Robust exponential stability of impulsive switched systems with switching delays: A Razumikhin approach

In this paper, we focus on the robust exponential stability of a class of uncertain nonlinear impulsive switched systems with switching delays. We introduce a novel type of piecewise Lyapunov-Razumikhin functions. Such functions can efficiently eliminate the impulsive and switching jump of adjacent Lyapunov functions at impulsive switching instants. By Razumikhin technique, the delay-independent criteria of exponential stability are established on the minimum dwell time. Finally, an illustrative numerical example is presented to show the effectiveness of the obtained theoretical results.     

Robust exponential stability of switched delay interconnected systems under arbitrary switching

The problem of robust exponential stability for a class of switched nonlinear dynamical systems with uncertainties and unbounded delay is addressed. On the assumption that the interconnected functions of the studied systems satisfy the Lipschitz condition, by resorting to vector Lyapunov approach and M-matrix theory, the sufficient conditions to ensure the robust exponential stability of the switched interconnected systems under arbitrary switching are obtained. The proposed method, which neither require the individual subsystems to share a Common Lyapunov Function (CLF), nor need to involve the values of individual Lyapunov functions at each switching time, provide a new way of thinking to study the stability of arbitrary switching. In addition, the proposed criteria are explicit, and it is convenient for practical applications. Finally, two numerical examples are given to illustrate the correctness and effectiveness of the proposed theories.     

Robust exponential stability of discrete‐time uncertain impulsive neural networks with time‐varying delay

The purpose of this paper is to investigate the robust exponential stability of discrete‐time uncertain impulsive neural networks with time‐varying delay. By using Lyapunov functions together with Razumikhin technique, some new robust exponential stability criteria are presented. The obtained results show that the robust stability can be retained under certain impulsive perturbations for the neural network, which has the robust stability property. The obtained results also show that impulses can robustly stabilize the neural network, which does not have the robust stability property. Some examples, together with their simulations, are also given to show the effectiveness and the advantage of the presented results. It should be noted that the impulsive robust exponential stabilization result for discrete‐time neural network with time‐varying delay is given for the first time. Copyright © 2012 John Wiley & Sons, Ltd.     

Exponential synchronization of chaotic systems subject to uncertainties in the control input

A sliding mode control technique is introduced for exponential synchronization of chaotic systems. These systems are described by a general form including matched and unmatched nonlinear functions. A new hitting-free switching surface of proportional-integral type is proposed. This type of switching surface is without the hitting process if the attraction of sliding manifold is ensured. This property makes it easy to exponentially synchronize the master-slave chaotic systems. Based on this switching surface, a robust sliding mode controller (SMC) is derived to guarantee the attraction of sliding manifold even when the system is subjected to input uncertainties. An example is included to illustrate the results developed in this paper.     

Stability of neutral stochastic delayed systems with switching and distributed-delay dependent impulses

This paper is devoted to discuss the exponential stability in mean square of neutral stochastic delayed systems (NSDDs) with switching and distributed-delay dependent impulses. By using multiple Lyapunov functions and average dwell time (ADT), we provide some sufficient conditions for the exponential stability in mean square for NSDDs with switching and distributed-delay dependent impulses. Compared with the existing related works, we consider not only the influences of switches and neutral type on the stability of NSDDs with switching and distributed-delay dependent impulses but also the influences of both the stable continuous dynamics case and the stable discrete dynamics case. Finally, we provide two examples to illustrate the effectiveness of the theory.     

Switching design for exponential stability of a class of nonlinear hybrid time-delay systems

This paper addresses the exponential stability for a class of nonlinear hybrid time-delay systems. The system to be considered is autonomous and the state delay is time-varying. Using the Lyapunov functional approach combined with the Newton–Leibniz formula, neither restriction on the derivative of time-delay function nor bound restriction on nonlinear perturbations is required to design a switching rule for the exponential stability of nonlinear switched systems with time-varying delays. The delay-dependent stability conditions are presented in terms of the solution of algebraic Riccati equations, which allows computing simultaneously the two bounds that characterize the stability rate of the solution. A simple procedure for constructing the switching rule is also presented.     

Delay-range-dependent exponential stability criteria and decay estimation for switched Hopfield neural networks of neutral type

This paper is concerned with the problem of delay-range-dependent global exponential stability and decay estimation for a class of switched Hopfield neural networks (SHNNs) of neutral type. An average dwell time method is introduced into switched Hopfield neural networks. By constructing a new Lyapunov–Krasovskii functional and designing a switching law, some criteria are proposed for guaranteeing exponential stability for a given system, while the exponential decay estimation is explicitly developed for the states. A numerical example is provided to demonstrate the effectiveness of the main results.     

Exponential Ergodicity for SDEs Driven by <Emphasis Type="Italic">α</Emphasis>-Stable Processes with Markovian Switching in Wasserstein Distances

In this paper, we consider the ergodicity for stochastic differential equations driven by symmetric α-stable processes with Markovian switching in Wasserstein distances. Some sufficient conditions for the exponential ergodicity are presented by using the theory of M-matrix, coupling method and Lyapunov function method. As applications, the Ornstein-Uhlenbeck type process and some other processes driven by symmetric α-stable processes with Markovian switching are presented to illustrate our results. In addition, under some conditions, an explicit expression of the invariant measure for Ornstein-Uhlenbeck process is given.     

