Moment Exponential Stability Research Articles

Formula omitted]th Moment Exponential Stability of Switched Discrete-Time Stochastic Systems: A Multiple Lyapunov Functions Method

This paper studies the pth moment exponential stability of switched discrete-time stochastic systems with stable and unstable subsystems. By using multiple Lyapunov functions and mode-dependent average dwell time, we provide a sufficient condition for the pth moment exponential stability with fast convergence rate for deterministic switched discrete-time stochastic systems with stable and unstable subsystems. Obviously, our result has a weaker conservatism than those previous results where the effect of random factor was ignored. In addition, the criteria for stability and instability of discrete-time stochastic systems without switching are provided. Finally, the correctness of the results is verified by two numerical examples. To the best of our knowledge, it is the first time that the stability of discrete time switching system with random disturbance has been discussed.

Inequalities and pth moment exponential stability of impulsive delayed Hopfield neural networks

In this paper, the pth moment exponential stability for a class of impulsive delayed Hopfield neural networks is investigated. Some concise algebraic criteria are provided by a new method concerned with impulsive integral inequalities. Our discussion neither requires a complicated Lyapunov function nor the differentiability of the delay function. In addition, we also summarize a new result on the exponential stability of a class of impulsive integral inequalities. Finally, one example is given to illustrate the effectiveness of the obtained results.

Stability analysis of switched stochastic delay system with unstable subsystems

This paper investigates the stability of switched stochastic delay systems (SSDSs) with unstable subsystems. By using the multiple Lyapunov–Krasovskii functionals (MLKFs) method combined with the mode-dependent average dwell time of stable subsystems (MDADTSS) and the mode-dependent average dwell time of unstable subsystems (MDADTUS) method, SSDSs with unstable subsystems are analyzed, and several novel sufficient conditions for finite-time stability and pth moment exponential stability are given. Compared with the existing related works, our work considers the influences of random disturbance and delays on the stability of SSDSs. Finally, we provide two examples to illustrate the feasibility of this method.

Formula omitted] th moment exponential stability of neutral stochastic pantograph differential equations with Markovian switching

In this paper we focus on the pth moment exponential stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEwMS). By means of the Lyapunov method, we develop some sufficient conditions on the pth moment exponential stability for NSPDEwMS. We analyze two examples to show the interest of the main results.

Delay feedback stabilisation of stochastic differential equations driven by G-Brownian motion

This paper aims to design the feedback control based on past states to stabilise a class of nonlinear stochastic differential equations driven by G-Brownian motion. By building up the connection between the delay feedback control and the control function of no-delay, sufficient conditions are established in terms of size of the time delay ensuring both the pth moment exponential stability and quasi-surely exponential stability of the delay feedback controlled system. Moreover, methods for determining the upper bound of the length of the time delay and further designing the delay feedback controller are provided. A numerical example is presented to demonstrate our new theory.

Stability analysis of neutral stochastic delay differential equations via the vector Lyapunov function method

In this paper, we study the pth moment exponential stability (p-ES) and the almost sure exponential stability (a-ES) of neutral stochastic delay differential equations (NSDDEs). By using the vector Lyapunov function (VLF) method, we can prove that the global solution of NSDDEs exists when the linear growth condition is removed, and we also get some stability criteria for NSDDEs. In the presented stability conditions, the established L-operator differential inequality is based on the VLF and allows cross item to exist. An example is given to verify the correctness of the proposed results.

