Finite-time synchronization for Cohen–Grossberg neural networks with mixed time-delays

Communicated by Dr. Derui Ding

Accepted Manuscript

Finite-time synchronization for Cohen-Grossberg neural networks with mixed time-delays Dongxue Peng, Xiaodi Li, Chaouki Aouiti, Foued Miaadi PII: DOI: Reference:

S0925-2312(18)30294-7 10.1016/j.neucom.2018.03.008 NEUCOM 19410

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

16 December 2017 2 February 2018 1 March 2018

Please cite this article as: Dongxue Peng, Xiaodi Li, Chaouki Aouiti, Foued Miaadi, Finite-time synchronization for Cohen-Grossberg neural networks with mixed time-delays, Neurocomputing (2018), doi: 10.1016/j.neucom.2018.03.008

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ACCEPTED MANUSCRIPT

Finite-time synchronization for Cohen-Grossberg neural networks with mixed time-delays ✩ Dongxue Penga , Xiaodi Lia,b,∗, Chaouki Aouitic , Foued Miaadic a School

of Mathematics and Statistics, Shandong Normal University, Ji’nan, 250014, PR China of Data Science and Technology, Shandong Normal University, Ji’nan, 250014, PR China c University of Carthage, Faculty of Sciences of Bizerta, Department of Mathematics, Research Units of Mathematics and Applications UR13ES47, Zarzouna 7021, Bizerta, Tunisia

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b Institute

Abstract

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This paper aims to study the finite-time synchronization (i.e., synchronization in finite-time sense) of Cohen-Grossberg neural networks with mixed time delays (both time-varying discrete delay and infinite-time distributed delay). By constructing LyapunovKrasovskii functional candidates and using inequality techniques, some new sufficient conditions are derived to design the discontinuous state feedback controllers such that the addressed neural networks can be synchronized in a finite settling time, where the upper bounds of the settling time of synchronization are estimated. The effects of unknown or known time-delay are seriously taken into account, respectively, which lead to two different delay-independent discontinuous state feedback controllers. Thus our results can be applied to the finite-time synchronization of neural networks whether the time delay can be measured or not. As some special cases, our results also improve some recent works. Simulation results show the applicability and the advantages of the proposed finite-time controllers. Keywords: Finite-time synchronization; Cohen-Grossberg neural networks; Mixed delays; Lyapunov-Krasovskii functional.

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1. Introduction

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In recent years, several artificial neural networks, including cellular neural networks, Hopfield neural networks and CohenGrossberg neural networks, have obtained the unprecedented development and have been widely studied in the dynamical behaviors such as stability [1–4], state estimations [5] and boundedness [6–8], periodicity [9, 10], and synchronization [11–14]. As one of the most popular and typical neural network models, Cohen-Grossberg neural network has attracted considerable attention due to its distinguished applications in classification, parallel computing, signal and image processing since its emergence in 1983 by Cohen and Grossberg [15]. This model includes a lots of classical models from evolutionary theory, population biology and neurobiology. Especially, it should be pointed out that the Cohen-Grossberg neural network encompasses the Hopfield neural network as a special case. In recent years, the dynamical behavior of Cohen-Grossberg neural network with or without time delays have been extensively studied, see [16–20] for instance.

Especially, the synchronization of Cohen-Crossberg neural networks with time delays has attracted much attention. As we ✩ This

work was supported by National Natural Science Foundation of China (11301308, 61673247), and the Research Fund for Distinguished Young Scholars and Excellent Young Scholars of Shandong Province (JQ201719, ZR2016JL024). The paper has not been presented at any conference. ∗ Corresponding Author. Email address: [email protected] (X.Li) Submitted to Neurocomputing

know, the study of synchronization for neural networks with time delay is not an easy work, since such model is a class of infinite dimensional systems and usually admits complicated structures [21–23]. Moreover, it was observed both experimentally and numerically in [24] that the effects of time delay on system dynamics are bilateral, i.e., it could induce instability, causing sustained oscillations which may be harmful to a system and it also, inversely, make an unstable system stable and achieve some desired performances. Until now, various synchronization control of Cohen-Crossberg neural networks with time delays have been presented, see [25–28] for recent works. In [25], the exponential synchronization of Cohen-Grossberg neural networks with time-varying delays were studied by designing a periodically intermittent controller. In [26], the adaptive synchronization of Cohen-Grossberg neural networks with mixed time-varying delays and stochastic perturbation were studied based on the LaSalle invariant principle. In [27], authors considered the lag synchronization for Cohen-Grossberg neural networks with mixed time-delays via periodically intermittent control. [28] dealt with the problem of function projective synchronization for a class of memristor-based CohenGrossberg neural networks with time-varying delays. Note that the above mentioned works on synchronization control are based on the fact that the trajectories between the response system and drive system can achieve the synchronization over the infinite horizon. In the application point of view, it is more realistic if the synchronization control could be reMarch 10, 2018

ACCEPTED MANUSCRIPT Manuscript by Dongxue Peng and Xiaodi Li norm | • | and Rn×m the n × m-dimensional real spaces. For any interval J ⊆ R, set S ⊆ Rk (1 ≤ k ≤ n), C(J, S) = {ϕ : J → S is continuous} and BC(J, S) = {ϕ : J → S is bounded and continuous}. In this paper, we consider a class of Cohen-Grossberg neural networks with time-varying discrete delays and infinite-time distributed delays described by  n X  h    ˙ x (t) = a (x (t)) − b (x (t)) + cij f j (x j (t))  i i i i i     j=1    n  X     + dij g j (x j (t − τij (t)))     j=1  (1)  Z t  n X     + vij Kij (t − s)h j (x j (s))ds     −∞  j=1   i     +Ii (t) ,      x(s) = φ(s) ∈ BC([−∞, 0], Rn ),

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alized in a finite-time [29–31]. Thus, the finite-time synchronization of neural networks has derived increasing attentions in recent years and many interesting results have been reported in the literature [32–36, 38–41]. In [32], a class of delayed feedback controller was designed to achieve the finite-time synchronization of Cohen-Grossberg neural networks. In [33, 34] the finite-time synchronization for memristor-based neural networks with time-varying delays were studied. Finite-time lag synchronization of neural networks with time delays was considered in [35] and finite-time synchronization of neural networks with one single time-varying delay coupling was studied in [36]. Note that the existing results, such as those in [32– 36], only handled with discrete delays including constant delays and time-varying delays. Recently, the design of discontinuous controllers deserves much attention [37]. Ref. [38] developed a proper Lyapunov-Krasovskii functional for finite-time synchronization, and a class of discontinuous control law was proposed in which the properties of sign function were employed. It is shown that neural networks with arbitrary delays including distributed delays can be finite-timely synchronized. Then based on the method, refs. [39, 40] further studied the effects of nonidentical perturbations on finite-time synchronization of coupled discontinuous neural networks and competitive neural networks, respectively.

