# Potential Effects of Delay on the Stability of a Class of Impulsive Neural Networks

Authors: Nan Zhan; Ailong Wu
Journal: Complexity
 (2022)

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## Abstract

Aiming at the interference of the delay term in continuous dynamics to the impulsive systems, we study the potential effects of time delay on the stability of a class of impulsive neural networks (INNs) in this paper. Two cases of delay are considered. For the case of small delay, a sufficient condition for the stability of delayed INNs is obtained by virtue of the average impulsive interval (AII) method. The derived results illustrate that within limits, the convergence rate of the system becomes larger with the increase of time delay. For another case, a strict comparison principle is proposed to prove that the impulsive system still maintains the original stability for any large but bounded delay under certain conditions. In particular, as an extension, the stability of delayed INNs for hybrid impulses containing both stabilizing and destabilizing impulses is also discussed. Finally, three examples are simulated to demonstrate the validity of the theoretical results.


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## Body

## 1. Introduction

As a mathematical model of information processing, neural network (NN) is one of the most active branches of computational intelligence and machine learning. There are many kinds of NNs, and as a special kind of NNs, impulsive neural networks (INNs) have unique research value. INN was first proposed by Alan Lloyd Hodgkin and Andrew Huxley in 1952. The simulation of its neurons is closer to reality because it characterizes the transient state mutation of neurons in neural networks at a certain moment. The impulsive system is a mixture of continuous dynamic system and discrete-time system, which is different from the pure continuous-time dynamic system and pure discrete-time system. It is suitable for studying a class of dynamic systems affected by sudden change or instantaneous disturbance [1]. Furthermore, impulsive phenomena exist in various fields such as secure communication, automatic control, and mechanical system. With the help of impulsive control, we can reduce a lot of application costs. So far, many interesting results have been obtained on INNs (see [2, 3] and their references).Time delay is known to exist in many complex networks and control systems due to the influence of some practical situations. Over the past decades, time-delay systems have been vigorously studied because of their wide applications in NNs, sampling data control, biological modeling, and other fields. Meanwhile, various types of delays are discussed in NNs, such as distributed delay [4], time-varying delay [5], and state-dependent delay [6]. However, in previous studies, time delay is generally considered to be an important source of poor system performance and system instability. Few researchers have noticed that time delay may be beneficial to system stability. This is because our impression of time delay is so rigid that we ignore the stabilizing effects of time delay. Actually, we can also extract the stabilizing information of time delay through some analysis methods. For instance, in [7], the authors make a point that the increase of time delay has a dual effect on the stability of the system, that is, it may stabilize a previously unstable system or destabilize a previously stable system.Combining the two points of time delay and impulsive effects, many scholars have done a lot of work on INNs with delay [8–12]. For example, in [8], Chen et al. utilized an auxiliary state variable to transform the impulsive delayed system into an equivalent augmented model. On the basis of this model, the stability criterion of the system was derived. In [9], Zhang et al. firstly designed an impulsive controller for the time-delay discrete system to guarantee that the system can achieve stability. In [11], Jiang et al. investigated the impacts of time delay in impulses on system stability through average impulsive delay and average impulsive interval (AII) methods. In [12], Li and Song focused on the stabilization of time-delay systems under impulsive control, and the results show that the delay term in impulses may be conducive to the stabilization of the system. Obviously, studying a system with both time delay and impulsive effects is challenging because we need to consider the interaction of the two on the system.Furthermore, it can be observed that both references [11, 12] have investigated the impacts of the delay term in impulses on stability of the system, but few papers have studied the latent effects of the delay term in continuous dynamics on stability of the impulsive system. Knowing that an impulsive system is a combination of continuous and discrete subsystems, it is interesting to think about the overall effects of the delay term in the continuous subsystem and the impulsive effect in the discrete subsystem on the system stability. In addition, looking back at the fact that time delays may facilitate the stability of systems, a natural problem emerges: under what conditions does the delay term in continuous dynamics play a positive role in the stability of systems?In view of the above discussion, this paper mainly studies the potential effects of delay term in continuous dynamics on the stability of a class of INNs. Compared with some existing results, this paper fully captures the information that time delay can enhance stability. With regard to small delay and large delay, the stability of INNs with delay is investigated by using AII condition, and the hidden role that delay plays in the stability of system is revealed. With regard to hybrid impulses, the AII condition is replaced by the dwell-time condition so as to deal with the impulsive parameters as a whole, and the stability criterion of INNs is also derived. On the whole, the main features of this paper can be generalized as follows:(1)
The time delay in two cases is considered. When the delay is small, we capture the stabilizing information of time delay by means of the impulsive delay inequality and then integrate it into the Lyapunov-based function. Finally, with the help of the AII condition, the stability criterion of a kind of general INNs is derived. The results show that in a certain range, the system converges more quickly when the delay value is larger.(2)
In order to handle the case of large delay, we adopt a strict comparison principle which is different from the comparison-like principle, and it is proved that these kinds of INNs are robust to any large but bounded delay.(3)
Considering the dual effects of impulses, we extend the ideas of the first two points to the hybrid INNs containing stabilizing and destabilizing impulses.This paper is organized as follows. In Section2, a kind of general INNs with delay is introduced, and some requisite definitions and assumptions are presented. In Section 3, the main theorem results of this paper are derived, which fully illustrate the latent effects of delay term in continuous dynamics on stability of a kind of INNs. In Section 4, three numerical examples are simulated to indicate the validity of the derived results. Finally, Sections 5 gives a brief conclusion and prospects of the feasibility of the future research.


## 2. Preliminaries

### 2.1. Notations

Letℝ, ℝ+, and ℝn stand for the set of real numbers, the set of nonnegative real numbers, and the set of n−dimensional real-valued vectors, respectively. Denote ℤ+ and ℤ+0 as the set of positive integer numbers and nonnegative integer numbers, respectively. For vector x=x1,x2,…,xnT∈ℝn, let x=∑i=1nxi. Denote PC−τ,0,ℝn as the set of piecewise right continuous function ϕ:−τ,0⟶ℝn, where ϕτ≜sup−τ≤θ≤0ϕθ. Denote the upper right-hand Dini derivative of function V as D+Vt=limh⟶0+supVt+h−Vt/h.


