# Globalμ-Stability of Impulsive Complex-Valued Neural Networks with Leakage Delay and Mixed Delays

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

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## Abstract

The impulsive complex-valued neural networks with three kinds of time delays including leakage delay, discrete delay, and distributed delay are considered. Based on the homeomorphism mapping principle of complex domain, a sufficient condition for the existence and uniqueness of the equilibrium point of the addressed complex-valued neural networks is proposed in terms of linear matrix inequality (LMI). By constructing appropriate Lyapunov-Krasovskii functionals, and employing the free weighting matrix method, several delay-dependent criteria for checking the globalμ-stability of the complex-valued neural networks are established in LMIs. As direct applications of these results, several criteria on the exponential stability, power-stability, and log-stability are obtained. Two examples with simulations are provided to demonstrate the effectiveness of the proposed criteria.


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## Body

## 1. Introduction

The nonlinear systems are ubiquitous in the real world [1–5]. As one of the most important nonlinear systems, the real-valued neural networks have been extensively studied and developed due to their extensive applications in pattern recognition, associative memory, signal processing, image processing, combinatorial optimization, and other areas [6]. In implementation of neural networks, however, time delays are unavoidably encountered [7]. It has been found that the existence of time delays may lead to instability and oscillation in a neural network [8]. Therefore, dynamics analysis of neural networks with time delays has received much attention. In [6–10], the exponential stability and asymptotic stability of delayed neural networks were investigated; some sufficient conditions for checking stability were given. In [11, 12], authors investigated the synchronization of chaotic neural networks with delay and obtained several criteria for checking the synchronization. In [13], the passivity of uncertain neural networks with both leakage delay and time-varying delay was considered. In [14], authors investigated the state estimation for neural networks with leakage delay and time-varying delays.However, besides delay effect, impulsive effects are also likely to exist in neural networks [15]. For instance, in implementation of electronic networks, the state of the networks is subject to instantaneous perturbations and experiences abrupt change at certain instants, which may be caused by switching phenomenon, frequency change. or other sudden noise; that is, it exhibits impulsive effects [16–18]. Therefore, it is necessary to consider both impulsive effect and delay effect on dynamical behaviors of neural networks. Some results on impulsive effect have been gained for delayed neural networks; for example, see [15, 16, 19] and the references therein.Recently, the power-rate global stability of the equilibrium is proposed in [20]. Moreover, in [21], the authors proposed a new concept of global μ-stability to unify the exponential stability, power-rate stability, and log-stability of neural networks. In [22], the authors investigated the global robust μ-stability in the mean square for a class of stochastic neural networks. In [23], the delayed neural systems with impulsion were considered, and the μ-stability criteria were derived by using Lyapunov-Krasovskii functional method. In [24], the multiple μ-stability of delayed neural networks was investigated, and several criteria for the coexistence of equilibrium points and their local μ-stability were derived.As an extension of real-valued neural networks, complex-valued neural networks with complex-valued state, output, connection weight, and activation function become strongly desired because of their practical applications in physical systems dealing with electromagnetic, light, ultrasonic, and quantum waves [25, 26]. It has been shown that such applications strongly depend on the stability of CVNNs [27]. Therefore, stability analysis of CVNNs has received much attention and various stability conditions have been obtained [28–33]. In [28], authors considered a discrete-time CVNNs and obtained several sufficient conditions for checking global exponential stability of a unique equilibrium. In [29, 30], the discrete-time CVNNs with linear threshold neurons were investigated, and some conditions for the boundedness, global attractivity, and complete stability as well as global exponential stability of the considered neural networks were also derived. In [31], the continuous-time CVNNs with delays were considered, and the boundedness, complete stability, and exponential stability were investigated. In [32, 33], the global stability was investigated for CVNNs on time scales, which is useful to unify the continuous-time and discrete-time CVNNs under the same framework. In [34], the authors investigated the μ-stability of delayed CVNNs and obtained several sufficient conditions to ensure the global μ-stability. To the best of our knowledge, there are no results on μ-stability of impulsive CVNNs with the three kinds of time delays including leakage delay, discrete delay, and distributed delay in the literature, and it remains as an open topic for further investigation.Motivated by the above discussions, in this paper, we will deal with the problem ofμ-stability for CVNNs with leakage delay, discrete delay, and distributed delay under impulsive perturbations. Based on the homeomorphism mapping principle of complex domain, a LMI condition for the existence and uniqueness of the equilibrium point of the addressed CVNNs is proposed. Several delay-dependent criteria for checking the global μ-stability of the CVNNs are obtained by constructing appropriate Lyapunov-Krasovskii functionals and employing the free weighting matrix method. The obtained results can also be applied to several special cases and we can get the exponential stability, power-stability, and log-stability of the CVNNs, correspondingly. Finally, two illustrative examples are provided to show the effectiveness of the proposed criteria.Notations. The notations are quite standard. Throughout this paper, let Z+ denote the set of positive integers. 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 and matrix conjugate transpose, respectively. For complex vector z∈Cn, let |z|=(|z1|,|z2|,…,|zn|)T be the module of the vector z, and let ∥z∥=∑k=1n‍|zk|2 be the norm of the vector z. I denotes the identity matrix with appropriate dimensions. 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. For any t≥0, zt is defined by zt=z(t+s), zt-=z(t-+s), s∈[-δ,0]. In addition, the notation ⋆ always denotes the conjugate transpose of block in a Hermitian matrix.