Stability for multi-links stochastic delayed complex networks with semi-Markov jump under hybrid multi-delay impulsive control

This paper is concerned with the problem of pth moment exponential stability for multi-links stochastic delayed complex networks with semi-Markov jump under hybrid multi-delay impulsive control. Different from the previous literature on multi-links systems, we take semi-Markov jump into account. The hybrid multi-delay impulses we addressed are composed of unstable delay-free impulses, stable delay-free impulses, unstable multi-delay impulses and stable multi-delay impulses. Average multi-delay impulsive gain is proposed to quantify the strength of the impulses. Serving as an extension of Halanay inequality, a novel multi-delay impulsive differential inequality is constructed. Combining the multi-delay impulsive differential inequality with Lyapunov method and graph theory, the sufficient criteria for pth moment exponential stability of the considered systems are given, which are closely related to average multi-delay impulsive gain, topological structure, stochastic disturbance strength and semi-Markov jump. Furthermore, the obtained theoretical results are applied to a class of stochastic delayed coupled oscillator systems. Finally, a corresponding numerical simulation is provided to demonstrate the effectiveness of the proposed theoretical results.

Stability Analysis of Discrete-Time Stochastic Delay Systems with Impulses

This paper is concerned with stability analysis of discrete-time stochastic delay systems with impulses. By using the sums average value of the time-varying coefficients and the average impulsive interval, two sufficient criteria for exponential stability of discrete-time impulsive stochastic delay systems are derived, which are more convenient to be applied than those Razumikhin-type conditions in previous literature. Both pth moment asymptotic stability and pth moment exponential stability are considered. Finally, two numerical examples to illustrate the effectiveness.

Stability equivalence among stochastic differential equations and stochastic differential equations with piecewise continuous arguments and corresponding Euler-Maruyama methods

In this paper, we consider the equivalence of the pth moment exponential stability for stochastic differential equations (SDEs), stochastic differential equations with piecewise continuous arguments (SDEPCAs) and the corresponding Euler-Maruyama methods EMSDEs and EMSDEPCAs. We show that if one of the SDEPCAs, SDEs, EMSDEs and EMSDEPCAs is pth moment exponentially stable, then any of them is pth moment exponentially stable for a sufficiently small step size h and τ under the global Lipschitz assumption on the drift and diffusion coefficients.

Stabilization by delay feedback control for highly nonlinear switched stochastic systems with time delays

AbstractThis paper firstly discusses the existence and boundedness of unique global solution for highly nonlinear switched stochastic systems (SSSs) with time delays, where the switching signal is deterministic switching signal. Although the existence and boundedness of unique global solution can be obtained under local Lipschitz condition and Khasminskii‐type condition, highly nonlinear SSSs with time delays may not be stable. Therefore, a delay feedback control function can be designed for studying the stability of highly nonlinear SSSs with time delays, whose coefficients do not satisfy the linear growth condition. By using the Mathematical induction, Lyapunov function approach and the average dwell time method, H∞ stability and the pth moment exponential stability are studied for highly nonlinear SSSs with time delays. An explanatory example is given to illustrate the effectiveness of the main results.

Asymptotic properties of solutions for impulsive neutral stochastic functional integro-differential equations

In this work, we are concerned with the asymptotic properties of solutions for an impulsive neutral stochastic functional integro-differential equation. By applying the theory of resolvent operators, the Banach fixed point principle theorem, and results on stochastic analysis, we study respectively the existence, uniqueness, and global attracting and quasi-invariant sets of mild solutions for the considered equation. We also derive some sufficient conditions of pth moment exponential stability and almost surely exponential stability of the mild solutions. An example is provided in the end to illustrate the applications of the obtained results.

Dynamics of the Exponential Population Growth System with Mixed Fractional Brownian Motion

This paper examines the dynamics of the exponential population growth system with mixed fractional Brownian motion. First, we establish some useful lemmas that provide powerful tools for studying the stochastic differential equations with mixed fractional Brownian motion. We offer some explicit expressions and numerical characteristics such as mathematical expectation and variance of the solutions of the exponential population growth system with mixed fractional Brownian motion. Second, we propose two sufficient and necessary conditions for the almost sure exponential stability and the kth moment exponential stability of the solution of the constant coefficient exponential population growth system with mixed fractional Brownian motion. Furthermore, we conduct some large deviation analysis of this mixed fractional population growth system. To the best of the authors’ knowledge, this is the first paper to investigate how the Hurst index affects the exponential stability and large deviations in the biological population system. It is interesting that the phenomenon of large deviations always occurs for addressed system when (1/2) < H < 1. Moreover, several numerical simulations are reported to show the effectiveness of the proposed approach.

