Impulsive Stochastic Differential Equations Research Articles

A note on generalized exponential stability of impulsive stochastic functional differential equations

This brief considers the generalized exponential stability of impulsive stochastic functional differential equations (ISFDEs). Initially, An extension of Halanay inequality is derived. Subsequently, by employing this generalized Halanay inequality, we establish the generalized exponential stability result for the given systems.

Almost sure exponential stability and stochastic stabilization of impulsive stochastic differential delay equations

In this paper, we mainly study the almost sure exponential stability of impulsive stochastic differential delay equations (ISDDEs) with bounded variable delays. The main technique is to compare ISDDEs with corresponding impulsive stochastic differential equations (ISDEs) without delay, to obtain the upper bound τ∗ of delays that ISDDEs can maintain stability by accurate calculation. The results show that if the corresponding ISDEs are almost surely exponentially stable, then the ISDDEs are also almost surely exponentially stable as long as these delays are less than τ∗. In addition, the stability theory established in this paper can be applied to noise stabilization based on sampled-data observations of a class of unstable impulsive systems. Taking impulsive Lurie systems (ILSs) as an example, we discuss the design of noise stabilization strategies.

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.

Stability of impulsive stochastic functional differential equations with delays

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.

Exponential contraction in impulsive stochastic differential equations: Analysis of exact and numerical solutions

Exponential contraction in impulsive stochastic differential equations: Analysis of exact and numerical solutions

Approximate controllability of second-order neutral stochastic differential evolution systems with random impulsive effect and state-dependent delay

<p>In this paper, we have discussed a class of second-order neutral stochastic differential evolution systems, based on the Wiener process, with random impulses and state-dependent delay. The system is an extension of impulsive stochastic differential equations, since its random effect is not only from stochastic disturbances but also from the random sequence of the impulse occurrence time. By using the cosine operator semigroup theory, stochastic analysis theorem, and the measure of noncompactness, the existence of solutions was obtained. Then, giving appropriate assumptions, the approximate controllability of the considered system was inferred. Finally, two examples were given to illustrate the effectiveness of our work.</p>

Some criteria on exponential stability of impulsive stochastic functional differential equations

This paper is mainly concerned with th moment exponential stability of impulsive stochastic functional differential equations. By using the Lyapunov direct method and mathematical induction, we obtain a new generalized impulsive Halanay inequality and some new criteria, which contain two scenarios regarding influence of impulses. On the one hand, when the system itself is unstable, a suitable unidirectional impulse or sequence of impulses can be added to make it stable. On the other hand, we discuss how much impulse perturbation the system can withstand without destabilizing the original system. Finally, two examples are illustrated to show the validity and universality of our results.

Stability of stochastic functional differential equations with impulses

The paper studies the stability of impulsive stochastic functional differential equations. At first, we obtained a new generalization of the Halanay inequality. Then, by applying the generalized Halanay inequality, the exponential stability results of impulsive stochastic functional differential equations are obtained.

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.

Ultimate boundedness of impulsive stochastic delay differential equations with delayed impulses

This paper is devoted to studying the ultimate boundedness of impulsive stochastic delay differential equations (SDDEs) with delayed impulses. Using suitable Lyapunov functionals and Itoˆ’s formula, some necessary criteria are met to guarantee the system’s pth moment global exponential ultimate boundedness (p-MGEUB). The results show that the unbounded stochastic delay differential equation can be transformed into a bounded system by impulses. Moreover, our results generalize and improve some results in the literature. Finally, we provide an illustration to clarify our findings.

Stability Analysis for a Class of Stochastic Differential Equations with Impulses

This paper is concerned with the problem of asymptotic stability for a class of stochastic differential equations with impulsive effects. A sufficient criterion on asymptotic stability is derived for such impulsive stochastic differential equations via Lyapunov stability theory, bounded difference condition and martingale convergence theorem. The results show that the impulses can facilitate the stability of the stochastic differential equations when the original system is not stable. Finally, the feasibility of our results is confirmed by two numerical examples and their simulations.

A NECESSARY CONDITION OF VIABILITY FOR IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION

Viability theory is a mathematical theory that offers mathematical metaphors of evolution of macrosystems arising in biology, economics, cognitive sciences, games, and similar areas, as well as in nonlinear systems of control theory. The viability problem has been studied by many authors by using various frameworks and techniques and is still one of the active directions of differential equations. The viability property in a stochastic framework was explored first by Aubin and Da Prato (Aubin & Da Prato, 1990). In this paper, we give a necessary condition for viability results of an impulsive stochastic functional differential equation driven by a fractional Brownian motion with Hurst parameter 1/2 < H <1.

