# Input-to-State Stability Analysis for Stochastic Mixed Time-Delayed Neural Networks with Hybrid Impulses

Authors: Chi Zhao; Yinfang Song; Quanxin Zhu; Kaibo Shi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper investigates the mean-square exponential input-to-state stability (MEISS) of stochastic mixed time-delayed neural networks with hybrid impulses. A generalized comparison principle is introduced and a new inequality about the solution of an impulsive differential equation is established. Moreover, by utilizing the proposed inequality and average impulsive interval approach based on different kinds of impulsive sequences, some novel criteria on MEISS are established. When the external input is removed, several conclusions on mean-square exponential stability (MES) are also derived. Unusually, the hybrid impulses including destabilizing and stabilizing impulses have been taken into account in the presented system. Finally, two simulation examples are provided to demonstrate the validity of our theoretical results.


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## Body

## 1. Introduction

Recently, neural networks have absorbed plenty of researchers’ interests owing to their extensive applications in a variety of areas including system pattern identification, wireless communications, optimization problems, and machine learning [1, 2]. Actually, the majority of the applications are related to the stability of equilibrium points. Hence, it is of great significance to analyze the stability of network systems. Moreover, stochastic disturbances exist inevitably, which affect the dynamic properties of systems. Meanwhile, in view of the limited switching velocity of the amplifier in the hardware implementation, it is often encountered with time delay, which leads to the instability and oscillations of associated systems. Subsequently, abundant results about stability analysis of stochastic network systems are attained by utilizing various methods [3–7].Since instantaneous perturbations or abrupt changes at certain moments appear unpredictably in the real environment, impulsive effects are incorporated to depict the phenomenon in neural networks. Generally, impulsive sequences can be classified into stabilizing impulses and destabilizing impulses. Impulsive sequences are called to be stabilizing impulses if they can promote the stability of differential systems, while destabilizing impulses can suppress the stability of differential systems. Stability and synchronization problems of a dynamical system with different categories of impulses have become an interesting research topic, and some results with respect to the problems have been reported in [8–16]. For instance, in [8], the asymptotic stability of impulsive recurrent neural networks with stochastic disturbances and time delays was discussed by virtue of the Lyapunov functional approach and LMI technique. In [9], the synchronization problem of stochastic memristor-based recurrent neural networks with impulsive effects was explored by utilizing the impulsive differential inequality. In the real world, some phenomena about hybrid impulses comprising stabilizing and destabilizing impulses simultaneously often occur. They can be observed in fishing management with impulsive harvesting and releasing, goods selling involving impulsive stocking and transferring, and ball motion process with impulsive accelerating and decelerating. Moreover, some stability results of neural network systems with hybrid impulses have been attained by employing comparison theory in [10, 11]. Subsequently, instead of general impulses, the exponential synchronization issue of complex networks with hybrid impulses was also tackled in [12, 13], where the approach of average impulsive gain was introduced based on the average impulsive interval method in [14].Additionally, the external inputs have great influences on the dynamic systems. To describe accurately how external perturbations impact the asymptotical properties of the control systems, the definition of input-to-state stability (ISS) was incorporated in [17]. Noting that the external disturbances or inputs also appear in the network systems, it is significant and meaningful to investigate the ISS of neural networks. Recently, many researchers paid considerable attention to the field, and plenty of achievements on ISS have emerged [18–26]. For instance, in [18], the ISS property for dynamical neural networks was analyzed by using the Lyapunov stability theory. In [19], a nice passive weight learning rule was designed for switched Hopfield neural networks, and some asymptotic stability and ISS results were proposed. Furthermore, some results about ISS analysis were extended to stochastic neural networks. Particularly, in [21, 22], MEISS of stochastic recurrent neural networks was discussed through the Lyapunov functional method. In [23], some algebraic conditions were derived to guarantee the mean-square stability ISS of stochastic network systems with Markovian switching on the basis of vector inequality methods and stochastic analysis techniques. Furthermore, robust input-to-state stability of stochastic neural networks with Markovian switching was examined in [24] under two circumstances by means of M-matrix theory. In [25], a novel criterion about MEISS of stochastic neural networks with multiproportional delays were established applying the variable transformations approach. More recently, the ISS of delay systems with hybrid impulses was studied in light of the Razumikhin method in [26]. As far as we know, up until now, although stability and synchronization problems of neural networks with multiple impulses have been resolved, the ISS properties of stochastic network systems with hybrid impulses have not been explored. Therefore, the analysis of influence for hybrid impulses and external input on system states becomes a significant topic.Motivated by the previous considerations, this paper focuses on the MEISS of stochastic neural networks with hybrid impulses. The essential innovations are summarized as below. First of all, a generalized comparison principle is introduced and a new inequality about the solution of the impulsive differential equation is achieved. Secondly, different from the existing works, hybrid impulses including destabilizing impulses, stabilizing impulses, and external input are taken into account simultaneously, which reflected reality more accurately and makes the addressed system more complex. Finally, some new criteria on MEISS and MES of stochastic neural networks with hybrid impulses are established by the average impulsive interval approach based on different kinds of impulsive sequences. The structure of our paper is arranged appropriately. Section2 proposes some preliminaries including mathematical models, assumptions, definitions, and lemmas. In Section 3, based on two proposed lemmas, several criteria on MEISS and MES of neural networks with hybrid impulses are established. Some simulation examples are provided in Section 4, and conclusions are drawn in the last section.Notations 1.
