# Mean Square Exponential Stability of Stochastic Complex-Valued Neural Networks with Mixed Delays

Authors: Xiaohui Xu; Jibin Yang; Yanhai Xu
Journal: Complexity
 (2019)

---

## Abstract

This paper investigates the mean square exponential stability problem of a class of complex-valued neural networks with stochastic disturbance and mixed delays including both time-varying delays and continuously distributed delays. Under different assumption conditions concerning stochastic disturbance term from the existing ones, some sufficient conditions are derived for assuring the mean square exponential stability of the equilibrium point of the system based on the vector Lyapunov function method andIto^ differential-integral theorem. The obtained results not only generalize the existing ones, but also reduce the conservatism of the previous stability results about complex-valued neural networks with stochastic disturbances. Two numerical examples with simulation results are given to verify the feasibility of the proposed results.


---

## Body

## 1. Introduction

In recent years, the dynamical behavior analysis of various neural networks models defined in complex number domain has attracted more and more attention due to their extensive applications in many fields, such as filtering, speech synthesis, remote sensing, and signal processing, which cannot be solved comprehensively with only their counterparts defined in real number domain [1, 2].It is well known that delays arise because of the processing of information in both biological and man-made neural networks in view of a practical point [3, 4]. Besides, the neural networks usually have a spatial extent due to the presences of a multitude of parallel pathway with a variety of axon sizes and lengths. There will be a distribution of conduction velocities along these pathways and a distribution of propagation be designed with discrete delays. Therefore, the more appropriate way is to incorporate continuously distributed delays [3–7]. In [8–11], the authors have studied several kinds of complex-valued neural networks with continuously distributed delays. Some significant results were obtained for assuring the stability of the proposed systems in [8–11].In fact, most real models of neural networks are affected by many external and internal perturbations which are of great uncertainty, such as impulsive disturbances [5, 9–15], Markovian jumping parameters [16–19], and parameter uncertainties [20–22]. As Haykin [23] points out, in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. One approach to the mathematical incorporation of such effects is to use probabilistic threshold models. In current papers, neural networks model with external random perturbations is to view as nonlinear dynamical systems with white noise (perturbations of Brownian motion) [3, 24]. Therefore, it is of real significance to consider that stochastic effects to the stability of neural networks with continuously distributed delays.As far as we know, the existing stability results concerning various neural networks are mainly applied to judge the neural networks in real number domain, such as [3, 7, 12–14, 16, 17, 24–31]. Recently, there have been some literatures with regard to the complex-valued neural networks with stochastic disturbances [32–34]. In [32], the authors established a class of stochastic memristor-based complex-valued neural networks with time varying delays. In [33], the robust state estimation problem was investigated for a class of complex-valued neural networks involving parameter uncertainties, constant time delays, and stochastic disturbances by resorting to the sampled data information from the available output measurements. However, the infinitely distributed delays have not been considered in the model of complex-valued neural networks studied in [32, 33]. In [34], the passivity analysis was conducted for a class of stochastic memristor-based complex-valued recurrent neural networks with discrete time delays and continuously distributed delays by applying both the scalar Lyapunov function method and the LMI method. The stochastic weighted coefficients in the model [34] were assumed to be constants in real number domain, which was conservatism and needed to be generalized in the complex number domain further. It is well known that the synchronization problem of chaotic neural networks can be translated into the stability problem of the corresponding error system of driving system and driven system. In [35], the authors concerned the problem of finite-time synchronization for a class of complex-valued neural networks with stochastic disturbances and mixed delays. However, the stochastic terms in the driving and driven systems were only assumed to be bounded directly, which was a shortage of generality. Besides, the results in [35] are unable to judge the stability of the corresponding systems.Separating the model of complex-valued neural networks into its real and imaginary parts is a routine approach to study the dynamical behavior of the systems, such as [2, 8, 11, 15, 32–35]. The complex-valued activation functions were supposed to be with existence, continuity, and boundedness of the partial derivatives of the activation functions about the real and imaginary parts of the state variables [2, 8, 11, 15, 32]. As pointed in [33, 36–38], the assumption concerning activation functions in [2, 8, 11, 15, 32] were with additional restriction that the partial derivatives of its real parts and imaginary parts were required to satisfy the existence and continuity.To the best of authors’ knowledge, there is no research concerning the exponential stability in the mean square sense for the complex-valued neural networks with mixed delays (including both time-varying delays and continuously distributed delays) and stochastic disturbances. Therefore, in this paper we will establish some new conditions for assuring the stability of the mentioned systems. The main advantages and contributions can be listed as follows. (a) The stochastic neural networks with mixed delays in complex number domain are proposed. (b) Different assumption conditions concerning stochastic disturbance term from the existing ones are given in this paper, which is of less conservatism. (c) Both stochastic disturbances and interval parameter uncertainties are considered in the addressed systems. (d) Some sufficient conditions with simple matrix forms are obtained for ensuring the mean square exponential stability and robust exponential stability in the mean square sense of the systems, which are easy to be checked in practice.


