# Robust Exponential Stability of Switched Complex-Valued Neural Networks with Interval Parameter Uncertainties and Impulses

Authors: Xiaohui Xu; Huanbin Xue; Yiqiang Peng; Quan Xu; Jibin Yang
Journal: Complexity
 (2018)

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## Abstract

In this paper, dynamic behavior analysis has been discussed for a class of switched complex-valued neural networks with interval parameter uncertainties and impulse disturbance. Sufficient conditions for guaranteeing the existence, uniqueness, and global robust exponential stability of the equilibrium point have been obtained by using the homomorphism mapping theorem, the scalar Lyapunov function method, the average dwell time method, and M-matrix theory. Since there is no result concerning the stability problem of switched neural networks defined in complex number domain, the stability results we describe in this paper generalize the existing ones. The effectiveness of the proposed results is illustrated by a numerical example.


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## Body

## 1. Introduction

It is of great importance to study complex-valued neural networks because of their extensive application in many fields, such as filtering, speech synthesis, remote sensing, signal processing, and others, which cannot be analyzed comprehensively with only their real-valued counterparts [1, 2]. Complex-valued neural networks are not only the simple extensions of real-valued systems due to their more complicated properties and research methods for dynamic behavior analysis, including stability, synchronization, and periodicity of system states. Therefore, there have been a great number of important studies published in the last few years that investigate dynamic behavior analysis of complex-valued neural networks.Ever since the authors in [3, 4] studied the stability problems of several kinds of neural networks defined in complex number domain, scholars have presented plenty of significant conclusions about the stability of various complex-valued neural networks. Considering that time delay is unavoidable in the practice model of artificial neural networks, which could lead to instability of systems, Hu [5] established a class of delayed complex-valued neural networks and obtained sufficient conditions for assuring the equilibrium point stability of the system with two classes of complex-valued activation functions. With respect to complex-valued neural networks with various delays, the corresponding achievements can be referred in [6–12]. The impulsive effect inevitably exists in practical neural networks. For instance, system states are subject to instantaneous perturbations and experience abrupt changes in certain instances, which may be caused by the switching phenomenon, a frequency change or another sudden noise, and exhibits an impulsive effect [13]. Authors [13] established some sufficient conditions for assuring the exponential stability of a class of delayed complex-valued neural networks with impulsive disturbance by using the vector Lyapunov function method and the mathematical induction method. In [14], the global exponential convergence of T-S fuzzy complex-valued neural networks with time-varying delays and impulsive effects was discussed, and some sufficient conditions were obtained in terms of a complex-valued linear matrix inequality for assuring the global exponential convergence of the system states. Song [15, 16] also considered the impulsive effect in two tapes of complex-valued neural networks with time delays.Recently, fruitful results on dynamic behavior analysis of nonlinear systems including neural networks with switching signals have been reported in the existing literatures [17–29]. As pointed out in [17], the nodes of a neural network in reality may encounter a connection failure or a new connection build-up; the structure of a neural network is inevitably designed to change abruptly with some internal or external incentives. This usually obligates neural networks to exhibit a special characteristic, network mode switching. Modeling this behavior by using the technique of the Markov process [2] or the switched system [18, 19] is more natural and generally accepted in the practical application of neural networks. However, the stability results proposed in [17–27] are only suitable to neural networks defined in the real number domain. In [2], a class of delayed complex-valued neural networks with Markova jumping parameters was established. The existence, uniqueness, and stochastic exponential robust stability of the equilibrium point of the addressed system were studied by coupling the M-matrix theory with the vector Lyapunov function method. Thus far, there are no stability results with respect to complex-valued neural networks with switching signal. In addition to the impulsive effects, parameter uncertainties have been considered in the form of delayed neural networks including real-valued systems [20, 30, 31] and complex-valued systems [2, 32–34], which were named interval systems. With respect to the stability problem of switched neural networks, there are two main problems that need to be generalized further. The switched neural networks defined in the real number domain can be extended to switched systems defined in the complex number domain. Moreover, the switched activation functions and self-feedback coefficients were not considered in the models of switched real-valued neural networks in the most of existing results, such as [19, 20, 26], which can be assumed to be switch signals.Motivated by the above discussion, in this paper we will investigate the robust exponential stability of a class of switched complex-valued neural networks with interval parameter uncertainties and impulsive disturbance. The advantages and contributions can be described as follows. (1) The switched neural networks are defined in the complex number domain which including real-valued neural networks. (2) The interconnected matrices, self-feedback coefficients, and activation functions in the model are with different forms in different switched subsystems. (3) Both impulsive disturbances and interval parameter uncertainties are considered in the systems. (4) The sufficient conditions for ensuring the robust exponential stability of the equilibrium point are expressed in terms of the simple forms of inequalities, which are easy to check in practice.This paper is organized as follows. In Section2, a model description and preliminaries are given, including definitions, assumed conditions, and lemmas. In Section 3, a theorem and several corollaries are proposed for the stability judgment of the complex-valued neural networks addressed in this paper. In Section 4, a numerical example with simulation results is given to verify the effectiveness of the main results. Finally, the conclusion is drawn and future work is proposed in Section 5.


