Adaptive synchronization of memristive neural networks with time-varying delays and reaction–diffusion term

Applied Mathematics and Computation 311 (2017) 118–128

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Adaptive synchronization of memristive neural networks with time-varying delays and reaction–diffusion termR Zhengwen Tu a, Nan Ding b,∗, Liangliang Li a, Yuming Feng a, Limin Zou a, Wei Zhang b a

Key Laboratory of Intelligent Information Processing and Control, School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404100, China b Key Laboratory of Intelligent Information Processing and Control, School of Computer Science and Engineering, Chongqing Three Gorges University, Wanzhou 404100, China

a r t i c l e

i n f o

Keywords: Memristor Neural network Synchronization Adaptive control Reaction–diffusion

a b s t r a c t This paper focuses on the synchronization control methodology for a class of delayed reaction–diffusion memristor-based neural networks. Adaptive controllers are designed such that the considered model can realize asymptotical and exponential synchronization goal under the framework of inequality techniques and Lyapunov method. The results obtained in this paper consider the effect of time delays as well as the reaction–diffusion terms, which generalize and improve some existing results. The derived synchronization criteria are presented in the form of algebraic, which can be easily verified. Finally, numerical example and its simulations are given to show the correctness of the obtained results. © 2017 Elsevier Inc. All rights reserved.

1. Introduction In 2008, a team led by R.Stanley Williams from the Hewlett-Packard Company announced the realization of a nanometersize memristor (a contraction for memory resistor) [1]. On the basis of the symmetric and logical properties observed in mathematical models of some electronic circuits, the theoretical existence of such basic device was postulated by Chua [2]. This passive electronic device has generated unprecedented worldwide interests because of its potential applications, from new high speed low-power processors [3], filters [4] to new biological models for associative memory [5] and learning models of simple organisms [6]. From the system theoretic point of view, memristor-based neural networks can be seen as a class of state-dependent nonlinear systems, which is a charming topic. With the developments and applications of memristor, plenty of scholars from various fields show great enthusiasm to this kinds of model [7–18]. The dissipativity analysis of delayed memristive neural networks was considered in [7]. Yang et al. paid attention to the synchronization problem of memristive Cohen–Grossberg neural networks [8]. And the anti-synchronization analysis of memristor-based system were considered in [9,10]. R This work was jointly supported by the National Natural Science Foundation of China under Grant No. 11601047, the Program for Innovation Team Building at Institutions of Higher Education in Chongqing No. CXTDX201601035, the Chongqing Municipal Key Laboratory of Institutions of Higher Education No. C16, the Scientific and Technological Research Program of Chongqing Municipal Education Commission under Grant Nos. KJ1401013, KJ1501002, KJ1601002, KJ1601009. ∗ Corresponding author. E-mail addresses: [email protected] (Z. Tu), [email protected] (N. Ding).

http://dx.doi.org/10.1016/j.amc.2017.05.005 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

