Impulsive Interval Research Articles

Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks

<p style='text-indent:20px;'>In this article, we present several results on Finite-Time Stability (FTS) of impulsive differential inclusion. In order to investigate the FTS problem, a new concept of Finite-Time Stable Function Pair (FTSFP) is proposed. By virtue of average impulsive interval and FTSFP, two unified criteria on FTS of impulsive differential inclusion are obtained, which are effective for both the destabilizing impulses and the stabilizing impulses. In addition, the settling-time depends not only on the initial value, but also on the information of impulsive sequence. As an extension, a delay-independent FTS result of impulsive delayed differential inclusion is presented. Finally, the obtained results are applied to study the FTS of discontinuous impulsive neural networks.

Stability of stochastic discrete-time piecewise homogeneous Markov jump systems with time delay and impulsive effects

This paper considers the stability of stochastic nonlinear discrete-time impulsive systems with time delay and Markov jumps. The transition probabilities of Markov jumps are finite piecewise homogeneous and the variations in the finite set are contingent on a higher-level transition probability matrix. By Lyapunov functional method and the discrete average impulsive interval approach, several novel stability criteria are obtained, which can loosen the constraint on impulsive intervals and thus reduce the conservativeness compared with previous results that take the upper/lower bound of all impulsive intervals. Also, our results are suitable for both stable impulses and unstable impulses. As illustrations, the obtained results are applied to stochastic impulsive neural networks and stochastic impulsive oscillator model, respectively. The simulations are also presented to support and validate the theoretical results.

Inner synchronisation of stochastic impulsive multi-links coupled systems via discrete-time state observations control

In this paper, the inner synchronisation of stochastic impulsive coupled systems with multi-links (SISM) is investigated. The feedback control based on discrete-time state observations, impulsive effects and multiple links are added to stochastic coupled systems simultaneously for the first time. Instead of the mean-square exponential synchronisation, the pth () moment exponential synchronisation is studied. A main theorem is proposed by average impulsive interval approach, graph theory and Lyapunov method to ensure the pth moment exponential synchronisation of SISM. Additionally, we apply the theoretical results to stochastic impulsive Rössler-like circuits with multi-links, which is the first attempt. Finally, a numerical example is provided to illustrate the effectiveness of the developed results.

Asymptotical Stability of Logic Dynamical Systems With Random Impulsive Disturbances

This article presents an investigation of the asymptotical set stability (ASS) of logic dynamical systems (LDSs) with impulsive disturbances at random instants. Unlike ordinary probabilistic Boolean networks, such an impulsive LDS is not Markovian in general. In this article, an impulsive LDS is described by using a hybrid index model and the sequence of intervals between adjacent impulsive instants is assumed to be independent and identically distributed. Under this assumption, the subsequence of a solution obtained by sampling at impulsive instants is a Markov chain. The initial distribution, the transition probability matrix, and the reachable set of the Markovian subsequence are calculated. Calculations of different invariant subsets are discussed and a necessary and sufficient condition for the convergence of finite Markov chains to subsets is obtained. Based on these results, necessary conditions for the ASS of impulsive LDSs in the hybrid and time domains, respectively, are obtained. In addition, we prove that the necessary conditions become sufficient if the impulsive intervals and the initial impulsive instant are with bounded expectations and variances. An example is provided to show the application of the proposed results to node synchronization of logic dynamical networks.

Impulsive-Interaction-Driven Synchronization in an Array of Coupled Neural Networks

This paper deals with the problem of globally exponential synchronization and bipartite synchronization of coupled neural networks with impulsive interactions. Impulsive interaction means that a number of neural networks only communicate with each other at impulsive instants, while the array of neural networks are independent from each other at the remaining time. The advantage of the scheme is that communication cost can be largely reduced when only discrete communication is required. Moreover the communication links between nodes can be either positive or negative at impulsive instants. Using the Lyapunov method combined with some mathematical analysis and average impulsive interval, some efficient criteria are obtained to guarantee synchronization of impulsive coupled neural networks. Finally, the validity of our theoretical results is demonstrated by two numerical examples.