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where i ∈ Λ = {1, 2, · · · , n}, n ≥ 2, n corresponds to the number of neuron in a neural network. xi (t) = (x1 (t), x2 (t), · · · , xn (t)) ∈ Rn . ai (·) represents an amplification function, bi (·) is an appropriately behaved function, cij , dij and vij denote the connection strengths of the jth neuron on the ith neuron. f (x(t)) = ( f1 (x1 (t)), · · · , fn (xn (t)))T , g(x(t − τ(t))) = (g1 (x1 (t − τi1 (t))), · · · , gn (xn (t − τin (t))))T , h(x(s)) = (h1 (x1 (s)), · · · , hn (xn (s)))T are the neuron activation functions. τij (t) is the time-varying discrete delay of jth neuron from the ith neuron. Kij (t) is the non-negative bounded scalar function defined on [0, +∞) describing the delay kernel of the infinitetime distributed delay along the axon of the jth neuron from the ith neuron. I(t) = (I1 (t), I2 (t), · · · , In (t))T ∈ Rn is the external input vector. φ(s) = (φ1 (s), · · · , φn (s))T is the initial value. BC([−∞, 0], Rn ) denotes the Banach space of all bounded continuous functions from [−∞, 0] to Rn with the n nX o norm kφk = sup |φi (s)| .

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In this paper, we consider a class of Cohen-Grossberg neural networks with mixed time delays including discrete timevarying delays and infinite distributed delays. The finite-time synchronization is achieved by designing two different discontinuous state feedback controllers. Moreover, the settling time of synchronization is estimated. The main contributions of this paper can be summarized as follows. First, the estimation for settling time can be derived for neural networks subject to infinite distributed delays, which improves the results in [38–40] in which the estimation for settling time was just valid for finite distributed delays. Second, the effects of unknown or known time-delay are seriously taken into account, respectively, which lead to two different delay-independent discontinuous state feedback controllers. That is to say, our designed controllers can be applied to the cases whether the time delay can be measured or not. Third, when our model reduces to a special case such as Hopfield neural networks, our results also improve and extend some existing results and can be applied to many cases not covered in existing results. The outline of the paper is organized as follows. In Section 2, the models and some necessary mathematical preliminaries are introduced. In Section 3, we present the main results in which the known and unknown time-delays are taken into account, respectively. Simulation examples are given in Section 4, and conclusions follow in Section 5.

−∞≤s≤0

i=1

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Based on the concept of drive-response synchronization, we take (1) as the drive system. The corresponding response system is constructed as follows:  n X  h    ˙ y (t) = a (y (t)) − b (y (t)) + cij f j (y j (t))  i i i i i     j=1    n  X     + dij g j (y j (t − τij (t)))     j=1  (2)  Z t  n X     + vij Kij (t − s)h j (y j (s))ds     −∞  j=1   i     +Ii (t) + Ui (t),      y(s) = ϕ(s) ∈ BC([−∞, 0], Rn ),

2. Preliminaries

where y(t) = (y1 (t), y2 (t), · · · , yn (t))T ∈ Rn is the state vector of the response system at time t, ϕ(s) = (ϕ1 (s), · · · , ϕn (s))T is the initial value and Ui (t) is the control input to be designed.

Notations. Let R denotes the set of real numbers, Z+ the set of positive integer numbers, R+ the set of positive numbers, Rn the n-dimensional real spaces equipped with the Euclidean 2

ACCEPTED MANUSCRIPT Manuscript by Dongxue Peng and Xiaodi Li In order to investigate the problem of finite-time synchronization between systems (1) and (2), we define the synchronization error signal ei (t) = yi (t) − xi (t), ψ(s) = φ(s) − ϕ(s), ψ(s) ∈ BC([−∞, 0], Rn ) and ψ(s) = (ψ1 (s), · · · , ψn (s))T .

proper Lyapunov-Krasovskii functional in which the information of the time-delays is needed; while for the latter case, we shall construct a class of Lyapunov functions avoiding the requirement on the information of the time-delays.

The following assumptions are needed in this paper.

3.1. Time-delays are known and continuously differentiable

(H1 ) For each i ∈ Λ, function ai (·) : R → R+ is continuous and there exist positive constants ai and ai such that ai ≤ ai (u) ≤ ai for all u ∈ R.

Consider the following controllers which involve the sign function:

(H2 ) For each i ∈ Λ, function bi (·) : R → R is continuous and bi (u) − bi (v) > there exists a positive constant βi such that u−v βi for u, v ∈ R and u , v.

where i ∈ Λ, ξi > 0 are the control gains to be determined and δi > 0 are some constants.

f+

g−

g+

and lhj , such that +

g−

lj ≤ −

lhj ≤

f j (u) − f j (v) u−v

g j (u) − g j (v) u−v h j (u) − h j (v) u−v

f+

≤ lj ,

(3)

ξi ≥ ai (−βi +

g+

≤ lj , ≤ lhj , f−

f+

g

for all u, v ∈ R, and let L j = max{|l j |, |l j |}, L j = g

+

−

max{|l j |, |l j |}, and Lhj = max{|lhj |, |lhj |}. (H4 ) For each j ∈ Ghj such that

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For each i, j ∈ Λ, there exist positive constants τi j and µi j < 1 such that 0 < τi j (t) ≤ τi j , τ˙ i j (t) ≤ µi j .

