# Robust Exponential Stability Analysis of Switched Neural Networks with Interval Parameter Uncertainties and Time Delays

Authors: Xiaohui Xu; Huanbin Xue; Yiqiang Peng; Jiye Zhang
Journal: Complexity
 (2018)

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## Abstract

In this paper, the stability of switched neural networks (SNNs) with interval parameter uncertainties and time delays is investigated. First, the conditions for the existence and uniqueness of the equilibrium point of the system are discussed. Second, the average dwell time approach and M-matrix property are employed to obtain conditions to ensure the globally exponential stability of the delayed SNNs under constrained switching. Third, by resorting to inequality technique and the idea of vector Lyapunov function, sufficient condition to ensure the robust exponential stability of the delayed SNNs under arbitrary switching is derived. The form of the constructed Lyapunov functions is simple, which has certain commonality in studying delayed SNNs, and the proposed results not only are explicit but also reveal the relationship between the constrained switching and the arbitrary switching of the SNNs. Finally, two numerical examples are presented to illustrate the effectiveness and less conservativeness of the main results compared with the existing literature.


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## Body

## 1. Introduction

In the past years, neural networks have been widely studied and successfully applied to various realms such as dynamic optimization, associative memory, and pattern recognition and to solve nonlinear algebraic equations and so on [1–5]. In the real world, the connections among different nodes of the networks are not always fixed or consistent, which frequently result in link failure and new link creation. Therefore, the abrupt changes in the structures and parameters of the neural networks often occur, which bring about switchings among certain different topologies and the instability of the networks [6]. In application’s point of view, a fundamental problem of applying neural networks is stability. This is a prerequisite for ensuring that the developed networks can work normally [7–10]. Thus, a popular topic about the stability analysis and stabilization of SNNs has been considered in [11–24].A switched neural network is a hybrid system, which is essentially composed of a family of subnetworks and a switching signal which defines a specially designated subnetwork being activated at each instant of time. SNNs have attracted significant attention and have been successfully applied to many fields such as artificial intelligence, high-speed signal processing, and gene selection in DNA microarray analysis [25–28]. Generally, a switching system can be described by the following differential equation:
(1)ẋ=fσx,where fp:p∈P is a family of functions parameterized by some index set P and switching signal σ is a piecewise constant and right continuous function of time mapping from 0,+∞ to P. The original motivation for the study on switched systems comes partly from that switching among different systems may cause many nonlinear system behaviors such as chaos and multiple limit cycles [29]. In recent years, switched systems have gained increasing attention because many practical systems (for example, constrained robotics, computer-controlled systems, and automated highway systems) can be modeled as switched systems. Furthermore, from the point of view of control, multicontroller switching is an effective way to deal with complex systems. It is well-known that time delays are inevitable in a practical control design which usually leads to unsatisfactory performances and the stability of the dynamic systems may even be destroyed with the increase of delays [30–35]. Attributing to the interaction among the discrete dynamics, continuous dynamics, and time delays, the behaviors of delayed SNNs are very complicated. Besides, due to many inevitable factors such as modelling errors and external perturbations, the models certainly contain uncertainties which can have a serious effect on the dynamical behavior of the systems. To analyze the robustness of the SNNs, one feasible method is to assume that the parameters are included in certain intervals [36]. Therefore, the robust stability analysis of SNNs with interval parameter uncertainties and time delays is of practical and theoretical importance.For switched dynamical systems, the unpredictable change of system dynamics, such as abrupt perturbation of external environment or sudden change of the system structure due to the failure of a component, may cause the sudden change of the switching signal. In these cases, in order to keep the system working, the system should be stable under arbitrary switching. A typical approach for the stability analysis of switched dynamical systems with arbitrary switching signal is to search for a suitable common Lyapunov function (CLF)Vx such that the rate of the decrease of Vx along the trajectories of systems is not affected by switching (see, e.g., [37–40] and the references therein). If the CLF for the systems does not exist or is not known, in this case, we can study the stability of the system by using multiple Lyapunov functions (MLFs) Vpx,p∈P, (see [37, 41, 42]). However, it is worth noting that to apply this MLF method, one needs to know some information of the state at each switching time. This is to be contrasted with the Lyapunov second method, which do not need to know the knowledge of the solutions. For example, Wu et al. studied the exponential stability of delayed SNNs by using a linear matrix inequality approach and an average dwell time method [12]; based on the piecewise Lyapunov function technique and average dwell time approach, the problem of the exponential stability of SNNs with constant and time-varying delays was investigated, respectively, in [43] and in [44]; by resorting to a novel delay division method, the stability analysis for uncertain SNNs with mixed time-varying delays was addressed. A common feature in these articles is that they all resort to scalar Lyapunov function (or functional). In this paper, the stability of the delayed SNNs with switching signal will be studied by using the idea of vector Lyapunov function with simple forms, which have certain commonality in studying SNNs, and this is the main reason why the obtained results in this paper have less conservativeness. By using the M-matrix property and average dwell time approach, the differential inequalities with time delays will be constructed. By the stability analysis of the differential inequalities, the sufficient conditions to ensure the robust exponential stability of the SNNs under arbitrary switching and constrained switching will be obtained.Compared with the existing results on SNNs, the contributions of this paper are listed as follows: (a) the forms of the constructed Lyapunov functions are simple, which have certain commonality in studying delayed SNNs under arbitrary switching; (b) unlike asymptotic stability, we analyze the exponential stability of SNNs which include uncertainty and time delays, and the exponential convergence rate can also be obtained; (c) the obtained results not only have less conservativeness but also reveal the relationship between the constrained switching and the arbitrary switching of the delayed SNNs; and (d) comparing with most of the previous results obtained by linear matrix inequalities approach (to apply LMIs approach, one has to determine too many unknown parameters), the proposed criteria are straightforward, which are conducive to practical applications.Notation.
Letx=x1,⋯,xnT denote a column vector of Rn (the symbol “T” denotes transpose), x denote x=x1,x2,⋯,xnT, and x denote a vector norm defined by x=x12+x22+⋯+xn21/2. For x,y∈Rn, x>y means that each pair of the corresponding elements of x and y satisfies the inequality “>.” For matrix A=aijn×n, A denote A=aijn×n. C−τ,t0;Rn denotes the set of continuous functions mapping from −τ,t0 to Rn.


