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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 p th moment asymptotic stability and p th moment exponential stability are considered. Finally, two numerical examples to illustrate the effectiveness.


Introduction
Over the past decades, the impulsive phenomena have been intensively investigated since their significance and applications in areas such as economics, mechanics, chemical, biological phenomena, population dynamics, see other works and the references therein [1][2][3][4][5][6]. On the other hand, time delays occur frequently in many evolution processes and it is the inherent feature of many physical processes. Therefore, the study of impulsive delay systems has attracted great attention over the past few years [7][8][9][10][11]. For example, in [7], the authors have studied the pth moment exponential stability of a class of impulsive delay stochastic functional differential systems, by using the Lyapunov functions and Razumikhin techniques, some stability results have been given. Li and Song [8] have proposed an impulsive delay inequality and studied the stabilization problem of delayed systems via impulsive control.In particular, Liu [9] have studied the pth moment asymptotical stability for impulsive stochastic differential equations by employing Lyapunov function method and Itô's formula. Recently, some important and interesting results for stability of impulsive systems have been obtained, see [12][13][14][15][16][17] and the references therein. Hence, it is necessary to investigate the exponential stability for discrete-time stochastic systems with impulses.
Formally speaking, discrete-time systems is more challenging than continuous-time systems [18,19]. Hence the study on discrete-time systems will become more and more important, and attract a lot of researchers' attention [20][21][22][23]. In [20], the global exponential stability results for discrete-time delay systems with impulsive controllers have been considered. By using Razumikhin technique, the robust exponential stability results for discrete-time neural network with uncertainty have been given in [21]. The discrete-time Markovian jump delay systems with impulses have been investigated in [22], where the impulses act as perturbations. The analysis and synthesis problems for stability of discretetime impulsive systems have been extensively studied in recent years [24][25][26][27][28][29]. However, both of the above results require that the time delay in system is always greater than time delays in impulses, which leads to very conservative results. Hence, the existing methods and tools on stability of discrete-time stochastic systems with impulsive control are very insufficient. Therefore, it is more usefully to consider the stability or stabilization problem for more general discrete-time stochastic systems with impulsive control.
Based on the above discussion, the aim of the present paper is to establish exponential stability criterion on discrete-time stochastic delay systems, the criterion of this paper allows stabilizing impulses and destabilizing impulses is simultaneously effective under average impulsive interval condition. This paper mainly employ the sums average value of time-varying coefficients and the average impulsive interval, which are quite different from existing results in literature [30][31][32][33][34].
The rest of the paper is structured as follows. In Section 2, we introduce some basic definitions and notations. In Section 3, some criteria for exponential stability of discrete-time stochastic time-varying systems under impulsive control are obtained. In Section 4, two examples and their simulations are presented to illustrate the effectiveness of the proposed results. Finally, some conclusions are given in Section 5.

Preliminaries
Throughout this paper, let (Ω, F , {F n } n≥0 , P) be a complete probability space with a filtration {F n } n≥0 satisfying the usual conditions (F 0 contains all P-null sets). Let R = (−∞, +∞), R + = [0, +∞), R d the d-dimensional Euclidean space, and R d×r the space of d × r real matrices. For a set A ⊂ R, we denote I A be the family of all integer in A.
For the purpose of stability, we assume the functions f (n, 0) = g(n, 0) ≡ 0 and I k (n, 0) ≡ 0, k ∈ N + , which implies that x(n) ≡ 0 is an equilibrium solution. Definition 1. The trivial solution of system (1) is said to be pth moment stable, if for any ε > 0, there exists δ = δ(ε) such that E|x(n)| p < ε, n ≥ n 0 for any initial value ξ p G b < δ.

Definition 2.
The trivial solution of system (1) is said to be pth moment asymptotically stable, if it is pth moment stable, and for any ε > 0, there exists δ = δ(ε) and T = T(ε) such that Definition 3. The trivial solution of system (1) is said to be pth(p > 0) moment exponentially stable if there is a pair of positive constants λ and C such that for any initial value ξ p G b < δ. When p = 2, it is usually said to be exponentially stable in mean square.

Definition 4.
The impulsive sequence ζ = {n k : k ∈ N + } is said to have an average impulsive interval T a if there exist N 0 ≥ 0 and T a > 0 such that for any n ≥ s ≥ 0, where N(n, s) denotes the number of impulsive times of the impulsive sequence ζ on the interval (s, n]. The N 0 is said to be the elasticity number.