Some results on the existence and stability of impulsive delayed stochastic differential equations with Poisson jumps

<abstract><p>This paper is concerned with the existence, uniqueness and exponential stability of mild solutions for a class of impulsive stochastic differential equations driven by Poisson jumps and time-varying delays. Utilizing the successive approximation method, we obtain the criteria of existence and uniqueness of mild solutions for the considered impulsive stochastic differential equations. Then, the exponential stability in the $ p $th moment of the mild solution is also devised for considered equations by establishing an improved impulsive-integral inequality, which improves some known existing ones. Finally, an example and numerical simulations are given to illustrate the efficiency of the obtained theoretical results.</p></abstract>

APPROXIMATE CONTROLLABILITY OF IMPULSIVE STOCHASTIC SYSTEMS DRIVEN BY ROSENBLATT PROCESS AND BROWNIAN MOTION

In this paper we consider a class of impulsive stochastic functional differential equations driven simultaneously by a Rosenblatt process and standard Brownian motion in a Hilbert space. We prove an existence and uniqueness result and we establish some conditions ensuring the approximate controllability for the mild solution by means of the Banach fixed point principle. At the end we provide a practical example in order to illustrate the viability of our result.

Stochastic fractional differential equations with generalized Caputo's derivative and impulsive effects

In this paper, impulsive stochastic fractional differential equations (ISFDEs) in Lp (p> 2) space are introduced. We present a general framework for finding solution for ISFDEs. Then, by using the Burkholder - Davis - Gundy inequality and Holder's inequality, we prove the existence and uniqueness of solution to ISFDE by fixed point theorem. We also investigate Lipschitz continuity of solutions with respect to initial values by using Gronwall inequality. Finally, we provide an application to illustrate the results we obtained.

Impulsive stochastic differential equations involving Hilfer fractional derivatives

Impulsive stochastic differential equations involving Hilfer fractional derivatives

Sensitivity analysis for a fractional stochastic differential equation with $$S^{p}$$-weighted pseudo almost periodic coefficients and infinite delay

In this paper, we study the parameter sensitivity analysis a class of fractional impulsive stochastic differential equation with \(S^{p}\)-weighted pseudo almost periodic coefficients and infinite delay in Hilbert spaces. Firstly, a more appropriate concept of piecewise \(S^{p}\)-weighted pseudo almost periodic in distribution for stochastic processes of class \(\infty \) is introduced. The existence of piecewise weighted pseudo almost periodic mild solutions in distribution is presented by using \(\alpha \)-order fractional resolvent operator theory, the stochastic analysis techniques and a fixed-point theorem. Secondly, the sensitivity properties of these infinite delay systems under non-Lipschitz conditions is also obtained. Finally, two examples are given for demonstration.

Quasi-Synchronization of Delayed Stochastic Multiplex Networks via Impulsive Pinning Control

The quasi-synchronization problems of the delayed multiplex networks with stochastic perturbations are investigated in this article. Different from previous studies on complete synchronization for the delayed multiplex networks, quasi-synchronization, which is more practical and more challenging than complete synchronization, is discussed. To adopt a better cost-saving control, an impulsive pinning scheme is designed. That is, the key nodes in the multiplex networks are pinned and controlled at impulsive instants. At the same time, a virtual leader is introduced. Based on the theory of stochastic impulsive differential equations and the Lyapunov–Krasovskii functional method, the sufficient conditions are derived for the error systems to converge into a bounded region in the sense of the mean square. Therefore, stochastic quasi-synchronization is guaranteed. Finally, the two-layer Newman–Watts (NW) small-world networks and two-layer Barabási–Albert (BA) scale-free networks are provided to illustrate the effectiveness of the theoretical results.

An Averaging Principle for Stochastic Fractional Differential Equations Driven by fBm Involving Impulses

In contrast to previous research on periodic averaging principles for various types of impulsive stochastic differential equations (ISDEs), we establish an averaging principle without periodic assumptions of coefficients and impulses for impulsive stochastic fractional differential equations (ISFDEs) excited by fractional Brownian motion (fBm). Under appropriate conditions, we demonstrate that the mild solution of the original equation is approximately equivalent to that of the reduced averaged equation without impulses. The obtained convergence result guarantees that one can study the complex system through the simplified system. Better yet, our techniques dealing with multi-time scales and impulsive terms can be applied to improve some existing results. As for application, three examples are worked out to explain the procedure and validity of the proposed averaging principles.

Exponential ultimate boundedness and stability of stochastic differential equations with impulese

AbstractThe paper mainly studies globally pth moment exponentially ultimate boundedness and pth moment exponential stability of impulsive stochastic functional differential equations. By using the Lyapunov direct method of Razumikhin‐type condition and the principle of comparison, some sufficient conditions for globally pth moment exponentially ultimate boundedness and globally pth moment exponential stability are presented. Theorems require the linear coefficients of the upper bound of Lyapunov differential operators are time‐varying functions; this generalizes the previous results. When the time delay is not considered in the system, a unified criterion is given to achieve boundedness and stability when the system is disturbed by stabilizing impulse and destabilizing impulse. It shows that the stochastic differential equation may be unbounded or unstable, and it can be bounded or stable by adding appropriate impulsive perturbation. Finally, we use two examples to illustrate the validity of our results.