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In this paper, we consider the global asymptotical stability of stochastic functional differential equations with impulsive effects. First, by constructing the Lyapunov function, some stability criteria of impulsive stochastic functional differential equations are established. Second, we propose an application to investigate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper considers the problems on the existence and uniqueness, the p th(p ≥ 1)-moment and the almost sure stability with a general decay for the global solution of stochastic delay differential equations with Markovian switching, when the drift term and the diffusion term satisfy the locally Lipschitz condition and the monotonicity condition. By using the Lyapunov function approach, the Barbalat Lemma and the nonnegative semi-martingale convergence theorem, some sufficient conditions are proposed to guarantee the existence and uniqueness as well as the stability with a general decay for the global solution of such equations. It is mentioned that, in this paper, the time-varying delay is a bounded measurable function. The derived stability results are more general, which not only include the exponential stability but also the polynomial stability as well as the logarithmic one. At last, two examples are given to show the effectiveness of the theoretical results obtained. [ABSTRACT FROM AUTHOR]

• A network epidemic model with nonlinear incidence rate and time delay is regarded. • Stability is proved in a strict mathematical way. • Four different control strategies are investigated and compared in this paper. • Numerical simulations provide a new insight into epidemic diffusion on complex networks. In this paper, we establish a susceptible-infected-susceptible (SIS) epidemic model with nonlinear incidence rate and time delay on complex networks. Firstly, according to the existence of a positive equilibrium point, we work out the threshold values R 0 and λ c of disease propagation. Secondly, we demonstrate the stabilities of the disease-free equilibrium point and the disease-spreading equilibrium point by constructing Lyapunov function and applying delay differential equations theorem. Thirdly, four different control strategies are investigated and compared, including uniform immunization control, acquaintance immunization control, active immunization control and optimal control. Finally, we perform representative numerical simulations to illustrate the theoretical results and further discover that the nonlinear incidence rate can more accurately reflect individual psychological activities when a certain disease outbreaks at a high level. [ABSTRACT FROM AUTHOR]

In this paper, we determine sufficient conditions for the boundedness, uniformly asymtotically stability of the solutions to a certain fourth-order non-autonomous differential equations with bounded delay by considering second method of Lyapunov. The results obtain essentially improve, include and complement the consequences in the current literature. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov's second method. The results obtained essentially improve, include and complement the results in the literature. [ABSTRACT FROM AUTHOR]

This paper deals with positivity and stability of linear implicit difference delay equations. Being different from the Lyapunov function approach commonly used in stability analysis, the method employed in this paper gives a way to solve the exponential stability of linear implicit difference equations with time-varying delay. By decomposition state-space and mathematical induction method, new necessary and sufficient conditions for positivity and stability of such systems are derived. Numerical examples are given to illustrate the proposed results. [ABSTRACT FROM AUTHOR]

The objective of this study was to get some sufficient conditions which guarantee the asymptotic stability and uniform boundedness of the null solution of some differential equations of third order with the variable delay. The most efficient tool for the study of the stability and boundedness of solutions of a given nonlinear differential equation is provided by Lyapunov theory. However the construction of such functions which are positive definite with corresponding negative definite derivatives is in general a difficult task, especially for higher-order differential equations with delay. Such functions and their time derivatives along the system under consideration must satisfy some fundamental inequalities. Here the Lyapunov second method or direct method is used as a basic tool. By defining an appropriate Lyapunov functional, we prove two new theorems on the asymptotic stability and uniform boundedness of the null solution of the considered equation. Our results obtained in this work improve and extend some existing well-known related results in the relevant literature which were obtained for nonlinear differential equations of third order with a constant delay. We also give an example to illustrate the importance of the theoretical analysis in this work and to test the effectiveness of the method employed. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this work is to study the dynamical behavior of a virus dynamics model with general incidence rate and two delays. The first delay represents the time from the virus entry to the production of new viruses and the second delay corresponds to the time necessary for a newly produced virus to become infectious. Lyapunov functionals are constructed and LaSalle invariance principle for delay differential equations is used to establish the global asymptotic stability of the disease-free and the chronic infection equilibria. The results obtained show that the global dynamics are completely determined by the value of a certain threshold parameter called the basic reproduction number and under some assumptions on the general incidence function. Our results extend the known results on delay virus dynamics considered in the other papers and suggest useful methods to control virus infection. These results can be applied to a variety of possible incidence functions that could be used in virus dynamics model as well as epidemic models. [Copyright &y& Elsevier]

