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Journal of Inequalities and Applications
 
J Inequal Appl. 2017; 2017(1): 215.
Published online 2017 September 11. doi:  10.1186/s13660-017-1490-0
PMCID: PMC5594064

Robust stability analysis of impulsive complex-valued neural networks with time delays and parameter uncertainties

Abstract

The present study considers the robust stability for impulsive complex-valued neural networks (CVNNs) with discrete time delays. By applying the homeomorphic mapping theorem and some inequalities in a complex domain, some sufficient conditions are obtained to prove the existence and uniqueness of the equilibrium for the CVNNs. By constructing appropriate Lyapunov-Krasovskii functionals and employing the complex-valued matrix inequality skills, the study finds the conditions to guarantee its global robust stability. A numerical simulation illustrates the correctness of the proposed theoretical results.

Keywords: complex-valued neural networks, robust stability, delay, impulsive

Introduction

Robustness is the ability of maintaining the performance of the controlling system under certain parameter perturbations. When these inner structural disturbances result in the instability of the system, additional control mechanisms should be used to improve these flimsy properties. When the control method is introduced, the uncertainty and error brought by the control itself also become another disturbance factor of the system. In real life and engineering practices, perturbations of system characteristics or parameters are often unavoidable. Perturbations exist for two main reasons. One is that the actual measurement is not accurate and usually deviates from its designed value. The other is slow drift of characteristics or parameters, which is influenced by the environmental factors in the running process of the system. When these uncertainties or random disturbances exist, the questions how and in what range to control the quality of the system or maintain its characteristics are of great importance. Therefore, robustness has become an important research topic in control theory, and it is also a basic problem that must be considered in the design of almost all kinds of control systems, such as image and signal processing, combinatorial optimization problems, pattern recognition, etc. It has attracted great attention of the scholars that work with neural networks [16].

For many applications of neural networks, on the one hand, the states change rapidly at a fixed time, and the duration of these abrupt changes is often neglected, assuming that they are caused by jumps. Such processes are studied by impulsive differential equations (for the relative theorems, we refer to [7]), and there are numerical applications of such equations in science and technology, mass services, etc. [811]. On the other hand, due to the neural processing and signal transmission, a time delay often occurs, which may cause instability and a poor performance of the system [12]. Generally, delays may be caused by the measuring process and therefore the effect of time delay is common. Also many efforts are being made as regards the delay-dependent stability analysis of neural networks [1323].

In the application of neural networks, complex signals are usually preferable [2429], so it is necessary to analyze complex-valued neural networks (CVNNs), which deal with the complex-valued date, weight and neuron activation functions. However, much work mainly focuses on the boundedness, μ-stability, power-stability, exponent-stability, etc. [3, 18, 3032], and little work considers the robust stability of neural networks with time delay and impulse in the complex domain. In [2, 6], the authors investigated a kind of recurrent CVNNs with time delays, but the activation functions are separated by real and imaginary parts and the analysis methods are also similar to those in their real domain. Therefore, the stability criteria cannot be applied if the activation functions cannot be expressed by separating their real and imaginary parts or if they are unbounded. Moreover, compared to real-valued neural networks, the advantage of CVNNs is that they can directly deal with two-dimensional data, which can also be processed by real-valued neural networks but then require double neurons. Consequently, as a class of complex-valued systems, CVNNs have undergone a growing number of studies that explore the application of neural networks. Therefore, the present study considers the robust stability of CVNNs with discrete time delay and impulse, which is valid regardless of whether the activation functions are separated or not. The relative results are extension of those in reference [2, 6]. Therefore, the robust stability for impulsive CVNNs with time delays is considered in this paper. Furthermore, compared with [2] and [6], the robust stability criteria for the addressed neural networks are valid regardless of whether the activation functions can be separated or not.

The structure of this paper is arranged as follows. Section 2 gives some preliminaries, including some notations and important lemmas, introducing the complex-valued recurrent networks model. The existence and uniqueness of the equilibrium are proved by using the homeomorphism mapping principle in Section 3. In Section 4, the global robust stability of the neural networks is investigated by building the proper Lyapunov functions. An example is given to illustrate the correction of our results. In Section 5, we conclude the paper.

