Exponential synchronization of memristor-based neural networks with time-varying delay and stochastic perturbation

Communicated by Sakthivel Rathinasamy

Accepted Manuscript

Exponential synchronization of memristor-based neural networks with time-varying delay and stochastic perturbation Xin Wang, Kun She, Shouming Zhong, Jun Cheng PII: DOI: Reference:

S0925-2312(17)30383-1 10.1016/j.neucom.2017.02.059 NEUCOM 18133

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

26 July 2016 1 December 2016 20 February 2017

Please cite this article as: Xin Wang, Kun She, Shouming Zhong, Jun Cheng, Exponential synchronization of memristor-based neural networks with time-varying delay and stochastic perturbation, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.02.059

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ACCEPTED MANUSCRIPT 1

Exponential synchronization of memristor-based neural networks with time-varying delay and stochastic perturbation

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Abstract—This paper deals with the stochastic exponential synchronization problem of memristor-based neural networks with time-varying delay. Firstly, considering the state-dependent properties of the memristor, less conservative of model is constructed to analyze the complicated memristor-based neural networks. Then, by applying the stochastic differential inclusions theory and Lyapunov functional approach, sufficient verifiable conditions that depend on the time-varying delay and stochastic perturbation are obtained. It is shown that synchronization can be realized by linear feedback control and adaptive feedback control. The obtained results complement and improve the previously known results. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results. Index Terms—Exponential synchronization, Memristor-based neural networks, Time-varying delay, Stochastic system

I. I NTRODUCTION

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Xin Wang, Kun She, Shouming Zhong, and Jun Cheng

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HE originally theoretical concept of memristor was derived from [1], it is well known that there are three basic circuit elements: 1) resistance; 2) capacitance; and 3) inductance. These elements are used to describe the relations between fundamental electrical quantities: 1) voltage; 2) current; 3) charge; 4) flux. Resistance relates voltage and current (R = dϑ/di), capacitance relates charge and current C = dq/dϑ, and inductance relates flux and current (L = dϕ/di), respectively. However, Chua [1] realized that there was a missing link between flux ϕ and charge q, as shown in Fig.1, which he called memristance; it could describe the nonlinear relation between flux and charge (M = dϕ/dq). Almost 40 years later, the practical memristor was realized by scientists at Hewlett-Packard Laboratories and was published in [2]. It is shown in [2] that the relation between current and voltage exhibits the so-called pinched hysteresis loops as the signature of memristor (see Fig.2). In recent years, increasing research attention from different branches of science and application fields has been paid to memristor, since it can be used to neural networks to mimic the memory and forgetting effect in the human memory [3-9].

Fig.1 Relations between four fundamental circuit elements (Chua, 1971).

This research was financially supported by the National Natural Science Foundation of China (Nos. 61273015 and 61533006). X. Wang and K. She are with the School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu Sichuan 611731, PR China (e-mail: [email protected]). S.M. Zhong is with the School of Mathematics Sciences, University of Electronic Science and Technology of China, Chengdu Sichuan 611731, PR China. J. Cheng is with the School of Science, Hubei University for Nationalities, Hubei 445000, PR China.

Fig.2 Current-voltage characteristics of the memristor (Strukov et al., 2008).

In addition, synchronization in coupled networks is one of the most important collective dynamical behaviors [10-11]. Some works dealing with the synchronization phenomena of the memristor-based neural networks have also appeared [1226]. In [12], based on the drive-response concept, fuzzy theory, and Lyapunov functional method, some new criteria are established to guarantee the exponential synchronization of coupled memristor-based neural networks. Li et al. [13] considered the lag synchronization problem of memristor-based coupled neural networks with or without parameter mismatch using two different algorithms. Chandrasekar et al. [14] extended the notion of synchronization of the memristor-based neural networks with two delay components based on second-order reciprocally convex approach. Moreover, the synchronization’s

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matrix with n dimensions. Y (X ≥ Y ) means that the matrix X −Y is positive definite (nonnegative). The symbol diag{. . .} denotes the block diagonal matrix. λmax (X) and λmin (X) denote the maximum and minimum eigenvalues of X. The superscript T denotes matrix or vector transposition. kxk indicates the 2-norm of a vector x, i.e., n P kxk = ( x2i )1/2 . Let (Ω, F , {Ft }t≥0 , P ) be a complete i=1

