Moment Exponential Stability Research Articles

Stabilization of hybrid stochastic systems in the presence of asynchronous switching and input delay

This paper studies the pth moment and almost sure stabilization of a class of nonlinear hybrid stochastic systems with input delays in both the switching signal and the state. By using Lyapunov functional approach, sufficient conditions are established in terms of sizes of the delays ensuring the pth moment exponential stability of the systems. By developing a composite mode-dependent Lyapunov function approach, we derive novel almost sure exponential stability criteria for a delay-free hybrid stochastic system without imposing the stability condition on the subsystems. We further prove that the almost sure stability property of the hybrid system with delayed feedback controller is preserved as long as the corresponding delay-free system is almost sure exponential stabilization and the delays are small enough. Numerical examples demonstrate the theoretical results.

Stochastic Dynamics of Cholera Epidemic Model: Formulation, Analysis and Numerical Simulation

In this paper, we describe the two different stochastic differential equations representing cholera dynamics. The first stochastic differential equation is formulated by introducing the stochasticity to deterministic model by parametric perturbation technique which is a standard technique in stochastic modeling and the second stochastic differential equation is formulated using transition probabilities. We analyse a stochastic model using suitable Lyapunov function and Ito formula. We state and prove the conditions for global existence, uniqueness of positive solutions, stochastic boundedness, global stability in probability, moment exponential stability, and almost sure convergence. We also carry out numerical simulation using Euler-Maruyama scheme to simulate the sample paths of stochastic differential equations. Our results show that the sample paths are continuous but not differentiable (a property of Wiener process). Also, we compare the numerical simulation results for deterministic and stochastic models. We find that the sample path of SIsIaR-B stochastic differential equations model fluctuates within the solution of the SIsIaR-B ordinary differential equation model. Furthermore, we use extended Kalman filter to estimate the model compartments (states), we find that the state estimates fit the measurements. Maximum likelihood estimation method for estimating the model parameters is also discussed.

Convergence and stability of the one-leg θ method for stochastic differential equations with piecewise continuous arguments

The equivalent relation is established here about the stability of stochastic differential equations with piecewise continuous arguments(SDEPCAs) and that of the one-leg ? method applied to the SDEPCAs. Firstly, the convergence of the one-leg ? method to SDEPCAs under the global Lipschitz condition is proved. Secondly, it is proved that the SDEPCAs are pth(p 2 (0; 1)) moment exponentially stable if and only if the one-leg ? method is pth moment exponentially stable for some sufficiently small step-size. Thirdly, the corollaries that the pth moment exponential stability of the SDEPCAs (the one-leg ? method) implies the almost sure exponential stability of the SDEPCAs (the one-leg ? method) are given. Finally, numerical simulations are provided to illustrate the theoretical results.

Almost sure exponential stability of stochastic differential delay equations

This paper mainly studies whether the almost sure exponential stability of stochastic differential delay equations (SDDEs) is shared with that of the stochastic theta method. We show that under the global Lipschitz condition the SDDE is pth moment exponentially stable (for p 2 (0; 1)) if and only if the stochastic theta method of the SDDE is pth moment exponentially stable and pth moment exponential stability of the SDDE or the stochastic theta method implies the almost sure exponential stability of the SDDE or the stochastic theta method, respectively. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same results. Hence, our new theory enables us to consider the almost sure exponential stability of the SDDEs using the stochastic theta method, instead of the method of Lyapunov functions. That is, we can now perform careful numerical simulations using the stochastic theta method with a sufficiently small step size ?t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ? (0,1), we can then deduce that the underlying SDDE is almost sure exponentially stable. Our new theory also enables us to show the pth moment exponential stability of the stochastic theta method to reproduce the almost sure exponential stability of the SDDEs.

Asymptotic properties of a stochastic Gilpin–Ayala model under regime switching

In this paper, a stochastic Gilpin–Ayala population model with regime switching and white noise is considered. All parameters are influenced by stochastic perturbations. The existence of global positive solution, asymptotic stability in probability, pth moment exponential stability, extinction, weak persistence, stochastic permanence and stationary distribution of the model are investigated, which generalize some results in the literatures. Moreover, the conditions presented for the stochastic permanence and the existence of stationary distribution improve the previous results.