Problems formulation and preliminaries

Some notations of this paper are presented here firstly. i denotes the imaginary unit, i.e., i=1. n, m×n, and m×n represent the set of n-dimensional complex vectors, m × n real matrices, and complex matrices, respectively. The subscripts T and * denote matrix transposition and matrix conjugate transposition, respectively. For complex vector z ∈ ℂn, let |z| = (|z1|,|z2|,…,|zn|)T be the module of the vector z and z=i=1n|zi|2 be the norm of the vector z. For complex matrix A = (aij)n×n ∈ ℂn×n, let |A| = (|aij|)n×n denote the module of the matrix A and A=i=1nj=1n|aij|2 denote the norm of the matrix A. I denotes the identity matrix with appropriate dimensions. The notation X ≥ Y (or X > Y) means that X − Y is positive semi-definite (or positive definite). In addition, λmax(P) and λmin(P) are defined as the largest and the smallest eigenvalue of positive definite matrix P, respectively.

Motivated by [2], we consider the following impulsive CVNN model with time delays:

{z˙i(t)=cizi(t)+j=1naijfj(zj(t))z˙i(t)=+j=1nbijfj(zj(tτj))+Ji,t0,ttk,Δzi(tk)=Iik(zi(tk)),k=1,2,,i=1,2,,n,
1

where n is the number of neurons, zi(t) ∈ ℂ denotes the state of neuron i at time t, fj(t) is the complex-valued activation function, τj (j = 1, 2, …, n) are constant time delays and satisfy 0 ≤ τj ≤ ρ, ci ∈ ℝ with ci > 0 is the self-feedback connection weight, aij ∈ ℂ and bij ∈ ℂ are the connection weights, and Ji ∈ ℂ is the external input. Here Iik is a linear map, Δzi(tk)=zi(tk+)zi(tk) is the jump of zi at moments tk, and 0 < t1 < t2 <  ⋯  is a strictly increasing sequence such that limk→∞tk =  + ∞.

We rewrite (1) in the equivalent matrix-vector form

{z˙(t)=Cz(t)+Af(z(t))+Bf(z(tτ))+J,Δz(tk)=I(z(tk)),k=1,2,,
2

where z(t) = (z1(t),z2(t),…,zn(t))T ∈ ℂn, C = diag(c1c2, …, cn), A = (aij)n×n ∈ ℂn×n, B = (bij)n×n ∈ ℂn×n, f(z(t)) = (f1(z1(t)),f2(z2(t)),…,fn(zn(t)))T, f(z(t − τ)) = (f1(z1(tτ1)),f2(z2(tτ2)),…,fn(zn(tτn)))T, J = (J1,J2,…,Jn)T ∈ ℂn, Δz(tk) = (Δz1(tk),Δz2(tk),…,Δzn(tk))T, and I(z(tk))=(I1k(z(tk)),I2k(z(tk)),,Ink(z(tk)))T.

Assume that system (1) or (2) is supplemented with the initial values given by

zi(s) = φi(s),  s ∈ [−ρ, 0], i = 1, 2, …, 
3

or in the equivalent vector form

z(s) = φ(s),  s ∈ [−ρ, 0], 
4

where φi( ⋅ ) is a complex-valued continuous function defined on [−ρ, 0] and φ(s) = (φ1(s),φ2(s),…,φn(s))T ∈ C([−ρ, 0], ℂn) with the norm φ(s)=sups[ρ,0]i=1n|φi(t)|2.

The following assumptions will be needed in the study:

(H1) The parameters C = diag(c1c2, …, cn), A = (aij)n×n, B = (bij)n×n, and J = (J1,J2,…,Jn)T in neural system (1) are assumed to be norm-bounded and satisfy

CI=[C_,C]={C=diag(c1,c2,,cn):0<c_icici,i=1,2,,n},AI=[A_,A]={A=(aij)n×n:a_ijRaijRaijR,a_ijIaijIaijI,i,j=1,2,,n},BI=[B_,B]={B=(bij)n×n:b_ijRbijRbijR,b_ijIbijIbijI,i,j=1,2,,n},JI=[J_,J]={J=(J1,J2,,Jn)T:J_iRJiRJiR,J_iIJiIJiI,i=1,2,,n},

where aij=aijR+iaijI, bij=bijR+ibijI, Ji=JiR+iJiI, C_=diag(c_1,c_2,,c_n), C=diag(c1,c2,,cn), A_=(a_ij)n×n, A=(aij)n×n, B_=(b_ij)n×n, B=(bij)n×n, J_=(J_1,J_2,,J_n)T, J=(J1,J2,,Jn)T with a_ij=a_ijR+ia_ijI, aij=aijR+iaijI, b_ij=b_ijR+ib_ijI, bij=bijR+ibijI, J_i=J_iR+iJ_iI, and Ji=JiR+iJiI.