## 2. Preliminaries

The model of a delayed SNNs can be described by the delayed differential equations as follows:(2)dwitdt=−eiσtwit+∑j=1naijσtgjσtwjt+∑j=1nbijσtgjσtwjt−τijσt+Jiσt,wis+t0=ϕis,s∈−τ,0,where i=1,2,…,n, n is the number of neurons, wit is the state of neuron i at time t, σt:0,+∞→Σ=1,2,⋯,m is the switching signal, which is a piecewise constant and right continuous function of time, and σt=k∈Σ means that the kth subnetwork is activated. Ek=diage1k,e2k,⋯,enk denotes the neuron self-feedback coefficient matrix of the k subnetwork, and eik>0 represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the networks and external inputs; gkwt=g1kw1t,g2kw2t,⋯,gnkwntT is the activation functions of neurons at time t; Ak=aijkn×n and Bk=bijkn×n are the connection weight matrices of the kth subnetwork, and aijk and bijk denote the connection strengths of the jth neuron on the ith neuron at time t and t−τijk, respectively; the delay τijk≥0 is the bounded function with τk=max1≤i,j≤nτijk≥0 and τ=max1≤k≤mτk; Jk=J1k,J2k,⋯,JnkT is the constant external input vector of the kth subnetwork. wis+t0=ϕis is the initial condition of the system, where ϕi∈C−τ,t0,R,i=1,2,…,n.We assume that the switching signalσt is unknown a priori. Corresponding to the switching signal σt, we have a switching sequence t0,i0⋯tk,ik⋯∣ik∈Σ,k=0,1,⋯, which means that the ikth subsystem is activated when t∈tk,tk+1. We also assume that there is only finite switching in any finite interval and satisfy the following conditions.Assumption 1.
Each activation functiongik⋅ in the delayed SNNs (2) is assumed to satisfy
(3)L¯ik≤giku−gikvu−v≤L¯ik,for any u,v∈R,u≠v,i=1,2,…,n,k∈Σ, where L¯ik and L¯ik are known constant scalars and L¯ik<L¯ik.Remark 1.
Assumption1 was first proposed in [45]. The constants L¯ik and L¯ik in this assumption are allowed to be any real number (positive, negative, or zero). Therefore, the activation functions can be nonmonotonic, which are more general than commonly used Lipschitz conditions and sigmoid functions. Such assumption is very useful to obtain less conservative results.
To facilitate the following analysis, letLk=diagL1k,L2k,⋯,Lnk with Lik=maxL¯ik,L¯ik. In order to study the stability of SNNs under parameter uncertainties, for k∈Σ, the matrices are intervalized as follows:
(4)EkI=Ek=diageikn×n:E¯k≤Ek≤E¯k,i.e.,0<e¯ik≤eik≤e¯ik,AkI=Ak=aijkn×n:A¯k≤Ak≤A¯k,i.e.,a¯ijk≤aijk≤a¯ijk,BkI=Bk=bijkn×n:B¯k≤Bk≤B¯k,i.e.,b¯ijk≤bijk≤b¯ijk.
Define(5)Ek∗=diage¯1k,e¯2k,⋯,e¯nk,Ak∗=aijk∗n×nwithaijk∗=maxa¯ijk,a¯ijk,Bk∗=bijk∗n×nwithbijk∗=maxb¯ijk,b¯ijk,Bk⋆=bijk⋆n×nwithbijk⋆=minb¯ijk,b¯ijk.Definition 1.
For the delayed SNNs (2), the equilibrium point w∗=w1,w2,⋯,wnT is said to be robustly exponentially stable if for each E∈EkI, A∈AkI, and B∈BkI, there exist constants λ>0 and M>1 such that
(6)wt−w∗≤Mϕ−w∗t0exp−λt−t0,t≥t0,where ϕ−w∗t0=∑i=1nsups∈−τ,t0ϕis−wi∗21/2.


## 3. Existence and Uniqueness of the Equilibrium Point

The purpose of the present section is to give a sufficient condition which ensures that the equilibrium point of each subsystem satisfies the existence and uniqueness, which implies that for any initial conditionϕ∈C−τ,t0;Rn, system (2) admits a solution wt,t0,ϕ which exists in a maximal interval −τ,t0+K, where 0<K≤∞.Definition 2.
A realn×n matrix A=aijn×n is said to be an M-matrix if aij≤0,i,j=1,2,…,n,i≠j, and all successive principal minors of A are positive.Lemma 1
([46]).
LetA=aij be an n×n matrix with nonpositive off-diagonal elements. Then the following statements are equivalent:
(i)
A is an M-matrix
(ii)
There exists a vectorξ>0 such that Aξ>0.Definition 3.
A mappingℋ:ℝn→ℝn is a homeomorphism of Rn onto itself if ℋ∈C0, ℋ is one to one, ℋ is onto, and the inverse mapping ℋ−1∈C0, where C0 denotes the set of continuous functions.Lemma 2
([47]).
Ifℋu∈C0 satisfies the following conditions:
(i)
ℋu is injective on ℝn
(ii)
ℋu→∞ as u→∞
Thenℋu is a homeomorphism of Rn.Theorem 1.
Under Assumption1, if for all k∈Σ, Ck∗=Ek∗−Ak∗+Bk∗Lk are nonsingular M-matrices, then for each specified switching signal σt, system (2) has a unique equilibrium point.Proof 1.
Because the equilibrium point of subsystems,(7)dwitdt=−eikwit+∑j=1naijkgjkwjt+∑j=1nbijkgjkwjt−τijk+Jiksatisfies the following equation:
(8)−eikwit+∑j=1naijk+bijkgjkwjt+Jik=0,for i=1,2,…,n and k∈Σ. Let
(9)ℋkwt=ℋ1kwt,ℋ2kwt,⋯,ℋnkwtT,where
(10)ℋikwt=−eikwit+∑j=1naijk+bijkgjkwjt+Jik,for i=1,2,…,n. In the following, we will give a proof that ℋkwt are homeomorphisms of Rn onto itself.
First, we prove thatℋkwt are injective mappings on Rn. Actually, if there exist vectors x=x1,x2,⋯,xnT, y=y1,y2,⋯ynT, and x≠y such that ℋkx=ℋky; then
(11)−eikxi−yi+∑j=1naijk+bijkgjkxj−gjkyj=0,for i=1,2,…,n and k=1,2,…,m. From Assumption 1, it can be derived that
(12)−eikxi−yi+∑j=1naijk+bijkLjkxj−yj≥0,for i=1,2,…,n. That is,
(13)Ek−Ak+BkLkx−y≤0.
LetCk=Ek−Ak+BkLk. Obviously, Ck have nonpositive off-diagonal entries and Ck≥Ck∗ which implies that Ck are nonsingular M-matrices. From Theorem 2.3 of [48], we can get x=y. That is,
(14)xi=yi,i=1,2,…,n,which is a contradiction. As a result, ℋkwt are injective mappings on ℝn.
Next, we prove thatℋkwt→∞ as wt→∞.
BecauseCk are nonsingular M-matrices, we know that there exist positive diagonal matrices Dk=diagd1k,d2k,⋯,dnk, which make matrices DkCk+CkTDk positively definite. Let
(15)ℋ~kwt=ℋ~1kwt,ℋ~2kwt,⋯,ℋ~nkwtT,where
(16)ℋ~ikwt=−eikwit+∑j=1naijk+bijkgjkwjt−gjk0,for i=1,2,…,n. Calculate
(17)w1,w2,⋯,wnDkℋ~kwt=∑i=1nwidikℋ~ikwt=∑i=1n−eikdikwi2t+∑j=1naijk+bijkdikwitgjkwjt−gjk0≤∑i=1n−eikdikwi2t+∑j=1naijk+bijkdikLjkwitwjt=−w1t,w2t,⋯,wntDkCkw1t,w2t,⋯,wntT=−12wtTDkCk+CkTDkwt≤−12λminDkCk+CkTDkwt2.
Using Schwartz inequality, we have(18)wt⋅Dk⋅ℋ~kwt≥12λminDkCk+CkTDkwt2.
Whenwt≠0, we get
(19)ℋ~kwt≥12λminDkCk+CkTDkwtDk,which implies ℋ~kwt→∞ as wt→∞.
Sinceℋ~kwt→∞ implies ℋkwt→∞, by Lemma 2, we know that ℋkwt are homeomorphisms of ℝn. So each subnetwork has a unique equilibrium point. Therefore, for each specified switching signal σt, system (2) has a unique equilibrium point. The proof is completed.