Synchronization of nonlinearly coupled complex networks: Distributed impulsive method

Abstract This paper is devoted to discussing the exponential synchronization of a class of nonlinearly coupled complex networks (NCCNs) with time-varying delays and stochastic disturbance. In consideration of the complex networks would subject to certain impulse disturbances, a kind of distributed controller combining with pinning control and impulsive control schemes is designed. Based on the concept of average impulsive interval and comparison principles, sufficient conditions for achieving the exponential synchronization are derived. In addition, the exponential convergence velocities are obtained, respectively, with considering different functions of the impulsive inputs. Finally, a numerical simulation is presented to demonstrate the feasibility of the theoretical deduction.

Quasi sure exponential stabilisation of impulsive stochastic complex networks under a sublinear expectation framework

In this paper, we explore quasi sure exponential stabilisation of impulsive stochastic complex networks under a sublinear expectation framework – impulsive stochastic complex networks driven by G-Brownian motion (G-ISCNs). In contrast with the existing literature, by combining graph theory, Lyapunov method and average impulsive interval method, some sufficient conditions for quasi sure exponential stabilisation of G-ISCNs are obtained, which removes the commonly premised assumption that the system is moment stable. Moreover, in virtue of the method based on supremum and infimum of impulsive interval, quasi sure exponential stabilisation criterion of G-ISCNs is established under discussing the following three cases, respectively: (1) the continuous dynamics is stable; (2) the impulsive dynamics is stable; (3) the impulse is inactive. Then the theoretical results are applied to impulsive coupled oscillators systems driven by G-Brownian motion, and the corresponding criterion is gained, which is substantiated with a numerical example finally.

Robust finite-time control of impulsive positive systems under ℒ1-gain performance

This paper is concerned with robust finite-time ℒ1-gain control problem for impulsive positive systems (IPSs). By adopting the average impulsive interval technique, sufficient conditions ensuring the finite-time boundedness of IPSs under ℒ1-gain characterization are formulated. The design of a feedback controller is also addressed to make the closed-loop system be positive, finite-time bounded (FTB), and have ℒ1-gain characterization. Results are presented in the form of linear programming (LP) inequalities. Finally, a numerical example is given to demonstrate the efficiency of the proposed design.

Finite-Time Stability and Stabilization of Impulsive Stochastic Delayed Neural Networks With Rous and Rons

This paper mainly tends to investigate finite-time stability and stabilization of impulsive stochastic delayed neural networks with randomly occurring uncertainties (ROUs) and randomly occurring nonlinearities (RONs). Firstly, by constructing the proper Lyapunov-Krasovskii functional and employing the average impulsive interval method, several novel criteria for ensuring the finite-time stability of impulsive stochastic delayed neural networks are obtained by means of linear matrix inequalities (LMIs). Then, some conditions about the state feedback controller are derived to ensure the finite-time stabilization of impulsive stochastic delayed neural networks with ROUs and RONs. Finally, numerical examples are provided to demonstrate the effectiveness and feasibility of the proposed results.

A Unified Mean Square Consensus Criterion for Stochastic Multi-Agent Systems With ROUs and RONs Under Impulse Time Windows

This paper probes into the leader-following mean square consensus problem of stochastic multi-agent systems with randomly occurring uncertainties (ROUs) and randomly occurring nonlinearities (RONs) under impulse time windows (ITWs). A new concept named average left endpoint interval (ALEI) related to ITWs is proposed, which is inspired by the average impulsive interval. Based on ALEI and Lyapunov stability theory, a unified mean square consensus criterion for both stabilizing and interference impulses is attained. The introduction of ALEI allows for a larger upper bound of left endpoint interval and smaller lower bound of left endpoint interval, which makes the preset of ITWs more flexible and less conservative. The validity of the theoretical results is verified by the numerical simulation.

