# Stability of Stochastic Differential Switching Systems with Time-Delay and Impulsive Effects

Authors: Zhuang Fang; Xiaozhong Huang; Xuegang Tan
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper studies the stability of hybrid impulsive and switching stochastic neural networks. First, a new type of switching signal is constructed. The stochastic differential switching systems are steerable under the work of the switching signals. Then, using switching Lyapunov function approach, Itô formula, and generalized Halanay’s inequality, some global asymptotical and global exponential stability criteria are derived. These criteria improve the existing results on hybrid systems without noises. An example is given to demonstrate the effectiveness of the results.


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## Body

## 1. Introduction

A switching system consists of several subsystems and a rule that orchestrates the switching among them. Switching plays an important role in many fields, such as optimal control, electrical circuit control, and other practical applications [1, 2]. However, switching is also a main source of instability and often deteriorates system performances. It is important and necessary to study the stability of switching systems [3].Linear switching systems are primary and the earliest researched switching systems [4]. The model can be written as(1)x˙=Aikxt,t∈tk,tk+1,k=0,1,…,where Aik∈Rn×n, x(t)∈Rn is the trivial solution. Switching signal σ:R+→{1,2,…,N} is represented by {ik} according to [tk,tk+1)→ik∈{1,2,…,N}. In general, switching signal σ is a piecewise constant function. In [4], based on a valid Lyapunov function for each subsystem of (1) in some subregion of Rn, the authors presented a state dependent switching rule to stabilize linear system (1) if the systems exist in a Hurwitz linear convex combination, H=∑i=1NβiAi, ∑i=1Nβi=1,  βi∈(0,1). Kim et al. studied the stability of a finite number of linear ordinary differential equation systems with time delay [5].Recently, more and more researchers focus on the nonlinear switching ordinary differential equation systems (see, e.g., [6–12] and references therein). The general nonlinear form is given as follows:(2)x˙=fikxt,t∈tk,tk+1,k=0,1,….Fruitful results on the stability of the switching nonlinear systems are reported. For instance, Wang et al. discussed the finite-time stability problem for a class of impulsive switching systems with nonlinear perturbations and provided a sufficient condition for finite-time stability of perturbed switching systems [7]. In [8], some stability results were given by an average dwell-time approach for nonlinear switching systems. Hu and Michel analyzed a dwell-time scheme for local asymptotic stability of nonlinear switching systems with the activation time being used as the dwell time [10]. Lee and Lim considered continuous-time nonlinear switching systems and provided the stability results for nonlinear time-varying polytypic systems [11].In addition, delay is also considered in many models. In [13, 14], the authors established several criteria on exponential stability for nonlinear systems with time delay by using multiple Lyapunov function technique and a dwell-time approach. Recently, Pu et al. considered the problem of stability of switching delay systems in [1, 15].As is known, practical systems are always affected by external disturbances, which may degrade the system performances. The chief problem that needs solving is how to eliminate the negative effects of the external disturbances as far as possible. With the development of stochastic analysis theory, stochastic systems are proposed to solve this kind of problems. These stochastic systems are adequate mathematical models for many processes and phenomena [16, 17]. Since the stochastic models have much more substance and are also important for many branches of science and engineering [18, 19], many researchers studied the stability of stochastic systems (see [20–25] and references therein). For instance, Li et al. [20] considered the stability of a class of impulsive stochastic neural networks with delays by resorting new integral inequalities and using the properties of spectral radius of nonnegative matrix. In their methods, some global p-exponential stability criteria of periodic solution for impulsive stochastic neural networks with delays were given. Furthermore, in [21–24], stability of some kinds of stochastic systems were discussed and some pth moment stability criteria were obtained for impulsive stochastic systems with Markovian switching.Although the stability of stochastic systems stirred some initial research interest, few works discussed the stability of nonlinear switching differential systems including stochastic, time-delay, and impulsive effects simultaneously. The stability analysis is more involved, since time-delay, impulsive, and stochastic effects are all considered in switching system. Many criteria of stability for differential dynamical systems (such as the systems of paper [26–32]) may be ineffective.To stabilize the systems, it is necessary to construct a suitable switching signal to reduce the negative effects of time delay and external disturbances. It is no doubt that the three factors make the systems more complex, and finding the stability conditions is quite a challenging task. Motivated by the above discussions, in this paper we will establish the stability criteria for hybrid impulsive and switching stochastic neural networks. The model is written as follows:(3)dxt=Aδkxt+fδkt,xt-τdt+gδkt,xt,xt-τdωt,t∈tk-1,tk,Δxtk=Bktk-,xtk-,t=tk,xt0,δt0=ξt0+s,s∈-τ,0,xtk=Zδk,where t∈R+, x∈Rn is the state variable, t0≥0 is the initial time, δ(t):R+→I, I={1,2,…,m}, R+ is the positive real number set, and the time sequence tk satisfies 0≤t0<t1<⋯<tk<⋯, limk→∞⁡tk=∞.In fact, if we do not consider the stochastic factor of systems (3) (i.e., gδ(k)(t,x(t),x(t-τ))=0), the systems reduce to ordinary differential equation systems. For example, the model become a hybrid impulsive and switching NN model without stochastic effects (see [13–16, 26]). If fδ(k)(t,x(t-τ))=0, the model is a linear switching system considered in [3, 4]. If fδ(k)(t,x(t-τ))=Cδ(k)x(t-τ), the model is described in [5]. In addition, systems (3) reduce to an impulsive stochastic differential equation without switching in [27], when δ(k) is a constant function and gδ(k)(t,x(t),x(t-τ))≠0. If Bk=0, the system is a stochastic differential switching system without impulsive effects. By the discussion above, we draw a conclusion that many existing systems are included in systems (3).This paper is organized as follows: The hybrid impulsive and switching stochastic neural networks and some kinds of stability are defined in Section2. Section 3 constructs a type of switching signal and establishes several criteria for global asymptotical and exponential stability of a kind of impulsive stochastic differential switching systems with time delay. In Section 4, a numerical example is given to illustrate the theoretical results. Finally, Section 5 contains some conclusions.Notation.Unless otherwise specified, we employ the definition as follows: Let (Ω,F,{Ft}t≥0,P) be a complete probability space with filtration {Ft}t≥0 satisfying the usual conditions (i.e., F0 contains all p-null sets and Ft is right continuous) and · denote the Euclidean norm of Rn. Let E[·] be the expectation operator with respect to the probability space. ω(t)=(ω1(t),…,ωn(t))T represents n-dimensional Brownian with E(dω(t))=0,E((dω(t))2)=dt.