Z. Tu et al. / Applied Mathematics and Computation 311 (2017) 118–128

119

While, the afore-mentioned conclusions ignored the effects of the reaction terms. If the dynamical behavior of a system depends only on the time, the model is an ordinary differential equation. Strictly speaking, the diffusion phenomena can not be ignored in neural networks and electric circuits once electrons transport in a nonuniform eletromagnetic field. For example, the multilayer cellular neural networks (MCNNs) are arrays of nonlinear and simple computing elements characterized by local interactions between cells, therefore, the paradigm are well suited to describe locally interconnected simple dynamical systems showing a lattice-like structure [11]. In other words, the whole structure and dynamic behavior of MCNNs are seriously dependent on the evolution time of each variables and its position (space), but also intensively dependent on its interactions deriving from the space-distributed structure of whole networks. Thus, it is essential to consider the state variables varying with the time and spaces. On the other hand, there are a large number of reaction–diffusion phenomena in nature and many discipline fields, particularly in chemistry and biology domain. The reaction–diffusion effects, therefore, should be seriously considered in both biological and artificial neural networks. As a result, it is natural to consider the states of the neuron vary in space as well as in time. The passivity and synchronization of a class of coupled dynamical networks with reaction–diffusion terms were investigated via inequality techniques in [12]. Wang et al. considered the synchronization problem of complex neural networks with reaction–diffusion effects by employing pinning control technique [13]. In the course of studying neural networks, it is found that the delays frequently arises due to the information processing and the limited transfer speed. In fact, in the electronic implementation of analog neural networks, time delay often occurs in the communication and response of neurons. And the time delay likely results in poor performance, such as oscillations, bifurcation, instability etc. However, it has been proved that some linear time-invariant fractional exponential delay systems can only be stable with certain delay and they are unstable if without any delay [14]. In view of this, it is interesting and rewarding to study the dynamics of delayed neural networks [15–22]. The uniform stability analysis for the delayed neural networks fractional-order complex-valued neural networks was given out in [16]. Based on the differential inclusion theory and Lyapunov theory, Cao and Li provided some sufficient criteria to guarantee the stability of memristive reaction–diffusion neural networks [20]. Scholars discussed the asymptotical synchronization and robust synchronization for delayed coupled partial differential systems in [21]. The problem of exponential synchronization for delayed memristor-based recurrent neural networks was investigated in [22]. Synchronization, which means that the dynamical behaviors of coupled systems achieve the same time spatial state. In the real word, synchronization of chaotic systems is very important in consideration of its potential applications in many different areas including secure communication, biology systems, optics, and information processing. In addition, by employing synchronization to communication system, one may transmit digital signals by transform between synchronization and anti-synchronization continuously, which can strengthen the security and secrecy. Therefore, the synchronization problem is an important area of study [23–35]. By virtue of sampled data control technique, Li and Wei paid attention to the synchronization of delayed memristive model [23]. Bao et al. investigated the problem of exponential synchronization and anti-synchronization of memristrive neural networks with the help of matrix measure approach [26]. Bao et al. [27] concentrated on the exponential synchronization of delayed stochastic memristor-based neural networks, and the synchronization problem for fractional-order complex-valued neural networks was also considered in [28]. The adaptive synchronization of Cohen–Grossberg neural networks with stochastic perturbation was investigated in [34], and an adaptive controller was established to ascertain the exponential synchronization goal. The fixed-time synchronization of delayed memristive model was addressed as well in [35]. Besides, the control problem of nonlinear Markovian-jump systems was investigated in [36,37]. Some adaptive laws were designed to discuss the synchronization problem of chaotic systems in [38]. And the quasi-synchronization of impulsive neural networks was considered in [39]. Motivated by the above discussion, the aim of this paper is to further develop some simple sufficient conditions for the synchronization problem of delayed reaction–diffusion memristive system. When opposed to some related works, the main contribution of the present paper can be concluded in the following lines: (i) The reaction–diffusion term is considered in this paper, which is more close to the actual model; (ii) The well known Hardy–Poincarè inequality was carried out to evaluate the upper bound of the reaction–diffusion term, which makes the derived conditions contain much more factors that do affect the dynamic behaviors of the target system; (iii) Instead of employing the differential inclusion theorem, to obtain the error system, the master-slave system are treated as two models with unmatched parameters. The organization of this paper is arrayed as follows. Some preliminaries are introduced in Section 2. In Section 3, we provide several new sufficient conditions to check the synchronization issues of memristive system. In Section 4, a numerical example is presented to show the validity of the proposed results. Conclusions are drawn in Section 5. Notation: R denotes the set of real numbers, Rn denotes the n-dimensional Euclidean space.  ⊂ Rn is an open domain including the origin with smooth boundary ∂  and radially bounded by β , and mes denotes the measure of . Let C = C ([−τ , 0] × Rm , Rn ) be the Banach space of continuous functions which maps [−τ , 0] × Rm into Rn with the topology of  uniform converge. Let  ·  is the Euclidean vector norm or the induced matrix norm, which denoted as y = ( ni=1 y2i )1/2 .  n T n 2 1 / 2 For any y(t, x ) = (y1 (t, x ), y2 (t, x ), . . . , yn (t, x )) ∈ R , y(t, x) is defined as y(t, x ) = (  i=1 yi (t, x )dx ) . Moreover, Let ai j = max{sup |a´ i j (y )|, sup |a` i j (y )|} and bi j = max{sup |b´ i j (y )|, sup |b` i j (y )|}, for i, j = 1, 2, . . . , n. y∈R

y∈R

y∈R

y∈R

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Z. Tu et al. / Applied Mathematics and Computation 311 (2017) 118–128

2. Preliminaries In this sequence, we consider a class of delayed memristor-based recurrent neural networks with reaction–diffusion terms, which can be described by the following equations: l n  ∂ yi (t, x )  ∂  ∂ yi (t, x )  = Dik − ci yi (t, x ) + ai j (yi (t, x )) f j (y j (t, x )) ∂t ∂ xk ∂ xk j=1 k=1