(H2) For i = 1, 2, …, n, the neuron activation function fi is continuous and satisfies

|fi(z1)−fi(z2)| ≤ γi|z1 − z2|, 

for any z1z2 ∈ ℂ, where γi is a real constant. Furthermore, define Γ = diag(γ1γ2, …, γn).

Definition 1

A function z(t) ∈ C([τ,  + ∞), ℂn) is a solution of system (1) satisfying the initial value condition (3), if the following conditions are satisfied:

  • (i)
    z(t) is absolutely continuous on each interval (tktk+1) ⊂ [−τ,  + ∞), k = 1, 2, … ,
  • (ii)
    for any tk ∈ [0,  + ∞), k = 1, 2, … , z(tk+) and (z(tk)) exist and z(tk+)=z(tk).

Definition 2

The neural network defined by (1) with the parameter ranges defined by (H1) is globally asymptotically robust stable if the unique equilibrium point zˇ=(zˇ1,zˇ2,,zˇn)T of the neural system (1) is globally asymptotically stable for all C ∈ CI, A ∈ AI, B ∈ BI, and J ∈ JI.

Lemma 1

[10]

For any ab ∈ ℂn, if P ∈ ℂn×n is a positive definite Hermitian matrix, then abba ≤ aPabP−1b.

Lemma 2

See [10]

A given Hermitian matrix

S=(S11S12S21S22)<0,

where S11=S11, S12=S21, and S22=S22, is equivalent to any of the following conditions:

  • (i)
    S22 < 0 and S11S12S221S21<0,
  • (ii)
    S11 < 0 and S22S21S111S12<0.

Lemma 3

[10]

If H(z):ℂn → ℂn is a continuous map and satisfies the following conditions:

  • (i)
    H(z) is injective on n,
  • (ii)
    limz∥→∞ ∥ H(z) ∥  = ∞,

then H(z) is a homeomorphism of n onto itself.

Lemma 4

Suppose A ∈ AI. Let R and S be real positive diagonal matrices. The function fi (i = 1, 2, …, n) satisfies (H2). Then, for any z=(z1,z2,,zn)T,z˜=(z˜1,z˜2,,z˜n)TCn, the following inequalities hold:

zAAz|z|AˆAˆ|z|,
5

zRASARz|z|RAˆSAˆR|z|,
6

(f(z)f(z˜))AA(f(z)f(z˜))|zz˜|ΓAˆAˆΓ|zz˜|,
7

where Aˆ=(aˆij)n×n, aˆij=max{|a_ij|,|aij|}, and f(z) = (f1(z1),f2(z2),…,fn(zn))T.

Proof

It should be noted that |aij|aˆij since A ∈ AI. Then we calculate directly that

zAAz=i=1n|j=1naijzj|2i=1n(j=1n|aij||zj|)2i=1n(j=1naˆij|zj|)2=|z|AˆAˆ|z|.

Hence inequality (5) holds.

Next we prove inequality (6). Let S = diag(s1s2, …, sn) and S˜=diag(s1,s2,,sn). Then S=S˜2. It is obvious that |Rz| = R|z| since R is a real positive diagonal matrix. From A ∈ AI, it follows that a_ijRaijRaijR and a_ijIaijIaijI for all ij = 1, 2, …, n. Then sia_ijRsiaijRsiaijR and sia_ijIsiaijIsiaijI, which means S˜AS˜AI. Hence siaˆij=max{|sia_ij|,|siaij|}. Noting that S˜Aˆ=(siaˆij)n×n, by inequality (5), we infer

zRASARz=(Rz)(S˜A)(S˜A)(Rz)|Rz|(S˜Aˆ)(S˜Aˆ)|Rz|=|z|RAˆS˜S˜AˆR|z|=|z|RAˆSAˆR|z|.