LetΩ,ℱ,ℱt,ℙ be a complete probability space with a filtration ℱtt≥t0 satisfying the usual conditions, we set ℝ+=0,+∞, ℕ+=1,2,…. PCa,b;ℝn represents the family of all piecewise continuous functions from a,b to ℝn. Lℱt02t0−τ0,t0;ℝn denotes the family of all ℱt0− measurable PCt0−τ0,t0;ℝn -valued random processes φ=φs|t0−τ0≤s≤t0 such that supt0−τ0≤s≤t0Eφs2<∞. λmax⋅ stands for the largest eigenvalue of a matrix. u∞=supt≥t0ut denotes the infinite norm of the input function ut. Dini-derivative D+⋅ is defined by D+vt=lim¯ε⟶0+vt+ε−vt/t.


## 2. Preliminaries

Consider the following class of neural networks with stochastic disturbances and mixed delays:(1)dxt=−Axt+ℬfxt+Cgxt−τt+D∫t−τ0thxsdsdt+ϱt,xt,xt−τtdwt,xs=φs,t0−τ0≤s≤t0,where xt=x1,x2,…,xnT∈ℝn denotes the state vector of the neurons. A=diaga1,a2,…,an>0 is the self-feedback matrix. ℬ=bijn×n, C=cijn×n, and D=dijn×n represent the connection weight strength matrices. fxt=f1xt,f2xt,…,fxntT, gxt=g1xt,g2xt,…,gxntT, and hxt=h1xt,h2xt,…,hxntT are the activation vector functions. Additionally, wt denotes an n-dimensional independent standard Wiener process, and function ϱt,xt,xt−τt∈ℝn×n stands for the noise intensity. We suppose that τt satisfies that 0≤τt≤τ0. Moreover, some assumptions are imposed on the neuron activation functions and noise intensity functions.Assumption 1.
Suppose that activation functionsfix,gix, and hix with fi0=0,gi0=0,hi0=0 satisfy the globally Lipschitz continuous condition, i.e., there exist some positive constants li, pi, qi, and i=1,2,…,n such that(2)fiι1−fiι2≤liι1−ι2,giι1−giι2≤piι1−ι2,hiι1−hiι2≤qiι1−ι2,hold for any ι1,ι2∈ℝn, which indicate that(3)diagf1ι1−f1ι2,f2ι1−f2ι2,…,fnι1−fnι2≤Lι1−ι2;diagg1ι1−g1ι2,g2ι1−g2ι2,…,gnι1−gnι2≤Pι1−ι2;diagh1ι1−h1ι2,h2ι1−h2ι2,…,hnι1−hnι2≤Qι1−ι2,where L=diagl1,l2,…,ln, P=diagp1,p2,…,pn, and Q=diagq1,q2,…,qn are the diagonal matrices.Assumption 2.
Suppose that the noise intensity functionϱt,xt,xt−τt satisfies the globally Lipschitz condition ϱt,0,0=0. Furthermore, there exist two symmetric real matrices M1,M2 such that(4)tracϱTt,xt,xt−τtϱt,xt,xt−τt≤xTtM1xt+xTt−τtM2xt−τt.
By incorporating impulsive jumps and external input functionut=u1t,u2t,…,untT∈PCt0,+∞;ℝn, system (1) is rewritten as follows:(5)dxt=−Axt+ℬfxt+Cgxt−τt+D∫t−τ0thxsds+utdt+ϱt,xt,xt−τtdwt,t≠tk,xtk=γ1kxtk−+γ2kutk−,t=tk,xs=φs,t0−τ0≤s≤t0,where γ1k,γ2k are bounded constants. Moreover, we assume that there exists a positive constant γ0 satisfies γ2k≤γ0.Definition 1.