## 2. Notations and Model Descriptions

In this paper, we consider a class of complex-valued neural networks as follows:(1)dzkt=-dkzkt+∑j=1nakjfjzjt+bkjfjzjt-τkjt+pkj∫-∞tθkjt-sfjzjsds+Jkdt+∑j=1nσkjzjt,zjt-τkjtdwjt,In (1), zk∈C represents thek-th neuron state, k=1,2,......,n, where C denotes a complex number set, and n denotes the neuron number. A=(akj)n×n∈Cn×n, B=(bkj)n×n∈Cn×n, and P=(pkj)n×n∈Cn×n represent the connection weighted matrices, respectively. J=(J1,J2,......,Jn)T∈Cn is the external input vector, where (.)T denotes the transpose of the vector. f(z(t))=(f1(z1(t)),f2(z2(t)),...…,fn(zn(t)))T∈Cn represents the activation function. D=diag(d1,d2,...…,dn)∈Rn×n with dk>0(k=1,2,......,n) denotes the neuron self-feedback coefficient matrix, where R denote a real number set. σkj(j,k=1,2,......,n) denotes the weighted function of stochastic disturbances and w(t)=[w1(t),w2(t),...…,wn(t)]T is the Brown motion defined on a complete probability space (Ω,F,{Ft}t≥0,Ρ) with a natural filtration {Ft}t≥0 generated by {w(s):0≤s≤t}.It is assumed thatzk(s)=φk(s) is the initial condition of system (1), where φk(s) is continuous function mapping from (-∞,0] to C, k=1,2,......,n.Letz#=(z1#,z2#,...…,zn#)T be the equilibrium point of (1), where zk#=xk#+iyk# (k=1,2,......,n) and i denotes the imaginary unit; i.e., i=-1.Letzk=Re(zk)+Im(zk)i=xk+iyk, where Re(.)and Im(.) represent the real part and imaginary part of a complex number, respectively. Furthermore, the activation functions fk(zk) are assumed to be expressed as (2) by separating them into their real parts and imaginary parts, (2)fkzk=fkRxk,yk+ifkIxk,yk,k=1,2,…,n,where fkR(xk,yk):R2→R and fkI(xk,yk):R2→R represent the real part and the imaginary part of fk(zk), respectively.Next, some assumptions concerning (1) are given to obtain the stability results.Assumption 1.
Suppose thatτkj(t) (k,j=1,2,......,n) is a bounded function with τ=max1≤k,j≤nsupt≥0τkj(t).Assumption 2.
Suppose thatθkj(.): [0,+∞)→[0,+∞) are piecewise continuous functions and satisfy (3)∫0+∞exp⁡βsθkjsds=μkjβ,k,j=1,2,…,n,where μkj(β) is continuous on [0,δ), and μkj(0)=1.Let|.| denote the module of a complex number.Assumption 3.
It is assumed thatfk(.) with the form of (2) satisfies the mean value theorem of multivariable functions. That is, for any given xk,xk′,yk,yk′∈R, there exist positive constants lkRR, lkRI, lkIR, and lkII such that(4)fkRxk,yk-fkRxk′,yk′≤lkRRxk-xk′+lkRIyk-yk′,(5)fkIxk,yk-fkIxk′,yk′≤lkIRxk-xk′+lkIIyk-yk′.LetLRR=diag(l1RR,l2RR,......,lnRR), LRI=diag(l1RI,l2RI,......,lnRI), LIR=diag(l1IR,l2IR,......,lnIR), LII=diag(l1II,l2II,......,lnII).Remark 4.
In [2, 8, 11, 15, 32], the activation functions were supposed to satisfy that the partial derivatives of their real parts and imaginary parts were required to be with the existence and continuity. In fact it is an additional restriction on the activation functions in complex number domain. In this paper the mentioned restriction is no longer required.Assumption 5.
Assume that the weighted functions of stochastic disturbancesσkj (k,j=1,2,......,n) can be separated into their real parts and imaginary parts with following forms:(6)σkjR=σkjRxjt,yjt,xjt-τkjt,yjt-τkjt:R4→R,σkjI=σkjIxjt,yjt,xjt-τkjt,yjt-τkjt:R4→R,where σkjR(0,0,0,0)=σkjI(0,0,0,0)=0. Suppose that there exist nonnegative constants ωkjRR, ωkjRI, ωkjIR, and ωkjII such that inequalities (5) and (7) hold:(7)σkjRxjt,yjt,xjt-τkjt,yjt-τkjt2-σkjRx′jt,y′jt,x′jt-τkjt,y′jt-τkjt2≤ωkjRR2xjt-x′jt2+xjt-τkjt-x′jt-τkjt2+ωkjRI2yjt-y′jt2+yjt-τkjt-y′jt-τkjt2,(8)σkjIxjt,yjt,xjt-τkjt,yjt-τkjt2-σkjIx′jt,y′jt,x′jt-τkjt,y′jt-τkjt2≤ωkjIR2xjt-x′jt2+xjt-τkjt-x′jt-τkjt2+ωkjII2yjt-y′jt2+yjt-τkjt-y′jt-τkjt2.LetWRR=ωkjRR2n×n, WRI=ωkjRI2n×n, WIR=ωkjIR2n×n, WII=ωkjII2n×n.Next, we will separate (1) into its real part and imaginary part as follows:(9)dxkt=-dkxkt+∑j=1nakjRfjRxjt,yjt-akjIfjIxjt,yjt+∑j=1nbkjRfjRxjt-τkjt,yjt-τkjt-bkjIfjIxjt-τkjt,yjt-τkjt+∑j=1n∫-∞tθkjt-spkjRfjRxjs,yjs-pkjIfjIxjs,yjsds+JkRdt+∑j=1nσkjRxjt,yjt,xjt-τkjt,yjt-τkjtdwjt,(10)dykt=-dkykt+∑j=1nakjRfjIxjt,yjt+akjIfjRxjt,yjt+∑j=1nbkjRfjIxjt-τkjt,yjt-τkjt+bkjIfjRxjt-τkjt,yjt-τkjt+∑j=1n∫-∞tθkjt-spkjRfjIxjs,yjs+pkjIfjRxjs,yjsds+JkIdt∑j=1nσkjIxjt,yjt,xjt-τkjt,yjt-τkjtdwjt,where k=1,2,......,n. Let AR=(akjR)n×n and AI=(akjI)n×n, BR=(bkjR)n×n and BI=(bkjI)n×n, PR=(pkjR)n×n and PI=(pkjI)n×n be the real part and imaginary part of matrices A, B, and P, respectively. Let JR=(J1R,J2R,......,JnR)T and JI=(J1I,J2I,......,JnI)T be the real part and imaginary part of external input J.For a complex number vectorz∈Cn, let |z|=(|z1|,|z2|,......,|zn|)T and z=∑k=1n|zk|2.Definition 6.
The equilibrium pointz#=(z1#,z2#,...…,zn#)T of (1) is said to be exponentially stable in the mean square sense if, for all J∈Cn, there exist constants λ>0 and Γ>0 such that (11)Ezt-z#2≤Γsups∈-∞,0Eφs-z#2exp⁡-λt,t≥0,where (12)Εzt-z#2=Εz1t-z1#2,Εz2t-z2#2,…,Εznt-zn#2T;Εφs-z#2=Εφ1s-z1#2,Εφ2s-z2#2,…,Εφns-zn#2T.Lemma 7 (see [8]).
LetA=(akj)n×n be a real number matrix with akj≤0, (k≠j). The following statements are equivalent:
(i)A is an M-matrix.
(ii) The real parts of all eigenvalues ofA are positive.
(iii) There exists a positive vectorς∈Rn such that Aς>0.Lemma 8 (see [39]).
Lety(t) be an Ito^ process given by (13)dyt=udt+vdwt.Let V(t,y):0,∞×R→R be again an Ito^ process with second derivative with respect toy; then (14)dVt,y=∂V∂tdt+∂V∂ydy+12∂2V∂y2dy2,where dy2=dydy is computed according to the rules dt.dt=dt.dw(t)=dw(t).dt=0 and dw(t).dw(t)=dt.Remark 9.
By substituting (13) into (14) and considering the computation rule in Lemma 8, we can obtain the following equation, which will be used in the process of the proof for theorems follow-up. (15)dVt,y=∂V∂t+∂V∂yu+12∂2V∂y2v2dt+∂V∂yvdwt.