## 2. Problems Formulation and Preliminaries

Consider the following complex-valued neural networks with leakage delay and mixed delays under impulsive perturbations by a nonlinear differential equation of the form(1)z˙(t)=-Dz(t-δ)+Af(z(t))+Bf(z(t-τ1))+C∫t-τ2t‍f(z(s))ds+h,t≠tk,t>0,Δz(tk)=z(tk)-z(tk-)=Jk(z(tk-),ztk-),k∈Z+,
where the impulse times tk satisfy 0=t0<t1<⋯<tk<⋯ and lim⁡k→∞tk=+∞; z(t) = (z1(t),z2(t),…,zn(t))T∈Cn is the state vector of the neural networks; D=diag⁡{d1,d2,…,dn}∈Rn×n is the self-feedback connection weight matrix with dj>0 (j=1,2,…,n); A,B, and C∈Cn×n are, respectively, the connection weight matrix, the discretely delayed connection weight matrix, and distributively delayed connection weight matrix; f(z(t)) = (f1(z1(t)),f2(z2(t)),…,fn(zn(t)))T∈Cn represents the neuron activation function; h=(h1,h2,…,hn)T∈Cn is the external input vector; δ≥0, τ1≥0, and τ2≥0 are the leakage time delay, the discrete time delay, and the distributed time delay, respectively; Jk is the impulsive function.In the analysis of complex-valued neural networks, it is usually assumed that the activation functions are differentiable [31]. However, in this paper, we adopt the following assumption on the activation functions in which the differentiability is not be required:(H1)
the neuron activation functionsfj are continuous and satisfy
(2)∥fj(z1)-fj(z2)∥≤γj∥z1-z2∥, for any z1, z2∈C, j=1,2,…,n, where γj is a constant. Moreover, we define Γ=diag⁡{γ12,γ22,…,γn2}.Remark 1.
Note that the assumptions on activation functions are weaker than those generally used in the literature. Namely, the boundedness and differentiability of the activation functionsfj are not required in this paper.Next we introduce some definitions and lemmas to be used in the stability analysis.Definition 2.
Letz^ be an equilibrium point of system (1). Suppose that z(t) is an arbitrary solution of system (1); μ(t) is a positive continuous function and satisfies μ(t)→∞ as t→∞. If there is a positive constant M such that
(3)∥z(t)-z^∥≤Mμ(t),t≥0,
then the equilibrium point z^ is said to be μ-stable.In particular, if takingμ(t) in Definition 2 to exponential function, power function, and logarithmic function, we can get the definitions of exponential stability, power-stability, and log-stability, correspondingly.Definition 3.
Letz^ be an equilibrium point of system (1). Suppose that z(t) is an arbitrary solution of system (1). If there are two positive constants ɛ and M such that
(4)∥z(t)-z^∥≤Meɛt,t≥0,
then the equilibrium point z^ is said to be exponentially stable.Definition 4.
Letz^ be an equilibrium point of system (1). Suppose that z(t) is an arbitrary solution of system (1). If there are two positive constants ɛ and M such that
(5)∥z(t)-z^∥≤Mtɛ,t≥0,
then the equilibrium point z^ is said to be power-stable.Definition 5.
Letz^ be an equilibrium point of system (1). Suppose that z(t) is an arbitrary solution of system (1). If there are two positive constants ɛ and M such that
(6)∥z(t)-z^∥≤Mln⁡(ɛt+1),t>0,
then the equilibrium point z^ is said to be log-stable.Lemma 6.
IfH(z):Cn→Cn is a continuous map and satisfies the following conditions: (i)
H(z) is injective on Cn,(ii)
lim⁡∥z∥→∞∥H(z)∥=∞, then H(z) is a homeomorphism of Cn onto itself.Proof.
Letz=x+iy and α=(xT,yT)T, where x,y∈Rn. Define a homeomorphism I:Cn→R2n by
(7)I(z)=α.
Obviously, I is invertible. Let L=I∘H∘I-1. Then L is injective on R2n, since H and I are injective. In addition, ∥L(α)∥→∞ when ∥α∥→∞, since lim⁡∥z∥→∞∥I(z)∥=∞ and lim⁡∥z∥→∞∥H(z)∥=∞. Therefore, L is a homeomorphism of R2n onto itself. Then H=I-1∘L∘I is a homeomorphism of Cn onto itself. The proof is completed.Lemma 7.
For anya,b∈Cn, if P∈Cn×n is a positive definite Hermitian matrix, then a*b+b*a≤a*Pa+b*P-1b.Proof.
SinceP is a positive definite Hermitian matrix, there exists an invertible matrix Q∈Cn×n, such that P=Q*Q. For any a,b∈Cn, it follows from Cauchy inequality that
(8)a*b+b*a=(Qa)*[(Q*)-1b]+[(Q*)-1b]*(Qa)=2Re{(Qa)*[(Q*)-1b]}≤2∥(Qa)*[(Q*)-1b]∥≤2∥Qa∥∥(Q*)-1b∥≤(Qa)*(Qa)+[(Q*)-1b]*[(Q*)-1b]=a*Pa+b*P-1b.
The proof is completed.Lemma 8 (see [33]).
For any constant matrixW∈Cn×n and W>0, a scalar function ω:[a,b]→Cn with scalars a<b such that the integration concerned are well defined, then
(9)(b-a)∫ab‍ω*(s)Wω(s)ds≤(∫ab‍ω(s)ds)*W(∫ab‍ω(s)ds).Lemma 9.
A given Hermitian matrix(10)S=(S11S12S21S22)<0,
where S11*=S11, S12*=S21 and S22*=S22, is equivalent to any one of the following conditions: (1)
S22<0 and S11-S12S22-1S21<0,(2)
S11<0 and S22-S21S11-1S12<0.Proof.
(
1
) Note that
(11)S=(S11S12S21S22)=(IS12S22-10I)(S11-S12S22-1S2100S22)(IS12S22-10I)*.
Therefore, S<0⇔S22<0 and S11-S12S22-1S21<0.
(
2
) Note that
(12)S=(S11S12S21S22)=(I0S21S11-1I)(S1100S22-S21S11-1S12)(I0S21S11-1I)*.
Therefore, S<0⇔S11<0 and S22-S21S11-1S12<0. The proof is completed.


