fixed-time synchronization of delayed memristive reaction-diffusion neural networks

ARTICLE IN PRESS

JID: NEUCOM

[m5G;November 5, 2019;21:18]

Neurocomputing xxx (xxxx) xxx

Contents lists available at ScienceDirect

Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Brief papers

Finite/fixed-time synchronization of delayed memristive reaction-diffusion neural networksR Shiqin Wang a, Zhenyuan Guo a,∗, Shiping Wen b, Tingwen Huang c, Shuqing Gong a,d a

College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China c Science Program, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar d School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China b

a r t i c l e

i n f o

Article history: Received 21 January 2019 Revised 12 April 2019 Accepted 13 June 2019 Available online xxx Communicated by Haijun Jiang Keywords: Finite/fixed-time synchronization Reaction–diffusion Memristive neural network State feedback controller

a b s t r a c t This paper is concerned with the finite/fixed-time synchronization (FFTS) problems of two delayed memristive reaction–diffusion neural networks (MRDNNs). By designing appropriate state feedback controllers, utilizing the Lyapunov function method and inequality techniques, several sufficient criteria are derived to guarantee the FFTS of the drive-response MRDNNs. Taking into account both the influences of time and space, the model, described as a state-dependent switching system here, is more complex and closer to practical applications than those in the existing results. Finally, an example is presented to substantiate the effectiveness of the theoretical results.

1. Introduction Artificial neural network (ANN) is a nonlinear dynamic system connected by a large number of simple neurons. Its working principle and functional characteristics are similar to human brain. It can self-identify, summarize the scale of operations, and complete self-control. These features of ANN attract experts from different professions to undertake research and development of it. So far, some achievements have been made on the basis of ANN, such as electric load forecasting in [1], rainfall-runoff process modeling in [2] and business failure prediction in [3]. Artificial neural network is a new discipline and there are still many blanks in theory, model, algorithm, application and implementation, etc., which need to be explored, studied and developed. In a general neural network model, connection weights are usually obtained from resistors which are used to model the biological synapses. It is known to us that biological synapses have longterm memory, but unfortunately, the conventional resistors do not possess this function. In order to meet the needs of scientific experiments, Chua first postulated the existence of a new fundamental circuit element which can remember its own historical state R Research supported by National Natural Science Foundation of China (61573003), Natural Science Foundation of Hunan (2019JJ40022) and NPRP grants: NPRP 9-466-1-103 from Qatar National Research Fund. ∗ Corresponding author. E-mail address: [email protected] (Z. Guo).

© 2019 Elsevier B.V. All rights reserved.

and called it memristor from the abbreviation of memory and resistor [4]. In 2008, the researchers at the Hewlett-Packard Laboratory announced that they had built memristors successfully [5]. Their experimental results showed that the value of the memristor (i.e., memristance) depends on the magnitude and polarity of the voltage applied to the memristor and the length of the time that the voltage has been applied [6]. With the emergence of memristor, a large number of research results have come on the heels. For example, the Lagrange stability [7], attractivity [8], exponential stabilization [9], passivity and passification [10], exponential synchronization [11], robust stability [12] and robust synchronization [13] of memristor-based neural networks (NNs) have been investigated. Normally, the neural network model is described by ordinary differential equations. That is to say, only the effects of time on the states of neurons have been considered. It should be pointed out that, in electric circuits, the moving electrons have diffusion behaviors in a nonuniform electromagnetic field [14]. Thus, it is essential to consider the state variables varying with time and space meanwhile. This prompts us to use partial differential equations to describe the neural network model. Up to now, many results have been obtained on the synchronization problem of reaction-diffusion neural networks (RDNNs), such as [15–20]. The authors in [15] combined Lyapunov functional differential method with Green’s formula and inequality techniques to investigate the synchronization problem of delayed RDNNs. By utilizing theories of partial differential equations, [16] studied the synchronization

https://doi.org/10.1016/j.neucom.2019.06.092 0925-2312/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: S. Wang, Z. Guo and S. Wen et al., Finite/fixed-time synchronization of delayed memristive reaction-diffusion neural networks, Neurocomputing, https://doi.org/10.1016/j.neucom.2019.06.092

JID: NEUCOM 2

ARTICLE IN PRESS

[m5G;November 5, 2019;21:18]

S. Wang, Z. Guo and S. Wen et al. / Neurocomputing xxx (xxxx) xxx

issue of RDNNs with Dirichlet boundary conditions and infinite delays. Impulsive control synchronization problems of RDNNs were discussed in [17] and [18]. For coupled RDNNs with directed topology, [19] designed some novel edge-based adaptive laws on coupling weights for realizing synchronization and [20] proposed an adaptive strategy to achieve synchronization based on pinning control. Synchronization, as one of the most important cluster behaviors of NNs, has various forms, such as, asymptotical synchronization in [21–23], exponential synchronization in [24–26], robust synchronization in [27–29] and so on. Through careful analysis and summarization, it is not difficult to find that most types of synchronization belong to infinite time synchronization. But in practical applications, due to objective constraints, we usually want to achieve the synchronization of drive-response pair of NNs in a limited time [30–32]. Compared with infinite time synchronization, finite time synchronization essentially requires faster convergence speed. More importantly, the states of the drive system and the response system can be exactly the same after a finite time which is called settling-time. But the only thing of finite time synchronization that matters is that the settling-time depends on the initial states of the drive-response systems. However, the initial conditions of many practical systems are difficult to be adjusted or even impossible to be estimated, which leads to the inaccessibility of the final settling-time of the drive-response systems, and it shall make the performance of the system worse. In order to overcome this difficulty of finite-time control, Polyakov proposed a new concept: fixed-time convergence [30]. Subsequently, the concept of fixed-time synchronization came into being. Different from finitetime synchronization, fixed-time synchronization can directly calculate the settling-time without considering the initial state error of the drive-response systems. Therefore, it has a better application prospect in practice [33]. So far, many useful results have been achieved for FFTS. In [34– 36], finite-time synchronization of second-order nonlinear multiagent system, fuzzy cellular neural networks with time-varying delays and multi-link complex networks were studied respectively. Fixed-time synchronization of uncertain complex-valued neural networks, coupled discontinuous neural networks and coupled delayed continuous neural networks were investigated in [37,38] and [39], respectively. The authors in [40–42] considered the FFTS of inertial memristive neural networks, discontinuous complex networks and multi-proportional delayed inertial neural networks respectively. However, to the best of our knowledge, up to now, there are few results on the FFTS of delayed MRDNNs. Generally speaking, there are many results on the synchronization of neural network models at present, but less research has been done on their FFTS, and few achievements have been made on the FFTS of partial differential neural network models. Let alone the studies on the FFTS of state-dependent partial differential switching neural network models. Based on the aforementioned concern, in this paper, we try to investigate the FFTS of delayed MRDNNs subject to Dirichlet boundary conditions. The contributions of this paper are summarized as follows: (i) the drive-response pair of MRDNNs is studied, which generalizes some existing neural networks for FFTS. It should be pointed out that MRDNNs are different from conventional NNs because MRDNNs are discontinuous, in which the influences of time and space are considered. Whereas conventional NNs are continuous where only the influence of time is considered; (ii) several FFTS criteria are derived based on Lyapunov function method, Divergence theorem and inequality techniques, and the settling-time for synchronization is estimated, which is valuable for adjusting the synchronization time of the drive-response systems. The rest of this paper is organized as follows. In Section 2, some notations, definitions and lemmas which will be used in this paper

