General decay synchronization of memristor-based Cohen–Grossberg neural networks with mixed time-delays and discontinuous activations

Accepted Manuscript

General decay synchronization of memristor-based Cohen-Grossberg neural networks with mixed time-delays and discontinuous activations Abdujelil Abdurahman, Haijun Jiang, Cheng Hu PII: DOI: Reference:

S0016-0032(17)30386-1 10.1016/j.jfranklin.2017.08.013 FI 3096

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

31 December 2016 29 May 2017 3 August 2017

Please cite this article as: Abdujelil Abdurahman, Haijun Jiang, Cheng Hu, General decay synchronization of memristor-based Cohen-Grossberg neural networks with mixed time-delays and discontinuous activations, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.08.013

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General decay synchronization of memristor-based Cohen-Grossberg neural networks with mixed time-delays and discontinuous activations Abdujelil Abdurahman∗, Haijun Jiang, Cheng Hu

College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, Xinjiang, P.R. China

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Abstract: This paper investigates the general decay synchronization (GDS) of memristorbased Cohen-Grossberg neural networks (MCGNNs) with discontinuous neuron activation functions and mixed time-delays. Based on the concept of Filippov solution and theory of differential inclusion, introducing suitable Lyapunov-Krasovskii functionals and employing useful inequality techniques, some novel criteria ensuring the GDS of considered Cohen-Grossberg neural networks are established via two types of nonlinear controls. In addition, the feasibility of the obtained theoretical results is validated via two numerical examples and their simulations. The polynomial synchronization, asymptotical synchronization, and exponential synchronization can be seen the special cases of the GDS. To the authors’ knowledge, the results established in the paper are the only available results on the synchronization of neural networks, connecting the three main characteristics, i.e., memristor, discontinuous activation functions and mixed time-delays. Key words: Memristor, Cohen-Grossberg neural network, General decay synchronization, Discontinuous activation, Mixed delay

Introduction

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In the past few years, memristor-based neural networks (MNNs) have been extensively investigated based on the belief that it will profoundly improve human’s capability on the neural processing using memory devices, which is presumed to be an essential step towards replicate complex learning via artificial neural networks [1]. The main difference between MNNs and conventional neural networks is that the strengths of synapses among the neurons are modeled by memristors in MNNs, while the resistors are employed in the conventional neural networks to model the strengths of synapses among neurons. Unlike resistors, the conductance of memristors can response to the changes of applied current or voltage just like a biological synapse [2], and they possess very nice device characteristics including nonvolatility, multiple memory states, and nanometer geometries that can be shrunk to the ultimate physical dimensions [3–6]. These superior features of memristors enable them to be successfully applied in nonvolatile memory [7], artificial neural networks [2, 8–13], composite circuits [14, 15], and so on. When we investigate the dynamical behaviours of neural networks, synchronization is one of the major concerns in which the primary attention should be given. This is rooted in the fact that the synchronization is unique in nature and can play an extremely vital ∗

E-mail: [email protected]; [email protected].

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role in many fields of science including biology, climatology, sociology, ecology, and so on. In [16], via substituting Chua’s diode with a memristor, Prof. Chua firstly got several memristor-based oscillators. By employing the same method, so far, researchers introduced a plenty of memristor-based chaotic systems [14,17,18]. Therefor, it is of great importance to study the chaotic behaviour and synchronization control for MNNs [19–29]. In [19], by employing the second-order reciprocally convex approach, the authors concerned with the synchronization problem for memristor-based recurrent neural networks (MRNNs) with two delay components. In [20], the exponential synchronization of a class of MRNNs was studied via fuzzy theory and Lyapunov approach. In [21], based on the classical theory of differential equations with discontinuous right-hand sides, the authors investigated the adaptive general synchronization problem for MNNs with time-varying delays. [22] dealt with the global exponential synchronization problem for two types of MRNNs with time delays through static or dynamic coupling. In [23], by using the Lyapunov function approach and extended Halanay differential inequality, the authors concerned with the exponential synchronization problem for a class of linearly coupled MRNNs with impulses and time-varying delays. In [24], the exponential lag synchronization problem of MNNs was studied by employing fuzzy method and obtained results were applied for generating pseudorandom numbers. It is worthy of noting that the majority of the published works on the synchronization of neural networks were focused on the hypothesis that the neuron activation functions are continuously bounded or even globally Lipschitz. However, a shortly review of many classical neural network models unveils that discontinuous neural networks are essential and do frequently appear in practise [30–34]. For example, the standard assumption used in the primitive Hopfield neural networks under graded response neurons is that the activation functions employed in high-gain limit can proximately be treated as discontinuous functions. Hence, it is our intention in this paper to remove the possible restrictions. In addition, as mentioned in [35], the study of the ideal discontinuous systems can enable us to find some crucial characteristics of the nonlinear systems, such as finite-time convergence to the trivial solution and the existence of sliding modes on the switching surfaces [36]. For these reasons, when studying the synchronization of neural networks, many scholars turn their attention from continuous neural networks to the networks with discontinuous neuron activations even though the synchronization aim is not easy to be achieved due to the discontinuity of system dynamics [37–40]. In [37], the complete synchronization problem for a class of discontinuous neural networks with constant delays was studied by using the classical theory of differential inclusions and Lyapunov function approach. In [38], the authors investigated the synchronization problem for a class of linearly coupled dynamical networks with discontinuous right-hand sides, and as applications, they proposed a method to realize the synchronization of coupled switching dynamical systems. In [39], under the concept of Filippov differential inclusion, the problem of exponential synchronization of delayed neural networks with discontinuous neuron activations was considered by designing a type of discontinuous controller and using some analytic techniques. In [40], the finite-time synchronization (FTS) problem for a type of delayed neural networks with discontinuous activation functions was studied via Lyapunov-Krasovskii functional methods. To the authors’ knowledge, however, so far the synchronization of general type of networks, where the discontinuous neuron activation are appeared in a memristor-based 2

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neural network model, especially those with mixed time-varying delays, has not been investigated in the literature. As mentioned above, studying the synchronization, especially GDS, of these networks is of great significance in the understanding of a wide types of dynamical behaviors which can be encountered in both biological and man-made neural networks. In addition, when studying the synchronization of chaotic systems, it is a very interesting and useful topic to find estimate of the convergent rate of synchronization. However, in some cases, the convergence rate of the synchronization can not be shown or it is very difficult to estimate. This motivate us to define a new type of convergence rate, such as convergence with general decay. Recently, in [41, 42], the authors investigated the synchronization problem for two classes of chaotic NNs by introducing new concept of synchronization, namely GDS. However, there are some limitations on the results of [41, 42] mainly because they archived GDS via designing a type of hybrid switching controller which consists of linear and nonlinear terms. In fact, the linear term ωi ei (t) in the controllers of [41, 42], which is introduced to cope the unmatched parameters, is extra and it can be removed. Inspired from the above discussions, in the paper, we investigate the GDS for a class of MCGNNs with discontinuous activation functions and mixed time-varying delays. The main contributions of the paper can be stated as follows:

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1. By applying the concept of Filippov solution and theory of differential inclusion [43–45], the drive-response MCGNNs with mixed time-delays and discontinuous activations are transformed to differential inclusions with unmatched parameters.

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2. Based on the switching feature of memristor, two different types of nonlinear feedback controllers are developed.

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3. By constructing suitable Lyapunov-Krasovskii functionals and applying the novel analysis methods, some simple but useful criteria are introduced to ensure the GDS for considered MCGNNs.

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4. Two examples and their numerical simulations are given to validate the correctness of the theoretical findings.