## 3. Existence and Uniqueness of Equilibrium Point

Now we study the existence and uniqueness of the equilibrium point of system (1). As usual, we denote an equilibrium point of the system (1) by the constant complex vector z^∈Cn, where z^ satisfies
(13)-Dz^+Af(z^)+Bf(z^)+τ2Cf(z^)+h=0,Jk(z^,z^)=0,k∈Z+.
In this paper, it is assumed that the constant complex vector z^ satisfies Jk(z^,z^)=0 for k∈Z+, if z^ satisfies -Dz^+Af(z^)+Bf(z^)+τ2Cf(z^)+h=0. Hence, to prove the existence and uniqueness of a solution of (13), it suffices to show that the following map H:Cn→Cn has a unique zero point:
(14)H(z)=-Dz+(A+B+τ2C)f(z)+h.
In the following, we will give some conditions for checking that H is a homeomorphism on Cn, that is, for assuring the existence and uniqueness of the equilibrium point of system (1).Theorem 10.
Under condition (H1), the system (1) has a unique equilibrium point, if there exist two complex matrices U1,U2 and a real positive diagonal matrix R, such that the following LMI holds:
(15)(Π11Π12⋆Π22)<0,
where Π11=Γ*RΓ-U1D-DU1*, Π12=U1(A+B+τ2C)-DU2*, Π22=U2(A+B+τ2C) + (A*+B*+τ2C*)U2*-R.Proof.
In the following, we will prove thatH(z) is a homeomorphism of Cn onto itself.
First, we prove thatH(z) is an injective map on Cn. Suppose that there exist z,z′∈Cn with z≠z′, such that H(z)=H(z′). Then
(16)-D(z-z′)+(A+B+τ2C)(f(z)-f(z′))=0.
Multiply both sides above by (z-z′)*U1+(f(z)-f(z′))*U2; we get
(17)0=-(z-z′)*U1D(z-z′)+(z-z′)*U1(A+B+τ2C)(f(z)-f(z′))-(f(z)-f(z′))*U2D(z-z′)+(f(z)-f(z′))*U2(A+B+τ2C)(f(z)-f(z′)).
Taking the conjugate transpose of (17), we get that
(18)0=-(z-z′)*DU1*(z-z′)+(f(z)-f(z′))*(A*+B*+τ2C*)U1*(z-z′)-(z-z′)*DU2*(f(z)-f(z′))+(f(z)-f(z′))*×(A*+B*+τ2C*)U2*(f(z)-f(z′)).
Let Σ1=R-U2(A+B+τ2C)-(A*+B*+τ2C*)U2*, Σ2=-U1D-DU1* + [U1(A+B+τ2C) − DU2*]Σ1-1[(A*+B*+τ2C*)U1*-U2D]. Then Σ1 is a positive definite Hermitian matrix from Lemma 9 and LMI (15). Summing (17) and (18), applying Lemma 7, we have
(19)0=-(z-z′)*(U1D+DU1*)(z-z′)+(z-z′)*[U1(A+B+τ2C)-DU2*](f(z)-f(z′))+(f(z)-f(z′))*×[(A*+B*+τ2C*)U1*-U2D](z-z′)+(f(z)-f(z′))*×[U2(A+B+τ2C)+(A*+B*+τ2C*)U2*]×(f(z)-f(z′))≤-(z-z′)*(U1D+DU1*)(z-z′)+(z-z′)*[U1(A+B+τ2C)-DU2*]×Σ1-1[(A*+B*+τ2C*)U1*-U2D](z-z′)+(f(z)-f(z′))*Σ1(f(z)-f(z′))+(f(z)-f(z′))*×[U2(A+B+τ2C)+(A*+B*+τ2C*)U2*]×(f(z)-f(z′))=(z-z′)*Σ2(z-z′)+(f(z)-f(z′))*R(f(z)-f(z′)).
Since R is a positive diagonal matrix, from condition (H1), we can get
(20)(f(z)-f(z′))*R(f(z)-f(z′))≤(z-z′)*Γ*RΓ(z-z′).
It follows from (19) and (20) that
(21)0≤(z-z′)*Σ3(z-z′),
where Σ3=Γ*RΓ+Σ2=Γ*RΓ-U1D-DU1*+[U1(A+B+τ2C)-DU2*]Σ1-1[(A*+B*+τ2C*)U1*-U2D]. From Lemma 9 and LMI (15), we can know Σ3<0. Then z-z′=0 from (21). Therefore, H(z) is an injective map on Cn.
Second, we prove∥H(z)∥→∞ as ∥z∥→∞. Let H~(z)=H(z)-H(0). By Lemma 7, we have
(22)[z*U1+(f(z)-f(0))*U2]H~(z)+H~(z)*[z*U1+(f(z)-f(0))*U2]*=-z*(U1D+DU1*)z+z*[U1(A+B+τ2C)-DU2*](f(z)-f(0))+(f(z)-f(0))*[(A*+B*+τ2C*)U1*-U2D]z+(f(z)-f(0))*×[U2(A+B+τ2C)+(A*+B*+τ2C*)U2*]×(f(z)-f(0))≤-z*(U1D+DU1*)z+z*[U1(A+B+τ2C)-DU2*]×Σ1-1[(A*+B*+τ2C*)U1*-U2D]z+(f(z)-f(0))*Σ1(f(z)-f(0))+(f(z)-f(0))*×[U2(A+B+τ2C)+(A*+B*+τ2C*)U2*]×(f(z)-f(0))=z*Σ2z+(f(z)-f(0))*R(f(z)-f(0)).
It follows from condition (H1) that
(23)(f(z)-f(0))*R(f(z)-f(0))≤z*Γ*RΓz.
Then from (22), (23), and Σ3=Γ*RΓ+Σ2<0, we obtain
(24)[z*U1+(f(z)-f(0))*U2]H~(z)+H~(z)*[z*U1+(f(z)-f(0))*U2]*≤z*Σ3z≤-ρmin⁡(-Σ3)∥z∥2,
which can imply that
(25)λmin⁡(-Σ3)∥z∥2≤2∥[z*U1+(f(z)-f(0))*U2]∥∥H~(z)∥≤2(∥U1∥+∥Γ*U2∥)∥z∥∥H~(z)∥.
When z≠0, we have
(26)∥H~(z)∥≥λmin⁡(-Σ3)∥z∥2(∥U1∥+∥Γ*U2∥).
Therefore, ∥H~(z)∥→∞ as ∥z∥→∞ which implies ∥H(z)∥→∞ as ∥z∥→∞. From Lemma 6, we know that H(z) is a homeomorphism of Cn. Thus, the system (1) has a unique equilibrium point.Remark 11.
It should be noted that Theorem10 is independent of leakage time delay and initial conditions. So, the time delays in the leakage terms do not affect the existence and uniqueness of the equilibrium point.


