Dynamics of complex-valued neural networks with variable coefficients and proportional delays

Communicated by Dr Hu Jun

Accepted Manuscript

Dynamics of complex-valued neural networks with variable coefficients and proportional delays Qiankun Song, Qinqin Yu, Zhenjiang Zhao, Yurong Liu, Fuad E. Alsaadi PII: DOI: Reference:

S0925-2312(17)31800-3 10.1016/j.neucom.2017.11.041 NEUCOM 19106

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

27 August 2017 5 November 2017 19 November 2017

Please cite this article as: Qiankun Song, Qinqin Yu, Zhenjiang Zhao, Yurong Liu, Fuad E. Alsaadi, Dynamics of complex-valued neural networks with variable coefficients and proportional delays, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.11.041

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ACCEPTED MANUSCRIPT 1

Dynamics of complex-valued neural networks with variable coefficients and proportional delays Qiankun Song1,∗ , Qinqin Yu2 , Zhenjiang Zhao3 , Yurong Liu4,5 , Fuad E. Alsaadi5 1 2

Department of Mathematics, Chongqing Jiaotong University, Chongqing 400074, China

School of Economic and Management, Chongqing Jiaotong University, Chongqing 400074, China

5

Department of Mathematics, Huzhou University, Huzhou 313000, China

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3 4

Department of Mathematics, Yangzhou University, Yangzhou 225002, China

Communication Systems and Networks (CSN) Research Group, Faculty of Engineering,

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King Abdulaziz University, Jeddah 21589, Saudi Arabia

Abstract

In this paper, the dynamics including boundedness and stability for a general class of complex-valued neural networks with variable coefficients and proportional delays are investigated. By employing inequality techniques and mathematical analysis method, some sufficient criteria to guarantee boundedness and global exponential stability are established for the considered neural networks. As a special case that the coefficients of networks are constants,

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sufficient criteria are also derived to guarantee the existence, uniqueness and global exponential stability of the equilibrium point. This work generalizes and improves previously known results, and the obtained criteria can be

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tested and applied easily in practice. An illustrative example demonstrates the feasibility of the proposed results.

Keywords: Complex-valued neural networks; boundedness; stability; equilibrium point; variable coefficient; pro-

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portional delays

I. Introduction

The nonlinear systems are everywhere in the real world [1]- [8], while complex networks are a class of nonlinear

CE

systems [9]- [10]. As one of the most important complex networks, neural networks have been paid much attentions over the past three decades by reason of their applicability in many fields such as signal processing, image processing,

AC

pattern recognition, associative memory and optimization [11]. In these applications, stability is a very crucial topic [12]. When implementing neural networks, time delay is ubiquitous due to finite switching speeds of the amplifiers, and often becomes sources of oscillation or instability [13]. Therefore, it is necessary to analyze the stability of the neural networks with time delays. Many interesting papers on stability of neural networks with time delays have been published, for example, see [11]- [20] and the references therein. In [11]- [13], neural networks with constant delay were investigated, and several sufficient criteria for checking stability were provided. Authors in [14]- [16] took into account time-varying delays in neural networks and supplied some sufficient conditions to assure stability of networks. In addition, in [17]- [19], the authors studied the stability of neural networks with *E-mail address of author: [email protected] (Q. Song)

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distributed delays. Further, mixed time delays, including time-varying delays and distributed delays were introduced into neural networks [20], and sufficient criterion on stability was acquired for neural networks with mixed time delays. On the other hand, delays may occur in the form of proportion [21]. As a kind of many time delay types, the proportional delay may occur in some situations. For instance, in Web quality of service routing decision, the proportional delay is demanded and it would be advantageous to control the running time of network according to the network allowed delays [22]. Different from constant delay, bounded time-varying delays and unbounded

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distributed delay, the proportional delay τ(t) = (1 − q)t is unbounded time-varying delay, where q is a constant and satisfies 0 < q < 1. Besides, it is a monotonically increasing function along the time t > 0. Consequently, it is

valuable to research stability of the neural networks with proportional delays. Recent years, the stability of neural networks with proportional delays has been probed which can be seen in [23]- [27].