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probability space with filtration {Ft }t≥0 satisfying the usual conditions(i.e., the filtration contains all P -null sets and is right continuous) and E{·} denotes the expectation operator with respect to the given probability measure P . II. NETWORKS MODEL AND PRELIMINARIES In this section, we consider the following memristor-based neural network with time-varying delay and stochastic perturbation:  n X   dxi (t) =[−di (xi (t))xi (t) +  bij (xi (t))fj (xj (t))     j=1    n  X   cij (xi (t))fj (xj (t − τ (t))) + Ii ]dt + (1) j=1     + σi (t, xi (t), xi (t − τ (t)))dwi (t),      i = 1, 2, . . . , n     xi (t) =φˆi (t), ∀t ∈ [−τ, 0]

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results in [27] and [28] show that memristor-based nonlinear system plays an important role in secure communication due to the special feature of memristor. Therefore, it is necessary to study the synchronization of memristor-based neural networks. On the other hand, the reason that time-delay is significant to the study of neural networks lies in the communication time between neurons and the finite switching speed of amplifiers in hardware implementations of neural networks. As is well known, the time-delay may lead to oscillation, instability, and poor performance of the dynamical systems [29-32]. Therefore, it is crucial to investigate neural networks subject to delays. Moreover, in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. Thus, noise is unavoidable in actual applications of neural networks. Meanwhile, stochastic perturbations may take important effects on the dynamic behaviors of delayed system, and a neural network could be stabilized or destabilized by certain stochastic inputs [33-34]. Recently, a stochastic memristor-based neural network is proposed in [35-36] and the global exponential stability in the mean square for this system is considered. Because of taking stochastic disturbances into consideration, it plays an important role in the quality of synchronization. However, very few works have been done on exponential synchronization of memristor-based neural networks with time-varying delay and stochastic perturbation. For example, the control schemes for delayed memristorbased neural networks obtained in [12,37-39] cannot be applied and extended directly to the synchronization problem of memristor-based neural networks with stochastic perturbation. Above all, this motivates us to study the synchronization of delayed memristor-based neural networks that depends on the time-varying delay and stochastic effects. Motivated by the above discussions, this paper investigates the stochastic exponential synchronization issue of memristorbased neural networks with time-varying delay. The main contributions of this paper are listed as follows: First, considering the state-dependent properties of the memristor, less conservative of model is constructed to analyze the complicated memristor-based neural networks. Second, by applying the stochastic differential inclusions theory and Lyapunov functional method, sufficient verifiable conditions that depend on the time-varying delay and stochastic perturbation are obtained. It is shown that exponential synchronization can be realized by linear feedback control and adaptive feedback control. Third, the obtained results complement and improve the previously known results. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results. The rest of this paper is organized as follows. In section II, the stochastic memristor-based neural network is introduced and some necessary definition and lemmas are given. In section III, the linear feedback control and the adaptive feedback control are designed, and the corresponding mean square exponential synchronization conditions are derived, respectively. In section IV, a example is provided to illustrate the effectiveness of the theoretical results. Finally, section V concludes this paper. Notation: Throughout this paper, In denotes the identity

where fj (·) is activation function, τ (t) is time-varying transmission delay, and it satisfies 0 < τ (t) ≤ τ , τ˙ (t) ≤ µ < 1 for some positive constants τ and µ, σi (·, ·, ·) : R+ × Rn × Rn → Rn is the noise intensity function matrix, and wi (·) is an ndimensional weiner process defined on (Ω, F , {Ft }t≥0 , P ) satisfying E{dw(t)} = 0, Ii is the external input, φˆi (t) ∈ C([−τ, 0]; Rn ), (i = 1, 2, . . . , n) are the initial values are associated with system (1), and the connection weight coefficients satisfy ( d´i , |xi (t)| ≤ Ti di (xi (t)) = d`i , |xi (t)| > Ti bij (xi (t)) =

cij (xi (t)) =

(

´bij , |xi (t)| ≤ Ti `bij , |xi (t)| > Ti , ( c´ij , |xi (t)| ≤ Ti c`ij , |xi (t)| > Ti

The switching jumps Ti > 0, d´i > 0, d`i > 0, ´bij , `bij , c´ij , c`ij , (i, j = 1, 2, . . . , n) are constants. Remark 1: When we let ( d1i = min(d´i , d`i ) d2i = max(d´i , d`i ), (

b1ij = min(´bij , `bij ) b2ij = max(´bij , `bij ),

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c1ij = min(´ cij , c`ij )

where the response system (3) has the similar structure as the drive system (2), ui (t) (i = 1, 2, . . . , n) are the controllers to be designed. Suppose that ei (t) = yi (t) − xi (t) is the error state, then the error dynamical system between (2) and (3) is given as follows:  2 X 2 X 2 X    de (t) = θ1`1 (t)θ2`2 (t)θ3`3 (t){[−d`1 i ei (t) i     `1 =1 `2 =1 `3 =1    n  X    + b`2 ij gj (ej (t))  

c2ij = max(´ cij , c`ij ).