## 2. Model Description and Preliminaries

Notations, herein defined, will be used throughout the whole paper. LetR, C, and Z+ denote a real number set, a complex number set, and a positive integer set, respectively. Assuming that u=x+yi; here x,y∈R and i denotes the imaginary unit. Let |u| be the module; i.e., |u|=x2+y2. Denote u- as the conjugate complex number of u. For complex number vector u∈Cn, let |u|=(|u1|,|u2|,…,|un|)T, when (·)T denotes the transpose of the vector. Define the norm of the complex number vector u as u=∑j=1n|uj|2. For the complex number matrix A=(ajm)n×n∈Cn×n, let |A|=(|ajm|)n×n∈Rn×n, where |ajm|=[Re(ajm)]2+[Im(ajm)]2 and Re(·) and Im(·) represent the real and the imaginary parts of a complex number, respectively.A class of switched complex-valued neural networks with interval parameter uncertainties and impulse disturbance is considered in this paper. It can be described as follows:(1)u˙t=-Wσtut+Aσtfσtut+Bσtfσtut-τ+O,t≥t0utk+=Δktk,utk,k∈1,2,…us+t0=φs,s∈-τ,0,For analysis convenience, (1) is separated into the following forms:(2)u˙jt=-ωjσtujt+∑m=1najmσtfmσtumt+bjmσtfmσtumt-τjm+Oj,t≥t0ujtk+=Δjktk,ujtk,k∈1,2,…ujs+t0=φjs,s∈-τ,0,In (2), uj(t)∈C represents the neuron state, j=1,2,…,n. It is assumed that the impulsive effect only occurs at the switching instant. The neuron state is right continuous at the instant of impulsive occurrence, i.e., uj(tk)=ujtk+ and uj(tk-)=limt→tk-⁡uj(t).σ(t) is the switching signal or switching law, which is a piecewise constant function. Let σ(t) be [0,∞)→Ξ={1,2,…,N}. It is assumed that σ(t) is unknown switching signal. The following switching sequence can be obtained according to the switching signal σ(t), which means that the h-th subsystem is active when t∈(tk,tk+1],(3)t0,j0,t1,j1,…,tk,jh,…∣jh,h∈Ξ,k=1,2,….Wh=diagω1h,ω2h,…,ωnh∈Rn×n denotes the neuron self-feedback coefficient matrix of theh-th subsystem, where ωjh>0,h∈Ξ, j=1,2,…,n. Ah=(ajmh)n×n∈Cn×n and Bh=(bjmh)n×n∈Cn×n are the connection weight matrices of theh-th subsystem. fh(u(t))=f1h(u1(t)),f2h(u2(t)),…,fnh(un(t))T∈Cn represents the activation function of theh-th subsystem. O(t)=O1(t),O2(t),…,On(t)T∈Cn is the external input vector. Δjk(tk,uj(tk)) is a bounded sequence depending on tk and uj(tk). The delay τjm≥0 is a positive number with τ=max1≤j,m≤n⁡{τjm}. uj(s+t0)=φj(s) is the initial condition of system; here φj(s) is continuous function mapping from [-τ,0] to C, j=1,2,…,n.Assumption 1.
Each functionfjh(·) is globally Lipschitz with Lipschitz constant ljh>0; i.e., the inequality |fjh(uj(t))-fjh(vj(t))|≤ljh|uj(t)-vj(t)| holds for all uj(t),vj(t)∈C, j=1,2,…,n,h∈Ξ. Let Lh=diag(l1h,l2h,…,lnh),h∈Ξ.Remark 2.
