Stochastically Globally Exponential Stability of Stochastic Impulsive Differential Systems with Discrete and Infinite Distributed Delays Based on Vector Lyapunov Function

.is paper deals with stochastically globally exponential stability (SGES) for stochastic impulsive differential systems (SIDSs) with discrete delays (DDs) and infinite distributed delays (IDDs). By using vector Lyapunov function (VLF) and average dwell-time (ADT) condition, we investigate the unstable impulsive dynamics and stable impulsive dynamics of the suggested system, and some novel stability criteria are obtained for SIDSs with DDs and IDDs. Moreover, our results allow the discrete delay term to be coupled with the nondelay term, and the infinite distributed delay term to be coupled with the nondelay term. Finally, two examples are given to verify the effectiveness of our theories.


Introduction
During the past few years, stochastic differential systems (SDSs) have been paid a great deal of attention in various fields. For example, there have been a number of works on the stability of SDSs (e.g., see [1][2][3][4][5][6]), and such a topic is of great significance in many practical applications. On the other hand, delay problems are often encountered in the chemical industry system, neural network system, and other systems. As a result, the stability analysis problem of SDSs with delays was studied in [2][3][4][5][6][7][8][9][10][11][12], where the delays were divided into constant delays, time-varying delays, and distributed delays. For a kind of SDSs with DDs and IDDs, there are many researchers to pay much attention on this topic, SDSs with DDs and IDDs were applied to study neural network systems [13][14][15], especially. However, these modes do not include some dynamic phenomena with impulses even if they are indeed important in practice.
Systems with impulses have been widely applied in practice. For instance, they are often used to describe dynamic processes that mutate at successive times [7,8,11,12,[16][17][18][19][20]. In the past few decades, addressed system was studied extensively (e.g., see [16,19]) and it was found that impulsive systems can contribute the exponential stability of SDSs (e.g., see [8,11,12]). e impulse one can not only cause complex dynamic behaviors such as instability, but also stabilize the unstable dynamic system. How to use the appropriate impulse control to stabilize the unstable SDSs or let impulses play a negative role on the stable system is of great significance. is paper aims to study these interesting topics for SIDSs with DDs and IDDs.
In order to deal with the stability of systems with DDs and IDDs, there have appeared many methods such as the fixed point theory, Lyapunov-Krasovskii function or the scalar Lyapunov function. For example, the Lyapunov -Krasovskii function and matrix inequality method were used in [13]; Chen et al. in [15] employed the fixed point theorem; Huang and Cao in [14] applied the Lyapunov functional method and the semimartingale convergence theorem. However, until now, there have been essentially no results to deal with SIDSs with DDs and IDDs by using the vector Lyapunov method. To investigate the stability issue, two new methods were recently proposed: the VLF method [9,12,[19][20][21][22][23] and the ADT method [18,24]. e ADT effectively limits the impulse and could promote the stability of the system. By using VLF, inequality techniques, and impulse conditions, some useful exponential stability criteria are obtained. As a feasible alternative to scalar Lyapunov function, VLF has attracted more and more attention in recent years (e.g., see [9]). In [25], VLF was first introduced and widely used in various fields owing to its outstanding advantages. In terms of construction, the theory of VLF provides a more flexible method for dealing with the complexity of SIDSs (e.g., see [19,26]). e real reason is that the theory of VLF can reduce the dimension and reduce the requirement of system component (e.g., see [26]). erefore, there are many related results reported on the vector Lyapunov function method (e.g., see [9,12,[19][20][21][22][23]). However, the joint system with IDDs of stochastic impulsive and DDs has not been solved, which greatly limits the effectiveness of VLF.
Motivated by the above discussions, we study SGES of SIDS with DDs and IDDs by using ADT condition and VLF. We consider two cases: unstable impulse dynamics and stable impulsive dynamics. For these two cases, some sufficient conditions are established for SIDS with DDs and IDDs based on the strength of VLF and ADT condition. Moreover, the results show that continuous SDSs with DDs and IDDs are stable and the impulsive one is unstable, according to the relationship between ADT and impulse, a lower bound of ADT is given to the mixed system is exponential stability. When continuous SIDS with DDs and IDDs are not stable, the impulsive effect can stabilize the system successfully under the upper bound condition of the given ADT.
ere are three contributions to the paper. (1) To the best of our knowledge, there have been no studies on the stability of SIDS with DDs and IDDs by VLF. (2) e discrete delay term is coupled with the nondelay term and the infinitely distributed delay term is coupled with the nondelay term. It should be mentioned that the comparison principle was used [19,20,23] and the components of VLF were separate, but the coupling of distributed delay term with nondelay term was not considered in [9,12]. (3) e third is infinitely distributed delay: due to its infinite nature, we deal with it by the construction formula n j�1 ∞ 0 k ij (s)exp(η i s)α j ds and n j�1 ∞ 0 k ij (s)exp(−η i s)α j ds.
us, our results are innovative than those in [9,19,20,23]. e remainder of this paper is organized as follows. In the second part, the model and the preliminary knowledge are introduced. Two novel stability criteria are established for the stochastic impulsive systems with DDs and IDDs in the third part. In the fourth part, two examples are given to verify the correctness of our results.