Then, di (xi (t)) =θ11 (t)d1i + θ12 (t)d2i =

2 X

θ1` (t)d`i ,

2 X

θ2` (t)b`ij ,

`=1

bij (xi (t)) =θ21 (t)b1ij + θ22 (t)b2ij =

j=1

               

`=1

cij (xi (t)) =θ31 (t)c1ij + θ32 (t)c2ij =

2 X

θ3` (t)b`ij ,

`=1

where θk` (t) ≥ 0 (k = 1, 2, 3)(` = 1, 2),

2 P

θk` (t) = 1.

`=1

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`1 =1 `2 =1 `3 =1 n X

+

+

θ1`1 (t)θ2`2 (t)θ3`3 (t){[−d`1 i xi (t)

b`2 ij fj (xj (t))

j=1 n X j=1

ED

2 2 X 2 X X

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=

c`3 ij fj (xj (t − τ (t))) + Ii ]dt

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                                   

n X j=1

c`3 ij gj (ej (t − τ (t))) + ui (t)]dt

+ σi (t, ei (t), ei (t − τ (t)))dwi (t)}, i = 1, 2, . . . , n

ei (t) =ϕi (t), ∀t ∈ [−τ, 0]

(4) where gj (ej (t)) = fj (yj (t)) − fj (xi (t)), gj (ej (t − τ (t))) = fj (yj (t − τ (t))) − fj (xj (t − τ (t))), σi (t, ei (t), ei (t − τ (t))) = σi (t, yi (t), yi (t − τ (t))) − σi (t, xi (t), xi (t − τ (t))) and ϕi (t) = φˇi (t) − φˆi (t).

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Hence, we can obtain the following stochastic memristorbased neural network  2 X    dx (t) =[− θ1` (t)d`i xi (t) i     `=1    n 2 X  X    θ2` (t)b`ij fj (xj (t)) +     `=1 j=1     2 X n  X    + θ3` (t)c`ij fj (xj (t − τ (t))) + Ii ]dt     j=1 `=1     + σi (t, xi (t), xi (t − τ (t)))dwi (t),  

+

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(

To obtain the main results, the following assumptions, definition and lemmas are introduced. Assumption 2.1 For ∀x, y ∈ R, x 6= y, the neuron activation function fi (·) is bounded and satisfies L− i ≤

fi (x) − fi (y) ≤ L+ i , i = 1, 2, . . . , n. x−y

Assumption 2.2 σi (·, ·, ·) : R+ × R × R → R, and there exist two n × n matrices Γ1 > 0 and Γ2 > 0 such that trace{σ T (·, x, y)σ(·, x, y)} ≤ xT Γ1 x + y T Γ2 y

(5)

holds for all x, y ∈ R . Definition 2.1 The drive system (2) and the response system (3) is said to be mean square exponentially synchronized if there exist a pair of positive constants M and α such that n

E(ke(t)k2 ) ≤ M

sup Ekϕ(s)k2 e−αt , t ≥ 0.

−τ ≤s≤0

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+ σi (t, xi (t), xi (t − τ (t)))dwi (t)}, i = 1, 2, . . . , n lemma 2.1 [40] Let X and Y be arbitrary n-dimensional real ˆ xi (t) =φi (t), ∀t ∈ [−τ, 0] vectors, R be a positive definite matrix, and P ∈
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2 ( ∂∂xVi(x,t) ∂xj )n×n .

Remark 2: The memristor-based neural network (1) is a special model of stochastic neural network (2), and when d´i = d`i , ´bij = b`ij , c´ij = c`ij and Ti is a arbitrary positive constant, then the (2) changes as common stochastic recurrent neural network [33]. Moreover, compared to the system (1), the range of solutions of the system (2) was enlarged, so the study of network (1) can be converted to investigate the network (2). III. MAIN RESULTS

A. Mean square exponential synchronization of memristorbased neural networks via linear feedback control

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where hi > 0 is the feedback gain. Then, we can get the following error system under the control (7),  2 2 X 2 X X    θ1`1 (t)θ2`2 (t)θ3`3 (t){[−d`1 i ei (t) de (t) = i     ` =1 ` =1 ` =1 3 2 1    n  X    + b`2 ij gj (ej (t))   c`3 ij gj (ej (t − τ (t))) − hi ei (t)]dt