+

n 

bi j (yi (t, x )) f j (y j (t − τ (t ), x )) + Ii ,

t ≥ 0,

i = 1, 2, . . . , n,

(1)

j=1

where x = (x1 , x2 , . . . , xl )T ∈ , which is a bounded compact set with smooth boundary ∂  and mes > 0 in space Rl ; yi (t, x) corresponds to the state of the ith neural unit at time t and in space x; aij (yi (t, x)) and bij (yi (t, x)) are the connection strength and the time-varying delay connection weight of the jth neuron on the ith neuron, respectively; fj ( · ) is the activation function; Ii is the external input on the ith neuron; τ (t) is the time-varying delay with 0 < τ (t) ≤ τ , smooth function Dik = Dik (t, x, y ) ≥ 0 corresponds to the transmission diffusion operator along the ith neuron. According to the characteristics of current and voltage of memristor, one can get



ai j (yi (t, x )) =

a´ i j , a` i j ,

|yi (t, x )| ≤ Ti , |yi (t, x )| > Ti ,

b´ i j , b` i j ,

|yi (t, x )| ≤ Ti , |yi (t, x )| > Ti ,

 bi j (yi (t, x )) =

in which, the switching jumps Ti > 0, a´ i j , a` i j , b´ i j , b` i j , i, j = 1, 2, . . . , n, are constants. The boundary and initial conditions of the drive system (1) are given by

yi (t, x )|∂  = 0,

(t, x ) ∈ [−τ , +∞ ) × ,

i = 1, 2, . . . , n,

(2)

and

yi (s, x ) = φi (s, x ), (s, x ) ∈ [−τ , +∞ ) × , i = 1, 2, . . . , n,

(3)

where φi (s, x ), i = 1, 2, . . . , n, are bounded and continuous on [−τ , +∞ ) × . In this paper, the following hypothesis is necessary. (H) [35]: There exist unknown positive constants li , Fi , i = 1, 2, . . . , n, such that

| fi (s1 ) − fi (s2 )| ≤ li |s1 − s2 |, | fi (· )| ≤ Fi , ∀s1 , s2 ∈ R. To investigate the synchronization behavior of (1), the corresponding response system of (1) is designed as l n  ∂ zi (t, x )  ∂  ∂ zi (t, x )  = Dik − ci zi (t, x ) + ai j (zi (t, x )) f j (z j (t, x )) ∂t ∂ xk ∂ xk j=1 k=1

+

n 

bi j (zi (t, x )) f j (z j (t − τ (t ), x )) + Ii + ui (t, x ), t ≥ 0, i = 1, 2, . . . , n,

(4)

j=1

where ui (t, x) is the controller to be appropriately designed for a certain control objective. Similarly, the parameters can be read as:



ai j (zi (t, x )) =

a´ i j , a` i j ,

|zi (t, x )| ≤ Ti |zi (t, x )| > Ti ,

b´ i j , b` i j ,

|zi (t, x )| ≤ Ti |zi (t, x )| > Ti .

 bi j (zi (t, x )) =

The boundary conditions and initial conditions of (4) are given as

zi (t, x )|∂  = 0,

(t, x ) ∈ [−τ , +∞ ) × ,

i = 1, 2, . . . , n,

(5)

and

zi (s, x ) = ϕi (s, x ), (s, x ) ∈ [−τ , +∞ ) × , i = 1, 2, . . . , n, where ϕi (s, x ), i = 1, 2, . . . , n, are bounded and continuous on [−τ , +∞ ) × . Definition 2.1. [40] The reaction–diffusion system (1) and (4) are said to be asymptotically synchronized if

lim zi (t, x ) − yi (t, x ) = 0,

t→∞

t ≥ 0,

i = 1, 2, . . . , n.