## 4. Globalμ-Stability Results

In the preceding section, we have shown the existence and uniqueness of the equilibrium point for system (1). In this section, we will further investigate the global μ-stability of the unique equilibrium point. For this purpose, the impulsive function Jk which is viewed as a perturbation of the equilibrium point z^ of system (1) without impulses is defined by
(27)Jk(z(tk-),ztk-)=Ek[z(tk-)-z^-D∫tk-δtk‍(z(s)-z^)ds],
where k∈Z+, Ek∈Cn×n. It is obvious that Jk(z^,z^)=0.Remark 12.
The type of impulse such as (27) describes the fact that the instantaneous perturbations are not only related to the state of neurons at impulse times tk but also related to the state of neurons in recent history, which reflects more realistic dynamics [19].Theorem 13.
Under the conditions of Theorem10, the equilibrium point of system (1) is globally μ-stable, if there exist six positive definite Hermitian matrices P1, P2, P3,P4, P5, and P6, two real positive diagonal matrices R1 and R2, three complex matrices Q1, Q2, and Q3, a positive differential function μ(t), and three constants α≥0, β>0, and T>0 such that
(28)bbbbbbbbbbbbbbbbbbbb0≤μ˙(t)μ(t)≤α,min⁡{μ(t-δ),μ(t-τ1),μ(t-τ2)}μ(t)≥β,bbbbbbbbbbbbbbbbbbbbb∀t∈[T,∞),
and the following LMIs hold:(29)(Ω11Ω12P1Dβ2τ1P5Ω15000⋆Ω22Q1*DQ3-P1D-Q1*A-Q1*B-Q1*C⋆⋆Ω33DQ3-DP1D-Q2*A-Q2*B-Q2*C⋆⋆⋆Ω440-Q3*A-Q3*B-Q3*C⋆⋆⋆⋆-β2δP3000⋆⋆⋆⋆⋆Ω6600⋆⋆⋆⋆⋆⋆-R20⋆⋆⋆⋆⋆⋆⋆-β2τ2P6)<0,(30)(P1(I+Ek)P1⋆P1)>0,
where Ω11=2αP1-DP1-P1D+2αP2+δP3+P4-(β2/τ1)P5+R1Γ, Ω12=P1+P2, Ω15=DP1D − 2αP1D, Ω22=Q1*+Q1+τ1P5, Ω33=Q2*D+DQ2, Ω44=-β2P4-(β2/τ1)P5+R2Γ, and Ω66=τ2P6-R1.Proof.
Under the condition of Theorem10, system (1) has a unique equilibrium point z^. Then we shift the equilibrium point of (1) to the origin by the translation z~(t)=z(t)-z^ and obtain
(31)z~˙(t)=-Dz~(t-δ)+Ag(z~(t))+Bg(z~(t-τ1))+C∫t-τ2t‍g(z~(s))ds,t≠tk,t>0,Δz~(tk)=z~(tk)-z~(tk-)=Ek[z~(tk-)-D∫tk-δtk‍z~(s)ds],k∈Z+.
Consider the following Lyapunov-Krasovskii functional candidate:
(32)V(t)=V1(t)+V2(t)+V3(t)+V4(t),
where
(33)V1(t)=μ2(t)[z~(t)-D∫t-δt‍z~(s)ds]*×P1[z~(t)-D∫t-δt‍z~(s)ds],V2(t)=μ2(t)z~*(t)P2z~(t)+∫0δ‍∫t-ut‍μ2(s)z~*(s)P3z~(s)dsdu,V3(t)=∫t-τ1t‍μ2(s)z~*(s)P4z~(s)ds+∫0τ1‍∫t-ut‍μ2(s)z~˙*(s)P5z~˙(s)dsdu,V4(t)=∫0τ2‍∫t-ut‍μ2(s)g*(z~(s))P6g(z~(s))dsdu.
Calculating the upper right derivative of V along the solution of (31), applying Lemma 8, we get