## 4. Exponential Stability of the Delayed SNNs

### 4.1. Exponential Stability under Constrained Switching

In this section, we will give a sufficient condition ensuring the global exponential stability of delayed SNNs (2) by using the average dwell time method. Let Q be an M-matrix; we denote
(20)ΛQ≜ξ∈ℝn∣Qξ>0,ξ>0.Definition 4
(see [37]).
LetNσt1,t2 denote the number of discontinuities of a switching signal σ on an interval t1,t2. T>0 is called the average dwell time, if for any t2≥t1≥0 and N0≥0,
(21)Nσt1,t2≤N0+t2−t1Thold.Theorem 2.
Under Assumption1, if for all k∈Σ, Ck∗=Ek∗−Ak∗+Bk∗Lk are nonsingular M-matrices, then for all Ek∈EkI, Ak∈AkI, Bk∈BkI and any external input Jk, the delayed SNN (2) is robustly exponentially stable for any switching signal with the average dwell time satisfying
(22)T>T∗=lnϑε,where ε>0 is determined by inequalities
(23)−e¯ik+εexpετkξik+∑j=1nξjkLjkajik∗+bjik∗<0,for some given ξk=ξ1k,ξ2k,⋯,ξnkT∈ΛCk∗ and ϑ=max1≤i≤n,1≤ik−1≤mηiik−1,ηiik−1βiik−1≥1 with
(24)ηiik−1=max1≤i≤nξiikmin1≤i≤nξiik−1,βiik−1=max1≤j≤nexpετijikLjikbijik∗min1≤j≤nexpετijik−1Ljik−1bijik−1⋆.Proof 2.
According to Theorem1, we know that if Ck∗=Ek∗−Ak∗+Bk∗Lk are M-matrices, then the system has a unique equilibrium point for each specified switching signal. Let w∗=w1∗,w2∗,⋯,wn∗T be an equilibrium point of system (2) and wt=w1t,w2t,⋯,wntT be any solution of system (2). Denote xit=wit−wi∗, fjkxjt=gjkxjt+wj∗−gjkwj∗, and ψis=ϕis−wi∗; then system (2) can be rewritten as
(25)dxitdt=−eiσtxit+∑j=1naijσtfjσtxjt+∑j=1nbijσtfjσtxjt−τijσt,xis+t0=ψis,s∈−τ,0,with i=1,2,…,n.
Due toCk∗ being M-matrices, by Lemma 1 (ii), we know that there exist ξik>0 and δik>0,k∈Σ,i=1,2,⋯,n such that
(26)−e¯ikξik+∑j=1nξjkLjkajik∗+bjik∗=−δik<0.
Consider a Lyapunov functional candidate(27)Vx,t=∑i=1nξiσtexpεtxi+∑j=1nLjσtbijσt∫t−τijσttexpεs+τijσtxjsds.
Calculating the upper right derivativeD+V of V along the solutions of (25), we get
(28)D+Vx,t=∑i=1nξiσtexpεtsgnxidxidt+εexpεtxi+∑j=1nLjσtbijσtexpεt+τijσtxjt−expεtxjt−τijσt=∑i=1nξiσtexpεtsgnxi−eiσtxit+∑j=1naijσtfjσtxjt+∑j=1nbijσtfjσtxjt−τijσt+εexpεtxi+∑j=1nLjσtbijσtexpεt+τijσtxjt−expεtxjt−τijσt≤∑i=1nξiσtexpεt−eiσtxit+∑j=1naijσtfjσtxjt+∑j=1nbijσtfjσtxjt−τijσt+εexpεtxi+∑j=1nLjσtbijσtexpεt+τijσtxjt−expεtxjt−τijσt≤∑i=1nξiσtexpεt−eiσtxit+∑j=1nLjσtaijσtxjt+∑j=1nLjσtbijσtxjt−τijσt+εexpεtxi+∑j=1nLjσtbijσtexpεt+τijσtxjt−expεtxjt−τijσt=∑i=1nexpεtξiσt−eiσt+εxit+∑j=1naijσt+expετijσtbijσtLjσtxjt≤expεt∑i=1n−eiσt+εξiσt+expετσt∑j=1nξjσtLjσtajiσt+bjiσtxit≤expεt+τσt∑i=1n−e¯iσt+εexpετσtξiσt+∑j=1nξjσtLjσtajiσt∗+bjiσt∗xit=expεt+τσt∑i=1n−δiσt+−e¯iσt+εexpετσt+e¯iσtξiσtxit.