are introduced, and the models of delayed MRDNNs are presented. In Section 3, we will give several sufficient conditions for ensuring the FFTS of MRDNNs. In Section 4, an illustrative example and its simulations will be provided to substantiate the effectiveness of the obtained algebraic criteria. Finally, conclusions are drawn in Section 5. 2. Preliminaries In this section, we first introduce some notations which will be used in the rest of this paper, and then briefly describe the model of MRDNNs. The definitions and lemmas used to deduce FFTS will be given lastly. 2.1. Notations Here we introduce some conventional notations. let R =

(−∞, +∞ ) and Rn be the n-dimensional Euclidean space.  = {x = (x1 , x2 , . . . , xl )T ||xk | < hk , k = 1, 2, . . . , l } is a bounded compact set with smooth boundary ∂  and mes > 0 where mes is the measure of . C0 stands for the set of continuous functions, and B  C0 ([−τ , 0 ) ×  → Rn ) describes the Banach space of all continuous functions φ (t, x ) = (φ1 (t, x ), φ2 (t, x ), . . . , φn (t, x ))T : [−τ , 0 ) ×  → Rn equipped with the norm defined by φ (t, · ) =   1 {  ni=1 |φi (t, x )|2 dx} 2 . D+ v(t ) and D− v(t ) describe the upper right Dini-derivation and the left Dini-derivation of v(t) respectively. 2.2. Model description In a drive-response scheme, there are two identical NNs with different initial conditions. Here we first introduce the driven system which can be described as the following delayed MRDNNs: l n  ∂ zi (t, x )  ∂ 2 zi (t, x ) = dik − c z ( t, x ) + ai j (z ) f j (z j (t, x )) i i ∂t ∂ x2k j=1 k=1

+

n 

bi j (z )g j (z j (t − τ , x )) + Ii ,

(1)

j=1

where i, j ∈ I  {1, 2, . . . , n} and n is the number of neurons in this system; zi (t, x) corresponds to the state of the ith neuron at time t and in space x ∈  ⊆ Rl ; dik ≥ 0 is the transmission diffusion coefficient along the ith neuron; ci > 0 represents the neural self-inhibitions; Ii ∈ Rn is the external input or bias; τ > 0 is the transmission delay among neurons; fj (zj (t, x)) and g j (z j (t − τ , x )) correspond to neuronal activation function of jth neuron without and with time delay, respectively; The feedback connection weight aij (z) is the function of fi j (t, x ) = f j (z j (t, x )) − zi (t, x ), and the delayed feedback connection weight bij (z) is the function of gi j (t, x ) = g j (z j (t − τ , x )) − zi (t, x ). They are defined as



ai j ( z ) = and

bi j ( z ) =

aˆi j , aˇi j , ai j (t − , x ),

⎧ ⎨bˆ i j , bˇ , ⎩bi j (t − , x ), ij

D− fi j (t, x ) > 0; D− fi j (t, x ) < 0; D− fi j (t, x ) = 0, D− gi j (t, x ) > 0; D− gi j (t, x ) < 0; D− gi j (t, x ) = 0,

where D− fi j (t, x ) and D− gi j (t, x ) are the left Dini-derivation of fij (t, x) and gij (t, x) with respect to time t respectively; aˆi j , aˇi j , bˆ i j and bˇ i j are known constants; ai j (t − , x ) and bi j (t − , x ) are the left limit of aij and bij with respect to time t. For convenience, we denote ai j = max{aˆi j , aˇi j }, a = min{aˆi j , aˇi j }, bi j = max{bˆ i j , bˇ i j } and ij

Please cite this article as: S. Wang, Z. Guo and S. Wen et al., Finite/fixed-time synchronization of delayed memristive reaction-diffusion neural networks, Neurocomputing, https://doi.org/10.1016/j.neucom.2019.06.092

ARTICLE IN PRESS

JID: NEUCOM

[m5G;November 5, 2019;21:18]

S. Wang, Z. Guo and S. Wen et al. / Neurocomputing xxx (xxxx) xxx

bi j = min{bˆ i j , bˇ i j }. Meanwhile, we denote diaj = ai j − ai j and dibj =

+

bi j − bi j . For any i ∈ I, the boundary condition and initial value of system (1) are given by

(t, x ) ∈ [−τ , +∞ ) × ∂ ,

zi (t, x ) = 0,

zi (s, x ) = ψi (s, x ),

(2)

(s, x ) ∈ [−τ , 0 ) × ,

(3)

where ψi (s, x ) ∈ B is a real valued function. Assumption 1. The activation functions fi and gi are globally Lipschitz continuous and bounded, namely, there exist positive constants li , ρ i , Fi and Gi such that

3



n 

bi j (z˜) g j (z˜ j (t − τ , x )) − g j (z j (t − τ , x ))



j=1

+Ui (t, x ).

(7)

Correspondingly, the boundary condition and initial value of NNs (7) are given as follows:

ei (t, x ) = 0,

(t, x ) ∈ [−τ , +∞ ) × ∂ ,

ei (s, x ) = ϕi (s, x ) − ψi (s, x ),

(8)

(s, x ) ∈ [−τ , 0 ) × .