(H6 ) For each i, j ∈ Λ, there exist positive constants ki j such Z +∞ that Ki j (u)du ≤ ki j .

j=1

n

X 1 g |d ji |Li + Lhi k ji ). (5) 1 − µ ji j=1

where

n X

Z

sgn(ei (t))

−∞

and

( S(e) =

Definition 1. ([38, 41]). The response system (2) is said to be synchronized with the drive system (1) in finite time if, under a suitable designed feedback controller U(t) = (U1 (t), U2 (t), · · · , Un (t))T , there exists a constant T∗ > 0 such that ke(T∗ )k1 = 0 and ke(t)k1 ≡ 0 for t > T∗ , where ke(t)k1 = Σni=1 |ei (t)|. T is called the settling time of synchronization if T = inf{T∗ ≥ 0 : x(t) = y(t) f or t ≥ T∗ }.

yi (t)

ds , xi (t) ai (s) i=1 Z t n X n X  1 g V2 (t) = |dij |L j |e j (s)|ds · S(e(t)), 1 − µij t−τi j (t) i=1 j=1 Z Z n n 0 t XX  V3 (t) = |vij |Lhj Kij (−s)|e j (u)|duds · S(e(t)), V1 (t) =

i=1 j=1

CE

0

j=1

n X

V(t) = V1 (t) + V2 (t) + V3 (t),

f g Λ, there exist positive constants L j , G j , and g f j satisfies (3) and | g j (u) |≤ G j , | h j (u) |≤

Ghj , for all u ∈ R. (H5 )

+

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g

−

f

|c ji |Li +

P ROOF. Since the time-delays are known and continuously differentiable satisfying (H5 ), one may consider the following Lyapunov-Krasovskii functional candidate:

+

f

n X

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f−

(4)

Theorem 1. Assume that (H1 ) − (H3 ), (H5 ), and (H6 ) hold. Then under the controller (4), the response system (2) is synchronized with the drive system (1) in finite time if for any i ∈ Λ, the following condition is satisfied:

−

(H3 ) For each j ∈ Λ, there exist constants l j , l j , l j , l j , lhj ,

lj ≤

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f−

Ui (t) = −ξi sgn(ei (t))|ei (t)| − δi sgn(ei (t)),

t+s

1, ke(t)k1 , 0, 0, ke(t)k1 = 0.

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In the following, we show that there exists T∗ ∈ (0, +∞) such that ke(T∗ )k1 = 0 and ke(t)k1 ≡ 0, ∀ t ≥ T∗ .

3. Main results In this section, in order to design the controllers ensuring the finite-time synchronization between system (1) and system (2), we consider two different cases, i.e., time-varying discrete delays (short for time-delays) are known and continuously differentiable, and they are unknown or discontinuous are considered, respectively. For the former case, we shall construct a 3

(6)

First, it follows from the definitions of V(t) and S(e(t)) that for any given t ∈ (0, +∞), if ke(t)k1 = 0, then V1 (t) = 0, V2 (t) = 0 and V3 (t) = 0, which implies that V(t) = 0. Thus, we know that if V(t) > 0, then ke(t)k1 > 0. Then one may transfer the discussion of ke(t)k1 satisfying (6) to the discussion of V(t) satisfying: there exists T∗ ∈ (0, +∞) such that V(T∗ ) = 0 and V(t) ≡ 0 for every t ≥ T∗ . Firstly, we show that there exists a constant T∗ ∈ (0, +∞) such that V(T∗ ) = 0. Suppose on the contrary that for any t ∈ (0, +∞), V(t) > 0, then ke(t)k1 > 0, which implies that

ACCEPTED MANUSCRIPT Manuscript by Dongxue Peng and Xiaodi Li S(e(t)) = 1. In this case, calculating the time derivatives of V1 (t), V2 (t) and V3 (t) with S(e(t)) = 1, respectively, and applying conditions (H1 ) − (H3 ), (H5 ), and (H6 ), we have i=1

+ +

i=1 n X i=1

+

n X

di j (g j (y j (t − τi j (t))) − g j (x j (t − τi j (t)))) vi j

i=1

+

+

Z |vi j |Lhj

i=1 j=1

−

n X δλ

i i

ai

i=1

t

(7)

,

(

On the other hand, we get n X n X i=1 j=1

1 g |di j |L j |e j (t − τi j (t))|(1 − τ˙ i j (t)) 1 − µi j

i=1 j=1

n X n X

1 g |di j |L j |e j (t)| 1 − µi j

AC

≤

n X n X

CE

−

1 g |di j |L j |e j (t)| 1 − µi j

PT

V˙ 2 (t) =

i=1 j=1

−

V˙ 3 (t) =

n X n X i=1 j=1

n X n X i=1 j=1

− ≤

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1, |ei (t)| , 0, 0, otherwise.

λi =

i=1 j=1

i=1 j=1

Z

|vi j |Lhj |e j (t)|

n X n X

n X n X

g

|di j |L j |e j (t − τi j (t))|,

Z |vi j |Lhj

i i

i=1

n X δλ

i i

i=1

ai

(10)

ai

, t ≥ 0.

(11)

t→+∞

tradicts V(t) > 0, for t ∈ (0, +∞). Hence there exists T∗ ∈ (0, +∞) such that V(T∗ ) = 0, which implies that ke(T∗ )k1 = 0. Next, we show that ke(t)k1 ≡ 0 for ∀t ≥ T∗ . Or else, suppose that there exists T1 > T∗ such n o that ke(T1 )k1 > 0. Let ∗ Ts = sup t ∈ [T , T1 ] : ke(t)k1 = 0 . It then follows from the fact ke(T∗ )k1 = 0 that Ts is nonempty. It is easy to see that Ts < T1 , ke(Ts )k1 = 0 and ke(t)k1 > 0 for all t ∈ (Ts , T1 ]. In this case, S(e(t)) = 1, t ∈ (Ts , T1 ]. Then V2 (t) and V3 (t) are differentiable on (Ts , T1 ]. We claim that there ˙ t=T2 > 0, otherwise, exists T2 ∈ [Ts , T1 ] such that V(t)| ˙ ∀ t ∈ [Ts , T1 ], V(t) ≤ 0. Then V(t) is monotone increasing, we get 0 = V(Ts ) ≥ V(T1 ) > 0, which is a contradiction. ˙ t=T2 > 0. Therefore, there exists T2 ∈ (Ts , T1 ] such that V(t)| On the other hand, note that ke(T2 )k1 > 0 implies that i=1

Ki j (−s)ds

> 0.

It then follows from (11) that lim V(t) = −∞, which con-

i i

ai

> 0.

˙ 2 ) < 0, which is It follows from the inequality (10) that V(T also a contradiction. Hence ke(t)k1 ≡ 0, ∀t ≥ T∗ . Therefore, according to Definition 1, the response system (2) is synchronized with the drive system (1) in finite time under the controller (4). The proof of Theorem 1 is completed. 