(6)

Z. Tu et al. / Applied Mathematics and Computation 311 (2017) 118–128

121

Definition 2.2. [41] Drive-response systems (1) and (4) are said to be globally exponentially synchronized under an adaptive controller, if there exist constants ρ ≥ 1 and ε > 0 such that

    z(t, x ) − y(t, x ) ≤ ρ φ − ϕ e−εt , f or al l (t, x ) ∈ [0, +∞ ) × ,

and constant ε is said to be the degree of exponential synchronization. Lemma 2.1. [43] Let scalar s > 0, x, y ∈ Rn , and Q ∈ Rn×n , then

2xT Qy ≤ sxT Q Q T x + s−1 yT y. Lemma 2.2. ([44] Hardy–Poincarè inequality) Let G ∈ Rm , (m > 2 ) be a bounded open set containing the origin, then



G

|∇υ|2 dx ≥

2

( m − 2 )2

RG

4

υ22 + 2



G

υ2 dx, |x|2

∀υ ∈ H01 (G ),



where υ (x ) is a real-valued function belonging to H01 (G ) = υ ∈ L2 (G ), Di υ = ∂∂υ ∈ L2 (G ), υ|∂ G = 0, 1 ≤ i ≤ m , υ22 = xi  2 G υ (x )dx, 2 = 5.783... is the first eigenvalue of the following Dirichlet Laplacian



− υ = λυ , υ = 0,

υ ∈ G∗, υ ∈ ∂ G∗,

where G ∗ is the unite disk in R2 , and RG is the radius of the ball G ∗∗ ⊆ Rm centered at the origin having the same measure as G. Before moving on, set ei (t, x ) = zi (t, x ) − yi (t, x ), i = 1, 2, . . . , n be the synchronization error, then, one can read that

 l n   ∂ ei (t, x )  ∂  ∂ ei (t, x )  = Dik − ci ei (t, x ) + ai j (zi (t, x )) f j (z j (t, x )) − ai j (yi (t, x )) f j (y j (t, x )) ∂t ∂ xk ∂ xk j=1 k=1  n   + bi j (zi (t, x )) f j (z j (t − τ (t ), x )) − bi j (yi (t, x )) f j (y j (t − τ (t ), x )) + ui (t, x ) j=1

=

 n  ∂  ∂ ei (t, x )  Dik − ci ei (t, x ) + ai j (zi (t, x ))g j (e j (t, x )) + ai j (zi (t, x )) ∂ xk ∂ xk j=1 k=1 

  n −ai j (yi (t, x )) f j (y j (t, x )) + bi j (zi (t, x ))g j (e j (t − τ (t ), x )) + bi j (zi (t, x )) l 

j=1



−bi j (yi (t, x )) f j (y j (t − τ (t ), x )) + ui (t, x ),

t ≥ 0, i = 1, 2, . . . , n,

(7)

where g j (e j (·, x )) = f j (z j (·, x )) − f j (y j (·, x )). Remark 2.1. From the above assumption (H), it is obviously that

|g j ( s1 )| ≤ L j |s1 |, |g j ( · )| ≤ G j , ∀s1 ∈ R. 3. Main results In this section, some synchronization conditions are established on the strength of adaptive control scheme, which can be appropriately designed to reduce the control gains. By using some analysis techniques and designing new Lyapunov– Krasovskii functionals, synchronization criteria are formulated in the form of algebraic inequalities. Theorem 3.1. Suppose that the assumptions (H) holds, then the drive-response delayed neural networks (1) and (4) are asymptotically synchronized under the following adaptive controller



ui (t, x ) = −αi ei (t, x ) − βi sign(ei (t, x )), α˙ i = εi e2i (t, x ), β˙ i = ηi |ei (t, x )|, i = 1, 2, . . . , n,

(8)

where ε i and ηi are arbitrary positive constants. Proof. In order to obtain the asymptotically synchronized goal, the following Lyapunov functional is adopted:

V (t ) =



n 1 1 e2i (t, x ) + (αi − μi )2 + (Mi − βi )2 dx,  i=1

εi

ηi

(9)

122

Z. Tu et al. / Applied Mathematics and Computation 311 (2017) 118–128

where the constants μi and Mi are subjected as

1 2

μi ≥ − ϑ − c i + n 

Mi ≥

n n 1 1 a¯ i j + a¯ ji li2 , 2 2 j=1

b¯ i j G j +

j=1

n  

j=1



sup |a´ i j (v ) − a` i j (v )| + sup |bi j (v ) − bi j (v )| Fj , v∈R

j=1

v∈R

D (m − 2 ) 2D2 + , D = min{Dik : i = 1, 2, . . . , n; k = 1, 2, . . . , m} > 0. 2β 2 R2 2

ϑ=

(10)

Consider the time derivative of V(t) along the trajectory of (7), then one has



n ∂ ei (t, x ) 2ei (t, x ) + 2(αi − μi )e2i (t, x ) − 2(Mi − βi )|ei (t, x )| dx ∂t 