### 2.2. Model

In this paper, based on relevant work in reference [13], we consider a class of INNs, the main form of which is as follows:(1)x˙it=−aixit+∑j=1nbijfjxjt+∑j=1ncijgjxjt−τ,t≠tk,t≥t0≥0,Δxitk=Uk,xitk−,k∈ℤ+,xit0+θ=ϕiθ,θ∈−τ,0,where ai>0,i=1,2,…,n are constants, n is the number of neurons, xit represents the state variable of the ith neuron at time t, x˙it represents the derivative of xit, τ is the transmission delay, fjxjt and gjxjt−τ are the neuron activation functions at time t and t−τ, respectively, bij and cij are real constants representing the connection weight, tk is the impulse sequence satisfying 0≤t0<t1<⋯<tk<⋯ and limk⟶+∞tk=+∞, and Δxitk≜xitk+−xitk−, where xit+=limt⟶tk+xit and xit−=limt⟶tk−xit. Generally, we suppose that the solution of network (1) is right continuous, that is, xitk+=xitk, and the sequence tk,Uk,xitk− is called the impulsive control rule. As a matter of convenience, we let hkxitk−≜xitk−+Uk,xitk−, which means xitk+=hkxitk−,k∈ℤ+. Define the solution of network (1) through t0,ϕ as xt=xt,t0,ϕ, where ϕ∈PC−τ,0,ℝn represents the initial state.For subsequent needs, we give some requisite definitions and assumptions as follows.Definition 1 (see [14]).
Suppose that there exist positive constantsN0 and Ta such that(2)Nt2,t1≥t2−t1Ta−N0,∀t2≥t1≥t0,where Nt2,t1 represents the number of impulses in the interval t1,t2. Then, N0 is called the elasticity number, and Ta denotes the AII constants.Remark 1.
The concept of AII is proposed to handle various types of impulses. In fact, AII condition (2) allows an upper bound, that is, Nt2,t1≤t2−t1/Ta+N0. Particularly, at least one impulse is required for each interval of length Ta in the case of N0=1. For AII constant Ta, it can be observed that it contains more impulsive instant sequences when the elasticity number N0 is larger.Definition 2.
For any given initial valueϕ∈PC−τ,0,ℝn, if there exist positive numbers M and λ such that(3)xt,t0,ϕ≤Mϕτe−λt−t0,∀t≥t0,holds for every sequence tk∈J∗Ta,N0, then we can say that the network (1) is globally uniformly exponentially stable (GUES) over the class J∗Ta,N0.Remark 2.
J∗Ta,N0 mentioned in the above definition represents a collection of impulsive instant sequences tk that satisfy AII condition (2).Assumption 1.
The functionsf⋅,g⋅∈ℝn satisfy f0=0, g0=0, and function hk:ℝ⟶ℝ satisfies hk0=0.Remark 3.
Clearly, Assumption1 guarantees that x=0 is an equilibrium point to network (1).Assumption 2.
The functionsf⋅,g⋅ are all Lipschitz continuous and meet(4)fjθ1−fjθ2≤γjθ1−θ2,gjθ1−gjθ2≤γj¯θ1−θ2,for all θ1,θ2∈ℝ,j=1,2,…,n, where γj,γ¯j>0 are constants.Assumption 3.
The impulsive operatorhk meets(5)hkθ1−hkθ2≤qθ1−θ2,for all θ1,θ2∈ℝ,k∈ℤ+, where q>0 is Lipschitz constant.In order to facilitate the subsequent expression, we make(6)α1=max1≤i≤n−ai+∑j=1nbjiγi,α2=max1≤i≤n∑j=1ncjiγ¯i.


## 2.1. Notations

Letℝ, ℝ+, and ℝn stand for the set of real numbers, the set of nonnegative real numbers, and the set of n−dimensional real-valued vectors, respectively. Denote ℤ+ and ℤ+0 as the set of positive integer numbers and nonnegative integer numbers, respectively. For vector x=x1,x2,…,xnT∈ℝn, let x=∑i=1nxi. Denote PC−τ,0,ℝn as the set of piecewise right continuous function ϕ:−τ,0⟶ℝn, where ϕτ≜sup−τ≤θ≤0ϕθ. Denote the upper right-hand Dini derivative of function V as D+Vt=limh⟶0+supVt+h−Vt/h.


## 2.2. Model

In this paper, based on relevant work in reference [13], we consider a class of INNs, the main form of which is as follows:(1)x˙it=−aixit+∑j=1nbijfjxjt+∑j=1ncijgjxjt−τ,t≠tk,t≥t0≥0,Δxitk=Uk,xitk−,k∈ℤ+,xit0+θ=ϕiθ,θ∈−τ,0,where ai>0,i=1,2,…,n are constants, n is the number of neurons, xit represents the state variable of the ith neuron at time t, x˙it represents the derivative of xit, τ is the transmission delay, fjxjt and gjxjt−τ are the neuron activation functions at time t and t−τ, respectively, bij and cij are real constants representing the connection weight, tk is the impulse sequence satisfying 0≤t0<t1<⋯<tk<⋯ and limk⟶+∞tk=+∞, and Δxitk≜xitk+−xitk−, where xit+=limt⟶tk+xit and xit−=limt⟶tk−xit. Generally, we suppose that the solution of network (1) is right continuous, that is, xitk+=xitk, and the sequence tk,Uk,xitk− is called the impulsive control rule. As a matter of convenience, we let hkxitk−≜xitk−+Uk,xitk−, which means xitk+=hkxitk−,k∈ℤ+. Define the solution of network (1) through t0,ϕ as xt=xt,t0,ϕ, where ϕ∈PC−τ,0,ℝn represents the initial state.For subsequent needs, we give some requisite definitions and assumptions as follows.Definition 1 (see [14]).
Suppose that there exist positive constantsN0 and Ta such that(2)Nt2,t1≥t2−t1Ta−N0,∀t2≥t1≥t0,where Nt2,t1 represents the number of impulses in the interval t1,t2. Then, N0 is called the elasticity number, and Ta denotes the AII constants.Remark 1.
The concept of AII is proposed to handle various types of impulses. In fact, AII condition (2) allows an upper bound, that is, Nt2,t1≤t2−t1/Ta+N0. Particularly, at least one impulse is required for each interval of length Ta in the case of N0=1. For AII constant Ta, it can be observed that it contains more impulsive instant sequences when the elasticity number N0 is larger.Definition 2.
For any given initial valueϕ∈PC−τ,0,ℝn, if there exist positive numbers M and λ such that(3)xt,t0,ϕ≤Mϕτe−λt−t0,∀t≥t0,holds for every sequence tk∈J∗Ta,N0, then we can say that the network (1) is globally uniformly exponentially stable (GUES) over the class J∗Ta,N0.Remark 2.
J∗Ta,N0 mentioned in the above definition represents a collection of impulsive instant sequences tk that satisfy AII condition (2).Assumption 1.
The functionsf⋅,g⋅∈ℝn satisfy f0=0, g0=0, and function hk:ℝ⟶ℝ satisfies hk0=0.Remark 3.
Clearly, Assumption1 guarantees that x=0 is an equilibrium point to network (1).Assumption 2.
The functionsf⋅,g⋅ are all Lipschitz continuous and meet(4)fjθ1−fjθ2≤γjθ1−θ2,gjθ1−gjθ2≤γj¯θ1−θ2,for all θ1,θ2∈ℝ,j=1,2,…,n, where γj,γ¯j>0 are constants.Assumption 3.
The impulsive operatorhk meets(5)hkθ1−hkθ2≤qθ1−θ2,for all θ1,θ2∈ℝ,k∈ℤ+, where q>0 is Lipschitz constant.In order to facilitate the subsequent expression, we make(6)α1=max1≤i≤n−ai+∑j=1nbjiγi,α2=max1≤i≤n∑j=1ncjiγ¯i.


