Finite-time synchronization for competitive neural networks with mixed delays and non-identical perturbations

Author’s Accepted Manuscript Finite-time synchronization for competitive neural networks with mixed delays and nonidentical perturbations Yingchun Li, Xinsong Yang, Lei Shi www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(15)02017-2 http://dx.doi.org/10.1016/j.neucom.2015.11.094 NEUCOM16582

To appear in: Neurocomputing Received date: 15 July 2015 Revised date: 8 September 2015 Accepted date: 27 November 2015 Cite this article as: Yingchun Li, Xinsong Yang and Lei Shi, Finite-time synchronization for competitive neural networks with mixed delays and nonidentical perturbations, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.11.094 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1

Finite-time synchronization for competitive neural networks with mixed delays and nonidentical perturbations Yingchun Li1 , Xinsong Yang2,∗ and Lei Shi1 1 2

Department of Mathematics, Honghe University, Mengzi 661199, China

School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

Abstract. This paper considers the drive-response synchronization in finite time of competitive neural networks (CNNs) with different time scales, time-varying and infinite-time distributed delays (mixed delays), as well as uncertain nonlinear perturbations. The drive and response systems are disturbed by different uncertain nonlinear perturbations. The effects of the nonidentical uncertain nonlinear perturbations are suppressed by designing some simple controllers. Moreover, by designing new Lyapunov-Krasovskii functionals, sufficient conditions are obtained to guarantee that the CNNs can be synchronized in a setting time without using existing finite-time stability theorem. Furthermore, the setting time is explicitly estimated for CNNs with bounded distributed delay and without delay. It is shown that the setting time is dependent on the time delays and the initial values of the coupled CNNs. Some results on synchronization of CNNs are essentially extended. Finally, numerical examples are provided to illustrate the effectiveness of the presented synchronization scheme. Keywords. Competitive neural networks, Mixed delays, Finite-time synchronization, Nonidentical perturbations.

I. I NTRODUCTION The CNN model was first proposed by Cohen and Grossberg in 1983. After that, Meyer-Ba ¨se extended the CNNs to the case that CNN with different time scales in [1], which have two types of state variables, i.e., short-term memory (STM) describing the fast neural activity and long-term memory (LTM) describing the slow ∗ Corresponding author. E-mail addresses: [email protected] (Y. Li); [email protected] (X. Yang); [email protected] (L. Shi).

2

unsupervised synaptic modifications. As a result of the broad spectrum of applications in image processing, signal processing, pattern recognition and control theory and so on, increasing attention has been attracted to the dynamics of CNNs with different time scales. In the past decades, chaos synchronization and control [2–10] have been extensively studied due to their important applications in science and engineerings such as secure communication, biological system, information processing [11–14]. Recently, synchronization of CNNs with different time scales has attracted increasing interests in various fields. By using instantaneous state feedback control technique, the authors in [15] investigated the exponential synchronization for a class of CNNs with time-varying delays. The authors in [16] studied complete synchronization of coupled delayed CNNs with different time scales and stochastic perturbations by designing an adaptive feedback controller. In [17], Yang et al. investigated the problem of adaptive lag synchronization of CNNs with mixed delays and uncertain hybrid perturbations by designing a simple but robust adaptive controller. Based on new multiple Lyapunov-Krasovkii functionals, free-weighting matrix method, Newton-Leibniz formulation, as well as the invariance principle of stochastic differential equations, exponential synchronization of switched stochastic CNNs with both interval time-varying delays and distributed delays was considered in [18]. By utilizing Lyapunov stability theory and parameter identification technique, Gan et al. [19] studied the adaptive synchronization problem of delayed CNNs with different time scales and unknown parameters. In [20], authors considered the synchronization problem for CNNs with mixed time-varying delays and stochastic perturbations by designing an adaptive feedback controller. However, as far as the authors know, most of existing results concerning synchronization of CNNs are asymptotical, from which the synchronization of CNNs with different time scales can be guaranteed only when time goes to infinity. Since the life spans of machines and human being are finite, asymptotic synchronization is inapplicable in practice. Hence, much efforts have been devoted to finite-time synchronization [21]. Finite-time synchronization implies that the synchronization can be realized in a setting time and the setting time can be computed from the parameters of the coupled chaotic systems. Therefore, compared with asymptotic synchronization, finitetime synchronization is optimal. Moreover, it was reported that finite-time control technique has better robust and disturbance rejection property than asymptotic control technique [22]. This advantage is also studied in this paper. Therefore, much attention has been attracted to finite-time synchronization. For instance, the authors in [23–26] addressed finite-time synchronization issue of multi-agent systems, and finite-time synchronization of coupled chaotic systems was investigated in [27–29]. In [30], Yang et al. considered the finite-time synchronization of linearly coupled complex networks with discontinuous state on right-hand sides. However, most of existing results on finite-time synchronization of chaotic systems did not consider time delay. Actually, most of existing results on finite-time synchronization are derived under the framework of the finitetime stability theorem in [31]. Unfortunately, it can be known from [32] that it is difficult to apply the finite-time stability theorem in [31] to time-delay system. In [33], finite-time stability of a special linear system with constant delay was investigated. But the results in [33] are not usable in practice for studying the finite-time stabilization problem. This is due to the fact that it is extremely difficult to find a Lyapunov-function satisfying the assumptions in [33]. Discrete time delay is unavoidable in practice due the finite signal transmission speed and congestion of bandwidth [15, 34]. In addition, distributed delay was widely investigated [17, 18, 35, 36] due to the presence of

3

an amount of parallel pathways with a variety of axon size and length. However, to the best of our knowledge, few results concerning finite-time synchronization of CNNs with both discrete and distributed delays are published. Therefore, it is urgent to develop some special methods to study the finite-time synchronization of CNNs with delays, which is challenging. On the other hand, uncertain disturbances are unavoidable in many practical situations which can destroy a system’s stability or make synchronization more difficult to be realized. Taking account of this, many researches have focused their attention to the synchronization of neural networks with external perturbations [16–19, 37– 40]. It should be noted that perturbations in most of existing papers are stochastic and identical in driving and response systems. However, external perturbations to the driving and response systems may be not identical. Obviously, when the driving and response CNNs are affected by nonidentical perturbations, realizing their finitetime synchronization is more difficult than that of CNNs with or without identical perturbations. Based on the above analysis, finite-time synchronization of drive-response CNNs with mixed delays and nonidentical uncertain external nonlinear perturbations is considered in the present paper. The main contributions of this paper are three-fold: (a) By designing a set of simple controllers, the effects of nonidentical uncertain external perturbations to the CNNs are overcome; (b) Without using the finite-time stability theorem in [31], new method is utilized to study the finite-time synchronization of the CNNs with delays; (c) sufficient conditions are obtained to guarantee that the CNNs with infinite-time distributed delays can be synchronized in finite time. Moreover, the setting time is estimated when there are bounded delays or without delay. Different from asymptotic synchronization, it is shown that both the STM and LTM should be controlled for finite-time synchronization. Results of this paper can be easily extended to general nonlinear time-delay systems. Numerical simulations demonstrate the effectiveness of our results. The rest of this paper is organized as follows. In Section II, the model of CNNs with mixed delays and nonidentical uncertain nonlinear perturbations is presented. Some necessary assumptions and definitions are also given in this section. Our main results and their rigorous proofs are described in Section III. In Section IV, simulation examples are provided to show the effectiveness of theresults. Conclusions are given in Section V.