PT

j=1

+ σi (t, ei (t), ei (t − τ (t)))dwi (t)}, i = 1, 2, . . . , n

ei (t) =ϕi (t), ∀t ∈ [−τ, 0]

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(8) Theorem 3.1 Suppose that the Assumptions 2.1-2.2 hold, if there exist a positive constant µ, diagonal positive define matrices L, H and positive define matrices Γ1 , Γ2 such for any `i = 1, 2 (i = 1, 2, 3), the following inequality satisfies: 1 [(|C`3 |L)(|C`3 |L)T + Γ1 + (1 − µ)−1 (In + Γ2 )] 2 − D`1 + |B`2 |L − H < 0

(9)

then the drive system (2) and the response system (3) are mean square exponentially synchronized by the linear feedback controller (8). Proof. Consider the following Lyapunov-Krasovskii functional candidate: Z t n n X X V (t) = eTi (t)ei (t) + ri eTi (η)ei (η)dη i=1

i=1

θ1`1 (t)θ2`2 (t)θ3`3 (t)

`1 =1 `2 =1 `3 =1 n n X n X X ×[ eTi (t) − d`1 i ei (t) + eTi (t)b`2 ij gj (ej (t)) i=1

+

i=1 j=1

n X n X i=1 j=1

n X i=1

+

n X i=1

eTi (t)c`3 ij gj (ej (t − τ (t))) −

n X

hi eTi (t)ei (t)]

i=1

trace[σiT (t, ei (t), ei (t − τ (t)))σi (t, ei (t), ei (t − τ (t)))] ri eTi (t)ei (t) − (1 − τ˙ (t))

n X i=1

ri eTi (t − τ (t))ei (t − τ (t))

t−τ (t)

where ri (i = 1, 2, . . . , n) is positive constant to be determined below.

(10)

For one thing, from Assumption 2.1, we have n n X X eTi (t)(b`2 ij (ei (t)))gj (ej (t)) i=1 j=1

≤

(7)

ui (t) = −hi ei (t), i = 1, 2, . . . , n

+

2 X 2 X 2 X

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In this subsection, synchronization of the drive-response system will be realized by using the linear feedback control. The linear feedback controller ui is designed as

               

L V (t) = 2

+

In this section, our purpose is to design the proper controller ui (t) (i = 1, 2, . . . , n) such that the drive system and response system achieve synchronization

j=1 n X

Computing the weak infinitesimal operator L along the trajectory of the error system (8) yields

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(x,t) ∂V (x,t) , ∂x2 , . . . , ∂V∂x(x,t) ), Vxx (x, t) = where Vx (x, t) = ( ∂V∂x 1 n

≤

and

n X n X i=1 j=1 n X n X i=1 j=1 T

|ei (t)||(b`2 ij (ei (t)))||gj (ej (t))| |ei (t)||(b`2 ij (ei (t)))|Li |ej (t)|

=e (t)|B`2 |Le(t)

n n X X i=1 j=1

≤ ≤

(11)

eTi (t)c`3 ij gj (ej (t − τ (t))) n X n X

i=1 j=1 n X n X i=1 j=1 T

|ei (t)||c`3 ij ||gj (ej (t − τ (t)))| |ei (t)||c`3 ij |Li |ej (t − τ (t))|

=e (t)|C`3 |Le(t − τ (t))

(12)

On the other hand, based on lemma 2.1, we obtain 2eT (t)|C`3 |Le(t − τ (t)) ≤eT (t)(|C`3 |L)(|C`3 |L)T e(t) + eT (t − τ (t))e(t − τ (t))

(13)

+ max(|L− i |, |Li |),

where Li = L = diag{L1 , L2 , . . . , Ln }, |B`2 | = (|b`2 ij |)n×n , |C`3 | = (|c`3 ij |)n×n , e(t) = (|e1 (t)|, |e2 (t)|, . . . , |en (t)|)T , e(t − τ (t)) = (|e1 (t − τ (t))|, |e2 (t − τ (t))|, . . . , |en (t − τ (t))|)T . Then, under the Assumptions 2.2 and combining (11)-(13) results in 2 X 2 X 2 X L V (t) ≤2 θ1`1 (t)θ2`2 (t)θ3`3 (t) `1 =1 `2 =1 `3 =1

1 {eT (t)(−D`1 + |B`2 |L + (|C`3 |L)(|C`3 |L)T 2 1 1 + Γ1 + R − H)e(t)} 2 2 + eT (t − τ (t))[I + Γ2 − (1 − µ)R]e(t − τ (t)) (14)