Preliminaries
rough the paper, no special instructions, we will use the following instructions.
w(t) is an R m -valued Brownian motion defined on a complete probability space (Ω, F, F t t≥t 0 , P). N >0 represents the set of positive integer and R denotes the real number. For a give t 0 ≥ 0, let Given a vector or matrix C, its transpose is denoted by C T . A\D ≔ z: z ∈ A, z∈D { } for two given sets A with D. tr [B] represents the trace of the matrix B, where B � B T ∈ R n×n . Let |.| be the Euclidean norm and E present the vector that all the components are 1. I means identity matrix. For a given function p: R ≥t 0 ⟶ R n and the initial time −∞ < t ≤ t 0 , define ‖p‖ ∞ ≔ sup t≤t 0 |p(t)|. Given a function g: is of class K for each fixed t ≥ 0 and decreases to zero as t ⟶ ∞ for each fixed s ≥ 0. e inverse of the function β is denoted by β − 1 .
We will consider the following SIDSs with DDs and IDDs: where x(t) ∈ R n is the system state, x t ≔ x(t − τ), τ is a bounded and positive constant. T ≔ t 0 , t 1 , . . . , t n is a impulsive time sequence satisfying 0 ≤ t 0 ≤ t 1 ≤ · · · ≤ t n . e initial function ξ: [−∞, t 0 ] ⟶ R n is a F t − 0 adapted continuous stochastic variable with finite E[‖ξ‖ p τ ]. For all k ∈ N >0 , the function f: R ≥t 0 × R n × R n ⟶ R n , g: R ≥t 0 × R n × R n ⟶ R n×m , and h k : R ≥t 0 × R n ⟶ R n are assumed to be Lipschitz and Borel measurable. For the aim of stability, we assume that f(t, 0, 0) ≡ 0 and g(t, 0, 0) ≡ 0, and thus x(t) ≡ 0 is a trivial solution of system (1). As a usual, we assume with no emphasis on conditions that there exists a unique global solution x(t, ξ) for the initial value ξ.
Definition 2 (see [11,24]): For an impulsive sequence t k k∈N , N(t, s) shows the number of impulses that occur in for N 0 > 0, τ c > 0, then τ c and N 0 are called the average dwell-time (ADT) and the elasticity number, respectively.
Definition 3 (see [9]): Let C 1,2 denote the family of the nonnegative functions V i (t, x): R ≥0 × R n ⟶ R ≥0 that are continuously twice differentiable in x and once in t, n is bounded. en, for any C 1,2 , the operator of Definition 4 (see [28]) Let off-diagonal elements of the matrix A � (a ij ) n×n be nonpositive. If each of the following statements holds, then A is a nonsingular M-matrix.
(1) If the diagonal elements of A are all positive, then there exists a positive vector a such that Aa > 0 or . . , c n and ρ(·) is the spectral radius of the matrix (·).

Main Results
In this section, we will establish the SGES of system (1) with destabilizing impulses.
Theorem 1. For system (1), assume that there is locally n×n is continuous function and k ij (s) ≥ 0 such that the following conditions: Complexity 3 where Proof. By taking expectation on both sides of (5) and (6) and using Holder inequality in [6], we have Next, we divide the proof into three steps.
Step 1. We first prove the existence of λ.
For any given i, let Obviously, it follows from (11) that H i (0) > 0 and On the other hand, we have us, there is a unique constant η i > 0 such that H i (η i ) � 0. Letting λ � max 1≤i≤n η i > 0, there is a constant λ such that λ ∈ (0, λ).
Step 2. Next, we need to prove that for i ∈ ι and λ ∈ (0, λ), where It is easy to check that (13) is equivalent to the following: 4 Complexity Now suppose that (13) is not valid in some interval, then there are two cases: Case 1. (13) is not true at the nonimpulse point of the certain interval; Case 2. (13) does not hold at the impulse point of the certain interval. For Case 1, there exists a k such that is not true for i ∈ ι and t ∈ (t k , t k+1 ). Define Noting that U i (t) and W i (t − t 0 ) are continuous for t ∈ R ≥t 0 \T, there exist i and t * such that where Δt > 0 is arbitrarily small. Hence, it follows from (16) and (17) that By the definition of λ, we have which together with (19) yields Complexity 5