The neural networks studied above is called as real-valued neural networks(RVNNs) since the states, connection weights, activation functions and external input in networks are all real-valued. Although RVNNs have been applied

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in many fields, they have limitations [28]. For instance, in applications such as optoelectronics, speech synthesis, and information flow, we need to deal with the complex-valued signals [29]. Thus complex-valued neural network arises at the historic moment [30]. As a general extension of RVNNs, complex-valued neural networks (CVNNs) including the complex-valued states, connection weights, activation functions and external input have more complex properties than RVNNs [31]. In recent years, CVNNs with constant delay [29]- [33], CVNNs with time-varying delays [34]- [37] and CVNNs with mixed time delays including time-varying delays and distributed delays [38]-

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[41] were discussed, and the stability for the considered CVNNs was researched. As far as we know, up to now, there are few papers published on the stability of CVNNs with proportional delays.

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Based on the above analysis, it is significant to deliberate on the dynamics for CVNNs with proportional delays. This paper aims at studying the boundedness and global exponential stability of CVNNs with variable coefficients and proportional delays.

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Notations: In this paper, i stands for the imaginary unit, that is, i =

√

−1. Let z = x + iy, then |z| =

p

x2 + y2 denotes

the module of z. E shows the identity matrix with corresponding dimensions. C, Cn and Cn×m represents the the set

CE

of all complex numbers, the set of all n-dimensional complex-valued vectors and the set of all n×m complex-valued matrices, respectively. For a complex-valued matrix A, A denotes the conjugate of A and A∗ shows the conjugate transpose of A. For a complex-valued vector u = (u1 , u2 , · · · , un )T ∈ Cn , |u| represents for the module vector provided

AC

by |u| = (|u1 |, |u2 |, · · · , |un |)T , kuk denotes the ∞-norm of u. Given a complex-valued matrix A = (ai j )n×n ∈ Cn×n , |A| stands for the module matrix provided by |A| = (|ai j |)n×n . ρ(A) is the spectral radius of matrix A. II. Preliminaries and model description

The following CVNNs with variable coefficients and proportional delays are considered in this paper. Z t n n n X X X z˙i (t) = −di (t)zi (t + ai j (t)h j (z j (t)) + bi j (t)h j (z j (qi j t)) + ci j (t) θi j (t − s)h j (z j (s))ds + Ji (t) j=1

j=1

j=1

(1)

−∞

for t ≥ 0, i = 1, 2, · · · , n, where zi (t) ∈ C denotes the i-th state variable at time t; h j (z j (t)) ∈ C is the activation function at time t; qi j (i, j = 1, 2, · · · , n) is a proportional delay factor and satisfies 0 < qi j < 1, qi j t = t − (1 − qi j )t, in which

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(1 − qi j )t is a time-varying delay function. D(t) = diag{d1 (t), d2 (t), · · · , dn (t)} ∈ Rn×n is the self-regulating parameter

connection matrix, and di (t) > 0; A(t) = (ai j (t))n×n ∈ Cn×n , B(t) = (bi j (t))n×n ∈ Cn×n and C(t) = (ci j (t))n×n ∈ Cn×n are the connection weight matrices; J(t) = (J1 (t), J2 (t), · · · , Jn (t))T ∈ Cn represents for the input vector; θi j : [0, +∞) → [0, +∞) is the delay kernel function.

The initial condition for system (1) is in the form of zi (s) = ϕi (s), s ∈ (−∞, 0], where ϕi are continuous and

bounded on (−∞,0].

Remark 1. Model (1) is a very general model. For example, when di (t), ai j (t), bi j (t), ci j (t) and Ji (t) (i, j = 1, 2, · · · , n) (2)

(i, j = 1, 2, · · · , n) are real-valued constants, model (2) turns into the RVNNs model as following n n X X x˙i (t) = −di xi (t) + ai j h j (x j (t)) + bi j h j (x j (qi j t)) + Ji .