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where R = diag{r1 , r2 , . . . , rn }, H = diag{h1 , h2 , . . . , hn }, D`1 = diag{d`1 1 , d`1 2 , . . . , d`1 n }. Letting R = {(1 − µ)−1 (I + Γ2 )}, then we have 2 X 2 X 2 X

eεt Eke(t)k2 ≤ eεt EkV (t)k2 ≤ M Thus, we have

θ1`1 (t)θ2`2 (t)θ3`3 (t)

Eke(t)k2 ≤ M

`1 =1 `2 =1 `3 =1

1 eT (t)(−D`1 + |B`2 |L + (|C`3 |L)(|C`3 |L)T 2 1 1 −1 + Γ1 + (1 − µ) (I + Γ2 ) − H)e(t) 2 2 (15) From condition (9), we obtain L V (t) < 0. Next we prove that system (8) is exponentially synchronization in mean square. Let λ = λmin {−(−D`1 + |B`2 |L + 21 (|C`3 |L)(|C`3 |L)T + 1 1 −1 (I + Γ2 ) − H)} > 0, from (15), we get 2 Γ1 + 2 (1 − µ) E(L V (t)) ≤ −λke(t)k2

(16)

By the definitions of V (t), we have t

t−τ

ke(η)k2 dη (17)

Choose an appropriate ε > 0 such that

ε(1 + λmax (R)τ eετ ) ≤ λ

(18)

ED

Combining (16) and (17), we have Z t Z t E L W (s)ds ≤E eεs [(ε − λ)ke(s)k2 0 0 Z t + ελmax (R) ke(η)k2 dη]ds

(20)

PT

t−τ

By changing the integration sequence, we have Z t Z t Z t Z η+τ eεs ds ke(η)k2 dη ≤ dη eεs ke(η)k2 ds t−τ

−τ

CE

0

η

t

−τ

εη

2

e ke(η)k dη

(21)

AC

≤τ e

ετ

Z

Substituting the above inequality into (19) and by using (18), we obtain eεt EV (t) − EV (0) Z t ≤E eεη [ε(1 + λmax (R)τ eετ ) − λ]ke(η)k2 dη + ET (0) 0

≤ ET (0)

(22) R ετ 0

where T (0) = ελmax (R)τ e eεη ke(η)k2 dη. −τ Moreover, it is not difficult to see that there exists a positive constant M > 0 such that EV (0) + ET (0) ≤ M

sup Ekϕ(s)k −τ ≤s≤0

(25)

which implies that the drive system (2) and the response system (3) are mean square exponentially synchronized. The proof is complete. Remark 3: It is worth emphasizing that the activation function [37] needs to meet fi (±Ti ) = 0, which is a strong condition, and most of the functions do not have this property. However, the activation function in this paper only needs to satisfy the Lipschitz condition, even it does not need fi (0) = 0, which is suitable and more general. Remark 4: In general, the strength of linear feedback must be maximal, which is a kind of waste in practice to some extent. Compared with linear control [13,18,39], the control gains of adaptive control increase according to the adaptive laws. In the following, an adaptive strategy will be designed. B. Mean square exponential synchronization of memristorbased neural networks via the adaptive feedback control In this subsection, synchronization of the drive-response system will be realized by using the adaptive feedback control. The adaptive feedback controllers ui (t) is designed as

M

Define W (t) = eεt V (t), and applying the generalized Itˆo’s formula, gives Z t E L W (s)ds = eεt EV (t) − EV (0), t ≥ 0. (19) 0

sup Ekϕ(s)k2 e−εt . −τ ≤s≤0

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Z

ke(t)k2 ≤ V (t) ≤ ke(t)k2 + λmax (R)

sup Ekϕ(s)k2 . (24) −τ ≤s≤0

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L V (t) ≤2

It follows from (17), (22) and (23) that

2

(23)

ui (t) = −li (t)ei (t), l˙i (t) = hi eTi (t)ei (t), i = 1, 2, . . . , n.

(26)

where hi > 0 is the adaptive gain. Then, under the control (26) we can get the error system as follows  2 2 X 2 X X   de (t) =  θ1`1 (t)θ2`2 (t)θ3`3 (t){[−d`1 i ei (t) i     `1 =1 `2 =1 `3 =1    n  X    + b`2 ij gj (ej (t))   j=1

               

+

n X j=1

c`3 ij gj (ej (t − τ (t))) − li (t)ei (t)]dt

+ σi (t, ei (t), ei (t − τ (t)))dwi (t)}, i = 1, 2, . . . , n

ei (t) =ϕi (t), ∀t ∈ [−τ, 0]

(27) Theorem 3.2 Suppose that the Assumptions 2.1-2.2 hold, if there exist a positive constant µ, diagonal positive define matrices L, δ and positive define matrices Γ1 , Γ2 such for any `i = 1, 2 (i = 1, 2, 3), the following inequality satisfies: 1 [(|C`3 |L)(|C`3 |L)T + Γ1 + (1 − µ)−1 (In + Γ2 )] 2 − D`1 + |B`2 |L − δ < 0

(28)

then the drive system (2) and the response system (3) are mean square exponentially synchronized by the adaptive feedback controller (26).