The trivial solutions of system (1) are said to be MEISS, if for arbitrary φ∈Lℱ02t0−τ0,t0;ℝn and ut∈PCt0−τ0,∞;ℝn, there exist positive constants α0, β0, ε0 such that(6)Ext,φ2≤α0e−β0t−t0Eφ2+ε0u∞2.Lemma 1.
Assumeϑt,zt,ζ1t, and ζ2t are piecewise continuous functions. The time-varying delay τt and impulsive sequences tk satisfy that 0≤τt≤τ0 and tk+1−tk≥τ0. If there exist some constants αj, βk, and j=1,2,3, k∈ℕ+ satisfying the following inequalities:(7)D+ϑt≤α1ϑt+α2ϑt−τt+α3∫t−τ0tϑsds+ζ1t,t≠tk;ϑtk≤βkϑtk−+ζ2tk−,t=tk,,and(8)D+zt>α1zt+α2zt−τt+α3∫t−τ0tzsds+ζ1t,t≠tk;ztk≥βkztk−+ζ2tk−,t=tk,then ϑt≤zt for t0−τ0≤t≤t0 implies that ϑt≤zt,t>t0.Proof.
This proof procedure is completely parallel to Lemma1 in [15], so the process is omitted here.Lemma 2.
Letα1, α2≥0, α3≥0, βk≥0, and τ0≥0 be some constants. ζ˜1t and ζ˜2t are nonnegative bounded functions. Delay τt and impulsive sequences tk satisfy that 0≤τt≤τ0, tk+1−tk≥τ0. If the following impulsive differential equation with initial value xs=ϕs≥0,t0−τ0≤s≤t0 holds(9)z′t=α1zt+α2zt−τt+α3∫t−τ0tzsds+ζ˜1t,t≠tk,ztk=βkztk−+ζ˜2tk−,t=tk,k∈ℕ+,,then one can derive that(10)zt≤∏t0<tk≤tβkzt0eα1t−t0+∑t0<tk≤t∏tk<tm≤tβmζ˜2tkeα1t−tk+∫t0t∏s<tk≤tβkeα1t−sα2zs−τs+α3∫s−τ0szϑdϑ+ζ˜1sds.Proof.
Whent∈t0,t1, by (9), one gets that(11)e−α1t−t0zt′=e−α1t−t0α2zt−τt+α3∫t−τ0tzsds+ζ˜1t.
By integrating the above equation, it yields that(12)zt≤zt0eα1t−t0+∫t0teα1t−sα2zs−τs+α3∫s−τ0szϑdϑ+ζ˜1sds,which implies that (10) holds in the time interval t0,t1. Moreover, we assume that for t∈t0,tk, the assertion (10) holds. Therefore, for t∈tk,tk+1, we derive that(13)zt=ztkeα1t−tk+∫tktα2zs−τs+α3∫s−τ0szϑdϑ+ζ˜1sds≤βkztk−eα1t−tk+eα1t−tkζ˜2tk−+∫tktα2zs−τs+α3∫s−τ0szϑdϑ+ζ˜1sds≤βkeα1t−tk∏t0<tm<tkβmzt0eα1tk−t0+∑t0<tm<tk∏tm<ti<tkβiζ˜2tmeα1tk−tm+∫t0tk∏s<tm<tkβm×eα1tk−sα2zs−τs+α3∫s−τ0szϑdϑ+ζ˜1sds+eα1t−tkζ˜2tk−+∫tktα2zs−τs+α3∫s−τ0szϑdϑ+ζ˜1sds≤∏t0<tk≤tβkzt0eα1t−t0+∑t0<tk≤t∏tk<tm≤tβmζ˜2tkeα1t−tk+∫t0t∏s<tk≤tβkeα1t−sα2zs−τs+α3∫s−τ0szϑdϑ+ζ˜1sds.
We can get the assertion (10).Remark 1.
In Lemma2, when βk<1, the impulses are called to be stabilizing impulses since the absolute value of the state is reduced, and it convergences to the equilibrium point. When βk>1, the impulses are called to be destabilizing impulses since the absolute value of the state is enlarged and it is away from the equilibrium point. Hybrid impulses include stabilizing impulses and destabilizing impulses simultaneously. Recently, the stability and synchronization problems of neural networks or complex networks with mixed impulses have been discussed [11–14]. Furthermore, the ISS property of nonlinear delay systems with multiple impulses was examined by the Razumikhin method in the article [26]. Based on the existing results [26], this paper aims to investigate the MEISS of stochastic neural networks with hybrid impulses.
