying stochastic differential equation be negative definite, while most of the existing results do require this negative definite property essentially.     

Comparison principle and stability of Ito stochastic differential delay equations with Poisson jump and Markovian switching

In this paper the comparison principle for the nonlinear Itô stochastic differential delay equations with Poisson jump and Markovian switching is established. Later, using this comparison principle, we obtain some stability criteria, including stability in probability, asymptotic stability in probability, stability in the pth mean, asymptotic stability in the pth mean and the pth moment exponential stability of such equations. Some known results are generalized and improved.     

Stability Criteria for Solutions of Systems of Linear Deterministic or Stochastic Delay Difference Equations with Continuous Time

We give spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the Lyapunov asymptotic stability of solutions to systems of linear deterministic or stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For stochastic systems, mean-square asymptotic stability is studied. The Lyapunov function method is used. Our criteria on algebraic coefficients and our sufficient conditions are stated in terms of matrix Lyapunov equations (for deterministic systems) and matrix Sylvester equations (for stochastic systems).     

Global asymptotic stability for a periodic delay hematopoiesis model with impulses

In this paper, sufficient conditions for the global asymptotic stability of a broad family of periodic impulsive scalar delay differential equations are obtained. These conditions are applied to a periodic hematopoiesis model with multiple time-dependent delays and linear impulses, in order to establish criteria for the global asymptotic stability of a positive periodic solution. The present results are discussed within the context of recent literature. In conclusion, when compared with previous works, not only sharper stability criteria are obtained here, even for models without impulses, but also the usual constraints imposed on the linear impulses are relaxed.