(9)

2.3. Definitions and lemmas

| f i ( ξ1 ) − f i ( ξ2 ) | ≤ l i | ξ 1 − ξ2 | , Definition 1 [43]. The drive-response systems (1) and (4) are said to be finite-time synchronized, if for a suitable designed controller and any initial state e0 , there exists a time T(e0 ) such that limt→T(e0 ) e(t, x ) = 0, and e(t, x) ≡ 0, for t ≥ T(e0 ).

| g i ( ξ1 ) − g i ( ξ2 ) | ≤ ρi | ξ 1 − ξ2 | , and

| fi (u )| ≤ Fi , |gi (u )| ≤ Gi hold for any ξ1 , ξ2 , u ∈ R, i ∈ I. The corresponding response system of (1) can be designed as the following partial differential equations: l 

∂ z˜i (t, x ) ∂ 2 z˜i (t, x ) = dik − ci z˜i (t, x ) + ∂t ∂ x2k k=1 +

n 

n 

ai j (z˜) f j (z˜ j (t, x ))

j=1

bi j (z˜)g j (z˜ j (t − τ , x )) + Ii + Ui (t, x ),

(4)

j=1

(t, x ) ∈ [−τ , +∞ ) × ∂ ,

z˜i (s, x ) = ϕi (s, x ),

(s, x ) ∈ [−τ , 0 ) × ,

(5) (6)

where ϕi (s, x ) ∈ B is a real valued function. Define the synchronization error ei (t, x ) = z˜i (t, x ) − zi (t, x ), then the error dynamics system of systems (1) and (4) can be expressed by the following form:

∂ ei (t, x ) = ∂t

l 

∂ ei (t, x ) − ci ei (t, x ) ∂ x2k k=1

n  + ai j (z˜) f j (z˜ j (t, x )) − ai j (z ) f j (z j (t, x )) 2

dik

j=1

+

n 

Definition 3 [44]. A function V (x ) : Rn → R is C-regular if V(x) is: (i) regular in Rn ; (ii) positive definite, i.e.,V(x) > 0 for x = 0 and V (0 ) = 0; (iii) radially unbounded, i.e., V(x) → ∞ as x → ∞. Lemma 1 [45]. Let h(x) be a real-valued continuous function defined on  satisfying: h(x )|∂  = 0, then



where Ui (t, x) is the external control input that will be properly designed to realize FFTS; For any i ∈ I, the boundary condition and initial value of system (4) are given by

z˜i (t, x ) = 0,

Definition 2 [30]. The drive-response systems (1) and (4) are said to be fixed-time synchronized, if they are finite-time synchronized and the settling-time function T(e0 ) is bounded, i.e., there exists a constant Tmax > 0 such that T(e0 ) < Tmax .

bi j (z˜)g j (z˜ j (t − τ , x )) − bi j (z )g j (z j (t − τ , x ))



l 

∂ 2 ei (t, x ) = dik − ci ei (t, x ) ∂ x2k k=1

n  + ai j (z˜) f j (z˜ j (t, x )) − f j (z j (t, x )) j=1

+

n  j=1

+

n  j=1



ai j (z˜) − ai j (z ) f j (z j (t, x ))



bi j (z˜) − bi j (z ) g j (z j (t − τ , x ))



(

∂h 2 ) dx. ∂ xk

Lemma 2 [30, 46]. Suppose that V (x ) : Rn → R is C-regular and that x(t ) : [0, ∞ ) → Rn is absolutely continuous on any compact subinterval of [0, ∞). Let K (V ) = k1V α + k2V β where k1 , k2 > 0. If D+V (x ) ≤  V (0 ) 1 −K (V ) and T = 0 dσ < ∞, then, K (σ ) 1 −α

1

0 ≤ β < 1 < α.

2

Lemma 3 [47]. Suppose that the function V(t) is continuous and positive definite, and it satisfies the following differential inequality :

V˙ (t ) ≤ −lV α (t ) + kV (t ),



1 −β

1. V(t) will reach zero in a finite time T ≤ min{ Vk (1−(α0 )) , Vk (1−(β0 )) } 1 2 when 0 ≤ α , β < 1; 1 1 1 1 2. V(t) will reach zero in a fixed time T ≤ k α −1 + k 1−β when

l k

∀t > t0 , V 1−α (t0 ) < ,

(10)

where α ∈ (0, 1), l and k are positive constants. Then, V(t) ≡ 0, ∀t ≥ t1 , where the settling-time t1 is given as follows:

t1 = t0 +

ln(1 − kl V 1−α (t0 )) k (α − 1 )

j=1

+Ui (t, x )



h2 (x )dx ≤ h2k

.

3. Main results In this section, we will design a state feedback controller and derive the sufficient algebraic criteria to ensure FFTS of systems (1) and (4). First, we propose the following state feedback controller:

Ui (t, x ) = − pi1 ei (t, x ) − pi2 sign(ei (t, x )) − pi3 −



n 

e(t, · )−1 + e(t, · )2α−2 ei (t, x )

|bi j |ρ j sign(ei (t, x ))|e j (t − τ , x )|,

(11)

j=1

Please cite this article as: S. Wang, Z. Guo and S. Wen et al., Finite/fixed-time synchronization of delayed memristive reaction-diffusion neural networks, Neurocomputing, https://doi.org/10.1016/j.neucom.2019.06.092

ARTICLE IN PRESS

JID: NEUCOM 4

where i ∈ I; e(t, x ) = (e1 (t, x ), e2 (t, x ), . . . , en (t, x ))T ; pi1 , pi2 , pi3 > 0 and α ≥ 0 are constants which should be determined later. Remark 1. It should be pointed out that controller (11) can not be realized when e(t, · ) = 0 in circuit implementation, because it contains the negative power of e(t, · ). But if we design it as a piecewise function:

Ui (t, x )

⎧ −p e (t, x ) − pi2 sign(ei (t, x )) ⎪ ⎨ i 1 i −pi3 e(t, · )−1 + e(t, · )2α −2 ei (t, x ) = (12) n ⎪ ⎩− j=1 |bi j |ρ j sign(ei (t, x ))|e j (t − τ , x )|, e(t, · ) = 0; 0, e(t, · ) = 0, the controller will be feasible. In fact, in the circuit implementation, what we use is the controller (12). But for convenience, we will write the controller in a non-piecewise function form as (11) throughout this paper. From the later analysis, we can see that this treatment does not affect the theoretical results. If the controller Ui (t, x) in system (4) is designed as (11), we will obtain our first sufficient criterion of FFTS as follows. Theorem 1. Let Assumption 1 hold. For any i ∈ I, if the control parameters in (11) satisfy l n 

dik 1  pi1 ≥ − − c + ai j l j + a ji li i 2 h2k j=1

(13)

k=1

Based on the boundary condition (8), by applying Divergence theorem, we obtain

  n

l 

∂ 2 ei (t, x ) dx ∂ x2k  i=1 k=1   l    n ∂ ei (t, x ) = 2ei (t, x ) ∇ · dik dx ∂ xk  i=1 k=1   l   n  ∂ ei (t, x ) = 2 ∇ · ei (t, x )dik dx ∂ xk  i=1 k=1   l   n  ∂ ei (t, x ) − 2 ∇ ei (t, x ) · dik dx ∂ xk  i=1 k=1   l  n  ∂ ei (t, x ) = 2 dik ei (t, x ) · n dx ∂ xk ∂  i=1 k=1  2  n  l  ∂ ei (t, x ) − 2 dik dx ∂ xk  i=1 k=1  2  n  l  ∂ ei (t, x ) =− 2 dik dx, ∂ xk  i=1 k=1 2ei (t, x )

dik

n  



(14)