(3)

j=1

j=1

j=1

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are complex-valued constants, model (1) can be expressed as following CVNNs model Z t n n n X X X z˙i (t) = −di zi (t) + ai j h j (z j (t)) + bi j h j (z j (qi j t)) + ci j θi j (t − s)h j (z j (s))ds + Ji . −∞

j=1

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To the best of our knowledge, model (2) has not been considered. Further, when ci j = 0, and di , ai j , bi j and Ji

j=1

The initial condition for system (3) is in the form of xi (s) = ϕi (s), s ∈ [ min {qi j }, 1], where ϕi are continuous and 1≤i≤n

bounded on [ min {qi j }, 1]. Model (3) has been extensively studied in [21]- [26]. 1≤i≤n

To obtain the main result, the following assumptions are established.

(H1). There exists a positive diagonal matrix L = diag{l1 , l2 , · · · , ln } satisfying

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|hi (u1 ) − hi (u2 )| ≤ li |u1 − u2 | for all u1 , u2 ∈ C, i = 1, 2, · · · , n.

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(H2). The delay kernel θi j : [0, +∞) → [0, +∞) is real-valued nonnegative continuous function and meets Z +∞ eβs θi j (s)ds = ζi j (β), 0

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where ζi j (β) is continuous function in [0,δ), δ > 0, and ζi j (0) = 1, i, j ∈ {1, 2, · · · , n}.

(H3). The variable coefficients di (t), ai j (t), bi j (t), ci j (t) and Ji (t) are bounded for i, j ∈ {1, 2, · · · , n}, t ≥ 0.

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Definition 1. Model (1) is said to be bounded if for any i = 1, 2, · · · , n, there is positive constant Mi > 0 satisfying |zi (t)| ≤ Mi

AC

for t ≥ 0, where z(t) = (z1 (t), z2 (t), · · · , zn (t))T is any solution of model (1).

Definition 2. Model (1) is globally exponentially stable if there exist constants ε > 0 and α > 0 satisfying kz(t) − z˜(t)k ≤ αkϕ − ψke−εt

for t ≥ 0, where z(t) = (z1 (t), z2 (t), · · · , zn (t))T and z˜(t) = (˜z1 (t), z˜2 (t), · · · , z˜n (t))T are any solutions of model (1)

under initial conditions ϕ(s) = (ϕ1 (s), · · · , ϕn (s))T and ψ(s) = (ψ1 (s), · · · , ψn (s))T , s ∈ (−∞, 0] and kϕ − ψk = max sup |ϕi (s) − ψi (s)|.

1≤i≤n s∈(−∞,0]

Definition 3. The equilibrium point z˜ = (˜z1 , z˜2 , · · · , z˜n )T of system (2) is globally exponentially stable, if there

exist constants ε > 0 and α > 0 satisfying

kz(t) − z˜k ≤ αkϕ − z˜ke−εt

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for all t ≥ 0, where z(t) = (z1 (t), z2 (t), · · · , zn (t))T is one arbitrary solution of model (2) under initial condition

ϕ(s) = (ϕ1 (s), · · · , ϕn (s))T , s ∈ (−∞, 0], kϕ − z˜k = max sup |ϕi (s) − z˜i |. 1≤i≤n s∈(−∞,0]

III. Main results In this section, we will establish several sufficient criteria to guarantee the stability of CVNNs with proportional delays. for j = 1, 2, · · · , n, such that max sup

n X ξ j l j (|ai j (t)| + |bi j (t)| + |ci j (t)|)

ξi di (t)

1≤i≤n t≥0 j=1

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Theorem 1. Assume that the (H1)-(H3) hold, model (1) is bounded if there exist n constants ξ j where ξ j > 0

< 1.

(4)

Proof. Suppose that z(t) = (z1 (t), z2 (t), · · · , zn (t))T is any a solution of model (1) under initial condition ϕ(s) =

(ϕ1 (s), · · · , ϕn (s))T . Let

then v˙ i (t) =

t ≥ 0,

i = 1, 2, · · · , n,

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vi (t) = |zi (t)|,

    1 1 z˙i (t)zi (t) + zi (t)˙zi (t) = Re z˙i (t)zi (t) . q |zi (t)| 2 zi (t)zi (t)

Based on assumption (H1), we get that

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|hi (zi (t))| ≤ li |zi (t)| + |hi (0)| It is easy to compute that

(5)

(6)

(7)