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It is believed that our approaches and obtained results may bring some new guidance for the synchronization study of MNNs do to the fact that they extend the previous works on synchronization of MNNs with continuous activations and synchronization of discontinuous neural networks without memristor. Also, these approaches can be used for synchronization analysis of other types of memristor-based chaotic systems with discontinuous activation functions. We organized the rest of the paper as follows. In Section 2, we will introduce the driveresponse MCGNN models. In addition, we will present some hypotheses, definitions, and lemmas used in the paper. In section 3, we will investigate the GDS of chaotic MCGNNs with discontinuous activations by designing two novel nonlinear controls. In Section 4, we will illustrate the effectiveness of the obtained results by giving two numerical examples and their simulations. Finally, we will draw some general conclusions in Section 5.

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2

Preliminaries

In this paper, we consider a class of MCGNNs with mixed delays and discontinuous activation functions described by the following equation  n X x˙ i (t) = qi (xi (t)) − di (xi (t))xi (t) + aij (xj (t))fj (xj (t)) + +

bij (xj (t j=1 n Z t X

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j=1

n X

− τj (t)))fj (xj (t − τj (t)))

(1)



cij (xj (s))fj (xj (s))ds + Ii ,

j=1

t−ρj(t)

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where i ∈ I , {1, 2, · · · , n}, n ≥ 2 denotes the number of neurons in the neural network; xi (t) corresponds to the voltage of the capacitor Ci ; qi (·) represents an amplification function; fj (·) is the feedback function; τj (·) and ρj (·)denote the discrete-time delay and distributed delay satisfying 0 ≤ τj (t) ≤ τj and 0 ≤ ρj (t) ≤ ρj , respectively; Ii denotes the external bias on the ith unit; di (·), aij (·), bij (·) and cij (·) are the memristor-based weights given by   n X Maij 1  1  di (xi (t)) = (Maij + Mbij + Mcij ) × sgnij + , aij (xj (t)) = × sgnij , Ci Ri Ci j=1  Mbij Mcij 1, i = j, bij (xj (t − τj (t))) = × sgnij , cij (xi (s)) = × sgnij , sgnij = −1, i 6= j, Ci Ci

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a , Wb and Wc , where Maij , Mbij and Mcij denote the memductances of memristors Wij ij ij a respectively. Wij represents the memristor between the feedback function fj (xj (t)) and b represents the memristor between the feedback function f (x (t − τ (t))) and xi (t), Wij j j j c represents the memristor between the feedback function f (x (s)) and x (t), xi (t), Wij j j i and Ri represents the parallel-resistor corresponding to the capacitor Ci . In light of the switching characteristics of memristor, in the paper we use the following mathematical model for memristor-based weights   a ˆij , |xj (t)| ≤ Tj , dˆi , |xi (t)| ≤ Ti , di (xi (t)) = aij (xj (t)) = ˇ a ˇij , |xj (t)| > Tj , di , |xi (t)| > Ti ,   ˆbij , |xj (t − τj (t))| ≤ Tj , cˆij , |xj (s)| ≤ Tj , bij (xj (t − τj (t))) = ˇ cij (xj (s)) = cˇij , |xj (s)| > Tj , bij , |xj (t − τj (t))| > Tj , (2)

where Ti > 0 are switching jumps, and dˆi , dˇi , a ˆij , a ˇij , ˆbij , ˇbij , cˆij and cˇij are constant numbers. For more explanations on the construction of MNNs, please refer to the works [1, 11, 14, 20]. Assume Rn be the space of n−dimensional real column vectors. For any x ∈ Rn , kxk
1 Pn 2 2 or kxk = max denotes a vector norm defined by kxk = i∈I |xi |. Given set i=1 |xi | ¯ we mean the closure of E and by µ(E) we mean the Lebesgue measure of E ∈ Rn , by E E in Rn . Moreover, K[E] denotes the closure of the convex hull of set E, co{a, b} denotes the closure of the convex hull generated by real numbers a and b. Let σ = max{τ, ρ}, τ = 4

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maxj∈I {τj }, ρ = maxj∈I {ρj }, then C = C([−σ, 0], Rn ) denotes the Banach space of all continuous functions mapping [−σ, 0] into Rn with the norm kϕk = sup−σ≤s≤0 kϕ(s)k. If for L ∈ (0, +∞], x(t) : [−σ, L) → Rn is continuous, then xt ∈ C is defined by xt (θ) = x(t + θ), θ ∈ [−σ, 0] for any t ∈ [0, L). The initial condition associated with system (1) is given by xt (θ)|t=0 = ϕ(θ), θ ∈ [−σ, 0], (3)

q i ≤ qi (v) ≤ q i

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where ϕ ∈ C. Throughout the paper, for model (1), we introduce the following assumptions. A1 : For each i ∈ I , function qi (·) : R → R+ is continuous and there exist positive constants q i and q i such that for all v ∈ R.

sup ηi ∈K[fi (u)],γi ∈K[fi (v)]

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A2 : For each i, fi (·) is continuous on R expect a countable set of isolated points ρki , where there exist fi+ (ρik ) and fi− (ρik ), respectively. A3 : For each i, there exist a nonnegative constants Li and Ni such that |ηi − γi | ≤ Li |u − v| + Ni ,

u, v ∈ R,

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  where K[fi (s)] = min{fi− (s), fi+ (s)}, max{fi− (s), fi+ (s)} for s ∈ R. A4 : Time-varying delays τj (t) and ρj (t) are differentiable, and there exist positive constants 0 ≤ κj < 1 and 0 ≤ ιj ≤ 1/2 such that 0 ≤ τ˙j (t) ≤ κj ,

0 ≤ ρ˙ j (t) ≤ ιj .

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For i, j ∈ I , we let d¯i = (1/2)(dˆi + dˇi ), di = (1/2)(dˆi − dˇi ), a ¯ij = (1/2)(ˆ aij +ˇ aij ), aij = cij + cˇij ), cij = (1/2)(ˆ aij − a ˇij ), ¯bij = (1/2)(ˆbij + ˇbij ), bij = (1/2)(ˆbij − ˇbij ), c¯ij = (1/2)(ˆ M = max{d , −d }, aM = |a |, bM = |b | and (1/2)(ˆ cij − cˇij ), dm i i ij ij i = min{di , −di }, di ij ij cM ij = |cij |. Then the MCGNN (1) can be written in the following form:

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 n X
x˙ i (t) = qi (xi (t)) − d¯i + d˜i (xi (t)) xi (t) + (¯ aij + a ˜ij (xj (t))) fj (xj (t)) + +

j=1

n X


¯bij + ˜bij (xj (t − τj (t))) fj (xj (t − τj (t)))

j=1

t−ρj(t)

j=1 n Z t X

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 (¯ cij + c˜ij (xj (s))) fj (xj (s))ds + Ii ,

where

d˜i (xi (s)) =



˜bij (xj (s)) =



di , −di ,

bij , −bij ,

|xi (s)| ≤ Ti , |xi (s)| > Tj ,

a ˜ij (xj (s)) =

|xj (s)| ≤ Tj , |xj (s)| > Tj ,

c˜ij (xj (s)) =





aij , −aij ,

cij , −cij ,

|xj (s)| ≤ Tj , |xj (s)| > Tj , |xj (s)| ≤ Tj , |xj (s)| > Tj .

