# Globalμ-Stability of Complex-Valued Neural Networks with Unbounded Time-Varying Delays

Authors: Xiaofeng Chen; Qiankun Song; Xiaohui Liu; Zhenjiang Zhao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The complex-valued neural networks with unbounded time-varying delays are considered. By constructing appropriate Lyapunov-Krasovskii functionals, and employing the free weighting matrix method, several delay-dependent criteria for checking the globalμ-stability of the addressed complex-valued neural networks are established in linear matrix inequality (LMI), which can be checked numerically using the effective LMI toolbox in MATLAB. Two examples with simulations are given to show the effectiveness and less conservatism of the proposed criteria.


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## Body

## 1. Introduction

The nonlinear system is everywhere in real life [1–4], while complex network is one of the kinds of it [5–7]. And neural network is one of the most important of complex networks [8, 9]. In recent years, there have been increasing research interests in analyzing the dynamic behaviors of complex-valued neural networks due to their widespread applications in filtering, imaging, optoelectronics, speech synthesis, computer vision, and so on. For example, see [10–16] and the references therein. In these applications, the stability of complex-valued neural networks plays a very important role.In real-valued neural networks, the activation function is usually chosen to be smooth and bounded. However, in the complex domain, according to Liouville’s theorem [17], every bounded entire function must be constant. Thus, if the activation function is entire and bounded in the complex domain, then it is a constant. This is not suitable. Therefore, the problem of choosing activation functions is the main challenge for complex-valued neural networks. In [18–20], authors considered three kinds of activation functions, respectively. In [18], authors considered a class of continuous-time recurrent neural networks with two types of activations functions and gave several sufficient conditions for existence, uniqueness, and global stability of the equilibrium point. In [19], a class of generalized discrete complex-valued neural networks were studied. The existence of a unique equilibrium pattern is discussed and a sufficient condition of global exponential stability was given. In [20], authors discuss a class of discrete-time recurrent neural networks with complex-valued linear threshold neurons. The boundedness, global attractivity, and complete stability of such networks were investigated. Some conditions for those properties were also derived. In [21], the discrete-time delayed neural networks with complex-valued linear threshold neurons were investigated, and several criteria on boundedness and global exponential stability were obtained. In [22–24], authors considered different kinds of time delays in complex-valued neural networks. In [22], the boundedness and complete stability of complex-valued neural networks with time delay were studied. Some conditions to guarantee the boundedness and complete stability for the considered neural networks were derived by using the local inhibition and the energy minimization method. In [23], authors investigated the problem of the dynamic behaviors of a class of complex-valued neural networks with mixed time delays. Some sufficient conditions for assuring the existence, uniqueness, and exponential stability of the equilibrium point of the system are derived using the vector Lyapunov function method, homeomorphism mapping lemma, and the matrix theory. In [24], the complex-valued neural networks with both leakage time delay and discrete time delay as well as two types of activation