## 3. Main Results on Mean Square Exponential Stability

For simplification, letx=[x1,x2,......,xn]T, y=[y1,y2,......,yn]T, α=[xT,yT]T.In what follows, we will study the mean square exponential stability of the equilibrium point of (1). For analysis convenience, we translate the coordinate of (1). Let z~k=zk-zk#=x~k+iy~k, k=1,2,......,n.By the translation, (1) can be changed as follows, where k,j=1,2,......,n,(16)dx~kt=-dkx~kt+∑j=1nakjRgjRx~jt,y~jt-akjIgjIx~jt,y~jt+∑j=1nbkjRgjRx~jt-τkjt,y~jt-τkjt-bkjIgjIx~jt-τkjt,y~jt-τkjt+∑j=1n∫-∞tθkjt-spkjRgjRx~js,y~js-pkjIgjIx~js,y~jsdsdt+∑j=1nσ~kjRx~jt,y~jt,x~jt-τkjt,y~jt-τkjtdwjt,(17)dy~kt=-dky~kt+∑j=1nakjRgjIx~jt,y~jt+akjIgjRx~jt,y~jt+∑j=1nbkjRgjIx~jt-τkjt,y~jt-τkjt+bkjIgjRx~jt-τkjt,y~jt-τkjt+∑j=1n∫-∞tθkjt-spkjRgjIx~js,y~js+pkjIgjRx~js,y~jsdsdt+∑j=1nσ~kjIx~jt,y~jt,x~jt-τkjt,y~jt-τkjtdwjt,where(18)α~=x~T,y~TT;gjRx~j,y~j=fjRxj,yj-fjRxj#,yj#;gjIx~j,y~j=fjIxj,yj-fjIxj#,yj#;σ~kjRx~jt,y~jt,x~jt-τkjt,y~jt-τkjt=σkjRxjt,yjt,xjt-τkjt,yjt-τkjt-σkjRxj#t,yj#t,xj#t-τkjt,yj#t-τkjt;σ~kjIx~jt,y~jt,x~jt-τkjt,y~jt-τkjt=σkjIxjt,yjt,xjt-τkjt,yjt-τkjt-σkjIxj#t,yj#t,xj#t-τkjt,yj#t-τkjt.Moreover, the initial conditions of (1) are of the forms ψk(s)=φk(s)-zk#, k=1,2,......,n.Obviously, if the zero solution of the system being composed of (16) and (17) is exponentially stable in the mean square sense, then the equilibrium point of the system being composed of (9) and (10) is also exponentially stable in the mean square sense.Next, we will give some sufficient conditions for assuring the mean square exponential stability of the equilibrium pointz# of (1). First of all, we let (19)D~=D00D,L~=LRR+LRI00LIR+LII,L^=LRRLRILIRLII,W=WRRWRIWIRWII,Q~=AR+BR+PR00AI+BI+PI,Q^=AR+BR+PRAI+BI+PIAI+BI+PIAR+BR+PR.Theorem 10.
If Assumptions1–5 are satisfied and the matrix 2D~-Q~L~-Q^L^-2W is an M-matrix, then the equilibrium point z# of (1) is of mean square exponential stability for any external input J∈Cn.Proof.
Because2D~-Q~L~-Q^L^-2W is an M matrix, according to Lemma 7 there exists a positive vector Λ=(ς1,ς2,...,ς2n) such that the following inequalities hold, where ςk′=βk and k=k′, when 1≤k′≤n; ςk′=γk and k=k′-n when n+1≤k′≤2n: (20)-2dk+∑j=1nakjR+bkjR+pkjRljRR+ljRI+akjI+bkjI+pkjIljIR+ljIIβk+∑j=1nakjR+bkjR+pkjRljRR+akjI+bkjI+pkjIljIR+2ωkjRR2βj+∑j=1nakjR+bkjR+pkjRljRI+akjI+bkjI+pkjIljII+2ωkjRI2γj<0,-2dk+∑j=1nakjR+bkjR+pkjRljIR+ljII+akjI+bkjI+pkjIljRR+ljRIγk+∑j=1nakjR+bkjR+pkjRljIR+akjI+bkjI+pkjIljRR+2ωkjIR2βj+∑j=1nakjR+bkjR+pkjRljII+akjI+bkjI+pkjIljRI+2ωkjII2γj<0.
Construct the following functions:(21)FkRε=-2dk+ε+∑j=1nakjR+bkjR+pkjRljRR+ljRI+akjI+bkjI+pkjIljIR+ljIIβk+∑j=1nakjR+bkjRexp⁡ετ+pkjRμkjεljRR+akjI+bkjIexp⁡ετ+pkjIμkjεljIR+ωkjRR2+exp⁡ετωkjRR2βj+∑j=1nakjR+bkjRexp⁡ετ+pkjRμkjεljRI+akjI+bkjIexp⁡ετ+pkjIμkjεljII+exp⁡ετωkjRI2+ωkjRI2γj,FkIε=-2dk+ε+∑j=1nakjR+bkjR+pkjRljIR+ljII+akjI+bkjI+pkjIljRR+ljRIγk+∑j=1nakjR+bkjRexp⁡ετ+pkjRμkjεljIR+akjI+bkjIexp⁡ετ+pkjIμkjεljRR+exp⁡ετωkjIR2+ωkjIR2βj+∑j=1nakjR+bkjRexp⁡ετ+pkjRμkjεljII+akjI+bkjIexp⁡ετ+pkjIμkjεljRI+exp⁡ετωkjII2+ωkjII2γj.
BecauseFkR(ε) and FkI(ε) are continuous with respect to ε, and FkR(0)<0, FkI(0)<0, there exist constants λ1>0 and λ2>0, such that FkR(λ1)<0 and FkI(λ2)<0, k=1,2,…,n. Obviously, there exists a positive constant λ>0 satisfying λ≤min⁡{λ1,λ2} such that FkR(λ)<0 and FkI(λ)<0 hold; that is,(22)FkRλ=-2dk+λ+∑j=1nakjR+bkjR+pkjRljRR+ljRI+akjI+bkjI+pkjIljIR+ljIIβk+∑j=1nakjR+bkjRexp⁡λτ+pkjRμkjλljRR+akjI+bkjIexp⁡λτ+pkjIμkjλljIR+ωkjRR2+exp⁡λτωkjRR2βj+∑j=1nakjR+bkjRexp⁡λτ+pkjRμkjλljRI+akjI+bkjIexp⁡λτ+pkjIμkjλljII+exp⁡λτωkjRI2+ωkjRI2γj<0,(23)FkIλ=-2dk+λ+∑j=1nakjR+bkjR+pkjRljIR+ljII+akjI+bkjI+pkjIljRR+ljRIγk+∑j=1nakjR+bkjRexp⁡λτ+pkjRμkjλljIR+akjI+bkjIexp⁡λτ+pkjIμkjλljRR+exp⁡λτωkjIR2+ωkjIR2βj+∑j=1nakjR+bkjRexp⁡λτ+pkjRμkjλljII+akjI+bkjIexp⁡λτ+pkjIμkjλljRI+exp⁡λτωkjII2+ωkjII2γj<0.
Choose a candidate vector Lyapunov function in what follows:(24)Vk′t,α~k′t=12exp⁡λtα~k′t2,k′=1,2,…,2n;i.e., (25)Vk′t,α~k′t=12exp⁡λtx~k′t2=12exp⁡λtx~kt2,1≤k′≤n,k′=kVk′t,α~k′t=12exp⁡λty~k′-nt2=12exp⁡λty~kt2,n+1≤k′≤2n,k=k′-n,where k=1,2,......,n.