Abstract: In this paper, the global stability problem for a general discrete Cohen–Grossberg neural network with finite and infinite delays is investigated. A simple criterion ensuring the global asymptotical stability is established, by applying the Lyapunov method and graph theory. Finally, an example showing the effectiveness of the provided criterion is given. [Copyright &y& Elsevier]

Abstract: This paper studies the exponential stability problem for a class of networked control systems (NCSs) with time delays and packet dropouts. The state-feedback closed-loop NCS is modeled as a new discrete-time switched system. Through using the parameter-dependent Lyapunov function, a sufficient condition is obtained for the exponential stability of the NCS under a packet-dropout rate. A linear matrix inequality approach is employed to design the state-feedback controller. An illustrative example is presented to demonstrate the effectiveness of the proposed method. [Copyright &y& Elsevier]

This paper investigates the problems of stability analysis and stabilization for a class of discrete-time Takagi-Sugeno fuzzy systems with time-varying state delay. Based on a novel fuzzy Lyapunov–Krasovskii functional, a delay partitioning method has been developed for the delay-dependent stability analysis of fuzzy time-varying state delay systems. As a result of the novel idea of delay partitioning, the proposed stability condition is much less conservative than most of the existing results. A delay-dependent stabilization approach based on a nonparallel distributed compensation scheme is given for the closed-loop fuzzy systems. The proposed stability and stabilization conditions are formulated in the form of linear matrix inequalities (LMIs), which can be solved readily by using existing LMI optimization techniques. Finally, two illustrative examples are provided to demonstrate the effectiveness of the techniques proposed in this paper. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates control for a class of nonlinear systems with time-varying delays and convex polytopic uncertainties. A new type of parameter-dependent Lyapunov–Krasovskii functional is introduced to derive delay-dependent sufficient conditions for the optimal control with exponential stability. All the conditions developed in this paper are formulated in terms of linear matrix inequalities. Finally, a numerical example shows the effectiveness of the proposed methodology. [Copyright &y& Elsevier]

Abstract: The purpose of this note is twofold: First we highlight the importance of an implicit assumption in [Murat Adıvar, Youssef N. Raffoul, Stability and periodicity in dynamic delay equations, Computers and Mathematics with Applications 58 (2009) 264–272]. Second we emphasize one consequence of the bijectivity assumption which enables ruling out the commutativity condition on the delay function. [Copyright &y& Elsevier]

In this paper, some theorems of uniform stability and uniform asymptotic stability for impulsive functional differential equations with infinite delay are proved by using Lyapunov functionals and Razumikhin techniques. An example is also proved at the end to illustrate the application of the obtained results. [ABSTRACT FROM AUTHOR]

Abstract: This Letter considers a robust stability analysis method for delayed neural networks of neutral type. By constructing a new Lyapunov functional, a novel delay-dependent criterion for the stability is derived in terms of LMIs (linear matrix inequalities). A less conservative stability criterion is derived by using nonlinear properties of the activation function of the neural networks. Two numerical examples are illustrated to show the effectiveness of the proposed method. [Copyright &y& Elsevier]

Sufficiency criteria are established to ensure the asymptotic stability and boundedness of solutions to third-order nonlinear delay differential equations of the form x⃛ (t) + e(x(t), ẋ(t), ẍ(t))ẍ(t) + g(x(t - r), ẋ(t - r)) + ψ(x(t - r)) = p(t, x(t), x(t - r), x'(t), x'(t - r), x"(t)). By using Lyapunov's functional approach, we obtain two new results on the subject, which include and improve some related results in the relevant literature. Two examples are also given to illustrate the importance of results obtained. [ABSTRACT FROM AUTHOR]