Therefore, inequality (6) holds.

Next we prove inequality (7). For simplicity, let wi=ziz˜i, gi=fi(zi)fi(z˜i) (i = 1, 2, …, n), w = (w1,w2,…,wn)T, and g = (g1,g2,…,gn)T. Then |gi| ≤ γi|wi| due to assumption (H2). So we calculate directly that

gAAg=i=1n|j=1naijgj|2i=1n(j=1n|aij||gj|)2i=1n(j=1nγjaˆij|wj|)2=|w|ΓAˆAˆΓ|w|.

Accordingly, inequality (7) holds. The proof is completed.

Existence and uniqueness of equilibrium point

In this section, we will give the sufficient conditions to prove the existence and uniqueness of equilibrium for system (1). An equilibrium solution of (1) is a constant complex vector zˇCn, which satisfies

Czˇ+Af(zˇ)+Bf(zˇ)+J=0
8

and Ik(zˇ)=0, k = 1, 2, … , when ž is the impulsive jump.

Hence, proving the existence and uniqueness of solution (8) is equivalent to proving the existence of a unique zero point of map ℋ:ℂn → ℂn, where

ℋ(z) = −CzAf(z) + Bf(z) + J.
9

We have the following theorem.

Theorem 1

For the CVNN defined by (1), assume that the network parameters and the activation function satisfy assumptions (H1) and (H2), respectively. Then the neural network (1) has a unique equilibrium point for every input vector J = (J1,J2,…,Jn)T ∈ ℂn, if there exist two real positive diagonal matrices U and V such that the following linear matrix inequality (LMI) holds:

(2UC_+ΓVΓU(Aˆ+Bˆ)(Aˆ+Bˆ)UV)<0,
10

where Aˆ=(aˆij)n×n, Bˆ=(bˆij)n×n, aˆij=max{|a_ij|,|aij|}, and bˆij=max{|b_ij|,|bij|}.

Proof

We will use the homeomorphism mapping theorem on the complex domain to prove the theorem, that is, to show the map ℋ(z) is a homeomorphism of n onto itself.

First, we prove that ℋ(z) is an injective map on n. Let z,z˜Cn with zz˜, such that H(z)=H(z˜). Then

H(z)H(z˜)=C(zz˜)+(A+B)(f(z)f(z˜))=0.
11

Multiplying both sides of (11) by (zz˜)U, we obtain

0=(zz˜)UC(zz˜)+(zz˜)U(A+B)(f(z)f(z˜)).
12

Then taking the conjugate transpose of (12) leads to

0=(zz˜)CU(zz˜)+(f(z)f(z˜))(A+B)U(zz˜).
13

From (12), (13) and Lemmas 1 and 4, we have

0=(zz˜)(UC+CU)(zz˜)+(zz˜)U(A+B)(f(z)f(z˜))+(f(z)f(z˜))(A+B)U(zz˜)(zz˜)(UC+CU)(zz˜)+(zz˜)U(A+B)V1(A+B)U(zz˜)+(f(z)f(z˜))V(f(z)f(z˜))|zz˜|[2UC_+U(Aˆ+Bˆ)V1(Aˆ+Bˆ)U]|zz˜|+(f(z)f(z˜))V(f(z)f(z˜)).
14

Since V is a positive diagonal matrix, from assumption (H2) we get

(f(z)f(z˜))V(f(z)f(z˜))(zz˜)ΓVΓ(zz˜)=|zz˜|ΓVΓ|zz˜|.
15

It follows from (14) and (15) that

0|zz˜|Ω|zz˜|,
16

where Ω=2UC_+ΓVΓ+U(Aˆ+Bˆ)V1(Aˆ+Bˆ)U. From Lemma 2 and the LMI (10), we know Ω < 0. Then zz˜=0 due to (16). Therefore, ℋ(z) is an injective map on n.

Secondly, we prove  ∥ ℋ(z) ∥  → ∞ as  ∥ z ∥  → ∞. Let H˜(z)=H(z)H(0). By Lemmas 1 and 4, we have

zUH˜(z)+H˜(z)Uz=z(UC+CU)z+zU(A+B)(f(z)f(0))+(f(z)f(0))(A+B)Uzz(UC+CU)z+zU(A+B)V1(B+C)Uz+(f(z)f(0))V(f(z)f(0))|z|[2UC_+U(Aˆ+Bˆ)V1(Aˆ+Bˆ)U]|z|+|z|ΓVΓ|z||z|Ω|z|λmin(Ω)z2.