functions on time scales were considered. Several delay-dependent criteria for checking the global stability of the addressed complex-valued neural networks are established in linear matrix inequality by constructing appropriate Lyapunov-Krasovskii functionals and employing the free weighting matrix method. However, the obtained results were based on the assumption that the time delay is bounded. As we know, time delays occur and vary frequently and irregularly in many engineering systems, and sometimes they depend on the histories heavily and may be unbounded [25–27].How to guarantee the desirable stability if the time delays are unbounded? Recently, the authors proposed two new concepts, power stability andμ-stability [28, 29], which can be applied to dynamical systems with unbounded time-varying delays. In [28], authors considered the dynamical systems with unbounded time-varying delays. Two approaches were developed to derive sufficient conditions ensuring the existence, uniqueness of the equilibrium, and its global stability. Moreover, under mild conditions, authors proved that the dynamical systems with unbounded time-varying delays were globally power stable. In [29], authors discussed the μ-stability of dynamical systems with unbounded time-varying delays and explored that under mild conditions; the delayed system with very large time delays has a unique equilibrium, which is μ-stable globally. In [30], authors investigated the global robust stability for uncertain stochastic neural networks with unbounded time-varying delays and norm-bounded parameter uncertainties. A new concept of global robust μ-stability in the mean square for neural networks was given first, and then by means of the linear matrix inequality approach, stability criteria were presented. To the best of our knowledge, however, few authors have considered the problem on μ-stability of complex-valued neural networks with unbounded time-varying delays.Motivated by the above discussions, the objective of this paper is to study the globalμ-stability of complex-valued neural networks with unbounded time-varying delays by employing a combination of Lyapunov-Krasovskii functionals and the free weighting matrix method. The obtained sufficient conditions do not require the existence of the partial derivatives of the activation functions, and are expressed in linear matrix inequality (LMI), which can be checked numerically using the effective LMI toolbox in MATLAB. Two examples are given to show the effectiveness and less conservatism of the proposed criteria.In the literature, the usual method analyzing complex-valued neural networks is to separate it into real part and imaginary part and then to recast it into equivalent real-valued neural networks [18, 24]. However, this separation is not always expressible in an analytical form if the real and imaginary parts of activation functions cannot be separated. In this paper, we do not separate the complex-valued neural networks and directly consider its properties on ℂn.Notations. The notations are quite standard. Throughout this paper, let i denote the imaginary unit; that is, i=-1. ℂn, ℝm×n, and ℂm×n denote, respectively, the set of n-dimensional complex vectors, m×n real matrices, and complex matrices. The subscripts T and * denote matrix transposition, matrix complex conjugation and transposition, respectively. For complex vector z∈ℂn, let |z|=(|z1|,|z2|,…,|zn|)T be the module of the vector z and ∥z∥=∑k=1n‍|zk|2 the norm of the vector z. The notation X≥Y (resp., X>Y) means that X-Y is positive semidefinite (resp., positive definite). λmax⁡(P) and λmin⁡(P) are defined as the largest and the smallest eigenvalue of positive definite matrix P, respectively. Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.