(34)D+V1(t)=2μ(t)μ˙(t)[z~(t)-D∫t-δt‍z~(s)ds]*×P1[z~(t)-D∫t-δt‍z~(s)ds]+μ2(t)[z~˙(t)-Dz~(t)+Dz~(t-δ)]*×P1[z~(t)-D∫t-δt‍z~(s)ds]+μ2(t)[z~(t)-D∫t-δt‍z~(s)ds]*×P1[z~˙(t)-Dz~(t)+Dz~(t-δ)]≤μ2(t)[2αz~*(t)P1z~(t)-2αz~*(t)P1D∫t-δt‍z~(s)ds-2α∫t-δt‍z~*(s)dsDP1z~(t)ds+2α∫t-δt‍z~*(s)dsDP1D∫t-δt‍z~(s)ds+z~˙*(t)P1z~(t)-z~˙*(t)P1D∫t-δt‍z~(s)ds-z~*(t)DP1z~(t)+z~*(t)DP1D∫t-δt‍z~(s)ds+z~*(t-δ)DP1z~(t)-z~*(t-δ)DP1D∫t-δt‍z~(s)ds+z~*(t)P1z~˙(t)-∫t-δt‍z~*(s)dsDP1z~˙(t)-z~*(t)P1Dz~(t)+∫t-δt‍z~*(s)dsDP1Dz~(t)+z~*(t)P1Dz~(t-δ)-∫t-δt‍z~*(s)dsDP1Dz~(t-δ)]=μ2(t)[∫t-δt‍z~*(t)(2αP1-DP1-P1D)z~(t)+z~*(t)(DP1D-2αP1D)∫t-δt‍z~(s)ds+∫t-δt‍z~*(s)ds(DP1D-2αDP1)z~(t)+2α∫t-δt‍z~*(s)dsDP1D∫t-δt‍z~(s)ds+z~˙*(t)P1z~(t)+z~*(t)P1z~˙(t)-z~˙*(t)P1D∫t-δt‍z~(s)ds-∫t-δt‍z~*(s)dsDP1z~˙(t)+z~*(t-δ)DP1z~(t)+z~*(t)P1Dz~(t-δ)-z~*(t-δ)DP1D∫t-δt‍z~(s)ds-∫t-δt‍z~*(s)dsDP1Dz~(t-δ)],(35)D+V2(t)=2μ(t)μ˙(t)z~*(t)P2z~(t)+μ2(t)z~˙*(t)P2z~(t)+μ2(t)z~*(t)P2z~˙(t)+δμ2(t)z~*(t)P3z~(t)-∫0δ‍μ2(t-u)z~*(t-u)P3z~(t-u)du≤2αμ2(t)z~*(t)P2z~(t)+μ2(t)z~˙*(t)P2z~(t)+μ2(t)z~*(t)P2z~˙(t)+δμ2(t)z~*(t)P3z~(t)-∫t-δt‍μ2(s)z~*(s)P3z~(s)ds≤2αμ2(t)z~*(t)P2z~(t)+μ2(t)z~˙*(t)P2z~(t)+μ2(t)z~*(t)P2z~˙(t)+δμ2(t)z~*(t)P3z~(t)-β2μ2(t)∫t-δt‍z~*(s)P3z~(s)ds≤μ2(t)[∫t-δt‍z~*(t)(2αP2+δP3)z~(t)+z~*(t)P2z~˙(t)+z~˙*(t)P2z~(t)-β2δ∫t-δt‍z~*(s)dsP3∫t-δt‍z~(s)ds],(36)D+V3(t)=μ2(t)z~*(t)P4z~(t)-μ2(t-τ1)z~*(t-τ1)P4z~(t-τ1)+τ1μ2(t)z~˙*(t)P5z~˙(t)-∫0τ1‍μ2(t-u)z~˙*(t-u)P5z~˙(t-u)du=μ2(t)z~*(t)P4z~(t)-μ2(t-τ1)z~*(t-τ1)P4z~(t-τ1)+τ1μ2(t)z~˙*(t)P5z~˙(t)-∫t-τ1t‍μ2(s)z~˙*(s)P5z~˙(s)ds≤μ2(t)[β2τ1∫t-τ1t‍z~*(t)P4z~(t)-β2z~*(t-τ1)P4z~(t-τ1)+τ1z~˙*(t)P5z~˙(t)-β2τ1∫t-τ1t‍z~˙*(s)dsP5∫t-τ1t‍z~˙(s)ds]=μ2(t)[β2τ1z~*(t)P4z~(t)-β2z~*(t-τ1)P4z~(t-τ1)+τ1z~˙*(t)P5z~˙(t)-β2τ1(z~*(t)-z~*(t-τ1))×P5(z~(t)-z~(t-τ1))β2τ1],(37)D+V4(t)=τ2μ2(t)g*(z~(t))P6g(z~(t))-∫0τ2‍μ2(t-u)g*(z~(t-u))P6g(z~(t-u))du=τ2μ2(t)g*(z~(t))P6g(z~(t))-∫t-τ2t‍μ2(s)g*(z~(s))P6g(z~(s))ds≤τ2μ2(t)g*(z~(t))P6g(z~(t))-β2μ2(t)∫t-τ2t‍g*(z~(s))P6g(z~(s))ds≤μ2(t)[β2τ2∫t-τ2t‍τ2g*(z~(t))P6g(z~(t))-β2τ2∫t-τ2t‍g*(z~(s))dsP6∫t-τ2t‍g(z~(s))ds].
Combining (34), (35), (36), and (37), we can deduce that
(38)D+V(t)≤μ2(t)[β2δ∫t-δt‍z~*(t)(β2τ12αP1-DP1-P1D+2αP2+δP3+P4-β2τ1P5)z~(t)+z~*(t)(P1+P2)z~˙(t)+z~˙*(t)(P1+P2)z~(t)+z~*(t)P1Dz~(t-δ)+z~*(t-δ)DP1z~(t)+β2τ1z~*(t)P5z~(t-τ1)+β2τ1z~*(t-τ1)P5z~(t)+z~*(t)(DP1D-2αP1D)∫t-δt‍z~(s)ds+∫t-δt‍z~*(s)ds(DP1D-2αDP1)z~(t)-z~˙*(t)P1D∫t-δt‍z~(s)ds-∫t-δt‍z~*(s)dsDP1z~˙(t)-z~*(t-δ)DP1D∫t-δt‍z~(s)ds-∫t-δt‍z~*(s)dsDP1Dz~(t-δ)-β2δ∫t-δt‍z~*(s)dsP3∫t-δt‍z~(s)ds-z~*(t-τ1)(β2P4+β2τ1P5)z~(t-τ1)+τ2g*(z~(t))P6g(z~(t))-β2τ2∫t-τ2t‍g*(z~(s))dsP6∫t-τ2t‍g(z~(s))ds].
