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

Authors: Xiaoyan Liu; Quanxin Zhu
Journal: Complexity
 (2020)

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## Abstract

This 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.


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## Body

## 1. 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 [1–6]), 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 [2–12], 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 [13–15], 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, 16–20]. 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]). The 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. This 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, 19–23] and the ADT method [18, 24]. The 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]). The real reason is that the theory of VLF can reduce the dimension and reduce the requirement of system component (e.g., see [26]). Therefore, there are many related results reported on the vector Lyapunov function method (e.g., see [9, 12, 19–23]). 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.There 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) The 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) The third is infinitely distributed delay: due to its infinite nature, we deal with it by the construction formula ∑j=1n∫0∞kijsexpηisαjds and ∑j=1n∫0∞kijsexp−ηisαjds. Thus, our results are innovative than those in [9, 19, 20, 23].The 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.


## 2. Preliminaries

Through the paper, no special instructions, we will use the following instructions.wt is an ℝm-valued Brownian motion defined on a complete probability space Ω,ℱ,ℱtt≥t0,P. N>0 represents the set of positive integer and ℝ denotes the real number. For a give t0≥0, let ℝ≥t0≔t0,+∞. Given a, b∈ℝn, a>b if ai>bi, for all i∈ι=1,…,n. Define a,b≔aT,bTT. Given a vector or matrix C, its transpose is denoted by CT. A\D≔z:z∈A,z∈D for two given sets A with D. trB represents the trace of the matrix B, where B=BT∈ℝn×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≥t0⟶ℝn and the initial time −∞<t≤t0, define p∞≔supt≤t0pt. Given a function g:ℝ⟶ℝ, denote gt−≔limsups⟶0−gt+s. A function α:ℝ≥0⟶ℝ≥0 is of class CK∞ if it is of class K∞ and concave. A function γ:ℝ≥0×ℝ≥0⟶ℝ≥0 is of class Kℒ, if γs,t is of class K for each fixed t≥0 and decreases to zero as t⟶∞ for each fixed s≥0. The inverse of the function β is denoted by β−1.We will consider the following SIDSs with DDs and IDDs:(1)dxt=ft,xt,∫0∞ksxt−sdsdt+gt,xt,∫0∞ksxt−sdsdwt,t∈ℝ≥t0∖T,Δxt/t=tk=xtk−xtk−=hktk,xtk−,t∈T,k∈N>0,xt=ξt,t∈−∞,t0,where xt∈ℝn is the system state, xt≔xt−τ, τ is a bounded and positive constant. T≔t0,t1,…,tn is a impulsive time sequence satisfying 0≤t0≤t1≤⋯≤tn. The initial function ξ:−∞,t0⟶ℝn is a ℱt0− adapted continuous stochastic variable with finite Eξτp. For all k∈N>0, the function f:ℝ≥t0×ℝn×ℝn⟶ℝn, g:ℝ≥t0×ℝn×ℝn⟶ℝn×m, and hk:ℝ≥t0×ℝn⟶ℝn are assumed to be Lipschitz and Borel measurable. For the aim of stability, we assume that ft,0,0≡0 and gt,0,0≡0, and thus xt≡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 xt,ξ for the initial value ξ.In this paper, we always assume thatK≔∫0∞kijdsn×n is meaningful. That is, the delay kernels kij:0,+∞→0,+∞ are real-valued nonnegative continuous functions and satisfy ∫0∞eμskijsds<∞, where μ is a positive number.Definition 1.
(see [6]): For arbitrary ε∈0,1, if there is ϱ∈Kℒ, xt0∈ℝn, t∈ℝ≥t0, M≥0, b>0 such that(2)Pxt≤ϱxt0,t−t0≥1−ε.
Then, system (1) is stochastically globally asymptotically stable. Furthermore, if ϱa,t≔Ma2e−bt, then system (1) is stochastically globally exponentially stable (SGES).Definition 2.
(see [11, 24]): For an impulsive sequence tkk∈N, Nt,s shows the number of impulses that occur in the half-open interval s,t. If(3)t−sτc−N0≤Nt,s≤t−sτc+N0,for N0>0, τc>0, then τc and N0 are called the average dwell-time (ADT) and the elasticity number, respectively.Definition 3.
(see [9]): Let C1,2 denote the family of the nonnegative functions Vit,x:ℝ≥0×ℝn⟶ℝ≥0 that are continuously twice differentiable in x and once in t, n is bounded. Then, for any C1,2, the operator of ℒVit,xt,∫0∞ksxt−sds:ℝ≥0×ℝn×ℝn⟶ℝ≥0, i∈ι is defined as(4)ℒVit,xt,∫0∞ksxt−sds=∂Vit,xt∂t+∂Vit,xt∂xft,xt,∫0∞ksxt−sds+12trgTt,xt,∫0∞ksxt−sds∂Vit,xt∂x2gt,xt,∫0∞ksxt−sds.
DefineVt,xt≔V1t,xt,…,Vnt,xt. In particular, if n=1, then V becomes a scalar.
LetΨ:ℝ≥t0⟶ℝ≥t0 be a continuous function. Then, the upper Dini derivative of Ψt is defined as D+Ψt≔limsups⟶0+Ψt+s−Ψt/s. In particular, if EℒVit,xt,xt is continuous, then it follows from [27] that EℒVit,xt,xt=D+EVit,xt,xt.Definition 4.
(see [28]) Let off-diagonal elements of the matrix A=aijn×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 vectora such that Aa>0 or ATa>0.(2)
A=C−M, ρC−1M<1, where M≥0, C=diagc1,c2,…,cn and ρ⋅ is the spectral radius of the matrix ⋅.
For a nonsingular M-matrix A, it can denoteΩMA≜z∈Rn|Az>0,z>0.