## 2. Problems Formulation and Preliminaries

In this section, we will recall some definitions and lemmas which will be used in the proofs of our main results.In this paper, we consider the following complex-valued neural networks with time-varying delays:(1)u˙(t)=-Cu(t)+Df(u(t))+Ef(u(t-τ(t)))+H
for t>0, where u(t)=(u1(t),u2(t),…,un(t))T∈ℂn is the state vector of the neural networks with n neurons at time t. C=diag⁡{c1,c2,…,cn}∈ℝn×n is the self-feedback connection weight matrix with ck>0 (k=1,2,…,n). D=(dkj)n×n∈ℂn×n and E=(ekj)n×n∈ℂn×n are the connection weight matrix and the delayed connection weight matrix, respectively. f(u(t))=(f1(u1(t)),f2(u2(t)),…,fn(un(t)))T∈ℂn denotes the neuron activation at time t. H=(h1,h2,…,hn)T∈ℂn is the external input vector.The initial condition associated with neural network (1) is given by
(2)u(s)=ψ(s),s∈(-∞,0],
where ψ(s)=(ψ1(s),ψ2(s),…,ψn(s))T is continuous in (-∞,0].As we know, the activation functions play a very important role in the study of global stability of neural networks. However, in the complex domain, the activation functions cannot be both entire and bounded. In this paper, we will consider the following two types of activation functions.(H1) Letu=x+iy. fj(u) can be expressed by separating into its real and imaginary part as
(3)fj(u)=fjR(x,y)+ifjI(x,y)
for j=1,2,…,n, where fjR(·,·):ℝ2→ℝ and fjI(·,·):ℝ2→ℝ. Suppose there exist positive constants ξjR1, ξjR2, ξjI1, and ξjI2 such that
(4)|fjR(x1,y1)-fjR(x2,y2)|≤ξjR1|x1-x2|+ξjR2|y1-y2|,|fjI(x1,y1)-fjI(x2,y2)|≤ξjI1|x1-x2|+ξjI2|y1-y2|
for any x1,x2,y1,y2∈ℝ, j=1,2,…,n. Moreover, we define Γ=diag⁡{γ1,γ2,…,γn}, where γj=2[(ξjR)2+(ξI)j2], ξjR=max⁡{ξjR1,ξjR2}, and ξjI=max⁡{ξjI1,ξjI2}.(H2) The real and imaginary parts offj(·) cannot be separated, but fj(·) is bounded and satisfies the following condition:
(5)|fj(z1)-fj(z2)|≤ηj|z1-z2|
for any z1,z2∈ℂ, j=1,2,…,n, where ηj is a constant. Moreover, we define Π=diag⁡{η12,η22,…,ηn2}.For the time-varying delaysτ(t), we give the following assumption.(H3)τ(t) is nonnegative and differentiable and satisfies τ˙(t)≤ρ<1, where ρ is a nonnegative constant.As usual, a vectoru-=(u-1,u-2,…,u-n)T is said to be an equilibrium point of neural network (1) if it satisfies
(6)-Cu-+(D+E)f(u-)+H=0.Throughout the paper, the equilibrium of neural network (1) is assumed to exist. For notational convenience, we will always shift an intended equilibrium point u- of neural network (1) to the origin by letting z(t)=u(t)-u-. It is easy to transform neural network (1) into the following form:
(7)z˙(t)=-Cz(t)+Dg(z(t))+Eg(z(t-τ(t))),z(s)=ϕ(s),s∈(-∞,0],
for t>0, where g(z(·))=g(z(·)+u-)-g(u-) and ϕ(s)=ψ(s)-u-.Next we introduce some definitions and lemmas to be used in the stability analysis.Definition 11.
Suppose thatμ(t) is a positive continuous function and satisfies μ(t)→∞ as t→∞. If there exist scalars M>0 and T≥0 such that
(8)∥z(t)∥≤Mμ(t),t≥T,
then neural network (1) is said to be μ-stable.Definition 2.
If there exist scalarsM>0, ε>0, and T≥0 such that
(9)∥z(t)∥≤Meεt,t≥T,
then neural network (1) is said to be exponentially stable.Definition 33.
If there exist scalarsM>0, ε>0, and T≥0 such that
(10)∥z(t)∥≤Mtε,t≥T,
then neural network (1) is said to be power stable.Remark 44.
It is obvious that botheεt and tε approaches +∞, when t approaches +∞. Therefore, neural network (1) is μ-stable if it is exponentially stable or power stable. In other words, exponential stability and power stability are two special cases of μ-stability.Lemma 5 (see [22]).
Given a Hermitian matrixQ, then Q<0 is equivalent to
(11)(QRQI-QIQR)<0,
where QR=Re(Q) and QI=Im(Q).