Razumikhin-type theorem for pth exponential stability of impulsive stochastic functional differential equations based on vector Lyapunov function

This paper is concerned with the pth moment exponential stability of stochastic functional differential equations with impulses. Based on average dwell-time method, Razumikhin-type technique and vector Lyapunov function, some novel stability criteria are obtained for impulsive stochastic functional differential systems. Two examples are given to demonstrate the validity of the proposed results.

An Enlarged Framework of Event-Triggered Control for Stochastic Systems

This article aims to enlarge the applicability of event-triggered control for stochastic systems, and is particularly concerned with event-triggered stabilization for stochastic systems with control-dependent diffusion term. Roughly treating stochastic noises as unfavorable factors for system stability, the related works are based on Lyapunov-type conditions for ISS property with respect to sampling error as those in the nonstochastic context. The conditions imply the existence of a twice-differentiable Lyapunov function with negative-definite infinitesimal under continuous-time control, and indeed exclude many familiar stochastic systems with control-dependent diffusion term which are continuously stabilizable while no such Lyapunov function exists. As the main contribution of this article, an enlarged framework of event-triggered stabilization is established for stochastic systems with control-dependent diffusion term, starting from the system stability under continuous-time control, rather than from Lyapunov-type conditions for ISS property. Specifically, with moment exponential stability assumed under continuous-time control, elementary event-triggering mechanisms are proposed with an enforced minimum inter-execution interval, under which the event-triggered controller can render the system exponentially stable both in the sense of expectation and in the almost sure sense. Particularly, a distinctive analysis of closed-loop stability is performed via delicate comparison between the solutions under the event-triggered, and continuous-time controls. Within the framework, inclusive Lyapunov-type conditions incorporating the underlying positive effects of stochastic noises are presented for event-triggered exponential stabilization of the stochastic systems, where the candidate Lyapunov function involved is weakened to be nondifferentiable at the origin and, even, to allow non-negative infinitesimal under continuous-time control.

The pth moment exponential stability of uncertain differential equation with jumps

Under the axiom system of uncertainty theory, the paper mainly introduce the new definition of the pth moment exponential stability for uncertain differential equation with jumps. For illustrating the concept, some examples and counterexamples are given. Furthermore, we obtain a necessary and sufficient condition of stability in pth moment exponential for the linear uncertain differential equation with jumps. Also, the conclusion condition is illustrated very clearly by two examples.

Dynamic behaviours for semi-discrete stochastic Cohen-Grossberg neural networks with time delays

By using semi-discretization technique, a discrete analogue of stochastic Cohen-Grossberg neural networks is formulated. Firstly, the existence of pth mean almost periodic sequence solutions and pth moment global exponential stability of the above semi-discrete stochastic system are investigated with the help of Krasnoselskii’s fixed point theorem and some analysis techniques in stochastic theory. Secondly, the exponential stability and stochastic stabilization for a special discrete stochastic Cohen-Grossberg neural networks are studied. These findings show that stochastic disturbances and small discrete step length have negative effects on the existence of mean almost periodic solutions and moment exponential stability. But they have positive effects on the exponential stability and exponential non-stability for some special models, respectively. Besides, this paper also finds that some unstable neural networks should become exponentially stable by stochastic disturbances. In the end, some examples and computer simulations are given to demonstrate the effectiveness of the theoretical results.