From assumption (H1), we have
(39)|gj(z~j(t))|≤γj|z~j(t)|
for j=1,2,…,n. Let R1=diag⁡(r1,r2,…,rn)>0. It follows from (39) that
(40)rjμ2(t)gj*(z~j(t))gj(z~j(t))-rjμ2(t)γj2z~j*(t)z~j(t)≤0
for j=1,2,…,n. Hence
(41)μ2(t)g*(z~(t))R1g(z~(t))-μ2(t)z~*(t)R1Γz~(t)≤0.
Also, we can get that
(42)μ2(t)g*(z~(t-τ1))R2g(z~(t-τ1))-μ2(t)z~*(t-τ1)R2Γz~(t-τ1)≤0.
From (31), we have that
(43)0=μ2(t)[Q1z~˙(t)+Q2z~(t-δ)+Q3z~(t-τ1)]*×[∫t-τ2t‍z~˙(t)+Dz~(t-δ)-Ag(z~(t))-Bg(z~(t-τ1))-C∫t-τ2t‍g(z~(s))ds]+μ2(t)[∫t-τ2t‍z~˙(t)+Dz~(t-δ)-Ag(z~(t))-Bg(z~(t-τ1))-C∫t-τ2t‍g(z~(s))ds]*×[Q1z~˙(t)+Q2z~(t-δ)+Q3z~(t-τ1)].
It follows from (38), (41), (42), and (43) that
(44)D+V(t)≤μ2(t)w*(t)Ωw(t),
where w(t)=(z~*(t),z~˙*(t),z~*(t-δ),z~(t-τ1),∫t-δt‍z~*(s)ds,g*(z~(t)),g*(z~(t-τ1)),∫t-τ2t‍g*(z~(s))ds)* and(45)Ω=(Ω11Ω12P1Dβ2τ1P5Ω15000⋆Ω22Q1*DQ3-P1D-Q1*A-Q1*B-Q1*C⋆⋆Ω33DQ3-DP1D-Q2*A-Q2*B-Q2*C⋆⋆⋆Ω440-Q3*A-Q3*B-Q3*C⋆⋆⋆⋆-β2δP3000⋆⋆⋆⋆⋆Ω6600⋆⋆⋆⋆⋆⋆-R20⋆⋆⋆⋆⋆⋆⋆-β2τ2P6)with Ω11=2αP1-DP1-P1D+2αP2+δP3+P4-(β2/τ1)P5+R1Γ, Ω12=P1+P2, Ω15=DP1D-2αP1D, Ω22=Q1*+Q1+τ1P5, Ω33=Q2*D+DQ2, Ω44=-β2P4-(β2/τ1)P5+R2Γ, Ω66=τ2P6-R1. Thus, we get from (29) and (44) that
(46)D+V(t)≤0,t∈[tk-1,tk)∩[T,∞),k∈Z+.
In addition, we note that
(47)(P1(I+Ek)P1⋆P1)>0⟺(I00P1-1)(P1(I+Ek)P1⋆P1)(I00P1-1)>0⟺(P1(I+Ek)⋆P1-1)>0⟺P1-(I+Ek)*P1(I+Ek)>0
in which the last equivalent relation is obtained by Lemma 9. Thus, it yields
(48)V1(tk)=μ2(tk)[z~(tk)-D∫tk-δtk‍z~(s)ds]*×P1[z~(tk)-D∫tk-δtk‍z~(s)ds]=μ2(tk-)[z~(tk-)-D∫tk-δtk‍z~(s)ds]*×(I+Ek)*P1(I+Ek)×[z~(tk)-D∫tk-δtk‍z~(s)ds]≤μ2(tk-)[z~(tk-)-D∫tk-δtk‍z~(s)ds]*×P1[z~(tk)-D∫tk-δtk‍z~(s)ds]=V1(tk-).
Hence, we can deduce that
(49)V(tk)≤V(tk-),k∈Z+.
From (46) and (49), we know that V is monotonically nonincreasing for t∈[T,∞), which implies that
(50)V(t)≤V(T),t≥T.
From the definition of V(t) in (32), we obtain that
(51)μ2(t)λmin⁡(P2)∥z~(t)∥2≤V(t)≤V0<∞,t≥0,
where V0=max⁡0≤s≤TV(s). It implies that
(52)∥z~(t)∥≤Mμ(t),t≥0,
where M=V0/λmin⁡(P2). The proof is completed.Corollary 14.