## 2. Preliminaries

To begin with, we introduce some conditions, basic definitions, and lemmas. Throughout this paper, it is assumed that the solution of systems (3) is unique and existing and satisfies the following conditions in any bounded interval 0<tk-tk-1≤T:(C1)
Bk(tk-,x(tk-)):R+×Rn→Rn(k≥1) are all continuous functions, and B1(t,0)≡0 for any t∈R+.(C2)
There exist nonnegative constant sequences{Eδ(k)} and {Fδ(k)} such that
(i)
fδkt,xt+gδkt,xt≤Eδk1+xt,(ii)
fδ(k)(t,x)-fδ(k)(t,y)+gδ(k)(t,x)-gδ(k)(t,y)≤Fδkx-y.(C3)
Letx(tk)=Zδ(k) be a random variable which is independent of the δ-algebra F∞(m) generated by ω(·) (ω(·) is an m-dimensional normal Brownian motion), and EZδk2≤M<∞.LetTi(t0,t)=∑k∈Λ[min⁡(t,tk)-tk-1] denote the working time of the ith subsystem during the interval [t0,t],  μ(Ti(t0,t)) denote the Lebesgue measure of the set Ti(t0,t),  k0=min⁡{k∈Z,tk≥k},  Λ={1,2,…,k0}∩{k∈Z,δ(k)=i}. Then systems (3) can be written as(4)dxs=Aixs+fis,xs-τds+gis,xs,xs-τdωs,s∈Tit0,t,Δxs=Bks-,xs-,s=tk,xt0,δt0=ξt0+l,l∈-τ,0,xsk=Zi,where i∈I and ⋃i=1NTi(t0,t)=[t0,t], fi(0,x(0))=gi(0,x(0))=0. Let Γ={ti:0≤t1<t2<⋯<ti<⋯} and Rδ={x∈R:x≥δ}, where δ∈R is a given constant, and C2,1(Rn×Rδ×I;R+) denote a family of all nonnegative functions V(x(t),t,i) on Rn×Rδ×I that are twice continuously differentiable in x and once in t. If V(x(t),t,i)∈C2,1(Rn×Rδ×I;R+), define an operator L associated with systems (3) and (4) from Rn×Rδ×I to R by(5)LVxt,t,i=Vtxt,t,i+Vixt,t,iAixt+fit,xt-τ+12tracegiTt,xt,xt-τVxxxt,t,igit,xt,xt-τ,where(6)Vtxt,t,i=∂Vxt,t,i∂t,Vxxxt,t,i=∂2Vxt,t,i∂xi∂xjn×n,Vxxt,t,i=∂Vxt,t,i∂x1,∂Vxt,t,i∂x2,…,∂Vxt,t,i∂xn.Definition 1.
(i) Asymptotically stable in mean square: the trivial solution of systems (3) and (4) is said to be asymptotically stable in mean square, for ε>0, if there exists δ=δ(ε,t0)>0 such that(7)Ext02≤δ,limt→∞Ext2=0,t≥t0.(ii) Exponentially stable in mean square: the trivial solution of systems (3) and (4) is said to be mean square exponentially stable if there exists positive constant α>0 such that(8)Ext2≤Ext02e-αt-t0,t>t0≥0.Definition 2.
D + f ( x ) = lim Δ x → 0 + ⁡ f ( x + Δ x ) - f ( x ) / Δ x if f(x) is differentiable at its right side.Furthermore, we introduce the following lemmas.Lemma 3 (a generalized Halanay’s inequality [33]).
Letω(t) be a nonnegative function defined on the interval [t0-τ,∞) and be continuous on the interval [t0,∞). Assume that(9)ω˙t≤-atωt+btωt-τ,t≥t0,where a(t) and b(t) are nonnegative functions satisfying a(t)>b(t). Then(10)ωt≤ω¯0exp⁡-∫t0tλds,where ω¯=supt0-τ≤θ≤t0⁡ω(θ) and λ>0 satisfying(11)λ-at+bteλτ=0.Lemma 4.
For anyt1<t2, if  V(x,t,i)∈C2,1(Rn×Rδ×S;R+), then(12)EVxt2,t2,δt2=EVxt1,t1,δt1+E∫t1t2LVxs,s,δsds.Lemma 5.
For anyx,y∈Rn, there exist nonnegative functions hi(t), such that(13)fiTt,xy≤hitxTy,i∈1,2,…,N,t≤t0.Lemma 6.
For alli∈I, if a(t)≥α(t)>α0(t)>υ(t)≥b(t), then there exist λ and λ0 such that λ-≥λ>λ0>0 satisfying λ-α(t)+υ(t)eλτ=0 and λ0-α0(t)+υ(t)eλ0τ=0, where a(t), b(t), and λ- are defined in Lemma 3.Lemma 7.
Letω(t) be a nonnegative function defined on the interval [t0-τ,∞) and be continuous on the interval [t0,∞). Assume that(14)ω˙t≤-aωt+bωt-τ,t≥t0,where a and b are nonnegative constants satisfying a>b. Then(15)ωt≤ω¯0exp⁡-∫t0tλds,where ω¯=supt0-τ≤θ≤t0⁡ω(θ) and λ>0 satisfying(16)λ-a+beλτ=0.