Taking the stabilizing and destabilizing impulses into account, we suppose that the values of stabilizing impulsive strengthsβk belong to one finite set δi|δi<1,i=1,2,…,l¯1 while the values of destabilizing impulsive strengths βk belong to the other finite set ηj|ηj>1,j=1,2,…,l¯2. The following assumption is further introduced.Assumption 3.
Suppose that there exist positive constantsN0≥0, λi>0, and μj>0i=1,2,…,l¯1,j=1,2,…,l¯2 such that χ1iT,t≥T−t/λi−N0, χ2jT,t≤T−t/μj+N0, where χ1iT,t and χ2jT,t respectively denote the amount of the stable impulsive sequence with strength δi and destabilizing impulsive sequence with impulsive strength ηj on the time interval t,T. Besides, the impulsive activation times tk satisfy that inftk−tk−1=κ≥τ0>0.Remark 2.
From the above assumption, we can find thatλi and μj stand for the average dwell-time of stabilizing and destabilizing impulsive sequences, respectively. δi and ηj represent the impulsive strength of stabilizing and destabilizing impulsive sequences, respectively. Let t^ik and tˇjk denote the activation moments of the stabilizing impulses and the destabilizing impulses. If maxt^ik−t^ik−1=λi and inftˇjk−tˇjk−1=μji=1,2,…,l¯1,j=1,2,…,l¯2, where l¯1 and l¯2 represent the finite positive integers, then we have that χ1iT,t≥T−t/λi−1, χ2jT,t≤T−t/μj+1.


## 3. Main Results

In this section, according to the proposed lemmas, several criteria on MEISS and MES of stochastic delay hybrid impulses neural networks are established by utilizing the comparison principle and stochastic Lyapunov function approach.Theorem 1.
Letβk=2γ1k2 represent the impulsive strengths, which take values from two finite sets δi|0<δi<1,i=1,2,…,l¯1 and ηj|ηj>1,j=1,2,…,l¯2. Under Assumptions 1, 2, and 3, if there are some parameters ϵi>0,i∈1,2,3,4 satisfying the following inequality,(14)α0−q0α2−q0α3τ0>0,then system (1) is MEISS, where α0=−∑i=1l¯1lnδi/λi+∑j=1l¯2lnηj/μj+α1, α1=λmax−A+AT+ϵ1ℬℬT+1/ϵ1LTL+ϵ2CCT+ϵ3DDT+ϵ4I+M1, α2=λmax1/ϵ2PTP+M2, α3=λmaxτ0/ε3QTQ, and q0=∏i=1l¯11/δi∏j=1l¯2ηjN0.Proof.
The following Lyapunov function is chosen:(15)Vt=xTtxt.
By the Ito^ formula, one can derive that(16)ℒVt=2xTt−Axt+ℬfxt+Cgxt−τt+D∫t−τ0thxsds+ut+tracϱTt,xt,xt−τtϱt,xt,xt−τt≤−xTtA+ATxt+2xTtℬfxt+2xTtCgxt−τt+2xTtD∫t−τ0thxsds+2xTtut+xTtM1TM1xt+xTt−τtM2TM2xt−τt.
By making use of the inequality2xTy≤ϵxTx+1/ϵyTy,x,y∈Rn,ϵ>0, we can obtain that(17)2xTtℬfxt≤2ℬTxtTfxt≤ϵ1xTtℬℬTxt+1ϵ1xTtLTLxt.
Similarly, one has that.(18)2xTtCgxt−τt≤ϵ2xTtCCTxt+1ϵ2xTt−τtPTPxt−τt2xTtD∫t−τ0thxsds≤ϵ3xTtDDTxt+1ϵ3∫t−τ0thxsdsT∫t−τ0thxsds≤ϵ3xTtDDTxt+τ0ϵ3∫t−τ0txTsQTQxsds2xTtut≤ϵ4xTtxt+1ϵ4uTtut.
Then, we derive that(19)ℒVt≤xTt−A+AT+ϵ1ℬℬT+1ϵ1LTL+ϵ2CCT+ϵ3DDT+ϵ4I+M1TM1xt+xTt−τt1ϵ2PTP+M2TM2xt−τt+τ0ϵ3∫t−τ0txTsQTQxsds+1ϵ4uTtut≤α1Vt+α2Vt−τt+α3∫t−τ0tVsds+ζ1t,where α1=λmax−A+AT+ε1ℬℬT+1/ε1LTL+ε2CCT+ε3DDT+ε4I+M1, α2=λmax1/ϵ2PTP+M2, and α3=λmaxτ0/ϵ3QTQ, ζ1t=1/ϵ4uTtut. Moreover, for t≠tk, one can see that(20)D+EVt≤EℒVt≤α1EVt+α2EVt−τt+α3∫t−τ0tEVsds+ζ1t.