0 −∞

, t ∈ (0, +∞].

n X δλt

n X δλ

(8)

0 −∞

i=1

Note that ke(t)k1 > 0, then it indicates that

Ki j (t − s)|e j (s)|ds

where λi is defined by

i i

ai

V(t) − V(0) ≤ −

− τi j (t))|

−∞

n X δλ

Integrating both sides of the above inequality from 0 to t gets the following inequality:

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i=1 j=1 n X n X

(9)

n X i δ i λi Lhi k ji |ei (t)| − , ai i=1

˙ V(t) ≤−

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+

g |di j |L j |e j (t

Kij (t − s)|e j (s)|ds.

which, together with condition (5) yields that

Ki j (t − s)(h j (y j (s)) − h j (x j (s)))ds

i 1 + (−ξi sgn(ei (t))|ei (t)| − δi sgn(ei (t))) ai (yi (t)) n n X n X X ξi f ≤ [−(βi + )|ei (t)|] + |ci j |L j |e j (t)| a i i=1 i=1 j=1 n X n X

n X j=1

t −∞

−∞

˙ V(t) =V˙ 1 (t) + V˙ 2 (t) + V˙ 3 (t) n h n n X X X ξi f g ≤ − (βi + ) + |c ji |Li + |d ji |Li ai i=1 j=1 j=1

ci j ( f j (y j (t)) − f j (x j (t)))

Z

t

It then follows from (7)-(9) that

h sgn(ei (t)) − (bi (yi (t)) − bi (xi (t)))

n X

i=1 j=1

Z |vij |Lhj

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V˙ 1 (t) =

n X

−

n X n X

Ki j (−s)|e j (t + s)|ds

|vi j |Lhj ki j |e j (t)| 4

When the drive-response systems without infinite-time distributed delays are concerned, i.e., vij = 0, i, j ∈ Λ in (1) and

ACCEPTED MANUSCRIPT Manuscript by Dongxue Peng and Xiaodi Li (2), one may only need to consider the V1 (t) and V2 (t) as the Lyapunov-Krasovskii functional candidate. By the similar discussion in Theorem 1, the following Corollary can be derived.

exists a constant t1 ∈ (0, T1 ] such that V(t1 ) = 0. Suppose on the contrary that for all t ∈ (0, T1 ], V(t) > 0, then ke(t)k1 > 0 for t ∈ (0, T1 ]. In this case, S(e(t)) = 1, t ∈ (0, T1 ]. Using the same proof procedure as Theorem 1 yields that

Corollary 1. Assume that (H1 ) − (H3 ), and (H5 ) hold. Then under the controller (4), the response system (2) is synchronized with the drive system (1) in finite time if for any i ∈ Λ, the following condition is satisfied:

j=1

f

|c ji |Li +

n X j=1

j=1

1 g |d ji |Li ), i ∈ Λ. 1 − µ ji

which indicates that it is necessary that n X j=1

Then V(T1 ) ≤ V(0)−

|b ji |Fi for some i ∈ Λ.

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PT

where δ = min

s

∈Λ .

s

We show that there exists t1 ∈ (0, T1 ] such that k e(t1 ) k1 = 0 and k e(t) k1 ≡ 0, ∀t ∈ [t1 , T1 ]. Firstly, we show that there

j=1

f

|c ji |Li ), i ∈ Λ.

n

δi ai , i

o

∈Λ .

If ai (xi (t)) ≡ 1 for all t ≥ 0 and bi (u) ≡ bi u, i ∈ Λ, then the drive system (1) is reduced to the following Hopfield neural networks:  n X    ˙ x (t) = −b x (t) + cij f j (x j (t))  i i i     j=1     n X     + dij g j (x j (t − τij (t)))   (14)  j=1    Z  n t  X     + vij Kij (t − s)h j (x j (s))ds + Ii (t),    −∞  j=1     x(s) = φ(s) ∈ BC([−∞, 0], Rn ).

o

−∞

n X

i=1

Z 0 n n n 1 h X |ei (0)| X X 1 g + |di j |L j |e j (s)|ds δ ai 1 − µi j −τij i=1 i=1 j=1 Z 0 Z 0 n X n X i + |vi j |Lhj Ki j (−s)|e j (u)|duds . (13) i=1 j=1

Corollary 3. Assume that (H1 ) − (H3 ), and (H5 ) hold. Then under the controller (4), the response system (2) is synchronized with the drive system (1) in finite time if for any i ∈ Λ, the following condition is satisfied:

n

P ROOF. Consider the auxiliary constant T1 =

i i

1 h X |ei (0)| i , T≤ δ ai

CE

AC

where δ = min

T1 .

Moreover, the settling time is estimated by

Z 0 n n n 1 h X |ei (0)| X X 1 g T≤ |di j |L j + |e j (s)|ds δ ai 1 − µi j −τij i=1 i=1 j=1 Z 0 Z 0 n X n X i h + |vi j |L j Ki j (−s)|e j (u)|duds , (12) δi ai , i

i=1

n X δλ

ξi ≥ ai (−βi +

Corollary 2. Under Theorem 1, the settling time T of the finitetime synchronization of (1) and (2) can be estimated, which satisfies

n

i i

ai

When dij = vij = 0, i, j ∈ Λ in systems (1) and (2), one may only consider the V1 (t) as the Lyapunov function. Since V1 (t) = 0 if and only if ke(t)k1 = 0, we have the following Corollary.

The following Corollary is given to estimate settling time for finite-time synchronization between systems (1) and (2).

−∞

n X δλ

T1 < 0, which contradicts V(t) > ai i=1 0, ∀t ∈ (0, T1 ]. Thus there exists t1 ∈ (0, T1 ] such that V(t1 ) = 0, which indicates that ke(t1 )k1 = 0. Then using the same proof in Theorem 1, we can obtain ke(t)k1 ≡ 0 for t ∈ [t1 , T1 ]. Hence, the settling time T can be estimated, which satisfies T ≤ T1 , i.e., (12) holds. The proof is completed. 

In this paper, it follows from Corollary 1 that the above restriction is completely removed and there is no strict restriction on βi . Hence, the controllers designed in this paper are more general than that in [32].

i=1 j=1

, t ∈ (0, T1 ].