Remark 1.
For most real-world impulsive signals, the occurrence of impulses is not uniformly distributed. For this impulsive signals, the lower bound of the impulsive intervals is small, meanwhile the upper bound of the impulsive intervals is quite large. Hence, for non-uniformly distributed impulsive signals, taking sup k∈N {n k+1 − n k } (upper bound) or inf k∈N {n k+1 − n k } (lower bound) to characterize the frequency of the impulses' occurrence would make the obtained results very conservative.

Asymptotically Stable
In this subsection, we will consider the pth moment globally asymptotically stable for discrete-time impulsive stochastic systems (1) by using average impulsive interval.

Proof. For any initial date
and write x(n; n 0 , ξ) = x(n) and V(n, x(n)) = V(n) simply. For any φ > 0, construct an auxiliary function where u(n) the solution of (3) with initial condition u(s) holds for any n ∈ N.
For n ∈ I[n 0 , n 1 ), we have From (5) and condition (iii), we have Next, it is similar to the proof of (5), for any n ∈ I[n 1 , n 2 ), we can prove By the induction principle, for any n ∈ I[n k , n k+1 ), k ∈ N + , we have Then it can be deduced that where K(n, s) = ∏ (1 +α(j)) for any n > s ≥ n 0 . We only need to consider the following two possible cases.
Combining (17) and condition (iii), we obtain that It is similar to the proof of (20), one can prove that for any n ∈ I[n 1 , n 2 )

EV(n) ≤ u(n).
From condition (iii), we get By a simple derivation, we can prove in general that Finally, we show that system (1) is pth moment asymptotical stability. By using Definition 4, we get that ∏ n k ∈I[n 0 ,n) Ta (n−n 0 ) , 0 < ρ < 1. , n ∈ N, Combining this with the condition (iv) yields lim n→∞ e n ∑ j=n 0 (α(j)+α which implies that the trivial solution of system (1) is pth moment asymptotical stability. The proof is complete.

Remark 2.
The parameters ρ in condition (iii) describe the influence of impulses on the stability of the underlying discrete-time systems. If ρ > 1, the impulses are destabilizing, which means that the impulses do not occur too frequently. If ρ < 1, the impulses are stabilizing, which means that the impulses act appropriately frequently.

Remark 3.
Both destabilizing and stabilizing impulses are discussed with the aid of average impulsive interval technique. Compared with [18,23], we obtained results have a greater range of applications.

Almost Sure Exponential Stability
At the end of this section, under an irrestrictive condition, we shall establish a theorem about the almost exponentially stable of system (1).

Remark 4.
In [22], Dai and Xu have investigated the exponentially stable for discrete-time delayed systems by using Razumikhin-type method. However, Theorem 2 proposed in this paper removes this restriction and is applicable to discrete-time delayed systems with impulsive control including both ρ ≥ 1 and ρ < 1, which makes it suitable for a broader scope of applications than the results in [22].
Consider the special case with α(n) ≡ α and β(n) ≡ β, here α and β are constants. Similar to proof of Theorem 2, we can obtain the following results.

Remark 5.
Since the Razumikhin-type method is more conservative than the Lyapunov function method, the Razumikhin-type theorem in [26] is also not convenient to be applied to this system since it is not easy to find an appropriate constant to satisfy the Razumikhin-type condition. Remark 6. Since the system without impulses is exponentially stable and the impulses are destabilizing, the existing results in [35,36] cannot be applied to system (37). From Theorem 1 in this paper, we see that the coefficients α(n), β(n) have wider range. Example 2. Consider the following discrete-time stochastic delay system where ω(n) obeys Gaussian distribution N (0, 1). f (n, x(n)) = e 1 2 (sin n−1) x(n − 1), g(n − τ(n), x(n − τ(n)) = cos n √ 1+n 2 x(n − τ(n)) and τ(n) = cos 2 n + τ, τ ∈ N + .
Similar to the proof of Theorem 1, we can prove that (sin(s−1)−1) , n ∈ N.

Remark 7.
In Example 2, the impulses are used to stabilize an unstable system. In this case, the impulses must be frequent enough, and their amplitude must be suitably related to the growth rate of the continuous flow.

Conclusions
This paper has studied the stability analysis of discrete-time stochastic delay systems with impulses. The stability analysis is achieved with the help of the sums average value of the time-varying coefficients and the average impulsive interval. Some examples were also presented to illustrate the efficiency of the obtained results.
Author Contributions: T.C., Writing-original draft; P.C., Writing-review & editing. All authors have read and agreed to the published version of the manuscript.