LetVk′(t) denote Vk′(t,α~k′(t)) if there is no confusion, k′=1,2,......,2n.
(i) When1≤k′≤n, according to Ito^ formula (15) and considering the real parts (16) of (1), we get(26)dVk′t,α~k′t=d12exp⁡λtx~kt2=∑j=1nσ~kjRx~jt,y~jt,x~jt-τkjt,y~jt-τkjt212λexp⁡λtx~kt2+exp⁡λtx~kt∑j=1nσ~kjRx~jt,y~jt,x~jt-τkjt,y~jt-τkjt2-dkx~kt+∑j=1nakjRgjRx~jt,y~jt-akjIgjIx~jt,y~jt+∑j=1nx~jt-τkjt,y~jt-τkjtbkjRgjRx~jt-τkjt,y~jt-τkjt-bkjIgjIx~jt-τkjt,y~jt-τkjt+∑j=1n∫-∞tθkjt-spkjRgjRx~js,y~js-pkjIgjIx~js,y~jsds∑j=1n∫-∞tθkjt-s+12exp⁡λt∑j=1n,y~jt-τkjtσ~kjRx~jt,y~jt,x~jt-τkjt,y~jt-τkjt2∑j=1nx~jt,y~jt,x~jt-τkjtσ~kjRdt+exp⁡λtx~kt∑j=1nx~jt-τkjt,y~jt-τkjtσ~kjRy~jt-τkjtx~jt,y~jt,x~jt-τkjt,y~jt-τkjtdwjt.
Next, we will transformIto^ differential form of (26) into Ito^ integral form. By using stochastic differential equation theory [39], we can integrate on both sides of (26) from t to t+δ for any δ>0 and compute the expectation of (26): (27)
Ε
V
k
′
t
+
δ
-
Ε
V
k
′
t
=
Ε
∑
j
=
1
n
σ
~
k
j
R
x
~
j
t
,
y
~
j
t
,
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
2
∫
t
t
+
δ
∑
j
=
1
n
σ
~
k
j
R
x
~
j
t
,
y
~
j
t
,
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
2
1
2
λ
exp
⁡
λ
t
x
~
k
t
2
+
exp
⁡
λ
t
x
~
k
t
∑
j
=
1
n
a
k
j
R
g
j
R
x
~
j
t
,
y
~
j
t
-
a
k
j
I
g
j
I
x
~
j
t
,
y
~
j
t
-
d
k
x
~
k
t
+
∑
j
=
1
n
a
k
j
R
g
j
R
x
~
j
t
,
y
~
j
t
-
a
k
j
I
g
j
I
x
~
j
t
,
y
~
j
t
+
∑
j
=
1
n
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
b
k
j
R
g
j
R
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
-
b
k
j
I
g
j
I
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
+
∑
j
=
1
n
∫
-
∞
t
θ
k
j
t
-
s
p
k
j
R
g
j
R
x
~
j
s
,
y
~
j
s
-
p
k
j
I
g
j
I
x
~
j
s
,
y
~
j
s
d
s
∑
j
=
1
n
∫
-
∞
t
θ
k
j
t
-
s
+
1
2
exp
⁡
λ
t
∑
j
=
1
n
σ
~
k
j
R
t
-
τ
k
j
t
x
~
j
t
,
y
~
j
t
,
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
2
∑
j
=
1
n
σ
~
k
j
R
x
~
j
t
,
y
~
j
t
,
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
2
d
t
+
Ε
∫
t
t
+
δ
exp
⁡
λ
t
x
~
k
t
∑
j
=
1
n
y
~
j
t
-
τ
k
j
t
σ
~
k
j
R
t
-
τ
k
j
t
x
~
j
t
,
y
~
j
t
,
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
d
w
j
t
∫
t
t
+
δ
exp
⁡
λ
t
x
~
k
t
.
It is well known that the expectation ofIto^ process is a continuous function. According to the properties of Ito^ integral [39] and Dini derivation, (27) can be deduced as follows: (28)
D
+
Ε
V
k
′
t
=
Ε
∑
j
=
1
n
σ
~
k
j
R
x
~
j
t
,
y
~
j
t
,
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
2
1
2
λ
exp
⁡
λ
t
x
~
k
t
2
+
exp
⁡
λ
t
x
~
k
t
∑
j
=
1
n
σ
~
k
j
R
x
~
j
t
,
y
~
j
t
,
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
2
-
d
k
x
~
k
t
+
∑
j
=
1
n
a
k
j
R
g
j
R
x
~
j
t
,
y
~
j
t
-
a
k
j
I
g
j
I
x
~
j
t
,
y
~
j
t
+
∑
j
=
1
n
b
k
j
R
g
j
R
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
-
b
k
j
I
g
j
I
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
+
∑
j
=
1
n
∫
-
∞
t
θ
k
j
t
-
s
p
k
j
R
g
j
R
x
~
j
s
,
y
~
j
s
-
p
k
j
I
g
j
I
x
~
j
s
,
y
~
j
s
d
s
+
1
2
exp
⁡
λ
t
∑
j
=
1
n
σ
~
k
j
R
x
~
j
t
,
y
~
j
t
,
x
~
j
t
-
τ
k
j
t
,
y
~
j
t
-
τ
k
j
t
2
.
Considering Assumptions1–5, (28) can be deduced furthermore as follows: (29)D+ΕVk′t≤-2dk+λ+∑j=1nakjR+bkjR+pkjRljRR+ljRI+akjI+bkjI+pkjIljIR+ljIIΕVkt+∑j=1nakjRljRR+akjIljIR+ωkjRR2ΕVjt+∑j=1nakjRljRI+akjIljII+ωkjRI2ΕVj+nt+∑j=1nbkjRljRR+bkjIljIR+ωkjRR2exp⁡λτΕVjt-τkjt+∑j=1nbkjRljRI+bkjIljII+ωkjRI2exp⁡λτΕVj+nt-τkjt+∑j=1n∫-∞tθkjt-sexp⁡λt-spkjRljRR+pkjIljIRΕVjsds+∑j=1n∫-∞tθkjt-sexp⁡λt-spkjRljRI+pkjIljIIΕVj+nsds,k=1,2,…,n.
The detailed deducing of (29) can be referred in the Appendix part of this paper.
Define the curve(30)ς=ηχ:ηk′=ςk′χ,χ>0,k′=1,2,…,2nand the sets(31)Ωη=h:0≤h≤η,η∈ς.
It is obvious that we haveΩ(η(χ))⊃Ω(η(χ′)) as χ>χ′.
In what follows, we will prove the mean square exponential stability of (16). That is, there exist constants Γ>0 and λ>0 such that |x~(t)|≤Γψexp(-λt) hold, where ψ=sups∈(-∞,0]ψ(s), t≥0.
Letπ-=max1≤k≤n{βk,γk}, π_=min1≤k≤n{βk,γk}, and χ0=δψ/π_, where δ>1 is a constant; then (32)ΕVs:ΕVs=12exp⁡λsΕα~s2,-∞<s≤0⊂Ωη0χ0;i.e., when -∞<s≤0, we have ΕVj(s)=1/2exp(λs)Εxj(s)2<βjχ0, ΕVj+n(s)=1/2exp(λs)Εy~j(s)2<γjχ0, j=1,2,......,n.