## 3. Main Results

We will discuss the stability of INNs from the following three aspects in current section. Firstly, we consider the stability of a kind of INNs with small delay (the delay does not exceed any two consecutive impulsive time intervals, i.e.,τ≤tk−tk−1). Besides, the latent effects of time delay are also explored. Secondly, the stability of INNs with arbitrarily finite delay is considered. Compared with small delay, we use large delay (which implies that the delay may be greater than a certain impulsive time interval, namely, τ≤tk−tk−1 may not be true) to represent relatively larger delay (which is collectively referred to as arbitrarily finite delay here). For arbitrarily finite delay, we analyze the robustness of the stability of INNs with delay and verify that the system can remain stable for any large but bounded delay under certain conditions. Finally, in view of the fact that the impulsive effects may promote or suppress system stability, we extend the ideas of the first two points to delayed INNs with hybrid impulses.


### 3.1. INNs with Small Delay

In what follows, we will discuss the case where the delay is small. We capture the stabilizing information of time delay with the help of the impulsive delay inequality and then integrate it into the Lyapunov-based function. Finally, through the AII method, we can derive the stability criterion of INNs.Theorem 1.
If there exist constantsq∈0,1, η∗=maxα1+α2,η0 and the following conditions hold:(7)α1+α2>0,lnqTa+η∗<0,where(8)α1+α2qe−η0τ−η0=0,then under Assumptions 1–3, network (1) is GUES over the class J∗Ta,N0.Proof.
Construct a functionVt≜Vt,xt=xt=∑i=1nxit and make V¯t0=supt0−τ≤s≤t0Vs. For any ε>0, let η=η∗+ε, and design an auxiliary function(9)Lt=Vte−ηt−tk,t∈tk,tk+1,Vt,t0−τ≤t≤t0.
To start with, letΩk=qkV¯t0, and then we will confirm that(10)Vt≤Ωkeηt−t0,t∈tk,tk+1,k∈ℤ+0.
Together with (7) and (8),(11)Lt≤Ωkeηtk−t0,t∈tk,tk+1.
Firstly, we demonstrate that (9) is true for k=0, namely, Lt≤Ω0=V¯t0,t∈t0,t1. Note that Lt0=Vt0≤V¯t0=Ω0. If the above statement is incorrect, then there is an instant t∗∈t0,t1 such that(12)Lt∗=Ω0,Lt≤Ω0,t∈t0,t∗andD+Lt∗≥0.
Whens∈t0−τ,t0, it is apparent that Ls=Vs≤V¯t0=Ω0, and in combination with (10), we derive Ls≤Ω0,t∈t0−τ,t∗. For t∈tk,tk+1,k∈ℤ+0, we can calculate that(13)D+Vt=∑i=1nsgnxit−aixit+∑j=1nbijfjxjt+∑j=1ncijgjxjt−τ≤∑i=1n−ai|xit|+∑i=1n∑j=1nbijγjxjt+∑i=1n∑j=1ncijγ¯jxjt−τ≤max1≤i≤n−ai+∑j=1nbjiγixt+max1≤i≤n∑j=1ncjiγ¯ixt−τ=α1Vt+α2Vt−τ.
Whent∗−τ∈t0−τ,t∗, combining the definition of Lt and conditions (6), (10), and (11), one has(14)D+Lt|t=t∗=e−ηt∗−t0D+Vt|t=t∗−ηe−ηt∗−t0Vt∗≤e−ηt∗−t0α1Vt∗+α2Vt∗−τ−ηe−ηt∗−t0Vt∗=α1Lt∗+α2Lt∗−τe−ητ−ηLt∗≤α1+α2e−ητ−ηΩ0<0.
Obviously, it could be observed that it contradictsD+Lt∗≥0, namely, (9) holds for k=0.
Next, through mathematical induction method, we assume that (9) is true for k≤p,p∈ℤ+0, i.e.,(15)Lt≤ΩkeηtK−t0,t∈tk,tk+1,k≤p,which means(16)Lt≤Ωpeηtp−t0,t∈tp,tp+1.
Subsequently, we demonstrate that(17)Lt≤Ωp+1eηtp+1−t0,t∈tp+1,tp+2.
Recall (7) and (14), and we obtain(18)Ltp+1=Vtp+1=xtp+1=hp+1xtp+1−−hp+10≤qLtp+1−eηtp+1−tp=Ωp+1eηtp+1−t0.
Thus, (15) holds for t=tp+1. On the contrary, it is assumed that there is an instant t∗∈tp+1,tp+2 which makes(19)Lt∗=Ωp+1eηtp+1−t0,Lt≤Ωp+1eηtp+1−t0,t∈tp+1,t∗andD+Lt∗≥0.
Ift∗−τ≥tp+1, referring to (12), we derive that(20)D+Lt|t=t∗≤α1Lt∗+α2Lt∗−τe−ητ−ηLt∗≤α1+α2e−ητ−ηΩp+1eηtp+1−t0<0.
Similarly, what calls for special attention is that whens∈tp,tp+1, it follows from (14) that(21)Ls≤Ωpeηtp−t0=Ωp+1qeηtp+1−t0e−ηtp+1−tp=Lt∗qe−ηtp+1−tp≤Lt∗qe−ητ.
At this time, ift∗−τ<tp+1, on account of tp≤tp+1−τ≤t∗−τ<tp+1, and together with (6), (16), and (17), we could compute that(22)D+Lt|t=t∗≤α1Lt∗+α2Lt∗−τe−ητ−ηLt∗≤α1Lt∗+α2Lt∗qe−ητ−ηLt∗=α1+α2qe−ητ−ηΩp+1eηtp+1−t0<0,which contradicts D+Lt∗≥0, namely, (15) holds.
Hence, we can get(23)Lt≤Ωkeηtk−t0,∀t∈tk,tk+1,k∈ℤ+0,which means (8) is true, namely, Vt≤Ωkeηt−t0=Ωkeη∗+εt−t0,t∈tk,tk+1.
Letε⟶0+, and then we obtain(24)Vt≤Ωkeη∗t−t0,∀t∈tk,tk+1,k∈ℤ+0.
Sincetk∈J∗Ta,N0, the AII method further yields that(25)Vt≤qNt,t0eη∗t−t0V¯t0≤qt−t0/Ta−N0eη∗t−t0V¯t0≤q−N0elnq/Ta+η∗t−t0V¯t0,∀t≥t0,where Nt,t0 represents the number of impulses in the interval t0,t.
According to (18) and the definition of Vt, we have(26)xt≤Me−λt−t0ϕτ,∀t≥t0,where M=q−N0,λ=−lnq/Ta+η∗>0. Until now, we have done the proof.Remark 4.
From (6) and (8) in Theorem 1, we can notice the latent impacts of time delay τ on the decay rate of Lyapunov function Vt. In particular, if η0>α1+α2, then we have η∗=η0. Meanwhile, the implicit function η∗τ is determined by α1+α2/qe−η∗τ−η∗=0, which decreases as τ increases. Obviously, the result derived from the above theorem shows that the convergence rate λ is related to parameter η∗, and in view of the relationship between η∗ and delay τ, it can be concluded that the convergence rate of the system will become larger with the increase of delay, which means that we have captured the stabilizing effects of time delay. In addition, what needs special attention is that in the majority of the available literature about the stability of delayed INNs, we can see that the stability of the system tends to be destroyed as delay increases, but different results are obtained in this paper.Remark 5.
It should be noted that the conclusion of the relationship between time delay and system stability derived from Remark4 is based on η∗=η0, so the results may be conservative to some extent. Furthermore, the conclusion of Theorem 1 is a sufficient condition rather than a necessary condition; then, it is possible for the system to be stable when τ is small and does not meet the conditions of Theorem 1.