## 3. Stability Analysis

Before stating our main results, we need to construct switching signals for systems (3) and (4). In fact, if there exist positive-definite and symmetric matrices Pi, such that(17)Q=∑i=1NαiAiTPiT+PiAi,where Q is a negative matrix, αi∈[0,1], and ∑i=1Nαi=1, then the solution of system (4) is stable. Thus, for any x≠0, we obtain that(18)∑i=1NαixTAiTPiT+PiAix=xTQx≤0.Note αi∈[0,1]. Based on formula (18), we know that there at least exist αi, i∈{1,2,…,N} such that(19)αixTAiTPiT+PiAix<0.We construct switching signal area Ωi as follows:(20)Ωi=x∈Rn:xTAiTPiT+PiAix≤xTQx.Theorem 8.
Assume thatΩi has the same form as formula (20), then(21)Rn=⋃i=1NΩi.Proof.
Suppose that formal (21) does not hold, then there exists a set D⊂Rn, such that D=Rn∖⋃i=1NΩi. Thus, for any i∈{1,2,…,N}, x∈D, we obtain that(22)xTAiTPiT+PiAix>xTQx.For any αi∈[0,1], ∑i=1Nαi=1, we have(23)∑i=1NαixTAiTPiT+PiAix>∑i=1NαixTQx=xTQx.However, based on formula (18), formula (23) is impossible to hold. This completes the proof.Based on Theorem8, we can find N subdomain of Rn for systems (3) and (4). Several new subdomains Ω-i(Ω-i⊂Ω-j,i≤j,i,j=1,2,…,N) are structured as follows:(24)Ω-i=⋃j=1iΩj,i=1,2,…,N.Theorem 9.
Assume that systems (3) and (4) satisfy the following conditions:(C4)
There exist symmetric and positive-definite matricesPi, such that Q=∑i=1Nαi(AiTPiT+PiAi), where αi∈[0,1], ∑i=1Nαi=1, and Q is negative definite matrix.(C5)
For anyx,y∈Rn, there exist continuous functions hi(t)≥0 such that(25)fiTt,yy≤hitxTy,i∈1,2,…,N,t≥t0.(C6)
trace[giT(t,x(t,x(t-τ)))gi(t,x(t),x(t-τ))]≤λi1xt2+λi2xt-τ2.(C7)
λmax(Pi-1)(λmax(Q)+λi(1)λmax(Pi)+(λmax(PiTSiPi))hi(t))+αi(t)<0,t≥t0,i∈{1,2,…,N}.(C8)
∑j=1k-1lnλmax(Pi)/λmin(Pi)+2ln⁡1/λmin(E-Bj)-∑i=1N(λi(tj-tj-1))≤φ(t0,t),t∈[tk-1,tk), where λi is the unique positive root of the equation -αi(t)+νi(t)eλiτ+λi=0.
Then, it follows thatφ(t0,t)≤-c(t-t0), c>0, t≥t0, implies that trivial solution of systems (3) and (4) is globally exponentially stable in mean square and limt→∞⁡φ(t0,t)=-∞ implies that the trivial solution of systems (3) and (4) is globally asymptotically stable in mean square.Proof.
Suppose(26)Ωi=x∈Rn:xTAiTPT+PiAix≤xTQx,i∈1,2,…,N,where Pi are positive-definite matrices. We have Rn=⋃i=1NΩi and construct the Ω-i as formula (24). Construct the switching Lyapunov function by(27)Vit=xTtPixt,t∈tk-1,tk,and set the switching rule as follows:(28)δ=N-i+1,i∈1,2,…,N,where x(t)∈Ω-N∖Ω-N-i and there at least exists t∗ such that x(t∗)∈Ω-N-i-1, Ω-0≜Ø. When x(t)∈Ω-N∖Ω-i-1,  t∈[ti-1,ti), there exists t∗ such that x(t∗)∈Ω-i. Some x(t) have access to Ω-i∖Ω-i-1 and switching signal δ=i under the impulse effects. Using Itô’s formula in (27) yields(29)dVit=LVitdt+git,xt∂Vit∂tdωt.Based on formulas (5) and (6), we obtain that(30)LVit=xTtPiAixt+fit,xt-τ+Aixt+fit,xt-τTPixt+tracegiTt,xt,xt-τgit,xt,xt-τ.Thus(31)LVit=xTtAiTPixt+2fiTt,xt-τPixt+xTPiAixt+12tracegiTt,xt,xt-τPigit,xt,xt-τ⩽xTtAiTPi+PiAixt+2hitxTt-τPixt+λi1xTtPixt+λi2xTt-τPixt-τ⩽xTtAiTPi+PiAixt+hitxTt-τSi-1xt-τ+xTtPiTSiPixt+λi1xTtPixt+λi2xTt-τPixt-τ⩽xTtλmaxQi+λmaxPiTSiPhit+λi1λmaxPixt+xTt-τSi-1hit+λi2Pixt-τ⩽λmaxPi-1λmaxQi+λmaxPiTSiPhit+λi1λmaxPixTtPixt+λmaxSi-1hit+λi2PiPi-1xTt-τPixt-τ⩽-αitVit+νitVit-τ.Applying Lemma 3 to (31) yields(32)EVit≤EVitk-1exp-λitk-tk-1,t∈tk,tk-1,where λi satisfies -αit+νiteλiτ+λi=0.