Activation functions in complex number domains that satisfy Assumption1 are actually the extensions of the real-valued functions that satisfy the Lipschitz continuity condition. As demonstrated in [2, 7, 14, 34, 35], it is easy to verify that assumptions concerning the conditions of the decomposition of a complex-valued activation function into its real and imaginary parts in [8, 10, 13] are strong constraints, and that, in this case, shows a special case of Assumption 1. In order to unify the research approach using real-valued systems and complex-valued systems, it is assumed that complex-valued activation functions are not explicitly expressed by separating real and imaginary parts, but by satisfying the Lipschitz continuity condition.Assumption 3.
It is assumed that, for alluj(t),vj(t)∈C, there exists a positive number ρjk>0 such that |Δjk(tk,uj(tk+))-Δjk(tk,vj(tk+))|≤ρjk|uj(tk-)-vj(tk-)| holds, and j=1,2,…,n, k=1,2,….It is obvious that Assumption3 implies that the impulsive function Δjk(tk,uj(tk+)) is bounded.After considering parameter uncertainties in (2), the interval weight matrices in the complex number domain are defined as follows for every switching subsystemh (h∈Ξ):(4)Ah∈AhI=Ah=ajmhn×n:A~h≤Ah≤A~h,i.e.a~jmh≤ajmh≤a~jmh,(5)Bh∈BhI=Bh=bjmhn×n:B~h≤Bh≤B~h,i.e.b~jmh≤bjmh≤b~jmh.Letu#=(u1#,u2#,…,un#)T∈Cn be the equilibrium point of (2).Definition 4.
The equilibrium pointu# of (2) is defined to be robustly exponentially stable if there exists M≥1 and λ>0 such that the following inequality holds for all Ah∈AhI,Bh∈BhI(h∈Ξ) and t≥t0≥0,(6)ut-u#≤Mexp⁡-λt-t0sups∈-τ,0⁡φs-u#.Definition 5 (see [36, 37]).
It is supposed thatNσ(t1,t2) denotes the switching numbers on time-interval (t1,t2), here t1>0. If there exists Γ>0 such that, for any t2≥t1≥0 and N0≥0, the inequality Nσ(t1,t2)≤N0+t2-t1/Γ holds, then Γ is called the average dwell time.
In order to obtain results, the following lemmas are given in this paper.Lemma 6 (see [6]).
LetQ=(qjm)n×n∈Rn×n be a matrix with qjm≤0(j,m=1,2,⋯,n,j≠m). The following statements are equivalent: (i) Q=(qjm)n×n is an M-matrix; (ii) the real parts of all eigenvalues of Q are positive; and (iii) there exists a positive vector ξ∈Rn such that Qξ>0.
DenoteΩ(Q)={ξ∈Rn∣Aξ>0,ξ>0} if Q is an M-matrix.Lemma 7 (see [6]).
IfH(z) is a continuous function on Cn and satisfies the following conditions, (i) Hz is univalent injective on Cn, and (ii) limz→∞⁡H(z)→∞, then H(z) is a homeomorphism of Cn into itself.