V(T1 ) − V(0) ≤ −

|b ji |Fi − ai ≤ 0 for some i ∈ Λ,

ai ≥

i=1

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n X

i i

ai

Integrating both sides of the above inequality from 0 to T1 and considering (13), we get

Remark 1. In [32], the finite-time synchronization of delayed Cohen-Grossberg neural networks has been studied based on a class of delayed feedback controllers. However, note that those controllers in [32] are based on the fact that βi =

n X δλ

CR IP T

ξi ≥ ai (−βi +

n X

˙ V(t) ≤−

5

ACCEPTED MANUSCRIPT Manuscript by Dongxue Peng and Xiaodi Li The response neural network (2) is replaced by  n X     ˙ y (t) = −b y (t) + ci j f j (y j (t)) i i i     j=1     n  X    + di j g j (y j (t − τi j (t)))      j=1   Z t n  X     + v Ki j (t − s)h j (y j (s))ds ij    −∞  j=1      +Ii (t) + Ui (t),      y(s) = ϕ(s) ∈ BC([−∞, 0], Rn ).

unknown. Hence, in the following, we shall present another class of controllers for finite-time synchronization in which the time-delays may be unknown or discontinuous. 3.2. Time-delays are unknown (or discontinuous) In this section, we design the controllers as follows: (15)

n X n X h g Ui (t) = − ξi sgn(ei (t))|ei (t)| − δi + 2ai |dij |G j i=1 j=1

i=1 j=1

In this case, assumption (H1 ) holds with ai = ai = 1 for every i ∈ Λ. Then the following result can be derived by Theorem 1.

j=1

Moreover, the settling time is estimated by Z

n X n X

0

i=1 j=1

−∞

j=1

f

|c ji |Li , i ∈ Λ.

n

T≤

s

where δ = min{δi , i ∈ Λ}.

n X

Moreover, the settling time T is estimated by

M

1 g |di j |L j |e j (s)|ds δ 1 − µi j −τij i=1 i=1 j=1 Z 0 Z 0 n X n X i h + Ki j (−s)|e j (u)|duds , |vij |L j |ei (0)| +

ξi ≥ −βi ai + ai

ED

T≤

n 1h X

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ξi ≥ −βi +

 1 g f |d ji |Li + |v ji |Lhi k ji , i ∈ Λ. |c ji |Li + 1 − µ ji

(16)

Theorem 2. Assume that the time-delays are unknown (or discontinuous) and assumptions (H1 ), (H2 ), (H4 ), and (H6 ) hold. Then under the controller (16), the drive-response system (1) and (2) are synchronized in finite time if there exist constants ξi , δi , i ∈ Λ, such that

Corollary 4. Assume that (H2 ), (H3 ), (H5 ), and (H6 ) hold. Then under the controller (4), the response system (15) is synchronized with the drive system (14) in finite time if for any i ∈ Λ, the following condition is satisfied: n  X

i |vij |Ghj kij sgn(ei (t)), i ∈ Λ.

CR IP T

+ 2ai

n n X X

where δ = min

n

δi ai , i

i=1

o

∈Λ .

P ROOF. Consider the following Lyapunov function V(t) =

Remark 2. In [38, 39], the finite-time synchronization of neural networks (14) and (15) have been extensively studied and some discontinuous control law were proposed to guarantee the finite-time synchronization. However, the results are inapplicable to estimate the settling time for the neural networks subject to unbounded time delays. In this paper, based on the designed controller (4), Corollary 4 provides some sufficient conditions for estimating the settling time whether the time delays are bounded or not.

1 h X |ei (0)| i , δ ai

n X

Z sgn(ei (t))

PT

i=1

yi (t) xi (t)

ds . ai (s)

Calculating the time derivative of V(t) and apply (H1 ), (H2 ), (H4 ), and (H6 ), we get

CE

˙ V(t) =

n X i=1

AC

+

h sgn(ei (t)) − (bi (yi (t)) − bi (xi (t)))

n X j=1

Remark 3. It follows from the proof of Theorem 1 that the auxiliary function S(e(t)) which is introduced in the construction of Lyapunov-Krasovskii functional V plays an important role in achieving the FTS. In fact, it provides an effective way to transfer the discussion of ke(t)k1 satisfying (6) to the discussion of V(t) satisfying: there exists T∗ ∈ (0, +∞) such that V(T∗ ) = 0 and V(t) ≡ 0 for every t ≥ T∗ .

+

n X j=1

+

n X j=1

cij ( f j (y j (t)) − f j (x j (t))) dij (g j (y j (t − τij (t))) − g j (x j (t − τij (t)))) Z vij

t −∞

Kij (t − s)(h j (y j (s)) − h j (x j (s)))ds n

+

The above discussions are based on the fact that the timedelays are known and continuously differentiable. However, sometimes the time-delays are time-varying and cannot be exactly measured, and the information of the time-delays may be

n

XX  1  g |dij |G j − ξi sgn(ei (t))|ei (t)| − δi + 2ai ai (yi (t)) i=1 j=1

+ 2ai

n X n X i=1 j=1

6

 i |vij |Ghj kij sgn(ei (t))

ACCEPTED MANUSCRIPT Manuscript by Dongxue Peng and Xiaodi Li ≤

n h X i=1

≤−

n

n

X Xδλ i ξi f i i − (βi + ) + |c ji |Li |ei (t)| − ai a i j=1 i=1

n X δλ

i i

i=1

ai

and τ1 (t) = τ2 (t) =

et 0.1 . , a1 (x) = a2 (x) = 0.7 + t 1+e 1 + x2 f

,

In this case, it is easy to see that 0.7 ≤ ai (x) ≤ 0.8, Li = = 1 for i = 1, 2 and β1 = 1.4, β2 = 0.4, τ j = 1, µ j = 0 < 1, j = 1, 2. Hence, assumptions (H1 ) − (H3 ) and (H5 ) are satisfied. g Li

where λi is defined by ( λi =

1, |ei (t)| , 0, 0, otherwise.

a 2.5

||e(t)|| 1

Then one may claim that there exists T ∈ (0, +∞) such that ke(T∗ )k1 = 0 and ke(t)k1 ≡ 0, ∀t ≥ T∗ . The proof is similar to Theorem 1 and thus is omitted here. Then using the same proof in Corollary 2, we can estimate the settling time T. The proof is completed. 

1.5

1

0.5

0

Remark 4. In [32–36, 38–41], the finite-time synchronization of neural networks with various time delays have been studied. However, those existing results are based on the fact that the time-delays are exactly measured, even continuously differentiable. In this paper, the designed controller (16) in Theorem 2 drops the restriction and can be applied to the case that the information of the time-delays is unknown.

0

1

b 0.6 0.5

2

3

||e(t)|| 1

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0.4

CR IP T

2

∗

4

5

6

7

8

9

10

t

4

5

6

7

8

9

10

t

0.3 0.2 0.1

0

Remark 5. Note that assumption (H3 ) is used in Theorem 1 for the case that time-delays are known and continuously differentiable. While assumption (H4 ) is used in Theorem 2 for the case that time-delays are unknown (or discontinuous). In addition, one may observe from Theorem 2 that the controller (16) depends on the bounds of the nonlinear functions g and h, but the settling time is independent of them, which implies that the bounds of the nonlinear functions will affect the control performance but not change the settling time of finite-time synchronization.

