Impulsive Stochastic Research Articles

Exponential Stability of Impulsive Stochastic Neutral Neural Networks with Lévy Noise Under Non-Lipschitz Conditions

In this paper, the exponential stability issue of stochastic impulsive neutral neural networks driven by Lévy noise is explored. By resorting to the Lyapunov-Krasovskii function that involves neutral time-delay components, the properties of the Lévy process, as well as various inequality approaches, some sufficient exponential stability criteria in non-Lipschitz cases are obtained. Besides, the achieved results depend on the time-delay, noise intensity, and impulse factor. At the end of the paper, two numerical examples with simulations are presented to demonstrate the effectiveness and feasibility of the addressed results

Existence and Hyers–Ulam Stability for Random Impulsive Stochastic Pantograph Equations with the Caputo Fractional Derivative

In this paper, we study the existence, uniqueness and Hyers–Ulam stability of a class of fractional stochastic pantograph equations with random impulses. Firstly, we establish sufficient conditions to ensure the existence of solutions for the considered equations by applying Schaefer’s fixed point theorem under relaxed linear growth conditions. Secondly, we prove the solution for the considered equations is Hyers–Ulam stable via Gronwall’s inequality. Moreover, the previous literature will be significantly generalized in our paper. Finally, an example is given to explain the efficiency of the obtained results.

Pth Moment Exponential Stability of Impulsive Stochastic Differential Equations with Unbounded Delays

Stochastic impulsive differential equations with Markovian switching have been applied extensively in various areas including ecology systems, neural networks, and control systems, and stability analysis is one fundamental premise of their applications. For two categories of Markovian switched impulsive stochastic differential functional equations with unbounded delays, this paper investigates the pth moment exponential stability by adopting stochastic Lyapunov stability theory and stochastic analysis approach. Several criteria on pth moment exponential stability have been acquired. In the proposed model, the time-varying coefficients and the hybrid impulsive effects are considered simultaneously. It can be seen that the criteria derived in this paper are more concise and the conditions are easier to verify compared with those existing results based on Razumikhin theory. Finally, two examples are illustrated to show the effectiveness of the theoretical findings.

Observer-Based Integral Sliding Mode Control With Switching Gains for Fuzzy Impulsive Stochastic Systems

Observer-Based Integral Sliding Mode Control With Switching Gains for Fuzzy Impulsive Stochastic Systems

Convergence Theorems for Stochastic Impulsive Systems with Application to Discrete-Time Stochastic Feedback Control

Convergence Theorems for Stochastic Impulsive Systems with Application to Discrete-Time Stochastic Feedback Control

Existence and Controllability Results for Impulsive Stochastic Integrodifferential Systems with State-Dependent Delay

Existence and Controllability Results for Impulsive Stochastic Integrodifferential Systems with State-Dependent Delay

Novel Criteria on Finite-Time Stability of Impulsive Stochastic Nonlinear Systems

Dear Editor, This letter considers the finite-time stability (FTS) problem of generalized impulsive stochastic nonlinear systems (ISNS). By employing the stochastic Lyapunov and impulsive control approach, some novel criteria on FTS are presented, where both situations of stabilizing and destabilizing impulses are considered. Furthermore, new impulse-dependent estimation strategies of stochastic settling time (SST) are proposed. These estimation strategies establish quantitative relationships between the impulsive effects and the mathematical expectation of SST, which can directly assess the influence of impulses on the system performance. Finally, an example is given to validate the effectiveness of the presented results.

Open-source Software Reliability Modeling with Stochastic Impulsive Differential Equations

In reality, sudden updates of software, attacks of hackers, influence of the Internet market, etc. can cause a surge in the number of open-source software (OSS) faults (this moment is the time when impulse occurs), which results in impulsive phenomenon. For the existing software reliability models, dynamic process of software fault is considered to be continuous when assessing reliability, but continuity of the process can be disrupted with appearance of random impulses. Thus, to more accurately assess software reliability, we proposed an OSS reliability model with SIDE. In the model, dynamic process of software fault is divided into a continuous and a skipped part, described the continuous part of the process with SDE, and described destruction of the continuity caused by unpredictable random events with random impulses. Finally, the proposed model is verified with two datasets from real OSS project, and the results show that the proposed model is more in line with reality and has better fitting effect than the existing models.