### 3.2. INNs with Arbitrarily Finite Delay

For the case of arbitrarily finite delay, based on strict comparison principle and the concept of AII, a stability criterion of INNs is also derived.Lemma 1 (see [13]).
Letα1∈ℝ,α2≥0 and q>0. Suppose that at,bt∈Ctk−1,tk,ℝ+ meet(27)D+at≤α1at+α2at−τ,t≥t0,t≠tk,at≤qat−,t=tk,,(28)D+bt=α1bt+α2bt−τ,t≥t0,t≠tk,vt=qbt−,t=tk,,for all k∈ℤ+. Then, at≤bt,∀t0−τ≤t≤t0 implies that at≤bt,∀t≥t0.Theorem 2.
If there exists constantq∈0,1 such that(29)α1+α2qN0+lnqTa<0,then under Assumptions 1–3, network (1) is GUES over the class J∗Ta,N0 for arbitrarily finite delay τ.Proof.
Construct a functionVt=xt=∑i=1nxit, and let V¯t0=supt0−τ≤s≤t0Vs.
Next, similar to (11), we have(30)D+Vt≤α1Vt+α2Vt−τ,∀t∈tk−1,tk,k∈ℤ+.
Whent=tk, according to Assumptions 1–3, we could get(31)Vtk=xtk=hkxtk−=hkxtk−−hk0≤qxtk−=qVtk−.
Introduce an impulsive delayed system withut as its unique solution:(32)D+ut=α1ut+α2ut−τ,t≥t0,t≠tk,ut=qut−,t=tk,ut=V¯t0,t∈t0−τ,t0.
Apparently,Vt≤V¯t0=ut when t∈t0−τ,t0. In accordance with Lemma 1, we have(33)0≤Vt≤ut,∀t≥t0.
From the variable parameter formula, we obtain(34)ut=Kt,t0ut0+∫t0tKt,sα2us−τds,t≥t0,where Kt,s denotes the Cauchy matrix of the following system:(35)D+ut=α1ut,t≥t0,t≠tk,ut=qut−,t=tk,k∈ℤ+.
According to the properties of the Cauchy matrix, combining0<q<1 and tk∈J∗Ta,N0, we can obtain(36)Kt,s=eα1t−sΠs<tk≤tq≤eα1t−sqt−s/Ta−N0=q−N0eα1+lnq/Tat−s=q−N0e−ct−s,where c=−α1+lnq/Ta, and it is evident that c>0 by using condition (21).
Reviewing (26) and (28), we have(37)ut≤q−N0e−ct−t0ut0+∫t0tq−N0e−ct−sα2us−τds,∀t≥t0.
Sincec>0,N0>0, and 0<q<1, this implies(38)ut=V¯t0=ut0≤ut0e−ct−t0<q−N0ut0e−λt−t0,t∈t0−τ,t0,where λ=c−q−N0α2eλτ. In fact, by using (21), we can obtain λ>0. Then, we shall confirm that(39)ut<q−N0ut0e−λt−t0,t≥t0.
On the contrary, suppose (31) is untenable; then, there exists an instant t∗>t0 which makes(40)ut∗≥q−N0ut0e−λt∗−t0,(41)ut<q−N0ut0e−λt−t0,t<t∗.
Subsequently, from (29) and (33), one has(42)ut∗≤q−N0e−ct∗−t0ut0+∫t0t∗q−N0e−ct∗−sα2us−τds<q−N0e−ct∗−t0ut0+∫t0t∗q−N0e−ct∗−sα2q−N0ut0e−λs−τ−t0ds=q−N0ut0e−ct∗−t0+q−N0α2eλτe−ct∗+λt0∫t0t∗ec−λsds=q−N0ut0e−λt∗−t0,which is in contradiction with (32). Thus, we can derive that (31) holds. Finally, combining (25), we have(43)Vt≤ut<q−N0ut0e−λt−t0,∀t≥t0,i.e.,(44)xt≤q−N0e−λt−t0ϕτ,∀t≥t0.
So far, we have done the proof.Remark 6.
It can be observed from Theorems1 and 2 that the AII constant Ta should be small enough to meet conditions (5) and (21) in the case of 0<q<1. The AII constant means the frequency of impulsive control. The smaller Ta is, the higher the impulsive frequency will be. In addition, it should be noted that the result derived from Theorem 2 involves elasticity number N0, and equation (21) may not hold when N0 is sufficiently large. However, for delay-free system (see [14]), the derived result does not involve N0, so we cannot obtain such a conclusion. Moreover, we can obverse that the elasticity number N0 is not involved in the result of Theorem 1, which is the difference between Theorems 1 and 2. Therefore, under the condition of AII, Theorem 2 further illustrates the internal relationship between large delay and system stability.Remark 7.
Although Theorems1 and 2 are proposed for small and large delays, respectively, this does not mean that Theorem 2 and Theorem 1 are mutually exclusive. Actually, Theorem 2 is a supplement to Theorem 1 because the so-called large delay in Theorem 2 just means that τ≤tk−tk−1 may not be true, in which case it covers the case of small delay. Therefore, Theorem 2 is also applicable to the case of small delay.