In addition, we have that(21)EVtk≤2γ1k2EVtk−+2γ2k2utk−2≤βkEVtk−+ζ2tk−,where βk=2γ1k2,ζ2t=2γ2k2ut2. Together with (20) and (21), we construct the following comparison system:(22)z′t=α1zt+α2zt−τt+α3∫t−τ0tzsds+ζ1t+θ,t≠tk,ztk=βkztk−+ζ2tk−,t=tk,zs=φs2,t0−τ0≤s≤t0,where θ is a sufficiently small positive constant. By virtue of Lemma 1, it can be concluded that EVt≤zt,t≥0. Furthermore, applying Lemma 2 to the system (22), we can derive that(23)EVt≤zt≤∏t0<tk≤tβkH0eα1t−t0+∑t0<tk≤t∏tk<tm≤tβmζ2tk−eα1t−tk+∫t0t∏s<tk≤tβkeα1t−sα2EVs−τ1s+α3∫s−τ0sEVudu+ζ1s+θds,where H0=sup−τ0≤s≤0Exs2. Let q0=∏i=1l¯11/δi∏j=1l¯2ηjN0, α0=−∑i=1l¯1lnδi/λi+∑j=1l¯2lnηj/μj+α1. From Assumption 3, it easily follows that t−t0/λi−N0≤χ1i and t−t0/μj+N0≥χ2j, where χ1i and χ2j represent the jump times of stabilizing impulses and destabilizing impulses. We have that(24)∏t0<tk≤tβkeα1t−t0≤∏i=1l¯1δiχ1i∏j=1l¯2ηjχ2jeα1t−t0≤∏i=1l¯1δit−t0/λi−N0∏j=1l¯2ηjt−t0/μj+N0eα1t−t0≤q0e∑i=1l¯1lnδi/λi+∑j=1l¯1lnηj/μj+α1t−t0≤q0e−α0t−t0.
Accordingly, we acquire that(25)∏tk<tm≤tβmζ2tk−eα1t−tk≤q0ζ2tk−e−α0t−tk,∏s<tk≤tβkeα1t−s≤q0e−α0t−s.
Hence, it follows that(26)EVt≤zt≤q0H0e−α0t−t0+q0∑t0<tk≤tζ2tk−e−α0t−tk+∫t0tq0e−α0t−sα2EVs−τ1s+α3∫s−τ0sEVvdv+ζ1s+θds.
Considering the following equation:(27)ξ−α0+α2q0eξτ0+α3q0eξτ0−1ξ=0.
LetΦξ=ξ−α0+α2q0eξτ0+α3q0eξτ0−1/ξ. Since Φ0+=−α0−q0α2−q0α3τ0<0, Φ+∞>0, and Φξ is a continuous function in the time interval 0,+∞, there is a root satisfying (27). Besides, it is obvious that Φ′ξ>0. Thus, there exists a unique positive root σ such that the above equation holds. Subsequently, we will claim that(28)EVt≤q0H0e−σt−t0+q0ζ1t∞+θα0−q0α2−q0α3τ0+α0q0ζ2t∞1−e−κα0α0−q0α2−q0α3τ0,t∈−τ0,∞.
Whent∈t0−τ0,t0, it can be easily verified that what assertion (28) holds. Let H1=q0ζ1t∞+θ/α0−q0α2−q0α3τ0+α0q0ζ2t∞/1−e−κα0α0−q0α2−q0α3τ0. For t>t0, if inequality (3.6) is not true, on the contrary, then there exists a t¯ such that(29)EVt¯≥q0H0e−σt¯−t0+H1,and for t∈−τ0,t¯(30)EVt≤q0H0e−σt−t0+H1.
Then, we obtain that(31)EVt¯≤q0H0e−α0t¯−t0+q0∑t0<tk≤t¯ζ2tk−e−α0t¯−tk+∫t0t¯q0e−α0t¯−sα2EVs−τ1s+α3∫s−τ0sEVudu+ζ1s+θds≤q0H0e−α0t¯−t0+q0ζ2t∞1−e−κα0+∫t0t¯q0e−α0t¯−sα2EVs−τ1s+α3∫s−τ0sEVudu+ζ1s+θds.