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Based on the theory of measurable selection in [48], there exist measurable functions ∗ ∗ ∗ ∗ di (t) ∈ co{−di , di }, aij (t) ∈ co{−aij , aij }, bij (t) ∈ co{−bij , bij }, cij (t) ∈ co{−cij , cij },

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and ηj (t) ∈ K[fj (xj (t))] for a.e.t ∈ [0, L) such that "

n X
∗ ∗ x˙ i (t) = qi (xi (t)) − d¯i + di (t) xi (t) + (¯ aij + aij (t)) ηj (t) j=1

j=1

+


∗ ¯bij + bij (t) ηj (t − τj (t))

n Z X j=1

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+

n X

(¯ cij + cij (s)) ηj (s)ds + Ii .

t−ρj(t)

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Any function η satisfying (6) is called an output solution associated to the state x. Remark 1. Suppose that the hypotheses A1 − A3 hold, then the growth condition (8) in Theorem 1 in [49] is satisfied. Therefore, system (1) has at least one Filippov solution x(t) on the interval [0, +∞). The response system of driver system (1) is given by  n X aij (yj (t))fj (yj (t)) y˙ i (t) = qi (yi (t)) − di (yi (t))yi (t) + j=1

+

bij (yj (t j=1 n Z t X j=1

− τj (t)))fj (yj (t − τj (t)))

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cij (yj (s))fj (yj (s))ds + Ii + ui (t),

t−ρj(t)

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+

n X

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where di (yi (t)), aij (yj (t)), bij (yj (t − τj (t))) and cij (yi (t)) are defined similar to (2). ui (t) is the controller to be designed. Generally, the initial conditions of (7) are different from those of (1) and denoted by y(t) = φ(t), φ ∈ C([−σ, 0], Rn ). Similar to (4), the MCGNN (7) can be written as follows:  n X
y˙ i (t) = qi (yi (t)) − d¯i + d˜i (yi (t)) yi (t) + (¯ aij + a ˜ij (yj (t))) fj (yj (t))

+

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+

j=1

n X


¯bij + ˜bij (yj (t − τj (t))) fj (yj (t − τj (t)))

j=1

t−ρj(t)

j=1 n Z t X

(8)

 (¯ cij + c˜ij (yj (s))) fj (yj (s))ds + Ii + ui (t),

where d˜i (·), a ˜ij (·), ˜bij (·) and c˜ij (·) are the same as defined in (5). It can be obtained from (8) that  n n X X

∗∗ ∗∗ ∗∗ ¯bij + bij (t) γj (t − τj (t)) y˙ i (t) = qi (yi (t)) − d¯i + di (t) yi (t) + (¯ aij + aij (t)) γj (t) + +

j=1

n Z t X j=1



∗∗

j=1

(¯ cij + cij (s)) γj (s)ds + Ii + ui (t),

t−ρj(t)

(9) ∗∗

∗∗

∗∗

where measurable functions di (t) ∈ co{−di , di }, aij (t) ∈ co{−aij , aij }, bij (t) ∈ co{−bij , bij }, ∗∗ cij (s) ∈ co{−cij , cij }, and γj (t) ∈ K[fj (yj (t))] for a.e.t ∈ [0, +∞). 6

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In addition, during the proof of the main theorems we
shall need the following lemmas. Lemma 1. Let Ωi (xi , yi ) = K[d˜i (yi )]yi −K[d˜i (xi )]xi for i ∈ I , where xi and yi ∈ R. Then for any ωi (xi , yi ) ∈ Ωi (xi , yi ), the following inequality is satisfied M −sign(yi − xi )ωi (xi , yi ) ≤ −dm i |yi − xi | + 2Ti di .

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(i) When |xi | < Ti and |yi | < Ti , we have ωi (xi , yi ) = −sign(yi − xi )(di yi − di xi )

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Proof. For any given i ∈ I and xi , yi ∈ Rn , the proof will be divided into four cases.

M = −di |yi − xi | ≤ −dm i |yi − xi | + 2Ti di .

(ii) When |xi | > Ti and |yi | > Ti , we have

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ωi (xi , yi ) = −sign(yi − xi )(−di yi − (−di )xi )

M ≤ −(−di )|yi − xi | ≤ −dm i |yi − xi | + 2Ti di . ∗

(iii) When |xi | < Ti and |yi | ≥ Ti , there exits a di ∈ K[d˜i (yi )] = co{−di , di } such that ∗

ωi (xi , yi ) = −sign(yi − xi )(di yi − di xi ) ∗

∗

= −sign(yi − xi )[di (yi − xi ) + (di − di )xi ]

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M ≤ −dm i |yi − xi | + 2Ti di .

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(iv) When |yi | ≤ Ti and |xi | ≥ Ti , there exit di ∈ K[d˜i (yi )] = co{−di , di } and di ∈ K[d˜i (xi )] = co{−di , di } such that ∗

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∗

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= −sign(yi − xi )[di (yi − xi ) + (di − di )yi ]

M ≤ −dm i |yi − xi | + 2Ti di .

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ωi (xi , yi ) = −sign(yi − xi )(di yi − di xi )

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This completes the proof of the Lemma 1. Lemma 2. Let

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Υaij (xj , yj ) = K[˜ aij (yj )]K[f (yj )] − K[˜ aij (xj )]K[f (xj )] ,
Υbij (xj , yj ) = K[˜bij (yj )]K[f (yj )] − K[˜bij (xj )]K[f (xj )]
, Υcij (xj , yj ) = K[˜ cij (yj )]K[f (yj )] − K[˜ cij (xj )]K[f (xj )] ,

a (x , y ) ∈ Υa (x , y ), ω b (x , y ) ∈ where i, j ∈ I , xj and yj ∈ R. Then for any ωij j j ij j j ij j j c (x , y ) ∈ Υc (x , y ), the following inequalities are true Υbij (xj , yj ) and ωij j j j j ij a (x , y )| ≤ aM L |y − x | + aM (N + H ), |ωij j j j j j ij j j ij b (x , y )| ≤ bM L |y − x | + bM (N + H ), |ωij j j j j j ij j j ij c (x , y )| ≤ cM L |y − x | + cM (N + H ), |ωij j j j j j ij j j ij

where Hj = 2(Lj Tj + Nj + Hj∗ ), Hj∗ = |K[fj (0)]| with |k[fj (0)]| = sup{|νj | : νj ∈ k[fj (0)]}. Proof. given i ∈ I and xi , yi ∈ Rn , the proof will be divided into four cases. 7

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(i) When |xj | < Tj and |yj | < Tj , from A3 there exist γj ∈ K[f (yj )] and ηj ∈ K[f (xj )] such that a |ωij (xj , yj )| = |aij ||γj − ηj | ≤ aM ij (Lj |yj − xj | + Nj ) M ≤ aM ij Lj |yj − xj | + aij (Nj + Hj ).

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(ii) Similarly, when |xj | > Tj and |yj | > Ti the statement is satisfied. ∗

(iii) When |xj | < Tj and |yj | ≥ Tj , there exits a aij ∈ K[˜ aij (yj )] = co{−aij , aij } such that ∗

a |ωij (xj , yj )| = |aij γj − aij ηj | ∗

∗

= |aij ||γj − ηj | + |aij − aij ||ηj | ∗

≤ aM ij |γj − ηj | + |aij − aij ||ηj |

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≤ aM ij (Lj |yj − xj | + Nj ) ∗

+ |aij − aij |(Lj |xj | + Nj + Hj )

M ≤ aM ij Lj |yj − xj | + aij (Nj + Hj ). ∗

(iv) When |xj | ≥ Tj and |yj | ≤ Tj , there exits aij ∈ K[˜ aij (xj )] = co{−aij , aij } and ∗∗ aij ∈ K[˜ aij (yj )] = co{−aij , aij } such that ∗∗

∗

M

a |ωij (xj , yj )| = |aij γj − aij ηj | ∗

∗∗

∗

= |aij ||γj − ηj | + |aij − aij ||γj | ∗∗

∗

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≤ aM ij |γj − ηj | + |aij − aij ||γj |

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≤ aM ij (Lj |yj − xj | + Nj ) ∗∗

∗

+ |aij − aij |(Lj |yj | + Nj + Hj )

M ≤ aM ij Lj |yj − xj | + aij (Nj + Hj ).