## 3. Main Results

In this section, we are going to give several criteria for checking the globalμ-stability of the complex-valued neural networks (1).Theorem 66.
Assume that assumptions (H1) and (H3) hold. Neural network (1) is globally μ-stable, if there exist two positive definite Hermitian matrices P1=P1R+iP1I and P2=P2R+iP2I, two real positive diagonal matrices R1>0 and R2>0, two complex matrices Q1=Q1R+iQ1I, Q2=Q2R+iQ2I, a positive differential function μ(t) defined on [0,∞), and three constants α≥0, β>0, and T>0 such that
(12)μ˙(t)μ(t)≤α,μ(t-τ(t))μ(t)≥β,∀t∈[T,∞),
and the following LMI holds
(13)(ΩRΩI-ΩIΩR)<0,
where
(14)ΩR=(Ω11RΩ12R0Ω14RΩ15RΩ21RΩ22R0Ω24RΩ25R00Ω33R00Ω41RΩ42R0-R10Ω51RΩ52R00-R2),ΩI=(Ω11IΩ12I0Ω14IΩ15IΩ21IΩ22I0Ω24IΩ25I00Ω33I00Ω41IΩ42I000Ω51IΩ52I000),
in which
(15)Ω11R=2αP1R+P2R+CQ1R+(Q1R)TC+R1Γ,Ω12R=P1R+(Q1R)T+CQ2R,Ω14R=-(Q1R)TDR-(Q1I)TDI,Ω15R=-(Q1R)TER-(Q1I)TEI,Ω21R=P1R+Q1R+(Q2R)TC,Ω22R=Q2R+(Q2R)T,Ω24R=-(Q2R)TDR-(Q2I)TDI,Ω25R=-(Q2R)TER-(Q2I)TEI,Ω33R=-β2(1-ρ)P2R+R2Γ,Ω41R=-(DR)TQ1R-(DI)TQ1I,Ω42R=-(DR)TQ2R-(DI)TQ2I,Ω51R=-(ER)TQ1R-(EI)TQ1I,Ω52R=-(ER)TQ2R-(EI)TQ2I,Ω11I=2αP1I+P2I+CQ1I-(Q1I)TC,Ω12I=P1I-(Q1I)T+CQ2I,Ω14I=(Q1I)TDR-(Q1R)TDI,Ω15I=(Q1I)TER-(Q1R)TEI,Ω21I=P1I+Q1I-(Q2I)TC,Ω22I=Q2I-(Q2I)T,Ω24I=(Q2I)TDR-(Q2R)TDI,Ω25I=(Q2I)TER-(Q2R)TEI,Ω33I=-β2(1-ρ)P2I,Ω41I=(DI)TQ1R-(DR)TQ1I,Ω42I=(DI)TQ2R-(DR)TQ2I,Ω51I=(EI)TQ1R-(ER)TQ1I,Ω52I=(EI)TQ2R-(ER)TQ2I.Proof.
Consider the following Lyapunov-Krasovskii functional candidates:(16)V(t)=μ2(t)z*(t)P1z(t)+∫t-τ(t)t‍μ2(s)z*(s)P2z(s)ds.
Calculate the derivative of V(t) along the trajectories of neural network (7). We obtain
(17)V˙(t)=2μ˙(t)μ(t)z*(t)P1z(t)+μ2(t)z˙*(t)P1z(t)+μ2(t)z*(t)P1z˙(t)+μ2(t)z*(t)P2z(t)-μ2(t-τ(t))z*(t-τ(t))P2z(t-τ(t))×(1-τ˙(t))≤μ2(t)[2αz*(t)P1z(t)+z˙*(t)P1z(t)+z*(t)P1z˙(t)+z*(t)P2z(t)-β2(1-ρ)×z*(t-τ(t))P2z(t-τ(t))],t≥T.
In deriving inequality (17), we have made use of condition (12) and assumption (H3).
From assumption (H1), we get(18)|gjR(xj,yj)|≤ξjR1|xj|+ξjR2|yj|,|gjI(xj,yj)|≤ξjI1|xj|+ξjI2|yj|
for all j=1,2,…,n, where xj=Re(zj), yj=Im⁡(zj). Hence,
(19)|gj(zj(t))|2≤[(ξjR)2+(ξI)j2](|xj|+|yj|)2≤2[(ξjR)2+(ξI)j2]|zj(t)|22=γj|zj(t)|2,
where ξjR=max⁡{ξjR1,ξjR2}, ξjI=max⁡{ξjI1,ξjI2}, γj=2[(ξjR)2+(ξI)j2], j=1,2,…,n. Let R1=diag⁡(r1,r2,…,rn)>0. It follows from (19) that
(20)rjμ2(t)gj*(zj(t))gj(zj(t))-rjμ2(t)γjzj*(t)zj(t)≤0
for j=1,2,…,n. Thus
(21)μ2(t)g*(z(t))R1g(z(t))-μ2(t)z*(t)R1Γz(t)≤0.
Also, we can get that(22)μ2(t)g*(z(t-τ(t)))R2g(z(t-τ(t)))-μ2(t)z*(t-τ(t))R2Γz(t-τ(t))≤0.
From (7), we have that
(23)0=μ2(t)[z˙(t)+Cz(t)-Dg(z(t))-Eg(z(t-τ(t)))]*×[Q1z(t)+Q2z˙(t)]+μ2(t)[Q1z(t)+Q2z˙(t)]*×[z˙(t)+Cz(t)-Dg(z(t))-Eg(z(t-τ(t)))].
It follows from (17), (21), (22), and (23) that
(24)V˙(t)≤μ2(t)w*(t)Ωw(t),t≥T,
where w(t)=(z(t),z˙(t),z(t-τ(t)),g(z(t)),g(z(t-τ(t))))* and