By means of (30), it is noted that(32)∫t0t¯q0e−α0t¯−sα2EVs−τ1sds≤q0∫t0t¯α2e−α0t¯−sq0H0e−σs−τ1s−t0+H1ds≤q02H0α2eστ0eσt0−α0t¯∫t0t¯eα0−σsds+q0α2H11−e−α0t¯−t0α0≤q02H0α2eστ0α0−σe−σt¯−t0−e−α0t¯−t0+q0α2H11−e−α0t¯−t0α0.
Similarly, we have that(33)∫t0t¯q0e−α0t¯−sα3∫s−τ0sEVududs≤q0∫t0t¯α3e−α0t¯−s∫s−τ0sq0H0e−σv−t0+H1dvds≤q02H0α3eσt0−α0t¯eστ0−1σ∫t0t¯eα0−σsds+q0α3τ0H11−e−α0t¯−t0α0≤q02H0α3α0−σeστ0−1σe−σt¯−t0−e−α0t¯−t0+q0α3τ0H11−e−α0t¯−t0α0,∫t0t¯q0e−α0t¯−sζ1s+θds≤q0ζ1s∞+θ1−e−α0t¯−t0α0.
Therefore, one gets that(34)EVt¯≤q0H0e−α0t¯−t0+q0ζ2t∞1−e−κα0+q0H0α2q0eστ0+α3q0eστ0−1σe−σ0t¯−t0α0−σ−e−σt−t0α0−σ+q0α2+α3τ0H1+q0ζ1s∞+θ1−e−α0t¯−t0α0<q0H0e−α0t¯−t0+q0ζ2t∞1−e−κα0+q0H0α2q0eστ0+α3q0eστ0−1σe−σt¯−t0α0−σ−e−α0t¯−t0α0−σ+q0α2+α3τ0H1+q0ζ1s∞+θα0.
Noting thatσ−α0+α2q0eστ0+α3q0eστ0−1/σ=0 and H1=q0ζ1t∞+θ/α0−q0α2−q0α3τ0+α0q0ζ2t∞/1−e−κα0α0−q0α2−q0α3τ0, we have that(35)EVt¯<q0H0e−σt¯−t0+q0ζ2t∞1−e−κα0+1α0α0H1−q0α0ζ2t∞1−e−κα0,which implies that(36)EVt¯<q0H0e−σt¯−t0+H1.
It yields a contradiction with (29). Then, we have that(37)EVt≤q0H0e−σt−t0+q0ζ1t∞+θα0−q0α2−q0α3τ0+α0q0ζ2t∞1−e−κα0α0−q0α2−q0α3τ0.
Letθ⟶0, it yields that(38)EVt≤q0H0e−σt−t0+q0ζ1t∞α0−q0α2−q0α3τ0+α0q0ζ2t∞1−e−κα0α0−q0α2−q0α3τ0.
Sinceζ1t∞=1/ϵ4uTtut∞≤1/ϵ4ut∞2, ζ2t∞=2γ2k2ut2∞≤2γ02ut∞2, EVt=Ext2, one acquires that(39)Ext2≤q0H0e−σt−t0+q01−e−κα0+2γ02α0ϵ4ut∞2ϵ41−e−κα0α0−q0α2−q0α3τ0,which implies that system (1) is MEISS.Remark 3.
In [21, 22], the MEISS of stochastic recurrent neural networks and Cohen-Grossberg neural networks have been investigated. In this paper, impulsive sequences are composed of destabilizing impulses, stabilizing impulses, and external input, which makes our model complicated. Meanwhile, some novel ISS criteria are established by employing impulsive differential inequality and average impulsive interval approach based on different kinds of impulsive sequences. On the other hand, the ISS property of impulsive delay systems with multiple impulses was also analyzed by means of the Razumikhin method in [26], and some restrictive conditions are required such as βk=βN+k. It implies that the impulsive sequences are periodic, which is not necessary for our results. Besides, compared with the existing results [26], our results are applied to stochastic neural networks and the sufficient conditions are more easily verified.Remark 4.
It is worth pointing out that the method applied in [12, 13] is not suitable in this paper since the impulsive part includes the external input. Therefore, this paper constructs the new impulsive inequality to overcome the difficulties.
If the stochastic disturbances and distributed delays are removed (1) immediately, it yields the following system:(40)dxt=−Axt+ℬfxt+Cgxt−τt+utdt,t≠tk,xtk=γ1kxtk−+γ2kut−,t=tk,xs=φs,t−τ0≤s≤t0.