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This completes the proof of the Lemma 2. 2 Lemma 3 (Chain Rule [33, 44]). Suppose that V (z) : Rn 7→ R is C-regular, and that z(t) : [0, +∞) 7→ R is absolutely continuous on any compact interval of [0, +∞). Then, V (z(t)) : [0, +∞) 7→ R are differential for a.e. t ∈ [0, +∞), and we have dV (z(t)) = ξ(t)z(t), ˙ dt

∀ξ(t) ∈ ∂V (z(t)).

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Lemma 4 ( [47]). Suppose that the assumption A1 is satisfied, then the functions hi : R2 → R defined by Z z2 ds (12) hi (z1 , z2 ) = ,i ∈ I, z1 qi (s)

are regular in R2 and

   −1 1 T   , z1 6= z2 ,  sign(u2 − u1 ) q (z ) , q (z ) i 1 i 2  ∂hi (z1 , z2 ) =   1 1   (z1∗ , −z1∗ )T : |z1∗ | ≤ , z 1 = z2 . qi (z1 ) qi (z1 ) 8

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Also, if z1 (t), z2 (t) : R → R are absolutely continuous on any compact subinterval of [0, +∞). Then, hi (z1 (t), z2 (t)) defined by (12) is differentiable for a.e.t ∈ [0, +∞), and we have d hi (z1 (t), z2 (t)) = hζ(t), (z˙1 (t), z˙2 (t))i, dt where ζ(t) ∈ ∂hi (z1 (t), z2 (t)).

Main Results

In this section, we will obtain some sufficient condition to insure the GDS of systems (1) and (7). First, we give the definitions of ψ−type function and GDS. Definition 1. ( [41, 42]). A function ψ : R+ → [1, +∞) is said to be ψ−type function if it satisfies the following criteria:

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1) It is differentiable and nondecreasing; 2) ψ(0) = 1 and ψ(+∞) = +∞;

˜ = ψ(t)/ψ(t) ˙ ˜ < +∞, where ψ(t) ˙ 3) ψ(t) is nondecreasing and ψ ∗ = supt≥0 ψ(t) is the time derivative of ψ(t); 4) For any t, s ≥ 0, ψ(t + s) ≤ ψ(t)ψ(s).

M

It is not difficult check that functions ψ(t) = eαt and ψ(t) = (1 + t)α for any α > 0 satisfy the above four conditions, thus can be seen as ψ−type functions. Definition 2. ( [41,42]). The drive-response systems (1) and (7) are said to be general decay synchronized if there exist a constant ε > 0 such that

ED

log ky(t) − x(t)k ≤ ε, log ψ(t) t→+∞

lim sup

AC

CE

PT

where ε > 0 can be seen the convergence rate as synchronization error approaches to zero. Remark 2. The function ψ is employed as the decay function, so GDS is also said to be ψ−type synchronization. When ψ(t) = eαt and ψ(t) = (1 + t)α for any α > 0, GDS may be specialized as exponential synchronization and polynomial synchronization. ˜ A5 : For the functions ψ(t), ψ(t) given in Definition 1, there exist a function %(t) ∈ + C(R, R ) and a constant δ such that for any t ≥ 0 Z t ˜ ψ(t) ≤ 1, sup ψ δ (s)%(s)ds < +∞. (13) t∈[0,+∞) 0

Following lemma is essential to our later study, which can be proved analogously to [41, 42]. Lemma 5. Under the assumption A5 , assume that the synchronization error e(t) = y(t)−x(t) of driver-response systems (1) and (7) is satisfied the differential equation z(t) ˙ = g(t, zt ) with zt |t=0 = ϕ, ¯ where ϕ¯ ∈ C and g : R × C → Rn is measurable and essentially locally bounded. If there exist a differentiable functional V (t, et ) : R+ × C → R+ , and positive constants λ1 , λ2 such that for any (t, et ) ∈ R+ × C, (λ1 ke(t)k)2 ≤ V (t, et ), dV (t, et ) ≤ −δV (t, et ) + λ2 %(t), dt (4) 9

(14) (15)

ACCEPTED MANUSCRIPT

log ky(t) − x(t)k δ ≤− . log ψ(t) 2 t→+∞

lim sup

CR IP T

where x(t) and y(t) are Filippov solutions of systems (1) and (7) respectively, δ > 0 and %(t) are defined in A5 . Then the driver-response systems (1) and (7) are general decay synchronized in the sense of Definition 2, and the convergence rate is δ/2. Remark 3. If the functions %(t) = 0 and ψ(t) = et , then condition (13) is satisfied obviously. Thus, from Lemma 5, the drive-response systems (1) and (7) are exponential synchronized with the convergence rate of δ/2. (16)

Di = d¯i + dm i ,

Bij = (|¯bij | + bM ij )|Lj ,

Cij = (|¯ cij | + cM ij )Lj ,

n n o X  M  M ¯ + aij + bM cij |)Nj . ij + ρj cij (Nj + Hj ) + (πij + |bij | + ρj |¯ j=1

M

Ri =

2Ti dM i

AN US

In addition, in classical exponential synchronization theorem, the time-derivative of the Lyapunov function should not be greater than zero. However, since %(t) ∈ C(R, R+ ), the right side of condition (15) in Lemma 5 can be greater than zero. To summarize, Lemma 5 generalize the classical exponential synchronization. For convenience, first we introduce the following denotations ( ( (πii + aM max{¯ aii , 0}, j = i, ii )Li , j = i, πij = Aij = M |¯ aij |, j 6= i, (|¯ aij | + aij )Lj , j 6= i,

ED

Now, letting ei (t) = yi (t) − xi (t) for i ∈ I , and designing the controller ui (t) in response system (7) as follows: q¯i ξi ke(t)kei (t) , (ke(t)k + %(t))

i ∈ I,

(17)

PT

ui (t) = −βi sign(ei (t)) −

CE

where ξ = mini∈I {ξi }, βi and ξi for i ∈ I are control gains satisfying  n   X Bji 1   ξi > −Di + Aji + + τi Bji + ρ2i Cji + ρi Cji , (1 − κi ) 2 i=1   βi ≥ q¯i Ri .

(18)

AC

Then based on the discontinuous nonlinear controller (17), the following theorem can be obtained. Theorem 1. Suppose A1 − A5 hold, then the response network (7) can be general decay synchronized with the drive network (1) under the nonlinear controller (17) if, the control gains βi and ξi satisfy the inequality (18). Proof. Consider a non-smooth function given by Z yi (t) ds e˜i (t) , hi (xi (t), yi (t)) = . qi (s)

(19)

xi (t)

Then from assumption A1 , it is not difficult check that

q i e˜i (t) ≤ |ei (t)| ≤ q i e˜i (t). 10

(20)

ACCEPTED MANUSCRIPT


T 1 From Lemma 4, take ζi (t) = ϑi (t) qi (x−1 , with (t)) q (y (t)) i i i    sign(yi (t) − xi (t)), if yi (t) 6= xi (t), sign(γi (t) − ηi (t)), if yi (t) = xi (t), γi (t) 6= ηi (t), ϑi (t) =   0, if yi (t) = xi (t), γi (t) = ηi (t).