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where δ = diag{δ1 , δ2 , · · · , δn } and the other symbols are the same as those defined in Theorem 3.1. Proof. Consider the Lyapunov-Krasovskii functional as follows:

L V (t) = 2

2 X 2 X 2 X

θ1`1 (t)θ2`2 (t)θ3`3 (t)

`1 =1 `2 =1 `3 =1 n n X n X X ×[ eTi (t) − d`1 i ei (t) + eTi (t)b`2 ij gj (ej (t)) i=1

n X 1 (li (t) − δi )2 V1 (t) = eTi (t)ei (t) + h i i=1 i=1 Z t n X + ri eTi (η)ei (η)dη

i=1 j=1

−

t−τ (t)

i=1

+

n X

4 3 2

M

0 −1 −2 −3 −4 −5

−60

−40

−20

x1(t)

0

20

40

PT

−80

ED

x2(t)

1

=2

trace[σiT (t, ei (t), ei (t − τ (t)))σi (t, ei (t), ei (t − τ (t)))] ri eTi (t)ei (t) − (1 − τ˙ (t))

2 X 2 X 2 X

n X i=1

ri eTi (t − τ (t))ei (t − τ (t))

θ1`1 (t)θ2`2 (t)θ3`3 (t)

`1 =1 `2 =1 `3 =1 n n X n X X ×[ eTi (t) − d`1 i ei (t) + eTi (t)b`2 ij gj (ej (t)) i=1 i=1 j=1 n X n n X X + eTi (t)c`3 ij gj (ej (t − τ (t))) − δi eTi (t)ei (t)] i=1 j=1 i=1 n X trace[σiT (t, ei (t), ei (t − τ (t)))σi (t, ei (t), ei (t − τ (t)))] + i=1 n n X X ri eTi (t − τ (t))ei (t − τ (t)) ri eTi (t)ei (t) − (1 − τ˙ (t)) + i=1 i=1

CE AC

y1(t)

0

−20

−40

−60

−80

0.5

1

1.5

2

2.5

t

3

3.5

Then, similar to the proof of Theorem 3.1, we get the following result 2 X 2 X 2 X L V (t) ≤2 θ1`1 (t)θ2`2 (t)θ3`3 (t)

1 e (t)(−D`1 + |B`2 |L + (|C`3 |L)(|C`3 |L)T 2 1 1 + Γ1 + (1 − µ)−1 (I + Γ2 ) − δ)e(t) 2 2 (30)

x1(t)

20

0

(29)

T

40

x1(t), y1(t)

i=1

i=1

`1 =1 `2 =1 `3 =1

Fig.3 Phase trajectories of drive system (31).

−100

n X

n X (li (t) − δi )eTi (t)ei (t)

AN US

where ri and δi (i = 1, 2, . . . , n) are positive constants to be determined below. The operator L V along the trajectory of error system (28) gives

li (t)eTi (t)ei (t)] + 2

i=1

n X i=1

+

eTi (t)c`3 ij gj (ej (t − τ (t)))

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+

n X

i=1 j=1

n X n X

4

4.5

5

Fig.4 The state trajectories of variables x1 (t) and y1 (t).

where δ = diag{δ1 , δ2 , · · · , δn }. According to the condition (28), similar to the proof of Theorem 3.1, we can obtain the drive system (2) and the response system (3) are mean square exponentially synchronized. Therefore, the proof is complete. Remark 5: Different from some existing papers, the proposed conditions in this paper depend on the jump point Ti of memristor-based neural networks (1), we have added Figs.8-10 to further explain it. Thus the results have less conservativeness and expand the results in the existing literatures. Remark 6: In the existing papers, i.e. Refs.[19,25,36], the authors usually assumed that [d1i , d2i ]yi (t) − [d1i , d2i ]xi (t) ⊆ [d1i , d2i ](yi (t) − xi (t)) or |di (yi (t)) − di (xi (t))| ≤ di |yi (t) − xi (t)| (di = max{|d1i |, |d2i |}) is satisfied. However, in this paper, we don’t have to assume this condition but by