Defining functions,(29)ℱikz=−δik+−e¯ik+zexpzτk+e¯ikξik,i=1,2,⋯,n,k∈Σ.
Obviously,ℱik0=−δik<0. Since ℱikz are continuous functions, there exist εik>0,i=1,2,⋯,n, such that ℱikεik<0. Let ε=min1≤k≤m,1≤i≤nεik; we can get  ℱikε<0,i=1,2,⋯,n,k∈Σ. Combining it with inequality (28), we get
(30)D+Vx,t≤expεt+τσt∑i=1nℱiσtεxit≤0.
So fort∈tk,tk+1,
(31)Vx,t≤Vx,tk.
For convenience, we denoteσt=ik when t∈tk,tk+1,k=0,1,…,n. That is, the ikth subnetwork is activated for t∈tk,tk+1; then
(32)Vx,tk=∑i=1nξiikexpεtkxitk+∑j=1nLjikbijik∫tk−τijiktkexpεs+τijikxjsds=∑i=1nξiikexpεtk−xitk−+∑j=1nLjikbijik∫tk−−τijiktk−expεs+τijikxjsds≤∑i=1nηiik−1ξiik−1expεtk−xitk−+βiik−1∑j=1nLjik−1bijik−1∫tk−−τijik−1tk−expεs+τijik−1xjsds,where
(33)ηiik−1=max1≤i≤nξiikmin1≤i≤nξiik−1,βiik−1=max1≤j≤nexpετijikLjikbijik∗min1≤j≤nexpετijik−1Ljik−1bijik−1⋆.
Letϑ=max1≤i≤n,1≤ik−1≤mηiik−1,ηiik−1βiik−1≥1; we can get
(34)Vx,tk≤ϑ∑i=1nξiik−1expεtk−xitk−+∑j=1nLjik−1bijik−1∫tk−−τijik−1tk−expεs+τijik−1xjsds=ϑVx,tk−.
Combining (31) and (34) yields
(35)D+Vx,t≤Vx,tk≤ϑVx,tk−≤ϑVx,tk−1≤⋯≤ϑNσt0,tVx,t0.
Whent=t0, the i0th subnetwork is activated; then
(36)Vx,t0=∑i=1nξii0expεt0xit0+∑j=1nLji0biji0∫t0−τiji0t0expεs+τiji0xjsds=∑i=1nξii0expεt0wit0−wi∗+∑j=1nLji0biji0∫t0−τiji0t0expεs+τiji0wjs−wj∗ds≤∑i=1nξii0expεt0sups∈t0−τ,t0ϕis−wi∗+τi0∑j=1nLji0biji0expεt0+τi0sups∈t0−τ,t0ϕis−wj∗=expεt0∑i=1nξii0sups∈t0−τ,t0ϕis−wi∗+τi0expεt0∑i=1nξii0∑j=1nLji0biji0expετi0sups∈t0−τ,t0ϕis−wj∗≤expεt0∑i=1nξii0sups∈t0−τ,t0ϕis−wi∗+τi0expεt0∑i=1n∑j=1nξji0Lii0bjii0expετi0sups∈t0−τ,t0ϕis−wi∗=expεt0∑i=1nξii0+τi0∑j=1nξji0Lii0bjii0expετi0sups∈t0−τ,t0ϕis−wi∗≤expεt0Mi0∑i=1nsups∈t0−τ,t0ϕis−wi∗21/2=expεt0Mi0ϕ−wt0,where
(37)Mi0=nmax1≤i≤nξii0+τi0∑j=1nξji0Lii0bjii0expετi0.
Combining (27), (35), and (36) yields
(38)expεt∑i=1nξiikxit≤ϑNσt0,texpεt0Mi0ϕ−w∗t0.
Letξ=min1≤i≤n,1≤k≤mξik; (38) becomes
(39)wt−w∗≤ϑNσt0,tMi0ξϕ−w∗t0exp−εt−t0=ϑNσt0,texp−εt−t0Mi0ξϕ−w∗t0=expNσt0,tlnϑ−εt−t0Mi0ξϕ−w∗t0≤expN0lnϑ+t−t0Tlnϑ−εt−t0Mi0ξϕ−w∗t0=expt−t0Tlnϑ−εt−t0ϑN0Mi0ξϕ−w∗t0=exp−t−t0ε−lnϑTϑN0Mi0ξϕ−w∗t0.
LetM=ϑN0Mi0/ξ and λ=ε−lnϑ/T; we have
(40)wt−w∗≤Mϕ−w∗t0exp−λt−t0,when T>lnϑ/ε and λ>0. According to Definition 1, equilibrium point system (2) w∗ is robustly exponentially stable. The proof is completed.Remark 2.
For allk∈Σ, Ck∗ are M-matrices which mean that delayed SNN (2) is globally exponentially stable under constrained switching. From the definitions of ℱikz functions, we know that the value of ε relies on vector ξk=ξ1k,ξ2k,⋯,ξnkT∈ΛCk∗. So, for obtaining the maximum convergence rate λ∗ or the minimum average dwell time T∗, one can solve the optimization problem under constraint conditions ℱikλ,ξk<0,ξk∈ΛCk∗,i=1,2,⋯,n,k∈Σ.