Stability of hybrid stochastic functional differential equations

In this paper, a class of hybrid stochastic functional differential equations is investigated. Under some suitable assumptions, the pth moment stability, asymptotic stability and exponential stability are discussed by means of constructing an auxiliary functional differential equation and using the comparison principle. The proposed criteria removes some conditions in some earlier publications. Moreover, two examples and simulations are given to illustrate our results.

Implicit numerical solutions to neutral-type stochastic systems with superlinearly growing coefficients

In this paper, our main aim is to investigate the stability and strong convergence of an implicit numerical approximations for neutral-type stochastic differential equations with superlinearly growing coefficients. After providing moment boundedness and exponential stability for the exact solutions, we show that the backward Euler–Maruyama numerical method preserves stability and boundedness of moments, and the numerical approximations converge strongly to the true solutions for sufficiently small step size.

Novel aperiodically intermittent stability criteria for Markovian switching stochastic delayed coupled systems.

This paper concerns th moment exponential stability of stochastic coupled systems with multiple time-varying delays, and Markovian switching topologies via intermittent control. Compared with previous research results, the mathematical model of this kind of stochastic coupled systems with multiple time-varying delays and Markovian switching topologies is studied for the first time. The intermittent control designed in this paper is aperiodical, which is more general in practice. Moreover, the restriction between control width and time delays is removed. By constructing a new differential inequality on delayed dynamical systems with Markovian switching topologies and combining the graph-theoretic approach with M-matrix theory, two sufficient criteria are derived to guarantee th moment exponential stability of systems. Moreover, the exponential convergence rate has a close relationship with the maximum ratio of the rest width to the aperiodical time span (the sum of the control width and the rest width). Finally, we employ the theoretical results to study the exponential stability of stochastic coupled oscillators with multiple time-varying delays and Markovian switching topologies. Meanwhile, a numerical example is presented to illustrate the effectiveness and feasibility of the proposed results.

The pth moment exponential stability and almost surely exponential stability of stochastic differential delay equations with Poisson jump

This paper is devoted to study a class of nonlinear stochastic differential delay equations with Poisson jump. In comparison to the Brownian motion, the jump leads to the discontinuity of sample paths, which makes the analysis more difficult. We first introduce the local Lipschitz condition and a new nonlinear growth condition, which is weaker than those in the previous literature. Then by virtue of Lyapunov function and semi-martingale convergence theorem, we prove that the considered stochastic system has a unique global solution. Moreover, we also investigate the pth moment exponential stability and the almost surely exponential stability of solutions. Finally, an example is given to illustrate the effectiveness of our theoretical results.

A New Stability Criterion for Neutral Stochastic Delay Differential Equations with Markovian Switching

In this short paper, a new stability theorem for neutral stochastic delay differential equations with Markovian switching is investigated by applying stochastic analysis technique and Razumikhin stability approach. A novel criterion of thepth moment exponential stability is derived for the related systems. The feature of the criterion shows that the estimated upper bound for the diffusion operator of Lyapunov function is allowed to be indefinite, even if to be unbounded, which can loosen the constraints of the existing results. Last, an example is provided to illustrate the usefulness and significance of the theoretical results.

Discrete Razumikhin-type stability theorems for stochastic discrete-time delay systems

By using the Razumikhin-type technique, for stochastic discrete-time delay systems, this paper establishes the discrete Razumikhin-type theorems on the pth moment stability, the global pth moment stability and the pth moment exponential stability, respectively. The almost sure exponential stability is also investigated by using the pth moment exponential stability and the Borel–Cantelli lemma. As the applications of t he established theorems, stability of a special class of stochastic discrete-time delay systems, synchronization of the stochastic discrete-time delay dynamical networks and stabilization of a stochastic discrete-time linear delay time invariant system are examined.

Stability of a class of neutral stochastic functional differential equations with Markovian switching

This study investigates the stability of a class of neutral stochastic functional differential equations with Markovian switching. Some novel stability criteria are first established, including boundedness, pth moment exponential stability and almost sure exponential stability, based on multiple Lyapunov functions, generalised Ito formula and non-negative semi-martingale convergence theorem. Concretely, the authors generalise the existing results under the essential neutral term and improve the diffusion operators from being controlled by two auxiliary functions to other multiple auxiliary functions with not only constant coefficients but also time-varying coefficients. Some numerical examples are presented to illustrate the effectiveness of the obtained results.