ED

n n X     X   Re z˙i (t)zi (t) = −di (t)|zi (t)|2 + Re ai j (t)h j (z j (t))zi (t) + Re bi j (t)h j (z j (qi j t))zi (t) j=1

j=1

Z  Re ci j (t)

PT

+

n X j=1

2

CE

≤ −di (t)|zi (t)| + +

n X

AC

j=1

|ci j (t)

Z

t

−∞

n X j=1

   θi j (t − s)h j (z j (s))zi (t)ds + Re Ji (t)zi (t)

|ai j (t)h j (z j (t))zi (t)| +

t

−∞

n X j=1

|bi j (t)h j (z j (qi j t))zi (t)|

θi j (t − s)h j (z j (s))zi (t)ds| + |Ji (t)zi (t)|

n n X X h ≤ |zi (t)| − di (t)|zi (t)| + |ai j (t)||h j (z j (t))| + |bi j (t)||h j (z j (qi j t))|

+

n X j=1

|ci j (t)|

Z

j=1

t

−∞

j=1

i θi j (t − s)|h j (z j (s))|ds + |Ji (t)|

n n X X h ≤ |zi (t)| − di (t)|zi (t)| + l j |ai j (t)||z j (t)| + l j |bi j (t)||z j (qi j t)|

+

n X j=1

l j |ci j (t)|

Z

j=1

j=1

t

−∞

θi j (t − s)|z j (s)|ds + |Ji (t)| +

n X j=1

i (|ai j (t)| + |bi j (t)| + |ci j (t)|)|h j (0)| .

(8)

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In the derivation of (8), inequality (7) and assumption (H2) were been used. Thus Z t n n n X X X v˙ i (t) ≤ −di (t)|zi (t)| + l j |ai j (t)||z j (t)| + l j |bi j (t)||z j (qi j t)| + l j |ci j (t)| θi j (t − s)|z j (s)|ds + Ni (t) = −di (t)vi (t) + n P

j=1

j=1

l j |ai j (t)|v j (t) +

j=1 n X

l j |bi j (t)|v j (qi j t) +

j=1

j=1 n X j=1

l j |ci j (t)|

Z

−∞

t

−∞

θi j (t − s)v j (s)ds + Ni (t),

(9)

(|ai j (t)| + |bi j (t)| + |ci j (t)|)|h j (0)|.

From condition (4), we have that n X j=1

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where Ni (t) = |Ji (t)| +

j=1 n X

ξ j l j (|ai j (t)| + |bi j (t)| + |ci j (t)|) < ξi di (t)

for t ≥ 0, i ∈ {1, 2, · · · , n}. Hence, there is a sufficiently small positive constant λ > 0 such that n X ξ j l j (|ai j (t)| + |bi j (t)| + |ci j (t)|) < ξi (di (t) − λ) j=1

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for t ≥ 0, i ∈ {1, 2, · · · , n}.

(10)

(11)

According to (H3), we can know Ni (t) is bounded nonnegative function on [0, +∞). Let Ni = sup Ni (t), and

R=

(1+δ)kϕk min {ξi } 1≤i≤n

t≥0

(δ is a positive constant), then we can choose a sufficiently large constant W ≥ 1 satisfying Ni ≤W λξi R

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for i ∈ {1, 2, · · · , n}. Obviously,

(12)

vi (s) = |zi (s)| = |ϕi (s)| ≤ kϕk ≤ ξi R ≤ ξi RW = ξi γ

(13)

ED

for s ∈ (−∞, 0], i = 1, 2, · · · , n, where γ = RW. As for next, we will prove that vi (t) ≤ ξi γ,

t ≥ 0,

i = 1, 2, · · · , n.

(14)

CE

PT

As a matter of fact, if (14) is not true, then there would be several i0 and t0 > 0 satisfying    vi0 (t0 ) = ξi0 γ,      v˙ i0 (t0 ) ≥ 0,        v j (t) ≤ ξ j γ, t ∈ (−∞, t0 ], j = 1, 2, · · · , n.