Considering the discrete part of system (4), we can obtain that(33)λminPiExt2⩽EVit⩽1λmin2E-BjEVitk-1-exp-λitk-tk-1.Thus(34)Ext2⩽1λmin2E-BjλmaxPiλminPiExtk-1-2exp-λitk-tk-1⋮⩽∏j=1k-11λmin2E-BjλmaxPiλminPiexp-∑i=1Nλitj-tj-1Ext02⩽Ext02exp∑j=1k-12ln⁡1λminE-Bj+ln⁡λmaxPiλminPi-∑i=1Nλitj-tj-1⩽Ext02eφt0,t,t∈tk-1,tk.When φ(t0,t)≤-c(t-t0),  c>0, t≥t0, the trivial solution of systems (3) and (4) is globally asymptotically exponentially stable in mean square. When limt→∞⁡φ(t0,t)=-∞, the trivial solution of systems (3) and (4) is globally asymptotically stable.Remark 10.
Theorem8 gives general criteria for the asymptotic and exponential stability of systems (3) and (4) consisting of stable and/or unstable modes. If g(t,x(t),x(t-τ))≡0,  t∈[t0,+∞), the stochastic part has been missed. That is to say, the systems have become hybrid impulse and switching systems with delay, which have been discussed in [7–16]. If g(t,x(t),x(t-τ))≡0,  t∈[t0,+∞) and τ=0, the models have been studied in [11]. Furthermore, switching systems without impulses have been investigated in [13].Remark 11.
Ifhi=Li are all constants, then inequality (31) satisfies the Halanay inequality (Lemma 7). The inequality can be solved with relevance linear matrix inequality (LMI). The problem turns to solve an optimization problem as follows:(35)OPminαi,s.t.relevanceLMIholds,where αi satisfies inequality (33).Remark 12.
In Theorem9, inequality of (C8) characterizes the switching effect -∑i=1Nλitk-tk-1 and the impulse effect ∑j=1k-12ln⁡1/λminE-Bj+ln⁡λmaxPi/λminPi in an aggregated form. There is no particular limitation to the switching sequence and switching subinterval.Corollary 13.
For systems (3) and (4), suppose conditions ((C1)–(C4)) hold; if there exists positive constant α such that(36)2ln⁡1/λminE-Bk+ln⁡λmaxPi/λminPitk-tk-1⩽α,k=1,2,…,where λi≥α>0, then the trivial solution is globally exponentially stable.Proof.
Letλ=min1⩽i⩽N⁡λi. Based on (31), we have(37)∑j=1k-12ln⁡1λminE-Bj+ln⁡λmaxPiλminPi-∑i=1Nλitj-tj-1⩽∑j=1k-12ln⁡1λminE-Bj+ln⁡λmaxPiλminPi-λ∑j=1k-1tj-tj-1⩽α∑j=1k-1tj-tj-1-λ∑j=1k-1tj-tj-1⩽-λ-αt-t0,t∈tk-1,tk,k=1,2,….Then the trivial solution of systems (3) and (4) is globally exponentially stable. The proof of the corollary is completed.Remark 14.
Based on Corollary13, the trivial solution of systems (4) is globally asymptotically exponentially stable if there exists α such that 0⩽α<λi. Furthermore, the convergence rate of the trivial solution is min(λi)-α/2,i∈I.Corollary 15.
For alli∈I, if αi⩾α>α0>ν⩾νi, the trivial solution of systems (3) and (4) is globally exponentially stable provided that(38)2ln⁡1λminE-Bk+ln⁡λmaxPiλminPi-λ0tk-tk-1⩽0,i∈I,k∈1,2,…,where λi⩾λ>λ0>0 and λi satisfies -αit+νiteλiτ+λi=0.Proof.
According to Lemma6, there exist λ and λ0 such that λi≥λ>λ0>0 satisfying -αt+ηteλτ+λ=0 and -α0t+ηteλ0τ+λ0=0. Note that(39)2ln⁡1λminE-Bk+lnλmaxPiλminPi⩽λ0ti-ti-1,i∈I,k∈1,2,….Thus(40)∑j=1k-12ln⁡1λminE-Bj+ln⁡λmaxPiλminPi⩽λ0t-t0.Since λi⩾λ>λ0, we have(41)∑j=1k-12ln⁡1λminE-Bj+ln⁡λmaxPiλminPi-∑i=1Nλitj-tj-1⩽-λ-λ0t-t0.Then the proof is completed.Remark 16.