Under the conditions of Theorem10, the equilibrium point of system (1) is globally exponentially stable, if there exist six positive definite Hermitian matrices P1, P2, P3, P4, P5, and P6, two real positive diagonal matrices R1 and R2, three complex matrices Q1, Q2, and Q3, and a positive constant ɛ such that conditions (29) and (30) in Theorem 13 are satisfied, where α=ɛ, β=e-ɛmax⁡{δ,τ1,τ2}.Proof.
Letμ(t)=eɛt (t≥0); then μ˙(t)/μ(t)=ɛ, min⁡{μ(t-δ),μ(t-τ1),μ(t-τ2)}/μ(t) = min⁡{e-ɛδ,e-ɛτ1,e-ɛτ2}=e-ɛmax⁡{δ,τ1,τ2}. Take α=ɛ, β=e-ɛmax⁡{δ,τ1,τ2}, and T=0. Then it is obvious that condition (28) in Theorem 13 is satisfied. The proof is completed.Corollary 15.
Under the conditions of Theorem10, the equilibrium point of system (1) is globally power-stable, if there exist six positive definite Hermitian matrices P1, P2, P3, P4, P5, and P6, two real positive diagonal matrices R1 and R2, three complex matrices Q1, Q2, and Q3, and a positive constant ɛ such that conditions (29) and (30) in Theorem 13 are satisfied, where α=ɛ/2, β=2-ɛ.Proof.
Letμ(t)=tɛ. For all t≥2max⁡{1,δ,τ1,τ2}, we have μ˙(t)/μ(t)=ɛ/t≤ɛ/2, min⁡{μ(t-δ),μ(t-τ1),μ(t-τ2)}/μ(t) = (1-(max⁡{δ,τ1,τ2}/t))ɛ≥2-ɛ. Take α=ɛ/2, β=2-ɛ, and T=2max⁡{1,δ,τ1,τ2}. Then it is obvious that condition (28) in Theorem 13 is satisfied. The proof is completed.Corollary 16.
Under the conditions of Theorem10, the equilibrium point of system (1) is globally log-stable, if there exist six positive definite Hermitian matrices P1, P2, P3,P4, P5, and P6, two real positive diagonal matrices R1 and R2, three complex matrices Q1, Q2, and Q3, and a positive constant ɛ such that conditions (29) and (30) in Theorem 13 are satisfied, where α=ɛ/e, β=1/ln⁡(e+ɛmax⁡{δ,τ1,τ2}).Proof.
Letμ(t)=ln⁡(ɛt+1). For all t≥((e-1)/ɛ)+max⁡{δ,τ1,τ2}, we have μ˙(t)/μ(t) = ɛ/(ɛt+1)ln⁡(ɛt+1)≤ɛ/e, min⁡{μ(t-δ),μ(t-τ1),μ(t-τ2)}/μ(t) = (ln[ɛ(t-max{δ,τ_1,τ_2})+1])⁄(ln(ɛt+1)) ≥ 1/ln⁡(e+ɛmax⁡{δ,τ1,τ2}). Take α=ɛ/e, β=1/ln⁡(e+ɛmax⁡{δ,τ1,τ2}), and T=((e-1)/ɛ)+max⁡{δ,τ1,τ2}. Then it is obvious that condition (28) in Theorem 13 is satisfied. The proof is completed.Remark 17.
It is noted that LMIs (15), (29), and (30) are complex-valued, which cannot be directly handled via MATLAB LMI Toolbox. However, the authors in [33] give the result that a complex Hermitian matrix L satisfies L<0 if and only if
(53)(Re(L)Im⁡(L)-Im⁡(L)Re(L))<0.
Therefore, applying the result, the complex-valued LMIs (15), (29), and (30) can be turned into real-valued LMIs, which can be checked numerically using LMI toolbox in MATLAB.