## 3. Main Results

In this section, we will give several sufficient conditions for judging the robust exponential stability ofu#of (2).For analysis convenience, letu~(t)=u(t)-u#. Then, (2) is converted to the following forms by translation:(7)u~˙jt=-ωjσtu~jt+∑m=1najmσtgmσtu~mt+bjmσtgmσtu~mt-τjm,t≥t0u~jtk+=Δ~jktk,u~jtk,k∈1,2,…u~js+t0=φ~js,s∈-τ,0,where φ~j(s)=φj(s)-uj#, gmσ(t)(u~m(t))=fmσ(t)(u~m(t)+um#)-fmσ(t)(um#), and Δ~jk(tk,u~j(tk))=Δjk(tk,u~j(tk)+uj#)-Δjk(tk,uj#).Theorem 8.
Suppose that Assumptions1 and 3 hold. If the following conditions are satisfied, then the equilibrium point u# of (2) exists and is unique and robustly exponentially stable for all Ah∈AhI, Bh∈BhI, and arbitrary input O(t)∈Cn:
(I) Every matrixQh=(qjmh)n×n is a M-matrix, where (8)qjjh=2ωjh-∑m=1nlmha~jmh+b~jmh,qjmh=-2lmha~jmh+b~jmh,h∈Ξ,j,m=1,2,…,n,j≠m.
(II) For any givenξh∈Ω(Qh), the average dwell time satisfies Γ>Γ∗=ln⁡υ/ε; here ε>0 satisfies the following inequalities: (9)ξjhε-2ωjh+∑m=1nlmha~jmh+b~jmhexp⁡ετ+2ξmaxh∑m=1nlmha~jmh+b~jmh<0,where υ=max1≤j≤n,k∈Z+,jh,mh∈Ξ⁡{πjmh(ρjk)2,πjmhβjmh}≥1, ξmaxh=max1≤j≤n⁡{ξjh}, πjmh=max1≤j≤n⁡{ξjjh}/min1≤j≤n⁡{ξjmh}, and βjmh=max1≤m≤n⁡lmjhb~jmjh/min1≤m≤n⁡{lmmh|b~jmmh|}.Proof.
First, the existence and uniqueness ofu#of (2) will be proven for every subsystem by Lemmas 6 and 7.
The mapHh(u)=[H1h(u),H2h(u),…,Hnh(u)]T∈Cn which is associated with (2), is defined, i.e., for all j=1,2,…,n and h∈Ξ, (10)Hjhu=-ωjhuj+∑m=1najmh+bjmhfmhum+Oj.
It is well known that ifHh(u) is a homeomorphism map, then there is a unique equilibrium point u# for (2).
We prove thatHh(u) is a univalent injective on Cn under Assumption 1.
According to Lemma6, it can be concluded that there exists a positive vector ξh such that the following inequalities hold for all j=1,2,…,n,h∈Ξ:(11)ξjh-2ωjh+∑m=1nlmha~jmh+b~jmh+2∑m=1nlmhξmha~jmh+b~jmh<0.
Moreover, it follows from (9) that there exists a sufficient small positive number μ>0 such that the following inequalities hold for all j=1,2,…,n,h∈Ξ: (12)ξjh2ωjh-∑m=1nlmha~jmh+b~jmh-2ξmaxh∑m=1nlmha~jmh+b~jmh≥μ>0.
It is assumed that there existu,v∈Cn with u≠v such that Hjh(u)=Hjh(v), j=1,2,…,n,h∈Ξ; i.e., (13)-ωjhuj+∑m=1najmh+bjmhfmhum=-ωjhvj+∑m=1najmh+bjmhfmhvm.
Considering (4) and (5) as well as Assumption 1, for all j=1,2,…,n,h∈Ξ, we get(14)ωjhuj-vj≤∑m=1nlmha~jmh+b~jmhum-vm.
Furthermore, (14) can be rewritten as Q_h|u-v|≤0 where q_jjh=ωjh, q_jmh=-lmh|a~jmh|+|b~jmh|, j=1,2,…,n, j≠m. Because the matrix Qh is an M-matrix, it is obvious that the matrix Q_h=(q_jmh)n×n is also an M-matrix. We know that detQ_h>0 holds and (Q_h)-1 exists obviously. Furthermore, it can be concluded that |u-v|=0; i.e., u=v. It is an obvious contradiction with the assumption u≠v. Hence, Hh(u) is a univalent injective on Cn,h∈Ξ.
Next, we will prove thatlimu→∞⁡Hh(u)→∞.
LetH~jh(u)=Hjh(u)-Hjh(0), j=1,2,…,n,h∈Ξ; i.e.,(15)H~jhu=-ωjhuj-ωjh0+∑m=1najmh+bjmhfmhum-fmh0.