An application of the Cauchy-Schwarz inequality yields

λmin(Ω)z22zUH˜(z).

When z ≠ 0, we have

H˜(z)λmin(Ω)z2U.

Therefore, H˜(z) as  ∥ z ∥  → ∞, which implies  ∥ ℋ(z) ∥  → ∞ as  ∥ z ∥  → ∞. We know that ℋ(z) is a homeomorphism of n from Lemma 3, thus system (1) has a unique equilibrium point.

Global robust stability results

In this section, we will investigate the global robust stability of the unique equilibrium point for system (1). Firstly, the following assumption for the impulsive operators is needed: (H3) For i = 1, 2, …, n and k = 1, 2, … , Iik( ⋅ ) is such that

Iik(zi(tk))=δik(zi(tk)zˇi),

where δik ∈ [0, 2] is a real constant, and zˇi is the ith component of the equilibrium point zˇ=(zˇ1,zˇ2,,zˇn)T. Then we have the following global robust stability theorem.

Theorem 2

Suppose the conditions of Theorem  1 and (H3) hold. The equilibrium point of system (1) is globally robust stable, if there exist two real positive diagonal matrices P = diag(p1p2, …, pn) and Q = diag(q1q2, …, qn), such that the following linear matrix inequalities hold:

(C_P+ΓAˆAˆΓPPI)<0
17

and

(PC_+ΓQΓPBˆBˆPQ)<0,
18

where Aˆ=(aˆij)n×n, Bˆ=(bˆij)n×n, aˆij=max{|a_ij|,|aij|}, and bˆij=max{|b_ij|,|bij|}.

Proof

By Lemma 2, it follows from the LMI (17) that the following condition holds:

C_P+ΓAˆAˆΓ+PP<0.
19

By the LMI (18), according to Lemma 2, the following condition holds:

PC_+ΓQΓ+PBˆQ1BˆP<0.
20

Summing (19) and (20), we have the following matrix inequality:

C_PPC_+PP+ΓAˆAˆΓ+PBˆQ1BˆP+ΓQΓ<0.
21

Under the conditions of Theorem 1, system (2) has a unique equilibrium point ž. For convenience, we shift the equilibrium to the origin by letting z˜(t)=z(t)zˇ, and then system (2) can be transformed into

{z˜˙(t)=Cz˜(t)+Ag(z˜(t))+Bg(z˜(tτ)),Δz˜(t)=I˜(z˜(tk)),k=1,2,,
22

where g(z˜(t))=f(z(t))f(zˇ) and I˜(z˜(tk))=δikz˜i(tk). Meanwhile, the initial condition (4) can be transformed into

z˜(s)=φ˜(s),s[ρ,0],
23

where φ˜(s)=φ(s)zˇC([ρ,0],Cn).

Consider the following Lyapunov-Krasovskii functional candidate:

V(z˜(t))=V1(z˜(t))+V2(z˜(t)),
24

where

V1(z˜(t))=j=1npjz˜j(t)z˜j(t),
25

V2(z˜(t))=j=1nqjtτjtgj(z˜j(t))gj(z˜j(t))dt.
26

When t ≠ tk, k = 1, 2, … , calculating the upper right derivative of V along the solution of (22), applying Lemmas 1 and 4, we get

D+V1(z˜(t))=z˜˙(t)Pz˜(t)+z˜(t)Pz˜˙(t)=z˜(t)CPz˜(t)z˜(t)PCz˜(t)+g(z˜(t))APz˜(t)+z˜(t)PAg(z˜(t))+g(z˜(tτ))BPz˜(t)+z˜(t)PBg(z˜(tτ))z˜(t)(CP+PC)z˜(t)+g(z˜(t))AAg(z˜(t))+z˜(t)PPz˜(t)+g(z˜(tτ))Qg(z˜(tτ))+z˜(t)PBQ1BPz˜(t)|z˜(t)|(CPPC+PP+ΓAˆAˆΓ+PBQ1BP)|z˜(t)|+g(z˜(tτ))Qg(z˜(tτ))|z˜(t)|(C_PPC_+PP+ΓAˆAˆΓ+PBˆQ1BˆP)|z˜(t)|+g(z˜(tτ))Qg(z˜(tτ)),
27