AC

We obtain from (9), (11), (12) and (15) that Z n n n X X X v˙ i0 (t0 ) ≤ −di0 (t0 )vi0 (t0 ) + l j |ai0 j (t0 )|v j (t0 ) + l j |bi0 j (t0 )|v j (qi0 j t0 ) + l j |ci0 j (t0 )| ≤ −di0 (t0 )ξi0 γ +

=

h

j=1

n X j=1

l j |ai0 j (t0 )|ξ j γ +

− ξi0 (di0 (t0 ) − λ) +

< 0.

j=1

n X j=1

n X j=1

j=1

l j |bi0 j (t0 )ξ j γ +

n X j=1

l j |ci0 j (t0 )

i l j ξ j (|ai0 j (t0 )| + |bi0 j (t0 )| + |ci0 j (t0 )|) γ

Z

t0

−∞

t0

−∞

(15)

θi0 j (t0 − s)v j (s)ds + Ni0

θi0 j (t0 − s)ξ j γds + λξi0 RW

(16)

This contradicts (15). Hence, inequality (14) holds. That is |zi (t)| ≤ ξi γ,

t ≥ 0,

i = 1, 2, · · · , n.

(17)

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So model (1) is bounded. The proof is completed. Theorem 2. Assume that the (H1)-(H3) hold, system (1) is globally exponentially stable if there exist positive constants dˇ1 , dˇ2 , · · · , dˇn and ξ1 , ξ2 , · · · , ξn satisfying for t ≥ 0, di (t) > dˇi ,

i = 1, 2, · · · , n

(18)

and n X ξ j l j (|ai j (t)| + |bi j (t)| + |ci j (t)|)

ξi d˘i

1≤i≤n t≥0 j=1

< 1.

(19)

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max sup

Proof. Suppose that z(t) = (z1 (t), z2 (t), · · · , zn (t))T and z˜(t) = (˜z1 (t), z˜2 (t), · · · , z˜n (t))T are two solutions of system

(1) under initial condition ϕ(s) = (ϕ1 (s), · · · , ϕn (s))T and ψ(s) = (ψ1 (s), · · · , ψn (s))T , s ∈ (−∞, 0], respectively. Let η˙ i (t) = −di (t)ηi (t) + n X

+

ci j (t)

j=1

Z

n X j=1

ai j (t)[h j (z j (t)) − h j (˜z j (t))] +

t

−∞

n X j=1

bi j (t)[h j (z j (qi j t)) − h j (˜z j (qi j t))]

θi j (t − s)[h j (z j (s)) − h j (˜z j (s))]ds

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ηi (t) = zi (t) − z˜i (t), then

(20)

for t ≥ 0, i ∈ {1, 2, · · · , n}. Note that n n X X ai j (p)[h j (z j (p)) − h j (˜z j (p))] + bi j (p)[h j (z j (qi j p)) − h j (˜z j (qi j p))] η˙ i (p) + di (p)ηi (p) = +

n X

ci j (p)

j=1

Z

j=1

p

θi j (p − s)[h j (z j (s)) − h j (˜z j (s))]ds

M

j=1

−∞

R

(21)

p

ED

for p ≥ 0, i ∈ {1, 2, · · · , n}. Multiplying both sides of (21) by e 0 di (u)du , and integrating it on [0, t], we have that Z t Rt n n Rt X nX − 0 di (u)du ai j (p)[h j (z j (p)) − h j (˜z j (p))] + bi j (p)[h j (z j (qi j p)) − h j (˜z j (qi j p))] ηi (t) = ηi (0)e + e− p di (u)du 0

ci j (p)

j=1

p

j=1

j=1

o θi j (p − s)[h j (z j (s)) − h j (˜z j (s))]ds d p

(22)

PT

+

n X

Z

−∞

n X

ci j (p)

AC

+

CE

Application of assumption (H1) and condition (18) engenders that n n Z t R t Rt X nX − p di (u)du − 0 di (u)du |ηi (t)| ≤ ηi (0)e e ai j (p)[h j (z j (p)) − h j (˜z j (p))] + bi j (p)[h j (z j (qi j p)) − h j (˜z j (qi j p))] + j=1

ˇ

Z

≤ |ηi (0)e−di t | + +

n X j=1

p

−∞

Z

l j |ci j (p)|

t

for t ≥ 0, i ∈ {1, 2, · · · , n}.

j=1

j=1

o θi j (p − s)[h j (z j (s)) − h j (˜z j (s))]ds d p ˇ

e−di (t−p)