After multiplying by the conjugate complex numberu-j of uj on both sides of (15), then for all j=1,2,…,n,h∈Ξ, we get (16)H~jhuu-j=-ωjhuj-ωjh0u-j+u-j∑m=1najmh+bjmhfmhum-fmh0.
Taking the conjugate operation on both sides of (16), for all j=1,2,…,n,h∈Ξ, we have (17)H~-jhuuj=-ωjhu-j-ωjh0uj+uj∑m=1na-jmh+b-jmhf-mhum-f-mh0.
After combining (16) and (17) and considering (4), (5), Assumption 1, for all j=1,2,…,n,h∈Ξ, we obtain (18)ReH~jhuu-j=-Reωjhuj-ωjh0u-j+Reu-j∑m=1najmh+bjmhfmhum-fmh0≤-ωjhuj2+u-j∑m=1nlmha~jmh+b~jmhum.
After multiplying byξjh (j=1,2,…,n,h∈Ξ) on both sides of (18), taking the sum, and considering (12), we get (19)∑j=1nξjhRe⁡H~jhuu-j≤∑j=1nξjh-ωjhuj2+uj∑m=1nlmha~jmh+b~jmhum≤12∑j=1nξjh-2ωjhuj2+uj2∑m=1nlmha~jmh+b~jmh+∑m=1nlmha~jmh+b~jmhum2≤12∑j=1nξjh-2ωjh+∑m=1nlmha~jmh+b~jmhuj2+2∑j=1nξmaxh∑m=1nlmha~mjh+b~mjhuj2≤-0.5μ∑j=1nuj2.
Furthermore, it can be concluded that(20)0.5μ∑j=1nuj2<-∑j=1nξjhRe⁡H~jhuu-j≤ξmaxh∑j=1nH~jhuu-j≤0.5ξmaxh∑j=1nH~jhu2+0.5ξmaxh∑j=1nu-j2.
That isμ-ξmaxh∑j=1n|uj|2≤ξmaxh∑j=1n|H~jh(u)|2; i.e., μ-ξmaxhu2≤ξmaxhH~h(u)2. Therefore, u2≤μ-ξmaxh-1ξmaxhH~h(u)2 holds. Obviously, we have that H~h(u)→∞ as u→∞. It means that Hh(u)→∞ as u→∞.
In summation, it follows from Lemma7 that Hh(u) is a homeomorphism on Cn,h∈Ξ. Therefore, (2) has a unique equilibrium point u#.
Next, the robust exponential stability of the equilibrium pointu# will be proved by applying the scalar Lyapunov function method.
From (11), it can be obtained that (21)ξjσt-2ωjσt+∑m=1nlmσta~jmσt+b~jmσt+∑m=1nlmσtξmσta~jmσt+b~jmσt<0.
Define a function as follows:(22)Fjλ=ξjσtλ-2ωjσt+∑m=1nlmσta~jmσt+b~jmσtexp⁡λτ+∑m=1nlmσtξmσta~jmσt+b~jmσt.
Due toFj(0)<0 and Fj(·) is a continuous function, there exists εjσ(t)>0 such that Fj(εjσ(t))<0; i.e.,(23)Fjεjσt=ξjσtεjσt-2ωjσt+∑m=1nlmσta~jmσt+b~jmσtexp⁡εjσtτ+∑m=1nlmσtξmσta~jmσt+b~jmσt<0.
Choose a candidate scalar Lyapunov function as follows:(24)Vt,u~t=∑j=1nξjσtexp⁡εtu~jt2+∑m=1nlmσtb~jmσt∫t-τjmtexp⁡εs+τjmu~ms2ds.
Calculating the right upper derivationV(t,u~(t)) along the zero solution of (7), we obtain (25)D+Vt,u~t=∑j=1nξjσtεexp⁡εtujt2+2expεtReu~-jtu~˙jt+∑m=1nlmσtbjmσt×exp⁡εt+τjmu~mt2-exp⁡εtu~mt-τjm2=∑j=1nξjσtεexp⁡εtu~jt2+2exp⁡εtReu~-jt×-ωjσtu~jt+∑m=1najmσtgmσtu~mt+bjmσtgmσtu~mt-τjm+∑m=1nlmσtbjmσtexp⁡εt+τjmu~mt2-exp⁡εtu~mt-τjm2≤∑j=1nξjσtεexp⁡εtu~jt2-2exp⁡εtωjσtu~jt2+∑m=1nlmkexp⁡εta~jmσtu~jt2+u~mt2+b~jmσtu~jt2+u~mt-τjm2+∑m=1nlmσtb~jmσtexp⁡εt+τjmu~mt2-exp⁡εtu~mt-τjm2≤∑j=1nξjσtexpεtε-2ωjσt+∑m=1nlmσta~jmσt+b~jmσtu~jt2+∑m=1nlmσtexp⁡ετa~jmσt+b~jmσtu~mt2≤exp⁡εt+τ∑j=1nξjσtε-2ωjσt+∑m=1nlmσtajmσt+bjmσtexp⁡ετu~jt2+∑m=1nξjσtlmσta~jmσt+b~jmσtu~mt2.
Considering (23), we get(26)D+Vt,u~t≤exp⁡εjσtt+τ∑j=1nFjεjσtu~jt2≤0.
Becauseσ(t): [0,∞)→Ξ={1,2,…,N} is a piecewise constant function, taking ε=minh∈Ξ,1≤j≤n⁡{εjh}, we can get(27)D+Vt,u~t≤expεt+τ∑j=1nFjεu~jt2≤0.
Furthermore, whent∈[tk,tk+1), V(t,u~(t))≤V(tk+,u~(tk+)) holds.