Due to U i (t) � [EV i (t)] 1/r , then EV i (t) � [U i (t)] r , for
i ∈ ι. By virtue of Dini-derivation, (8), and the Ito formula, we obtain en, it follows from (22) that Combining (21) and (23), at is, (13) holds for all t ∈ (t k , t k+1 ). For Case 2, we have that (13) holds for all t ∈ [t 0 , t k ) and does not hold at t k . us, for some i ∈ ι, en, it follows from (9) that at the time instance t k , which contradicts with (24). erefore, (13) holds for t � t k .
erefore, by using the mathematical induction, we see that (13) is satisfied for all t ∈ [t 0 , ∞).
Step 3. Finally, we will prove that system (1) is SGES. In fact, it follows from condition (i) and Jensen's inequality in [29] that
□ Corollary 1. For system (1), assume that there is locally n×n is continuous function and k ij (s) ≥ 0 such that the following conditions hold.
Remark 1. In eorem 1, a difficulty is that condition ζE > E destabilizes system (1). To overcome this difficulty, we give a lower bound by using the relation between the ADT and impulses, and it may guarantee that the number of destabilizing impulses can be reduced. As a consequence, we can prove that system (1) is SGES. e next theorem will show that the impulses can promote the stability of system (1) even if system (1) without impulses may be unstable. (ζ 1 , ζ 2 , . . . , ζ n ), and non-
Proof. Like the proof of eorem 1, we divide the proof into three steps.
Step 1. We first prove the existence of λ. Since μ + P − Q + K is a nonsingular M-matrix, there is π ∈ Ω M (μ + P − Q + K), where min 1≤i≤n π i ≥ 1, such that For any given i, set Obviously, it follows from (36) that H i (0) > 0 and On the other hand, we have en H i ′ (ω) < 0, and so there is a unique constant ω i > 0 such that H i (ω i ) � 0. Setting λ � min 1≤i≤n (ω i ), it is clear that there exists a constant λ > λ > 0.
It is easy to check that (38) is equivalent to the following: Now suppose that (38) is not true for i ∈ ι and t ∈ (t k , t k+1 ). Define Since U i (t) and W i (t − t 0 ) are continuous for t ∈ R ≥t 0 \T, there exist i and t * such that where Δt > 0 is arbitrarily small. erefore, it follows from (41) and (42) that By the definition of λ, we have which together with (44) yields Complexity 9 Due to U i (t) � [EV i (t)] 1/r , then EV i (t) � [U i (t)] r , for i ∈ ι. By virtue of Dini-derivation and the Ito formula, we obtain D + EV i (t) � ELV i and erefore, it follows from (47) that Combining (46) and (48), which contradicts with (39). In the end, U i (t) ≤ W i (t − t 0 ) holds for all t ∈ (t k , t k+1 ). In other words, (38) holds for all t ∈ (t k , t k+1 ). For Case 2, (38) holds for all t ∈ [t 0 , t k ) and dose not hold at t k . en, for some i ∈ ι.
By taking expectation and using the Holder inequality, it follows from (A3) that at the impulsive instance t k , which contradicts with (49). is verifies that (38) holds for t � t k . erefore, by using the mathematical induction, we see that (38) is valid for all t ∈ [t 0 , ∞).
Remark 2. In eorem 2, the condition (ζ + O)E < E shows that the impulses can do the contribution of the stability of system (1). Although system (1) may not be stable, we can give an upper bound by using the relation between ADT and impulses and prove that SIDSs with DDs and IDDs are SGES.
Remark 3. When k ij � 0, r � 2, eorems 1 and 2 will be reduced to the case of stochastic differential systems with only DDS, which was studied in [12]. It should be mentioned that [12] only considered time-delay terms coupled with nondelay terms. However, we consider the effect of mixed delay terms including the DDs item and IDDs item coupled with delay-free item, which also appears in the LV-operator differential inequality. us, our results not only avoid to use elementary inequality to analyze crossterm problem but also is more representative.