## 5. Numerical Examples

The following two illustrative examples will demonstrate the effectiveness and superiority of our results.Example 1.
Consider the following two-neuron CVNNs with leakage, discrete, and distributed delays, described by(54)z˙(t)=-Dz(t-δ)+Af(z(t))+Bf(z(t-τ1))+C∫t-τ2t‍f(z(s))ds+h,t≠tk,t>0,Δz(tk)=z(tk)-z(tk-)=Ek[z(tk-)-D∫tk-δtk‍z(s)ds],k∈Z+,
where f1(u)=f2(u)=(1/20)(|u+1|-|u-1|), δ=0.1, τ1=0.2, τ2=0.2, h=(0,0)T, tk=0.4k, k∈Z+, and the parameter matrices D, A, B, C, and Ek are given as follows:
(55)D=(3003),A=(0.5+i-0.2-0.3i0.2+0.5i1-0.5i),B=(0.4+0.3i-0.1-i-1.2+i-0.4-0.3i),C=(0.1-0.5i0.6+0.4i0.2-0.5i0.2),Ek=(-0.3+0.2i00.1-0.6i-0.2+0.2i),k∈Z+.
It can be verified that the activation functions f1 and f2 satisfy condition (H1), and Γ=diag⁡{0.01,0.01}. Then LMI in Theorem 10 has the following feasible solution via the MATLAB LMI toolbox:
(56)U1=(19.0818+6.7535i-9.2220+2.2044i8.0728+0.4398i19.0186+9.3571i),U2=(4.7472-0.2821i-6.9954-6.3515i1.8858+0.8953i4.5109+1.3211i),R=(161.011700129.7777).
Also there exists a constant ɛ=0.1, and by employing MATLAB LMI Toolbox, we can find the solutions to the LMIs in Corollary 14 as follows:
(57)P1=(12.4935-0.6208-3.5072i-0.6208+3.5072i38.9645),P2=(16.5671-1.5742-1.2335i-1.5742+1.2335i27.6891),P3=(43.4484-1.3614-5.3324i-1.3614+5.3324i87.1860),P4=(12.8707-0.3721-2.5769i-0.3721+2.5769i30.8832),P5=(18.0542-0.3996+0.2524i-0.3996-0.2524i16.1543),P6=(49.8004-0.2065+3.8010i-0.2065-3.8010i48.9421),Q1=(-33.7168+0.0480i3.0570-2.0568i2.2459+1.1090i-39.3508+0.0752i),Q2=(-66.6664-1.5581i6.8518-15.0791i1.2207+10.1224i-73.9430+1.7006i),Q3=(-4.2664+0.1047i0.4653-0.9119i0.3881+1.2719i0.9145-0.0078i),R1=(260.795600276.7577),R2=(293.913300256.1810).
Hence, from Theorem 10 and Corollary 14, the unique equilibrium (0,0)T of system (54) is globally exponentially stable. Figure 1 depicts the real and imaginary parts of states of the considered system (54), where the initial condition is z1(t)=0.5-0.4i,z2(t)=-0.4+0.5i.State trajectories of system (54).
(a)
(b)Example 2.
Consider the following two-neuron CVNNs with leakage, discrete, and distributed delays, described by(58)z˙(t)=-Dz(t-δ)+Af(z(t))+Bf(z(t-τ1))+C∫t-τ2t‍f(z(s))ds+h,t≠tk,t>0,Δz(tk)=z(tk)-z(tk-)=Ek[z(tk-)-D∫tk-δtk‍z(s)ds-γ],k∈Z+,
where f1(u)=f2(u)=(1/4)[max⁡{0,Re(u)}+imax⁡{0,Im⁡(u)}], δ=0.2, τ1=0.6, τ2=1, h=(1+i,-1+i)T, γ=(0.0295+0.0623i,-0.0604+0.0397i)T, tk=0.8k, k∈Z+, and the parameter matrices D, A, B, C, and Ek are given as follows:
(59)D=(4004),A=(-1+3i-1+2i-2-2i1+0.5i),B=(0.8+ii-1+0.5i1+0.5i),C=(1+i1-2i-1+i0.5+i),Ek=(-10.5i0.5i-0.5),k∈Z+.
It can be verified that the activation functions f1 and f2 satisfy condition (H1), and Γ=diag⁡{1/16,1/16}. Then LMI in Theorem 10 has the following feasible solution via the MATLAB LMI toolbox:
(60)U1=(2.9099-5.3413i2.7078+7.2154i-2.5180+7.0990i2.6469+0.8628i),U2=(5.4524+4.8583i-11.7495+2.6580i-0.5273-7.2479i-0.8154-1.1665i),R=(79.40220037.6964).
Also there exists a constant ɛ=0.1, and by employing MATLAB LMI Toolbox, we can find the solutions to LMIs in Corollary 15 as follows:
(61)P1=(1.65370.0741+0.0987i0.0741-0.0987i2.5236),P2=(0.19840.0640-0.0118i0.0640+0.0118i0.2985),P3=(21.56590.7498+1.4712i0.7498-1.4712i32.7091),P4=(0.68520.0304+0.0669i0.0304-0.0669i1.1632),P5=(0.24190.0941-0.1415i0.0941+0.1415i0.5347),P6=(7.7723-0.2071-0.7161i-0.2071+0.7161i10.5317),Q1=(-0.7898+0.0341i-0.1269-0.5099i-0.1170+0.5513i-1.2707-0.0430i),Q2=(-2.5965+0.0654i-0.5192-2.1874i-0.5396+2.4027i-4.2140-0.0583i),Q3=(-0.0376+0.0030i-0.0029-0.0779i-0.0046+0.0854i-0.0596-0.0047i),R1=(19.98800032.1768),R2=(6.82680014.7991).
Hence, from Theorem 10 and Corollary 15, system (58) has a unique equilibrium, which is globally exponentially power-stable. Figure 2 depicts the real and imaginary parts of states of the considered system (58), where the initial condition is z1(t)=0.5-0.2i,z2(t)=-0.5+0.6i.State trajectories of system (58) with δ=0.2.
(a)
(b)Remark 18.
If we take leakage delayδ≥0.37 in system (58), one may check that LMIs in Corollary 15 do not have a feasible solution via MATLAB. In other words, our results cannot guarantee the stability of system (58) with δ≥0.37. From the simulations, it is easy to check that the unique equilibrium point of system (58) with δ=0.37 is not stable (see Figure 3); this implies that the effect of leakage delay on the dynamics of CVNNs cannot be ignored.State trajectories of system (58) with δ=0.37.
(a)
(b)


## 6. Conclusion

In this paper, theμ-stability of impulsive CVNNs with leakage delay, discrete delay, and distributed delay has been investigated. Several sufficient conditions to ensure the existence, uniqueness, and global μ-stability of the equilibrium point of the considered neural networks have been established in LMIs by applying the homeomorphism mapping principle of complex domain, constructing appropriate Lyapunov-Krasovskii functionals, and employing the free weighting matrix method. As direct applications of these results, several criteria on the exponential stability, power-stability and log-stability have been obtained. Two numerical examples are given to illustrate the effectiveness of the proposed theoretical results.



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*Source: 397532-2014-05-27.xml*