Accordingly, we have the following result.Corollary 1.
Letβk=2γ1k2 represent the impulsive strengths, which take values from two finite sets δi|0<δi<1,i=1,2,…,l¯1 and ηj|ηj>1,j=1,2,…,l¯2. Under Assumptions 1 and 3, if there exist positive constants ϵi,i∈1,2,31,2,3 such that(41)α0−q0α2>0,where α0=−∑i=1N1lnδi/λi+∑j=1N2lnηj/μj+α1, α1=λmax−A+AT+ϵ1ℬℬT+1/ϵ1LTL+ϵ2CCT+ϵ3I, α2=λmax1/ϵ2PTP, and q0=∏i=1l¯11/δi∏j=1l¯2ηjN0, then system (40) is exponentially ISS.In particular, whenl¯1=l¯2=1, the stabilizing impulsive strength and destabilizing impulse strength are reduced to be δ1 and η1, respectively. Moreover, similar to Remark 2, let maxt^1k−t^1k−1=λ1 and inftˇ1k−tˇ1k−1=μ1. It is equivalent to χ11T,t≥T−t/λ1−1, χ21T,t≤T−t/μ1+1, which leads to the following corollary.Corollary 2.
Suppose that two constantsδ1 and η1 satisfy 0<δ1<1,η1>1. βk=2γ1k2 denotes the impulsive strengths, which can be chosen by the set δ1,η1. Under Assumptions 1, 2, and 3, if(42)α0−q0α2−q0α3τ0>0,where α0=−lnδ/λ+lnη/μ+α1, α1=λmax−A+AT+ℬℬT+LTL+CCT+DDT+I+M1, α2=λmaxPTP+M2, α3=λmaxτ0QTQ, and q0=∏i=1l¯11/δi∏j=1l¯2ηj, then system (1) is MEISS.When external inputut=0, according to the above results, the following assertions might be derived immediately.Theorem 2.
Letβk=γ1k2 denote the impulsive strengths, which take values from two finite sets δi|0<δi<1,i=1,2,…,l¯1 and ηj|ηj>1,j=1,2,…,l¯2. Suppose that all the other conditions in Theorem 1 are on hold. If external input ut=0, then system (1) is MES.Corollary 3.
Letβk=γ1k2 represent the impulsive strengths, which take values from two finite sets δi|0<δi<1,i=1,2,…,l¯1 and ηj|ηj>1,j=1,2,…,l¯2. Suppose that all the other conditions in Corollary 1 are on hold. If external input ut=0, then system (40) is exponentially stable.Remark 5.
In [11], the exponential stability of delayed neural networks with hybrid impulses was discussed. This paper further explores the MEISS of mixed delayed neural networks with stochastic disturbances and hybrid impulses. Particularly, when stochastic terms, distributed delays end external input are removed, the addressed system is reduced to the system in [11]. In light of Remark 2, when maxt^ik−t^ik−1=λi and inftˇjk−tˇjk−1=μj, we can choose flexible constant N0=1 in Assumption 3. Let ϵ1=ϵ2=1. Applying Corollary 3 to this case, if α0=−∑i=1N1lnδi/λi+∑j=1N2lnηj/μj+α1, α1=λmax−A+AT+ℬℬT+LTL+CCT, α2=λmaxPTP, and q0=∏i=1l¯11/δi∏j=1l¯2ηj, then the considered system is exponentially stable, which accords with the result in [11].Remark 6.
Recently, different control strategies have been developed to deal with some nonlinear systems. For instance, in [27], a stochastic integral sliding mode control strategy for singularly perturbed Markov jump descriptor systems subject to nonlinear perturbation was proposed and a novel mode and switch-dependent integral switching surface were introduced. In [28], the problem of path following for the underactuated unmanned surface vehicles (USVs) subject to state constraints has been tackled, where one control scheme was presented by combining the backstepping technique, adaptive dynamic programming, and the event-triggered mechanism. In [29], quantized nonstationary filtering for networked Markov switching repeated scalar nonlinear systems was investigated based on a multiple hierarchical structure strategy. Moreover, these control strategies can be applied to many applicable systems in the real world. Actually, impulsive control is also one of the significant control strategies. In this paper, we establish some MEISS criteria about stochastic neural networks with hybrid impulses. In the future, we could further explore the control problems of some applicable systems combined with input-to-state stability theory.


## 4. Numerical Example

In this section, two examples have been provided to exhibit the validity of the theoretical results in Theorem1 and Theorem 2, in which hybrid impulsive effects are introduced.Example 1.