### 3.3. Extension to INNs with Hybrid Impulses

In recent years, hybrid impulse as an important topic has attracted wide attention, and numerous meaningful results have emerged. Particularly, in [15], considering the influence of hybrid impulses on the synchronization process, the authors designed an effective hybrid impulsive controller so as to achieve the quasi-synchronization of NNs. In this case, a sufficient delay-dependent criterion for quasi-synchronization is obtained. Meanwhile, in [15, 16], the authors adopted AII and average impulsive gain methods to deal with the hybrid impulses. In this paper, an improved dwell-time condition is introduced to treat the hybrid impulses. In view of the above discussion, we extend the results of the first two theorems in this section.First of all, in order to extend Theorem1, we put forward Theorem 3 by referring to the processing procedure of the small delay case.Theorem 3.
If Assumption3 is replaced by the following condition:(45)hkθ1−hkθ2≤e−ρkθ1−θ2,where ρk∈ℝ, then under Assumptions 1and 2 and (35), for given constants ρ¯>0,ρ∗>0,c>0,δ>0, the following conditions are satisfied:(46)A1α1+α2>0A2∑k=1Nt,t0ρk>ρ¯Nt,t0−ρ∗A3ρ¯Nt,t0≥ct−t0−δA4η∗−c<0,where η∗=maxα1+α2,η1, and η1 satisfies α1+α2eρsupe−η1τ−η1=0 with ρsup=supk∈ℤ+ρk<∞; then, network (1) is GES.Proof.
The same analysis method as Theorem1 is used here, except that the impulsive parameters are changed, so we shall take advantage of A1 and A4 to acquire the following statement:(47)Vt≤Πk=1Nt,t0e−ρkeη∗t−t0V¯t0,∀t≥t0.
Then, by usingA2 and A3, one has(48)Vt≤e−Σk=1Nt,t0e−ρkeη∗t−t0V¯t0≤e−ρ¯Nt,t0+ρ∗eη∗t−t0V¯t0≤e−ct−t0+δ+ρ∗eη∗t−t0V¯t0≤eδ+ρ∗e−c−η∗t−t0V¯t0,∀t≥t0.
That is,(49)xt≤eδ+ρ∗e−c−η∗t−t0ϕτ,∀t≥t0.
The proof is completed.
Next, referring to the analysis method of arbitrarily finite delay case, we obtain Theorem4 as an extension of Theorem 2.Theorem 4.
Suppose that the parameterq in Assumption 3 is replaced by e−ρk, then under Assumptions 1 and 2 and modified Assumption 3, for given constants ρ¯>0,ρ∗>0,c>0,δ>0, the following conditions are fulfilled:(50)A^1∑k=1Nt,t0ρk>ρ¯Nt,t0−ρ∗A^2ρ¯Nt,t0≥α1+ct−t0−δA^3α2eδ+ρ∗−c<0.
Then, network (1) is GES.Proof.
It is easy to prove this theorem by combining the analytical methods of Theorems2 and 3, so we leave out it for brevity.Remark 8.
The parametere−ρk in Theorems 3 and 4 is used to describe the variable of hybrid impulses in impulsive control systems. As you can see, q in Theorems 1 and 2 is required to be 0<q<1, but e−ρk in Theorem 3 and 4 satisfies ρk∈ℝ, that is, e−ρk<1 if ρk>0 and e−ρk>1 if ρk<0. It implies that stabilizing impulses and destabilizing impulses may exist at the same time. Furthermore, in order to handle these parameters overall, we propose conditions A2 and A^1, and in a sense, the parameter ρ¯ may be approximately regarded as the “average value” of ρk. In fact, condition A2 combined with A3 or A^1 combined with A^2 can be considered as an improvement of dwell-time condition.Remark 9.
Compared with reference [11], this paper studies the effects of time delay in continuous dynamics on system stability. Bear in mind that the relationship between time delay in continuous dynamics and the stability of impulsive systems is not easy to find, and the derived results based on AII method contain both time delay and the AII constant Ta, which is not obtained in previous results. Furthermore, we extend the results to systems with hybrid impulses. In addition, when discussing the delay effects in reference [12], the authors limit the time delay to be less than any two consecutive impulsive time intervals. However, we loosen the condition of time delay in this paper, that is, the time delay can be smaller than any two consecutive impulsive intervals, or it can be greater than any two consecutive impulsive time intervals.Remark 10.
On the basis of this paper, we could also investigate the case of neural network models with time-varying delays. Actually, the results of this paper are still valid after the constant delay is replaced by time-varying delay in the model. We will continue to explore this question in depth in future studies.