## 3. 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 Lipschitz Lyapunov function Vi:ℝ≥0×ℝn⟶ℝ≥0, i∈ι, r>1, ζE≥E, ν∈K∞, ς∈CK∞, positive matrices μ≔diagμ1,μ2,…,μn, ζ≔diagζ1,ζ2,…,ζn and nonnegative matrices P≔pijn×n, Q≔qijn×n, O≔oijn×n, where K≔∫0∞kijsdsn×n is continuous function and kijs≥0 such that the following conditions:(i)
i∈ι, t∈ℝ≥t0, νxt≤Vit≤ςxt.(ii)
i∈ι, t∈ℝ≥t0\T,(5)ℒVit,xt,xt,∫0∞ksxt−sds≤−μiVit+∑j=1npijVit1/r′Vjt1/r+sup−τ≤θ≤0∑j=1nqijVit1/r′Vjt+θ1/r+∑j=1n∫0∞kijsVit1/r′Vjt−s1/rds,whereμ−P−Q−K is a nonsingular M-matrix, θ∈−τ,0,r′=1−r−1−1.(iii)
i∈ι and k∈N>0,(6)Vitk≤ζiVitk−+∑j=1noijVitk−1/r′Vjtk−1/r.(iv)
τc>maxi∈ιlnζi+∑j=1noij/λ, where λ∈0,λ¯,Vit=Vit,xt,(7)λ¯≔maxηi>0:rαiηi−μiαi+∑j=1npijαj+∑j=1nqijαjexpτηi+∑j=1n∫0∞kijsexpηisαjds<0,where i∈ι, α∈ΩMμ−P−Q−K, with min1≤i≤nαi≥1, α¯=maxαi.
Then, system (1) is SGES.Proof.
By taking expectation on both sides of (5) and (6) and using Holder inequality in [6], we have(8)EℒVit,xt,xt,∫0∞ksxt−sds≤−μiEVit+∑j=1npijEVit1/r′EVjt1/r+sup−τ≤θ≤0∑j=1nqijEVit1╱r′EVjt+θ1/r+∑j=1n∫0∞kijsEVit1/r′EVjt−s1/rds,i∈ι,t∈ℝ≥t0\T.(9)EVitk≤ζiEVitk−+∑j=1noijEVitk−1/r′EVjtk−1/r,i∈ιandk∈N>0.
Next, we divide the proof into three steps.Step 1. We first prove the existence ofλ.Ifμ−P−Q−K is a nonsingular M-matrix, there is α∈ΩMμ−P−Q−K with min1≤i≤nαi≥1 such that(10)μiαi−∑j=1npijαj−∑j=1nqijαj−∑j=1n∫0∞kijsαjds>0,i∈ι.For any giveni, let(11)Hiη=−rαiη+μiαi−∑j=1npijαj−∑j=1nqijαjexpτη−∑j=1n∫0∞kijsexpηsαjds.Obviously, it follows from (11) that Hi0>0 and Hi∞=−∞, where η⟶∞. On the other hand, we have(12)Hi′η=−rαi−τ∑j=1nqijαjexpτη−∑j=1n∫0∞skijsexpηsαjds<0Thus, there is a unique constantηi>0 such that Hiηi=0. Letting λ¯=max1≤i≤nηi>0, there is a constant λ such that λ∈0,λ¯.Step 2. Next, we need to prove that fori∈ι and λ∈0,λ¯,(13)EVit≤EVt0μ¯rNt,t0α¯rexp−rλt−t0,whereVt0=∑i=1nVit0, μ¯=maxi∈ιζi+∑j=1noij and α¯≔maxi∈ιαi. Clearly, (13) holds for t=t0.For simplicity, letWit−t0=EVt01/rμ¯Nt,t0αiexp−λt−t0, Uit=EVit1/r.It is easy to check that (13) is equivalent to the following:(14)EVit1/r≤maxi∈ιEVt01/rμ¯Nt,t0αiexp−λt−t0≕maxi∈ιWit−t0.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 ak such that Uit≤Wit−t0 holds for all t∈t0,tk and i∈ι, and Uit≤Wit−t0 is not true for i∈ι and t∈tk,tk+1. Define(15)t∗≔inft∈tk,tk+1:Uit>Wit−t0.Noting thatUit and Wit−t0 are continuous for t∈ℝ≥t0\T, there exist i¯ and t∗ such that(16)Ui¯t∗=Wi¯t∗−t0,(17)Uit>Wit−t0,t∈t∗,t∗+Δt,whereΔt>0 is arbitrarily small. Hence, it follows from (16) and (17) that(18)D+Ui¯t∗>D+Wi¯t∗−t0.