-0.1

M

ED

Consider the the response system in the form of

2 X j=1

(18)

Ui (t) = −αsign(ei (t))|ei (t)|η − δi sign(ei (t))|ei (t − τi (t))|,

for i = 1, 2, with η = 0.8, α = 2.5, δ1 = 2.2, and δ2 = 3.1. The corresponding numerical simulation can be found in [32] and here we omit it. Note that the designed controller in [32] is based on the fact that there is at least βi ≤ 0, where βi = Σnj=1 |b ji |Fi −ai . Thus it is only applicable to some special cases. For example, when β1 = 1.4 and β2 = 0.4, it is easy to see that the controllers Ui , i = 1, 2 in [32] is invalid. In this case, based on Theorem 1 in this paper, one may derive a controller like (4). Choose ξ1 = 3.68, ξ2 = 2.56, and δ1 = δ2 = 0.1, i.e.,

j=1

(17)

for i = 1, 2, where f j (x) = g j (x) = tanh(x), j = 1, 2, b1 (x1 (t)) = 1.4x1 (t), b2 (x2 (t)) = 0.4x2 (t), " # " # 1.8 −0.1 −1.7 −0.6 C= , D= , −2 0.4 0.5 −2.5

i dij g j (y j (t − τ j (t))) + Ui (t),

where ai , bi , cij , dij , f j , g j , τ j are defined in system (17). When β1 = 1.4, β2 = 1.6, the finite-time synchronization between systems (17) and (18) has been studied in [32], where the controller for finite-time synchronization was designed by

CE

AC j=1

Trajectory of the synchronization error ke(t)k1 with the controller.

+

2 X h x˙ i (t) =ai (xi (t)) − bi (xi (t)) + ci j f j (x j (t))

i di j g j (x j (t − τ j (t))) ,

3

j=1

Example 1. Consider the following Cohen-Grossberg neural networks with variable delays ([32]):

+

2

2 X h y˙ i (t) =ai (yi (t)) − bi (yi (t)) + cij f j (y j (t))

In this section, two numerical examples and their simulations are given to show the effectiveness and advantages of our theoretical results.

2 X

1

Fig.1. (a). Trajectory of the synchronization error ke(t)k1 without the controller; (b).

PT

4. Examples

0

U1 (t) = − 3.68sgn(e1 (t))|e1 (t)| − 0.1sgn(e1 (t)),

U2 (t) = − 2.56sgn(e2 (t))|e2 (t)| − 0.1sgn(e2 (t)), 7

ACCEPTED MANUSCRIPT Manuscript by Dongxue Peng and Xiaodi Li then it follows from Theorem 1 that the response system (18) can be synchronized with the drive system (17) in finite-time. The simulations are shown in Figs.1 (a-d). In this case, the settling time T ≤ 4.757.

a 8 7 6

||e(t)|| 1

5 4 3

c 0.5 x

0.4

2 1

y1

0.3

1 0

0.1

0

0.5

-0.2

0.8

0

1

2

3

4

5

6

7

8

9

||e(t)|| 1

-0.3

t

10

0.6 0.4

1.5 y2

0.2

x2

1

2.5

3

3.5

4

3.5

4

t

0

0.5

1

1.5

2

2.5

3

t

Fig.2. (a). Trajectory of the synchronization error ke(t)k1 without the controller; (b).

0

Trajectory of the synchronization error ke(t)k1 with the controller.

-0.5

0

1

2

3

4

5

6

7

8

9

10

t

Fig.1. (c). Trajectories of x1 and y1 ; (d). Trajectories of x2 and y2 .

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x 2 (t),y 2 (t)

0 0.5

-1

2

1

-0.1

d

1.5

b 1.2

0

-0.4

1

CR IP T

x 1 (t),y 1 (t)

0.2

It is easy to get that τij = 1 and µij = 0 < 1, kij = 2, i, j = f

j=1

Z vi j

j=1

j=1

t −∞

2 X

di j g j (x j (t − τi j (t)))

Ki j (t − s)h j (x j (s))ds + Ii (t),

(19)

CE

However, it is difficult to estimate the settling time based on the results in [38] since the system involves the infinite-time distributed time-delays. In this paper, based on Corollary 4, the settling time can be estimated by

AC

y˙ i (t) = − bi yi (t) + +

2 X j=1

Z vi j

j=1

ci j f j (y j (t)) +

j=1

2

T≤

di j g j (y j (t − τi j (t)))

2

2

XX 1 1h X g |ei (0)| + |dij |L j δ 1 − µij i=1

+

2 X 2 X i=1 j=1

i=1 j=1

Z

|vij |Lhj

0 −∞

Z

0 s

Z

0 −τi j

|e j (s)|ds

i Kij (−s)|e j (u)|duds ,

where δ = min{δ1 , δ2 }. If let δ1 = 1 and δ2 = 3, then the settling time can be estimated by T ≤ 3.54, see in Figs.2 (a-d).

t −∞

j=1

 1 g |d j1 |Li + |v ji |Lhi k ji . 1 − µ ji

f

|c ji |Li +

for any positive constants δ1 and δ2 .

When the initial value is given by x(t) = (x1 (t), x2 (t))T = (0.4, 0.6)T , ∀t ∈ [−3, 0] and x(t) = 0 for t < −3, the trajectory of (19) has been shown in [38] and here we omit it. The response system is considered by 2 X

2  X

Ui (t) = −ξi sgn(ei (t))|ei (t)| − δi sgn(ei (t)), i = 1, 2,

for i = 1, 2, where f j (x) = g j (x) = h j (x) = tanhx, τi j (t) = 1, Ki j (t) = e−0.5t , i, j = 1, 2, and " # " # " # 1.2 0 3 −0.3 −1.4 0.1 B= , C= , D= , 0 1 4 5 0.3 −8 " # " # −1.2 0.1 1 V= , I(t) = . −2.8 −1 1.2

2 X

ξi ≥ −βi +

Choose ξ1 = 15.5 and ξ2 = 14.6 such that the neural networks (19) and (20) can be finite-timely synchronized under the controller (4), i.e.,