(25)Ω=(Ω11Ω120-Q1*D-Q1*EΩ21Q2*+Q20-Q2*D-Q2*E00Ω3300-D*Q1-D*Q20-R10-E*Q1-E*Q200-R2)
with Ω11=2αP1+P2+CQ1+Q1*C+R1Γ,  Ω12=P1*+Q1*+CQ2, Ω21=P1+Q1+Q2*C,  Ω33=-β2(1-ρ)P2+R2Γ. From (14), we have that Ω=ΩR+iΩI. It follows from (13) and Lemma 5 that Ω<0. Thus, we get from (24) that
(26)V˙(t)≤0,t≥T,
which means V(t) is monotonically nonincreasing for t≥T. So we have
(27)V(t)≤V(T),t≥T.
From the definition of V(t) in (16), we obtain that
(28)μ2(t)λmin⁡(P1)∥z(t)∥2≤V(t)≤V0≤∞,t≥T,
where V0=max⁡0≤s≤TV(s). It implies that
(29)∥z(t)∥≤Mμ(t),t≥T,
where M=V0/λmin⁡(P1). The proof is completed.Theorem 77.
Assume that assumptions (H2) and (H3) hold. Neural network (1) is globally μ-stable, if there exist two positive definite Hermitian matrices P1=P1R+iP1I and P2=P2R+iP2I, two real positive diagonal matrices R1>0 and R2>0, two complex matrices Q1=Q1R+iQ1I, Q2=Q2R+iQ2I, a positive differential function μ(t) defined on [0,∞), and three constants α≥0, β>0, and T>0 such that conditions (12), (13), and (14) in Theorem 6 are satisfied, where Ω11R=2αP1R+P2R+CQ1R+(Q1R)TC+R1Π and Ω33R=-β2(1-ρ)P2R+R2Π.Proof.
From assumption (H2), we get(30)|gj(zj(t))|≤ηj|zj(t)|
for j=1,2,…,n. Let R1=diag⁡(r1,r2,…,rn)>0. It follows from (30) that
(31)rjμ2(t)gj*(zj(t))gj(zj(t))-rjμ2(t)ηj2zj*(t)zj(t)≤0
for j=1,2,…,n. Hence
(32)μ2(t)g*(z(t))R1g(z(t))-μ2(t)z*(t)R1Πz(t)≤0.
Also, we can get that
(33)μ2(t)g*(z(t-τ(t)))R2g(z(t-τ(t)))-μ2(t)z*(t-τ(t))R2Πz(t-τ(t))≤0.
Consider the Lyapunov-Krasovskii functional (16). From (17), (23), (32), and (33), we can get that
(34)V˙(t)≤μ2(t)w*(t)Ωw(t),t≥T,
where w(t)=(z(t),z˙(t),z(t-τ(t)),g(z(t)),g(z(t-τ(t))))* and
(35)Ω=(Ω11Ω120-Q1*D-Q1*EΩ21Q2*+Q20-Q2*D-Q2*E00Ω3300-D*Q1-D*Q20-R10-E*Q1-E*Q200-R2)
with Ω11=2αP1+P2+CQ1+Q1*C+R1Π,  Ω12=P1*+Q1*+CQ2, Ω21=P1+Q1+Q2*C,  Ω33=-β2(1-ρ)P2+R2Π. From the proof of Theorem 6, we know that neural network (1) is μ-stable. The proof is completed.Corollary 88.
Assume that assumptions (H1) and (H3) hold andτ(t)≤τ. Neural network (1) is globally exponentially stable, if there exist two positive definite Hermitian matrices P1=P1R+iP1I and P2=P2R+iP2I, two real positive diagonal matrices R1>0 and R2>0, two complex matrices Q1=Q1R+iQ1I, Q2=Q2R+iQ2I, and a constant ε>0 such that conditions (13) and (14) in Theorem 6 are satisfied, where Ω11R=2εP1R+P2R+CQ1R+(Q1R)TC+R1Γ,  Ω11I=2εP1I+P2I+CQ1I-(Q1I)TC,  Ω33R=-e-2ετ(1-ρ)P2R+R2Γ,  Ω33I=-e-2ετ(1-ρ)P2I.Proof.
Letμ(t)=eεt (t≥0); then μ˙(t)/μ(t)=ε, μ(t-τ(t))/μ(t)=e-ετ(t)≥e-ετ. Take α=ε, β=e-ετ, and T=0. It is obvious that condition (12) in Theorem 6 is satisfied. The proof is completed.Corollary 9.