(19)D+Wi¯t∗−t0=−λαi¯μ¯Nt∗,t0EVt01/rexp−λt∗−t0.By the definition ofλ, we have(20)−λαi>1r−μiαi+∑j=1npijαj+∑j=1nqijαjexpτλ+∑j=1n∫0∞kijsexpλsαjds,which together with (19) yields(21)D+Wi¯t∗−t0>1r−μi¯αi¯+∑j=1npi¯jαj+∑j=1nqi¯jαjexpτλ+∑j=1n∫0∞ki¯jsexpλsαjds×μ¯Nt∗,t0EVt01/rexp−λt∗−t0=1r−μi¯αi¯μ¯Nt∗,t0EVt01/rexp−λt∗−t0+1r∑j=1npi¯jαjμ¯Nt∗,t0EVt01/rexp−λt∗−t0+1r∑j=1nqi¯jαjexpτλμ¯Nt∗,t0EVt01/rexp−λt∗−t0+1r∑j=1n∫0∞ki¯jsexpλsαjdsμ¯Nt∗,t0EVt01/rexp−λt∗−t0≥1r−μi¯EVi¯t∗1/r+∑j=1npi¯jEVj¯t∗1/r+∑j=1nqi¯jsup−τ≤θ≤0expλτ+θμ¯Nt∗,t∗,t0EVi¯t∗1/r+∑j=1n∫0∞ki¯jsEVj¯t∗−s1/rds.Due toUit=EVit1/r, then EVit=Uitr, for i∈ι. By virtue of Dini-derivation, (8), and the Ito^ formula, we obtain(22)D+EVit=rUitr−1D+Uit≤−μiUitr+∑j=inpijUitr/r′Ujt+∑j=inqijsup−τ≤θ≤0Uitr/r′Ujt+θ+∑j=in∫0∞kijsUitr/r′Ujt−sds.Then, it follows from (22) that(23)D+Uit≤1r−μiUit+∑j=inpijUjt+∑j=inqijsup−τ≤θ≤0Ujt+θ+∑j=in∫0∞kijsUjt−sds.Combining (21) and (23), we get D+Ui¯t∗≤D+Wi¯t∗−t0, which contradicts with (18). Thus, Uit≤Wit−t0 holds for all t∈tk,tk+1. That is, (13) holds for all t∈tk,tk+1.For Case 2, we have that (13) holds for all t∈t0,tk and does not hold at tk. Thus, for some i∈ι,(24)EVit>EVt0μ¯rNt,t0α¯rexp−rλt−t0.Then, it follows from (9) that at the time instance tk,(25)EVitk≤ζiEVitk−+∑j=1noijEVitk−1/rEVjtk−1/r≤ζiEVt0μ¯rNtk−,t0α¯rexp−rλtk−−t0+∑j=1noij×EVt0μ¯rNtk−,t0α¯rexp−rλtk−−t01/r′EVt0μ¯rNtk−,t0α¯rexp−rλtk−−t01/r≤ζi+∑j=1noijEVt0μ¯rNtk−,t0α¯rexp−rλtk−−t0≤EVt0μ¯rNtk−,t0α¯rexp−rλtk−−t0,which contradicts with (24). Therefore, (13) holds for t=tk.Therefore, by using the mathematical induction, we see that (13) is satisfied for all t∈t0,∞.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(26)EVt0μ¯rNt,t0α¯rexp−rλt−t0≤EVt0μ¯rN0+rt−t0/τcα¯rexp−rλt−t0=EVt0μ¯rN0α¯rexp−rλ+r/τclnμ¯t−t0≤ςEξτμ¯rN0α¯rnexp−rλ+r/τclnμ¯t−t0.
According to (iv), we obtain−rλ+r/τclnμ¯<0. This fact together with and (26) gives(27)EVit≤ςEξτμ¯rN0α¯rnexp−rλ+r/τc×lnμ¯t−t0.
By using Markov inequality in [29] and (27), we have that for arbitrary ε∈0,1 and i∈ι, PVit≤ε−1ςEξτμ¯rN0α¯rnexp−rλ+r/τc×lnμ¯t−t0≥1−ε, which together with (i) yields(28)Pxt≤ϱEξτ,t−t0≥1−ε,where ϱa,t=ν−1ε−1ςaEξτμ¯rN0α¯rnexp−rλ+r/τc×lnμ¯t. This verifies that system (1) is SGES.Corollary 1.