Suppose that parameters of the neural network (9) meet that(43)A=2.7003;ℬ=0.30.50.80.9;C=0.40.20.10.3;D=0.70.80.90.5;ϱt,xt,xt−τt=0.4x1t0.3x1t−τt0.5x2t0.2x2t−τt,fxt=0.8tanhx1t0.9tanhx2t;gxt=0.24tanhx1t0.2tanhx2t;hxt=0.9tanhx1t0.85tanhx2t;ut=1.8sint1.8cost;γ1k=0.5,k=2j−1,0.8,k=2j,j∈Z,γ2k=0.8,k∈Z.
Letλ1=0.5,μ1=0.5, and time-varying τt  =  0.2|cost|. By choosing ϵi=1,i=1,2,3,4, we derive α1=λmax−A+AT+ε1ℬℬT+1/ε1LTL+ε2CCT+ε3DDT+ε4I+M1=0.2146, α2=λmax1/ϵ2PTP+M2=0.1476, and α3=λmaxτ0/ϵ3QTQ=0.162. Moreover, we calculate that δ=0.5,η=1.28,α0=0.6780,q0=2.56, and it can be easily verified that α0−q0α2−q0α3τ0=0.2172>0. Therefore, all the conditions are satisfied in Theorem 1, then system (9) is MEISS (See Figure 1). While ut=0, according to Theorem 2, the system is MES (Figure 2).Figure 1
2nd moment of the system statesx1t,x2t in Example 1 with input ut=1.8sint,1.8costT, and initial value −40,36T.Figure 2
2nd moment of the system statesx1t,x2t in Example 1 input ut=0, and initial value −40,36T.Example 2.
Assume that parameters of the neural network (9) meet that(44)A=2.1002.4;ℬ=0.2−0.150.320.1;C=0.650.25−0.180.4;D=0.120.160.35−0.3;ϱt,xt,xt−τt=0.4x1t0.32x2t−τt0.6x2t0.3x1t−τt.fxt=0.36sinx1t0.24sinx2t;gxt=0.12sinx1t0.15sinx2t;hxt=0.65sinx1t0.5sinx2t;ut=5sint2cost;γ1k=0.9,k=2j−1,0.6,k=2j,;γ2k=0.7.
Letλ1=0.4,μ1=0.4, and time-varying τt  =  0.1cost+0.15. By choosing εi=1,i=1,2,3,4, we derive α1=λmax−A+AT+ϵ1ℬℬT+1/ϵ1LTL+ϵ2CCT+ϵ3DDT+ϵ4I+M1=−2.3470, α2=λmax1/ϵ2PTP+M2=0.1249, and α3=λmaxτ0/ϵ3QTQ=0.1056. Moreover, we calculate that δ=0.72,η=1.62,α0=1.9622,q0=2.25, and it can be easily verified that α0−q0α2−q0α3τ0=1.6218>0. Therefore, all the conditions hold in Theorem 1, then system (9) is MEISS (shown in Figure 3). While ut=0, according to Theorem 2, the system is MES (shown in Figure 4).Figure 3
2nd moment of the system statesx1t,x2t in Example 2 with input ut=5sint,2costT and initial value 18,26T.Figure 4
2nd moment of the system statesx1t,x2t in Example 2 with input ut=0 and initial value 18,26T.Remark 7.
From the above two examples, it could be seen that stabilizing impulses and destabilizing impulses coexist simultaneously. In Example1, the stabilizing impulses are dominant, which plays an important role in stabilizing systems. While in Example 2, the destabilizing impulses are dominant, which could be seen as impulsive perturbations. Under this circumstance, the original system can tolerate the impulsive perturbations and maintain input-to-state stability. Different from the theoretical results in [26], the sufficient conditions on MEISS criteria are more easily verified by virtue of the average impulsive interval approach.


## 5. Conclusions

This article investigates the issues of MEISS for stochastic neural networks with hybrid impulses. A generalized comparison principle is introduced and a new inequality about the solution of the impulsive differential equation is derived. Moreover, combining the impulsive differential inequality and average impulsive interval approach based on different kinds of impulsive sequences, some novel criteria are constructed to guarantee that the system is MEISS. When external inputut=0, the addressed system is MES. Since hybrid impulses and external input at each impulsive moment are incorporated, our model becomes more general. Consequently, our theoretical achievements also improve the previous results. In the future, we will further explore some control problems of applicable systems by virtue of ISS theory.



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