On ν-Level Interval of Fuzzy Set for Fractional Order Neutral Impulsive Stochastic Differential System

The main concern of this paper is to investigate the existence and uniqueness of a fuzzy neutral impulsive stochastic differential system with Caputo fractional order driven by fuzzy Brownian motion using fuzzy numbers with bounded ν-level intervals that are convex, normal and upper-semicontinuous. Fuzzy Itô process, Grönwall’s inequality and the Banach fixed-point theorem are employed to probe the local and global existence. An analytical example is provided to examine the theoretical results.

Stability Analysis for Impulsive Stochastic Time-Varying Systems

The average impulsive interval is widely used to describe the frequency of impulsive occurrence (FIO), where the occurrence number of impulses is bounded by a linear function of time interval length. However, the linear relationship may insufficiently or excessively characterize the occurrence number of impulses to stabilize impulsive time-varying systems. In this article, the impulsive density is introduced to describe a time-varying FIO, such that the occurrence number of impulses can be characterized more explicitly. Under the impulsive density, the asymptotical stability is considered for impulsive stochastic time-varying systems, where the continuous dynamics of systems, impulsive strengths, and instants are all assumed to be time-varying. In addition, the exponential stability is also investigated for impulsive stochastic time-varying systems with time-delay, which can extend some existing results. Two examples, including one example of the consensus for impulsive time-varying multiagent systems with time-delay, are presented to demonstrate the effectiveness of the proposed results.

Asymptotical Stability and Exponential Stability in Mean Square of Impulsive Stochastic Time-Varying Neural Network

The effect of impulse on stability of neural network is evident not only in performance, that is, impulsive control and impulsive interference. The amount of impulse has a certain impact on stability of neural network. Unlike traditional average impulsive interval, a new strategy is applied in this paper, namely, impulsive density. Based on this strategy, by constructing Lyapunov function, we establish sufficient conditions for mean square asymptotical stability of impulsive stochastic time-varying neural network without time delay. As well as, under this strategy and uniformly asymptotically stable function, by combining trajectory based approach and improved Razumikhin method, mean square exponential stability criterion of impulsive stochastic time-varying neural network with time delay is established. Finally, to demonstrate the viability of our theoretical findings, some instances are provided.

Practical Exponential Stability of Impulsive Stochastic Food Chain System with Time-Varying Delays

This paper studies the practical exponential stability of an impulsive stochastic food chain system with time-varying delays (ISOFCSs). By constructing an auxiliary system equivalent to the original system and comparison theorem, the existence of global positive solutions to the suggested system is discussed. Moreover, we investigate the sufficient conditions for the exponential stability and practical exponential stability of the system, which is given by Razumikhin technique and the Lyapunov method. In addition, when Razumikhin’s condition holds, the exponential stability and practical exponential stability of species are independent of time delay. Finally, numerical simulation finds the validity of the method.

Approximate Controllability of Non-Instantaneous Impulsive Stochastic Evolution Systems Driven by Fractional Brownian Motion with Hurst Parameter H∈(0,12)

This paper initiates a study on the existence and approximate controllability for a type of non-instantaneous impulsive stochastic evolution equation (ISEE) excited by fractional Brownian motion (fBm) with Hurst index 0<H<1/2. First, to overcome the irregular or singular properties of fBm with Hurst parameter 0<H<1/2, we define a new type of control function. Then, by virtue of the stochastic analysis theory, inequality technique, the semigroup approach, Krasnoselskii’s fixed-point theorem and Schaefer’s fixed-point theorem, we derive two new sets of sufficient conditions for the existence and approximate controllability of the concerned system. In the end, a concrete example is worked out to demonstrate the applicability of our obtained results.

Stability of stochastic delayed semi-Markov jump systems with stochastic mixed impulses: A novel stochastic impulsive differential inequality

In this paper, the pth moment exponential stability of stochastic delayed systems is studied. It is worth mentioning that semi-Markov jump and stochastic mixed impulses are considered simultaneously in the stochastic delayed systems. Besides, a new impulsive differential inequalities with semi-Markov jump and stochastic mixed impulses are established. The method of graph theory, stochastic analysis technique and Lyapunov method are cleverly combined, which contributes to propose stability criteria for stochastic delayed semi-Markov jump systems with stochastic mixed impulses. Finally, the theoretical results are applied to a kind of oscillator systems. The simulation results verify the effectiveness of the theoretical results.