Assume that assumptions (H1) and (H3) hold and(t)≤δt(0<δ<1). Neural network (1) is globally power stable, if there exist two positive definite Hermitian matrices P1=P1R+iP1I and P2=P2R+iP2I, two real positive diagonal matrices R1>0 and R2>0, two complex matrices Q1=Q1R+iQ1I, Q2=Q2R+iQ2I, and two constants ε>0 and T>0 such that conditions (13) and (14) in Theorem 6 are satisfied, where Ω11R=(2ε/T)P1R+P2R+CQ1R+(Q1R)TC+R1Γ,  Ω11I=(2ε/T)P1I+P2I+CQ1I-(Q1I)TC, Ω33R=-(1-δ)2ε(1-ρ)P2R+R2Γ, Ω33I=-(1-δ)2ε(1-ρ)P2I.Proof.
Letμ(t)=tε (t≥T); then μ˙(t)/μ(t)=ε/t≤ε/T, μ(t-τ(t))/μ(t)=(1-τ(t)/t)ε≥(1-δ)ε. Take α=ε/T, β=(1-δ)ε. It is obvious that condition (12) in Theorem 6 is satisfied. The proof is completed.It is similar to the proof of Corollaries8 and 9, and we have the following results.Corollary 1010.
Assume that assumptions (H2) and (H3) hold andτ(t)≤τ. Neural network (1) is globally exponentially stable, if there exist two positive definite Hermitian matrices P1=P1R+iP1I and P2=P2R+iP2I, two real positive diagonal matrices R1>0 and R2>0, two complex matrices Q1=Q1R+iQ1I, Q2=Q2R+iQ2I, and a constant ε>0 such that conditions (13) and (14) in Theorem 7 are satisfied, where Ω11R=2εP1R+P2R+CQ1R+(Q1R)TC+R1Π, Ω11I=2εP1I+P2I+CQ1I-(Q1I)TC, Ω33R=-e-2ετ(1-ρ)P2R+R2Π, Ω33I=-e-2ετ(1-ρ)P2I.Corollary 1111.
Assume that assumptions (H2) and (H3) hold andτ(t)≤δt (0<δ<1). Neural network (1) is globally power stable, if there exist two positive definite Hermitian matrices P1=P1R+iP1I and P2=P2R+iP2I, two real positive diagonal matrices R1>0 and R2>0, two complex matrices Q1=Q1R+iQ1I, Q2=Q2R+iQ2I, and two constants ε>0 and T>0 such that conditions (13) and (14) in Theorem 7 are satisfied, where Ω11R=(2ε/T)P1R+P2R+CQ1R+(Q1R)TC+R1Π, Ω11I=(2ε/T)P1I+P2I+CQ1I-(Q1I)TC,  Ω33R=-(1-δ)2ε(1-ρ)P2R+R2Π, Ω33I=-(1-δ)2ε(1-ρ)P2I.Remark 12.
In [18, 23], the authors studied the global stability of complex-valued neural networks with time-varying delays, and the activation function was supposed to be
(36)fj(z)=fR(x,y)+ifI(x,y),
where z=x+iy and fjR(·,·), fjI(·,·) are real-valued for all j=1,2,…,n, and the partial derivatives of fjR(·,·), fjI(·,·) with respect to x and y were required to exist and be continuous. In this paper, both the real parts and the imaginary parts of the activation functions are no longer assumed to be derivable.Remark 1313.
In [21], the authors investigated boundedness and stability for discrete-time complex-valued neural networks with constant delay, and the activation function was supposed to be
(37)fj(z)=max⁡{0,Re(z)}+imax⁡{0,Im⁡(z)}.
Also, the authors gave a complex-valued LMI criterion for checking stability of the considered neural networks. But a feasible way to solve the given complex-valued LMI is not provided. In this paper, the criteria for checking stability of the complex-valued neural networks are established in the real LMI, which can be checked numerically using the effective LMI toolbox in MATLAB.


