Pth Moment Stability Research Articles

Partial moment stability of stochastic differential equations with a general decay rate

In the first part of this paper, we discuss the problem of partial pth moment stability of stochastic differential equation with general decay rate when the origin is an equilibrium point. The results are obtained by using the method of Lyapunov and generalised integral inequalities. In the second part, we develop the problem of partial pth moment stability with general decay rate when the origin is not an equilibrium point. Some necessary numerical examples are also provided to show the effectiveness of the addressed method.

The [formula omitted]th moment stability of stochastic functional differential equations with mixed time-varying delay and its applications

This paper investigates the pth moment stability with general decay rate for stochastic mixed time-varying delay functional differential equations with semi-Markov switching and Lévy noise (SMDFDEs-SMS-LN). By employing multiple degenerate Lyapunov functionals, the non-negative martingale convergence theorem, and functional Itô’s formula, a criterion for assessing the pth moment stability with general decay rate is derived. Auxiliary functions with multiple time-varying coefficients are introduced to regulate the diffusion operator, which enhances the system’s adaptability to high-order nonlinear coefficients. These conditions are applied to a stochastic mixed-delay neural network with semi-Markov switching and Lévy noise (SMDNN-SMS-LN) and a scalar non-autonomous stochastic mixed time-varying delay functional differential equations with semi-Markov switching and Lévy noise system with non-global Lipschitz conditions. Using vertex method and linear matrix inequalities, the pth moment stability criteria with general decay rate for the above systems are respectively derived. Finally, the effectiveness of the obtained results is validated through two numerical examples.

Existence and exponential stability in pth moment of non-autonomous stochastic integro-differential equations

This study is concerned with the existence and exponential stability of solutions of non-autonomous neutral stochastic integro-differential equations with delay; the linear part of this equation is dependent on time and generates a linear evolution system. The obtained results are applied to some neutral stochastic integro-differential equations. These kinds of equations arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences. Finally, we provide an example to illustrate the results.

Stability analysis for positive Markov jump systems in [formula omitted]th moment sense: Necessary and sufficient conditions

This paper focuses on the stability analysis and synthesis of positive Markov jump systems (PMJSs) in pth moment sense, including stability, strong stability, observability, and detectability. First, by the ℋ-representation technique, the extended system method is constructed. With the help of the extended system method, stability analysis of PMJSs in pth moment sense are transformed into those of the extended systems. Furthermore, the necessary and sufficient conditions for stability, observability, and detectability in pth moment sense are obtained. Moreover, the definitions of pth moment strong stability (PMSS) and pth moment stability threshold (PMST) are given for the first time, and the sufficient/necessary conditions for these are derived, respectively. In addition, the problems of stabilization and strong stabilization in pth moment sense are considered. Finally, several examples are presented to show the effectiveness of the obtained results.

Formula omitted]th moment stability of discrete-time Markov jump systems by extended system method

This paper studies the pth moment stability of discrete-time Markov jump systems by utilizing the extended system method, which is composed of the operator spectrum technique and the ℋ-representation technique. Firstly, some extended systems are constructed by the operator spectrum technique and the ℋ-representation technique to transform the discrete-time Markov jump systems into the extended systems. Furthermore, the relationship in pth moment stability between the extended systems and the stochastic systems is discussed. Next, with the help of the extended systems, the necessary and sufficient conditions for the even-order moment stability, and several sufficient conditions/necessary conditions for the odd-order moment stability are obtained, respectively. Additionally, as applications of the extended system method, observability and detectability in pth moment sense are discussed, and the necessary and sufficient conditions for observability and detectability in even-order moment are obtained, respectively. Finally, several examples are given to illustrate the validity of the results.

The dynamical behavior of a class of stochastic vegetation models

We investigate the influence of random fluctuations in environmental parameters (e.g. rainfall and grazing rate) on the dynamical behavior of a class of ecological models for vegetation biomass in semi-arid regions. Firstly, we show the global existence and uniqueness of positive solutions for the number of vegetation biomass B(t). Then some sufficient conditions are presented for extinction and persistence of B(t) by using the stochastic linearization technique, the Hurwitz criterion and the Lyapunov function method. In particular, a threshold value R0s for the stochastic vegetation model is established in order to illustrate extinction and persistence of B(t). We also find that R0s=1 can be a warning signal for the emergence of desertification. In the case of R0s>1, we show that the stochastic vegetation system has a unique stationary distribution. Finally, we present conditions for persistence of B(t), and prove the pth moment stability and almost sure stability of the null solution in a general decay rate sense (including the exponential, polynomial and logarithmic rates) where the parameters in the stochastic vegetation system can also be dependent on time. These results are illustrated by numerical simulations.

