Adaptive synchronization of memristor-based BAM ...

Applied Mathematics and Computation 322 (2018) 100–110

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

Adaptive synchronization of memristor-based BAM neural networks with mixed delays Chuan Chen a, Lixiang Li a,∗, Haipeng Peng a, Yixian Yang a,b a

Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China b State Key Laboratory of Public Big Data, Guizhou 550025, China

a r t i c l e

i n f o

Keywords: Memristor BAM neural networks Mixed delays Synchronization Adaptive feedback controllers

a b s t r a c t This paper investigates the adaptive synchronization of memristor-based BAM neural networks (MBAMNNs) with discrete delay and distributed delay (mixed delays). We design two kinds of adaptive feedback controllers, under which the considered MBAMNNs can achieve asymptotic synchronization and exponential synchronization respectively. The adaptive feedback controllers can be utilized even when there is no perfect knowledge of the system parameters. Furthermore, computing algebraic conditions and solving linear matrix inequalities are not needed to determine suitable control gains. Numerical simulations illustrate the effectiveness of the theoretical results. © 2017 Elsevier Inc. All rights reserved.

1. Introduction In 1971, memristor was first proposed by Chua [1] as the fourth fundamental circuit element besides resistor, inductor and capacitor. However, not until a prototype of memristor was manufactured by the scientists of HP Lab [2] in 2008 did memristor attract much attention of researchers. Memristor, which is the abbreviation of memory resistor, reflects the relationship between flux and charge. The memristance of memristor varies with the quantity of charge that has passed through the memristor [3]. Since memristor can remember the quantity of the passed charge, it is believed that memristor has the function of memory. In the circuit implementation of neural network, the synapses between neurons are usually simulated by resistors. It is well-known that the synapses play a key role in the memory formation, but the conventional resistors don’t have the function of memory. If the resistors used in the circuit implementation of neural network are replaced by memristors, the usual artificial neural network becomes a memristor-based neural network [4], which is the good candidate for simulating the human brain [5]. In recent years, the synchronization control [6–11] of neural networks has attracted considerable attention due to its great application prospect in the different fields such as associative memory [12], secure communications [13] and image encryption [14]. Since 2008, the research on the synchronization control of neural networks has been extended to memristor-based neural networks. Up to now, a lot of results about the synchronization control of memristor-based neural networks have been obtained, see [15–24] and references therein. On the other hand, traffic jams and finite transmission speed make discrete delays inevitably exist in neural networks [25,26]. Furthermore, since there exist parallel pathways

∗

Corresponding author. E-mail address: [email protected] (L. Li).

https://doi.org/10.1016/j.amc.2017.11.037 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

C. Chen et al. / Applied Mathematics and Computation 322 (2018) 100–110

101

with different axon lengths and sizes in neural networks, distributed delays [27] should also be considered in the studies of neural networks. Bidirectional associative memory (BAM) neural network, which was introduced by Kosko in 1988 [28], is deemed as one of the most important neural networks. Unlike other neural networks, BAM neural network is consisted of neurons distributed in two layers. The neurons distributed in one layer are fully interconnected with the neurons distributed in the other layer, while there is no interconnection between the neurons distributed in the same layer. So far, there have been some results about the synchronization control of BAM neural networks [29–31], and the synchronization control of MBAMNNs has also been investigated in [32–35]. In [32], impulsive control was used to synchronize MBAMNNs with discrete delay and random nonlinearities. In [33], feedback control was utilized to achieve the asymptotical antisynchronization of MBAMNNs with discrete delay and different memductance functions. In [35], non-fragile feedback control was applied to realize the non-fragile synchronization of MBAMNNs with discrete delay. As we know, the control gains of feedback controllers are usually much larger than those needed in practical applications because of the conservativeness of theoretical analysis, while adaptive feedback controllers can avoid the high control gains effectively. What is more, adaptive control [36,37] can be used even when there is no perfect knowledge of the system parameters. However, there exist some difficulties in studying the adaptive synchronization of MBAMNNs, such as the complex connection structure of BAM neural networks, the parameters mismatch problem and the existence of time delays. To the best of our knowledge, adaptive control has not been used to study the synchronization control of MBAMNNs until now. Motivated by the above analysis, in this paper we attempt to investigate the synchronization control of MBAMNNs with mixed delays via adaptive feedback control. The contributions of this paper can be summarized as follows: (1) MBAMNN is modeled with both time-varying discrete delay and time-varying distributed delay, what is more, the rates of neuron self-inhibition are also state-dependent. (2) In studying the synchronization control of MBAMNNs, the complex connection structure of BAM neural networks and the parameters mismatch problem are the main difficulties, which can be solved effectively by means of our Lemmas 3 and 4. (3) We design two kinds of adaptive feedback controllers, under which the considered MBAMNNs can achieve asymptotic synchronization and exponential synchronization respectively. The remainder of this paper is organized as follows. In Section 2, some necessary preliminaries are presented. The main theoretical results are derived in Section 3. In Section 4, numerical simulations are given to verify the effectiveness of the theoretical results. Section 5 draws the conclusion. 2. Preliminaries In this paper, we consider the following MBAMNN with mixed delays:

 ⎧ x˙ i (t ) = −σi (xi (t ))xi (t ) + mj=1 a ji (xi (t )) f j (y j (t )) ⎪ t  ⎨ m + j=1 b ji (xi (t )) f j (y j (t − τ (t ))) + mj=1 p ji (xi (t )) t −ρ (t ) f j (y j (s ))ds, n t ) = −ξ j (y j (t ))y j (t ) + i=1 ci j (y j (t ))gi (xi (t )) ⎪y˙ j ( ⎩ t  + ni=1 di j (y j (t ))gi (xi (t − τ (t ))) + ni=1 qi j (y j (t )) t −ρ (t ) gi (xi (s ))ds,

(1)

 , respectively; σ ( · ) > 0 and i = 1, 2, . . . , n, j = 1, 2, . . . , m, where xi (t) and yj (t) represent the voltages of capacitors Ci and C j i ξ j ( · ) > 0 denote the rates of neuron self-inhibition; fj ( · ) and gi ( · ) are the activation functions; τ (t) and ρ (t) are discrete delay and distributed delay respectively, which satisfy 0 ≤ τ (t) ≤ τ 1 and 0 ≤ ρ (t) ≤ τ 2 ; σ i (xi (t)), aji (xi (t)), bji (xi (t)), pji (xi (t)), ξ j (yj (t)), cij (yj (t)), dij (yj (t)) and qij (yj (t)) are memristive connection weights, which are given by





m 1

1 σi (xi (t )) = (M ji + M∗ji + M∗∗ , ji ) × sign ji + Ci Ri j=1

M ji a ji (xi (t )) = × sign ji , Ci M∗∗ ji p ji (xi (t )) = × sign ji , Ci



b ji (xi (t )) =



sign ji =

M∗ji Ci

× sign ji ,

1, j = i, −1, j = i,



n 1  ∗ + M ∗∗ ) × signi j + 1 , ξ j (y j (t )) =  ( Mi j + M ij ij j C j i=1 R

ci j (y j (t )) =

i j M × signi j , j C

qi j (y j (t )) =

∗∗ M ij × signi j , j C

di j (y j (t )) =

∗ M ij × signi j , j C (2)

 ,M ∗ , M ∗∗ denote the memductances of memristors R , R∗ , R∗∗ , R  ,R ∗ , R ∗∗ , respectively. Moreover, in which M ji , M∗ji , M∗∗ ,M ij ji ij ji ij ij ji ji ij ij ∗ Rji represents the memristor between fj (yj (t)) and xi (t), R ji represents the memristor between f j (y j (t − τ (t ))) and xi (t), R∗∗ ji

102

C. Chen et al. / Applied Mathematics and Computation 322 (2018) 100–110

t

 represents the memristor between g (x (t)) and y (t), R ∗ f j (y j (s ))ds and xi (t), R ij i i j ij t ∗∗  represents the memristor between gi (xi (t − τ (t ))) and yj (t), Ri j represents the memristor between t −ρ (t ) gi (xi (s ))ds and  stand for the parallel-resistors. The interested readers can refer to the relevant papers [38,39], which gave y (t), R and R represents the memristor between

j

t −ρ (t )

j

i

the detailed explanations about how to construct memristor-based neural networks. Based on the properties of memristor, we set



  a ji , |ν| ≤ Ti , b ji , |ν| ≤ Ti , σi , |ν| ≤ Ti , σi ( ν ) = a (ν ) = b (ν ) = σi , |ν| > Ti , ji aji , |ν| > Ti , ji bji , |ν| > Ti ,   p ji , |ν| ≤ Ti , Tj , ci j , |ν| ≤  Tj , ξ , |ν| ≤  p ji (ν ) = ξ j (ν ) = j c ( ν ) = i j  p ji , |ν| > Ti , ξ j , |ν| >  Tj , cij , |ν| >  Tj ,  di j , |ν| ≤  Tj , qi j , |ν| ≤  Tj , di j ( ν ) = qi j ( ν ) = dij , |ν| >  Tj , qi j , |ν| >  Tj ,