Abstract: A sex-structured mathematical model for heterosexual transmission of HIV/AIDS with explicit incubation period is presented as a system of discrete delay differential equations. The epidemic threshold and equilibria for the model are determined and stabilities are examined. The disease-free equilibrium is shown to be locally and globally stable when the basic reproductive number is less than unity. We use the Lyapunov functional approach to show that the endemic equilibrium is locally asymptotically stable. Further comprehensive qualitative analysis of the model including persistence and permanence are investigated. [Copyright &y& Elsevier]

Abstract: Considering two kinds of delays accounting, respectively, for (i) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and (ii) a virus production period for new virions to be produced within and released from the infected cells, we develop and analyze a mathematical model for HIV-1 therapy by fighting a virus with another virus. For the different values of the basic reproduction number and the second basic reproduction number, we investigate the stability of the infection-free equilibrium, the single-infection equilibrium and the double-infection equilibrium. We conclude that increasing delays will decrease the values of the basic reproduction number and the second basic reproduction number. Our results have potential applications in HIV-1 therapy. The approach we use here is a combination of analysis of characteristic equations, Fluctuation Lemma and Lyapunov function. [Copyright &y& Elsevier]

Abstract: In this work, we investigate the stability of a class of impulsive functional differential equations. Some general stability theorems are obtained. Our results can be applied to finite delay impulsive systems or infinite delay impulsive systems or impulsive systems involving both finite and infinite delays, in a unified way. Examples are also given to illustrate that applying our theorems yields better conclusions than the results in the literature. [Copyright &y& Elsevier]

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution xε(·) of the underlying delay equation for ε > 0 small enough. Perturbed Liapunov function method is used in the analysis. [ABSTRACT FROM AUTHOR]

This paper studies stability problems of impulsive control systems with time delay. Several criteria on asymptotic stability are established using the method of Lyapunov functions and Lyapunov functionals. It is shown that by using impulsive feedback control, a linear delay differential system can be stabilized even if it contains no stable matrix. Some examples are also discussed to illustrate the theorems. [Copyright &y& Elsevier]

In the paper, employing the variational Lyapunov method, stability and instability properties in terms of two measures for impulsive delay differential equations with fixed moments of impulsive effects are discussed. Some stability and instability criteria are obtained. These results much generalize the known ones. Some examples are given to illustrate the advantages of them as well. [Copyright &y& Elsevier]

By using the variational Liapunov method, stability properties in terms of two measures for delay differential equations are discussed. In the case that the unperturbed systems are ordinary differential systems, employing auxiliary measure h*(t,x), criteria on nonuniform and uniform stability in terms of two measures for delay differential equations are established. [ABSTRACT FROM AUTHOR]

First, we set up in an appropriate way the initial value problem for nonlinear delay differential equations with a Riemann-Liouville (RL) fractional derivative. We define stability in time and generalize Mittag-Leffler stability for RL fractional differential equations and we study stability properties by an appropriate modification of the Razumikhin method. Two different types of derivatives of Lyapunov functions are studied: the RL fractional derivative when the argument of the Lyapunov function is any solution of the studied problem and a special type of Dini fractional derivative among the studied problem. [ABSTRACT FROM AUTHOR]

In this paper, by using Liapunov functional, some sufficient conditions are obtained for the stability of the equilibrium of a neural network model with delay of the type [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to linear parameter systems under linear fractional representations (LFR) of parameter-dependent nonlinear systems with real–rational nonlinearities and point-delayed dynamics. The robust global asymptotic stability of the system either independent of or dependent on the delay sizes is investigated. The associate matrix inequalities are related to the time-derivatives of appropriate Lyapunov functions at all the vertices of the polytope which contains the parameterized uncertainties. [Copyright &y& Elsevier]