 ∂ e (t, x ) l i

∂ xk

j=1

then, (i) systems (1) and (4) can achieve finite-time synchronization when 0 ≤ α < 1, and the settling-time is estimated

1 2

T ≤ min{2V (0 ),

as  n

V 1 −α ( 0 ) 1−α }

·

1

where

min1≤i≤n {2 pi3 }

V (0 ) =  i=1 e2i (0, x )dx; (ii) systems (1) and (4) can achieve fixed-time synchronization when α > 1, and the settling-time is estimated as T ≤ min 2{α2−1 . p } (α −1 ) 1≤i≤n

i3

Proof. Consider the following Lyapunov function:

V (t ) =

  n  i=1

e2i (t, x )dx.

(15)

Calculate the upper right Dini-derivative of V(t) along the trajectories of error system (7), then we have

D+V (t ) =

  n



l 

j=1

+



bi j (z˜) g j (z˜ j (t − τ , x )) − g j (z j (t − τ , x ))



2

n  l 

 dik

i=1 k=1

∂ ei (t, x ) ∂ xk

2 ≤−

  n  l 2dik 2 e (t, x )dx. (18) h2k i  i=1 k=1

From the Lipschitz continuity and boundedness of the activation function in Assumption 1, according to the conditions (13) and (14), we have

D+V (t )       n l n  n  dik 2 ≤ 2 − − c e ( t, x ) dx + 2|ei (t, x )| i i h2k  i=1  i=1 j=1 k=1





  n  i=1



(19)



2 − pi1 e2i (t, x ) − pi2 |ei (t, x )| dx



 i=1 j=1



  n





l n   dik 1  ≤ 2 − − ci − pi1 + ai j l j + a ji li e2i (t, x )dx 2 2 h  i=1 j=1 k=1 k

j=1

+

j=1

×g j (z j (t − τ , x )) − pi1 ei (t, x ) − pi2 sign(ei (t, x ))

e(t, · )−1 + e(t, · )2α−2 ei (t, x )  n  − |bi j |ρ j sign(ei (t, x ))|e j (t − τ , x )| dx.

 ∂ e (t, x ) ∂ e (t, x ) ∂ ei (t, x ) T i i , ,..., . ∂ x1 ∂ x2 ∂ xl

−2 pi3 e(t, · ) + e(t, · )2α  

n  n − 2|bi j |ρ j |ei (t, x ))||e j (t − τ , x )| dx

n  + bi j (z˜) − bi j (z )

−pi3

 −

+

n  ai j (z˜) − ai j (z ) f j (z j (t, x )) n 

k=1

=

From Lemma 1, we have

dik

j=1

+

l

× ai j l j |e j (t, x )| + diaj Fj + dibj G j + bi j ρ j |e j (t − τ , x )| dx

∂ 2 ei (t, x ) − ci ei (t, x ) ∂ x2k  i=1 k=1

n  + ai j (z˜) f j (z˜ j (t, x )) − f j (z j (t, x )) 2ei (t, x )

2

dient operator, and

diaj Fj + dibj G j ,

(17)

where “ · ” is inner product, n is the outward pointing unit normal field of the boundary ∂ , ∇ = ( ∂∂x , ∂∂x , . . . , ∂∂x ) denotes the gra1

and

pi2 ≥

[m5G;November 5, 2019;21:18]

S. Wang, Z. Guo and S. Wen et al. / Neurocomputing xxx (xxxx) xxx



  n  i=1

(16)

2

n  

diaj Fj

+

dibj G j





− pi2

1≤i≤n

|ei (t, x )|dx

j=1

e(t, · ) + e(t, · )2α  1  ≤ − min {2 pi3 } V 2 (t ) + V α (t ) . −2 pi3





(19)

j=1

Please cite this article as: S. Wang, Z. Guo and S. Wen et al., Finite/fixed-time synchronization of delayed memristive reaction-diffusion neural networks, Neurocomputing, https://doi.org/10.1016/j.neucom.2019.06.092

ARTICLE IN PRESS

JID: NEUCOM

[m5G;November 5, 2019;21:18]

S. Wang, Z. Guo and S. Wen et al. / Neurocomputing xxx (xxxx) xxx

Based on Lemma 2, we can get the following results: 1. if 0 ≤ α < 1, the finite-time synchronization between system (1) and (4) can be achieved, and the settling-time is bounded by



V 1 −α ( 0 ) T ≤ min 2V (0 ), 1−α



1 2

·

1 ; min1≤i≤n {2 pi3 }

2. if α > 1, the fixed-time synchronization between system (1) and (4) can be achieved, and the settling-time is bounded by

T≤

2α − 1 . min1≤i≤n {2 pi3 }(α − 1 )

+

  n  i=1

Corollary 1. Let Assumption 1 hold and 0 ≤ α < 1. For any i ∈ I, if the controller Ui (t, x) in system (4) is designed as (11) and the control parameters in (11) satisfy



diaj Fj + dibj G j ,

(21)

j=1

then, systems (1) and (4) can achieve finite-time synchronization 1 when V (0 ) < min M2 , M 1−α , and the settling-time is estimated as

T ≤ max

Mi = 2 −

(0 ) M

M

where M =



1 2

min1≤i≤n {2 pi3 } M

l  k=1



,

ln 1 −

V 1−α (0 ) M



T ≤ max

with M = max1≤i≤n {Mi }, in which n 



ai j l j + a ji li



.

j=1

Proof. Similar to the proof of Theorem 1, we have

D V (t )     n l n   2 dik 1  ai j l j + a ji li ei (t, x )dx ≤ 2 − − ci − pi1 + 2 h2k  i=1 j=1 +

k=1

− pi2

|ei (t, x )|dx





V

1 2

(0 ) M

M

This proof is completed.

1

(22)





ln 1 −

,

V 1−α (0 ) M

 .