Next, we will prove the convergence of system state at the impulsive instant. Suppose thatσ(t)=mh∈Ξ as t∈[tk-1,tk), which means the subsystem mh is activated. When t∈[tk,tk+1), σ(t)=jh∈Ξ, this means that the subsystem jh is activated at tk+, k=1,2,…. Furthermore, we have(28)Vtk+,u~tk+=Vtk,u~tk=∑j=1nξjjhexp⁡εtk+u~jtk+2+∑m=1nlmjhb~jmjh∫tk+-τjmtk+exp⁡εs+τjmu~ms2ds≤∑j=1nξjjhexpεtk+ρjk2u~jtk-2+∑m=1nlmjhb~jmjh∫tk+-τjmtk+expεs+τjmu~ms2ds=∑j=1nξjjhξjmhξjmhexp⁡εtk-ρjk2u~jtk-2+∑m=1n∫tk--τjmtk-lmmhbjmmhlmmhbjmmhlmjhb~jmjhexp⁡ετjmexp⁡εsu~ms2ds.
According to the conditions given below (9) and Assumption 3, we have (29)Vtk+,u~tk+≤∑j=1nπjmhξjmhexp⁡εtk-ρjk2u~jtk-2+βjmh∑m=1nlmmhb~jmmh∫tk--τjmtk-exp⁡εsexp⁡ετjmu~ms2ds≤υ∑j=1nξjmhexp⁡εtk-u~jtk-2+∑m=1nlmmhbjmmh∫tk--τjmtk-exp⁡εs+τjmu~ms2ds=υVtk-,u~tk-.
Combining (27) and (29), we have V(t,u~(t))≤V(tk+,u~(tk+))≤υV(tk-,u~(tk-)); here υ≥1; i.e., (30)Vt,u~t≤υVtk,u~tk,t∈tk,tk+1.
Using this analogy, we haveV(t,u~(t))≤υNσ(t0,t)V(t0,u~(t0)), where Nσ(t0,t) denotes the switching number. Because the impulses only occur at the switching moment, Nσ(t0,t) also denotes the impulse number.
Furthermore, according to integral mean value theorem, we have(31)Vt0,u~t0≤exp⁡εt0∑j=1nξjσt0supt0-τ≤s≤t0⁡u~js2+∑m=1nlmσt0b~jmσt0τexp⁡ετsupt0-τ≤s≤t0⁡u~ms2=exp⁡εt0∑j=1nξjσt0supt0-τ≤s≤t0⁡u~js2+∑j=1n∑m=1nξjσt0ljσt0b~mjσt0τexp⁡ετsupt0-τ≤s≤t0⁡u~ms2.Let Mσt0=nmax1≤j≤n⁡ξjσt0+τexp⁡ετξjσt0∑m=1nljσt0b~mjσt0.
Then(32)Vt0,u~t0≤exp⁡εt0Mσt0∑j=1nsupt0-τ≤s≤t0⁡u~js2=exp⁡εt0Mσt0supt0-τ≤s≤t0⁡u~s2.
That is,V(t,u~(t))≤υNσ(t0,t)V(t0,u~(t0))≤exp(εt0)υNσ(t0,t)Mσ(t0)supt0-τ≤s≤t0⁡u~s2.
In addition, it is obvious thatV(t,u~(t))≥∑j=1nξjσ(t)exp(εt)|u~j(t)|2 holds. Let ξjh=min1≤j≤n⁡{ξjjh}, so we have ξjhexp(εt)∑j=1n|u~j(t)|2≤V(t,u~(t)).
That is,(33)ξjhexp⁡εt∑j=1nu~jt2≤Vt,u~t≤exp⁡εt0υNσt0,tMσt0supt0-τ≤s≤t0⁡u~s2,i.e., u~t2≤Mσ(t0)/ξjhυNσ(t0,t)exp(-ε(t-t0))supt0-τ≤s≤t0⁡u~s2. According to Nσ(t1,t2)≤N0+t2-t1/Γ in Definition 5, furthermore, we have(34)u~t2≤Mσt0ξjhexp⁡ln⁡υNσt0,texp⁡-εt-t0supt0-τ≤s≤t0⁡u~s2≤Mσt0ξjhexp⁡ln⁡υN0+t-t0/Γexp⁡-εt-t0supt0-τ≤s≤t0⁡u~s2=Mσt0ξjhυN0exp⁡-ε-ln⁡υΓt-t0supt0-τ≤s≤t0⁡u~s2.
Letθ2=maxjh∈Ξ⁡MjhυN0/ξjh; μ=ε-ln⁡υ/Γ; thus, we get u~t≤θexp⁡(-0.5μ(t-t0))φ~(s). It is obvious that θ>0 when Γ>ln⁡υ/ε.
According to Definition4, it can be concluded that the zero solution of (7) is robustly exponentially stable, which means the equilibrium point of (2) is robustly exponentially stable. The proof is completed.Remark 9.
Robust stability of impulsive complex-valued neural networks with time delays was studied by applying Lyapunov-Krasovskii functional method and the complex-valued matrix inequality skills in [34]. Switching signal was not considered in the model addressed in there. In addition, in most of papers concerning switched neural networks defined in real number domain, such as [19, 20, 26], only interconnected matrices were supposed to be switched signals. Obviously, the impulsive complex-valued system with switched signals including both self-feedback functions and activation functions in this paper includes the models established in [19, 20, 26, 34].