## 3.1. INNs with Small Delay

In what follows, we will discuss the case where the delay is small. We capture the stabilizing information of time delay with the help of the impulsive delay inequality and then integrate it into the Lyapunov-based function. Finally, through the AII method, we can derive the stability criterion of INNs.Theorem 1.
If there exist constantsq∈0,1, η∗=maxα1+α2,η0 and the following conditions hold:(7)α1+α2>0,lnqTa+η∗<0,where(8)α1+α2qe−η0τ−η0=0,then under Assumptions 1–3, network (1) is GUES over the class J∗Ta,N0.Proof.
Construct a functionVt≜Vt,xt=xt=∑i=1nxit and make V¯t0=supt0−τ≤s≤t0Vs. For any ε>0, let η=η∗+ε, and design an auxiliary function(9)Lt=Vte−ηt−tk,t∈tk,tk+1,Vt,t0−τ≤t≤t0.
To start with, letΩk=qkV¯t0, and then we will confirm that(10)Vt≤Ωkeηt−t0,t∈tk,tk+1,k∈ℤ+0.
Together with (7) and (8),(11)Lt≤Ωkeηtk−t0,t∈tk,tk+1.
Firstly, we demonstrate that (9) is true for k=0, namely, Lt≤Ω0=V¯t0,t∈t0,t1. Note that Lt0=Vt0≤V¯t0=Ω0. If the above statement is incorrect, then there is an instant t∗∈t0,t1 such that(12)Lt∗=Ω0,Lt≤Ω0,t∈t0,t∗andD+Lt∗≥0.
Whens∈t0−τ,t0, it is apparent that Ls=Vs≤V¯t0=Ω0, and in combination with (10), we derive Ls≤Ω0,t∈t0−τ,t∗. For t∈tk,tk+1,k∈ℤ+0, we can calculate that(13)D+Vt=∑i=1nsgnxit−aixit+∑j=1nbijfjxjt+∑j=1ncijgjxjt−τ≤∑i=1n−ai|xit|+∑i=1n∑j=1nbijγjxjt+∑i=1n∑j=1ncijγ¯jxjt−τ≤max1≤i≤n−ai+∑j=1nbjiγixt+max1≤i≤n∑j=1ncjiγ¯ixt−τ=α1Vt+α2Vt−τ.
Whent∗−τ∈t0−τ,t∗, combining the definition of Lt and conditions (6), (10), and (11), one has(14)D+Lt|t=t∗=e−ηt∗−t0D+Vt|t=t∗−ηe−ηt∗−t0Vt∗≤e−ηt∗−t0α1Vt∗+α2Vt∗−τ−ηe−ηt∗−t0Vt∗=α1Lt∗+α2Lt∗−τe−ητ−ηLt∗≤α1+α2e−ητ−ηΩ0<0.
Obviously, it could be observed that it contradictsD+Lt∗≥0, namely, (9) holds for k=0.
Next, through mathematical induction method, we assume that (9) is true for k≤p,p∈ℤ+0, i.e.,(15)Lt≤ΩkeηtK−t0,t∈tk,tk+1,k≤p,which means(16)Lt≤Ωpeηtp−t0,t∈tp,tp+1.
Subsequently, we demonstrate that(17)Lt≤Ωp+1eηtp+1−t0,t∈tp+1,tp+2.
Recall (7) and (14), and we obtain(18)Ltp+1=Vtp+1=xtp+1=hp+1xtp+1−−hp+10≤qLtp+1−eηtp+1−tp=Ωp+1eηtp+1−t0.
Thus, (15) holds for t=tp+1. On the contrary, it is assumed that there is an instant t∗∈tp+1,tp+2 which makes(19)Lt∗=Ωp+1eηtp+1−t0,Lt≤Ωp+1eηtp+1−t0,t∈tp+1,t∗andD+Lt∗≥0.
Ift∗−τ≥tp+1, referring to (12), we derive that(20)D+Lt|t=t∗≤α1Lt∗+α2Lt∗−τe−ητ−ηLt∗≤α1+α2e−ητ−ηΩp+1eηtp+1−t0<0.
Similarly, what calls for special attention is that whens∈tp,tp+1, it follows from (14) that(21)Ls≤Ωpeηtp−t0=Ωp+1qeηtp+1−t0e−ηtp+1−tp=Lt∗qe−ηtp+1−tp≤Lt∗qe−ητ.
At this time, ift∗−τ<tp+1, on account of tp≤tp+1−τ≤t∗−τ<tp+1, and together with (6), (16), and (17), we could compute that(22)D+Lt|t=t∗≤α1Lt∗+α2Lt∗−τe−ητ−ηLt∗≤α1Lt∗+α2Lt∗qe−ητ−ηLt∗=α1+α2qe−ητ−ηΩp+1eηtp+1−t0<0,which contradicts D+Lt∗≥0, namely, (15) holds.
Hence, we can get(23)Lt≤Ωkeηtk−t0,∀t∈tk,tk+1,k∈ℤ+0,which means (8) is true, namely, Vt≤Ωkeηt−t0=Ωkeη∗+εt−t0,t∈tk,tk+1.
Letε⟶0+, and then we obtain(24)Vt≤Ωkeη∗t−t0,∀t∈tk,tk+1,k∈ℤ+0.
Sincetk∈J∗Ta,N0, the AII method further yields that(25)Vt≤qNt,t0eη∗t−t0V¯t0≤qt−t0/Ta−N0eη∗t−t0V¯t0≤q−N0elnq/Ta+η∗t−t0V¯t0,∀t≥t0,where Nt,t0 represents the number of impulses in the interval t0,t.
According to (18) and the definition of Vt, we have(26)xt≤Me−λt−t0ϕτ,∀t≥t0,where M=q−N0,λ=−lnq/Ta+η∗>0. Until now, we have done the proof.Remark 4.
From (6) and (8) in Theorem 1, we can notice the latent impacts of time delay τ on the decay rate of Lyapunov function Vt. In particular, if η0>α1+α2, then we have η∗=η0. Meanwhile, the implicit function η∗τ is determined by α1+α2/qe−η∗τ−η∗=0, which decreases as τ increases. Obviously, the result derived from the above theorem shows that the convergence rate λ is related to parameter η∗, and in view of the relationship between η∗ and delay τ, it can be concluded that the convergence rate of the system will become larger with the increase of delay, which means that we have captured the stabilizing effects of time delay. In addition, what needs special attention is that in the majority of the available literature about the stability of delayed INNs, we can see that the stability of the system tends to be destroyed as delay increases, but different results are obtained in this paper.Remark 5.
It should be noted that the conclusion of the relationship between time delay and system stability derived from Remark4 is based on η∗=η0, so the results may be conservative to some extent. Furthermore, the conclusion of Theorem 1 is a sufficient condition rather than a necessary condition; then, it is possible for the system to be stable when τ is small and does not meet the conditions of Theorem 1.