Furthermore, it can be obtained thatΕVj(t)<βjχ0, ΕVj+n(t)<γjχ0, j=1,2,......,n. If they are not true, then there exist some k and t1>0 such that ΕVk′(t1)=βkχ0, D+Ε(Vk′(t1))≥0; ΕVj(t1)<βjχ0, ΕVj+n(t1)≤γjχ0, j≠k′, j=1,2,......,n. Substituting them into (29) and considering (22), we have(33)
D
+
Ε
V
k
′
t
1
≤
-
2
d
k
+
λ
+
∑
j
=
1
n
a
k
j
R
+
b
k
j
R
+
p
k
j
R
l
j
R
R
+
l
j
R
I
+
a
k
j
I
+
b
k
j
I
+
p
k
j
I
l
j
I
R
+
l
j
I
I
Ε
V
k
t
1
+
∑
j
=
1
n
a
k
j
R
l
j
R
R
+
a
k
j
I
l
j
I
R
+
ω
k
j
R
R
2
Ε
V
j
t
1
+
∑
j
=
1
n
a
k
j
R
l
j
R
I
+
a
k
j
I
l
j
I
I
+
ω
k
j
R
I
2
Ε
V
j
+
n
t
1
+
∑
j
=
1
n
b
k
j
R
l
j
R
R
+
b
k
j
I
l
j
I
R
+
ω
k
j
R
R
2
exp
⁡
λ
τ
Ε
V
j
t
1
-
τ
k
j
t
1
+
∑
j
=
1
n
b
k
j
R
l
j
R
I
+
b
k
j
I
l
j
I
I
+
ω
k
j
R
I
2
exp
⁡
λ
τ
Ε
V
j
+
n
t
1
-
τ
k
j
t
1
+
∑
j
=
1
n
∫
-
∞
t
1
θ
k
j
t
1
-
s
exp
⁡
λ
t
1
-
s
p
k
j
R
l
j
R
R
+
p
k
j
I
l
j
I
R
Ε
V
j
s
d
s
+
∑
j
=
1
n
∫
-
∞
t
1
θ
k
j
t
1
-
s
exp
⁡
λ
t
1
-
s
p
k
j
R
l
j
R
I
+
p
k
j
I
l
j
I
I
Ε
V
j
+
n
s
d
s
≤
-
2
d
k
+
λ
+
∑
j
=
1
n
a
k
j
R
+
b
k
j
R
+
p
k
j
R
l
j
R
R
+
l
j
R
I
+
a
k
j
I
+
b
k
j
I
+
p
k
j
I
l
j
I
R
+
l
j
I
I
β
k
χ
0
+
∑
j
=
1
n
a
k
j
R
l
j
R
R
+
a
k
j
I
l
j
I
R
+
ω
k
j
R
R
2
+
exp
⁡
λ
τ
b
k
j
R
l
j
R
R
+
b
k
j
I
l
j
I
R
+
ω
k
j
R
R
2
+
p
k
j
R
l
j
R
R
+
p
k
j
I
l
j
I
R
μ
k
j
λ
β
j
χ
0
+
∑
j
=
1
n
a
k
j
R
l
j
R
I
+
a
k
j
I
l
j
I
I
+
ω
k
j
R
I
2
+
exp
⁡
λ
τ
b
k
j
R
l
j
R
I
+
b
k
j
I
l
j
I
I
+
ω
k
j
R
I
2
+
p
k
j
R
l
j
R
I
+
p
k
j
I
l
j
I
I
μ
k
j
λ
γ
j
χ
0
<
0
.
This is a contradiction with the assumptionD+Ε(Vk′(t1))≥0; i.e., for all t≥0, we have ΕVk(t)<βkχ0, ΕVk+n(t)<γkχ0, k=1,2,......,n. That is, ΕVk′(t)<ςk′χ0, k′=1,2,......,n.
(ii) Whenn+1≤k′≤2n, according to Ito^ formula (15) and considering the imaginary parts (17) of (1), we get(34)dVk′t,α~k′t=∑j=1nσ~kjIx~jt,y~jt,x~jt-τkjt,y~jt-τkjt212λexp⁡λty~kt2+exp⁡λty~kt∑j=1nσ~kjIx~jt,y~jt,x~jt-τkjt,y~jt-τkjt2-dky~kt+∑j=1nakjRgjIx~jt,y~jt+akjIgjRx~jt,y~jt+∑j=1nx~jt-τkjt,y~jt-τkjtbkjRgjIx~jt-τkjt,y~jt-τkjt+bkjIgjRx~jt-τkjt,y~jt-τkjt+∑j=1n∫-∞tθkjt-sx~js,y~jspkjRgjIx~js,y~js+pkjIgjRx~js,y~jsds∑j=1n∫-∞tθkjt-s+12∑j=1nt-τkjtσ~kjIx~jt,y~jt,x~jt-τkjt,y~jt-τkjt2dt+exp⁡λty~kt∑j=1nt-τkjtσ~kjIt-τkjtx~jt,y~jt,x~jt-τkjt,y~jt-τkjtdwjt.
Next, we will transformIto^ differential form of (34) into Ito^ integral form. By using stochastic differential equation theory [39], we can integrate on both sides of (34) from t to t+δ for any δ>0 and compute the expectation of (34), we get(35)ΕVk′t+δ-ΕVk′t=Ε∫tt+δ∑j=1nakjRgjIx~jt,y~jt+akjIgjRx~jt,y~jt12λexp⁡λty~kt2+expλty~kt∑j=1nakjRgjIx~jt,y~jt+akjIgjRx~jt,y~jt-dky~kt+∑j=1nakjRgjIx~jt,y~jt+akjIgjRx~jt,y~jt+∑j=1nbkjRgjIx~jt-τkjt,y~jt-τkjt+bkjIgjRx~jt-τkjt,y~jt-τkjt+∑j=1n∫-∞tθkjt-spkjRgjIx~js,y~js+pkjIgjRx~js,y~jsds+12∑j=1nσ~kjIx~jt,y~jt,x~jt-τkjt,y~jt-τkjt2dt+Ε∫tt+δexpλty~kt∑j=1nσ~kjIx~jt,y~jt,x~jt-τkjt,y~jt-τkjtdwjt.
Noticing that the expectation ofIto^ process is a continuous function, according to the properties of Ito^ integral and Dini derivation, (35) can be deduced for all k=1,2,......,n, k′=n+1,n+2,......,2n as follows: (36)D+ΕVk′t=Ε∑j=1n∫-∞tθkjt-s12λexp⁡λty~kt2+exp⁡λty~kt∑j=1n∫-∞tθkjt-s-dky~kt+∑j=1nx~jt,y~jtakjRgjIx~jt,y~jt+akjIgjRx~jt,y~jt+∑j=1nx~jt-τkjt,y~jt-τkjtbkjRgjIx~jt-τkjt,y~jt-τkjt+bkjIgjRx~jt-τkjt,y~jt-τkjt+∑j=1n∫-∞tθkjt-spkjRgjIx~js,y~js+pkjIgjRx~js,y~jsds∑j=1n∫-∞tθkjt-s+12∑j=1nσ~kjIx~jt,y~jt,x~jt-τkjt,y~jt-τkjt2∑j=1n∫-∞tθkjt-s.
Considering Assumptions1–5, the following inequality can be obtained by adopting the similar analysis of (29): (37)D+ΕVk′t≤-2dk+λ+∑j=1nakjR+bkjR+pkjRljIR+ljII+akjI+bkjI+pkjIljRR+ljRIΕVk′t+∑j=1nakjRljIR+akjIljRR+ωkjIR2ΕVjt+akjRljII+akjIljRI+ωkjII2ΕVj+nt+∑j=1nexp⁡λτbkjRljIR+bkjIljRR+ωkjIR2ΕVjt-τkjt+bkjRljII+bkjIljRI+ωkjII2ΕVj+nt-τkjt+∑j=1n∫-∞tθkjt-sexp⁡λt-spkjRljIR+pkjIljRRΕVjs+pkjRljII+pkjIljRIΕVj+nsds.
By similar analysis with part (i) previously, we can conclude thatΕVk(t)<βkχ0 and ΕVk′(t)<γkχ0, where 1≤k≤n, n+1≤k′≤2n, t≥0.
In what follows, we will prove the mean square exponential stability of system (17). That is to say, there exist constants Γ>0 and λ>0 such that |y~(t)|≤Γψexp(-λt) hold, where ψ=sups∈(-∞,0]ψ(s), t≥0.