Constraint impulsive consensus of nonlinear multi-agent systems with impulsive time windows

In this paper, the constraint impulsive consensus problem of nonlinear multi-agent systems in directed network is discussed. Impulsive time windows are designed for solving consensus problem of multi-agent systems. Different from the traditional impulsive protocol with fixed impulsive intervals, the impulsive protocol with impulsive time windows, where the impulsive instants can be changed randomly, is more effective and flexible. In addition, saturation impulse is also considered to restrict the jumping value of impulse beyond the threshold. Based on algebraic graph theory, matrix theory, and convex combination analysis, some novel conditions of impulsive consensus have been proposed. Our main results indicate that constraint impulsive consensus of the multi-agent systems via impulsive time windows can be achieved if the nonlinear systems satisfy suitable conditions. Numerical simulations are presented to validate the effectiveness of theoretical results.

Input-to-state stability for nonlinear systems with stochastic impulses

In this study, the problem of input-to-state stability (ISS) is systematically investigated for nonlinear systems with stochastic impulses. First, the ISS problem is considered for systems, where impulsive strengths are assumed to be stochastic and impulsive intervals are confined by the average impulsive interval. Then, the ISS problem is analyzed for nonlinear systems with both stochastic impulsive intensity and density. Several different cases are considered, i.e., cases where the impulsive instants are satisfied by a renewal process, Poisson process, semi-Markov chain and Markov chain. An example of coordination of multi-agent systems is provided to illustrate the effectiveness of the proposed new stability criteria.

Finite time impulsive synchronization of fractional order memristive BAM neural networks

This paper is concerned with the finite time impulsive synchronization problem of fractional order memristive BAM neural networks (MBAMNNs) with switching jumps mismatch. By utilizing the properties of fractional calculus and comparison principle, a novel fractional order result about finite time impulsive stability is obtained. In the light of the double layer structure of fractional order MBAMNNs, two impulsive controllers are designed for the response fractional order MBAMNNs. By combing properties of Gamma functions and some analysis techniques of impulsive interval, several sufficient conditions are given which ensure finite time synchronization of fractional order MBAMNNs, and the upper bound of the setting time for synchronization is estimated, which is related to the fractional order of the system. Finally, numerical simulation is provided to demonstrate the effectiveness of the obtained results.

Bipartite formation problem of second-order nonlinear multi-agent systems with hybrid impulses

In this paper, bipartite formation of second-order nonlinear multi-agent systems (MASs) with hybrid impulses is investigated. At first, state-feedback controllers and impulsive protocols are designed for second-order MASs based on the definition of bipartite formation. Secondly, with the aid of average impulsive interval and average impulsive gain, the design criterion of bipartite formation control protocol is derived such that second-order nonlinear MASs with hybrid impulses can realize bipartite formation. Furthermore, adaptive control is also addressed for second-order MASs with unknown nonlinear terms and the corresponding result is derived. Finally, the validity of the theoretical results is illustrated by two numerical simulations.

A new Lyapunov-like functional approach to dwell-time dependent stability for impulsive systems

This paper investigates the dwell-time dependent stability for impulsive systems by employing a new Lyapunov-like functional that is of the second order in time t. In contrary to those built on [tk, t], a part of the impulsive interval [tk,tk+1], the Lyapunov-like functional is two-sided in the sense of employing the system information on [t,tk+1] as well as [tk, t]. To deal with the derivative of the two-sided Lyapunov-like functional, which involves integrals of the state and integrals coupled by [t,tk+1] and [tk, t], integral equations of the impulsive systems are introduced and an advanced inequality is employed. By the Lyapunov-like functional theory, new dwell-time dependent stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time are derived for periodic or aperiodic impulsive systems. The stability results turn out to be less conservative than some existing ones, which is illustrated by numerical examples.

Exponential synchronization for inertial coupled neural networks under directed topology via pinning impulsive control

This article focuses on the exponential synchronization of inertial coupled neural networks (ICNNs) with time-varying delays. An extended Halanay differential inequality, matrix decomposition theory and the average impulsive interval approach are used to derive some sufficient conditions. Furthermore, in the view of directed topology, the pinning impulsive control is designed so that the dynamical networks can achieve the exponential synchronization. Finally, some numerical examples are given to illustrate our theoretical results.