## 3.2. INNs with Arbitrarily Finite Delay

For the case of arbitrarily finite delay, based on strict comparison principle and the concept of AII, a stability criterion of INNs is also derived.Lemma 1 (see [13]).
Letα1∈ℝ,α2≥0 and q>0. Suppose that at,bt∈Ctk−1,tk,ℝ+ meet(27)D+at≤α1at+α2at−τ,t≥t0,t≠tk,at≤qat−,t=tk,,(28)D+bt=α1bt+α2bt−τ,t≥t0,t≠tk,vt=qbt−,t=tk,,for all k∈ℤ+. Then, at≤bt,∀t0−τ≤t≤t0 implies that at≤bt,∀t≥t0.Theorem 2.
If there exists constantq∈0,1 such that(29)α1+α2qN0+lnqTa<0,then under Assumptions 1–3, network (1) is GUES over the class J∗Ta,N0 for arbitrarily finite delay τ.Proof.
Construct a functionVt=xt=∑i=1nxit, and let V¯t0=supt0−τ≤s≤t0Vs.
Next, similar to (11), we have(30)D+Vt≤α1Vt+α2Vt−τ,∀t∈tk−1,tk,k∈ℤ+.
Whent=tk, according to Assumptions 1–3, we could get(31)Vtk=xtk=hkxtk−=hkxtk−−hk0≤qxtk−=qVtk−.
Introduce an impulsive delayed system withut as its unique solution:(32)D+ut=α1ut+α2ut−τ,t≥t0,t≠tk,ut=qut−,t=tk,ut=V¯t0,t∈t0−τ,t0.
Apparently,Vt≤V¯t0=ut when t∈t0−τ,t0. In accordance with Lemma 1, we have(33)0≤Vt≤ut,∀t≥t0.
From the variable parameter formula, we obtain(34)ut=Kt,t0ut0+∫t0tKt,sα2us−τds,t≥t0,where Kt,s denotes the Cauchy matrix of the following system:(35)D+ut=α1ut,t≥t0,t≠tk,ut=qut−,t=tk,k∈ℤ+.
According to the properties of the Cauchy matrix, combining0<q<1 and tk∈J∗Ta,N0, we can obtain(36)Kt,s=eα1t−sΠs<tk≤tq≤eα1t−sqt−s/Ta−N0=q−N0eα1+lnq/Tat−s=q−N0e−ct−s,where c=−α1+lnq/Ta, and it is evident that c>0 by using condition (21).
Reviewing (26) and (28), we have(37)ut≤q−N0e−ct−t0ut0+∫t0tq−N0e−ct−sα2us−τds,∀t≥t0.
Sincec>0,N0>0, and 0<q<1, this implies(38)ut=V¯t0=ut0≤ut0e−ct−t0<q−N0ut0e−λt−t0,t∈t0−τ,t0,where λ=c−q−N0α2eλτ. In fact, by using (21), we can obtain λ>0. Then, we shall confirm that(39)ut<q−N0ut0e−λt−t0,t≥t0.
On the contrary, suppose (31) is untenable; then, there exists an instant t∗>t0 which makes(40)ut∗≥q−N0ut0e−λt∗−t0,(41)ut<q−N0ut0e−λt−t0,t<t∗.
Subsequently, from (29) and (33), one has(42)ut∗≤q−N0e−ct∗−t0ut0+∫t0t∗q−N0e−ct∗−sα2us−τds<q−N0e−ct∗−t0ut0+∫t0t∗q−N0e−ct∗−sα2q−N0ut0e−λs−τ−t0ds=q−N0ut0e−ct∗−t0+q−N0α2eλτe−ct∗+λt0∫t0t∗ec−λsds=q−N0ut0e−λt∗−t0,which is in contradiction with (32). Thus, we can derive that (31) holds. Finally, combining (25), we have(43)Vt≤ut<q−N0ut0e−λt−t0,∀t≥t0,i.e.,(44)xt≤q−N0e−λt−t0ϕτ,∀t≥t0.
So far, we have done the proof.Remark 6.
It can be observed from Theorems1 and 2 that the AII constant Ta should be small enough to meet conditions (5) and (21) in the case of 0<q<1. The AII constant means the frequency of impulsive control. The smaller Ta is, the higher the impulsive frequency will be. In addition, it should be noted that the result derived from Theorem 2 involves elasticity number N0, and equation (21) may not hold when N0 is sufficiently large. However, for delay-free system (see [14]), the derived result does not involve N0, so we cannot obtain such a conclusion. Moreover, we can obverse that the elasticity number N0 is not involved in the result of Theorem 1, which is the difference between Theorems 1 and 2. Therefore, under the condition of AII, Theorem 2 further illustrates the internal relationship between large delay and system stability.Remark 7.
Although Theorems1 and 2 are proposed for small and large delays, respectively, this does not mean that Theorem 2 and Theorem 1 are mutually exclusive. Actually, Theorem 2 is a supplement to Theorem 1 because the so-called large delay in Theorem 2 just means that τ≤tk−tk−1 may not be true, in which case it covers the case of small delay. Therefore, Theorem 2 is also applicable to the case of small delay.


## 3.3. Extension to INNs with Hybrid Impulses

In recent years, hybrid impulse as an important topic has attracted wide attention, and numerous meaningful results have emerged. Particularly, in [15], considering the influence of hybrid impulses on the synchronization process, the authors designed an effective hybrid impulsive controller so as to achieve the quasi-synchronization of NNs. In this case, a sufficient delay-dependent criterion for quasi-synchronization is obtained. Meanwhile, in [15, 16], the authors adopted AII and average impulsive gain methods to deal with the hybrid impulses. In this paper, an improved dwell-time condition is introduced to treat the hybrid impulses. In view of the above discussion, we extend the results of the first two theorems in this section.First of all, in order to extend Theorem1, we put forward Theorem 3 by referring to the processing procedure of the small delay case.Theorem 3.
If Assumption3 is replaced by the following condition:(45)hkθ1−hkθ2≤e−ρkθ1−θ2,where ρk∈ℝ, then under Assumptions 1and 2 and (35), for given constants ρ¯>0,ρ∗>0,c>0,δ>0, the following conditions are satisfied:(46)A1α1+α2>0A2∑k=1Nt,t0ρk>ρ¯Nt,t0−ρ∗A3ρ¯Nt,t0≥ct−t0−δA4η∗−c<0,where η∗=maxα1+α2,η1, and η1 satisfies α1+α2eρsupe−η1τ−η1=0 with ρsup=supk∈ℤ+ρk<∞; then, network (1) is GES.Proof.
The same analysis method as Theorem1 is used here, except that the impulsive parameters are changed, so we shall take advantage of A1 and A4 to acquire the following statement:(47)Vt≤Πk=1Nt,t0e−ρkeη∗t−t0V¯t0,∀t≥t0.
Then, by usingA2 and A3, one has(48)Vt≤e−Σk=1Nt,t0e−ρkeη∗t−t0V¯t0≤e−ρ¯Nt,t0+ρ∗eη∗t−t0V¯t0≤e−ct−t0+δ+ρ∗eη∗t−t0V¯t0≤eδ+ρ∗e−c−η∗t−t0V¯t0,∀t≥t0.
That is,(49)xt≤eδ+ρ∗e−c−η∗t−t0ϕτ,∀t≥t0.
The proof is completed.
Next, referring to the analysis method of arbitrarily finite delay case, we obtain Theorem4 as an extension of Theorem 2.Theorem 4.
Suppose that the parameterq in Assumption 3 is replaced by e−ρk, then under Assumptions 1 and 2 and modified Assumption 3, for given constants ρ¯>0,ρ∗>0,c>0,δ>0, the following conditions are fulfilled:(50)A^1∑k=1Nt,t0ρk>ρ¯Nt,t0−ρ∗A^2ρ¯Nt,t0≥α1+ct−t0−δA^3α2eδ+ρ∗−c<0.
Then, network (1) is GES.Proof.
It is easy to prove this theorem by combining the analytical methods of Theorems2 and 3, so we leave out it for brevity.Remark 8.
The parametere−ρk in Theorems 3 and 4 is used to describe the variable of hybrid impulses in impulsive control systems. As you can see, q in Theorems 1 and 2 is required to be 0<q<1, but e−ρk in Theorem 3 and 4 satisfies ρk∈ℝ, that is, e−ρk<1 if ρk>0 and e−ρk>1 if ρk<0. It implies that stabilizing impulses and destabilizing impulses may exist at the same time. Furthermore, in order to handle these parameters overall, we propose conditions A2 and A^1, and in a sense, the parameter ρ¯ may be approximately regarded as the “average value” of ρk. In fact, condition A2 combined with A3 or A^1 combined with A^2 can be considered as an improvement of dwell-time condition.Remark 9.
Compared with reference [11], this paper studies the effects of time delay in continuous dynamics on system stability. Bear in mind that the relationship between time delay in continuous dynamics and the stability of impulsive systems is not easy to find, and the derived results based on AII method contain both time delay and the AII constant Ta, which is not obtained in previous results. Furthermore, we extend the results to systems with hybrid impulses. In addition, when discussing the delay effects in reference [12], the authors limit the time delay to be less than any two consecutive impulsive time intervals. However, we loosen the condition of time delay in this paper, that is, the time delay can be smaller than any two consecutive impulsive intervals, or it can be greater than any two consecutive impulsive time intervals.Remark 10.
On the basis of this paper, we could also investigate the case of neural network models with time-varying delays. Actually, the results of this paper are still valid after the constant delay is replaced by time-varying delay in the model. We will continue to explore this question in depth in future studies.