M (α − 1 )



If we do not use memristors to replace the resistors in ANN, systems (1) and (4) will not be state-dependent switching systems, and their connection weights will be constants. For any i, j ∈ I, denote their feedback connection weight as aij and delayed feedback connection weight as bij , then, their error system can be described as follows: l ∂ ei (t, x )  ∂ 2 ei (t, x ) = dik − ci ei (t, x ) ∂t ∂ x2k k=1

n  + ai j f j (z˜ j (t, x )) − f j (z j (t, x )) j=1

+

n 





bi j g j (z˜ j (t − τ , x )) − g j (z j (t − τ , x )) + Ui (t, x ).

j=1

(23) Correspondingly, design the controller Ui (t, x) in (23) as

Ui (t, x ) = − pi1 ei (t, x ) − pi3 n 



e(t, x )−1 + e(t, x )2α−2 ei (t, x )

|bi j |ρ j sign(ei (t, x ))|e j (t − τ , x )|,

(24)

j=1

Corollary 2. Let Assumption 1 hold. For any i ∈ I, if the control parameters in (24) satisfy

pi1 ≥ −

l n 

dik 1  − ci + ai j l j + a ji li , 2 2 hk j=1

(25)

k=1

then, (i) systems (1) and (4) can achieve finite-time chronization when 0 ≤α < 1, and the settling-time is mated

M (α − 1 )

dik 1 − ci − pi1 + 2 h2k



then, we can get the following corollary.

and

V

+



j=1

2 ln 1 −

(20)

k=1



dibj G j

Based on Lemma 3, if V (0 ) < min M2 , M 1−α , the local finite-

−

l n 

dik 1  pi1 < − − ci + ai j l j + a ji li 2 2 hk j=1

2 ln 1 −

diaj Fj

1≤i≤n

In Theorem 1, we dropped the first term of the reciprocal second inequality of (19), and obtained the results of global FFTS. In fact, we can also reserve this term. But in this case, we can only get the results of local finite-time synchronization according to Lemma 3 as follows.



2





Remark 3. Among the recent results on synchronization of the partial differential neural network models, most of what scholars have studied are infinite time synchronization problems, such as reference [19,48,49]. However, the finite and fixed time synchronization problem is investigated in this paper. Moreover, for memristive neural networks, a lot of achievements for many types of synchronization including FFTS have been made, see [50–52] and the references therein. However, the influence of space on the state is not considered.

pi2 ≥

n  

time synchronization between system (1) and (4) can be achieved, and the settling-time is bounded by

Remark 2. From Theorem 1, it is not difficult to find that the synchronization time of systems (1) and (4) is closely related to the controller parameters pi1 , pi2 , pi3 and α . Therefore, we can obtain the convergence results as best as possible by adjusting the controller parameters pi1 , pi2 , pi3 and α appropriately.

n  



e(t, · ) + e(t, · )2α  1  ≤ MV (t ) − min {2 pi3 } V 2 (t ) + V α (t ) 1≤i≤n  1  ≤ MV (t ) − min {2 pi3 } max V 2 (t ), V α (t ) . −2 pi3



The proof is completed.

5

as  n

1

T ≤ min 2V 2 (0 ), V

1 −α ( 0 )

1−α

·

1 min1≤i≤n {2 pi3 }

synestiwhere

V (0 ) =  i=1 e2i (0, x )dx; (ii) systems (1) and (4) can achieve fixed-time synchronization when α > 1, and the settling-time is estimated as T ≤ min 2{α2−1 . p } (α −1 ) 1≤i≤n

i3

Proof. Consider the Lyapunov function (15). Calculate the upper right Dini-derivative of V(t) along the trajectories of error system (23), then we have

D V (t ) = +

  n



l 

∂ 2 ei (t, x ) − ci ei (t, x ) ∂ x2k  i=1 k=1

n  + ai j f j (z˜ j (t, x )) − f j (z j (t, x )) 2ei (t, x )

dik

j=1

Please cite this article as: S. Wang, Z. Guo and S. Wen et al., Finite/fixed-time synchronization of delayed memristive reaction-diffusion neural networks, Neurocomputing, https://doi.org/10.1016/j.neucom.2019.06.092

ARTICLE IN PRESS

JID: NEUCOM

+

n 



bi j g j (z˜ j (t − τ , x )) − g j (z j (t − τ , x ))

j=1



e(t, · )−1 + e(t, · )2α−2 ei (t, x )  n  − |bi j |ρ j sign(ei (t, x ))|e j (t − τ , x )| dx ≤

 i=1

 2 −

l  k=1

×e2i (t, x )dx +

dik 1 − ci − pi1 + 2 h2k

  n  n



 i=1 j=1

n 



ai j l j + a ji li

 

5

4

4

3

3 2

1

j=1

  n



e (t,x)

−pi1 ei (t, x ) − pi3

1

j=1

2bi j ρ j |ei (t, x )||e j (t − τ , x )|dx

 i=1 j=1



1 10 0

(26)

1≤i≤n

The remaining is the same as that in Theorem 1, which is omitted here.  If the time delay is not considered in systems (1) and (4), their error system can be described as follows: l ∂ ei (t, x )  ∂ 2 ei (t, x ) = dik − ci ei (t, x ) ∂t ∂ x2k k=1

n  + ai j (z˜) f j (z˜ j (t, x )) − f j (z j (t, x )) j=1

n  + ai j (z˜) − ai j (z ) f j (z j (t, x )) + Ui (t, x ).

(27)



e(t, x )−1 + e(t, x )2α−2 ei (t, x ),



bi j =

then, we can get the following corollary.

l n 

dik 1  pi1 ≥ − − c + ai j l j + a ji li i 2 h2k j=1

ai j , ai j , ai j (t − , x ),

D− fi j (t, x ) > 0; D− fi j (t, x ) < 0; D− fi j (t, x ) = 0,

bi j , bi j , bi j (t − , x ),

D− gi j (t, x ) > 0; D− gi j (t, x ) < 0; D− gi j (t, x ) = 0,



Corollary 3. Let Assumption 1 hold. For any i ∈ I, if the control parameters in (28) satisfy

where



A = [ai j ]2×2



B = [bi j ]2×2 =

and n 

diaj Fj ,

(30)

j=1

then, (i) systems (1) and (4) can achieve finite-time synchronization when 0 ≤ α < 1, and the settling-time is estimated

as  n

1

T ≤ min{2V 2 (0 ), V

2 i=1 ei (0, x )dx;

1 −α ( 0 )

1−α

}·

1 min1≤i≤n {2 pi3 }

p11 ≥ −

Consider that the settling-time of fixed-time synchronization has nothing to do with the initial value of the drive-response systems, and it has a better application prospect than finite-time synchronization. Therefore, here, we only verify the theoretical results of fixed-time synchronization. In theory, when the parameters of