V˙ (t ) ≤

i=1

 n l  ≤ 2ei (t, x )

n  ∂  ∂ ei (t, x )  Dik − 2ci e2i (t, x ) + 2 a¯ i j |ei (t, x )||g j (e j (t, x ))| ∂ xk ∂ xk j=1 k=1

 i=1

+2

n 

b¯ i j |ei (t, x )||g j (e j (t − τ (t ), x ))| + 2

j=1

+2

n 

n 

|ei (t, x )||ai j (zi (t, x )) − ai j (yi (t, x ))|| f j (y j (t, x ))|

j=1

|ei (t, x )||bi j (zi (t, x )) − bi j (yi (t, x ))|| f j (y j (t − τ (t ), x ))| − 2αi e2i (t, x ) − 2βi |ei (t, x )|

j=1

+2(αi − μi )e2i (t, x ) − 2(Mi − βi )|ei (t, x )| dx.

(11)

Based on the hypothesis (H), inequality (11) undergoes the following estimation:

V˙ (t ) ≤

 n l  2ei (t, x )

n  ∂  ∂ ei (t, x )  Dik − 2ci e2i (t, x ) + 2 a¯ i j |ei (t, x )|l j |e j (t, x )| ∂ xk ∂ xk j=1 k=1

 i=1

+2

n  j=1

+2

n 

b¯ i j |ei (t, x )|G j + 2

n 

|ei (t, x )| sup |a´ i j (v ) − a` i j (v )|Fj

j=1

v∈R

|ei (t, x )| sup |b´ i j (v ) − b` i j (v )|Fj − 2μi e2i (t, x ) − 2Mi |ei (t, x )| dx. v∈R

j=1

(12)

Then, according to Lemma 2.1, the following inequality can be obtained

2

n  n 

a¯ i j |ei (t, x )|l j |e j (t, x )| ≤

i=1 j=1

 n  n   a¯ i j e2i (t, x ) + a¯ i j l 2j e2j (t, x ) i=1 j=1

=

n  n  



a¯ i j + a¯ ji li2 e2i (t, x ).

i=1 j=1

Recalling the boundary conditions (2) and (5), one has

∂  ∂ ei (t, x )  D dx ∂ xk ik ∂ xk  k=1  ∂ e (t, x ) l i =2 ei (t, x )∇ · Dik dx ∂ xk  k=1   ∂ e (t, x ) l ∂ ei (t, x ) l =2 ∇ · ei (t, x )Dik i dx − 2 Dik · ∇ ei (t, x )dx ∂ xk ∂ xk   k=1 k=1   ∂ e (t, x ) 2 l  ∂ ei (t, x ) l i =2 ei (t, x )Dik dx − 2 Dik dx ∂ xk ∂ xk ∂  k=1 k=1

2

=−2

ei (t, x )

l 

 k=1

l 

Dik

 ∂ e (t, x ) 2 i

∂ xk

dx,

(13)

Z. Tu et al. / Applied Mathematics and Computation 311 (2017) 118–128

where “·” is inner product, ∇ =

 Dik



∂ ∂ ∂ x1 , . . . , ∂ xm

123

 denotes the gradient operator, and

 ∂ e (t, x ) ∂ ei (t, x ) l ∂ ei (t, x ) T i = Di1 , . . . , Dil . ∂ xk ∂ x1 ∂ xl k=1

A straightforward manipulation from Lemma 2.2 gives a more precisely estimation, which can be found as

2



=−2

ei (t, x ) l 

 k=1

∂  ∂ ei (t, x )  D dx ∂ xk ik ∂ xk k=1

l 

Dik

 ∂ e (t, x ) 2 i

∂ xk

dx



(14)



D (m − 2 ) 2D2 e2i (t, x )dx − e2i (t, x )dx 2β 2 R2   −ϑ e2i (t, x )dx. 2

≤−



2D −2 )2 with ϑ = D(m + 2 2 , D = min{Dik : i = 1, 2, . . . , n; k = 1, 2, . . . , m} > 0. 2β 2 R Combining (13)with (14), the following inequality can be deduced from (12)

V˙ (t ) ≤

 n   i=1

+2

 n

− ϑ − 2 ci +

− Mi

a¯ i j +

n 

j=1

b¯ i j G j +

j=1



n 

n  j=1



a¯ ji li2 − 2μi e2i (t, x )

j=1

sup |a´ i j (v ) − a` i j (v )|Fj + v∈R

n  j=1

sup |b´ i j (v ) − b` i j (v )|Fj v∈R

|ei (t, x )| dx.