## 4. Illustrative Examples

Finally, for the purpose of verifying the above achievements, we put forward the following three examples in current section.Example 1.
Consider a 2-dimensional INN with small delay:(51)x˙1tx˙2t=−0.2000.3x1tx2t+0.20.40.030.01tanhx1ttanhx2t+0.250.250.050.02tanhx1t−τtanhx2t−τ,under impulsive control(52)x1tkx2tk=0.01000.01x1tk−x2tk−,where t≥0,k∈ℤ+,tk−tk−1≥τ>0.
Obviously,γ1=γ2=1,γ¯1=γ¯2=1,q=0.01. By calculating,(53)α1=max1≤i≤n−ai+∑j=1nbjiγi=0.11,α2=max1≤i≤n∑j=1ncjiγ¯i=0.3.
Here, we set the initial valueϕ=3,−0.4T and choose impulsive instants tk=5k,k∈ℤ+, which means Ta=5. When τ∈4,5, we can figure out η∗≤0.91 and lnq/Ta+η∗<0, and it can be tested that all conditions in Theorem 1 hold. Therefore, we can derive that systems (39) and (40) are GUES when τ∈4,5. In addition, according to Remark 4, it can be seen that the system may converge faster with the increase of delay, which corresponds to the simulation results in Figures 1–3. Moreover, we calculate its corresponding parameter η∗ and estimate its convergence rate for different time delays τ=4,4.5,5, which are shown in Table 1. More importantly, it also reveals the potential stabilizing effect of time delay.Figure 1
The state of (39) and (40) with τ=4.Figure 2
The state of (39) and (40) with τ=4.5.Figure 3
The state of (39) and (40) with τ=5.Table 1
The convergence rateλ for different time delays.
Time delayτParameterη∗Convergence rateλ40.90700.01404.50.82910.091950.76490.1561Example 2.
Consider another 2-dimensional INN with large delay:(54)x˙1tx˙2t=−0.1000.1x1tx2t+0.080.50.20.35tanhx1t2tanhx2t2+0.030.10.10.07tanhx1t−τ8tanhx2t−τ2,under impulsive control(55)x1tkx2tk=sinx1tk−5sinx2tk−5,where t≥0,k∈ℤ+ and τ=25,tk=2k. Here, we set the initial value ϕ=0.4,3T. As shown in Figure 4, the impulse-free system is unstable.
In addition, it is apparent thatγ1=γ2=1/2,γ¯1=1/8,γ¯2=1/2,q=1/5,Ta=2,N0=1. By calculating,(56)α1=max1≤i≤n−ai+∑j=1nbjiγi=0.325,α2=max1≤i≤n∑j=1ncjiγ¯i=0.085.
Therefore, we can conclude thatα1+α2/qN0+lnq/Ta≈−0.05<0. That is to say, the conditions in Theorem 2 are fulfilled, and it is deduced that systems (43) and (44) are GUES, which is well reflected in Figure 5. By the way, the time delay could be much larger and the system would still be stable. In what follows, we calculate the corresponding convergence rate for different time delays τ=25,60,100, which are shown in Table 2. Actually, the delay is not limited to 100, and it can even be greater than 100. As long as the delay is bounded under certain conditions, the initial stability of impulsive system can be guaranteed. Here we only calculate the convergence rate of the system when the delay increases to 100.Figure 4
The state of (41) without impulse.Figure 5
The state of (41) with impulsive effects (42).Table 2
Stability of systems with different time delays.
Time delayτStable or notConvergence rateλ25✓0.004560✓0.0019100✓0.0012Example 3.
Consider a 3-dimensional INN:(57)x˙t=−Axt+Bfxt+Cgxt−τ,under hybrid impulsive control(58)xtk=e−ρkxtk−,tk=8k,k∈ℤ+,where(59)A=0.60000.60000.6,B=0.24−0.3−0.2−0.2−0.240.3−0.40.70.9,C=0.010000.20000.01,(60)ρk=3.1,k=2l−1−0.9,k=2l,l∈ℤ+.
Suppose thatfx=f1x1,f2x2,f3x3T,gx=g1x1,g2x2,g3x3T and fixi=tanhxit/2,gixi=tanhxit,i=1,2,3. We give the initial value ϕ=0.5,0.6,−0.3T and take time delay τ=7. As shown in Figure 6, the impulse-free system is unstable.
Additionally, it is evident thatγ1=γ2=γ3=1/2,γ¯1=γ¯2=γ3¯=1,ρsup=3.1,ρ¯=1.1>0. By calculating,(61)α1=max1≤i≤n−ai+∑j=1nbjiγi=0.1,α2=max1≤i≤n∑j=1ncjiγ¯i=0.2.
Furthermore, we can figure outη1≈0.3899 by Matlab. That is, η∗=maxα1+α2,η1≈0.39 and c=0.4 in conditions A1−A4 of Theorem 3 are fulfilled. At the same time, a series of conditions in Theorem 3 are completely true. Therefore, systems (43) and (44) are GES, which is well illustrated in Figure 7. From the simulation results, the impulsive effects indeed have both stabilizing and destabilizing effects.Figure 6
The state of (43) without impulse.Figure 7
The state of (43) under hybrid impulsive control (44).


## 5. Conclusion

In this paper, we have discussed the stability of a kind of INNs with delay. Particularly, the internal relation between time delay and system stability has been revealed. Firstly, we have investigated the case where the delay is small. By constructing Lyapunov function, combining the impulsive delay inequality and AII condition, we have obtained a sufficient condition to assure the exponential stability of INNs. The results have shown that within limits, the system converges more quickly with the increase of time delay. Secondly, we have explored the case where the delay is arbitrarily large but bounded and derived a Lyapunov-based stability criterion by virtue of the strict comparison principle. Finally, as an extension, we have considered the case where INN is a system with hybrid impulses. In future studies, we may discuss the delay effects of a kind of INN with state-dependent delay.



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