### 4.2. Exponential Stability under Arbitrary Switching

Define the indicator function(41)γt=γ1t,γ2t,⋯,γmtT,where
(42)γkt=1,whenthekthsubnetworkisactivated,0,otherwise,with k=1,2,…,m. Therefore, delayed SNN system (25) can be described as follows:
(43)dxitdt=∑k=1mγkt−eikxit+∑j=1naijkfjkxjt+∑j=1nbijkfjkxjt−τijk,xis+t0=ψis,s∈−τ,0,i=1,…,n.For any switching signal, only one subnetwork is activated at any time, so it follows that∑k=1mγkt=1.Theorem 3.
Under Assumption1, the equilibrium point of delayed SNNs (2) is robustly exponentially stable for all Ek∈EkI, Ak∈AkI, Bk in BkI, and any switching signal if the following conditions are satisfied:
(i)
Ck∗=Ek∗−Ak∗+Bk∗Lk,k∈Σ, are nonsingular M-matrices
(ii)
χ=∩k=1mΛCk∗ is nonempty.
Moreover, the exponential convergence rate of system (2) is equal to λ, which is determined by
(44)−ξie¯ik−λ+∑j=1nξjLjkaijk∗+eλτkbijk∗<0,for a given vector ξ∈χ.Proof 3.
Consider Lyapunov function candidatesvit=xitexpλt−t0. Calculating the upper right derivative D+vi of vi along the solutions of (43), we get
(45)D+vit=expλt−t0sgnxi∑k=1mγkt−eikxit+∑j=1naijkfjkxjt+∑j=1nbijkfjkxjt−τijk+λexpλt−t0xit≤expλt−t0∑k=1mγkt−eik+λxit+∑j=1naijkfjkxjt+∑j=1nbijkfjkxjt−τijk≤expλt−t0∑k=1mγkt−eik+λxit+∑j=1nLjkaijkxjt+∑j=1nLjkbijkxjt−τijk≤expλt−t0∑k=1mγkt−e¯ik+λxit+∑j=1nLjkaijk∗xjt+∑j=1nLjkbijk∗xjt−τijk≤∑k=1mγkt−e¯ik+λexpλt−t0xit+∑j=1nLjkaijk∗expλt−t0xjt+expλτk∑j=1nLjkbijk∗expλt−τijk−t0xjt−τijk.
Substitutingvit=expλt−t0xit into the above inequality, we can get
(46)D+vit≤∑k=1mγkt−e¯ik+λvit+∑j=1nLjkaijk∗vjt+expλτk∑j=1nLjkbijk∗supt−τk≤s≤tvjs,for i=1,2,…,n.
SinceCk∗ are nonsingular M-matrices and χ is nonempty, from Lemma 1, we know that there exists at least one vector ξ=ξ1,ξ2,⋯,ξnT∈χ⊆ΛCk∗ such that
(47)−e¯ikξi+∑j=1naijk∗+bijk∗Ljkξj<0,for i=1,2,…,n,k∈Σ.
Consider functions(48)Gikzik=−ξie¯ik−zik+∑j=1naijk∗+expzikτkbijk∗Ljkξj,with i=1,2,…,n and k=1,2,…,m.
By inequality (47) and the definition of functions Gik, it is clear that Gikzik∈C0 and Gik0<0. Because dGikzik/dzik>0, there are constants λik>0 such that
(49)Gikλik=−ξie¯ik−λik+∑j=1naijk∗+expλikτkbijk∗Ljkξj=0.
Let0<λ<min1≤k≤m,1≤i≤nλik; then
(50)Gikλ=−ξie¯ik−λ+∑j=1naijk∗+expλτkbijk∗Ljkξj<0.for i=1,2,…,n and k∈Σ.
Letl0=ϕ−w∗t0/ξmin, where ξmin=min1≤i≤nξi. So
(51)vis=expλs−t0ϕis−wi∗<ξil0,t0−τ≤s<t0,i=1,2,…,n.
Fort≥t0, we claim that vit<ξil0,i=1,2,…,n. If this is not true, there exist some i and corresponding t′>0, which make vit′=ξil0, D+vit′≥0, and vjt<ξjl0 for t0≤t≤t′, j=1,2,…,n,j≠i. However, applying (44) and (46) leads to
(52)D+vit′≤∑k=1mγkt′−e¯ik+λvit′+∑j=1nLjkaijk∗vjt′+expλτk∑j=1nLjkbijk∗supt′−τk≤s≤t′vjs≤∑k=1mγkt′−e¯ik+λξil0+∑j=1nLjkaijk∗ξjl0+expλτk∑j=1nLjkbijk∗ξjl0<0.
This is a contradiction. Sovit<ξil0,i=1,2,…,n, for t≥t0. That is, for t≥t0,
(53)xi<ξil0exp−λt−t0=ξiξminϕ−w∗t0exp−λt−t0,i=1,2,…,n.
LetM=n⋅max1≤i≤nξi/ξmin; then we can get
(54)w−w∗<Mϕ−w∗t0exp−λt−t0,for t≥t0. From Definition 1, the equilibrium point of system (2) is robustly exponentially stable. Moreover, the exponential convergence rate is λ. The proof is completed.Remark 3.
The existence of exponential convergence rateλ has been proved, and from the definitions of functions Gikz and (44), we know that the value of λ relies on vector ξ=ξ1,ξ2,⋯,ξnT∈χ. So, for obtaining maximum convergence rate λ∗, one can solve the optimization problem under constraint conditions Gikλ,ξ<0,ξ∈χi=1,2,⋯,n,k∈Σ.
By virtue of Theorem3, it is easy to get the following result.Corollary 1.
Under Assumption1, if for k∈Σ, Ck∗=Ek∗−Ak∗+Bk∗Lk are nonsingular M-matrices, then for all Ek∈EkI, Ak∈AkI, and Bk∈BkI, the equilibrium point of system (2) is robustly exponentially stable for any switching signal with the average dwell time satisfying
(55)T>T∗=lnηmaxλ,where ηmax=max1≤i≤n,1≤k≤mξik/min1≤i≤n,1≤k≤mξik and λ>0 is determined by inequality
(56)ξik−e¯ik+λ+∑j=1naijk∗+expλτkbijk∗Ljkξjk<0,for some given ξk=ξ1k,ξ2k,⋯,ξnkT∈χCk∗.Proof 4.
Consider Lyapunov function candidatesvit=xitexpλt−tk. For convenience, we assume that the jkth subsystem is activated when t∈tk,tk+1. Calculating the upper right derivative D+vit of vit along the solutions of (25), we get
(57)D+vit=expλt−tksgnxi−eijkxit+∑j=1naijjkfjjkxjt+∑j=1nbijjkfjjkxjt−τijjkt+λexpλt−tkxit.
It can be known from Theorem3 that there exists λijk>0 such that
(58)Gijkλijk=−ξijke¯ijk−λijk+∑j=1naijjk∗+expλijkτjkbijjk∗Ljjkξjjk=0.
Let0<λ<min1≤i≤nλijk; then
(59)Gijkλ=−ξijke¯ijk−λ+∑j=1nbijjk∗+expλτjkcijjk∗Ljjkξjjk<0.for i=1,2,…,n,jk∈Σ.
By the proof of Theorem3, we know that
(60)xit<ξijkmin1≤i≤nξijkxtkexp−λt−tk,for t∈tk,tk+1. Since the system state is continuous, it follows from (60) that
(61)xit<ξijkmin1≤i≤nξijkxitkexp−λt−tk<⋯<exp∑l=0klnηjl−λt−t0xit0<expk+1lnηmax−λt−t0xit0=ηmaxexpNσt0,tlnηmax−λt−t0xit0≤ηmaxN0+1exp−λ−lnηmaxTt−t0xit0,where ηjv=ξijv/min1≤i≤nξijv and ηmax=max1≤jv≤mηjv. Let M=nηmaxN0+1 and ε=λ−lnηmax/T yield
(62)wt−w∗≤Mϕ−w∗t0exp−εt−t0.
WhenT>T∗=lnηmax/λ, ε>0. According to Definition 1, system (2) is robustly exponentially stable, and the exponential convergence rate is ε. The proof is completed.Remark 4.
Stability conditions in Theorems2 and 3 and Corollary 1 are explicit for SNNs, which are convenient to verify in practice. However, they have the disadvantage of neglecting the signs of entries in the connection weight matrices Ak and Bk, and thus, differences between excitatory and inhibitory effects might be ignored.Remark 5.
Theorems2 and 3 and Corollary 1 reflect the relationship between arbitrary switching and constrained switching of system (2). If Ck∗ are M-matrices for all k∈Σ, then system (2) would be exponentially stable at least under constrained switching. If χ=∩k=1mΛCk∗ is nonempty, then system (2) is exponentially stable for any switching signal.