(15)

Recalling the parameters given in (10), one can conclude that

V˙ (t ) ≤ 0.

(16)

Hence, by Lyapunov theorem in functional differential equations [45], the origin of the error system is asymptotically stable, which implies that the two delayed memristive neural networks (1) and (4) are asymptotically synchronized under the designed controller (8). The proof is thus completed.  Remark 3.1. Adaptive control strategy, as shown in (8), is one of the most important control methods. Compared with the traditional linear feedback controller, where the feedback strength is fixed, and thus must be maximal, and this brings waste of resource in practice. While, according to the above analysis, the variable feedback strength in the adaptive controller will be automatically adapted to a suitable strength depending on the initial values, which is more reasonable. Generally speaking, the exponentially synchronized system is with better dynamic characteristics than those of the asymptotically synchronized one. Thus, we shall make some efforts to discuss the exponential synchronization of the driveresponse system. Theorem 3.2. Under the assumptions (H), the drive-response delayed neural networks (1) and (4) are in global exponential synchronization under the following adaptive controller



ui (t, x ) = −δi ei (t, x ) − ζi sign(ei (t, x )), δ˙ i = γi e2εt e2 (t, x ), i = 1, 2, . . . , n,

(17)

i

where γ i is an arbitrary positive constant, and

ζi ≥

n 

b¯ i j G j +

j=1

n   j=1



sup |a´ i j (v ) − a` i j (v )| + sup |b´ i j (v ) − b` i j (v )| Fj . v∈R

v∈R

(18)

Proof. Consider the following Lyapunov function:

V (t ) =



n 1 e2εt e2i (t, x ) + (δi − ξi )2 dx,  i=1

(19)

γi

the proof matches mutatis mutandis a similar process in Theorem 1, thus, the following inequality can be established

V˙ (t ) ≤e2εt

 n

n  2ε e2i (t, x ) − 2ci e2i (t, x ) + 2 a¯ i j |ei (t, x )|l j |e j (t, x )|  i=1

j=1

124

Z. Tu et al. / Applied Mathematics and Computation 311 (2017) 118–128

+2

n  j=1

+2

n 

b¯ i j |ei (t, x )|G j + 2

+

v∈R

 |ei (t, x )| sup |bi j (v ) − bi j (v )|Fj − 2ζi |ei (t, x )| − 2ξi e2i (t, x ) dx v∈R

 n   i=1

n 

|ei (t, x )| sup |a´ i j (v ) − a` i j (v )|Fj

j=1

j=1

≤e2εt

n 

2 ε − 2 ci +

n 

a¯ i j +

j=1

v∈R



a¯ ji li2 − 2ξi e2i (t, x ) + 2

 n

j=1

sup |a´ i j (v ) − a` i j (v )|Fj +

j=1

n 

n  j=1

sup |b´ i j (v ) − b` i j (v )|Fj − ζi v∈R



b¯ i j G j

j=1

 |ei (t, x )| dx.

(20)

According to the retraction expressed in (18), and setting

ξi ≥ ε − c i +

n n 1 1 a¯ i j + a¯ ji li2 , 2 2 j=1

j=1

then, one can reach that

V˙ (t ) ≤ 0.

(21)

According to the discussion given above, one can get V(t) ≤ V(0), t ≥ 0. Now, let

θ (t ) =

n  1 i=1

γi

(δi − ξi )2 .

Similar to the estimation approach adopted in the [42], one has

V (0 ) =

 n   i=1





e2i (0, x ) + θ (0 ) dx ≤ 1 + θ (0 )

  φ − ϕ 2 , 2

and

V (t ) ≥ then

 n  i=1



2





2

e2εt z(t, x ) − y(t, x ) ≤ 1 + θ (0 ) 2



Let ρ = 1 + θ (0 )

2

2

e2εt e2i (t, x )dx = e2εt z(t, x ) − y(t, x ) ,

  φ − ϕ 2 . 2

, and ρ > 1, then

    z(t, x ) − y(t, x ) ≤ ρ φ − ϕ  e−εt , 2 2

t ≥ 0.