0

Z

0

n nX j=1

p −∞

l j |ai j (p)||η j (p)| +

o θi j (p − s)|η j (s)|ds d p

n X j=1

l j |bi j (p)||η j (qi j p)|

According to condition (19), we get that n X ξ j l j (|ai j (t)| + |bi j (t)| + |ci j (t)|) < ξi dˇi j=1

(23)

(24)

ACCEPTED MANUSCRIPT 7

for t ≥ 0, i = 1, 2, · · · , n. Let

n X

g(r) = −ξi (dˇi − r) +

j=1

ξ j l j |ai j (t)| +

n X j=1

ξ j l j |bi j (t)|e

r(1−qi j )t

+

Z

n X

ξ j l j |ci j (t)|

n X

ξ j l j |ci j (t)| < 0

j=1

+∞ 0

ers θi j (s)ds

(25)

for t ≥ 0, i = 1, 2, · · · , n. We get based on (24) and assumption (H3) that g(0) = −ξi dˇi +

n X j=1

ξ j l j |ai j (t)| +

n X

ξ j l j |bi j (t)| +

j=1

j=1

j=1

j=1

j=1

for t ≥ 0, i = 1, 2, · · · , n. Let Q =

(1+ρ)kϕ−ψk min {ξi }

(ρ > 0 is a constant). Obviously,

1≤i≤n

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for t ≥ 0, i = 1, 2, · · · , n. Accordingly, there is a sufficiently small constant µ > 0 satisfying Z +∞ n n n X X X µ(1−qi j )t ˇ −ξi (di − µ) + ξ j l j |ai j (t)| + ξ j l j |bi j (t)|e + ξ j l j |ci j (t)| eµs θi j (s)ds < 0

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|ηi (t)| ≤ ξi Qe−µt ,

(27)

0

|ηi (s)| = |ϕi (s) − ψi (s)| ≤ kϕ − ψk ≤ ξi Q ≤ ξi Qe−µs for s ∈ (−∞, 0], i = 1, 2, · · · , n. Then we will demonstrate that for t ≥ 0,

(26)

i = 1, 2, · · · , n.

(28)

(29)

If formula (29) is not true, there should exist some i0 and T > 0 satisfying |ηi0 (T )| = ξi Qe−µT

M

and

|η j (t)| ≤ ξ j Qe−µt

(30)

ED

(31)

for t ≤ T , j = 1, 2, · · · , n. It follows from (23), (27) and (31) that Z T n n X nX −dˇi0 T −dˇi0 (T −p) |ηi0 (T )| ≤ |ηi0 (0)e e |+ l j |ai0 j (p)||η j (p)| + l j |bi0 j (p)||η j (qi0 j p)| 0

PT +

n X

CE

j=1

l j |ci0 j (p)| ˇ

AC

< ξi Qe−di0 T + +

n X j=1

= ξi Qe +

j=1

Z

T

j=1

−∞ ˇ

−∞

−dˇi0 T

+ Qe

−dˇi0 T

≤ ξi Qe

= ξi Qe−µT .

Z

Z

+ Qe

+∞

0

−dˇi0 T

l j |ai0 j (p)|ξ j Qe−µp +

o θi0 j (p − s)ξ j Qe−µs ds d p T

0

ξ j l j |ci0 j (p)|

n nX j=1

p

o

θi0 j (p − s)|η j (s)|ds d p

e−di0 (T −p)

Z

j=1

p

0

l j |ci0 j (p)|

−dˇi0 T

n X

Z

Z

n nX j=1

j=1

ξ j l j |ai0 j (p)| +

o eµs θi0 j (s)ds d p

T 0

ˇ

e−(µ−di0 )p

n X

l j |bi0 j (p)|ξ j Qe−µqi0 j p

n X j=1

ξ j l j |bi0 j (p)|eµ(1−qi0 j )p

ˇ e−(µ−di0 )p ξi (dˇi0 − µ)d p

(32)

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This contradicts (30). Hence, inequality (29) holds. That is |zi (t) − z˜i (t)| ≤ ξi Qe−µt

(33)

for t ≥ 0, i = 1, 2, · · · , n. Thus kz(t) − z˜(t)k ≤ αkϕ − ψke−εt for all t ≥ 0, where α = Q max {ξi }. The proof is completed. 1≤i≤n

(4) and (19) degenerate into the following inequality max

1≤i≤n

which equals to the following inequalities n X j=1

ξi di

j=1

ξ j l j (|ai j | + |bi j | + |ci j |) > 0,

< 1,

i = 1, 2, · · · , n.