II. P RELIMINARIES Consider a CNN with both time-varying delays and infinite-time distributed delays as follows: ⎧ p n n    ⎪ ⎪ ⎪ εx˙i (t) = −ci xi (t) + aij fj (xj (t)) + bij fj (xj (t − τij )) + Ei mil (t)Fl + σix (t) ⎪ ⎪ ⎪ ⎪ j=1 j=1 l=1 ⎪ ⎪ ⎨  t n  dij Kij (t − s)fj (xj (s))ds, i = 1, 2, · · · , n, + ⎪ ⎪ −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ m ˙ il (t) = −αi mil (t) + βi Fl (xi (t)), l = 1, 2, · · · , p,

(1)

where the first equation denotes the STM, the second equation denotes the LTM, n is the number of neurons, p is the number of the constant external stimulus, x i (t) is the neuron current activity level, ci > 0 is the time constant of

4

neuron, fj (xj (t)) is the output of neurons, m il (t) is the synaptic efficiency, F = (F 1 , F2 , · · · , Fp )T is the constant external stimulus, aij represents the connection weight between the ith neuron and j th neuron, E i is the strength of the external stimulus, ε is the time scale of STM state, b ij and dij represent the synaptic weight of delayed feedback, τ ij (t) represents time-varying discrete delay of the j th unit from the ith unit, K ij (t) is non-negative bounded scalar function defined on [0, +∞) describing the delay kernel of the unbounded distributed delay along the axon of the j th unit from the ith unit, αi and βi denote disposable scaling constants with α i > 0, σix (t)  t σix (t, S(t), x(t), x(t − τij ), −∞ x(s)ds) represents the nonlinear perturbations, where S i (t) = pi=1 mil (t)Fl =

mTi (t)F, (i = 1, 2, · · · , n), S(t) = (S1 (t), S2 (t), · · · , Sn (t))T , x(t) = (x1 (t), x2 (t), · · · , xn (t))T .

One has from (1) that ⎧ n n   ⎪ ⎪ ⎪ εx˙ i (t) = −ci xi (t) + aij fj (xj (t)) + bij fj (xj (t − τij )) ⎪ ⎪ ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎨  t n  d Kij (t − s)fj (xj (s))ds + Ei Si (t) + σix (t), + ij ⎪ ⎪ −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S˙ i (t) = −αi Si (t) + βi |F |2 fi (xi (t)), i = 1, 2, · · · , n, where |F |2 = F12 + F22 + · · · + Fn2

(2)

is a constant. Without loss of generality, the input stimulus vector F is

assumed to be normalized with unit magnitude |F | 2 = 1. Then (2) can be rewritten as follows: ⎧ n n 1 1 1 ⎪ ⎪ ⎪ c x ˙ (t) = − x (t) + a f (x (t)) + bij fj (xj (t − τij )) i i i ij j j ⎪ ⎪ ε ε ε ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎨  t n 1 1 1 d Kij (t − s)fj (xj (s))ds + Ei Si (t) + σix (t), + ij ⎪ ε ε ε ⎪ −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S˙ i (t) = −αi Si (t) + βi fi (xi (t)), i = 1, 2, · · · , n.

(3)

The initial condition of (3) is given as x(t) = φx (t) ∈ C([−∞, 0], Rn ), S(t) = φs (t) ∈ C([−∞, 0], Rn ). Based on the concept of drive-response synchronization, the corresponding response system of (3) is given in the following form : ⎧ n n 1 1 1 ⎪ ⎪ ⎪ y˙ i (t) = − ci yi (t) + aij fj (yj (t)) + bij fj (yj (t − τij )) ⎪ ⎪ ε ε ε ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎨  t n 1 1 y 1 ˜ (t) + Ui (t), dij Kij (t − s)fj (yj (s))ds + Ei Ri (t) + σ + ⎪ ε ε ε i ⎪ −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ R˙ i (t) = −αi Ri (t) + βi fi (yi (t)) + Qi (t), i = 1, 2, · · · , n, ˜iy (t)  σ ˜iy (t, R(t), y(t), y(t − τij ), where σ (R1 (t), R2 (t), · · · , Rn

(t))T ,

t

−∞ y(s)ds)

(4)

represents the nonlinear perturbations to (4), R(t) =

y(t) = (y1 (t), y2 (t), · · · , yn (t))T , Ui (t) and Qi (t) are the controllers to be designed.

The initial condition of (4) is given as y(t) = φy (t) ∈ C([−∞, 0], Rn ), R(t) = φR (t) ∈ C([−∞, 0], Rn ).

5

Define error states ei (t) = yi (t) − xi (t) and zi (t) = Ri (t) − Si (t). Then the following error system can be obtained from (3) and (4). ⎧ n n 1 1 1 ⎪ ⎪ ⎪ e˙ i (t) = − ci ei (t) + aij gj (ej (t)) + bij gj (ej (t − τij )) ⎪ ⎪ ε ε ε ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎨  t n 1 1 y 1 1 ˜i (t) − σix (t) + Ui (t), dij Kij (t − s)gj (ej (s))ds + Ei zi (t) + σ + ⎪ ε ε ε ε ⎪ −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z˙i (t) = −αi zi (t) + βi gi (ei (t)) + Qi (t), i = 1, 2, · · · , n,

(5)

where gj (ej (·)) = fj (yj (·)) − fj (xj (·)). The following assumptions are needed in this paper. (H1 ) There exist unknown positive constants γ i (i = 1, 2, · · · , n) such that |fi (x) − fi (y)| ≤ γi |x − y|, ∀x, y ∈ R, x = y.

(H2 ) There are positive constants τ ij and uij < 1 such that 0 < τij (t) ≤ τ ij , τ˙ ij (t) ≤ uij , i, j = 1, 2, · · · , n.  +∞ (H3 ) There exist positive constants k ij such that 0 Kij (u)du = kij , i, j = 1, 2, · · · , n.

(H4 ) For any u, v, ω, s, u˜, v˜, ω ˜ , s˜ ∈ Rn , there exist positive constants M i1 and Mi2 such that |σix (t, u, v, ω, s)| ≤ σiy (t, u˜, v˜, ω ˜ , s˜)| ≤ Mi2 for t ∈ R+ , i = 1, 2, · · · , n. Mi1 , |˜

(H5 ) The parameters αi and Ei of error systems (5) satisfy αi ≥ 1ε Ei , i = 1, 2, . . . , n. Definition 1: The system (4) is said to be synchronized with (3) in finite time (or in a setting time) if, under suitable designed feedback controllers U i (t) and Qi (t), there exists a constant t 1 > 0 (t1 > 0 depends on the initial state vector error value and the time-delay) such that lim t→t1 (e(t)1 + z(t)1 ) = 0 and e(t)1 + z(t)1 ≡ 0 for ∀t ≥ t1 , where e(t)1 + z(t)1 = ni=1 |ei (t)| + ni=1 |zi (t)|, e(t) = (e1 (t), e2 (t), · · · , en (t))T , z(t) = (z1 (t), z2 (t), · · · , zn (t))T , t1 is called the setting time.