3.3 −0.9





6.8 , 6.2

2.9 3.2 , B = [bi j ]2×2 = 1.8 −1.2



2.7 . 1.6

2

d1 1  − c + a1 j l j + a j1 l1 = 10.1, 1 2 2 h j=1

p21

2

d2 1  a2 j l j + a j2 l2 = 12.3, ≥ − 2 − c2 + 2 h j=1

i3

4. Numerical simulation



7.3 3.9 , A = [ai j ]2×2 = 6.7 5.1

The controller Ui (t, x) in system (4) is designed as (11) with p11 = 12.12, p12 = 1.68, p13 = 0.30, p21 = 14.76, p22 = 1.68, p23 = 0.445 and α = 2. It is not difficult to verify that

where

V (0 ) =  (ii) systems (1) and (4) can achieve fixed-time synchronization when α > 1, and the settling-time is estimated as T ≤ min 2{α2−1 . p } (α −1 ) 1≤i≤n

4.5 = 5.5

(29)

k=1

t

0

Example 1. Consider systems (1) and (4) with τ = 0.01, n = 2, l = 1, h = 1(i.e.,  = {x| − 1 < x < 1} ), D = diag(d1 , d2 ) = diag(0.6, 0.5 ), C = diag(c1 , c2 ) = diag(0.2, 0.3 ), f i ( ξ ) = ( |ξ + 1| − |ξ − 1| )/2 and gi (ξ ) = tanh(ξ ), i=1,2. Obviously, the activation functions f1 , f2 , g1 and g2 satisfy Assumption 1 with l1 = l2 = ρ1 = ρ2 = 1 and F1 = F2 = G1 = G2 = 1. The weighted matrices A = [ai j ]2×2 and B = [bi j ]2×2 are given by

ai j =

(28)

-1

the systems satisfy the theoretical conditions, changing their initial values will not affect the settling-time. Next, we verify this conclusion.

Ui (t, x ) = − pi1 ei (t, x ) − pi2 sign(ei (t, x )) − pi3

t

0

5

x

Fig. 1. Evolution processes of e1 (t, x) and e2 (t, x) under the initial conditions: z1 (t, x ) = 20t cos( π2x ), z2 (t, x ) = 35t sin(π x ), z˜1 (t, x ) = 40t cos( π2x ) and z˜2 (t, x ) = 125t sin(π x ), for (t, x ) ∈ [−τ , 0 ) × .

j=1

Correspondingly, design the controller Ui (t, x) in (27) as

-1

10 0

5

x



1

-1

1

≤ − min {2 pi3 } V 2 (t ) + V α (t ) .

1

t=T t=T

-1

2

0

0

−2 pi3 e(t, · ) + e(t, · )2α   n  n − 2|bi j |ρ j |ei (t, x )||e j (t − τ , x )|dx

pi2 ≥

[m5G;November 5, 2019;21:18]

S. Wang, Z. Guo and S. Wen et al. / Neurocomputing xxx (xxxx) xxx

e 2(t,x)

6

p12 ≥

2  



d1a j Fj + d1b j G j = 1.4,

j=1

p22 ≥

2  



d2a j Fj + d2b j G j = 1.4.

j=1

From Theorem 1, systems (1) and (4) can achieve fixedtime synchronization, and the settling-time is bounded by T ≤

Please cite this article as: S. Wang, Z. Guo and S. Wen et al., Finite/fixed-time synchronization of delayed memristive reaction-diffusion neural networks, Neurocomputing, https://doi.org/10.1016/j.neucom.2019.06.092

ARTICLE IN PRESS

JID: NEUCOM

[m5G;November 5, 2019;21:18]

S. Wang, Z. Guo and S. Wen et al. / Neurocomputing xxx (xxxx) xxx 2α −1 2 min{ p13 ,p23 } (α −1 )

0

0 t=T

e (t,x)

2

-0.4

1

e (t,x)

-0.2

-0.6

5. Conclusion

-0.4 -0.6

-1

-0.8

1

t=T

-0.2

-0.8

1 10 0

x

10 0

5 -1

0

x

t

= 5. Figs. 1–4, respectively describe the evolution

processes of synchronization errors e1 (t, x) and e2 (t, x) under four different initial values, from which we can see that both the synchronization errors e1 (t, x) and e2 (t, x) reach to zero before T = 5. Therefore, the theoretical results of this paper are valid.

0.2

0.2

7

5 -1

0

t

Fig. 2. Evolution processes of e1 (t, x) and e2 (t, x) under the initial conditions: z1 (t, x ) = 35t sin(π x ), z2 (t, x ) = 20t cos( π2x ), z˜1 (t, x ) = 125t sin(π x ) and z˜2 (t, x ) = 40t cos( π2x ), for (t, x ) ∈ [−τ , 0 ) × .

In this paper, by designing appropriate state feedback controllers, we investigated the FFTS of delayed MRDNNs with the help of Lyapunov function method, Divergence theorem and inequality techniques. The model in this paper takes into account the characteristic of simulated objects in circuit implementation (i.e., the long-term memory function of biological synapses) and the effects of simulated experimental environments on state variables. But strictly speaking, the model in this paper is still not perfect. For example, when the state variables in the experiment are disturbed by white noise, in order to make the model closer to reality, we need to introduce random interference terms. Further investigations may aim at analyzing the FFTS of multi-coupled MRDNNs, and the dynamical behaviors of driving-response MRDNNs and multi-coupled MRDNNs with random disturbances. Declarations of interests

0.6

1

None.

0.4

0

2

e (t,x)

1

e (t,x)

References

0.2

0.5

0

t=T

-0.2 -0.4

t=T

-0.6

-0.5 1

1 10 0

x

10 0

5 -1

0

x

t

5 -1

0

t

Fig. 3. Evolution processes of e1 (t, x) and e2 (t, x) under the initial conditions: z1 (t, x ) = 10t (x2 − 1 ), z2 (t, x ) = 35t sin(π x ), z˜1 (t, x ) = 20t (x2 − 1 ) and z˜2 (t, x ) = 60t sin(π x ), for (t, x ) ∈ [−τ , 0 ) × .

0.6

1 0.8

0.4

e (t,x)

0.2

2

0.4

1

e (t,x)

0.6

0.2

0 0

t=T t=T

-0.2

-0.2

1

1 10 0

x

5 -1

0

t

10 0

x

5 -1

0

t

Fig. 4. Evolution processes of e1 (t, x) and e2 (t, x) under the initial conditions: z1 (t, x ) = 35t sin(π x ), z2 (t, x ) = 10t (x2 − 1 ), z˜1 (t, x ) = 60t sin(π x ) and z˜2 (t, x ) = 20t (x2 − 1 ), for (t, x ) ∈ [−τ , 0 ) × .