Effects of infinite occurrence of hybrid impulses with quasi-synchronization of parameter mismatched neural networks

This article is deeply concerned with the effects of hybrid impulses on quasi-synchronization of neural networks with mixed time-varying delays and parameter mismatches. Hybrid impulses allow synchronizing as well as desynchronizing impulses in one impulsive sequence, so their infinite time occurrence with the system may destroy the synchronization process. Therefore, the effective hybrid impulsive controller has been designed to deal with the difficulties in achieving the quasi-synchronization under the effects of hybrid impulses, which occur all the time, but their density of occurrence gradually decrease. In addition, the new concepts of average impulsive interval and average impulsive gain have been applied to cope with the simultaneous existence of synchronizing and desynchronizing impulses. Based on the Lyapunov method together with the extended comparison principle and the formula of variation of parameters for mixed time-varying delayed impulsive system, the delay-dependent sufficient criteria of quasi-synchronization have been derived for two separate cases, viz., Ta<∞ and Ta=∞. Finally, the efficiency of the theoretical results has been illustrated by providing two numerical examples.

A new unified input‐to‐state stability criterion for impulsive stochastic delay systems with Markovian switching

SummaryIn this paper, the problems of the input‐to‐state stability (ISS), the integral input‐to‐state stability (iISS), the stochastic input‐to‐state stability (SISS) and the eλt(λ&gt;0)‐weighted input‐to‐state stability (eλt‐ISS) are investigated for nonlinear time‐varying impulsive stochastic delay systems with Markovian switching. We propose one unified criterion for the stabilizing impulse and the destabilizing impulse to guarantee the ISS, iISS, SISS and eλt‐ISS for such systems. We verify that when the upper bound of the average impulsive interval is given, the stabilizing impulsive effect can stabilize the systems without ISS. We also show that the destabilizing impulsive signal with a given lower bound of the average impulsive interval can preserve the ISS of the systems. In addition, one criterion for guaranteeing the ISS of nonlinear time‐varying stochastic hybrid systems under no impulsive effect is derived. Two examples including one coupled dynamic systems model subject to external random perturbation of the continuous input and impulsive input disturbances are provided to illustrate the effectiveness of the theoretic results developed.

Synchronization for impulsive hybrid-coupled reaction-diffusion neural networks with time-varying delays

In this paper, the synchronization problem of impulsive hybrid-coupled reaction-diffusion neural networks with time-varying delays is investigated. By the formula for the variation of parameters and the comparison principle, several unified synchronization criteria are established. The proposed conditions are based on the inequalities involving the reaction-diffusion terms and the average impulsive interval, which are less conservative than related results in previous literature. The numerical example and simulations are provided to illustrate the effectiveness of proposed result.

Global Exponential Stability of Impulsive Delay Systems With Flexible Impulse Frequency

We establish global exponential stability theorems for impulsive delay differential systems with flexible impulse frequency. Since impulses often occur instantaneously, intermittently, or irregularly, it is possible that impulses occur with high frequency in some time-domains, and with low frequency in some others, which means that it is not always reasonable for existing results to make assumptions on common lower (or upper) bound of impulsive intervals in the analysis of stability. We propose a new approach which is based on impulsive control theory, the method of auxiliary functions, and linear matrix inequality technique to solve the problem of flexible impulse frequency. This paper mainly focuses on the stability analysis from the impulsive perturbations point of view, that is, the system subjects to destabilizing impulses. In this category, our developing results admit the existence of irregular impulsive intervals. It shows a fact that even there are some time intervals with high impulse frequency, the stability of delay system can still be guaranteed provided that the impulsive intervals with low impulse frequency satisfy certain conditions. Two numerical examples and their simulations are given to show the effectiveness of the proposed approach.