(21)

n X

j=1

n X


∗ aij (t)γj (t) − aij (t)ηj (t)

∗∗

j=1

|¯bij (γj (t − τj (t)) − ηj (t − τj (t)))| +

+ ϑi (t) + ϑi (t)

n Z X

t

|¯ cij (γj (s) − ηj (s))|ds

AN US

+

j=1,j6=i n X

|¯ aij (γj (t) − ηj (t))| + ϑi (t)

j=1

t−ρj (t)

n   X ∗∗ ∗ bij (t)γj (t − τj (t)) − bij (t)ηj (t − τj (t)) j=1 n Z t X j=1

∗∗

t−ρj (t)

∗

(cij (s)γj (s) − cij (s)ηj (s))ds + ϑi (t)

ui (t) . qi (yi (t))

M

+

CR IP T

Then ζi (t) ∈ ∂hi (xi (t), yi (t)), |ϑi (t)| ≤ 1 and   y˙ i (t) x˙ i (t) d e˜i (t) =ϑi (t) − dt qi (yi (t)) qi (xi (t))  ∗∗  ∗ = − d¯i |ei (t)| − ϑi (t) di (t)yi (t) − di (t)xi (t) + a ¯ii |γi (t) − ηi (t)|

Applying A3 , Lemmas 1 and 2, and inequality (18) yields

+

n X 

bM ij Lj |ej (t

j=1 n Z t X

CE

+

n X 

+

j=1

j=1,j6=i

n  X M M aij Lj |ej (t)| + aij (Nj + Hj ) + |¯bij ||[Lj |ej (t − τj (t))| + Nj ]

PT

j=1

ED

n X d M e˜i (t) ≤ − (d¯i + dm )|e (t)| + 2T d + π (t)[L |e (t)| + N ] + |¯ aij |[Lj |ej (t)| + Nj ] i i i i i i i i dt

t−ρj (t)

− τj (t))| +

j=1

bM ij (Nj



+ Hj ) +

n X j=1

|¯ cij |

Z

t

t−ρj (t)

[Lj |ej (s)| + Nj ]ds

 M  cij Lj |ej (s)| + cM ij (Nj + Hj ) ds

AC

  q¯i ξi ke(t)k|ei (t)| 1 βi + − ai (yi (t) (ke(t)k + %(t)) n n X X m M ¯ ≤ − (di + di )|ei (t)| + (πij + aij )Lj |ej (t)| + (|¯bij | + bM ij )Lj |ej (t − τj (t))| +

j=1

n X j=1

(|¯ cij | + cM ij )Lj n X

Z

j=1

t

t−ρj (t)

|ej (s)|ds + (Ri −

βi ξi ke(t)k|ei (t)| )− q¯i (ke(t)k + %(t))

ξi ke(t)k|ei (t)| (ke(t)k + %(t)) j=1 Z t n n X X + Bij |ej (t − τj (t))| + Cij |ej (s)|ds.

≤ − Di |ei (t)| +

j=1

Aij |ej (t)| −

j=1

t−ρj (t)

11

ACCEPTED MANUSCRIPT

Now construct the following Lyapunov-Krasovskii functional

V2 (t) =

˜i (t) and i=1 e n X n Z t X

t−τj (t)

i=1 j=1

V3 (t) =2

n X n X

Cij

n

Z

−ρj (t)

i=1 j=1

t

t+s

|ej (ς)|dςds +

n X n X

0

−τj

i=1 j=1

Z

0

Z

n

XX Bij |ej (s)|ds + 1 − κj

(22)

Cij

Z

t

t+ς

0

−ρj

i=1 j=1

Then from (20), there exists a scalar χ > 1 such that

Bij |ej (s)|dsdς,

Z

υ

0Z t

t+s

|ej (ς)|dςdsdυ.

Z t n n χ XX |ej (s)|ds ≤ V (t) ≤χ Bij |ei (t)| + $ t−τj i=1 j=1 i=1 Z 0 Z t n n χ XX + Cij |ej (ς)|dςds, $ −ρj t+s

q¯i

AN US

n X

n X |ei (t)| i=1

Z

CR IP T

where V1 (t) =

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

Pn

(23)

i=1 j=1

where $ = mini∈I {$i } with n  X $i =Di + ξi − Aji + i=1

Bji (1 − κi )



 n  X 1 2 − τi Bji + ρi Cji + 2ρi Cji > 0. 2

(24)

i=1

PT

ED

M

Since V1 , V2 , V3 are regular [47], the time derivative of V (t) can be evaluated by Lemma 3 ( n n n X X X Bij Bij (1 − τ˙j (t)) d d V (t) = e˜i (t) + |ej (t)| − |ej (t − τj (t))| dt dt (1 − κj ) (1 − κj ) i=1 j=1 j=1 Z t n n n X X X Cij 2 |ej (s)|ds + + Bij τj |ej (t)| − Bij ρ |ej (t)| 2 j t−τj j=1 j=1 j=1 Z 0 Z t n n X X − Cij |ej (ς)|dςds + 2 Cij ρj (t)|ej (t)| −ρj

j=1

Cij ρ˙ j (t)

CE

+2

n X

AC

≤

i=1

(

n X

Z

t

− Di |ei (t)| + Z

j=1

t−ρj (t)

j=1

n X

t+s

n X j=1

|ej (s)|ds − 2 Aij |ej (t)| +

n X j=1

n X j=1

Cij

Z

t

t−ρj (t)

|ej (s)|ds

)

Bij |ej (t − τj (t))| n

X Bij ξi ke(t)k|ei (t)| + |ej (t)| (ke(t)k + %(t)) (1 − κj ) t−ρ (t) j j=1 j=1 Z t n n n X X X (1 − τ˙j (t)) − Bij |ej (t − τj (t))| + Bij τj |ej (t)| − Bij |ej (s)|dse (1 − κj ) t−τj j=1 j=1 j=1 Z t n n n X X X Cij 2 +2 Cij ρj (t)|ej (t)| + ρ |ej (t)| − 2 Cij |ej (s)|ds 2 j t−ρj (t) j=1 j=1 j=1 ) Z 0 Z t Z t n n X X +2 Cij ρ˙ j (t) |ej (s)|ds − Cij |ej (ς)|dςds +

Cij

j=1

t

|ej (s)|ds −

t−ρj (t)

j=1

12

−ρj

t+s

ACCEPTED MANUSCRIPT

i=1

− − ≤

n X n X

Cij

j=1

Z

0

−ρj t+s i=1 j=1 n X ξi ke(t)k|ei (t)| i=1

n X i=1

−

− Di +

i Bji 1 2 Aji + + τi Bji + ρi Cji + 2ρi (t)Cji |ei (t)| (1 − κi ) 2 Z t Z t n X n X |ej (ς)|dςds − Bij |ej (s)|ds

n h X

(

i=1 j=1

t−τj

(ke(t)k + %(t))

n  X − (Di + ξi ) + Aji +

n X n X

Bij

Z

t

t−τj

i=1 j=1 n X

Bji + τi Bji + (1 − κi ) j=1 Z 0 Z t n X n X |ej (s)|ds − Cij i=1 j=1

−ρj

t+s

CR IP T

≤

n  X

1 2 ρ Cji + 2ρi Cji 2 i

) |ei (t)|

|ej (ς)|dςds

n X ξi ke(t)k|ei (t)| (ke(t)k + %(t)) i=1 i=1 ( ) n n  X X Bji ≤ − (Di + ξi ) + Aji + + τi Bji + ρ2i Cji + ρi Cji |ei (t)| (1 − κi ) i=1 j=1 Z t n X n X ke(t)k%(t) − + λ2 Bij |ej (s)|ds (ke(t)k + %(t)) t−τj i=1 j=1 Z 0 Z t n X n X − |ej (ς)|dςds, Cij −ρj

i=1 j=1

AN US

ξi |ei (t)| −

M

+

t+s

CE

PT

ED

where λ2 = maxi∈I {ξi } > 0. By using inequality (18) and inequality 0 ≤ ab/(a + b) ≤ a for any a > 0, b > 0 we get Z t n n X n X X d V (t) ≤λ2 %(t) − $i |ei (t)| − |ej (s)|ds Bij dt t−τj i=1 i=1 j=1 Z 0 Z t n X n X − Cij |ej (ς)|dςds. −ρj

i=1 j=1

t+s

AC

Taking a positive constant δ such that δχ ≤ $, then from (20) and (23), we have Z t n n X n X X d Bij V (t) + δV (t) ≤λ2 %(t) − $i |ei (t)| − |ej (s)|ds dt t−τ j i=1 i=1 j=1 Z 0 Z t n X n n  X X − Cij |ej (ς)|dςds + δ χ|ei (t)| i=1 j=1 n X

