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.     

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.     

Impulsive control for differential systems with delay

This paper deals with the impulsive control for a class of differential systems with delay. Using Lyapunov functions and the comparison principle, we present some sufficient conditions for the asymptotic stability and exponential stability of impulsive control systems with delay. Moreover, we give an estimate of the upper bound of impulse interval. The results in this paper extend and improve the earlier publications. Copyright © 2012 John Wiley & Sons, Ltd.     

Stability of impulsive system by perturbing Lyapunov functions

This paper focuses on the stability of the zero solution for the impulsive differential system at fixed times by the method of perturbing Lyapunov functions. Based on the method, some sufficient conditions for the above stability to hold are given.     

Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems

In this note, we study the exponential stability of impulsive functional differential systems with infinite delays by using the Razumikhin technique and Lyapunov functions. Several Razumikhin-type theorems on exponential stability are obtained, which shows that certain impulsive perturbations may make unstable systems exponentially stable. Some examples are discussed to illustrate our results.     

Nonuniform stability properties and perturbing lyapunov functions

In this paper, without assuming the conditions everywhere, we study nonuniform stability properties of differential systems by employing the method of perturbing Lyapunov functions and splitting the given system suitably     

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.     

脉冲积分-微分系统零解稳定性的新的Razumikhin型定理

为研究积分-微分系统的稳定性,运用Lyapunov函数直接方法并借助Razumikhin技巧的思想,通过减弱Lyapunov函数沿系统解的导数须常负或定负的限制条件,给出了判断脉冲积分-微分系统零解稳定性的新的直接判定定理.     

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.     

Global stability results for impulsive differential equations

Global results concerning stability properties of impulsive differential equations are established, employing piecewise continuous Lyapunov functions which are then applied for proving stability and boundedness properties     

EVENTUAL STABILITY OF IMPULSIVE DIFFERENTIAL SYSTEMS

In this article, criteria of eventual stability are established for impulsive differential systems using piecewise continuous Lyapunov functions. The sufficient conditions that are obtained significantly depend on the moments of impulses. An example is discussed to illustrate the theorem.     

Vector Lyapunov functions for practical stability of nonlinear impulsive functional differential equations

This paper studies the practical stability of the solutions of nonlinear impulsive functional differential equations. The obtained results are based on the method of vector Lyapunov functions and on differential inequalities for piecewise continuous functions. Examples are given to illustrate our results.     

Exponential stability analysis of impulsive stochastic functional differential systems with delayed impulses

This paper is concerned with the exponential stability analysis of impulsive stochastic functional differential systems with delayed impulses. Although the stability of impulsive stochastic functional differential systems have received considerable attention. However, relatively few works are concerned with the stability of systems with delayed impulses and our aim here is mainly to close the gap. Based on the Lyapunov functions and Razumikhin techniques, some exponential stability criteria are derived, which show that the system will stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous flows. The obtained results improve and complement ones from some recent works. Three examples are discussed to illustrate the effectiveness and the advantages of the results obtained.     

Control vector Lyapunov functions for large-scale impulsive dynamical systems

Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuous-time systems to address stability and control design of impulsive dynamical systems via vector Lyapunov functions. Specifically, we provide a generalized comparison principle involving hybrid comparison dynamics that are dependent on the comparison system states as well as the nonlinear impulsive dynamical system states. Furthermore, we develop stability results for impulsive dynamical systems that involve vector Lyapunov functions and hybrid comparison inequalities. Based on these results, we show that partial stability for state-dependent impulsive dynamical systems can be addressed via vector Lyapunov functions. Furthermore, we extend the recently developed notion of control vector Lyapunov functions to impulsive dynamical systems. Using control vector Lyapunov functions, we construct a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are then used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.     

Impulsive stabilization of delay differential systems via the Lyapunov–Razumikhin method

This work studies global exponential stability of impulsive delay differential systems. By employing the Razumikhin technique and Lyapunov functions, several global exponential stability criteria are established for general impulsive delay differential equations. Our results show that delay differential equations may be exponentially stabilized by impulses. An example and its simulation are also given to illustrate our results.     

脉冲泛函微分方程关于两个测度的实际稳定性

利用Lyapunov,函数方法,建立了脉冲时滞微分方程关于两个测度的实际稳定的充分条件。     

Stability and robustness of quasi-linear impulsive hybrid systems

Both hybrid dynamical systems and impulsive dynamical systems are studied extensively in the literature. However, impulsive hybrid systems are not yet well studied. Nonetheless, many physical systems exhibit both system switching and impulsive jump phenomena. This paper investigates stability and robust stability of a class of quasi-linear impulsive hybrid systems by using the methods of Lyapunov functions and Riccati inequalities. Sufficient conditions for stability and robust stability of those systems are established. Some examples are given to illustrate the applicability of our results.     

Strict stability of impulsive functional differential equations

Strict stability is the kind of stability that can give us some information about the rate of decay of the solutions. There are some results about strict stability of differential equations. In the present paper, we shall extend the strict stability to impulsive functional differential equations. By using Lyapunov functions and Razumikhin technique, we shall get some criteria for the strict stability of impulsive functional differential equations, and we can see that impulses do contribute to the system's strict stability behavior.     

Study on the stability of nonlinear differential equations with initial time difference

This paper establishes several criteria of stability for nonlinear differential dynamical systems relative to initial time difference by employing perturbing Lyapunov functions. The reported novel results are more easy to be tested than the existing results. As an application, these results are applied to a nonlinear dynamical system to get the stability properties relative to initial time difference.     

脉冲泛函微分方程实用稳定性的一个比较定理

考虑一类脉冲泛函微分方程的实用稳定性，利用锥值李亚普诺夫函数方法，建立了脉冲泛函数微分方程与脉冲常微分方程的实用稳定性之间的比较定理。