Following from Definition 2.2, one can read that the drive-response systems (1) and (4) with reaction–diffusion terms are globally exponentially synchronized. Thus completed the proof.  Remark 3.2. In comparison with previous literature [8–10,22], the diffusion effect has been taken into account in our models. And one knows that there are very few works concentrate on this topic, thus this is a new attempt. Moreover, by employing the well known Hardy–Poincarè inequality, a more accurate estimation of the reaction–diffusion term is presented, i.e., in some existing works, the following inequality is adopted to deal with the reaction–diffusion term



ψ 2 (x )dx ≤ q2k



   ∂ψ 2  ∂ x  dx, k

where |xk | ≤ qk , (k = 1, 2, . . . , m ). From which, we can easily read that the regional feature are neglected, as a result, this may increase the conservatism of the conclusions. As a correction of the related works, in this paper we adopted Hardy– Poincarè inequality, which makes the derived conclusions are associated with the reaction–diffusion coefficients as well as the regional feature. Remark 3.3. This paper pay attention to the dynamic behavior of a class of memristive neural networks described by the partial differential equation (PDE), while in real life, lots of models can be described by PDE, such as crane system [46], thus the results derived in this paper can also be employed to this kinds of models.

Z. Tu et al. / Applied Mathematics and Computation 311 (2017) 118–128

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4. A numerical example In this section, one example is offered to illustrate the effectiveness of the conclusions obtained above.

Example 1. For the sake of simplification, we consider a reaction–diffusion memristive neural networks described by the following differential equations: 3  ∂ y2 (t, x ) ∂ yi (t, x ) = Di i 2 − yi (t, x ) + ai j (yi (t, x )) f j (y j (t, x )) ∂t ∂x j=1

+

3 

bi j (yi (t, x )) f j (y j (t − τ (t ), x )) + Ii ,

(22)

t ≥ 0, i = 1, 2, 3,

j=1

where



−0.3, |y1 (t, x )| ≤ 1, 0.3, |y1 (t, x )| > 1,

a11 (y1 (t, x )) =



|y1 (t, x )| ≤ 1, |y1 (t, x )| > 1,

0.4, 0.5,

a13 (y1 (t, x )) =



0.3, 0.2,

a22 (y2 (t, x )) =

 a31 (y3 (t, x )) =

b11 (y1 (t, x )) =

−0.6, −0.5,

|y1 (t, x )| ≤ 1, |y1 (t, x )| > 1,

b13 (y1 (t, x )) =

b32 (y2 (t, x )) =

|y3 (t, x )| ≤ 1, |y3 (t, x )| > 1

0.8, 0.9,

|y1 (t, x )| ≤ 1, |y1 (t, x )| > 1,

−0.1, 0.1,

|y1 (t, x )| ≤ 1, |y1 (t, x )| > 1,

0.7, 0.71,

|y2 (t, x )| ≤ 1, |y2 (t, x )| > 1.

−1, −0.9,

|y3 (t, x )| ≤ 1, |y3 (t, x )| > 1,

0.6, 0.5,

|y3 (t, x )| ≤ 1, |y3 (t, x )| > 1.

 b22 (y2 (t, x )) =

−0.5, −0.51,

|y2 (t, x )| ≤ 1, |y2 (t, x )| > 1.

b31 (y3 (t, x )) =

0.2, −0.2,

|y3 (t, x )| ≤ 1, |y3 (t, x )| > 1.

b33 (y3 (t, x )) =



−0.5, −0.7,



|y2 (t, x )| ≤ 1, |y2 (t, x )| > 1,

b23 (y2 (t, x )) =

|y2 (t, x )| ≤ 1, |y2 (t, x )| > 1.



0.4, 0.43,



0.7, 0.6,



|y3 (t, x )| ≤ 1, |y3 (t, x )| > 1



|y2 (t, x )| ≤ 1, |y2 (t, x )| > 1,



−0.3, −0.6,

|y1 (t, x )| ≤ 1, |y1 (t, x )| > 1,

−0.4, −0.6,

a23 (y2 (t, x )) = a32 (y3 (t, x )) =



b21 (y2 (t, x )) =



|y3 (t, x )| ≤ 1, |y3 (t, x )| > 1,

a33 (y3 (t, x )) =

0.5, 0.6,

a21 (y2 (t, x )) =

0.8, 0.7,



b12 (y1 (t, x )) =

|y2 (t, x )| ≤ 1, |y2 (t, x )| > 1,

 a12 (y1 (t, x )) =





For the drive system (22), constructing the following response system: 3  ∂ z2 (t, x ) ∂ zi (t, x ) = Di i 2 − zi (t − 1, x ) + ai j (zi (t, x )) f j (z j (t, x )) ∂t ∂x j=1

+

3 

bi j (zi (t, x )) f j (z j (t − τ (t ), x )) + Ii + ui (t, x ), t ≥ 0, i = 1, 2, 3.