We claim thatΕVj(t)<βjχ0 and ΕVj+n(t)<γjχ0, j=1,2,......,n. If they are not true, then there exist some n+1≤k^≤2n and time instant t2>0 such that ΕVk^(t2)=γk^-nχ0, D+Ε(Vk^(t2))≥0; ΕVj(t2)<βjχ0, ΕVj+n(t2)≤γjχ0, j≠k^-n, j=1,2,......,n. Substituting them into (37) and considering (23), we have (38)D+ΕVk^t2≤∑j=1n∫-∞t2θkjt2-sexp⁡λt2-s-2dk+λ+akjR+bkjR+pkjRljIR+ljII+akjI+bkjI+pkjIljRR+ljRIΕVk^t2+∑j=1nakjRljIR+akjIljRR+ωkjIR2ΕVjt2+akjRljII+akjIljRI+ωkjII2ΕVj+nt2+∑j=1nexp⁡λτbkjRljIR+bkjIljRR+ωkjIR2ΕVjt2-τkjt2+bkjRljII+bkjIljRI+ωkjII2ΕVj+nt2-τkjt2+∑j=1n∫-∞t2θkjt2-sexp⁡λt2-spkjRljIR+pkjIljRRΕVjs+pkjRljII+pkjIljRIΕVj+nsds∑j=1n∫-∞t2θkjt2-sexp⁡λt2-s≤-2dk+λ+∑j=1nakjR+bkjR+pkjRljIR+ljII+akjI+bkjI+pkjIljRR+ljRIγk^-nχ0+∑j=1nakjRljIR+akjIljRR+ωkjIR2+exp⁡λτbkjRljIR+bkjIljRR+ωkjIR2+μkjλpkjRljIR+pkjIljRRβjχ0+∑j=1nakjRljII+akjIljRI+ωkjII2+exp⁡λτbkjRljII+bkjIljRI+ωkjII2+μkjλpkjRljII+pkjIljRIγjχ0<0.
This is a contradiction with the assumptionD+Ε(Vk^(t2))≥0. Therefore, for all t≥0, we have ΕVk(t)<βkχ0 and ΕVk′(t)<γk′χ0. That is, ΕVk′(t)<ςk′χ0, k′=1,2,......,2n.
To sum up, it follows from (i) and (ii) thatΕVk′(t)<ςk′χ0 hold for all k′=1,2,......,2n; i.e., 1/2exp(-λt)Ε|x~k(t)|2<βkχ0, 1/2exp(λt)Ε|y~k(t)|2<γkχ0, k=1,2,......,n.
Furthermore, fork=1,2,......,n, we get (39)Εx~kt2<2δβkexp⁡-λtΕψs2π_,Εy~kt2<2δγkexp⁡-λtΕψs2π_.
LetΓ=2δπ-/π_. We further have(40)Εx~kt2<Γexp⁡-λtΕψs2,Εy~kt2<ΓΕψs2exp⁡-λt,k=1,2,…,n.
According to Definition6, the zero solution of the complex-valued neural networks being composed of the real part (16) and the imaginary part (17) is exponentially stable in the mean square sense. Namely, the equilibrium point of the complex-valued neural networks being composed of the real part (9) and the imaginary part (10) is exponentially stable in the mean square sense. The proof is completed.Remark 11.
When there is no stochastic disturbance in the complex-valued neural networks described by (1), model (1) in this paper is the same as model (1) in [8]. Theorem 2 in [8] can be obtained directly by using the same analysis in Theorem 10. The obtained results of this paper include the ones in [8], which means that they are with more generality.Remark 12.
The authors in [32] studied a class of delayed memristor-based complex-valued neural networks with stochastic disturbances. The continuously distributed delays were not considered in [32]. Moreover, it can be seen from Assumption 5 of this paper that the stochastic weighted functions σR(.) and σI(.) are with regard to both the real part and the imaginary part of neuro state, respectively. Obviously, Assumption 2 concerning stochastic disturbances in [32] was with more constraint because σR(.) was supposed to the function only with regard to the real part of neuro state and σI(.) with regard to imaginary part of neuro state. Therefore, the research of this paper extends the corresponding work in [32].Motivated by the ideal of Assumption 2 in [32], some other assumption conditions on stochastic disturbances are proposed as follows.Assumption 13.
It is assumed that the weighted function of stochastic disturbancesσkj (k,j=1,2,......,n) can be separated into its real part and imaginary part with the following forms:(41)σkjR=σkjRxjt,xjt-τkjt:R2→R,σkjI=σkjIyjt,yjt-τkjt:R2→R,where σkjR(0,0)=σkjI(0,0)=0. Suppose that there exist nonnegative constants ωkjRR and ωkjII such that the following inequalities hold:(42)σkjRxjt,xjt-τkjt2-σkjRx′jt,x′jt-τkjt2≤ωkjRR2xjt-x′jt2+xjt-τkjt-x′jt-τkjt2,σkjIyjt,yjt-τkjt2-σkjIy′jt,y′jt-τkjt2≤ωkjII2yjt-y′jt2+yjt-τkjt-y′jt-τkjt2.
LetWRR=ωkjRR2n×n, WII=ωkjII2n×n, and W′=WRR00WII.Corollary 14.
If Assumptions1–3 and Assumption 13 are satisfied, and the matrix 2D~-Q~L~-Q^L^-2W′ is an M-matrix, then the equilibrium point z# of (1) is of mean square exponential stability for any external input J∈Cn.Remark 15.