−ρj

t+s

i=1

 Z 0 Z t n χ χX + Bij |ej (s)|ds + Cij |ej (ς)|dςds $ $ t−τj −ρj t+s j=1 j=1 Z t n n X n   X X δχ − 1 Bij |ej (s)|ds ≤λ2 %(t) + (δχ − $) |ei (t)| + $ t−τj i=1 i=1 j=1 Z 0 Z t n X n   X δχ + − 1 Cij |ej (ς)|dςds ≤ λ2 %(t). $ −ρj t+s Z

t

i=1 j=1

13

ACCEPTED MANUSCRIPT

Which means that

d V (t) + δV (t) ≤ λ2 %(t). dt Then, from Lemma 5, the drive-response systems (1) and (7) achieve GDS under the nonlinear controller (17). The convergence rate of e(t) approaching zero is δ/2. The proof is completed. 2

AN US

CR IP T

Remark 4. In Theorem 1, we used non-adaptive discontinuous control to realize the GDS between two chaotic MCGNNs with discontinuous neuron activations. In some instances, such as when the differences of switching values |dˆi − dˇi |, |ˆ aij − a ˇij |, |ˆbij − ˇbij | and |ˆ cij − cˇij | are too large, the control strengths βi of this controller must be maximal, which is a kind of waste in practice. In Theorem 2, we will cope this problem by designing a novel adaptive-type discontinuous controller. Theorem 2. Under the assumptions A1 −A5 , if the response system (7) is controlled with the following adaptive controller:  q 2i ki (t)ke(t)k2 ei (t)    u (t) = −l (t)sign(e (t) − ,  i i i   (ke(t)k2 + %(t))  (25) l˙i (t) = εi |ei (t)| − εi (li (t) − li ),    2 2  αi ke(t)k ei (t)   ,  k˙ i (t) = −αi (ki (t) − ki ) + (ke(t)k2 + %(t))

M

where i ∈ I , εi and αi are arbitrary positive constants, ki and li are positive constants determined in later. Then the response network (7) can be general decay synchronized with the drive network (1).

ED

Proof. Construct the following Lyapunov-Krasovskii functional: W (t) = W1 (t) + W2 (t) + W3 (t) + W4 (t)

with W1 (t) =

Pn

(26)

1 2 ˜i (t), i=1 2 e

∗ n X n Z 0 Z t X ∗ Bij 2 ej (s)ds + W2 (t) = Bij e2j (s)dsdς, 1 − κj i=1 j=1 −τj t+ς i=1 j=1 t−τj (t) Z Z Z 0 Z 0Z t n n n X n 0 t XX ∗ X ∗ 2 W3 (t) =2 Cij ej (ς)dςds + Cij e2j (ς)dςdsdυ,

t

CE

PT

n X n Z X

i=1 j=1

n X

(

−ρj (t)

t+s

i=1 j=1

)

−ρj

υ

t+s

1 1 (li (t) − li )2 + (ki (t) − ki )2 , 2εi q i 2αi i=1     ∗ ∗ where Bij = |¯bij | + bM cij | + cM ij Lj , Cij = |¯ ij Lj . Then from (20), there exists a scalar χ∗ > 1 such that Z t n n n n X X χ∗ X X ∗ e2i (t) ∗ 2 ei (t) + ∗ Bij e2j (s)ds 2 ≤ W (t) ≤χ $ 2¯ q t−τj i i=1 i=1 i=1 j=1 Z Z n n 0 t χ∗ X X ∗ + ∗ Cij e2j (ς)dςds (27) $ −ρ t+s j i=1 j=1  n  χ∗ X 1 2 2 (li (t) − li ) + (ki (t) − ki ) , + ∗ $ qi

AC

W4 (t) =

i=1

14

ACCEPTED MANUSCRIPT

where $∗ = mini∈I {$i∗ }, $i∗ is a positive constant determined in later, and ∗

n

Aij =

X 1

∗∗ 1 M ¯ πij + aM cij |ρj + cM πij + aM ij + |bij | + bij + |¯ ij Lj , Aij = ij Lj . 2q i 2q i j=1

CR IP T

Calculating the derivative of W1 (t), we get (   n

∗∗ ∗ dW1 (t) X = e˜i (t)ϑi (t) − d¯i + di (t) yi (t) − d¯i + di (t)) xi (t) dt i=1  n  X ∗∗ ∗ + (¯ aij + aij (t)) γj (t) − (¯ aij + aij (t)) ηj (t) j=1

n  n    X X ∗∗ ∗ ¯ ¯bij + bij (t) ηj (t − τj (t)) + bij + bij (t) γj (t − τj (t)) − j=1

j=1

t

t−ρj (t)

n X
∗∗ c¯ij + cij (s) γj (s)ds − j=1

t

t−ρj (t)


∗ c¯ij + cij (s) ηj (s)ds +

ui (t) qi (yi (t))

 n X 1 1 ¯ m 2 M M M ¯ ≤ − (di + di )ei (t) + πij + aij + |bij | + bij + |¯ cij |ρj + cij Lj e2i (t) q¯i 2q i j=1 ( n n X 1 Xh

2 2dM + πij + aM aij | + |¯bij | + |¯ cij |ρj Nj ij Lj ej (t) + i Ti + |¯ 2q i j=1 j=1 ) i l (t)
i M M M e˜i (t) + aij + bij + cij ρj (t) (Nj + Hj ) − qi Z t n n X X

1 ¯ 1 M 2 M |bij | + bij Lj ej (t − τj (t))) + |¯ cij | + cij Lj e2j (s)ds + 2q i 2q i t−ρj (t)

ED

M



Z

AN US

+

j=1

n Z X

j=1

PT

−

j=1

ki (t)ke(t)k2 e2i (t) (ke(t)k2 + %(t)) n

n

n

X ∗ X ∗∗ X ∗ Di 2 ei (t) + Aij e2i (t) + Aij e2j (t) + Bij e2j (t − τj (t)) q¯i j=1 j=1 j=1 ! Z t n X ∗ li (t) ki (t)ke(t)k2 e2i (t) + Cij e2j (s)ds − Ri − e˜i (t) − . qi (ke(t)k2 + %(t)) t−ρj (t)

AC

CE

≤−

j=1

Now, let the time derivative of W (t) can be evaluated by Lemma 3 ( n n n n X X X X ∗ ∗∗ ∗ d Di 2 Bij e2j (t − τj (t)) W (t) ≤ − ei (t) + Aij e2i (t) + Aij e2j (t) + dt q¯i i=1 j=1 j=1 j=1 ! Z n t X ∗ li ki ke(t)k2 e2i (t) + Cij e2j (s)ds + Ri − e˜i (t) − qi (ke(t)k2 + %(t)) t−ρj (t) j=1   n X ∗ (1 − τ˙j (t)) 2 1 2 + Bij e (t) − e (t − τj (t)) (1 − κj ) j (1 − κj ) j j=1

15

)

ACCEPTED MANUSCRIPT

+

n X

∗

Bij

j=1

+2

n X

∗

−



Z

t

t−τj

e2j (s)ds

˙ Cij ρj (t)e2j (t) + ρ(t)

j=1

n X

τj e2j (t)

(

Z

!