It is easy to see that the expectation of trivial solution of systems (3) and (4) satisfies Ext⩽Ke-λ-λ0/2t-t0, and the convergence rate of the trivial solution is λ-λ0/2.Theorem 17.
For anyt∈[tk-1,tk), suppose that conditions ((C1)–(C4)) hold, there are hit⩽Li, and(42)∑j=1k-12ln⁡1λminE-Bj+ln⁡λmaxPiλminPi-∑i=1Nλitj-tj-1⩽φt,t0,where αi, νi satisfy -αi+νieλτ+λ=0. Then limt→∞⁡φt,t0=-∞ implies that the trivial solution of systems (3) and (4) is globally asymptotically stable, and φt,t0=-ct-t0,  t⩾t0, c>0, which implies that the trivial solution of systems (3) and (4) is globally exponentially stable.Proof.
Reconstruct the switching Lyapunov function (27). Based on formula (30) and Lemma 7, we obtain that(43)LVit⩽λmaxQiPi-1+hitλmaxPiTSi+λi1xTtPixt+λi2+hitλmaxSi-1Pi-1xTt-τPixt-τ⩽λmaxQiPi-1+LiλmaxPiTSiT+λi1Vit+λi2+LiλmaxSi-1Pi-1Vit-τ⩽-αiVit+νiVit-τ.Thus(44)EVit⩽EVitk-1exp-λitk-tk-1,t∈tk-1,tk.Considering the discrete part of system (4), one obtains that(45)λminPiExt2⩽EVit⩽1λmin2E-BkEVitk-1-exp-λitk-tk-1.Therefore(46)Ext2⩽1λmin2E-BkλmaxPiλminPiExtk-1-2exp-λitk-tk-1⋮⩽∏j=1k-11λmin2E-BjλmaxPiλminPiexp∑i=1N-λitj-tj-1Ext02⩽Ext02exp∑j=1k-12ln⁡1λminE-Bj+ln⁡λmaxPiλminPi-∑i=1Nλitj-tj-1⩽Keφt0,t,t∈tk-1,tk.Then limt→∞⁡φt,t0=-∞ implies that the trivial solution of system (3) and (4) is globally asymptotically stable, and φt,t0=-ct-t0,  t⩾t0, c>0, implies that the trivial solution of systems (3) and (4) is globally exponentially stable.Remark 18.
Similarly, inequality (43) can be solved by linear matrix inequality (LMI) with αi. The problem turns to solve an optimization problem.Remark 19.
In Theorem17, a general criterion for asymptotic and exponential stability of stochastic systems (3) and (4) has been given. Inequality (41) includes the switching effect -∑i=1Nλi(tk-tk-1) and the impulse effect ∑j=1k-12ln⁡1/λminE-Bj+ln⁡λmaxPi/λminPi in an aggregated form. In order to keep φ(t,t0)→-∞ or φ(t,t0)=-c(t-t0), the impulse effect ∑j=1k-12ln⁡1/λminE-Bj+ln⁡λmaxPi/λminPi should be negative or bounded. Thus, we obtain the following corollaries.Corollary 20.
In Theorem17, if impulse effect 2ln⁡1/λminE-Bk+ln⁡λmaxPi/λminPi<0, then the trivial solution of stochastic systems (3) and (4) is globally exponentially stable.Proof.
Based on inequality (46), we have(47)∑j=1k-12ln⁡1λminE-Bj+lnλmaxPiλminPi-∑i=1Nλitj-tj-1⩽-∑i=1Nλitj-tj-1⩽-minλi∑j=1k-1tj-tj-1⩽-ct-t0,t∈tk-1,tk.Let φt,t0=-ct-t0,  c>0; the trivial solution of stochastic systems (3) and (4) is globally exponentially stable.Corollary 21.
In Theorem17, if impulse effect 2ln⁡1/λminE-Bk+ln⁡λmaxPi/λminPi<M(k+1-k)/k+1k, where M is a positive constant, then the trivial solution of stochastic systems (3) and (4) is globally asymptotically exponentially stable.Proof.
Similarly, we obtain that(48)∑j=1k-12ln⁡1λminE-Bj+ln⁡λmaxPiλminPi-∑i=1Nλitj-tj-1⩽∑j=1k-1Mk+1-kk+1k-∑i=1Nλitj-tj-1⩽1-1kM-minλi∑j=1k-1tj-tj-1⩽M-ct-t0,t∈tk-1,tk.Let φt,t0=M-ct-t0,c>0. The trivial solution of systems (3) and (4) is globally asymptotically stable.