III. M AIN

RESULTS

In this section, suitable controllers shall be designed for the finite-time stability of the zero solution of the error system (5), which is equivalent to the finite-time synchronization between (3) and (4). Several synchronization criteria are derived on the basis of 1-norm. Moreover, some important remarks are also given to show the differences of our results from existing ones. The controllers are designed as follows: Ui (t) = −li ei (t) − ρi sgn(ei (t)), Qi (t) = −ηi sgn(zi (t)),

i = 1, 2, · · · , n,

i = 1, 2, · · · , n,

where li , ρi , ηi are control gains to be determined, sgn(·) is the standard sign function. The following Theorem 1 is our main result.

(6) (7)

6

Theorem 1: Suppose that the assumptions (H 1 )-(H5 ) are satisfied. Then the CNN (4) with controllers (6) and (7) can be synchronized with (3) in a setting time if η i > 0 and li , ρi satisfy the following inequalities: n

li ≥ −

n

j=1

ρi >

n

 ci 1 1 1 + |βi |γi + |aji |γi + |bji |γi + |dji |γi k¯ji , ε ε ε(1 − uji ) ε j=1

(8)

j=1

1 Mi , ε

(9)

where Mi = max{Mi1 + Mi2 }. Define the following Lyapunov-Krasovskii functional: V (t) =

3 

Vi (t),

(10)

i=1

where V1 (t) =

n 

|ei (t)| +

i=1

n 

|zi (t)|,

i=1

 t n n 1  1 |bij |γj |ej (s)|ds, V2 (t) = ε 1 − uij t−τij i=1 j=1  0  t n n 1  |dij |γj Kij (−s)|ej (u)|duds. V3 (t) = ε −∞ t+s i=1 j=1

Evaluating the time derivative of V1 (t) along the trajectory of the error system (5), one obtains V˙ 1 (t) =

n 

n

1 1 sgn(ei (t))[− ci ei (t) + aij gj (ej (t)) ε ε i=1 j=1  t n n 1 1 bij gj (ej (t − τij )) + dij Kij (t − s)gj (ej (s))ds + ε ε −∞ j=1

j=1

1 y 1 1 ˜ (t) − σix (t) − li ei (t) − ρi sgn(ei (t))] + Ei zi (t) + σ ε ε i ε n  sgn(zi (t))[−αi zi (t) + βi gi (ei (t)) − ηi sgn(zi (t)). +

(11)

i=1

By (H1 ), one has

   t n n n 1 1 1 sgn(ei (t)) aij gj (ej (t)) + bij gj (ej (t − τij )) + dij Kij (t − s)gj (ej (s))ds ε ε ε −∞ j=1

j=1

+sgn(zi (t))βi gi (ei (t)) n n 1 1 |aij |γj |ej (t)| + |bij |γj |ej (t − τij )| ≤ ε ε j=1 j=1  t n 1 |dij |γj Kij (t − s)|ej (s)|ds + |βi |γi |ei (t)|. + ε −∞ j=1

j=1

(12)

7

It is obtained from (H4 ) that 1 y 1 sgn(ei (t))( σ ˜ (t) − σix (t)) ≤ ε i ε ≤ =

1 y (|˜ σ (t)| + |σix (t)|)|sgn(ei (t))| ε i 1 (M 1 + Mi2 )|sgn(ei (t))| ε i 1 Mi |sgn(ei (t))|. ε

(13)

When ei (t) = 0, it is found that −sgn(ei (t))ρi sgn(ei (t)) = −ρi , otherwise −sgn(ei (t))ρi sgn(ei (t)) = 0. Similarly, −sgn(zi (t))ηi sgn(zi (t)) = −ηi for zi (t) = 0 and −sgn(zi (t))ηi sgn(zi (t)) = 0 for zi (t) = 0. Therefore, −sgn(ei (t))ρi sgn(ei (t)) = −ρi λi ,

(14)

¯i, −sgn(zi (t))ηi sgn(zi (t)) = −ηi λ

(15)

¯ i = 1 if |zi (t)| = 0, otherwise λ ¯ i = 0. where λi = 1 if |ei (t)| = 0, otherwise λi = 0; λ

Substituting (12)-(15) into (11), it is derived that V˙ 1 (t) ≤

n 

n

ci 1 − ( + li )|ei (t)| + |βi |γi |ei (t)| + |aij |γj |ej (t)| ε ε i=1 j=1  t n n 1 1 |bij |γj |ej (t − τij )| + |dij |γj Kij (t − s)|ej (s)|ds + ε ε −∞ j=1 j=1

Mi 1 ¯ − ρi )λi − ηi λi . + Ei |zi (t)| − αi |zi (t)| + ( ε ε

(16)

It is obtained from V2 (t), V3 (t), (H2 ), and (H3 ) that n

V˙ 2 (t) =

n

n

n

1  1 1  1 |bij |γj |ej (t)| − |(1 − τ˙ ij (t))|bij |γj |ej (t − τij (t))| ε 1 − uij ε 1 − uij ≤

i=1 j=1 n  n 

1 ε

i=1 j=1

i=1 j=1 n  n 

1 1 |bij |γj |ej (t)| − 1 − uij ε

|bij |γj |ej (t − τij (t))|,

(17)

i=1 j=1

and  0 n n 1  Kij (−s)|ej (t)|ds − |dij |γj Kij (−s)|ej (t + s)|ds ε −∞ −∞ i=1 j=1 i=1 j=1  t n n n n 1  1  |dij |γj kij |ej (t)| − |dij |γj Kij (t − s)|ej (s)|ds. ε ε −∞ n

V˙ 3 (t) = =

n

1  |dij |γj ε

i=1 j=1



0

i=1 j=1

(18)

8

From (16)-(18), one has ⎡ ⎤ n n n n c      1 1 1 ⎣− i + li + |βi |γi + |aji |γi + |bji |γi + |dji |γi kji ⎦ |ei (t)| V˙ (t) ≤ ε ε ε(1 − uji ) ε i=1 j=1 j=1 j=1    n

n 

  1 Mi ¯i . Ei − αi |zi (t)| + − ρi λi − ηi λ (19) + ε ε i=1

i=1

When e(t)1 + z(t)1 = 0, one can obtain from (H 5 ), (8), (9), and (19) that V˙ (t) ≤ −

n 

¯ i ) ≤ −θ0 < 0, (δi λi + ηi λ

(20)

i=1

where δi = ρi − 1ε Mi > 0, θ0 = min{δi , ηi , i = 1, 2, . . . , n}. Because V (t) is positive definite and non-increasing, known from (20), there exists nonnegative constant V

∗

such that lim V (t) = V ∗

t→+∞

and

V (t) ≥ V ∗ ,

∀t ≥ 0.

(21)

Integrating both sides of the inequality (20) from 0 to t gets the following inequality: V (t) − V (0) ≤ −θ0 t.