(23)

j=1

|s−1| .6e The activation functions are taken as f (s ) = |s+1|+ , the time delay is given by τ (t ) = 01+ . It can be verified that Fj = 2 et G j = 1. Set D1 = 0.4, D2 = 0.2, D3 = 0.3. For numerical simulations, the initial conditions of above systems are chosen as y(t, x ) = (0.5eπ x , eπ x , 1.5eπ x )T , and z(t, x ) = (−0.3tanh(π x ), −0.6tanh(π x ), −0.9tanh(π x ))T for (t, x ) ∈ [−0.6, 0] × . Scheme 1. According to Theorem 3.1, the simulation results are presented in Figs. 1 and 2. Fig. 1 describes the time responses of synchronization errors ei (t, x ) = zi (t, x ) − yi (t, x ), i = 1, 2, 3, which trend to be zero quickly with respect to time and space. Fig. 1 depicts the trajectories of control parameters α i and β i (i = 1, 2, 3) of the adaptive controller. From Fig. 2, one can read that the control parameters turn out to be constants eventually. t

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Z. Tu et al. / Applied Mathematics and Computation 311 (2017) 118–128

a

b

c 1000

5000

1000 0

0 e (t,x)

1

3

−1000

2

e (t,x)

e (t,x)

0

−1000 −2000

−5000 −2000

−3000

−3000 10

−10000 10

−4000 10 0.5

0.5 5

5

0 0

t

−0.5

t

x

0.5 5

0 0

−0.5

0 t

x

0

−0.5

x

Fig. 1. Synchronization errors e1 (t, x), e2 (t, x), e3 (t, x) between systems(22) and (23) under the adaptive controller (8).

a

b

Fig. 2. Trajectories of control parameters α i , β i , i = 1, 2, 3 in the adaptive controller (8).

a

b

c

Fig. 3. The state trajectories of synchronization errors e1 (t, x), e2 (t, x), and e3 (t, x) between systems (22) and (23) under the controller (17).

Scheme 2. From Theorem 3.2, it can be verified that

ζ1 ≥

3 

b¯ 1 j G j +

j=1

ζ2 ≥

n 

j=1

b¯ 2 j G j +

j=1

ζ3 ≥

n  j=1

3  

3   j=1

b¯ 3 j G j +

3   j=1



sup |a´ 1 j (v ) − a` 1 j (v )| + sup |b´ 1 j (v ) − b` 1 j (v )| Fj = 2.8, v∈R

v∈R



sup |a´ 2 j (v ) − a` 2 j (v )| + sup |b´ 2 j (v ) − b` 2 j (v )| Fj = 2.1, v∈R

v∈R



sup |a´ 3 j (v ) − a` 3 j (v )| + sup |b´ 3 j (v ) − b` 3 j (v )| Fj = 3. v∈R

v∈R

Z. Tu et al. / Applied Mathematics and Computation 311 (2017) 118–128

127

20 18 16

δ

1

14

δ1,δ2,δ3

12 10 δ3

8 6 4

δ2

2 0

0

2

4

6

8

10

t Fig. 4. Trajectories of control parameters δ 1 , δ 2 and δ 3 in the controller (17).

The corresponding simulation results are given in Figs. 3 and 4. The synchronization errors ei (t, x) (i = 1, 2, 3) are given in Fig. 3, which turn to zero quickly. From Fig. 4, one can see that the control parameters δ i , i = 1, 2, 3, turn out to be constants eventually. The simulation results show that, by introducing the proposed controller, the synchronization goal between the driveresponse memristive networks can be almost surely achieved. 5. Conclusion This paper introduced the synchronization control of delayed memristive neural networks with reaction–diffusion terms. Two different adaptive control strategies were proposed to study the asymptotical and exponential synchronization issues of delayed reaction–diffusion memristive system. In particular, by means of Lyapunov functions, analytical techniques, as well as an appropriate controller, some easy-verified conditions were established to reach the synchronization goal of the target model. It is worth noting that the model derived in this paper are associate with the state variables as well as the effect of the reaction–diffusion terms, which make the results obtained in this paper are more general. Finally, one example and its simulations are given to show the effectiveness of the obtained results. In addition, we will attempt to discuss the synchronization of Markovian-jump systems and stochastic memristor-based neural networks by the approach adopted here. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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