Pth-moment stability of stochastic functional differential equations with Markovian switching and impulsive control

In this paper, we investigate the problem of pth-moment stability of stochastic functional differential equations with Markovian switching and impulsive control via comparison principle. Employing stochastic analysis theory and an impulsive delay differential inequality, we establish a new comparison principle for stochastic functional differential equations with Markovian switching and impulsive control. Using the comparison principle, we derive sufficient conditions for stochastic functional differential equations with Markovian switching and impulsive control by the stability of impulsive delay differential equations. An example is provided to show the effectiveness of the proposed results.

Stability analysis and synthesis of discrete-time semi-Markov jump singular systems

This paper investigates the σ-error pth 0) $ ]]> ( p > 0 ) -moment stability and synthesis problems for a class of discrete-time semi-Markov jump singular systems (DTSMJSSs). In view of the Lyapunov functions approach, we first derive the σ-error pth 0) $ ]]> ( p > 0 ) -moment stable behaviour of interested systems. Based on semi-Markov kernel approach and the techniques of eliminating power of matrices (EPMs), we then obtain several LMI conditions for σ-error mean square stability. Subsequently, due to the time-varying singular E-matrix in DTSMJSSs, two different forms of controller gain are exploited in order to design stabilising state-feedback strategies. In particular, computational analysis and comparison of the proposed control schemes are provided, which shows the numerical complexity of the proposed control schemes is vastly lower than that of the existing literature. Finally, an example has been performed to verify the obtained analytical results.

Two categories of new criteria of [formula omitted]th moment stability for switching and impulsive stochastic delayed functional differential equation with markovian switching

This note focuses on the pth moment stability for switching and impulsive stochastic delayed functional differential equation with Markovian switching. Under delay and Markovian switching, several novel stability criteria of pth moment input-to-state stability and pth moment integral input-to-state stability are enumerated by wielding Razumikhin/Krasovskii stability theory. Meanwhile, we show the correlation and differences for Razumikhin and Krasovskii method. The characteristic of the criteria requires that the time derivatives of the Razumikhin or Krasovskii functionals are enabled to be a function of time various, which can slacken guard in requirement. It is shown that, if the subsystems are stable in some interval but the discrete dynamics are not, the stochastic delayed hybrid system is stable. On the contrary, when the jump maps are stable with uncertain subsystems, the delayed-dependent impulses may overcome the destabilizing effects to achieve the stability. Finally, an example is given to show the effectiveness and practicability of the results.

Stabilization analysis of Markovian asynchronous switched systems with input delay and Lévy noise

The almost sure exponential stabilization and pth moment exponential stabilization for hybrid stochastic system (HSS) with control input delay, Lévy noise and asynchronous switching will be studied in this paper. Adequate conditions will be built to ensure the pth moment stability for the HSS which refer to the sizes of the delays by taking full advantage of Lyapunov functional approach. We investigate the almost sure exponential stability standard of nominal systems by using compound Lyapunov function method with modal dependence. In addition, we further study the almost sure exponential stability for HSS with Lévy noise and delayed feedback controller according to the comparison principle for sufficiently small delays as long as the nominal system is almost surely exponentially stable. An example and some simulations will propose to illustrate the results.

Stabilization of stochastic delayed systems: Event-triggered impulsive control

In this paper, we discuss the stability problem of impulsive stochastic delayed systems within the framework of event-triggered control strategy. Event-triggered impulsive control (ETIC) is proposed to guarantee system’s pth moment stability performance. In ETIC, the impulsive control inputs are generated by some pre-designed event triggering conditions and control inputs in the period of two triggered instants are not required. Then controller design problems for nonlinear impulsive stochastic delayed systems are addressed to show the validity of our theoretical results. Moreover, the involved non-Zeno problem is addressed in detail. Finally, two interesting examples with simulation results are performed to substantiate the validity and merits of our theoretical analysis.

Almost sure and pth moment stability of uncertain differential equations with time-varying delay

ABSTRACT Uncertain differential equations with time-varying delay are a special type of uncertain functional differential equation. Stability in measure and stability in mean for this type of differential equation have been investigated. For the sake of completeness, this article explores the other two types of stability. Firstly, the concepts of the almost sure stability and the pth moment stability of uncertain differential equations with time-varying delay are proposed. Next, sufficient conditions, in the form of theorems, for uncertain differential equations with time-varying delay to be stable almost surely and in pth moment are deduced. Finally, the relationships between several types of stability for uncertain differential equations with time-varying delay are discussed.

Stability of nonlinear stochastic Markov jump system with mode-dependent delays and applications

ABSTRACT The pth moment stability of stochastic Markov jump systems (SMJSs) with time-delays and polynomial growth was investigated in this paper. Both the involved time-delays and polynomial growth are mode dependent. By using the new property of the polynomial as well as the relation between the time-delay and the generator of the Markov-chain, sufficient pth moment stability criteria are proposed. Meanwhile, the robustness of the stable SMJSs is also considered, which reveals that how much perturbation the stable SMJSs can tolerate. As an application, the consensus of stochastic multi-agent systems with nonlinear dynamics is investigated. Finally, the effectiveness of our results is illustrated by examples.