(22)

Now we prove that there exists t 1 ∈ (0, +∞) such that lim (e(t)1 + z(t)1 ) = 0 and

t→t1

e(t)1 + z(t)1 ≡ 0,

∀t ≥ t1 .

(23)

Firstly, we prove that limt→t1 (e(t)1 + z(t)1 ) = 0. If e(t)1 + z(t)1 > 0 for all t ∈ [0, +∞), then there exists i0 ∈ {1, 2, · · · , n} or i01 ∈ {1, 2, · · · , n} such that |ei0 (t)| > 0 or |zi01 (t)| > 0, from this and (20) one can get that V˙ (t) ≤ −θ0 < 0. In this case, the inequality (22) means that lim t→+∞ V (t) = −∞ ,which contradicts (21). Therefore, there exist t 1 ∈ (0, +∞) such that lim t→t1 (e(t)1 + z(t)1 ) = 0. Next, we prove that e(t)1 + z(t)1 ≡ 0 for ∀t ≥ t1 . On the contrary, there exists t 2 > t1 such that e(t2 )1 + z(t2 )1 > 0. Let ts = sup{t ∈ [t1 , t2 ] : e(t)1 + z(t)1 = 0}, we can get e(ts )1 + z(ts )1 = 0

when ts < t2 , and e(t)1 + z(t)1 > 0 for all t ∈ [ts , t2 ]. Moreover, e(t)1 + z(t)1 is monotonously increasing on the interval [t s , t3 ], where t3 ∈ (ts , t2 ]. Therefore V (t) is also monotonously increasing on the interval [ts , t3 ], i.e., V˙ (t) > 0 for t ∈ (ts , t3 ]. On the other hand, known from e(t) 1 + z(t)1 > 0 for all t ∈ (ts , t3 ], there exists at least one i 0 ∈ {1, 2, · · · , n} or i01 ∈ {1, 2, · · · , n} such that |ei0 (t)| > 0 or |zi01 (t)| > 0 at any instant t ∈ (ts , t3 ]. By the argument as above, it follows that V˙ (t) ≤ −θ0 < 0 hold for the instant t ∈ (ts , t3 ], this contradicts V˙ (t) > 0 for t ∈ (ts , t3 ]. So e(t)1 + z(t)1 ≡ 0 for ∀t ≥ t1 . Based on the above discussions, one can obtain that the conditions in (23) hold. According to Definition 1, the CNN (4) is synchronized with CNN (3) in a finite time under the controllers (6) and (7). This completes the proof.

9

Remark 1. Theorem 1 is also applicable to finite-time synchronization of CNNs with identical external

perturbations. In [17], Yang et al. considered adaptive lag synchronization for CNNs with mixed delays as well as uncertain nonlinear external and stochastic perturbations, where external perturbations imposed onto driving and response systems are identical. In real applications, external perturbations on driving and response systems may be different. In this sense, Theorem 1 essentially extend the results in [17]. It is discovered that asymptotic synchronization can be achieved by controlling the STM of CNNs [15–19], while both the STM and LTM should be controlled for finite-time synchronization. Remark 2. When the initial value of (5) is not zero, the value of V 3 (t) is a positive constant when the

synchronization has been realized due to the infinite-time integral. However, the value of V 3 (t) cannot be estimated from the proof of Theorem 1. Hence, the setting time t 1 cannot be explicitly estimated when there are infinite-time distributed delays in CNNs. Remark 3. The finite-time stability theorem in [31] is based on the inequality V˙ (x) ≤ −αV η (x), where α > 0, 0 < η < 1 are constants, α1 (|x|) ≤ V (x) ≤ α2 (|x|) with K-class functions α1 (·), α2 (·). Theorem 1 is

derived on the basis of 1-norm, where the key step is to obtain the inequality (20). If the 2-norm based Lyapunov functions as those in [23–30] are used, the inequality (20) cannot be obtained. If σix (t) ≡ 0 and σ ˜iy (t) = 0 or σix (t) = 0 and σ ˜iy (t) ≡ 0, it is obvious that the controllers (6) and (7) can

˜iy (t) ≡ 0, then (5) turns into the following form: synchronize the CNNs. Especially, if σ ix (t) = σ

⎧ n n 1 1 1 ⎪ ⎪ ⎪ c e ˙ (t) = − e (t) + a g (e (t)) + bij gj (ej (t − τij )) i i i ij j j ⎪ ⎪ ε ε ε ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎨  t n  1 1 dij Kij (t − s)gj (ej (s))ds + Ei zi (t) + Ui (t), + ⎪ ε ε ⎪ −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z˙i (t) = −αi zi (t) + βi gi (ei (t)) + Qi (t), i = 1, 2, · · · , n.

(24)

Theorem 2: Suppose that the assumptions (H 1 )-(H3 ) and (H5 ) hold and σix (t) = σ ˜iy (t) ≡ 0. Then the CNN (4) under controllers (6) and (7) can be synchronized with (3) in a setting time if ρ i , ηi > 0 and li satisfy the following inequalities: n

n

n

j=1

j=1

j=1

 ci 1 1 1 li ≥ − + |βi |γi + |aji |γi + |bji |γi + |dji |γi k¯ji . ε ε ε(1 − uji ) ε

(25)

Define the Lyapunov-Krasovskii functional candidate as in Theorem 1. Evaluating the time derivative of V (t) along the trajectory of the error system (24), by the same procedure the proof of Theorem 1, one obtain V˙ (t) ≤ −

n 

¯ i ) ≤ −θ1 < 0, (ρi λi + ηi λ

i=1

where θ1 = min{ρi , ηi , i = 1, 2, . . . , n}. By using the same discussion method in the Theorem 1, one can easily finish this proof.

(26)

10

When the delay kernels satisfy the following condition: ⎧ ⎪ ⎪ ⎨0 Kij (t) =

⎪ ⎪ ⎩1

t > θij ,

(27) 0 ≤ t ≤ θij ,

where θij > 0 (i = 1, 2, · · · , n) are constants, then the model (3) becomes the following CNN with bounded distributed delays: ⎧ n n 1 1 1 ⎪ ⎪ ⎪ x˙ i (t) = − ci xi (t) + aij fj (xj (t)) + bij fj (xj (t − τij )) ⎪ ⎪ ε ε ε ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎨  t n 1 1 1 d fj (xj (s))ds + Ei Si (t) + σix (t), + ij ⎪ ε ε ε ⎪ t−θij ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S˙ i (t) = −αi Si (t) + βi fi (xi (t)), i = 1, 2, · · · , n.

(28)

Corresponding, the error systems (5) becomes the following form: ⎧ n n 1 1 1 ⎪ ⎪ ⎪ c e ˙ (t) = − e (t) + a g (e (t)) + bij gj (ej (t − τij )) i i i ij j j ⎪ ⎪ ε ε ε ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎨  t n  1 1 y 1 1 ˜i (t) − σix (t) + Ui (t), dij gj (ej (s))ds + Ei zi (t) + σ + ⎪ ε ε ε ε ⎪ t−θij ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z˙i (t) = −αi zi (t) + βi gi (ei (t)) + Qi (t).