## 4.1. Exponential Stability under Constrained Switching

In this section, we will give a sufficient condition ensuring the global exponential stability of delayed SNNs (2) by using the average dwell time method. Let Q be an M-matrix; we denote
(20)ΛQ≜ξ∈ℝn∣Qξ>0,ξ>0.Definition 4
(see [37]).
LetNσt1,t2 denote the number of discontinuities of a switching signal σ on an interval t1,t2. T>0 is called the average dwell time, if for any t2≥t1≥0 and N0≥0,
(21)Nσt1,t2≤N0+t2−t1Thold.Theorem 2.
Under Assumption1, if for all k∈Σ, Ck∗=Ek∗−Ak∗+Bk∗Lk are nonsingular M-matrices, then for all Ek∈EkI, Ak∈AkI, Bk∈BkI and any external input Jk, the delayed SNN (2) is robustly exponentially stable for any switching signal with the average dwell time satisfying
(22)T>T∗=lnϑε,where ε>0 is determined by inequalities
(23)−e¯ik+εexpετkξik+∑j=1nξjkLjkajik∗+bjik∗<0,for some given ξk=ξ1k,ξ2k,⋯,ξnkT∈ΛCk∗ and ϑ=max1≤i≤n,1≤ik−1≤mηiik−1,ηiik−1βiik−1≥1 with
(24)ηiik−1=max1≤i≤nξiikmin1≤i≤nξiik−1,βiik−1=max1≤j≤nexpετijikLjikbijik∗min1≤j≤nexpετijik−1Ljik−1bijik−1⋆.Proof 2.
According to Theorem1, we know that if Ck∗=Ek∗−Ak∗+Bk∗Lk are M-matrices, then the system has a unique equilibrium point for each specified switching signal. Let w∗=w1∗,w2∗,⋯,wn∗T be an equilibrium point of system (2) and wt=w1t,w2t,⋯,wntT be any solution of system (2). Denote xit=wit−wi∗, fjkxjt=gjkxjt+wj∗−gjkwj∗, and ψis=ϕis−wi∗; then system (2) can be rewritten as
(25)dxitdt=−eiσtxit+∑j=1naijσtfjσtxjt+∑j=1nbijσtfjσtxjt−τijσt,xis+t0=ψis,s∈−τ,0,with i=1,2,…,n.
Due toCk∗ being M-matrices, by Lemma 1 (ii), we know that there exist ξik>0 and δik>0,k∈Σ,i=1,2,⋯,n such that
(26)−e¯ikξik+∑j=1nξjkLjkajik∗+bjik∗=−δik<0.
Consider a Lyapunov functional candidate(27)Vx,t=∑i=1nξiσtexpεtxi+∑j=1nLjσtbijσt∫t−τijσttexpεs+τijσtxjsds.
Calculating the upper right derivativeD+V of V along the solutions of (25), we get
(28)D+Vx,t=∑i=1nξiσtexpεtsgnxidxidt+εexpεtxi+∑j=1nLjσtbijσtexpεt+τijσtxjt−expεtxjt−τijσt=∑i=1nξiσtexpεtsgnxi−eiσtxit+∑j=1naijσtfjσtxjt+∑j=1nbijσtfjσtxjt−τijσt+εexpεtxi+∑j=1nLjσtbijσtexpεt+τijσtxjt−expεtxjt−τijσt≤∑i=1nξiσtexpεt−eiσtxit+∑j=1naijσtfjσtxjt+∑j=1nbijσtfjσtxjt−τijσt+εexpεtxi+∑j=1nLjσtbijσtexpεt+τijσtxjt−expεtxjt−τijσt≤∑i=1nξiσtexpεt−eiσtxit+∑j=1nLjσtaijσtxjt+∑j=1nLjσtbijσtxjt−τijσt+εexpεtxi+∑j=1nLjσtbijσtexpεt+τijσtxjt−expεtxjt−τijσt=∑i=1nexpεtξiσt−eiσt+εxit+∑j=1naijσt+expετijσtbijσtLjσtxjt≤expεt∑i=1n−eiσt+εξiσt+expετσt∑j=1nξjσtLjσtajiσt+bjiσtxit≤expεt+τσt∑i=1n−e¯iσt+εexpετσtξiσt+∑j=1nξjσtLjσtajiσt∗+bjiσt∗xit=expεt+τσt∑i=1n−δiσt+−e¯iσt+εexpετσt+e¯iσtξiσtxit.
Defining functions,(29)ℱikz=−δik+−e¯ik+zexpzτk+e¯ikξik,i=1,2,⋯,n,k∈Σ.
Obviously,ℱik0=−δik<0. Since ℱikz are continuous functions, there exist εik>0,i=1,2,⋯,n, such that ℱikεik<0. Let ε=min1≤k≤m,1≤i≤nεik; we can get  ℱikε<0,i=1,2,⋯,n,k∈Σ. Combining it with inequality (28), we get
(30)D+Vx,t≤expεt+τσt∑i=1nℱiσtεxit≤0.
So fort∈tk,tk+1,
(31)Vx,t≤Vx,tk.
For convenience, we denoteσt=ik when t∈tk,tk+1,k=0,1,…,n. That is, the ikth subnetwork is activated for t∈tk,tk+1; then
(32)Vx,tk=∑i=1nξiikexpεtkxitk+∑j=1nLjikbijik∫tk−τijiktkexpεs+τijikxjsds=∑i=1nξiikexpεtk−xitk−+∑j=1nLjikbijik∫tk−−τijiktk−expεs+τijikxjsds≤∑i=1nηiik−1ξiik−1expεtk−xitk−+βiik−1∑j=1nLjik−1bijik−1∫tk−−τijik−1tk−expεs+τijik−1xjsds,where
(33)ηiik−1=max1≤i≤nξiikmin1≤i≤nξiik−1,βiik−1=max1≤j≤nexpετijikLjikbijik∗min1≤j≤nexpετijik−1Ljik−1bijik−1⋆.
Letϑ=max1≤i≤n,1≤ik−1≤mηiik−1,ηiik−1βiik−1≥1; we can get
(34)Vx,tk≤ϑ∑i=1nξiik−1expεtk−xitk−+∑j=1nLjik−1bijik−1∫tk−−τijik−1tk−expεs+τijik−1xjsds=ϑVx,tk−.
Combining (31) and (34) yields
(35)D+Vx,t≤Vx,tk≤ϑVx,tk−≤ϑVx,tk−1≤⋯≤ϑNσt0,tVx,t0.
Whent=t0, the i0th subnetwork is activated; then
(36)Vx,t0=∑i=1nξii0expεt0xit0+∑j=1nLji0biji0∫t0−τiji0t0expεs+τiji0xjsds=∑i=1nξii0expεt0wit0−wi∗+∑j=1nLji0biji0∫t0−τiji0t0expεs+τiji0wjs−wj∗ds≤∑i=1nξii0expεt0sups∈t0−τ,t0ϕis−wi∗+τi0∑j=1nLji0biji0expεt0+τi0sups∈t0−τ,t0ϕis−wj∗=expεt0∑i=1nξii0sups∈t0−τ,t0ϕis−wi∗+τi0expεt0∑i=1nξii0∑j=1nLji0biji0expετi0sups∈t0−τ,t0ϕis−wj∗≤expεt0∑i=1nξii0sups∈t0−τ,t0ϕis−wi∗+τi0expεt0∑i=1n∑j=1nξji0Lii0bjii0expετi0sups∈t0−τ,t0ϕis−wi∗=expεt0∑i=1nξii0+τi0∑j=1nξji0Lii0bjii0expετi0sups∈t0−τ,t0ϕis−wi∗≤expεt0Mi0∑i=1nsups∈t0−τ,t0ϕis−wi∗21/2=expεt0Mi0ϕ−wt0,where
(37)Mi0=nmax1≤i≤nξii0+τi0∑j=1nξji0Lii0bjii0expετi0.
Combining (27), (35), and (36) yields
(38)expεt∑i=1nξiikxit≤ϑNσt0,texpεt0Mi0ϕ−w∗t0.
Letξ=min1≤i≤n,1≤k≤mξik; (38) becomes
(39)wt−w∗≤ϑNσt0,tMi0ξϕ−w∗t0exp−εt−t0=ϑNσt0,texp−εt−t0Mi0ξϕ−w∗t0=expNσt0,tlnϑ−εt−t0Mi0ξϕ−w∗t0≤expN0lnϑ+t−t0Tlnϑ−εt−t0Mi0ξϕ−w∗t0=expt−t0Tlnϑ−εt−t0ϑN0Mi0ξϕ−w∗t0=exp−t−t0ε−lnϑTϑN0Mi0ξϕ−w∗t0.
LetM=ϑN0Mi0/ξ and λ=ε−lnϑ/T; we have
(40)wt−w∗≤Mϕ−w∗t0exp−λt−t0,when T>lnϑ/ε and λ>0. According to Definition 1, equilibrium point system (2) w∗ is robustly exponentially stable. The proof is completed.Remark 2.
For allk∈Σ, Ck∗ are M-matrices which mean that delayed SNN (2) is globally exponentially stable under constrained switching. From the definitions of ℱikz functions, we know that the value of ε relies on vector ξk=ξ1k,ξ2k,⋯,ξnkT∈ΛCk∗. So, for obtaining the maximum convergence rate λ∗ or the minimum average dwell time T∗, one can solve the optimization problem under constraint conditions ℱikλ,ξk<0,ξk∈ΛCk∗,i=1,2,⋯,n,k∈Σ.