Equivalence of [formula omitted]th moment stability between stochastic differential delay equations and their numerical methods

The equivalence of pth (p>0) moment stability between stochastic differential delay equations and their numerical methods is studied under the assumptions that the numerical methods are strongly convergent and have the bounded pth moment in the finite time.

Pth Moment Stability of a Stationary Solution for a Reaction Diffusion System with Distributed Delays

In this paper, the Sobolev embedding theorem, Holder inequality, the Lebesgue contrl convergence theorem, the operator norm estimation technique, and critical point theory are employed to prove the existence of nontrivial stationary solution for p-Laplacian diffusion system with distributed delays. Furthermore, by giving the definition of pth moment stability, the authors use the Lyapunovfunctional method and Kamke function to derive the stability of nontrivialstationary solution. Moreover, a numerical example illuminates the effectiveness of the proposed methods. Finally, an interesting further thought is put forward, which is conducive to the in-depth study of the problem.

Stability analysis of neutral stochastic differential delay equations driven by Lévy noises

This paper mainly analyzes the well-posedness, and the stability analysis for the global solution of neutral stochastic differential delay equations (NSDDEs) driven by Lévy noises. By using an integral lemma and a Lyapunov function approach, the existence and uniqueness theorem is proved. Then, by using the inequality technique and the stochastic analysis theory, the exponential stability in pth(p ≥ 2) moment of such equations is discussed. By using another integral lemma, and using the Baralat lemma as well as the stochastic analysis, the almost surely asymptotic stability is also studied. Finally, one example is given to check the effectiveness of the findings derived.

Pth Moment stability of fractional stochastic differential inclusions via resolvent operators driven by the Rosenblatt process and poisson jumps with impulses

ABSTRACT This study is concerned with the existence of mild solution for the analysis of the moment stability of fractional stochastic differential inclusions driven by the Rosenblatt process and Poisson jumps with impulses in a Hilbert space. The results are established by using the fixed point theorem, multi-valued functions, and fractional calculus. Finally, numerical simulation is implemented to verify the attained theoretical results.

Aperiodically intermittent control for stabilization of random coupled systems on networks with Markovian switching

Throughout this paper, the stabilization problem of random coupled systems on networks with Markovian switching (RCSNMS) is considered via aperiodically intermittent control. It is worth noting that aperiodically intermittent control is firstly utilized to stabilize random systems which are derived by more general noise. On the basis of Lyapunov method, graph theory together with some new stochastic analysis techniques, several stability criteria are derived. Different from most existing stability results on random systems in the literature which are mainly noise-to-state stability, we consider global asymptotical stability in probability and exponential stability in pth moment of RCSNMS. Then an application of the obtained results for a class of random coupled oscillators with Markovian switching via aperiodically intermittent control is presented. Finally, two numerical examples are concerned to demonstrate the validity and feasibility of the theoretical results.

Stability of Stochastic Functional Differential Systems with Semi-Markovian Switching and Lévy Noise and Its Application

This paper investigates the general decay stability on systems represented by stochastic functional differential equations with semi-Markovian switching and Levy noise (SFDEs-sMS-LN). Based on generalized multidimensional Ito’s formula and multiple Lyapunov functions, a new pth moment stability criterion with general decay rate is established. Meanwhile, as an applications of the presented stability criterion, we consider the stabilization problem of stochastic delayed neural networks with semi-Markovian switching and Levy noise (SDNN-sMS-LN). A vertex approach is proposed to design the controller in terms of binary diagonal matrices (BDMs) and linear matrix inequalities (LMIs). Finally, a numerical example is presented to demonstrate the effectiveness of the proposed results.

Intermittent control to stationary distribution and exponential stability for hybrid multi-stochastic-weight coupled networks based on aperiodicity

In this paper, the issue about the stationary distribution for hybrid multi-stochastic-weight coupled networks (HMSWCN) via aperiodically intermittent control is investigated. Specially, when stochastic disturbance gets to zero, the exponential stability in pth moment for hybrid multi-weight coupled networks (HMWCN) is considered. Under the framework of the Lyapunov method, M-matrix and Kirchhoff’s Matrix Tree Theorem in the graph theory, several sufficient conditions are derived to guarantee the existence of a stationary distribution and exponential stability. Different from previous work, the existing area of a stationary distribution is not only related to the topological structure of coupled networks, but also aperiodically intermittent control (the rate of control width and control duration). Subsequently, as an application to theoretical results, a class of hybrid multi-stochastic-weight coupled oscillators is studied. Ultimately, numerical examples are carried out to demonstrate the effectiveness of theoretical results and effects of the control schemes.