Self-Triggered State-Feedback Control for Stochastic Nonlinear Systems With Markovian Switching

This paper deals with a state-feedback control scheme for nonlinear stochastic systems with Markovian switching. First, we present the results on the practically $ {p}$ -moment exponential stability with respect to an additional disturbance and the practically $ {p}$ -moment asymptotic stability relying on a specific event. However, elements in this event are hard to be observed directly. The main aim of this paper is to develop a self-triggered sampling rule to overcome this difficulty. By applying the improved monotone growth condition, Ito’s formula, Fubini’s theorem, Gronwall inequality, and comparison lemma, we establish a novel lemma to estimate the lower bound and upper bound of second-moment for state and error, respectively. Moreover, we also establish the practically asymptotic stability in mean square with the help of Jensen’s inequality technique, properties of $ {\mathcal {K}}$ -function and result on $ {p}$ -moment input-to-state stability. Furthermore, from Lipschitz continuity and monotonicity of functions, we obtain the value of the maximum triggering interval based on lasted-observed state. Finally, we give some remarks and discussions to show the significance of our results by comparing with those in the previous literature.

P-Moment Mittag–Leffler Stability of Riemann–Liouville Fractional Differential Equations with Random Impulses

Fractional differential equations with impulses arise in modeling real world phenomena where the state changes instantaneously at some moments. Often, these instantaneous changes occur at random moments. In this situation the theory of Differential equations has to be combined with Probability theory to set up the problem correctly and to study the properties of the solutions. We study the case when the time between two consecutive moments of impulses is exponentially distributed. In connection with the application of the Riemann–Liouville fractional derivative in the equation, we define in an appropriate way both the initial condition and the impulsive conditions. We consider the case when the lower limit of the Riemann–Liouville fractional derivative is fixed at the initial time. We define the so called p-moment Mittag–Leffler stability in time of the model. In the case of integer order derivative the introduced type of stability reduces to the p–moment exponential stability. Sufficient conditions for p–moment Mittag–Leffler stability in time are obtained. The argument is based on Lyapunov functions with the help of the defined fractional Dini derivative. The main contributions of the suggested model is connected with the implementation of impulses occurring at random times and the application of the Riemann–Liouville fractional derivative of order between 0 and 1. For this model the p-moment Mittag–Leffler stability in time of the model is defined and studied by Lyapunov functions once one defines in an appropriate way their Dini fractional derivative.

Delay Tolerance for Stable Stochastic Systems and Extensions

This article establishes a “robustness” type result, namely, delay tolerance for stable stochastic systems under suitable conditions. We study the delay tolerance for stable stochastic systems and delayed feedback controls of such systems, where the delay can be state-dependent or induced by the sampling-data. First, we consider systems with global Lipschitz continuous coefficients and show that when the original stochastic system without delay is pth moment exponentially stable, the system with small delays is still pth moment exponentially stable. In particular, when the pth moment exponential stability is based on Lyapunov conditions, we can obtain explicit delay bounds for moment exponential stability. Then, we consider a class of stochastic systems with nonglobal Lipschitz conditions and find a delay bound for almost sure and mean square exponential stability. As extension of the stability tolerance criteria, consensus, and tracking control of multiagent systems with measurement noises and nonuniform delays are studied.

Razumikhin method to stability of delay coupled systems with hybrid switching diffusions

Delay coupled systems have received a great deal of attention in recent years. However, delay coupled systems with hybrid switching diffusions (DCSHSD) have rarely been discussed and are still important. Thus, this paper is devoted to studying the stability of DCSHSD based on Razumikhin method. Then, combining Lyapunov method with graph theory, pth moment exponential stability of DCSHSD is studied and three kinds of stability criteria, including the Razumikhin-type theorem, the Lyapunov-type theorem and a coefficients-type theorem, are obtained. Wherein, an upper bound of Lyapunov exponent is estimated. In particular, as a practical application of our theoretical results, stability of delay coupled oscillators with hybrid switching diffusions is studied. Finally, a numerical example is provided to demonstrate the validity and feasibility of the theoretical results.