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IV. AN ILLUSTRATIVE EXAMPLE

d2 (x2 (t)) =

(

b11 (x1 (t)) =

(

1.2, |x1 (t)| > 1.2 0.2, |x2 (t)| ≤ 1.2

0.3, |x2 (t)| > 1.2

−0.5, |x1 (t)| ≤ 1.2

−0.6, |x1 (t)| > 1.2

b21 (x1 (t)) =

(

3, |x2 (t)| ≤ 1.2

3.2, |x2 (t)| > 1.2 5, |x1 (t)| ≤ 1.2

AC

c11 (x1 (t)) =

−0.4, |x2 (t)| > 1.2

c12 (x2 (t)) =

c21 (x1 (t)) =

c22 (x2 (t)) =

−0.1, |x1 (t)| ≤ 1.2

−0.2, |x1 (t)| > 1.2

(

(

(

5, |x2 (t)| ≤ 1.2

5.2, |x2 (t)| > 1.2 3, |x1 (t)| ≤ 1.2

3.1, |x1 (t)| > 1.2

−0.1, |x2 (t)| ≤ 1.2

−0.2, |x2 (t)| > 1.2,

and σ1 (t, x1 (t), x1 (t − τ (t))) =0.3x1 (t) + 0.2x1 (t − τ (t)),

σ2 (t, x2 (t), x2 (t − τ (t))) = − 0.4x2 (t) + 0.3x2 (t − τ (t))

y2(t)

2 1

x2(t), y2(t)

(

x2(t) 3

−0.5, |x2 (t)| ≤ 1.2

CE

(

4

5.1, |x1 (t)| > 1.2

PT

b12 (x2 (t)) =

(

b22 (x2 (t)) =

1, |x1 (t)| ≤ 1.2

M

d1 (x1 (t)) =

(

ED

where

AN US

In this section, an example is given to illustrate the effectiveness of the main results obtained in this paper. Example 1: Consider the following 2-D memristor-based neural networks:  2 X    dx (t) =[−d (x (t))x (t) + bij (xi (t))fj (xj (t))  i i i i    j=1     2 X (31) + cij (xi (t))fj (xj (t − τ (t))) + Ii ]dt    j=1      + σ i (t, xi (t), xi (t − τ (t)))dwi (t),    i = 1, 2.

Correspondingly, slave system can be written as  2 X    dy (t) =[−d (y (t))y (t) + bij (yi (t))fj (yj (t))  i i i i    j=1     2 X + cij (yi (t))fj (yj (t − τ (t))) + Ii + ui (t)]dt    j=1      + σi (t, yi (t), yi (t − τ (t)))dwi (t),    i = 1, 2. (32) The nonlinear activation functions are taken as f (x) = t (sinx1 , sinx2 )T , time-varying delay τ (t) = ete+1 , and external inputs I1 = I2 = 0. By simple calculation, it can be seen that L = In , τ˙ (t) ≤ µ = 14 < 1, and Γ1 = Γ2 = In /2, taking the control gains h1 = h2 = 20, so the conditions of the Theorem 3.1 are satisfied. In the following, we will verify the correctness by MATLAB simulation. The phase trajectory of states x1 (t) and x2 (t) with initial conditions x1 (s) = 1 and x2 (s) = −0.5 ∀s ∈ [−1, 0) are shown in Fig.3. The state trajectories of variables x1 (t) and y1 (t) with initial condition x1 (s) = 1, y1 (s) = −0.5, and ∀s ∈ [−1, 0) are shown in Fig.4. The state trajectories of variables x2 (t) and y2 (t) with initial condition x2 (s) = −0.5, y2 (s) = 0.5, and ∀s ∈ [−1, 0) are shown in Fig.5. Taking the above initial conditions, when there is no controller, the error trajectories of drive-response system are shown in Fig. 6, which can prove that the memristive-coupled networks cannot achieve synchronization. However, under the linear feedback control, the memristive drive-response system can achieve synchronization, and the error trajectories of drive-response system are described in Fig. 7. Moreover, the change in coefficients d1 (x1 (t)) and d1 (y1 (t)), b11 (x1 (t)) and b11 (y1 (t)), c12 (x1 (t)) and c12 (y1 (t)) are described in Figs. 8-10.

CR IP T

considering the state-dependent properties of the memristor and constructing a new model to analyze the complicated memristor-based neural networks. Hence, the results of this paper enrich and complement the earlier works.

0 −1 −2 −3 −4 −5

0

0.5

1

1.5

2

2.5

t

3

3.5

4

4.5

5

Fig.5 The state trajectories of variables x2 (t) and y2 (t).