## 4.2. Exponential Stability under Arbitrary Switching

Define the indicator function(41)γt=γ1t,γ2t,⋯,γmtT,where
(42)γkt=1,whenthekthsubnetworkisactivated,0,otherwise,with k=1,2,…,m. Therefore, delayed SNN system (25) can be described as follows:
(43)dxitdt=∑k=1mγkt−eikxit+∑j=1naijkfjkxjt+∑j=1nbijkfjkxjt−τijk,xis+t0=ψis,s∈−τ,0,i=1,…,n.For any switching signal, only one subnetwork is activated at any time, so it follows that∑k=1mγkt=1.Theorem 3.
Under Assumption1, the equilibrium point of delayed SNNs (2) is robustly exponentially stable for all Ek∈EkI, Ak∈AkI, Bk in BkI, and any switching signal if the following conditions are satisfied:
(i)
Ck∗=Ek∗−Ak∗+Bk∗Lk,k∈Σ, are nonsingular M-matrices
(ii)
χ=∩k=1mΛCk∗ is nonempty.
Moreover, the exponential convergence rate of system (2) is equal to λ, which is determined by
(44)−ξie¯ik−λ+∑j=1nξjLjkaijk∗+eλτkbijk∗<0,for a given vector ξ∈χ.Proof 3.
Consider Lyapunov function candidatesvit=xitexpλt−t0. Calculating the upper right derivative D+vi of vi along the solutions of (43), we get
(45)D+vit=expλt−t0sgnxi∑k=1mγkt−eikxit+∑j=1naijkfjkxjt+∑j=1nbijkfjkxjt−τijk+λexpλt−t0xit≤expλt−t0∑k=1mγkt−eik+λxit+∑j=1naijkfjkxjt+∑j=1nbijkfjkxjt−τijk≤expλt−t0∑k=1mγkt−eik+λxit+∑j=1nLjkaijkxjt+∑j=1nLjkbijkxjt−τijk≤expλt−t0∑k=1mγkt−e¯ik+λxit+∑j=1nLjkaijk∗xjt+∑j=1nLjkbijk∗xjt−τijk≤∑k=1mγkt−e¯ik+λexpλt−t0xit+∑j=1nLjkaijk∗expλt−t0xjt+expλτk∑j=1nLjkbijk∗expλt−τijk−t0xjt−τijk.
Substitutingvit=expλt−t0xit into the above inequality, we can get
(46)D+vit≤∑k=1mγkt−e¯ik+λvit+∑j=1nLjkaijk∗vjt+expλτk∑j=1nLjkbijk∗supt−τk≤s≤tvjs,for i=1,2,…,n.
SinceCk∗ are nonsingular M-matrices and χ is nonempty, from Lemma 1, we know that there exists at least one vector ξ=ξ1,ξ2,⋯,ξnT∈χ⊆ΛCk∗ such that
(47)−e¯ikξi+∑j=1naijk∗+bijk∗Ljkξj<0,for i=1,2,…,n,k∈Σ.
Consider functions(48)Gikzik=−ξie¯ik−zik+∑j=1naijk∗+expzikτkbijk∗Ljkξj,with i=1,2,…,n and k=1,2,…,m.
By inequality (47) and the definition of functions Gik, it is clear that Gikzik∈C0 and Gik0<0. Because dGikzik/dzik>0, there are constants λik>0 such that
(49)Gikλik=−ξie¯ik−λik+∑j=1naijk∗+expλikτkbijk∗Ljkξj=0.
Let0<λ<min1≤k≤m,1≤i≤nλik; then
(50)Gikλ=−ξie¯ik−λ+∑j=1naijk∗+expλτkbijk∗Ljkξj<0.for i=1,2,…,n and k∈Σ.
Letl0=ϕ−w∗t0/ξmin, where ξmin=min1≤i≤nξi. So
(51)vis=expλs−t0ϕis−wi∗<ξil0,t0−τ≤s<t0,i=1,2,…,n.
Fort≥t0, we claim that vit<ξil0,i=1,2,…,n. If this is not true, there exist some i and corresponding t′>0, which make vit′=ξil0, D+vit′≥0, and vjt<ξjl0 for t0≤t≤t′, j=1,2,…,n,j≠i. However, applying (44) and (46) leads to
(52)D+vit′≤∑k=1mγkt′−e¯ik+λvit′+∑j=1nLjkaijk∗vjt′+expλτk∑j=1nLjkbijk∗supt′−τk≤s≤t′vjs≤∑k=1mγkt′−e¯ik+λξil0+∑j=1nLjkaijk∗ξjl0+expλτk∑j=1nLjkbijk∗ξjl0<0.
This is a contradiction. Sovit<ξil0,i=1,2,…,n, for t≥t0. That is, for t≥t0,
(53)xi<ξil0exp−λt−t0=ξiξminϕ−w∗t0exp−λt−t0,i=1,2,…,n.
LetM=n⋅max1≤i≤nξi/ξmin; then we can get
(54)w−w∗<Mϕ−w∗t0exp−λt−t0,for t≥t0. From Definition 1, the equilibrium point of system (2) is robustly exponentially stable. Moreover, the exponential convergence rate is λ. The proof is completed.Remark 3.
The existence of exponential convergence rateλ has been proved, and from the definitions of functions Gikz and (44), we know that the value of λ relies on vector ξ=ξ1,ξ2,⋯,ξnT∈χ. So, for obtaining maximum convergence rate λ∗, one can solve the optimization problem under constraint conditions Gikλ,ξ<0,ξ∈χi=1,2,⋯,n,k∈Σ.
By virtue of Theorem3, it is easy to get the following result.Corollary 1.
Under Assumption1, if for k∈Σ, Ck∗=Ek∗−Ak∗+Bk∗Lk are nonsingular M-matrices, then for all Ek∈EkI, Ak∈AkI, and Bk∈BkI, the equilibrium point of system (2) is robustly exponentially stable for any switching signal with the average dwell time satisfying
(55)T>T∗=lnηmaxλ,where ηmax=max1≤i≤n,1≤k≤mξik/min1≤i≤n,1≤k≤mξik and λ>0 is determined by inequality
(56)ξik−e¯ik+λ+∑j=1naijk∗+expλτkbijk∗Ljkξjk<0,for some given ξk=ξ1k,ξ2k,⋯,ξnkT∈χCk∗.Proof 4.
Consider Lyapunov function candidatesvit=xitexpλt−tk. For convenience, we assume that the jkth subsystem is activated when t∈tk,tk+1. Calculating the upper right derivative D+vit of vit along the solutions of (25), we get
(57)D+vit=expλt−tksgnxi−eijkxit+∑j=1naijjkfjjkxjt+∑j=1nbijjkfjjkxjt−τijjkt+λexpλt−tkxit.
It can be known from Theorem3 that there exists λijk>0 such that
(58)Gijkλijk=−ξijke¯ijk−λijk+∑j=1naijjk∗+expλijkτjkbijjk∗Ljjkξjjk=0.
Let0<λ<min1≤i≤nλijk; then
(59)Gijkλ=−ξijke¯ijk−λ+∑j=1nbijjk∗+expλτjkcijjk∗Ljjkξjjk<0.for i=1,2,…,n,jk∈Σ.
By the proof of Theorem3, we know that
(60)xit<ξijkmin1≤i≤nξijkxtkexp−λt−tk,for t∈tk,tk+1. Since the system state is continuous, it follows from (60) that
(61)xit<ξijkmin1≤i≤nξijkxitkexp−λt−tk<⋯<exp∑l=0klnηjl−λt−t0xit0<expk+1lnηmax−λt−t0xit0=ηmaxexpNσt0,tlnηmax−λt−t0xit0≤ηmaxN0+1exp−λ−lnηmaxTt−t0xit0,where ηjv=ξijv/min1≤i≤nξijv and ηmax=max1≤jv≤mηjv. Let M=nηmaxN0+1 and ε=λ−lnηmax/T yield
(62)wt−w∗≤Mϕ−w∗t0exp−εt−t0.
WhenT>T∗=lnηmax/λ, ε>0. According to Definition 1, system (2) is robustly exponentially stable, and the exponential convergence rate is ε. The proof is completed.Remark 4.
Stability conditions in Theorems2 and 3 and Corollary 1 are explicit for SNNs, which are convenient to verify in practice. However, they have the disadvantage of neglecting the signs of entries in the connection weight matrices Ak and Bk, and thus, differences between excitatory and inhibitory effects might be ignored.Remark 5.
Theorems2 and 3 and Corollary 1 reflect the relationship between arbitrary switching and constrained switching of system (2). If Ck∗ are M-matrices for all k∈Σ, then system (2) would be exponentially stable at least under constrained switching. If χ=∩k=1mΛCk∗ is nonempty, then system (2) is exponentially stable for any switching signal.