D+V2(z˜(t))=g(z˜(t))Qg(z˜(t))g(z˜(tτ))Qg(z˜(tτ))|z˜(t)|ΓQΓ|z˜(t)|g(z˜(tτ))Qg(z˜(tτ)).
28

Combining (27) and (28), by (21) we deduce that

D+V(z˜(t))|z˜(t)|(C_PPC_+PP+ΓAˆAˆΓ+PBˆQ1BˆP+ΓQΓ)|z˜(t)|0.
29

When ttk, k = 1, 2, … , it should be noted that V2(tk)=V2(tk). Then we compute

V(z˜(tk))V(z˜(tk))=j=1npjz˜j(tk)z˜j(tk)j=1npjz˜j(tk)z˜j(tk)=j=1n(1δjk)2pjz˜j(tk)z˜j(tk)j=1npjz˜j(tk)z˜j(tk)0.
30

It follows from (29) and (30) that V(t) is non-increasing for t ≥ 0. Then, by the definition of V(t), we infer

V(t)V(0)=j=1npjz˜j(0)z˜j(0)+j=1nqjτj0gj(z˜j(t))gj(z˜j(t))dtj=1npj|φ˜j(0)|2+j=1nqjγj2τj0|φ˜j(t)|2dtj=1n(pj+ρqjγj2)supt[ρ,0]j=1n|φ˜j(t)|2=j=1n(pj+ρqjγj2)φ˜(t)2.
31

On the other hand, by the definition of V(t), we have

V(t)V1(t)j=1npjz˜(t)2,t0.
32

From (31) and (32), we obtain

z˜(t)j=1n(pj+τqjγj2)j=1npjφ˜(t),

from which it can be concluded that the origin of (22), or equivalently the equilibrium point of system (1), is globally asymptotically robust stable by the standard Lyapunov theorem. The proof is completed.

If the impulsive operator I( ⋅ ) ≡ 0 in (2), we get the following CVNN without impulses:

z˙(t)=Cz(t)+Af(z(t))+Bf(z(tτ))+J,
33

where C, A, B, J, and f( ⋅ ) are defined the same as in (2). Following Theorem 2, we obtain the following corollary on the global robust stability conditions of (33).

Corollary 1

Under the conditions of Theorem  1, the equilibrium point of system (33) is globally asymptotically robust stable, if there exist two real positive diagonal matrices P = diag(p1p2, …, pn) and Q = diag(q1q2, …, qn), such that the following linear matrix inequalities hold:

(C_P+ΓAˆAˆΓPPI)<0

and

(PC_+ΓQΓPBˆBˆPQ)<0,

where Aˆ=(aˆij)n×n, Bˆ=(bˆij)n×n, aˆij=max{|a_ij|,|aij|}, and bˆij=max{|b_ij|,|bij|}.

Remark 1

In [9, 13], some dynamic characteristics, such as exponential stability and exponential anti-synchronization, were investigated for real-valued neural networks. Compared to [9, 13], the neural networks model in this paper is complex-valued, which can be viewed as an extension of real-valued neural networks.

Remark 2

In [33, 34], the criteria for the stability of CVNNs are expressed in terms of complex-valued LMIs. As pointed out in [33], complex-valued LMIs cannot be solved by the MATLAB LMI Toolbox straightforwardly. A feasible approach is to convert complex-valued LMIs to real-valued ones but this could double the dimension of the LMIs. In this paper, we express the stability criteria for CVNNs directly in terms of real-valued LMIs, which can be solved by the MATLAB LMI Toolbox straightforwardly.

Remark 3

In [2], the authors investigated the problem of global robust stability of recurrent CVNNs with time delays and uncertainties. In Theorem 3.4 of [2], to check robust stability of CVNNs, the boundedness of activation function fi is required. However, in this paper, the boundedness condition is removed. In Example 1, in the next section, the activation function fi is unbounded.

A numerical example

The following example demonstrates the effectiveness and superiority of our results.