Formula omitted]th moment exponential stability of highly nonlinear neutral pantograph stochastic differential equations driven by Lévy noise

This paper is concerned to the pth moment exponential stability of highly nonlinear neutral pantograph stochastic differential equations driven by Lévy noise (NPSDE-LN). In the paper, the system model investigated in previous literature is generalized, exponential stability theorem and criterion for the underlying NPSDE-LN are directly established by the Lyapunov stability theory, stochastic analysis, and inequality technique by virtue of the drift dynamics f, diffusive dynamics g and h explicitly.

Stability analysis of stochastic delay differential equations with Lévy noise

This paper is concerned with pth moment exponential stability problem for a class of stochastic delay differential equations driven by Lévy processes. Several new stability theorems are obtained by developing a method—proof by contradiction. Moreover, the results are applied to investigate the pth moment exponential stability of stochastic neural networks with Lévy noise. In particular, the time-varying delay in our results is not required to be differentiable, even not continuous. The obtained results improve greatly some previous works given in the literature. In particular, our method can easily correct the incorrect proofs appeared in two recent papers. Finally, two examples are provided to show the effectiveness of the theoretical results.

Exponential stability for generalized stochastic impulsive functional differential equations with delayed impulses and Markovian switching

This paper is concerned with exponential stability for a class of generalized stochastic impulsive functional differential equations with delayed impulses and Markovian switching. A novel subsequence approach of the impulsive and switching time sequence is introduced to cope with the impulsive control problem with large and small delays. Based on the stochastic Lyapunov function and Razumikhin technique, a dwell time bound and related criteria are established to ensure the pth moment exponential stability, almost surely exponential stability and uniform stability of the trivial solutions. The main advantage of the proposed algorithm lies in that the delay bound and parameters are not necessarily required, which are commonly used to restrict the dwell-time bound and the decay rate of Lyapunov function. Finally, two examples are performed to demonstrate the usefulness of the main results.

On p-th Moment Exponential Stability for Stochastic Cellular Neural Networks with Distributed Delays

In this paper, a class of stochastic cellular neural networks with distributed delays are investigated. With the help of the method of variation parameter and inequality techniques, some sufficient conditions for the p-th moment exponential stability of the system are established. An example is given to illustrate the feasibility and effectiveness of our main results. Our results obtained in this paper improve and generalize some earlier works reported in the literature.

Exponential stability of stochastic cellular neural networks with mixed delays

ABSTRACTIn this paper, almost sure exponential stability and pth moment exponential stability of stochastic cellular neural networks with mixed delays are investigated. Employing the methods of stochastic analysis, the Lyapunov’s method, and useful inequality techniques, a sufficient condition ensuring the almost sure exponential stability and pth moment exponential stability is obtained. Two examples are given to illustrate this sufficient condition.

Pth moment and almost sure exponential stability of impulsive neutral stochastic functional differential equations with Markovian switching

ABSTRACTIn this paper, the problems on the pth moment and the almost sure exponential stability for a class of impulsive neutral stochastic functional differential equations with Markovian switching are investigated. By using the Lyapunov function, the Razumikhin-type theorem and the stochastic analysis, some new conditions about the pth moment exponential stability are first obtained. Then, by using the Borel–Cantelli lemma, the almost sure exponential stability is also discussed. The results generalise and improve some results obtained in the existing literature. Finally, two examples are given to illustrate the obtained results.

Moment exponential stability of stochastic nonlinear delay systems with impulse effects at random times

SummaryIn this paper, we investigate the pth moment exponential stability for a class of impulsive stochastic functional differential equations with impulses at random times. The impulsive times considered in this paper are random times that are different from those investigated in the existing literature. By using the stochastic process theory, stochastic analysis theory, Razumikhin technique, and Lyapunov method, we obtain some new criteria of the pth moment exponential stability for the related system. Finally, some examples are provided to show the effectiveness of the theoretical results.

Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach

In this article, we discuss a class of impulsive stochastic function differential equations driven by $G $-Brownian motion with delayed impulsive effects ($G $-DISFDEs, in short). Some sufficient conditions for $p$-th moment exponential stability of $G $-DISFDEs are derived by means of $G $-Lyapunov function method, average impulsive interval approach and Razumikhin-type conditions. An example is provided to show the effectiveness of the theoretical results.