(29)

It is obvious that the Theorem 1 is applicable to the system (29). The following corollary 1 shows that the synchronization time can also be estimated for the CNN (29) with bounded mixed delays and external perturbations. Corollary 1: Suppose that assumption conditions (H 1 ), (H2 ), (H4 ) (H5 ) are satisfied. Then the CNN (3) and (4) with the delay kernels K ij (t) satisfying (27) are synchronized in a finite-time under the controllers (6) and (7) when ηi > 0 and li , ρi satisfy the following inequalities:  n

ci γi  1 li ≥ − + |aji | + |bji | + |dji |θji + |βi |γi . ε ε 1 − uji

(30)

j=1

1 ρi > Mi . ε Moreover, the setting time is estimated as  n  0 n n 1  1  1 t1 ≤ |ei (0)| + |zi (0)| + |bij |γj |ej (s)|ds θ0 ε 1 − uij −τ ij i=1 i=1 j=1   0  0 n n 1  |dij |γj |ej (u)|duds − max{τ ij , θij , i, j = 1, 2, · · · , n}, + ε −θij s i=1 j=1

where θ0 = min{δi , ηi , i = 1, 2, · · · , n}, δi = ρi − 1ε Mi .

(31)

(32)

11

Define the Lyapunov-Krasovskii functional candidate as following: V (t) = V1 (t) + V2 (t) + V3 (t),

(33)

where V1 (t), V2 (t) are defined as those in (10) and n

n

1  V3 (t) = |dij |γj ε i=1 j=1



0

−θij



t

t+s

|ej (u)|duds.

Using the same discussion method in the Theorem 1 and the condition (30), (31), one can get that ˙ V (t) ≤ −θ0 ,

(34)

¯ i = 1 if |zi (t)| = 0, otherwise λ ¯ i = 0. where δi = ρi − 1ε Mi > 0, λi = 1 if |ei (t)| = 0, otherwise λi = 0; λ

Integrating both sides of the inequality (34) from 0 to t gets the following inequality: Vˆ (t) − Vˆ (0) ≤ −θ0 t.

(35)

Arguing as in the proof Theorem 1, there exists t 1 ∈ (0, +∞) such that lim (e(t)1 + z(t)1 ) = 0 and

t→t1

e(t)1 + z(t)1 ≡ 0,

∀t ≥ t1 .

(36)

From (33) and (36), it is obvious that there exists t 2 = t1 + max{τ ij , θij , i, j = 1, 2, · · · , n} such that lim Vˆ (t) = 0

t→t2

and

Vˆ (t) ≡ 0,

∀t ≥ t2 .

(37)

˙ From (34) and (37), we can get that V (t) ≤ −θ0 for t < t2 . Integrating both sides of the inequality from 0 to t2 obtains that: t2 ≤

V (0) . θ0

Hence, t1 ≤ t2 − max{¯ τij , θij , i, j = 1, 2, · · · , n}. This completes the proof. Remark 4. From Theorem 1 and Corollary 1, it can be found that the control gains l i , i = 1, 2, . . . , n are used to

stabilize the error system, while the control gains ρ i and ηi , i = 1, 2, · · · , n play an important role in overcoming the bad effects of nonidential external perturbations and finite-timely stabilize the error system to zero solution. Moreover, one can see from (32) that they can be used to tune the setting time. Generally, larger values of ρ i and ηi , i = 1, 2, · · · , n lead to smaller setting time. Numerical simulations in the next section demonstrate this discussion. If the delay τij = τ is a constant, we can obtain the following corollary from Theorem 1. Corollary 2: Suppose that the assumption conditions (H 1 ), (H3 )-(H5 ) are satisfied. Then the CNN (4) can be synchronized with (3) in a finite-time under the controllers (6) and (7) when η i > 0 and li , ρi satisfy the

12

following inequalities: li ≥ −

n

n

n

j=1

j=1

j=1

ci 1 1 1 + |βi |γi + |aji |γi + |bji |γi + |dji |γi k¯ji , ε ε ε ε

1 ρi > Mi . ε

(38)

(39)

Define the following Lyapunov-Krasovskii functional candidate: V¯ (t) = V1 (t) + Vˇ2 (t) + V3 (t),

(40)

where V1 (t), V3 (t) are defined as those in (10) and n

n

1  |bij |γj Vˇ2 (t) = ε i=1 j=1



t t−τ

|ej (s)|ds.

By using the same discussion methods as those in the proof of Theorem 1, one can complete the proof. In the case that bij = dij = 0, i, j = 1, 2, · · · , n in (3) and (4), the following corollary can be obtained from Corollary 1. Corollary 3: Let bij = dij = 0, i, j = 1, 2, · · · , n. Assume that the assumptions (H 1 ), (H4 ), (H5 ) hold, then the CNN (3) and (4) can be synchronized in a setting time under the controllers (6), (7) when η i > 0, and li , ρi , i = 1, 2, · · · , n satisfy the following inequalities n ci 1 li ≥ − + |βi |γi + |aji |γi , ε ε

(41)

j=1

1 ρi > Mi . ε Moreover, the setting time is estimated as t1 ≤

(42) n 1  [|ei (0)| + |zi (0)|] , θ0 i=1

where θ0 = min{δi , ηi , i = 1, 2, · · · , n}, δi = ρi − 1ε Mi . Define the following Lyapunov-Krasovskii functional candidate:  (t) = V1 (t), V

(43)

where V1 (t) is defined as those in (10). Similar to Theorem 1, evaluating the time derivative of V 1 (t) along the trajectory of the error system (5), one gets V˙ (t) ≤ −

n 

¯ i ). (δi λi + ηi λ

j=1

The rest step of the proof is same as those in the Corollary 1. This proof is completed.

(44)

13

IV. N UMERICAL E XAMPLES In this section, three numerical examples are given to demonstrate the effectiveness of the above theoretical results. Example 1: Consider the following 2-dimensional CNN with both mixed delay and uncertain nonlinear perturbation ⎧ n n 1 1 1 ⎪ ⎪ ⎪ x˙ i (t) = − ci xi (t) + aij fj (xj (t)) + bij fj (xj (t − τij )) ⎪ ⎪ ε ε ε ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎨  t n 1 1 1 + dij Kij (t − s)fj (xj (s))ds + Ei Si (t) + σix (t), ⎪ ε ε ε ⎪ −∞ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S(t) ˙ = −αi Si (t) + βi fi (xi (t)), i = 1, 2, · · · , n,

(45)

where x(t) = (x1 (t), x2 (t)), f (x(t)) = (tanh(x1 (t)), tanh(x2 (t)))T , τij (t) = 1, Kij (t) = e−0.5t , i, j = 1, 2, ε = 0.5, and