In order to compare the result to one in [32] from the computational point of view, we assume that σkjR=0.2xj(t)+0.4xj(t-τkj(t)) and σkjI=0.3yj(t)+0.1yj(t-τkj(t)). By computation, the parameters of Assumption 2 in [32] are δk2=0.12, ħk2=0.24, μk2=0.12, and δk2=0.04, respectively. The parameters in Assumption 13 of this paper are (ωkjRR)2=max{δk2,ħk2}=0.24 and (ωkjII)2=max{μk2,νk2}=0.12. Although the expression form of Assumption 2 in [32] is different from Assumption 13 in this paper, it can be seen from the proof of Theorem 10 about values of ak and bk that the obtained stability conditions concerning the stochastic terms in [32] are the same as ours. Besides, the matrix for judging stability in Corollary 14 in this paper is of more simple and compact expression, which is easier to calculate by MATLAB software.Remark 16.
There are some errors inevitably between theoretical and practical systems for the unexpected factors, such as sensing error, modeling error, parameter aging, and channel strength. Therefore, it is necessary to suppose that parameter uncertainties of system in given intervals can overcome those problems. We call the system with randomly occurring parameter uncertainties interval system. In [18, 20–22, 33, 40], the authors investigated some interval neural networks in complex number domain. However, the stochastic disturbances were not considered in [18, 20–22] and the continuously distributed delays were not considered in [33, 40].Next, we will give a theorem for judging the robust exponential stability in the mean square sense for a class of interval complex-valued neural networks with mixed delays and stochastic disturbances.Assumption 17.
The weighted matrices defined in complex number domain of (1) are supposed to be in the following intervals: (43)AI=A=akjn×n:A~≤A≤A~,i.e.a~kj≤akj≤a~kj,k,j=1,2,…,n,BI=B=bkjn×n:B~≤B≤B~,i.e.b~kj≤bkj≤b~kj,k,j=1,2,…,n,PI=P=pijn×n:P~≤P≤P~,i.e.p~kj≤pkj≤p~kj,k,j=1,2,…,n.
Let(44)Q~1=A~R+B~R+P~R00A~I+B~I+P~I,Q^2=A~R+B~R+P~RA~I+B~I+P~IA~I+B~I+P~IA~R+B~R+P~R.Definition 18.
The equilibrium pointz#=(z1#,z2#,...…,zn#)T of (1) is said to be robustly exponentially stable in the mean square sense if, for all A∈AI, B∈BI, P∈PI and J∈Cn, there exist constants λ>0 and Γ>0 such that (45)Ezt-z#2≤Γsups∈-∞,0⁡Eφs-z#2exp-λt,t≥0,where (46)Εzt-z#2=Εz1t-z1#2,Εz2t-z2#2,…,Εznt-zn#2T;Εφs-z#2=Εφ1s-z1#2,Εφ2s-z2#2,…,Εφns-zn#2T.Theorem 19.
If Assumptions1–5 and Assumption 17 are satisfied, and the matrix 2D~-Q~1L~-Q^2L^-2W is an M-matrix, then the equilibrium point z# of (1) is of robust exponential stability in the mean square sense for any external input J∈Cn.Proof.
Because2D~-Q~1L~-Q^2L^-2W is an M-matrix, according to Lemma 7 there exists positive vector Λ=(ς1,ς2,...,ς2n) such that the following inequalities hold, where ςk′=βk and =k′, when 1≤k′≤n; ςk′=γk and k=k′-n when n+1≤k′≤2n, (47)-2dk+∑j=1na~kjR+b~kjR+p~kjRljRR+ljRI+a~kjI+b~kjI+p~kjIljIR+ljIIβk+∑j=1na~kjR+b~kjR+p~kjRljRR+a~kjI+b~kjI+p~kjIljIR+2ωkjRR2βj+∑j=1na~kjR+b~kjR+p~kjRljRI+a~kjI+b~kjI+p~kjIljII+2ωkjRI2γj<0,-2dk+∑j=1na~kjR+b~kjR+p~kjRljIR+ljII+a~kjI+b~kjI+p~kjIljRR+ljRIγk+∑j=1na~kjR+b~kjR+p~kjRljIR+a~kjI+b~kjI+p~kjIljRR+2ωkjIR2βj+∑j=1na~kjR+b~kjR+p~kjRljII+a~kjI+b~kjI+p~kjIljRI+2ωkjII2γj<0.
The rest of proof is almost similar with the proof of Theorem10 only by replacing all parameters |akjR|,|bkjR|,|pkjR|,|akjI|,|bkjI|,|pkjI| by |a~kjR|,|b~kjR|,|p~kjR|,|a~kjI|,|b~kjI|,|p~kjI|, respectively, so it is omitted here. The proof is completed.When the stochastic disturbances satisfy Assumption13, it is easy to obtain the following Corollary 20 for (1) with the mentioned parameter uncertainties above.Corollary 20.
If Assumptions1–3 and Assumptions 13 and 17 are satisfied and the matrix 2D~-Q~1L~-Q^2L^-2W′ is an M-matrix, then the equilibrium point z# of (1) is of robust exponential stability in the mean square sense for any external input J∈Cn.Remark 21.
When there is only time-varying delays or continuously distributed delays in (1), some corresponding corollaries can be obtained directly by letting P=0 or B=0 in the matrices Q~,Q^,Q~1,Q^2 defined in Theorem 10, Corollary 14, Theorem 19, and Corollary 20, respectively.