For system (1), assume that there is locally Lipschitz Lyapunov function Vi:ℝ≥0×ℝn⟶ℝ≥0, i∈ι, r=1, ζE≥E, ν∈K∞, ς∈CK∞, positive matrices μ≔diagμ1,μ2,…,μn, ζ≔diagζ1,ζ2,…,ζn, and nonnegative matrices P≔pijn×n, Q≔qijn×n, O≔oijn×n, where K≔∫0∞kijsdsn×n is continuous function and kijs≥0 such that the following conditions hold.(A):i∈ι, t∈ℝ≥t0, νxt≤Vit≤ςxt;(B):i∈ι, t∈ℝ≥t0\T.(29)ℒVit,xt,∫0∞ksxt−sds≤−μiVit+∑j=1npijVjt+sup−τ≤θ≤0∑j=1nqijVjt+θ+∑j=1n∫0∞kijsVjt−sds,whereμ−P−Q−K is a nonsingular M-matrix, θ∈−τ,0.(C):i∈ι and k∈N>0,(30)Vitk≤ζiVitk−+∑j=1noijVjtk−.(D):τc>maxi∈ιlnζi+∑j=1noij/λ, where λ∈0,λ¯,Vit=Vit,xt.(31)λ¯≔supη>0:αiη−μiαi+∑j=1npijαj+∑j=1nqijαjexpτη+∑j=1n∫0∞kijsexpηsαjds<0,where i∈ι, α∈ΩMμ−P−Q−K, with min1≤i≤nαi≥1, α¯=maxαi.
Then, system (1) is SGES.Remark 1.
In Theorem1, 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.
The next theorem will show that the impulses can promote the stability of system (1) even if system (1) without impulses may be unstable.Theorem 2.
For system (1), assume that that there exist locally Lipschitz Lyapunov function Vi:ℝ≥0×ℝn⟶ℝ≥0, for all i∈ι , r>1, ν∈K∞, ς∈CK∞, positive matrices μ≔diagμ1,μ2,…,μn, ζ≔diagζ1,ζ2,…,ζn, and nonnegative matrices P≔pijn×n, Q≔qijn×n, O≔oijn×n, where K≔∫0∞kijsdsn×n be continuous function, kijs≥0 and ζ+OE<E:(A1)i∈ι, t∈ℝ≥t0, νxt≤Vit≤ςxt;(A2)i∈ι, t∈ℝ≥t0\T.(32)EℒVit,xt,∫0∞ksxt−sds≤μiEVit+∑j=1npijEVit1/rrEVjt1/r−sup−τ≤θ≤0∑j=1nqijEVit1/rrEVjt+θ1/r+∑j=1n∫0∞kijsEVit1/rrEVjt−s1/rds,whereμ+P−Q+K is a nonsingular M-matrix, θ∈−τ,0,r=1−r′−1.(A3)i∈ι and k∈N>0,(33)Vitk≤ζiVitk−+∑j=1noijVitk−1/rrVjtk−1/r.(A4)τc<mini∈ιlnζi+∑j=1noij/−λ, where λ>λ¯ and Vit=Vit,xt.(34)λ¯≔infω>0:−rπiω+μiπi+∑j=1npijπj−∑j=1nqijπjexpτω+∑j=1n∫0∞kijsexp−ωsπjds<0.where i∈ι, π∈ΩMμ+P−Q+K,min1≤i≤nπi≥1, λ¯=maxi∈ιωi, π¯=maxπi.
Then, system (1) is SGES.Proof.
Like the proof of Theorem1, we divide the proof into three steps.Step 1. We first prove the existence ofλ¯. Since μ+P−Q+K is a nonsingular M-matrix, there is π∈ΩMμ+P−Q+K, where min1≤i≤nπi≥1, such that(35)μiπi+∑j=1npijπj−∑j=1nqijπj+∑j=1n∫0∞kijsπjds>0,i∈ι.For any giveni, set(36)H¯iω=−rπiω+μiπi+∑j=1npijπj−∑j=1nqijπjexpτω+∑j=1n∫0∞kijsexp−ωsπjds.Obviously, it follows from (36) that H¯i0>0 and H¯i∞=−∞, where ω⟶∞. On the other hand, we