+

n X

∗

Cij

j=1

t

t−ρj (t)



1 2 2 ρ e (t) − 2 j j



e2j (s)ds − 2 )

n X j=1

∗

Cij

Z

Z

0

−ρj

Z

t

t−ρj (t)

t

t+s



e2j (ς)dςds )

e2j (s)ds

M

AN US

CR IP T

1 (li (t) − li )2 + (ki (t) − ki )2 2q i i=1 ( ∗ ) n n  X X ∗ ∗ ∗ ∗∗ Bji 1 2∗ ≤ − (Di + ki + + τi Bji + ρi Cji + 2ρi Cji e2i (t) Aij + Aji + (1 − κi ) 2 i=1 i=1 ! Z t n X n n n n X X X X ∗ li ki ke(t)k2 e2i (t) e ˜ (t) − + + ki e2i (t) − R − B e2j (s)ds i i ij (ke(t)k2 + %(t)) qi t−τj i=1 j=1 i=1 i=1 i=1 ( ) Z Z n n n t 0 X 1 XX ∗ e2j (ς)dςds − (li (t) − li )2 + (ki (t) − ki )2 − Cij q −ρj t+s i i=1 i=1 j=1 ( ∗  ) n n X ∗ X ∗∗ ∗ ∗ Bji 1 2∗ − (Di + ki ) + Aij + Aji + ≤ + τi Bji + ρi Cji + 2ρi Cji e2i (t) (1 − κi ) 2 i=1 i=1 ! Z t n X n n X X ∗ ke(t)k%(t) li e˜i (t) + max{ki } − Bij e2j (s)ds + Ri − i∈I qi (ke(t)k + %(t)) t−τ j i=1 j=1 i=1 ( ) Z Z n n n t 0 X 1 XX ∗ 2 2 2 ej (ς)dςds − − Cij (li (t) − li ) + (ki (t) − ki ) . qi −ρj t+s −

i=1

ED

i=1 j=1

PT

Choosing large enough ki and li such that  ∗  n  X  ∗ ∗ ∗ ∗ ∗∗ B  ji 2  −(D + k ) + A + A + + τ B + ρ C + 2ρ C  i ji i ji < 0, i i ij ji i ji  (1 − κi ) i=1  li     Ri − q < 0.

CE

i

∗

AC

Also letting λ2 = maxi∈I {ki } > 0 and by using the inequality 0 ≤ ab/(a + b) ≤ a for any ∀, a > 0, b > 0 again, we have n

n

n

X XX ∗ ∗ d W (t) ≤ − $i∗ e2i (t) + λ2 %(t) − Bij dt i=1

−

n X n X i=1 j=1

i=1 j=1

∗

Cij

Z

0

−τ

Z

t

t+s

e2j (ς)dςds −

Z

n X i=1

t

t−τ

(

e2j (s)ds

1 (li (t) − li )2 + (ki (t) − ki )2 qi

)

Now, we choose positive constants $i∗ as $i∗

n  X ∗ ∗∗ =(Di + ki ) − Aij + Aji + i=1

∗

∗ ∗ ∗ Bji + τi Bji + ρ2i Cji + 2ρi Cji (1 − κi )

16



> 0.

(28) .

ACCEPTED MANUSCRIPT

Taking a positive constant δ ∗ such that δ ∗ χ∗ ≤ $∗ , then from the inequalities (27) and (28), we have

AN US

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Z t n n X n X X ∗ ∗ d W (t) + δ ∗ W (t) ≤ λ2 %(t) − $i∗ e2i (t) − Bij e2j (s)ds dt t−τj i=1 i=1 j=1 ) ( Z Z n X n n 0 t X X ∗ 1 (li (t) − li )2 + (ki (t) − ki )2 − e2j (ς)dςds − Cij q −ρ t+s j i i=1 j=1 i=1 " n Z 0 Z t Z n n n n t X χ∗ X X χ∗ X X ∗ + δ∗ χ∗ e2i (t) + ∗ e2j (ς)dςds e2j (s)ds + ∗ Cij $ $ −τ t+s i=1 i=1 j=1 t−τ i=1 j=1 #   n χ∗ X 1 (li (t) − li )2 + (ki (t) − ki )2 + ∗ $ qi i=1 Z t n n X n  ∗ ∗ X X ∗ δ χ ∗ ∗ ∗ 2 ≤λ2 %(t) + (δ χ − $ ) ei (t) + −1 e2j (s)ds $∗ t−τ i=1 i=1 j=1 Z Z n n  0 t X X  δ ∗ χ∗ − 1 e2j (ς)dςds + %(t) + $∗ −τ t+s i=1 j=1  n   δ ∗ χ∗ X 1 2 2 + −1 (li (t) − li ) + (ki (t) − ki ) $∗ qi i=1

M

∗

≤λ2 %(t),

which means that

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∗ d W (t) + δ ∗ W (t) ≤ λ2 %(t). (29) dt Then, from Lemma 5, the drive-response systems (1) and (7) achieve GDS under the adaptive nonlinear controller (25). The convergence rate of e(t) approaching zero is δ ∗ /2. The proof is completed. 2

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By selecting different types of the functions ψ(t) and %(t), the hypothesis A5 can be clarified. If ψ(t) is chosen to be exponential function, then from Theorem 1, we have a following corollary. Corollary 1. Assume that A1 , A2 and A3 hold, if we let %(t) = e−t ,  > 0 and ψ(t) = et . Then the drive-response MNNs (1) and (7) can be exponential synchronized under the adaptive controller (25). Remark 5. The GDS investigated in this paper can only be achieved via nonlinear controls (17) and (25). From the proof of Theorem 1 and Theorem 2, it is not difficult to see that only exponential synchronization can be achieved if %(t) = 0, since the nonlinear terms of (17) and (25) are transferred into linear. Remark 6. Generally, the GDS is in a general and asymptotical synchronization, polynomial synchronization, exponential synchronization, and other synchronization can be seen as the special cases of it. Previous works along this line include [41, 42]. In [41], GDS of discontinuous neural networks with matched coefficients are studied. One drawback of the aforementioned works is that the designed controller requires an extra term G1 e(t), which is introduced to cope the unexpected parameters. Moreover, in [42], the neuron activation functions are required to be bounded and satisfy the extra condition fj (±Tj ) = 0 for j ∈ I . In this paper, by designing two types of nonlinear controllers and 17

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introducing suitable Lyapunov-Krasovskii functionals, we investigated the GDS for a class of MCGNNs with mixed-time-varying delays and discontinuous activation functions. The results of this paper can be seen as the improvement and extension of the results obtained in [41, 42].

Numerical Simulations

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In this section, two examples and their numerical simulations are given to illustrate the validity of our results developed in the paper. Example 1. For n = 2, consider the drive-response MCGNNs (1) and (7) with system parameters q1 (u) = 1.2 +

|v| ≤ 1.5, |v| > 1.5, |v| ≤ 1.5, |v| > 1.5, |v| ≤ 1.5, |v| > 1.5, |v| ≤ 1.5, |v| > 1.5, |v| ≤ 1.5, |v| > 1.5, |v| ≤ 1.5, |v| > 1.5, |v| ≤ 1.5, |v| > 1.5,



1.2 , 1 + u2

|v| ≤ 1.5, |v| > 1.5, |v| ≤ 1.5, |v| > 1.5, |v| ≤ 1.5, |v| > 1.5, |v| ≤ 1.5, |v| > 1.5, |v| ≤ 1.5, |v| > 1.5. |v| ≤ 1.5, |v| > 1.5, |v| ≤ 1.5, |v| > 1.5,

2.16,  2.04, −0.96, a12 (v) =  −0.72, 1.32, a22 (v) =  1.44, 1.20, b12 (v) =  0.96, −1.44, b22 (v) =  −1.32, 0.7, c12 (v) =  0.2, −2.0, c22 (v) = −1.4,

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d2 (v) =

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1.32, d1 (v) =  1.44, 1.212, a11 (v) =  1.176, 0.96, a21 (v) =  1.20, −1.68, b11 (v) =  −1.44, 0.96, b21 (v) =  1.20, −0.8, c11 (v) =  −0.4, 0.9, c21 (v) = 0.3,

q2 (u) = 0.9 +

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1.05 , 1 + u2

CE

PT

here v ∈ R. In addition, the activation functions and time-varying delays are chosen as follows, f1 (v) = 1.4 tanh(v) − 0.08sign(v), f2 (v) = 1.1 sin(v) + 0.06sign(v), τ1 (v) = 0.75 − 0.25 cos(v), τ2 (v) = 0.65 − 0.35 cos(v) and ρ1 = ρ2 = 1. The numerical simulation of system (1) in Example 1 with initial values x1 (s) = 2.22 and x2 (s) = 0.72 for s ∈ [−1, 0] is shown in Fig. 1. We can see that it has a chaotic attractor. 2.5

2

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2

x1

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−0.5

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−1

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x2

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t

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0 −1

−2 −2.5 −3

−2 −2

−1

0

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2

3

t

x1

Fig.1. The transient behavior of system (1) in Example 1.