PT

+

2 X

ci j f j (x j (t)) +

ED

2 X

M

Example 2. Consider ai (xi (t)) ≡ 1 for all t ≥ 0 and bi (u) ≡ bi u, i = 1, 2, Then we get the following 2D neural network with both discrete and infinite-time distributed delays ([38]): x˙ i (t) = − bi xi (t) +

g

1, 2, Li = Li = Lhi = 1. Therefore, assumptions (H1 ) − (H3 ), (H5 ), (H6 ) are satisfied. It then follows from Corollary 4 that

Ki j (t − s)h j (y j (s))ds + Ii (t) + Ui . (20)

In addition, based on Theorem 2, one may design a different

8

ACCEPTED MANUSCRIPT Manuscript by Dongxue Peng and Xiaodi Li been investigated. Two types of feedback controllers handling with unknown or known time-delay have been presented for response system to realize the synchronization with drive system. Moreover, the settling time has been estimated for the addressed models subject to mixed time delays. It should be noted that the criteria derived in this paper include some existing results as their special cases and are much less conservative than some existing results. Two examples and their numerical simulations have also given to illustrate the effectiveness. An object of our future investigations is to consider some other kinds of neural networks with mixed time delays such as Hopfield neural networks, Cellular neural networks, and BAM neural networks.

1.4 1.2 1

x 1 (t),y 1 (t)

0.8 0.6 0.4 0.2 0

x y

-0.2 -0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 1

1 t

0.9

d 1.4 x2

1.2

y

2

x 2 (t),y 2 (t)

1 0.8 0.6

References

0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 t

controller in the form of 2 X 2 X h g Ui (t) = − ξi sgn(ei (t))|ei (t)| − δi + 2ai |di j |G j i=1 j=1

i=1 j=1

i |vi j |Ghj ki j sgn(ei (t)), i = 1, 2,

M

+ 2ai

2 X 2 X

i.e.,

ED

U1 (t) = − 15.5sgn(e1 (t))|e1 (t)| − 30.8sgn(e1 (t)),

U2 (t) = − 14.6sgn(e2 (t))|e2 (t)| − 33.8sgn(e2 (t)),

CE

PT

Then the systems (19) and (20) can be finite-timely synchronized under the above controllers. Moreover, the settling time can be estimated by T ≤ 1.1. The numerical simulation is shown in Fig.3. 1.2 1

AC

||e(t)|| 1

0.8 0.6 0.4 0.2

0

0

0.2

0.4

0.6

0.8

1

1.2

[1] J. Cao, Y. Wan, Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays, Neural Networks 53 (2014) 165-172. [2] S. Arik, New criteria for global robust stability of delayed neural networks with norm-bounded uncertainties, IEEE Transactions on Neural Networks and Learning Systems 25 (2014) 1045-1052. [3] D. Ding, Q. Han, Y. Xiang, X. Ge, X. Zhang, A survey on security control and attack detection for industrial cyber-physical systems, Neurocomputing 275 (2018) 1674-1683. [4] I. Stamova, T. Stamov, X. Li, Global exponential stability of a class of impulsive cellular neural networks with Supremums, International Journal of Adaptive Control and Signal Processing 28 (2014) 1227-1239. [5] L. Wang, Z. Wang, G. Wei, Finite-time state estimation for recurrent delayed neural networks with component-based event-triggering protocol IEEE Transactions on Neural Networks and Learning Systems 99 (2017) 1-12. [6] J. Hu, Z. Wang, S. Liu, et al. A variance-constrained approach to recursive state estimation for time-varying complex networks with missing measurements, Automatica 64 (2016) 155-1621. [7] Y. Yu, H. Dong, Z. Wang, et al. Design of non-fragile state estimators for discrete time-delayed neural networks with parameter uncertainties, Neurocomputing 182 (2016) 18-24. [8] C. Aouiti, P. Coirault, F. Miaadi, E. Moulay, Finite time boundedness of neutral high-order Hopfield neural networks with time delay in the leakage term and mixed time delays, Neurocomputing 260 (2017) 378392. [9] X. Li, S. Song, Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays, IEEE Transactions on Neural Networks and Learning Systems 24 (2013) 868-877. [10] M. Hamdi, C. Aouiti, A. Touati, et al. Weighted pseudo almost-periodic solutions of shunting inhibitory cellular neural networks with mixed delays, Acta Mathematica Scientia 36 (2016) 1662-1682. [11] Z. Wu, P. Shi, H. Su, et al. Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled data, IEEE Transactions on Cybernetics 43 (2013) 1796-1806. [12] J. Lu, C. Ding, J. Lou, J. Cao, Outer synchronization of partially coupled dynamical networks via pinning impulsive controllers, Journal of the Franklin Institute 352 (2015) 5024-5041. [13] X. Zhang, X. Lv, X. Li, Sampled-data based lag synchronization of chaotic delayed neural networks with impulsive control, Nonlinear Dynamics 90 (2017) 2199-2207. [14] L. Wang, Z. Wang, Q. Han, G. Wei, Synchronization control for a class of discrete-time dynamical networks with packet dropouts: a codingdecoding-based approach, IEEE Transactions on Cybernetics 99(2017) 1-12. [15] M. A. Cohen, S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybern. 13 (1983) 815-826.

AN US

Fig.2. (c). Trajectories of x1 and y1 ; (d). Trajectories of x2 and y2 .

CR IP T

c

t

Fig.3. Trajectory of the synchronization error ke(t)k1 .

5. Conclusion In this paper, the finite-time synchronization for a class of Cohen-Grossberg neural networks with mixed time delays has 9

ACCEPTED MANUSCRIPT Manuscript by Dongxue Peng and Xiaodi Li chronized? Neurocomputing 143 (2014) 275-281. [39] X. Yang, Q. Song, J. Liang, et al. Finite-time synchronization of coupled discontinuous neural networks with mixed delays and nonidentical perturbations, Journal of the Franklin Institute 352 (2015) 4382-4406. [40] Y. Li, X. Yang, L. Shi, Finite-time synchronization for competitive neural networks with mixed delays and non-identical perturbations, Neurocomputing Volume 185 (2016) 242-253. [41] L. Wang, Y. Shen, Z. Ding, Finite time stabilization of delayed neural networks, Neural Networks 70 (2015) 74-80.