(34)

(35)

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ξi di −

n X ξ j l j (|ai j | + |bi j | + |ci j |)

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Remark 2. When di (t), ai j (t), bi j (t), ci j (t) and Ji (t) (i, j = 1, 2, · · · , n) are complex-valued constants, inequalities

Obviously, inequality (35) holds if and only if D − (|A| + |B| + |C|)L is a nonsingular M-matrix, where D = diag{d1 , d2 , · · · , dn }, |A| = (|ai j |)n×n , |B| = (|bi j |)n×n and |C| = (|ai j |)n×n .

From Remark 2 and the proof of Theorem 2 in [40], we have the following results. Corollary 1. Under assumptions (H1) and (H2), model (2) is bounded, and has a unique equilibrium point which

is globally exponentially stable if D − (|A| + |B| + |C|)L is a nonsingular M-matrix.

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Corollary 2. Under assumption (H1), model (3) is bounded, and has a unique equilibrium point which is globally

exponentially stable if D − (|A| + |B|)L is a nonsingular M-matrix.

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Remark 3. In [21]- [26], the asymptotic stability and exponential stability for RVNNs with proportional delays

were investigated by using the transformation of state variable as yi (t) = xi (et ). Different from the method in [21]-

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[26], the boundedness and stability criteria are established via the mathematical analysis method. IV. Example

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As for this section, it is shown that the results we obtained in this paper have less restriction compared with [23]. The result obtained by [23] is redescribed as following. Theorem 3. Assume the (H1) holds, the existence, uniqueness and global exponential stability for equilibrium

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point of RVNNs model

x˙i (t) = −di xi (t) +

n X

ai j h j (x j (t)) +

j=1

n X

bi j h j (x j (q j t)) + Ji ,

j=1

t≥0

(36)

can be ensured, if one of the following requirements is satisfied: (i). There is a positive constant σ > 1 such that ρ(K) < 1, where K = (ki j )n×n and ki j = (di − σ)−1 (|ai j |l j + |bi j |l j eστ j ), in which di − σ > 0, τ j = − ln p j .

(37)

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3.5 3 2.5 2

x

1.5

0.5 0

x1

-0.5 -1

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1

x2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

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t

Fig. 1. The state trajectories of model (36).

(ii). There are constants ξi > 0 (i ∈ {1, 2, · · · , n}) and σ > 1 satiafying ξi (di − σ) >

j=1

ξ j (|a ji |li + |b ji |li eστi ),

i = 1, 2, · · · , n,

(38)

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where τi = − ln pi .

n X

Example 1. Consider a two-neuron RVNNs (36), where the activations are described by

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1 hi (xi ) = (|xi + 1| − |xi − 1|), 2

i = 1, 2.

The connection weights are given by

b12 = −0.2,

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b11 = 0.1,

d2 = 2.8,

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d1 = 0.8,

a11 = −0.3,

b21 = 0.3,

a12 = 0.2,

b22 = −0.4,

J1 = 1,

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Clearly, assumption (H1) is satisfied with l1 = 1 and l2 = 1. And   0.4 −0.4 D − (|A| + |B|)L =  −1.3 1.4

a21 = 1, J2 = 8.5,

a22 = −1, q1 = 0.8,

q2 = 0.8.

   

is a nonsingular M-matrix. By using Corollary 2, we know that model (36) is bounded, and has a unique equilibrium point which is globally exponentially stable. Figure 1 shows the state variables of above model under initial value x1 (s) = 0.3 cos(0.2s), x2 (s) = −0.4 cos(6s) for s ∈ [0.8, 1]. It guarantees the global exponential stability of the

equilibrium point for above model. However, (37) and (38) are not satisfied. This signifies that the criteria in [23] are not feasible to demonstrate the stability of this model.