When there is no impulse effect in (2), the corresponding corollary can be stated as follows.Corollary 10.
Suppose that Assumption1 holds. If the following conditions are satisfied, then the equilibrium point u# of (2) is with existence, uniqueness, and robustly exponential stability for all Ah∈AhI, Bh∈BhI (h∈Ξ), and arbitrary input O(t)∈Cn:
(I) Every matrixQh=(qjmh)n×n is an M-matrix, where (35)qjjh=2ωjh-∑m=1nlmha~jmh+b~jmh,qjmh=-2lmha~jmh+b~jmh,j,m=1,2,…,n,j≠m.
(II) The average dwell time satisfiesΓ>Γ∗=ln⁡υ/ε, with υ=max1≤j≤n,jh,mh∈Ξ⁡{πjmhβjmh}≥1; here ε, πjmh, and βjmh are defined as Theorem 8.Because the proof of Corollary10 is similar to that of Theorem 8, it is omitted here.It is well known that complex-valued neural networks can be viewed as extensions of real-valued neural networks. In the following explanation, we will present some conditions guaranteeing the robust exponential stability of switched neural networks with impulses and delays under the condition that all variables and functions in (2) and Assumptions 1 and 3 are defined in the real number domain.Corollary 11.
Suppose that Assumptions1 and 3 hold. If the following conditions are satisfied, then the equilibrium point u# of (2) is with existence, uniqueness, and robustly exponential stability for all Ah∈AhI, Bh∈BhI and arbitrary input O(t)∈Rn:
(I) Every matrixQh=(qjmh)n×n is an M-matrix, where (36)qjjh=ωjh,qjmh=-lmha~jmh+b~jmh,h∈Ξ,j,m=1,2,…,n,j≠m.
(II) For any givenξh∈Ω(Qh), the average dwell time satisfies Γ>Γ∗=ln⁡υ/ε; here, ε>0 satisfies the following inequality:(37)ξjhε-ωjhexp⁡ετ+ξmaxh∑m=1nlmha~jmh+b~jmh<0,where(38)ξmaxh=max1≤j≤n⁡ξjh,πjmh=max1≤j≤n⁡ξjjhmin1≤j≤n⁡ξjmh,υ=max1≤j≤n,k∈Z+,jh,mh∈Ξ⁡πjmhρkjh,πjmhβjmh≥1,βjmh=max1≤m≤n⁡lmjhb~jmjhmin1≤m≤n⁡lmmhb~jmmh.Proof.
Choose the candidate scalar Lyapunov function as follows:(39)Vt,u~t=∑j=1nξjσtexp⁡εtu~jt+∑m=1nlmσtb~jmσt∫t-τjmtexp⁡εs+τjmu~msds.
By using similar analysis to that used as proof for Theorem8, the conclusion of Corollary 11 can be obtained directly. The proof is completed.When comparing Theorem8 and Corollary 11, it can be seen that the stability conditions for judging complex-valued neural networks are more complicated and conservative due to the more complicated properties of complex-valued systems.Remark 12.
In addition to impulse disturbances [13–15, 34, 35] and parameter uncertainties [2, 31, 34], stochastic disturbances [10, 38] also exist in real complex-valued neural networks. Additionally, the stability problem and the synchronization problem of fractional-order systems have become a hot research topic [39, 40]. The research in this paper can be extended further to the study of the stability of the mentioned systems.