[1] D.C. Park, M.A. El-Sharkawi, R.J. Marks, L.E. Atlas, M.J. Damborg, Electric load forecasting using an artificial neural network, IEEE Trans. Power Syst. 6 (2) (1991) 442–449. [2] K. Hsu, H.V. Gupta, S. Sorooshian, Artificial neural network modeling of the rainfall-runoff process, Water Resour. Res. 31 (10) (1995) 2517–2530. [3] B.S. Ahn, S.S. Cho, C.Y. Kim, The integrated methodology of rough set theory and artificial neural network for business failure prediction, Expert Syst. Appl. 18 (2) (20 0 0) 65–74. [4] L.O. Chua, Memristor–the missing circuit element, IEEE Trans. Circuit Theory 18 (1971) 507–519. [5] J.M. Tour, T. He, The fourth element, Nature 453 (2008) 42–43. [6] L.O. Chua, Resistance switching memories are memristors, Appl. Phys. A 102 (2011) 765–783. [7] A. Wu, Z. Zeng, Lagrange stability of memristive neural networks with discrete and distributed delays, IEEE Trans. Neural Netw. Learn. Syst. 25 (4) (2014) 690–703. [8] Z. Guo, J. Wang, Z. Yan, Attractivity analysis of memristor-based cellular neural networks with time-varying delays, IEEE Trans. Neural Netw. Learn. Syst. 25 (4) (2014) 704–716. [9] S. Ding, Z. Wang, N. Rong, H. Zhang, Exponential stabilization of memristive neural networks via saturating sampled-data control, IEEE Trans. Cybern. 47 (10) (2017) 3027–3039. [10] Z. Guo, J. Wang, Z. Yan, Passivity and passification of memristor-based recurrent neural networks with time-varying delays, IEEE Trans. Neural Netw. Learn. Syst. 25 (11) (2014) 2099–2109. [11] S. Gong, S. Yang, Z. Guo, T. Huang, Global exponential synchronization of inertial memristive neural networks with time-varying delay via nonlinear controller, Neural Netw. 102 (2018) 138–148. [12] Z. Guo, J. Wang, Z. Yan, A systematic method for analyzing robust stability of interval neural networks with time-delays based on stability criteria, Neural Netw. 54 (2014) 112–122. [13] S. Yang, Z. Guo, J. Wang, Robust synchronization of multiple memristive neural networks with uncertain parameters via nonlinear coupling, IEEE Trans. Syst. Man Cybern. Syst. 45 (7) (2015) 1077–1086. [14] Q. Ma, G. Feng, S. Xu, Delay-dependent stability criteria for reaction–diffusion neural networks with time-varying delays, IEEE Trans. Cybern. 43 (6) (2013) 1913–1920. [15] W. Li, T. Chen, D. Xu, K. Wang, Synchronization of delayed coupled reaction–diffusion systems on networks, Math. Methods Appl. Sci. 38 (2015) 2216–2228. [16] Y. Sheng, H. Zhang, Z. Zeng, Synchronization of reaction–diffusion neural networks with Dirichlet boundary conditions and infinite delays, IEEE Trans. Cybern. 47 (2017) 3005–3017. [17] C. Hu, H. Jiang, Z. Teng, Impulsive control and synchronization for delayed neural networks with reaction–diffusion terms, IEEE Trans. Neural Netw. 21 (1) (2010) 67–81. [18] X. Liu, K. Zhang, W. Xie, Pinning impulsive synchronization of reaction–diffusion neural networks with time–varying delays, IEEE Trans. Neural Netw. Learn. Syst. 28 (2017) 1055–1067.

Please cite this article as: S. Wang, Z. Guo and S. Wen et al., Finite/fixed-time synchronization of delayed memristive reaction-diffusion neural networks, Neurocomputing, https://doi.org/10.1016/j.neucom.2019.06.092

JID: NEUCOM 8

ARTICLE IN PRESS

[m5G;November 5, 2019;21:18]