## 4. Numerical Simulation

In this section, an example is presented to illustrate the main theoretical results.Consider systems (4) as follows:(49)dxt=A1xt+C1f1t,xt-τdt+g1t,xt,xt-τdωt,t∈kT,k+θT,Δxt=xt-xt-=B1xt,tk=k+θT,k=1,2,…,dxt=A2xt+C2f2t,xt-τdt+g2t,xt,xt-τdωt,t∈k+θT,k+1T,Δxt=xt-xt-=B2xt,tk=k+1T,k=1,2,…,where T=4, τ=1.2, B1=1.9614, B2=1.5437, A1=A2=-200-2, C1=0.65-0.430.10.63, C2=0.440.53-0.310.4,(50)fit,xt-τ=sin⁡tx+1-x-1.Let D1=0.65-0.430.10.63, D2=0.440.53-0.310.4,(51)git,xt,xt-τ=Disin⁡txt+sin⁡t-τxt-τ,i=1,2,t⩾t0.The switching sequence is as follows: subsystem1→subsystem2→subsystem1→⋯.Construct the switching Lyapunov function as formula (27), where xTt=x1t,x2t and(52)Pi=2.43470.03240.03242.7938.Based on Corollary 13, we obtain that 2ln⁡1/λminE-Bk+ln⁡λmaxPi/λminPi1/tk-tk-1=0.1137  ⩽ 0.5,  k=1,2,…. Thus, the trivial solution of systems (49) is exponentially stable, as shown in Figure 1.Figure 1
Trajectories of systems (49) with the initial value x(s)=[1.0,-1.48],  s∈[-0.4,0].The example results show that the stability of time-varying case can be guaranteed by our method. It is easy to see that model (49) and our conclusions can be degenerated into the general case if the αi(t) and νi(t) are constants. It is the extension of [1, 15, 26]. Thus, compared with the existing results discussed in [1, 15, 26], our results are less conservative. The tolerable delay upper bound of (49) is larger as shown in Table 1.Table 1
Maximum allowedτ.
Methods Our method Pu and Tan [1] Xing-cheng and Wei [15] Li et al. [26] τ 1.20 1.08 1.13 1.16


## 5. Conclusion

In this paper, we have discussed the stability of hybrid impulsive and switching stochastic neural networks. A type of switching signals is constructed. Using switching Lyapunov functions and stochastic analysis techniques, several general criteria for the asymptotic and exponential stability analysis of the new model are established. To illustrate the advantage of the results, a numerical example is given.



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