⎛

⎞

⎛

⎞

⎛

⎞

⎜ 2.4 0 ⎟ ⎜ 3 −0.3 ⎟ ⎜ −1.4 0.1 ⎟ ⎟,A = ⎜ ⎟,B = ⎜ ⎟, C=⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 8 8 5 0.3 −8 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ −1.2 0.1 ⎟ ⎜ 0.3 0 ⎟ ⎜ 0.2 0 ⎟ ⎟,E = ⎜ ⎟,α = ⎜ ⎟, D=⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ −2.8 −2 0 1.0 0 1.5 ⎛ ⎞ ⎛ ⎞ ⎜ 0.5 0 ⎟ ⎜ 0.01 ⎟ ⎟ , σ(x) = ⎜ ⎟. β=⎜ ⎝ ⎠ ⎝ ⎠ 0 0.3 0.1 In the case that initial conditions are chosen as x(t) = (0.4, 0.6) T , S(t) = (0.1, 0.6)T , ∀t ∈ [−1, 0], the trajectory of (45) is shown in Fig. 1. It is easy to get that τ¯ij = 1 and uij = 0 < 1, k¯ij = 2, i, j = 1, 2, γ1 =  n 1 γ2 = 1. Therefore, conditions (H 1 ) − (H4 ) are satisfied. Take l 1 = − cε1 + |β1 |γ1 + 1ε γ1 j=1 |aj1 | + 1−uji |bj1 | +    n |aj2 |+ 1 |bj2 |+ |dj2 |k¯j2 = 19.14. According to Theorem 1, |dj1 |k¯j1 = 37.1, l2 = − c2 + |β2 |γ2 + 1 γ2 ε

ε

j=1

1−uji

drive-response synchronization of the CNNs can be realized in a setting time under the controllers (6) and (7). In the simulations, the Euler scheme is used, the step size is taken as 0.0005. Take the system (45) as the drive system, the initial value is the same as that in Fig. 1. Choose the initial value of the response system as y(t) = (2, 1)T , R(t) = (0.3, 0.1)T , ∀t ∈ [−1, 0], respectively. Let σ(y) = (0.1, 1.0)T . By simple computation,

we get 1ε M1 = 1ε M2 = 0.22. Take ρ1 = ρ2 = 2.5 > 0.22, η1 = η2 = 3. Fig. 2 describes the time evolution of e(t)1 and z(t)1 with value of ρi and ηi , i = 1, 2, from which one can see that e(t) 1 reaches zero at 1.0

and z(t)1 at 1.15. Example 2: Now we consider the CNN

(28). Where x(t)

=

(x 1 (t), x2 (t)), f (x(t))

=

14

2.5

0.6

2

0.5

1.5 0.4 1 0.3 s2 (t)

x2 (t)

0.5 0 −0.5

0.2 0.1

−1 0 −1.5 −0.1

−2 −2.5 −1.5

−1

−0.5

0 x1 (t)

0.5

1

−0.2 −0.4

1.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

s1 (t)

Fig. 1. Trajectories of x(t) (a) and S(t) (b) of (45) with initial conditions x(t) = (0.4, 0.6)T , S(t) = (0.1, 0.6)T , ∀t ∈ [−1, 0]. 1

3 ρ1=ρ2=2.5

η1=η2=3

0.8

2.5

0.6 2 0.4 0.2 z(t)1

e(t)1

1.5 1

0 −0.2

0.5

−0.4 0 −0.6 −0.5 −1

−0.8 1

1.1

1.2

1.3

1.4

1.5

−1

1

1.1

t

1.2

1.3

1.4

1.5

t

Fig. 2. Time evolution of e(t)1 and z(t)1 with ρ1 = ρ2 = 2.5 and η1 = η2 = 3.

(tanh(x1 (t)), tanh(x2 (t)))T , τij (t) = 1, θij (t) = 0.3, i, j = 1, 2, ε = 2.5, and ⎛ ⎞ ⎛ ⎞ ⎛

⎞

⎜ 1.2 0 ⎟ ⎜ 3 −0.3 ⎟ ⎜ −1.4 0.1 ⎟ ⎟,A = ⎜ ⎟,B = ⎜ ⎟, C=⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 1 6 5 0.3 −8 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ −1.2 −0.1 ⎟ ⎜ 0.3 0 ⎟ ⎜ 2 0 ⎟ ⎟,E = ⎜ ⎟,α = ⎜ ⎟, D=⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ −2 −2 0 1.5 0 1.5 ⎛ ⎞ ⎛ ⎞ ⎜ 0.5 0 ⎟ ⎜ 0.01 ⎟ ⎟ , σ(x) = ⎜ ⎟. β=⎜ ⎝ ⎠ ⎝ ⎠ 0 0.3 0.1 In the case that initial conditions are chosen as x(t) = (0.4, 0.6) T , S(t) = (0.1, 0.6)T , ∀t ∈ [−1, 0], the trajectory of (28) is presented in Figure 3. It is easy to get that τ¯ij = 1 and uij = 0 < 1, θij = 0.3, i, j = 1, 2, γ1 = γ2 = 1. Take l1 = − cε1 +  n   n c2 1 1 1 |β1 |γ1 + 1ε γ1 j=1 |aj1 | + 1−uji |bj1 | + |dj1 |θj1 = 4.684, l2 = − ε + |β2 |γ2 + ε γ2 j=1 |aj2 | + 1−uji |bj2 | +  |dj2 |θj2 = 5.512. According to Corollary 1, the CNNs can be realized finite-time synchronization under the controllers (6), (7) for any positive constants ρ i > 1ε Mi and ηi , i = 1, 2. Take the system (28) as the drive

15

system, the initial value is the same as that in Fig. 3, choose the initial value of the response system as y(t) = (2, 1)T , R(t) = (0.3, 0.1)T , ∀t ∈ [−1, 0], respectively, and set σ(y) = (0.1, 1.0) T . Then we can get ρ 1 > 1 ε M1

1.1 = 0.044, ρ2 > 1ε M2 = 1.1 2.5  = 0.44, θ = 1 − 2.5 = 0.56, |e1 (0)| = 1.6, |e2 (0)| = 0.4, |z1 (0)| = 0 n 1 n n 1 + 0.2, |z2 (0)| = 0.5. Therefore, t1 ≤ θ10 i=1 |ei (0)| + |zi (0)| + ε i=1 j=1 1−uij |bij |γj −τ ij |ej (s)|ds  0 0 1 n n i=1 j=1 |dij |γj −θij s |ej (u)|duds − max{τ ij , θij , i, j = 1, 2, · · · , n} = 6.29. Fig. 4 describe the time ε

=

0.11 2.5

evolution of e(t)1 and z(t)1 with value of ρ1 = ρ2 = 1 and η1 = η2 = 1. Moreover, the synchronization can be realized before 6.29. The theoretical analysis of the Corollary 1 is verified. 5

0.6

4

0.5

3

0.4

2

s2 (t)

x2 (t)

0.3 1

0.2

0 0.1 −1 0

−2

−0.1

−3 −4 −0.2

−0.1

0

0.1

0.2 x1 (t)

0.3

0.4

0.5

−0.2 −0.04

0.6

−0.02

0

0.02

0.04 s1 (t)

0.06

0.08

0.1

0.12

Fig. 3. Trajectories of x(t) (a) and S(t) (b) of (28) with initial conditions x(t) = (0.4, 0.6)T , S(t) = (0.1, 0.6)T , ∀t ∈ [−1, 0].