Example 1

Assume that the network parameters of system (1) are given as follows:

C_=(0.3000.3),A_=(0.320.24i0.180.24i0.240.18i0.240.32i),A=(0.32+0.24i0.18+0.24i0.24+0.18i0.24+0.32i),B_=(0.240.18i0.16+0.12i0.180.24i0.240.18i),B=(0.24+0.18i0.12+0.16i0.18+0.24i0.24+0.18i),Γ=(0.2000.2),

and the impulsive operator I( ⋅ ) satisfies assumption (H3).

Using the above matrices A_, A, B_, and B, we have

Aˆ=(0.40.30.30.4),Bˆ=(0.30.20.30.3).

Then using YALMIP with the solver of LMILAB, the LMI (10) in Theorem 1, and the LMIs (17) and (18) in Theorem 2, we have the following feasible solutions:

U=(3.7733003.3139),V=(3.7733003.3139),P=(0.1854000.1833),Q=(0.8824000.8199).

Thus, the conditions of Theorems 1 and 2 are satisfied, and system (1) has a unique equilibrium point which is globally asymptotically robust stable. To simulate the results, let us choose C, A, and B from the proper intervals above, and obtain the following specific system:

{(z˙1(t)z˙2(t))=(0.3000.3)(z1(t)z2(t))+(0.30.2i0.15+0.2i0.20.1i0.2+0.3i)(f1(z1(t))f2(z2(t)))(z˙1(t)z˙2(t))=+(0.2+0.15i0.1+0.15i0.180.24i0.2+0.15i)(f1(z1(tτ1))f2(z2(tτ2)))(z˙1(t)z˙2(t))=+(0.102i0.20.1i),ttk,(Δz1(tk)Δz2(tk))=(δ1k[z1(tk)(0.41701.1278i)]δ2k[z1(tk)(0.48630.4654i)]),t=tk,k=1,2,,
34

where f1(u) = f2(u) = 0.2(eu − 1), δ1k = 1 + ½sin(1 + k), δ2k = 1 + ⅔cos(2k3), k = 1, 2, … , and t1 = 0.5, tktk−1 + 0.2k, k = 2, 3, … .

Figures 1 and and22 depict the real and imaginary parts of states of the considered system (34) with τ1τ2 = 0.5, where the initial conditions are with 10 random initial complex-valued points.

Figure 1
Real part of the state trajectories for system (34) with τ1τ2 = 0.5 .
Figure 2
Imaginary part of the state trajectories for system (34) with τ1τ2 = 0.5 .

Figures 3 and and44 depict the real and imaginary parts of states of the considered system (34) with τ1τ2 = 8, where the initial conditions are with 10 random initial complex-valued points.

Figure 3
Real part of the state trajectories for system (34) with τ1τ2 = 8 .
Figure 4
Imaginary part of the state trajectories for system (34) with τ1τ2 = 8 .

Remark 4

In Figures 1--4,4, we see that the equilibrium point of system (34) is asymptotically stable, whether the delay τ1τ2 = 0.5 or τ1τ2 = 8. It should be noted that the criteria (10), (17), and (18) in Theorems 1 and 2 are independent from the delays τ. Therefore, in system (34), the delays have no influence on the stability of the equilibrium point.

Conclusion

In this paper, we have investigated the existence and uniqueness of the equilibrium as well as its robust stability for an impulsive CVNN with discrete time delays, by applying the homeomorphic mapping theorem and some important inequalities in the complex domain. We have presented some sufficient conditions to guarantee the existence of a unique equilibrium point for the CVNN. In addition, by constructing appropriate Lyapunov-Krasovskii functionals and employing complex-valued matrix inequalities, we also have obtained sufficient conditions to guarantee the robust stability of the CVNN. Finally, a numerical simulation has illustrated the correctness of the proposed theoretical results. Moreover, the conditions in Theorems 1 and 2 are irrelevant to the parameter τ, which illustrates that the parameter τ has no effect on the uniqueness and existence, neither on the robust stability of system (1). The figures in the article confirm this result.

Footnotes

Funding

This work is supported by the National Natural Science Foundation of China under Grants (NSFCs:11631012, 11401060), and the Program of Chongqing Innovation Team Project in University under Grant (CXTDX201601022).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors conceived the study, participated in its design and coordination and read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Contributor Information

Yuanshun Tan, nc.ude.utjqc@synat.

Sanyi Tang, nc.ude.unns@gnatys.

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