have(37)H¯i′ω=−rπi−τ∑j=1nqijπjexpτω−∑j=1n∫0∞skijsexp−ωsπjds.ThenH¯i′ω<0, and so there is a unique constant ωi>0 such that H¯iωi=0. Setting λ¯=min1≤i≤nωi, it is clear that there exists a constant λ>λ¯>0.Step 2. Next, fori∈ι and λ>λ¯, then ζ+OE<E, we need to prove that(38)EVit≤EVt0μ¯rNt,t0π¯rexprλt−t0,whereVt0=∑i=1nVit0, μ¯=maxi∈ιζi+∑j=1noij, and π¯≔maxi∈ιπi. Clearly, (38) holds for t=t0.For convenience, defineWit−t0=EVt01/rμ¯Nt,t0πiexpλt−t0, Uit=EVit1/r. It is easy to check that (38) is equivalent to the following:(39)EVit1/r≤maxi∈ιEVt01/rμ¯Nt,t0πiexpλt−t0≕maxi∈ιWit−t0.Now suppose that (38) is not true in some interval, then there are two cases:Case 1. (38) is not true at the non-impulsive time of certain interval;Case 2. (38) is not true at the impulsive time of certain interval.For Case 1, there exists ak such that Uit≤Wit−t0 holds for all t∈t0,tk and i∈ι, and Uit≤Wit−t0 is not true for i∈ι and t∈tk,tk+1. Define(40)t∗≔inft∈tk,tk+1:Uit>Wit−t0.SinceUit and Wit−t0 are continuous for t∈ℝ≥t0\T, there exist i¯ and t∗ such that(41)Ui¯t∗=Wi¯t∗−t0,(42)Uit>Wit−t0,t∈t∗,t∗+Δt.whereΔt>0 is arbitrarily small. Therefore, it follows from (41) and (42) that(43)D+Ui¯t∗>D+Wi¯t∗−t0,(44)D+Wi¯t∗−t0=λπi¯μ¯Nt∗,t0EVt01/rexpλt∗−t0.By the definition ofλ, we have(45)λπi>1rμiπi+∑j=1npijπj−∑j=1nqijπjexpτλ+∑j=1n∫0∞kijsexp−λsπjds,which together with (44) yields(46)D+Wi¯t∗−t0>1rμi¯πi¯+∑j=1npi¯jπj−∑j=1nqi¯jπjexpτλ+∑j=1n∫0∞ki¯jsexp−λsπjds×μ¯Nt∗,t0EVt01/rexpλt∗−t0=1rμi¯πi¯μ¯Nt∗,t0EVt01/rexpλt∗−t0+1r∑j=1npi¯jπjμ¯Nt∗,t0EVt01/rexpλt∗−t0−1r∑j=1nqi¯jπjexpτλμ¯Nt∗,t0EVt01/rexpλt∗−t0+1r∑j=1n∫0∞ki¯jsexp−λsπjdsμ¯Nt∗,t0EVt01/rexpλt∗−t0≥1rμi¯EVi¯t∗1/r+∑j=1npi¯jEVjt∗1/r−1r∑j=1nqi¯jsup−τ≤θ≤0EVjt∗+θ1/r+1r∑j=1n∫0∞ki¯jsEVjt∗−s1/rds.Due toUit=EVit1/r, then EVit=Uitr, for i∈ι. By virtue of Dini-derivation and the Ito^ formula, we obtain D+EVit=EℒVi and(47)D+EVit=rUitr−1D+Uit≤μiUitr+∑j=inpijUitr/r′Ujt−∑j=inqijsup−τ≤θ≤0Uitr/r′Ujt+θ+∑j=in∫0∞kijsUitr/r′Ujt−sds.Therefore, it follows from (47) that(48)D+Uit≤1rμiUit+∑j=inpijUjt−∑j=inqijsup−τ≤θ≤0Ujt+θ+∑j=in∫0∞kijsUjt+sds.Combining (46) and (48), we get EVi¯t∗1/r=D+Ui¯t∗≤D+Wi¯t∗−t0, which contradicts with (39). In the end, Uit≤Wit−t0 holds for all t∈tk,tk+1. In other words, (38) holds for all t∈tk,tk+1.For Case 2, (38) holds for all t∈t0,tk and dose not hold at tk. Then, for some i∈ι.