1

1 η1=η2=1

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8 −1

η1=η2=1

0.8

z(t)1

z(t)1

0.8

−0.8 1

1.2

1.4

1.6

1.8

−1

2

1

1.2

1.4

t

1.6

1.8

2

t

Fig. 4. Time evolution of e(t)1 and z(t)1 with ρ1 = ρ2 = 1 and η1 = η2 = 1.

3

0.64

0.62

2

0.6 1

s2 (t)

x2 (t)

0.58 0

0.56 −1 0.54 −2

−3 −0.8

0.52

−0.6

−0.4

−0.2

0 x1 (t)

0.2

0.4

0.6

0.5 −0.02

0

0.02

0.04 s1 (t)

0.06

0.08

0.1

Fig. 5. Trajectories of x(t) (a) and S(t) (b) of (46) with initial conditions x(t) = (0.4, 0.6)T , S(t) = (0.1, 0.6)T , ∀t ∈ [−1, 0].

16

3

3 ρ1=ρ2=5

η1=η2=5 2.5

2

2

1.5

1.5 z(t)1

e(t)1

2.5

1 0.5

1 0.5

0

0

−0.5

−0.5

−1

1

1.05

1.1 t

1.15

1.2

−1

1

1.05

1.1 t

1.15

1.2

Fig. 6. Time evolution of e(t)1 and z(t)1 with ρ1 = ρ2 = 5 and η1 = η2 = 5.

Example 3: Consider the following CNN without delay ⎧ n ⎪ 1 1 1 1 ⎪ ⎪ aij fj (xj (t)) + Ei Si (t) + σix (t), ⎪ ⎨ x˙ i (t) = − ε ci xi (t) + ε ε ε j=1 ⎪ ⎪ ⎪ ⎪ ˙ ⎩ S(t) = −αi Si (t) + βi fi (xi (t)), i = 1, 2, · · · , n, where x(t) = (x1 (t), x2 (t)), f (x(t)) = (tanh(x1 (t)), tanh(x2 (t)))T , ε = 0.04, and ⎛ ⎞ ⎛ ⎞ ⎛

(46)

⎞

0 ⎟ ⎜ 2.4 0 ⎟ ⎜ 2.6 −0.3 ⎟ ⎜ 0.4 ⎟,A = ⎜ ⎟,E = ⎜ ⎟, C=⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 2.4 6 5 0 0.001 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 ⎟ ⎜ 1.6 ⎜ 0.5 0 ⎟ ⎜ 0.01 ⎟ ⎟,β = ⎜ ⎟ , σ(x) = ⎜ ⎟. α=⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0.06 0 0.3 0.08

In the case that initial conditions are chosen as x(t) = (0.4, 0.6) T , S(t) = (0.1, 0.6)T , the trajectory of (45) can be seen in Figure 5. It is easy to verify all the conditions of Corollary 3 are satisfied if take l 1 = − cε1 + |β1 |γ1 + 1ε γ1 ( nj=1 |aj1 |) = 165.7, l2 = − cε2 + |β2 |γ2 + 1ε γ2 ( nj=1 |aj2 |) = 72.8. Take the system (46) as the drive system, the initial value is

the same as that in Figure 6, choose the initial value of the response system as y(t) = (2, 1) T , R(t) = (0.3, 0.1)T ,

respectively, σ(y) = (0.1, 0.1)T . We can get ρ1 > 1ε M1 =

0.11 0.04

0.18 = 2.75, ρ2 > 1ε M2 = 0.18 0.04 = 4.5, θ0 = 5 − 0.04 = 0.5, |e1 (0)| = 1.6, |e2 (0)| = 0.4, |z1 (0)| = 0.2, |z2 (0)| = 0.5. Therefore, t1 ≤ θ10 ni=1 [|ei (0)| + |zi (0)|] = 5.4.

Figure 6 describes the time evolution of e(t) 1 and z(t)1 with value of ρ1 = ρ2 = 5 and η1 = η2 = 5. Moreover, the synchronization is realized before 5.4.

V. C ONCLUSION In this paper, the finite-time synchronization of CNNs with mixed delays and nonidentical uncertain nonlinear perturbations has been investigated. By designing a set of simple controllers, new analytical techniques and

17

Lyapunov-Krasovskii function, several synchronization criteria have been derived to guarantee the finite-time synchronization. Moreover, the setting time is also given for CNNs with bounded mixed delays and without delay. Finally, three illustrated examples with their simulations have been given to demonstrate the effectiveness of the theoretical results.

ACKNOWLEDGMENTS This work was jointly supported by the National Natural Science Foundation of China (NSFC) under Grants No. 61263020, the Program of Chongqing Innovation Team Project in University under Grant No. KJTD201308, the First Batch of Middle and Young Aged Academic Backbone of Honghe University under Grant No. 2014GG0102 and the Scientific Research Fund of Honghe University under Grant No. XJ15SX04.

R EFERENCES [1] A. Meyer-Ba ¨se, F. Ohl, H. Scheich, Singular perturbation analysis of competitive neural networks with different time-scales, Neural Computation 8 (8) (1996) 1731–1742. [2] J. Cao, G. Chen, P. Li, Global synchronization in an array of delayed neural networks with hybrid coupling, IEEE Transactions on Systems, Man and Cybernetics Part B: Cybernetics 38 (2) (2008) 488–498. [3] W. Yu, J. Cao, W. Lu, Synchronization control of switched linearly coupled neural networks with delay, Neurocomputing 73 (2010) 858–866. [4] G. Zhang, T. Wang, T. Li, S. Fei, Exponential synchronization for delayed chaotic neural networks with nonlinear hybrid coupling, Neurocomputing 85 (2012) 53–61. [5] J. Wang, H. Wu, Adaptitive output synchronization of complex delayed dynamical networks with output coupling, Neurocomputing 142 (2014) 174–181. [6] Q. Zhu, W. Zhou, D. Tong, J. Fang, Adaptitive synchronization for stochastic neural networks of neutral-type with mixed time-delays, Neurocomputing 99 (2013) 477–485. [7] J. Lu, D.W.C. Ho, Globally Exponential Synchronization and Synchronizability for General Dynamical Networks, IEEE Transactions on Systems, Man and Cybernetics. Part B: Cybernetics 40 (2) (2010) 350– 361. [8] J. Lu, J. Zhong, Y. Tang, T. Huang, J. Cao, J. Kurths, Synchronization in output-coupled temporal Boolean networks, Scientific Reports 6292 (4) (2014) DOI:10.1038/srep06292. [9] R. Lu, W. Yu, J. Lu, A. Xue, Synchronization on Complex Networks of Networks, IEEE Transactions on Neural Networks and Learning Systems 25 (11) (2014) 2110–2118. [10] B. Xu, Z. Shi, C. Yang, F. Sun, Composite Neural Dynamic Surface Control of a Class of Uncertain Nonlinear Systems in Strict-Feedback Form, IEEE Transactions on Cybernetics 44 (12) (2014) 2626–2634. [11] X. Yang, J. Cao, J. Lu, Synchronization of delayed complex dynamical networks with impulsive and stochastic effects, Nonlinear Analysis B: Real World Applications 12 (4) (2011) 2252–2266. [12] S. H. Strogatz, I. Stewart, Coupled oscillators and biological synchronization, Scientific American 269 (6) (1993) 102–109.