## 4. Example

In this section, an example with simulation results is given to illustrate the effectiveness of the sufficient conditions established in the preceding section.Consider a two-order system with three subsystems described by (2). In (2), we assume that the complex-valued weight matrices with parameter uncertainties are the same in all subsystems. It is supposed that the weight matrices are defined within the following interval:(40)A1I=A2I=A3I=-0.3,0.6+0,0.8i0.4,0.8+0.8,1i0.7,0.9+-1,-0.2i-0.3,0+0.1,0.6i,B1I=B2I=B3I=-1,1+-0.3,0.9i0,0.7+-0.1,2i0,0.5+0.1,0.6i0.2,0.4+-0.1,0.5i.By computation, the following matrices for every switched subsystem can be obtained:(41)A~=1.0001.2811.3450.671,B~=1.3451.5780.7810.640.Table1 shows the self-feedback coefficients, activation functions, and impulsive functions of three subsystems in (2), respectively.Table 1
Assumptions for (2) with j=1,2.
Functions Subsystem 1 Subsystem 2 Subsystem 3 ωj 4.5 9 8   f j ( u j ( t ) ) 0.25 u j ( t ) + 0.25 sin ⁡ ( u j ( t ) ) 0.75 u j ( t ) + 0.75 sin ⁡ ( u j ( t ) ) 1 - exp ⁡ ( - u - j ( t ) ) 1 + exp ⁡ ( - u - j ( t ) )   Δj(tk+,uj(tk+)) 2.2 u j ( t k - ) 1.8 u j ( t k - ) 2.1 u j ( t k - )It can be determined from Table1 that l11=l21=0.25, l12=l22=0.75, l13=l23=0.5, ρ1k=ρ2k=2.2, ρ1k=ρ2k=1.8, ρ1k=ρ2k=2.1, k=1,2,….Letε=1, ξ1=[1.05,0.95]T, ξ2=[1.1,1]T, and ξ3=[1,1.2]T. Through calculations, we can drive that Γ∗=ln⁡υ/ε=2.434s. According to the condition in Theorem 8, the average dwell time is taken as Γ=3s>Γ∗. Therefore, the switching time sequence (or impulse moment of occurrence) is described as tk∈{3s,6s,9s,…}. Assume that the external inputs of the three subsystems are zero. Let the delays be defined as τjm=0.6s, j,m=1,2.Furthermore, by using calculation, we can determine the following:(42)Q1=7.70-1.43-1.068.14,Q2=14.10-4.28-3.3215.42,Q3=13.40-2.86-2.1314.28.It follows from Lemma6 that matrices Q1, Q2, and Q3 are all M-matrix.Additionally, we need to verify (9) of condition (II) in Theorem 8. For the three subsystems, we obtain the calculated results in Table 2.Table 2
Calculation results of inequality (9).
Subsystem 1 Subsystem 2 Subsystem 3 Neuron 1 - 0.993 - 0.105 - 1.929  Neuron 2 - 1.924 - 2.254 - 3.174It can be determined from Table2 that inequalities (9) hold in all three subsystems.In conclusion, all conditions in Theorem8 are satisfied. According to Theorem 8, we can conclude that the equilibrium point of (2) with above assumptions possesses existence, uniqueness, and exponential robust stability.Tables3 and 4 show the initial conditions and the interconnected matrices of each of the three subsystems in (2), respectively. They will be used in simulation.Table 3
Initial conditions of (2).
Subsystem 1 Subsystem 2 Subsystem 3 u 1 ( s ) 2+1.5i 1.3+i 0.2-0.5i  u 2 ( s ) -1-1.2i -1-0.2i -0.7+1.1iTable 4
Interconnected matrices in (2) for numerical simulation.
Subsystem 1 Subsystem 2 Subsystem 3 A - 0.3 0.4 + 0.8 i 0.7 - i - 0.3 + 0.1 i 0.6 + 0.8 i 0.8 + i 0.9 - 0.2 i 0.6 i 0.3 + 0.4 i 0.6 + 0.9 i 0.8 - 0.6 i - 0.2 + 0.4 i    B - 1 - 0.3 i - 0.1 i 0.1 i 0.2 - 0.1 i 1 + 0.9 i 0.7 + 2 i 0.5 + 0.6 i 0.4 + 0.5 i 0.3 i 0.4 + i 0.3 + 0.6 i 0.3 - 0.1 iThe state curves of (2) with the three subsystems are, respectively, given in Figures 1–6. It follows from the simulation results that the equilibrium point of (2) is existent, unique, and stable, verifying the correctness of Theorem 8.Figure 1
State curves of subsystem 1 in (2).Figure 2
State curves of subsystem 2 in (2).Figure 3
State curves of subsystem 3 in (2).Figure 4
Real part curves and imaginary part curves of system (2).Figure 5
State module curves of system (2).Figure 6
State norm curves of system (2).


## 5. Conclusion and Future Work

The existence, the uniqueness, and the robust exponential stability of the equilibrium point of a class of switched complex-valued neural networks with interval parameter uncertainties and impulses are investigated in this paper. By employing homomorphism mapping theorem, the scalar Lyapunov function method, the average dwell time method, and M-matrix theory, several sufficient conditions are established for guaranteeing the existence, the uniqueness, and the robust exponential stability of the system. Finally, a numerical example with simulation results has been provided to show the effectiveness of the obtained results.Based on the research presented in this paper, we will try to study the dynamic behavior of a class of switched complex-valued neural networks with mixed delays (including both time-varying delays and continuously distributed delays) and impulsive effect which do not only occur at the switching moment between subsystems, but also when transferring inside subsystems.



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