Finally, we consider the synchronization of the above driveresponse system via adaptive feedback method proposed in Theorem 3.2. In simulation, choose hi = 0.5 i = 1, 2 and the other parameters are the same as those in the above. Fig. 11 describes the error trajectories of memristive-coupled networks. Compared with linear control, the control gains of adaptive control increase according to the adaptive laws. Thus, adaptive control is more flexible to study the synchronization of our system.

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40

2 d (x (t))

e1(t)

1

e (t) 2

20

1

d (y (t))

1.8

1

1

1.6

d1(x1(t)), d1(y1(t))

e(t)

0

−20

−40

1.4 1.2 1 0.8 0.6

−60

0

0.5

1

1.5

2

2.5

t

3

3.5

4

4.5

0.2

5

Fig.6 Error state trajectories without controller.

0

0.5

1

1.5

2

2.5

t

3

3.5

4

4.5

5

Fig.8 Change trajectories of coefficients d1 (x1 (t)) and d1 (y1 (t)).

1.5 e (t) 1

e2(t)

0.5

0

−0.5

AN US

1

e(t)

CR IP T

0.4 −80

b (x (t)) 11

1

b11(y1(t))

−0.2

0

0.5

1

1.5

2

2.5

t

3

3.5

4

4.5

5

M

−1.5

Fig.7 Error state trajectories under the linear controller.

ED

V. C ONCLUSION

−0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1

0

0.5

1

1.5

2

2.5

t

3

3.5

4

4.5

5

Fig.9 Change trajectories of coefficients b11 (x1 (t)) and b11 (y1 (t)).

ACKNOWLEDGMENT

6 c12(x1(t))

5.8

c12(y1(t))

5.6 5.4

c12(x1(t)), c12(y1(t))

AC

CE

PT

In this paper, the stochastic exponential synchronization problem of memristor-based neural networks with timevarying delay is studied. Firstly, considering the statedependent properties of the memristor, a new model is constructed to analyze the complicated memristor-based neural networks. Then, by applying the stochastic differential inclusions theory and Lyapunov functional approach, sufficient verifiable conditions are obtained for the exponential synchronization of delayed memristor-based neural networks. It is shown that synchronization can be realized by linear feedback control and adaptive feedback control. The obtained results complement and improve the previously known results. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results. Future work could be done on synchronization of memristor-based neural networks with parameter mismatch.

b11(x1(t)), b11(y1(t))

−0.3

−1

5.2 5 4.8 4.6

The authors would like to thank the editors and the reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the National Natural Science Foundation of China (Nos. 61273015 and 61533006).

4.4 4.2 4

0

0.5

1

1.5

2

2.5

t

3

3.5

4

4.5

5

Fig.10 Change trajectories of coefficients c12 (x1 (t)) and c12 (y1 (t)).

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e (t) 1

e (t) 2

1

e(t)

0.5

0

−0.5

−1

−1.5

0

0.5

1

1.5

2

2.5

t

3

3.5

4

4.5

5

Fig.11 Error state trajectories under the adaptive controller.

R EFERENCES

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V ITAE

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Xin Wangwas born in Sichuan, China. He received the B.S. degree from YiBin University, YiBin, China, in 2013. Now he is working towards the Ph.D. degree in School of Information and Software Engineering at the University of Electronic Science and Technology of China. His current research interests include time-delay systems, the robustness stability, and synchronization of complex systems.

Jun Chengwas born on August, 1987. He received B.S. degree from Hubei University for Nationalities. He is currently working towards the Ph.D. degree in School of Automation Engineering, University of Electronic Science and Technology of China. His research interests include Markov jump systems, switched systems, neural networks and time-delay systems. He is a very active reviewer for many international journals.

CE

PT

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M

Kun She received the Ph.D. degree from computer application, University of Electronic Science and Technology of China. He is currently a Professor of the School of Information and Software Engineering at University of Electronic Science and Technology of China. His research works focus on network safety, cloud computing and so on, and more than 100 scholar papers were published in these years.

AC

Shouming Zhongwas born on November 5, 1955. He graduated from University of Electronic Science and Technology of China, majoring applied mathematics on differential equation. He is a Professor of School of Mathematical Sciences, University of Electronic Science and Technology of China, on June 1997present. He is a Director of Chinese Mathematical Biology Society, the Chair of Biomathematics in Sichuan, Editors of Journal of Biomathematics. He has reviewed for many journals, such as Journal of Theory and Application on Control, Journal of Automation, Journal of Electronics, and Journal of Electronics Science. His research interest is stability theorem and its application research of the differential system, the robustness control, neural network and biomathematics.