(49)EVitk>EVt0μ¯rNtk,t0π¯rexprλtk−t0.By taking expectation and using the Holder inequality, it follows from (A3) that at the impulsive instancetk,(50)EVitk≤ζiEVitk−+∑j=1noijEVitk−1/r′EVjtk−1/r≤ζiEVt0μ¯rNtk−,t0π¯rexprλtk−−t0+∑j=1noij×EVt0μ¯rNtk−,t0π¯rexprλtk−−t01/r′EVt0μ¯rNtk−,t0π¯rexprλtk−−t01/r≤ζi+∑j=1noijEVt0μ¯rNtk−,t0π¯rexprλtk−−t0≤EVt0μ¯rNtk,t0π¯rexprλtk−t0,which contradicts with (49). This verifies that (38) holds for t=tk. Therefore, by using the mathematical induction, we see that (38) is valid for all t∈t0,∞.Step 3. Finally, we will prove that system (1) is SGES. According to A1 and Jensen’s inequality, we have(51)EVt0μ¯rNt,t0π¯rexprλt−t0≤EVt0μ¯rN0+rt−t0/τcπ¯rexprλt−t0=EVt0μ¯rN0π¯rexprλ+r/τclnμ¯t−t0≤ςEξτμ¯rN0π¯rexprλ+r/τclnμ¯t−t0.
It follows from condition (A4) thatrλ+r/τclnμ¯<0. The rest of the proof is similar to that in Theorem 1, and thus we omit it here. This completes the proof.Corollary 2.
For system (1), assume that there is locally Lipschitz Lyapunov function Vi:ℝ≥0×ℝn⟶ℝ≥0, i∈ι, r=1, ζ+OE<E, ν∈K∞, ς∈CK∞, positive matrices μ≔diagμ1,μ2,…,μn, ζ≔diagζ1,ζ2,…,ζn, and nonnegative matrices P≔pijn×n, Q≔qijn×n, O≔oijn×n. Where K≔∫0∞kijsdsn×n is continuous function and kijs≥0 such that the following conditions hold:(A):i∈ι, t∈ℝ≥t0, νxt≤Vit≤ςxt;(B):i∈ι, t∈ℝ≥t0\T,(52)ℒVit,xt,∫0∞ksxt−sds≤μiVit+∑j=1npijVjt−sup−τ≤θ≤0∑j=1nqijVjt+θ+∑j=1n∫0∞kijsVjt−sds,whereμ+P−Q+K is a nonsingular M-matrix, θ∈−τ,0;(C):i∈ι and k∈N>0,(53)Vitk≤ζiVitk−+∑j=1noijVjtk−.(D):λ+1/τclnμ¯<0, where μ¯=maxi∈ιζi+∑j=1noij and λ>λ¯ and Vit=Vit,xt.(54)λ¯≔infω>0:−πiω+μiπi+∑j=1npijπj−∑j=1nqijπjexpτω+∑j=1n∫0∞kijsexp−ωsπjds<0,where i∈ι, π∈ΩMμ+P−Q+K, with min1≤i≤nπi≥1, π¯=maxπi.
Then, system (1) is SGES.Remark 2.
In Theorem2, the condition ζ+OE<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.
Whenkij=0, r=2, Theorems 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. Thus, our results not only avoid to use elementary inequality to analyze crossterm problem but also is more representative.Remark 4.
Conditions (ii) and (A2) are the vector version of the Halandy inequality in [9], which is an important tool in the stability analysis of SDSs. Especially, they also play an important role in discussing stochastically perturbed neural networks and stochastically generalized ecological systems.