Stability of Delayed Markovian Switching Stochastic Neutral-type Reaction-diffusion Neural Networks

This paper is concerned about exponential stability in mean square of Markovian switching delayed reaction-diffusion neutral-type stochastic neural networks (RNSNNs). By Lyapunov function method, several novel stability criteria on exponential mean square stability of Markovian switching RNSNNs with time-varying delays are obtained. In the end, two examples are given to verify the feasibility of our findings.     

Stability of hybrid stochastic differential systems with switching and time delay

This article introduces a hybrid stochastic differential system with impulsive, switching and time-delay. Some stability criteria of p-moment global asymptotical stability, p-moment global exponential stability and mean square stability of this system are derived by using switching Lyapunov function approach, Itô formula, impulsive differential inequality method, and linear matrix equality techniques. Three examples are presented to demonstrate the efficiency of the obtained results.     

具有时变灰参数的随机时滞系统的p-阶矩指数鲁棒稳定性

利用灰矩阵的矩阵覆盖集的分解技术和Lyapunov函数法,研究了具有时变灰色参数的随机时滞系统的p-阶矩指数鲁棒稳定性问题,得到了该系统p-阶矩指数鲁棒稳定的时滞独立和时滞依赖的条件,并通过数值例子说明了判别条件的有效性和实用性.     

Stability analysis of impulsive delayed switched systems and applications

In this paper, a general class of impulsive delayed switched systems is considered. By employing the Lyapunov–Razumikhin method and some analysis techniques, we established several global asymptotic stability and global exponential stability criteria for the considered impulsive delayed switched systems, which improve and extend some recent works. As an application, the result of global exponential stability are used to study a class of uncertain linear switched systems with time‐varying delays. Several LMI‐based conditions are proposed to guarantee the global robust stability and global exponential stabilization. The designed memoryless state feedback controller can be easily checked by the LMI toolbox in Matlab. Moreover, the dwell time constraint is imposed for the switching law. Finally, two numerical examples and their simulations are given to show the effectiveness of our proposed results. Copyright © 2012 John Wiley & Sons, Ltd.     

New results of global robust exponential stability of neural networks with delays

In this paper, the global robust exponential stability of interval neural networks with delays is investigated. Employing homeomorphism techniques and Lyapunov functions, we establish some sufficient conditions for the existence, uniqueness, and global robust exponential stability of the equilibrium point for delayed neural networks. It is shown that the obtained results improve and generalize the previously published results.     

Stability of Linear Parameter Varying and Linear Switching Systems

We consider stability of families of linear time‐varying systems, that are determined by a set of time‐varying parameters which adhere to certain rules. The conditions are general enough to encompass on the one hand stability questions for systems that are frequently called linear parameter varying systems in the literature and on the other hand also linear switching systems, in which parameter variations are allowed to have discontinuities. Combinations of these two sets of assumptions are also possible within the framework studied here. Under the assumption of irreducibility of the sets of system matrices, we show how to construct parameter dependent Lyapunov functions for the systems under consideration that exactly characterize the exponential growth rate. It is clear that such Lyapunov functions do not exist in general. But every system of our class can be reduced to a finite number of subsystems for which irreducibility holds.     

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.     

Robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive stochastic systems with time-varying delay

This paper studies the problem of robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive stochastic systems with time-varying delay. The state variables on the impulses are assumed dependent on the present state variables as well as delayed state variables. Based on the Razumikhin techniques and Lyapunov functions, some robust mean-square exponential stability criteria are derived in terms of linear matrix inequalities. The results show that the system will stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous flows. Furthermore, the robust delayed-state-feedback controllers that mean-square exponentially stabilize the uncertain impulsive stochastic systems are proposed. Finally, several numerical examples are given to show the effectiveness of the results.     

Lyapunov Functions for General Nonuniform Dichotomies

For nonautonomous linear equations x′ = A(t)x, we give a complete characterization of the existence of exponential behavior in terms of Lyapunov functions. In particular, we obtain an inverse theorem giving explicitly Lyapunov functions for each exponential dichotomy. The main novelty of our work is that we consider a very general type of nonuniform exponential dichotomy. This includes for example uniform exponential dichotomies, nonuniform exponential dichotomies and polynomial dichotomies. We also consider the case of different growth rates for the uniform and the nonuniform parts of the dichotomy. As an application of our work, we establish in a very direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations.     

Admissibility for nonuniform exponential contractions

We study the relation between the notions of nonuniform exponential stability and admissibility. In particular, using appropriate adapted norms (which can be seen as Lyapunov norms), we show that if any of their associated Lp spaces, with p∈(1,∞], is admissible for a given evolution process, then this process is a nonuniform exponential contraction. We also provide a collection of admissible Banach spaces for any given nonuniform exponential contraction.     

Synchronization of different dimensional chaotic systems with time varying parameters, disturbances and input nonlinearities

In this paper, the problems of robust exponential generalized and robust exponential Q-S chaos synchronization are investigated between different dimensional chaotic systems. We consider the more practical and realistic cases when unknown time varying parameters with uncertainties, environmental disturbances, and nonlinearity of input control signals are present. The adaptive technique is employed to design the appropriate controllers and the validity of the proposed controllers are proved using Lyapunov stability theorem. Furthermore, numerical simulations are performed to show the efficiency of the presented scheme.     

Exponential stabilization for a class of hybrid systems with mixed delays in state and control

This paper proposes a switching design for the exponential stabilization problem of hybrid systems with mixed time-delays in both the state and control. By using an improved Lyapunov–Krasovskii functional, a memoryless switching controller for the exponential stabilization of the system is designed in terms of linear matrix inequalities. The approach also allows us to compute simultaneously the two bounds that characterize the exponential stability rate of the solution.