## 4. Numerical Examples

In this section, two numerical examples with simulation figures will be given to illustrate the obtained results of Theorems10 and 19, respectively.Example 1.
Consider the following system:(48)dzkt=-dkzkt+∑j=12akjfjzjt+bkjfjzjt-τkjt+pkj∫-∞tθkjt-sfjzjsds+Jkdt+∑j=12σkjzjt,zjt-τkjtdwjtk=1,2,
In (48), let D=diag(3,2); the interconnected matrices are(49)A=-0.3-0.2i0.8+0.5i-0.4-0.15i0.7+0.8i,B=0.9+0.5i-0.7+0.9i0.8-0.3i-0.3+0.6i,P=0.5+0.6i-0.2+0.3i0.7-0.7i-0.4+0.9i.The activation functions are f1(z1)=1.21-e-x1/1+e-x1+i0.8/1+e-y1 andf2(z2)=0.51-e-y2/1+e-y2+i1.5/1+e-x2. The real part and imaginary part of stochastic weighted function are σkjR=0.3xj(t)+0.2yj(t)+0.1xj(t-τkj(t))+0.15yj(t-τkj(t)) and σkjI=0.15xj(t)+0.1yj(t)+0.25xj(t-τkj(t))+0.2yj(t-τkj(t)), respectively.
By computing, we have(50)LRR=0.6000,LRI=0000.25,LIR=0000.2,LII=0.375000;WRR=0.2250.2250.2250.225,WRI=0.150.150.150.15,WIR=0.1750.1750.1750.175,WII=0.1220.1220.1220.122,D~=3000020000300002,L~=0.600000.2500000.37500000.2,L^=0.60000000.25000.375000.200,W=0.2250.2250.150.150.2250.2250.150.150.1750.1750.1220.1220.1750.1750.1220.122,Q~=1.61.7001.91.400001.31.7000.852.3,Q^=1.61.71.31.71.91.40.852.31.31.71.61.70.852.31.91.4.
It can be obtained that(51)2D~-Q~L~-Q^L^-2W=3.63-1.12-0.79-0.73-2.732.74-0.62-0.65-1.13-0.694.671.01-0.86-0.63-1.282.72.The eigenvalue matrix of the matrix 2D~-Q~L~-Q^L^-2W is diag(0.2552,3.1280,4.9570,5.4193). It follows from Lemma 7 that the matrix 2D~-Q~L~-Q^L^-2W is an M-matrix. According to Theorem 10, it can be concluded that the equilibrium point of (48) is exponentially stable in the mean square sense.Let the delays in (48) be τ1j=0.3+0.1sin⁡t and τ2j=0.2+0.1cos⁡t, t≥0. Let θkj(t-s)=exp(-(t-s)), k,j=1,2, t≥0, s∈(-∞,0]. It is assumed that the initial condition of (48) are z1(s)=1+2i and z2(s)=-3-4i,s∈(-∞,0].The numerical simulations of (48) are shown in Figures 1–5. In order to make comparison, Figures 1 and 3 show the state curves of the real part and the imaginary part of (48) without/with stochastic disturbances, respectively. Figures 2 and 4 show the amplitude curves of neuro states of (48) without/with stochastic disturbances, respectively. Figure 5 shows the norm curves of neuro states of (48) both with stochastic disturbance and without stochastic disturbances. From the simulation results, it can be seen that the equilibrium point of (48) is stable. The simulation results verify the correctness of Theorem 10 in this paper.Figure 1
State curves of the real parts and imaginary parts in (48) without stochastic term.Figure 2
State module curves of (48) without stochastic term.Figure 3
State curves of the real parts and imaginary parts in (48) with stochastic term.Figure 4
State module curves of (48) with stochastic term.Figure 5
State norm curves of (48) with/without stochastic term.Example 2.
Consider the following system:(52)dzkt=-dkzkt+∑j=12akjfjzjt+bkjfjzjt-τkjt+pkj∫-∞tθkjt-sfjzjsds+Jkdt+∑j=12σkjzjt,zjt-τkjtdwjtk=1,2,
In (52), let D=diag(8,5). The interval interconnected matrices are supposed to be (53)AI=-1.5,1+1.6,1.9i-2,1+1.7,2i1.5,1.8+-1.4,1i-1.4,1.5+1.9,2.2i,BI=1.4,1.8+1.8,2i-1.3,1.6+1,1.8i-1.3,1+1.1,1.6i1.7,1.9+-2,-1.2i,PI=0,1.5+1.1,1.6i0.2,1.4+-1.1,1.5i-1,2+-1.3,1.9i0,1.7+-1.1,2i.
The rest of the assumptions concerning the activation functions and stochastic weighted functions are the same as ones given in Example1.
By computation, we have(54)D~=8000050000800005,Q~1=4.85005.15.100005.55.3004.95.61,Q^2=4.855.55.35.15.14.95.615.55.34.85.04.95.615.15.1.
It can be obtained that(55)2D~-Q~1L~-Q^2L^-2W=9.79-2.76-2.361.55-6.577.15-2.14-1.58-3.65-1.3511.89-2.63-3.29-1.37-3.997.23.
Because the eigenvalue matrix of the matrix2D~-Q~1L~-Q^2L^-2W is diag(0.84,14.47,12.62,8.14), it follows from Lemma 7 that the matrix 2D~-Q~1L~-Q^2L^-2W is an M-matrix. Therefore, the equilibrium point of (52) is robustly exponentially stable in the mean square sense in accordance with Theorem 19.
In order to finish the simulation, let the delays in (52) be τ1j=0.8+0.2sin⁡t and τ2j=0.1+0.6cos⁡t, t≥0. Let θkj(t-s)=exp(-(t-s)), k,j=1,2, t≥0, s∈(-∞,0]. It is assumed that the initial conditions of (52) are z1(s)=2+5i and z2(s)=-4-3i,s∈(-∞,0].
Let(56)A=-1.5+1.9i-2+2i1.8-1.4i1.5+2.2i,B=1.8+2i1.6+1.8i-1.3+1.6i1.9-2i,P=1.5+1.6i1.4+1.5i2+1.9i1.7+2i.
It is obvious that the matricesA∈AI, B∈BI, and P∈PI.Figures6–8 show the state curves of the real part and the imaginary part, the amplitude curves of neuro states, and the norm curves of neuro states of (52), respectively. From the simulation results, it can be seen that the equilibrium point of (52) is stable. The simulation results verify the correctness of Theorem 19.Figure 6
State curves of the real parts and imaginary parts in (52) with stochastic term.Figure 7
State module curves of (52) with stochastic term.Figure 8
State norm curves of (52).


## 5. Conclusions and Future Work

Some sufficient conditions have been proposed for assuring the mean square exponential stability of a class of complex-valued neural networks with stochastic disturbances and mixed delays based on the vector Lyapunov function method andIto^ differential-integral theorem. The results established in this paper generalize the existing ones and reduce the conservatism of the previous results about complex-valued neural networks with stochastic disturbances. Two numerical examples are given at last to show the correctness and feasibility of the proposed results. Based on the study method presented in this paper and the idea of driving-response conception [35, 41], we will attempt to study the synchronization problem for a class of chaotic complex-valued neural networks with mixed delays and stochastic disturbances in the near future. Besides, numerous references concerning neural networks with impulsive effect have been emerged in the past two decades; see [5, 9–15, 22, 40, 42]. It will be an interesting research to consider impulsive effect from the view of both disturbance and control when scholars study the synchronization problem of various chaotic complex-valued neural networks.



---

*Source: 3429326-2019-06-20.xml*