## 4. Two Examples

In this section, two numerical examples are used to check the validity of our theories.Consider a two-neuron stochastically perturbed neural network with impulsive control.(55)dxt=Axt+Bfxt+Cfxt+Df∫06ksxt−sdsdt+σt,xtdwt,t∈ℝ≥t0\T,xtk=hxtk−,tk∈T,k=1,2,…n,where xt∈ℝ2, T is a given impulsive time sequence. Define fx=f1x1,f2x2=0.05∗tanhx1,0.05∗tanhx2, xt≔x1t−τ,x2t−τ, then we have(56)0≤fiz1−fiz2/z1−z2≤0.05,for all z1,z2∈ℝ and i=1,2.


### 4.1. Example I

Set(57)A=−2.400−2.8,B=1.63.50.30.4,C=0.10.60.40.1,D=0.40.50.30.1,σ=0.2∗x1t000.2∗x2t,h=1.5001.5,where ks=e−s/1−e−6.DefineVt,xt=V1t,xt,V2t,xt with Vit,xt=xitp,i=1,2p>1. Applying the Ito^ formula, we get(58)ℒV1t,xt≤−2.32p+0.02p2−px1tp+0.175px1tp−1x2t+0.05px1tp−1x1t−τ+0.03px1tp−1x2t−τ+0.02px1tp−1∫06e−s1−e−6x1t−sds+0.025px1tp−1∫06e−s1−e−6x2t−sds,ℒV2t,xt≤−2.78p+0.02p2−px2tp+0.015p|x2t|p−1x1t+0.02px2tp−1x1t−τ+0.005px2tp−1x2t−τ+0.015px2tp−1∫06e−s1−e−6x1t−sds+0.005px2t|p−1∫06e−s1−e−6x2t−sdsVitk≤1.5Vitk−i=1,2.Then, we can easily obtain(59)μ=2.32p−0.02p2−p002.78p−0.02p2−p,P=00.175p0.015p0,Q=00.05p0.03p0.02p0.05p,K=0.02p0.025p0.015p0.005p,ζ=1.5001.5.Thus, we have thatμ−P−Q−K is a nonsingular M-matrix, for any p∈1,112.4376. In particular, when p=4, α1=1, α2=1.5, τ=1, we can calculate λ¯=0.5453 and τc>2.9743, which verifies that all the conditions of Theorem 1 are satisfied. Thus, system (1) is SGES. Choosing τc=6 and a proper T, the initial state x0=0.7,−0.6, the sample path of solution is given in Figure 1.Figure 1
The solution ofτ=1, τc=6 in example I.


### 4.2. Example II

Set(60)A=−0.0400−0.01,B=1.63.50.50.4,C=−0.1−0.6−0.4−0.1,D=1001,σ=2∗x1t002∗x2t,h=0.75000.75,where ks=e−s/1−e−6.DefineVt,xt=V1t,xt,V2t,xt, where Vit,xt=xitp,i=1,2p>1. Applying the Ito^ formula, we get(61)ℒV1t,xt≤0.04p+2p2−px1tp+0.175px1tp−1x2t−0.005px1tp−1x1t−τ−0.03px1tp−1x2t−τ+0.05px1tp−1∫06e−s1−e−6x1t−sds,ℒV2t,xt≤0.01p+2p2−px2tp+0.025px2tp−1x1t−0.02px2tp−1x1t−τ−0.005px2tp−1x2t−τ+0.05px2tp−1∫06e−s1−e−6x2t−sds.By taking expectation on both sides of the inequality and using theHo˙lder inequality, we have(62)EℒV1t,xt≤0.04p+2p2−pEV1t+0.175pEV1tp−1/pEV2t1/p−0.005pEV1tp−1/pEV1t−τ1/p−0.03pEV1tp−1/pEV2t−τ1/p+0.05pEV1tp−1/p∫06e−s1−e−6EV1t−s1/pds,EℒV2t,xt≤0.01p+2p2−pEV2t+0.025pEV2tp−1/pEV1t1/p−0.02pEV2tp−1/pEV1t−τ1/p−0.005pEV2tp−1/pEV2t−τ1/p+0.05pEV2tp−1/p∫06e−s1−e−6EV2t1/pds,EVitk≤0.75EVitk−i=1,2.Then, it follows that(63)μ=0.04p+2p2−p000.01p+2p2−p,P=00.175p0.025p0,Q=0.005p0.03p0.02p0.005p,K=0.05p000.05p,ζ=0.75000.75.Then, we see thatμ+P−Q+K is a nonsingular M-matrix for any p∈1,+∞. Specially, when p=2, π1=1, π2=1.5, τ=1, we can calculate λ¯=1.9695 and τc<0.1661. Thus, all the conditions of Theorem 2 are true, which verifies that (1) is SGES. Choosing τc=0.16 and a proper T, the initial state x0=0.7,−0.6, the sample path of solution is presented in Figure 2.Figure 2
The solution ofτ=1, τc=0.16 in example II.