S. Wang, Z. Guo and S. Wen et al. / Neurocomputing xxx (xxxx) xxx

[19] H. Zhang, Y. Sheng, Z. Zeng, Synchronization of coupled reaction–diffusion neural networks with directed topology via an adaptive approach, IEEE Trans. Neural Netw. Learn. Syst. 29 (2018) 1550–1561. [20] J. Wang, H. Wu, T. Huang, S. Ren, J. Wu, Pinning control for synchronization of coupled reaction-diffusion neural networks with directed topologies, IEEE Trans. Syst. Man Cybern. Syst. 46 (2016) 1109–1120. [21] D. Tong, L. Zhang, W. Zhou, J. Zhou, Y. Xu, Asymptotical synchronization for delayed stochastic neural networks with uncertainty via adaptive control, Int. J. Control Autom. Syst. 14 (3) (2016) 706–712. [22] R. Rakkiyappan, S. Lakshmanan, C.P. Lim, Asymptotical synchronization of Lur’e systems using network reliable control, J. Dyn. Syst. Meas. Control 139 (1) (2017) 1–11. [23] D. Lin, J. Wu, J. Li, J. Cai, Asymptotical synchronization control of discrete-time neural networks with time-varying delays and controller nonlinearity, in: Proceedings of the Thirty Second Youth Academic Annual Conference of Chinese Association of Automation, 2017, pp. 998–1003. [24] J. Lu, D.W.C. Ho, Globally exponential synchronization and synchronizability for general dynamical networks, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 40 (2) (2010) 350–361. [25] Z. Guo, S. Gong, S. Yang, T. Huang, Global exponential synchronization of multiple coupled inertial memristive neural networks with time-varying delay via nonlinear coupling, Neural Netw. 108 (2018) 260–271. [26] Z. Guo, S. Yang, J. Wang, Global synchronization of memristive neural networks subject to random disturbances via distributed pinning control, Neural Netw. 84 (2016) 67–79. [27] C. Zheng, J. Cao, Robust synchronization of coupled neural networks with mixed delays and uncertain parameters by intermittent pinning control, Neurocomputing 141 (4) (2014) 153–159. [28] Y. Du, R. Xu, Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays, ISA Trans. 53 (4) (2014) 1015–1023. [29] Y. Tang, H. Gao, J. Kurths, Distributed robust synchronization of dynamical networks with stochastic coupling, IEEE Trans. Circuits Syst. I: Regul. Pap. 61 (5) (2014) 1508–1519. [30] A. Polyakov, Nonlinear feedback design for fixed-time stabilization of linear control systems, IEEE Trans. Autom. Control 57 (8) (2012) 2106–2110. [31] S.P. Bhat, D.S. Bernstein, Finite-time stability of continuous autonomous systems, Siam J. Control Optim. 38 (3) (20 0 0) 751–766. [32] X. Liu, H. Su, M.Z.Q. Chen, A switching approach to designing finite-time synchronization controllers of coupled neural networks, IEEE Trans. Neural Netw. Learn. Syst. 27 (2) (2016) 471–482. [33] Y. Wan, J. Cao, G. Wen, W. Yu, Robust fixed–time synchronization of delayed Cohen–Grossberg neural networks, Neural Netw. 73 (2016) 86–94. [34] H. Hou, Q. Zhang, Finite-time synchronization for second-order nonlinear multi-agent system via pinning exponent sliding mode control, ISA Trans. 65 (2016) 96–108. [35] A. Abdurahman, H. Jiang, Z. Teng, Finite-time synchronization for fuzzy cellular neural networks with time-varying delays, Fuzzy Sets Syst. 297 (2016) 96–111. [36] H. Zhao, L. Li, H. Peng, J. Xiao, Y. Yang, M. Zheng, S. Li, Finite-time synchronization for multi-link complex networks via discontinuous control, Optik 138 (2017) 440–454. [37] X. Ding, J. Cao, A. Alsaedi, F.E. Alsaadi, T. Hayat, Robust fixed-time synchronization for uncertain complex-valued neural networks with discontinuous activation functions, Neural Netw. 90 (2017) 42–55. [38] C. Hu, J. Yu, Z. Chen, H. Jiang, T. Huang, Fixed-time stability of dynamical systems and fixed-time synchronization of coupled discontinuous neural networks, Neural Netw. 89 (2017) 74–83. [39] H. L., W. He, Q. Han, C. Peng, Fixed-time synchronization for coupled delayed neural networks with discontinuous or continuous activations, Neurocomputing 314 (2018) 143–153. [40] R. Wei, J. Cao, A. Alsaedi, Finite-time and fixed-time synchronization analysis of inertial memristive neural networks with time-varying delays, Cognit. Neurodyn. 12 (1) (2018) 121–134. [41] G. Ji, C. Hu, J. Yu, H. Jiang, Finite-time and fixed-time synchronization of discontinuous complex networks: a unified control framework design, J. Frankl. Inst. 355 (2018) 4665–4685. [42] A.M. Alimi, C. Aouiti, E.A. Assali, Finite-time and fixed-time synchronization of a class of inertial neural networks with multi-proportional delays and its application to secure communication, Neurocomputing 332 (2019) 29–43. [43] S.P. Bhat, D.S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim. 38 (3) (20 0 0) 751–766. [44] M. Forti, M. Grazzini, P. Nistri, L. Pancioni, Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations, Phys. D Nonlinear Phenom. 214 (1) (2006) 88–99. [45] J.G. Lu, Global exponential stability and periodicity of reaction–diffusion delayed recurrent neural networks with Dirichlet boundary conditions, Chaos Solitons Fractals 35 (1) (2008) 116–125. [46] Y. Tang, Terminal sliding mode control for rigid robots, Automatica 34 (1) (1998) 51–56. [47] Y. Shen, X. Xia, Semi-global finite-time observers for nonlinear systems, Automatica 44 (2008) 3152–3156. [48] Y. Sheng, Z. Zeng, Impulsive synchronization of stochastic reaction–diffusion neural networks with mixed time delays, Neural Netw. 103 (2018) 83–93. [49] C. Zhang, F. Deng, X. Dai, S. Luo, Synchronization of reaction-diffusion stochastic complex networks, Int. J. Appl. Comput. Math. 5 (11) (2019) 1–15.

[50] Z. Guo, S. Gong, T. Huang, Finite-time synchronization of inertial memristive neural networks with time delay via delay-dependent control, Neurocomputing 293 (2018) 100–107. [51] R. Wei, J. Cao, A. Alsaedi, Finite-time and fixed-time synchronization analysis of inertial memristive neural networks with time-varying delays, Cognit. Neurodyn. 12 (2018) 121–134. [52] R. Wei, J. Cao, Fixed-time synchronization of quaternion-valued memristive neural networks with time delays, Neural Netw. 113 (2019) 1–10. Shiqin Wang received the B.S. degree in Mathematics and Applied Mathematics from Huanggang Normal University, Huanggang, China, in 2016. She was admitted to the College of Mathematics and Econometrics, Hunan University, Changsha, China, in 2016. Her current research interests include reaction–diffusion systems and memristive systems.

Zhenyuan Guo received the B.S. degree in mathematics and applied mathematics and the Ph.D. degree in applied mathematics from the School of Mathematics, Hunan University, Changsha, China, in 20 04 and 20 09, respectively. He was a Joint Ph.D. Student with the Department of Applied Mathematics, University of Western Ontario, London, ON, Canada, from 2008 to 2009. He was a Post-Doctoral Research Fellow with the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Hong Kong, from 2013 to 2015. He is a Professor with the School of Mathematics, Hunan University. His current research interests include the theory of differential equations with discontinuous right hand sides, and their applications to the dynamics of neural networks, memristive systems, and control systems. Shiping Wen received the M. Eng. degree in Control Science and Engineering, from the School of Automation, Wuhan University of Technology, Wuhan, China, in 2010, and received the Ph.D degree in Control Science and Engineering, from the School of Automation, Huazhong University of Science and Technology, Wuhan, China, in 2013. He is currently a Professor at the School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China. His current research interests include memristor-based circuits and systems, neural networks, and deep learning.

Tingwen Huang is a professor at Texas A & M UniversityQatar. He received his B.S. degree in mathematics from Southwest Normal University (now Southwest University), China, 1990, his M.S. degree in mathematics from Sichuan University, China, 1993, and his Ph.D. degree in mathematics from Texas A & M University, College Station, Texas, 2002. After graduated from Texas A & M University, he worked as a Visiting Assistant Professor there. Then he joined Texas A & M University at Qatar (TAMUQ) as an Assistant Professor in August 2003, then he was promoted to Professor in 2013. His research interests include neural networks based computational intelligence, distributed control and optimization, nonlinear dynamics and applications in smart grids. He has published more than three hundred peer-review reputable journal papers, including more than one hundred papers in IEEE Transactions. Currently, he serves as an associate editor for four journals including IEEE Transactions on Neural Networks and Learning Systems, IEEE Transactions on Cybernetics, and Cognitive Computation.

Shuqing Gong received the B.S. degree in mathematics from Anyang Normal University, Anying, China, in 2013, and the Ph.D. degree in applied mathematics from the School of Mathematics, Hunan University, Changsha, China, in 2019. She is currently a Lecture at the School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, China. Her research interests include neural networks, switched systems, memristive systems and networked control systems.

Please cite this article as: S. Wang, Z. Guo and S. Wen et al., Finite/fixed-time synchronization of delayed memristive reaction-diffusion neural networks, Neurocomputing, https://doi.org/10.1016/j.neucom.2019.06.092