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V. Conclusions In this paper, the boundedness and stability for CVNNs with variable coefficients and proportional delays have been investigated. Without employing the state variable transformation method in existing literature, several sufficient criteria to ensure the boundedness and global exponential stability were derived via mathematical analysis method and inequality techniques. As a special case that the coefficients of networks are constants, sufficient criteria to guarantee that the equilibrium point is unique and globally exponentially stable were also obtained. A illustrative example with simulations is also presented to illustrate the availability and less conservative of the proposed results.

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We would like to point out that it is possible to extend our main results to general nonlinear systems such as complex networks with proportional delays, Markov jump neural networks, fractional-order neural networks and fuzzy neural networks. The corresponding results will be carried out in the near future. Acknowledgements

The authors would like to thank the editor and the reviewers for their valuable suggestions and comments which

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have led to a much improved paper. This work is supported by the National Natural Science Foundation of China under Grants 61773004 and 61473332, and in part by the Program of Chongqing Innovation Team Project in University under Grant CXTDX201601022.

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Biography

Qiankun Song was born in 1964. He received the B.S. degree in Mathematics in 1986 from Sichuan Normal University, Chengdu, China, the M.S. degree in Applied Mathematics in 1996 from Northwestern Polytechnical University, Xi’an, China, and the Ph. D. degree in Applied Mathematics in 2010 from Sichuan University, Chengdu, China. From July 1986 to December 2000, he was with Department of Mathematics, Sichuan University of Science

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and Engineering, Sichuan, China. From January 2001 to June 2006, he was with the Department of Mathematics, Huzhou University, Zhejiang, China. In July 2006, he moved to the Department of Mathematics, Chongqing Jiaotong University, Chongqing, China. He is currently a Professor at Chongqing Jiaotong University. He is currently serving as an Editorial Board Member for Neurocomputing, Journal of Applied Mathematics, British Journal of Mathematics & Computer Science, ISRN Applied Mathematics, and a reviewer for Mathematical Reviews. He is the author or coauthor of more than 60 journal papers and two edited books. His current research interests include stability theory

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of neural networks and chaos synchronization.

Qinqin Yu was born in 1993. She received the B.S. degree in Applied Mathematics in 2016 and is working toward the M.E. degree in Management Science and Engineering, both from Chongqing Jiaotong University, Chongqing,

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China. Her current research interest is the stability theory of complex-valued neural networks.

Zhenjiang Zhao was born in 1962. He received the B.S. degree in Mathematics in 1984 from Kashgar Teachers College, Kashgar, China, the M.S. degree in Applied Mathematics in 1998 from Okayama University, Okayama,

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Japan. He is currently a Professor at Huzhou Teachers College. He is the author or coauthor of more than 30 journal

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papers. His current research interests include stability theory of neural networks.

Yurong Liu received his B.S. degree in Mathematics from Suzhou University, Suzhou, China, in 1986, the M.S. degree in Applied Mathematics from Nanjing University of Science and Technology, Nanjing, China, in 1989, and the Ph.D. degree in Applied Mathematics from Suzhou University, Suzhou, China, in 2000. Dr. Liu is

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currently a Professor in the Department of Mathematics at Yangzhou University, China. He has published more than 50 papers in refereed international journals. His current interests include neural networks, complex networks,

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nonlinear dynamics, time-delay systems, multiagent systems, and chaotic dynamics.

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Fuad E. Alsaadi received the B.S. and M.Sc. degrees in electronic and communication from King Abdulaziz University, Jeddah, Saudi Arabia, in 1996 and 2002. He then received the Ph.D. degree in Optical Wireless

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Communication Systems from the University of Leeds, Leeds, UK, in 2011. Between 1996 and 2005, he worked in Jeddah as a communication instructor in the College of Electronics & Communication. He was a lecturer in the Faculty of Engineering in King Abdulaziz University, Jeddah, Saudi Arabia in 2005. He is currently an assistant

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professor of the Electrical and Computer Engineering Department within the Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia. He published widely in the top IEEE communications conferences and journals and has received the Carter award, University of Leeds for the best PhD. He has research interests in optical systems

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and networks, signal processing, synchronization and systems design.