## 4.1. Example I

Set(57)A=−2.400−2.8,B=1.63.50.30.4,C=0.10.60.40.1,D=0.40.50.30.1,σ=0.2∗x1t000.2∗x2t,h=1.5001.5,where ks=e−s/1−e−6.DefineVt,xt=V1t,xt,V2t,xt with Vit,xt=xitp,i=1,2p>1. Applying the Ito^ formula, we get(58)ℒV1t,xt≤−2.32p+0.02p2−px1tp+0.175px1tp−1x2t+0.05px1tp−1x1t−τ+0.03px1tp−1x2t−τ+0.02px1tp−1∫06e−s1−e−6x1t−sds+0.025px1tp−1∫06e−s1−e−6x2t−sds,ℒV2t,xt≤−2.78p+0.02p2−px2tp+0.015p|x2t|p−1x1t+0.02px2tp−1x1t−τ+0.005px2tp−1x2t−τ+0.015px2tp−1∫06e−s1−e−6x1t−sds+0.005px2t|p−1∫06e−s1−e−6x2t−sdsVitk≤1.5Vitk−i=1,2.Then, we can easily obtain(59)μ=2.32p−0.02p2−p002.78p−0.02p2−p,P=00.175p0.015p0,Q=00.05p0.03p0.02p0.05p,K=0.02p0.025p0.015p0.005p,ζ=1.5001.5.Thus, we have thatμ−P−Q−K is a nonsingular M-matrix, for any p∈1,112.4376. In particular, when p=4, α1=1, α2=1.5, τ=1, we can calculate λ¯=0.5453 and τc>2.9743, which verifies that all the conditions of Theorem 1 are satisfied. Thus, system (1) is SGES. Choosing τc=6 and a proper T, the initial state x0=0.7,−0.6, the sample path of solution is given in Figure 1.Figure 1
The solution ofτ=1, τc=6 in example I.


## 4.2. Example II

Set(60)A=−0.0400−0.01,B=1.63.50.50.4,C=−0.1−0.6−0.4−0.1,D=1001,σ=2∗x1t002∗x2t,h=0.75000.75,where ks=e−s/1−e−6.DefineVt,xt=V1t,xt,V2t,xt, where Vit,xt=xitp,i=1,2p>1. Applying the Ito^ formula, we get(61)ℒV1t,xt≤0.04p+2p2−px1tp+0.175px1tp−1x2t−0.005px1tp−1x1t−τ−0.03px1tp−1x2t−τ+0.05px1tp−1∫06e−s1−e−6x1t−sds,ℒV2t,xt≤0.01p+2p2−px2tp+0.025px2tp−1x1t−0.02px2tp−1x1t−τ−0.005px2tp−1x2t−τ+0.05px2tp−1∫06e−s1−e−6x2t−sds.By taking expectation on both sides of the inequality and using theHo˙lder inequality, we have(62)EℒV1t,xt≤0.04p+2p2−pEV1t+0.175pEV1tp−1/pEV2t1/p−0.005pEV1tp−1/pEV1t−τ1/p−0.03pEV1tp−1/pEV2t−τ1/p+0.05pEV1tp−1/p∫06e−s1−e−6EV1t−s1/pds,EℒV2t,xt≤0.01p+2p2−pEV2t+0.025pEV2tp−1/pEV1t1/p−0.02pEV2tp−1/pEV1t−τ1/p−0.005pEV2tp−1/pEV2t−τ1/p+0.05pEV2tp−1/p∫06e−s1−e−6EV2t1/pds,EVitk≤0.75EVitk−i=1,2.Then, it follows that(63)μ=0.04p+2p2−p000.01p+2p2−p,P=00.175p0.025p0,Q=0.005p0.03p0.02p0.005p,K=0.05p000.05p,ζ=0.75000.75.Then, we see thatμ+P−Q+K is a nonsingular M-matrix for any p∈1,+∞. Specially, when p=2, π1=1, π2=1.5, τ=1, we can calculate λ¯=1.9695 and τc<0.1661. Thus, all the conditions of Theorem 2 are true, which verifies that (1) is SGES. Choosing τc=0.16 and a proper T, the initial state x0=0.7,−0.6, the sample path of solution is presented in Figure 2.Figure 2
The solution ofτ=1, τc=0.16 in example II.


## 5. Conclusion

In this paper, we have used the ADT condition and VLF to study SGES of SIDSs with DDs and IDDs under two cases: unstable impulse dynamics situation and stable impulse dynamics situation. By using VLF and ADT conditions, two sufficient stability criteria are established. One is that the lower bound of the mixed system relative to the average dwell-time is SGES when continuous SIDSs with DDs and IDDs is stable and the impulsive effect is unstable. The other is that the impulses can stabilize the system successfully under the upper bound condition of the given average dwell-time when continuous SIDSs with DDs and IDDs could not stochastic stable. Finally, two examples are provided to verify the effectiveness of our results. In future, we will consider apply our method to neural networks [30, 31] and semi-Markov switching systems [32–34].



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