## 4. Examples

Example 1.
Consider a two-neuron complex-valued neural network with constant delay(38)u˙(t)=-Cu(t)+Df(u(t))+Ef(u(t-τ))+H,
where
(39)C=(1001),D=(1+i-2+i1-i-1),E=(1+ii-1+i2),H=(2-i-2+i),τ=2,fjR(uj)=132(|Re(uj)+1|+|Re(uj)-1|),j=1,2,fjI(uj)=132(|Im⁡(uj)+1|+|Im⁡(uj)-1|),j=1,2.
It can be verified that the activation functionsfj satisfy assumption (H1), the time-varying delay τ(t) satisfies assumption (H3), and DR=(1-21-1), DI=(11-10), ER=(10-12), EI=(1110), Γ=(1/64001/64), ρ=0. There exists a constant ε=0.2, and by employing MATLAB LMI toolbox, we can find the solutions to the LMI under conditions (13) and (14) in Corollary 8 as follows:
(40)P1=(41.2118-6.8812-4.2410i-6.8812+4.2410i40.6762),P2=(26.0370-4.1385-0.7600i-4.1385+0.7600i25.9447),R1=(785.931900857.0773),R2=(560.516000617.1000),Q1=(-61.6337+1.1580i9.2494+0.6392i9.1464+0.1434i-61.4512-1.3717i),Q2=(-63.1096-0.5628i10.2662+2.7895i10.1047-3.3169i-62.0592+0.6291i).
Then all conditions in Corollary 8 are satisfied. Therefore the neural network (38) is globally exponentially stable. Figures 1 and 2 depict the real and imaginary parts of states of the considered neural network (38), where initial condition is z1(t)=0.5,z2(t)=-1-0.5i.Figure 1
State trajectories of neural network (38).Figure 2
State trajectories of neural network (38).Example 2.
Consider a two-neuron complex-valued neural network with unbounded time-varying delays(41)u˙(t)=-Cu(t)+Df(u(t))+Ef(u(t-τ(t)))+H,
where
(42)C=(1001),D=(1.5+3i-1+2i-2-2i2.5+0.5i),E=(0.5+ii-2+0.5i1+0.5i),H=(2+i-2+i),τ(t)=0.2t,f1(u1)=120(|u1+1|-|u1-1|),f2(u2)=120(|u2+1|-|u2-1|).
It can be verified that the activation functionsfj satisfy Condition (H2), the time-varying delay τ(t) satisfies assumption (H3), and DR=(1.5-1-22.5), DI=(32-20.5), ER=(0.50-21), EI=(110.50.5), Π=(0.01000.01), ρ=0.2. There exist two constants ε=1 and T=10, and by employing MATLAB LMI toolbox, we can find the solutions to the LMI under conditions (13) and (14) in Corollary 11 as follows:
(43)P1=(39.852414.7299-3.2336i14.7299+3.2336i31.5075),P2=(23.94719.7114-1.3426i9.7114+1.3426i18.8527),R1=103×(1.3534001.0135),R2=(584.627800429.3048),Q1=(-51.8770-1.5115i-16.7003+0.6185i-16.8517-0.9537i-40.7204+1.0687i),Q2=(-60.4708+0.2332i-21.2500+3.6232i-21.6247-3.5095i-48.2459-0.1731i).
Then all conditions in Corollary 11 are satisfied. Thus the neural network (41) is globally power stable. Figures 3 and 4 depict the real and imaginary parts of states of the considered network (41), where initial condition is x1(t)=0.5sin(0.5t)i, x2(t)=-0.5+1.5cos⁡(0.3t)i.Figure 3
State trajectories of neural network (41).Figure 4
State trajectories of neural network (41).


## 5. Conclusion

In this paper, the globalμ-stability of complex-valued neural networks with unbounded time delays has been investigated by employing a combination of Lyapunov-Krasovskii functionals and the free weighting matrix method. Several sufficient conditions for checking the global μ-stability of the considered complex-valued neural networks have been given in linear matrix inequality (LMI), which can be checked numerically using the effective LMI toolbox in MATLAB. As direct applications, we can get the exponential stability and power stability, with respect to different time-varying delays. Two examples have been provided to demonstrate the effectiveness and less conservatism of the proposed criteria.Remark 14.
Note that the activation functions in this paper are continuous. Recently, neural networks with discontinuous activations have received more and more attention [31–34]. Based on the results of this paper, we will consider μ-stability of complex-valued neural networks with discontinuous activations in future.



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*Source: 263847-2014-04-03.xml*