## 5. Numerical Examples

We present two examples to illustrate the main results.Example 1.
Consider a delayed SNNs with two subnetworks, and the relevant parameters of system (25) are given as follows [18, 21]:
(63)E1=200030002,A1=1.0−0.40.4−0.40.20.20.20.4−0.4,B1=0.30.20.30.20.20.20.30.20.2,E2=30002.80002,A2=1.5−0.30.4−0.40.30.40.20.3−0.5,B2=0.20.30.30.30.20.20.30.30.2.
Take the activation functions asf1x=1/2x1+sinx1,tanhx2,1/2x3+1−x3−1T and f2x=1/2x1+sinx1,1/2x2+sinx2,1/2x3+sinx3T.
Obviously,f1x=f2x satisfy Assumption 1 and L1=L2=diag1,1,1.Step 1.
Determine whetherCk∗=Ek∗−Ak∗+Bk∗Lk are M-matrices.
(64)C1∗=0.7−0.6−0.7−0.62.6−0.4−0.5−0.61.4,C2∗=1.3−0.6−0.7−0.72.3−0.6−0.5−0.61.3are both M-matrices, which imply that the considered SNN is at least exponentially stable under constrained switching. If χ is nonempty, then the system is exponentially stable under arbitrary switching.Step 2.
Determine whetherχ=∩k=12ΛCk∗ is nonempty.
Letξ=0.2146,0.1000,0.1288T; then we can get C1∗ξ>0 and C2∗ξ>0. That is, ∩k=12ΛCk∗≠∅. Therefore, the considered SNN is globally exponentially stable for any switching signal.Step 3.
Calculate the maximum exponential convergence rateλ.
By using LINGO solver, we can get the maximum convergence rateλ=0.6999 under the constraint conditions Gikλ,ξ<0,ξ∈χ=∩k=12ΛCk∗,k,i=1,2, and the corresponding vector ξ=0.2605,0.1205,0.1562T.The numerical simulations are given in Figures1–5. We can see that the state trajectories converge to the equilibrium point of the system, which is consistent with the conclusion of Theorem 3. On the other hand, from [18, 21], we know that when the average dwell time of switched signal is greater than or equal to 9.1936s and 0.8396s; then the considered neural network is exponentially stable. Table 1 shows that the stability criteria obtained in this paper are less conservative than those in [18, 21].Example 2.
Consider the second-order delayed SNNs in system (25) described by [49]: σt:0,+∞→∑=1,2, f1x=f2x=0.5x1+0.5sinx1,0.5x2+0.5sinx2T, τijkt=0.5+0.5sint, i,j,k=1,2, and
(65)E¯1=3.99002.99,E¯1=4.01003.02,E¯2=2.81003.60,E¯2=2.95003.72,A¯1=1.192.350.050.03,A¯1=1.212.410.060.04,A¯2=0.87−0.032.070.68,A¯2=1.010.102.280.80,B¯1=0.093.14−0.050.43,B¯1=0.113.320.130.54,B¯2=0.11−0.022.870.05,B¯2=0.350.103.000.13.
Obviously,f1x and f2x satisfy Assumption 1 and L1=L2=diag1,1,
(66)A1∗=3.99002.99,B1∗=1.212.410.060.04,C1∗=0.113.320.130.54,A2∗=2.81003.60,B2∗=1.010.102.280.80,C2∗=0.350.103.000.13.Step 1.
Determine whetherCk∗=Ek∗−Ak∗+Bk∗Lk,k=1,2, are M-matrices.
(67)C1∗=2.67−5.73−0.192.41,C2∗=1.45−0.20−5.282.67are both M-matrices, which mean that the considered system is at least globally exponentially stable under constrained switching.Step 2.
Determine whetherχ=∩k=12ΛCk∗,k=1,2, is nonempty.
As shown in Figure6, ∩k=12ΛCk∗=∅. Therefore, we can not claim that the considered system is stable under arbitrary switching.Step 3.
Calculate the average dwell timeT∗.
By using LINGO solver, we can get the maximum convergence rateλ=0.4387 under the constraint conditions Gikλ,ξk<0, ξk∈ΛCk∗, k,i=1,2, and the corresponding vectors ξ1=5.5719,1.4679T, ξ2=1.2981,4.1662T, and ηmax=4.2923. So we can get the average dwell time T∗=lnηmax/λ=3.3208s.Figure 1
State responses of subnetwork 1 of the considered SNNs for Example1 with the initial condition ψs=coss−0.4;sin2s+0.4;tanh2sT.Figure 2
State responses of subnetwork 2 of the considered SNNs for Example1 with the initial condition ψs=−coss;coss;−tanhs−0.2T.Figure 3
State norm responses of the subnetworks of the considered SNNs for Example1.Figure 4
State responses of the considered SNNs for Example1 with the initial condition ψs=3.5;3.8;−3.8T.Figure 5
State norm response of the considered SNNs for Example1 with switching signal.Table 1
Stability conditions are derived by different methods.
Methods
Switching signal
Average dwell time
[18]
Constrained
9.1936
[21]
Constrained
0.8396
Theorem3
Arbitrary
—Figure 6
ΛCk∗ of the considered SNNs for Example 2.For numerical simulation, letEk=Ek∗, Ak=Ak∗, and Bk=Bk∗, where k=1,2, and choose the initial value ψ1,ψ2T=cos2s−0.4,sin2s+0.4T,s∈−1,0. Figures 7–9 display the state responses and state norm responses of these two subnetworks. Figures 10–13 display the state responses and state norm responses of the delayed SNNs under two different switching signals. From Figures 10 and 11, we can see that with the dwell time T1=1s that is less than T∗, the trajectories can not converge to the equilibrium point of the system; Figures 12 and 13 show that with the dwell time T2=4s that is larger than T∗, the trajectories converge to the equilibrium point of the system. This is consistent with the conclusion of Corollary 1.Figure 7
State responses of subnetwork 1 of the considered SNNs for Example2.Figure 8
State responses of subnetwork 2 of the considered SNNs for Example2.Figure 9
State norm responses of the subnetworks of the considered SNNs for Example2.Figure 10
State responses of the considered SNNs for Example2 with average dwell time T1=1s<T∗.Figure 11
State norm response of the considered SNNs for Example2 with average dwell time T1=1s<T∗.Figure 12
State responses of the considered SNNs for Example2 with average dwell time T2=4s>T∗.Figure 13
State norm response of the considered SNNs for Example2 with average dwell time T2=4s>T∗.These two examples indicate the correctness and effectiveness of the results proposed in this paper.


## 6. Conclusion

The existence, uniqueness, and robust exponential stability of the equilibrium point of SNNs with time delays were investigated in this paper. For each specified switching signalσt, conditions for guaranteeing the existence and uniqueness of the delayed SNNs were obtained by resorting to the homomorphism mapping theorem and M-matrix theory. By using average dwell time approach, sufficient conditions to ensure the robust exponential stability of the delayed SNNs under constrained switching were derived, and inequality technique and the idea of vector Lyapunov function were employed to obtain conditions for ensuring the globally exponential stability of the delayed SNNs under arbitrary switching. The obtained results not only have less conservativeness but also reveal the relationship between the constrained switching and the arbitrary switching of the delayed SNNs. Finally, two numerical examples were presented to demonstrate the effectiveness and less conservativeness of the main results over existing literature.



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