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CE

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[16] X. Yang, J. Cao, W. Yu, Exponential synchronization of memristive Cohen-Grossberg neural networks with mixed delays, Cognitive neurodynamics 8 (2014) 239-249. [17] R. Rakkiyappan, A. Chandrasekar, S. Lakshmanan, et al. Exponential stability of Markovian jumping stochastic Cohen-Grossberg neural networks with mode-dependent probabilistic time-varying delays and impulses, Neurocomputing 131 (2014) 265-277. [18] A. Abdurahman, H. Jiang, The existence and stability of the anti-periodic solution for delayed Cohen-Grossberg neural networks with impulsive effects, Neurocomputing 149 (2015) 22-28. [19] L. Hien, T. Loan, B. Trang, et al. Existence and global asymptotic stability of positive periodic solution of delayed Cohen-Grossberg neural networks, Applied Mathematics and Computation 240 (2014) 200-212. [20] Y. Du, S. Zhong, Z. Zhou, et al. Exponential stability for stochastic CohenCGrossberg BAM neural networks with discrete and distributed timevarying delays, Neurocomputing 127 (2014) 144-151. [21] X. Li, J. Wu, Stability of nonlinear differential systems with statedependent delayed impulses, Automatica 64 (2016) 63-69. [22] W. Gong, J. Liang, J. Cao, et al. Global µ-stability of complex-valued delayed neural networks with leakage delay, Neurocomputing 168 (2015) 135-144. [23] X. Li, J. Cao, An impulsive delay inequality involving unbounded timevarying delay and applications, IEEE Transactions on Automatic Control 62 (2017) 3618-3625. [24] X. Li, S. Song, Stabilization of Delay Systems: Delay-dependent Impulsive Control, IEEE Transactions on Automatic Control 62 (2017) 406411. [25] J. Yu, C. Hu, H. Jiang, Z. Teng, et al. Exponential synchronization of Cohen-Grossberg neural networks via periodically intermittent control, Neurocomputing 74 (2011) 1776-1782. [26] C. Zhang, F. Deng, Y. Peng, et al. Adaptive synchronization of CohenGrossberg neural network with mixed time-varying delays and stochastic perturbation, Applied Mathematics and Computation 269 (2015) 792801. [27] A. Abdurahman, H. Jiang, Z. Teng, Lag synchronization for CohenGrossberg neural networks with mixed time-delays via periodically intermittent control, International Journal of Computer Mathermatics 94 (2017) 275-295. [28] A. Abdurahman, H. Jiang, K. Rahman, Function projective synchronization of memristor-based Cohen-Grossberg neural networks with timevarying delays, Cognitive neurodynamics 9 (2015). [29] E. Moulay, M. Dambrine, N. Yeganefar, W. Perruquetti, Finite-time stability and stabilization of time-delay systems, Systems & Control Letters 57 (2008 )561-566 [30] H. Shen, J. Park, Z. Wu, Finite-time synchronization control for uncertain Markov jump neural networks with input constraints, Nonlinear Dynamics 77 (2014) 1709-1720. [31] X. Liu, J. Cao, W. Yu, Q. Song, Nonsmooth finite-Time synchronization of switched coupled neural networks sign in or purchase, IEEE Transactions on Cybernetics 46 (2016) 2360 - 2371. [32] C. Hu, J. Yu, H. Jiang, Finite-time synchronization of delayed neural networks with Cohen-Grossberg type based on delayed feedback control, Neurocomputing 143 (2014) 90-96. [33] M. Liu, H. Jiang, C. Hu, Finite-time synchronization of memristor-based Cohen-Grossberg neural networks with time-varying delays, Neurocomputing 194 (2016) 1-9. [34] J. Cao, R. Li, Fixed-time synchronization of delayed memristor-based recurrent neural networks, Science China Information Sciences 60 (2017) 032201. [35] J. Huang, C. Li, T. Huang, et al. Finite-time lag synchronization of delayed neural networks, Neurocomputing 139 (2014) 145-149. [36] D. Li, J. Cao, Finite-time synchronization of coupled networks with one single time-varying delay coupling, Neurocomputing 166 (2015) 265270. [37] X. Liu, D. Ho, J. Cao, W. Xu, Discontinuous observers design for finitetime consensus of multiagent systems with external disturbances, IEEE Transactions on Neural Networks and Learning Systems 28 (2017) 2826 - 2830. [38] X. Yang, Can neural networks with arbitrary delays be finite-timely syn-

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ACCEPTED MANUSCRIPT Manuscript by Dongxue Peng and Xiaodi Li Authors’ Biographies

matics from the National Engineering School of Tunis, University Tunis El Manar in 1997 and 2003 respectively. He is currently an assistant professor in Applied Mathe-

***************************************************

matics at Faculty of Sciences of Bizerta, University of Carthage. His research interests include Neural networks, Dynamical systems, Differential equations, Almost periodic differential equation, Almost automorphic differential equation. He, currently, serves as a reviewer of several international journals and a Program Committee for various inter-

CR IP T

national conferences.

Dongxue Peng was born in Shandong Province, China, in 1995. She is currently a PhD student with control theory, School of Mathematics and Statistics, Shandong Normal University, Shandong, China. Her research interests include neural networks, stability and impulsive control theory.

Foued Miaadi was born in Bizerta, Tunisia, in 1991. He graduated in 3 year of university

AN US

study of Mathematics from the Faculty of Sciences of Bizerta (FSB) at the University of Carthage (Tunisia) in 2013. He obtained the master degree in Mathematics from Faculty of Sciences of Bizerta (FSB) at the University of Carthage (Tunisia) in 2015. Currently, he is working toward the Ph.D. degree in Mathematics at University of Carthage. His major research interests include delayed neural networks and their practical applications, finite time stability theory, Lyapunov stability analysis and stabilization, neutral differ-

ED

M

ential equations.

Xiaodi Li received the B.S. and M.S. degrees from Shandong Normal University, Jinan, China, in 2005 and 2008, respectively, and the Ph.D. degree from Xiamen University, Xiamen, China, in 2011, all in applied mathematics. He is currently a Professor with the

PT

School of Mathematics and Statistics, Shandong Normal University. From Nov. 2014 to Dec. 2016, he was a Visiting Research Fellow at Laboratory for Industrial and Applied Mathematics in York University, Canada, and the University of Texas at Dallas, USA. In 2017, he was working as Visiting Research Fellow at the Department of Mathematics,

CE

City University of Hong Kong, Hong Kong. He has received the May 4th Youth Award of Shandong province, China, in 2016. He has authored or coauthored more than 70 research papers. His current research interests include stability theory, delay systems,

AC

impulsive control theory, artificial neural networks, and applied mathematics.

Chaouki Aouiti graduated from the Faculty of Sciences of Tunis, University Tunis El Manar (1995). He obtained the M.S. degree and the Ph.D. degree in Applied Mathe-

11