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1.22

1.2 1.1

B21

A11

1.2

1

1.18 1.16

0.9 0

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−1.35

B22

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−1

50

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Fig.2. The evaluation of some memristor-based weights in Example 1.

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It is not difficult to check that q 1 = 1.20, q 1 = 2.25, q 2 = 0.90, q 2 = 2.10, L1 = 1.4, L2 = 1.1, N1 = 0.16, N2 = 0.12, d1 = 1.1, d2 = 1.6, a ˜11 = 1.01, a ˜12 = 0.8, a ˜21 = ˜ ˜ ˜ ˜ 1, a ˜22 = 1.2, b11 = 1.4, b12 = 1, b21 = 1 and b22 = 1.2. Thus, the assumptions A1 , A2 and A3 are satisfied. Letting %(t) = e−0.1t , ψ(t) = et and choosing ξ1 = 4.26, ξ2 = 3.85, β1 = 1.48 and β2 = 1.58. Then, condition (18) in Theorem 1 is also satisfied. Therefore, according to the Theorem 1, the drive response systems (1) and (7) can be achieved GDS under the controller (17). The time evolution of synchronization errors between systems (1) and (7) are demonstrated in Fig. 3 (left), where 5 different initial values are chosen for (7) by y1j (s) = 0.61 − 0.4j and y2j (s) = 0.86 − 0.45j for s ∈ [−1, 0] and j ∈ {−2, −1, 0, 1, 2, 3}. The synchronization curves between systems (1) and (7) with initial conditions x1 (s) = 0.85, x2 (s) = 1.1, y1 (s) = −0.59 and y2 (s) = −0.49 are shown in Fig. 3 (right).

−0.5 −1 0

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7

8

9

t

t

Fig.3. The evaluation of synchronization errors ei and synchronization curves of xi and yi in Example 1.

Example 2. For n = 3, consider the drive-response MCGNNs (1) and (7) with system parameters 1.4 1.6 1.2 , q2 (u) = 0.8 + , q3 (u) = 1 + , 2 2 1+u 1+u 1 + u2   1.820, |v| ≤ 1, 1.682, |v| ≤ 1, d1 (v) = d2 (v) = 1.541, |v| > 1, 1.964, |v| > 1,   2.24, |v| ≤ 1, 1.68, |v| ≤ 1, d3 (v) = a11 (v) = 2.38, |v| > 1, 1.96, |v| > 1,

q1 (u) = 0.6 +

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a23 (v) = a32 (v) = b11 (v) = b13 (v) = b22 (v) = b31 (v) = b33 (v) = c12 (v) = c21 (v) = c23 (v) = c32 (v) =























a22 (v) =



a31 (v) = a33 (v) = b12 (v) = b21 (v) = b23 (v) = b32 (v) =











−4.494, −5.068, 1.54, 1.68, −4.48, −3.64, 2.24, 2.52, 1.4, 1.12, 1.12, 1.4, −7.7, −8.4, 5.880, 5.488, 0.952, 0.784, 0.714, 0.630, −0.98, −0.896, 0.644, 1.288, 1.204, 1.036,

c11 (v) =

c13 (v) =

c22 (v) =

c31 (v) = c33 (v) =











|v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1. |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1. |v| ≤ 1, |v| > 1. |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1,

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a13 (v) =

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a21 (v) =

|v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1. |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1, |v| ≤ 1, |v| > 1,

M



−1.12, −0.84, 1.12, 1.40, −6.16, 4.76, 6.174, 5.180, −2.296, −2.548, −4.2, −2.8, −1.68, −1.54, −4.438, −4.396, −4.564, −3.90, 1.036, 0.924, −3.5, −2.1, −1.736, −0.434, −0.728, 0.532,

ED



a12 (v) =

PT

here v ∈ R. In addition, the activation functions and time-varying delays are chosen as follows, f1 (v) = 1.04 tanh(v) − 0.056sign(v), f2 (v) = 1.12 cos(v) + 0.04sign(v), f3 (v) = 0.9 sin(v) + 0.064sign(v), τ1 (v) = 0.52 − 0.28cos(v), τ2 (v) = 0.44 + 0.36cos(v), τ3 (v) = 0.784 ev /(1 + ev ) and ρj = 1 for j = 1, 2, 3.

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Fig.4. The transient behavior of system (1) in Example 2.

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b31

a31

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Fig.5. The evaluation of some memristor-based weights in Example 2.

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The numerical simulation of system (1) in Example 2 with initial values x1 (s) = 2.10, x2 (s) = 0.84 and x3 (s) = −1.68 for s ∈ [−1, 0] is shown in Fig. 4. We can see that, under the system parameters of Example 2, system (1) has a chaotic attractor. 5

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Fig.6. The evaluation of synchronization errors ei and synchronization curves of xi and yi in Example 2. 9

4.5 4

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8 7

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k1 k2

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Fig.7. Time evolution of the adaptive feedback gains ki and li .

It is not difficult to check that q 1 = 0.6, q 1 = 2.0, q 2 = 0.8, q 2 = 2.4, q 3 = 1.0, q 3 = 2.2, L1 = 1.04, L2 = 1.12, L3 = 0.9, N1 = 0.112, N2 = 0.08, N3 = 0.128. Thus, the assumptions A1 , A2 and A3 are satisfied. Therefore, according to the Theorem 2, the drive-response systems (1) and (7) can be achieved GDS under the discontinuous adaptive controller (25). Letting %(t) = e−0.1t , ψ(t) = e0.5t and taking 21

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y(s) = [2.10, 0.84, −1.68]T (s ∈ [−1, 0]), K(0) = [1.448, 0.476, 2.044]T , L(0) = [0.231, 1.054, 0.495]T , K = [6, 7, 8], L = [2.6, 3.2, 3.9], αi = 0.08 and ρi = 0.6 i ∈ {1, 2, 3}, then the time evolution of synchronization errors and synchronization curves between systems (1) and (7) are presented in Fig. 6, respectively. The adaptive feedback gains ki and li are given in Fig. 7.

Conclusion

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References

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In this paper, for the first time, we studied the GDS problem for a class of MCGNNs with discontinuous neuron activations and mixed time-varying delays. By employing the analysis technique and differential inclusion theory for discontinuous systems, we introduced two kinds of nonlinear controls strategies to guarantee the GDS of considered driveresponse systems. Finally, we validated the effectiveness and feasibility of the theoretical results by giving two numerical examples. We believe that our approach and obtained results have a great significance in the construction of synchronized MNN circuits relating to discontinuous activations and time-varying delays. In the future, following issues deserve further investigation. 1) can we the optimize the control laws by using other types of controllers, such as impulsive control, sample-date control, intermittent control and so on; 2) how to study the projective synchronization, lag synchronization and FTS of MCGNNs with discontinuous activations. Acknowledgements This work was supported by the National Natural Science Foundations of P.R. China (Grant No. 61473244) and by the Natural Science Foundation of Xinjiang University (Starting research fund for the Xinjiang University doctoral graduates, Grant Nos. BS160201 and BS150202).

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