Delay-Dependent Stability of Impulsive Stochastic Systems with Multiple Delays

This paper associates with stability analysis of linear impulsive stochastic delay systems (ISDSs). Although many conclusions about the stability of ISDSs have been obtained based on Lyapunov’s method, relatively few research theories about delay-dependent stability with less conservativeness have been established. Therefore, we introduce an appropriate Lyapunov-Krasovskii functional (LKF) to work out this problem, and a novel delay-dependent exponential stability theorem is first deduced. On the other hand, when mean-square stability is considered, we present delay-dependent stability conditions, it is of interest to note that the proposed conditions do not depend on the size of delays in the diffusion term, which solves the problems of determining the mean-square stability of ISDSs for which the diffusion term delays are not available. In the end, two numerical examples are carried out to verify the feasibility of our conclusions.

Improved Input-to-State Stability Analysis of Impulsive Stochastic Systems

This article is concerned with the input-to-state stability (ISS) and stochastic input-to-state stability (SISS) of impulsive stochastic nonlinear systems. By constructing a general SISS-Lyapunov function, which takes the exponential SISS-Lyapunov function as a special case, the ISS criteria are obtained, respectively, for destabilizing and stabilizing impulses. It should be stressed that we impose the average dwell-time (ADT) condition for destabilizing impulses and reverse average dwell-time (RADT) condition for stabilizing impulses to restrain the occurrence of impulses, which extends the fixed dwell-time condition in recent literature. Moreover, the ADT and RADT conditions in this article correspond to the average impulsive interval method in most existing results under the exponential ISS-Lyapunov condition. As a subsequent result, the ISS and SISS of impulsive stochastic interconnected systems on networks are also studied by the Lyapunov method and graph-theoretic technique. Finally, several illustrative examples together with their numerical simulations are provided to demonstrate the theoretical results.

Stability of Impulsive Stochastic Delay Systems with Markovian Switched Delay Effects

In this paper, we investigate the pth moment exponential stability of impulsive stochastic delay systems with Markovian switched delay effects. The model we consider here is rather different from the models in the existing literature. In particular, the delay is a Markov chain, which is quite different from the traditional deterministic delay. By using the Markov chain theory, stochastic analysis theory, Razumikhin technology and the Lyaponov method, we derive a criterion of pth moment exponential stability for the suggested system. Finally, an example is provided to illustrate the effectiveness of the obtained result.

Stability for multi-linked stochastic delayed complex networks with stochastic hybrid impulses by Dupire Itô’s formula

In this paper, the mean-square exponential stability is investigated for multi-linked stochastic delayed complex networks with stochastic hybrid impulses. Distinct from the existing literature, we study the MSDCNs on the basis of the multi-linked stochastic functional differential equations that consider the impact of a certain past interval on the present. Moreover, the stochastic hybrid impulses we discuss possess stochastic impulsive moments and impulsive gain, which make the impulses fit better to the real-world demands for control. Also, a novel concept of average stochastic impulsive gain is proposed to measure the intensity of the stochastic hybrid impulses. By the use of Dupire Itô’s formula, based on Lyapunov method, graph theory and stochastic analysis techniques, two sufficient criteria for the mean-square exponential stability are derived, which are closely related to average stochastic impulsive gain, stochastic disturbance strength as well as the topological structure of the network itself. Finally, an application about neural networks is discussed and corresponding numerical example is presented to demonstrate the feasibility and effectiveness of the theoretical results.

Asymptotic Properties of the Solutions to a Class of Perturbed Stochastic Impulsive Functional Differential Equations

The asymptotic properties of the solutions to a class of perturbed stochastic impulsive functional differential equations were investigated. Through comparison of the solution to the perturbed equation with the solution to the corresponding unperturbed one, the sufficient conditions for these solutions to be close in a finite time interval were derived. Then, when small perturbations approach zero and the length of the time interval approaches infinity, the 2 solutions will still be close to each other. Finally, an example illustrates the effectiveness of the results.

Almost Periodic Solutions to Impulsive Stochastic Delay Differential Equations Driven by Fractional Brownian Motion With 12 < H < 1

In this article, we study the existence and uniqueness of square-mean piecewise almost periodic solutions to a class of impulsive stochastic functional differential equations driven by fractional Brownian motion. Moreover, the stability of the mild solution is obtained. To illustrate the results obtained in the paper, an impulsive stochastic functional differential equation driven by fractional Brownian motion is considered.