18

[13] T. Liao, N. Huang. An observe-based approach for chaotic synchronization with applications to secure communications, IEEE Transactions on Automatic Control 46 (9) (1999) 1144–1150. [14] J. Cao, Z. Wang, Y. Sun, Synchronization in an array of linearly stochastically coupled networks with time delays, Physica A 385 (2) (2008) 718–728. [15] X. Lou, B. Cui, Synchronization of competitive neural networks with different time-scale, Physica A 380 (2007) 563–576. [16] H. Gu, Adaptitive synchronization for competitive neural networks with different time scales and stochastic perturbation, Neurocomputing 73 (1-3) (2009) 350–356. [17] X. Yang, J. Cao, Y. Long, W. Rui, Adaptive lag synchronization for competitive neural networks with mixed delays and uncertain hybrid perturbations, IEEE Transactions on Neural Networks 21 (10) (2010) 1656–1667. [18] X. Yang, C. Huang, J. Cao, An LMI approach for exponential synchronization of switched stochastic competitive neural networks with mixed delays, Neural Computing and Applications 21 (8) (2012) 2033– 2047. [19] Q. Gan, R. Xu, X. Kang, Synchronization of unknown chaotic delayed competitive neural networks with different time scales based on adaptive control and parameter identification, Nonlinear Dynamics 67 (2012) 1893–1902. [20] Q. Gan, R. Hu, Y. Liang, Adaptive synchronization for stochastic competitive neural networks with timevarying delays, Communications in Nonlinear Science and Numerical Simulation 17 (9) (2012) 3708–3718. [21] X. Yang, D. Ho, J. Lu, Q. Song, Finite-time cluster synchronization of TS fuzzy complex networks with discontinuous subsystems and random coupling delays, IEEE Transactions on Fuzzy Systems, 10.1109/TFUZZ.2015.2417973. [22] S. Bhat, D. Bernstein, Finite-time stability of continuous autonomous systems, SIAM Journal on Control and Optimization 38 (3) (2000) 751–766. [23] Q. Hui, W. M. Haddad, S. P. Bhat, Finite-time semistability and consensus for nonlinear dynamical networks, IEEE Transactions on Automatic Control 53 (8) (2008) 1887–1900. [24] J. Corte´s, Finite-time conveergent gradient flows with applications to network consensus, Automatica 42 (11) (2006) 1993–2000. [25] F. Xiao, L. Wang, J. Chen, Y. Gao, Finite-time formation control for multi-agent systems, Automatica 45 (11) (2009) 2605–2611. [26] L. Wang, F. Xiao, Finite-time consensus problems for networks of dynamic agents, IEEE Transactions on Automatic Control 55 (4) (2010) 950–955. [27] X. Yang, J. Cao, Finite-time stochastic synchronization of complex networks, Applied Mathematical Modelling 34 (11) (2010) 3631–3641. [28] J. Shen, J. Cao, Finite-time synchronization of coupled neural networks via discontinuous controllers, Cognitive Neurodynamics 5 (4) (2011) 373–385. [29] M. P. Aghababa, H. P. Aghababa, Synchronization of mechanical horizontal platform systems in finite time, Applied mathematical Modelling 36 (10) (2012) 4579–4591. [30] X. Yang, Z. Wu, J. Cao, Finite-time synchronization of complex networks with nonidentical discontinuous activations, Nonlinear Dynamics 73 (4) (2013) 2431–2439.

19

[31] Y. Tang, Terminal sliding mode control for rigid robots, Automatica 34 (1) (1998) 51–56. [32] D. Efimov, A. Polyakov, E. Fridman, W. Perruquetti, J. Richard, Comments on finite-time stability of timedelay systems, Automatica 50 (7) (2014) 1994–1997. [33] E. Moulay, M. Dambrine, N. Yeganefar, W. Perruquetti, Finite-time stability and stabilization of time-delay systems, Systems and Control Letters 57 (7) (2008) 561–566. [34] X. Yang, J. Cao, J. Lu, Synchronization of randomly coupled neural networks with markovian jumping and time-delay, Circuits and Systems I: Regular Papers, IEEE Transactions on 60 (2) (2013) 363–376. [35] T. Li, S. Fei, K. Zhang, Synchronization control of recurrent neural networks with distributed delays, Physica. Section A: Statistical and Theoretical Physics 387 (4) (2008) 982–996. [36] T. Li, S. Fei, Y. Guo, Synchronization control of chaotic neural networks with time-varying delays and distributed delays, Nonlinear Analysis: Theory, Methods & Applications 71 (5-6) (2009) 2372–2384. [37] X. Yang, J. Cao, Stochastic synchronization of coupled neural networks with intermittent control, Physics Letters A 373 (36) (2009) 3259–3273. [38] Y. Sun, J. Cao, Z. Wang, Exponential synchronization of stochastic perturbed chaotic delayed neural networks, Neurocomputing 70 (13-15) (2007) 2477–2485. [39] J. Lu, D. W. C. Ho, L. Wu, Exponential stabilization in switched stochastic dynamical networks, Nonlinearity 22 (2009) 889–911. [40] J. Cai, C. Wen, H. Su, Z. Liu, Robust Adaptive Failure Compensation of Hysteretic Actuators for a Class of Uncertain Nonlinear Systems, IEEE Transactions on Automatic Control 58(9) (2013) 2388–2394.

Yingchun Li received the B.S. degree in mathematics in 2001 from Shaoyang University, Shaoyang, China. She received the M.S. degree in applied mathematics from Changsha University of Science & Technology, Changsha, China, in 2008. She is currently a lecturer with the Department of Mathematics, Honghe University, Yunan, China. Her current research interests include synchronization and control, discontinuous dynamical systems, and neural networks.

Xinsong Yang received the B.S. degree in mathematics from Huaihua Normal University, Hunan, China, in 1992, and the M.S. degree in mathematics from Yunnan University, Yunnan, China, in 2006. From 2006 to 2012, he was with Honghe University, Honghe State, Yunnan. He is currently a professor with the Department of Mathematics, Chongqing Normal University, Chongqing. From 2008 to 2009, he was a visitings cholar with the Department of Mathematics, Southeast University, China. He was a visiting research fellow of City University of Hong Kong and the University Of Hong Kong in 2014 and 2015, respectively. He serves as a reviewer of several

20

international journals. He is the author or coauthor of more than 60 papers inrefereed international journals. His current research interests include collective behavior in complex dynamical networks, multi-agent system, chaos synchronization, control theory, discontinuous dynamical systems and neural networks.

Lei Shi received the B.S. degree in mathematics from Yunnan University, Kunming, China, in 2007, and the M.S. degree in mathematics from Yunnan University, Kunming, China, in 2010. He is currently a lecturer with the Department of Mathematics, Honghe University, Yunan, China. His current research interests include chaos synchronization, discontinuous dynamical systems, and neural networks. Shi serves as a reviewer for many international journals.