(3)

for i = 1, 2, . . . , n, j = 1, 2, . . . , m, where σi , σi , aji , aji , bji , bji , pji , pji , ξ j , ξ j , ci j , cij , di j , dij , qi j and qi j are known constants. Let x(t ) = (x1 (t ), x2 (t ), . . . , xn (t ))T , y(t ) = (y1 (t ), y2 (t ), . . . , ym (t ))T . The initial conditions of MBAMNN (1) are x(s ) = ϕ1 (s ) ∈ C ([−τ , 0], Rn ) and y(s ) = ϕ2 (s ) ∈ C ([−τ , 0], Rm ), where τ = max{τ1 , τ2 }. Throughout this paper, set σ i = min{σi , σi }, a+ji = max{|aji |, |aji |}, b+ji = max{|bji |, |bji |}, p+ji = max{| pji |, | pji |},

ξ j = min{ξ j , ξ j }, ci+j = max{|ci j |, |cij |}, di+j = max{|di j |, |dij |}, q+i j = max{|qi j |, |qi j |}, for i = 1, 2, . . . , n, j = 1, 2, . . . , m.

In this paper, MBAMNN (1) is referred to as the drive system, the corresponding response system is described by

⎧ 

x˙ i (t ) = −σi ( xi (t )) xi (t ) + mj=1 a ji ( xi (t )) f j ( y j (t )) ⎪ ⎪ t  ⎨ m + j=1 b ji ( xi (t )) f j ( y j (t − τ (t ))) + mj=1 p ji ( xi (t )) t −ρ (t ) f j ( y j (s ))ds + ui (t ), n ˙ j (t ) = −ξ j (



y y ( t )) y ( t ) + c ( y ( t )) g ( x ( t )) ⎪ j j j i i i=1 i j ⎪ t  ⎩ n + i=1 di j ( y j (t ))gi ( xi (t − τ (t ))) + ni=1 qi j ( y j (t )) t −ρ (t ) gi ( xi (s ))ds + v j (t ),

(4)

i = 1, 2, . . . , n, j = 1, 2, . . . , m, where ui (t) and v j (t ) are the appropriate controllers. Let x(t ) = ( x1 (t ), x2 (t ), . . . , xn (t ))T ,

y(t ) = ( y1 (t ), y2 (t ), . . . , ym (t ))T . The initial conditions of MBAMNN (4) are x(s ) = φ1 (s ) ∈ C ([−τ , 0], Rn ) and y ( s ) = φ2 ( s ) ∈ C ([−τ , 0], Rm ). y We define the synchronization errors as exi (t ) = xi (t ) − xi (t ), i = 1, 2, . . . , n, e j (t ) = y j (t ) − y j (t ), j = 1, 2, . . . , m. From 1 and 4, we have



e˙ xi (t ) = −σi ( xi (t )) xi (t ) + σi (xi (t ))xi (t ) + Fi (t ) + ui (t ), e˙ yj (t ) = −ξ j ( y j (t )) y j (t ) + ξ j (y j (t ))y j (t ) + G j (t ) + v j (t ),

(5)

i = 1, 2, . . . , n, j = 1, 2, . . . , m, where

Fi (t ) =

m

a ji ( xi (t )) f j ( y j (t )) −

j=1

+

m

a ji (xi (t )) f j (y j (t ))

j=1

m

b ji ( xi (t )) f j ( y j (t − τ (t ))) −

m

j=1

+

m

p ji ( xi (t ))



j=1

G j (t ) =

n

t

t −ρ (t )

f j ( y j (s ))ds −

n

n

i=1

p ji (xi (t ))



n

t

t −ρ (t )

f j (y j (s ))ds,

(6)

ci j (y j (t ))gi (xi (t ))

i=1

di j ( y j (t ))gi ( xi (t − τ (t ))) −

i=1

+

m

j=1

ci j ( y j (t ))gi ( xi (t )) −

i=1

+

b ji (xi (t )) f j (y j (t − τ (t )))

j=1

qi j ( y j (t ))



n

di j (y j (t ))gi (xi (t − τ (t )))

i=1 t

t −ρ (t )

gi ( xi (s ))ds −

n

qi j (y j (t ))

i=1

y

y



t

t −ρ (t )

y

gi (xi (s ))ds.

Let ex (t ) = (ex1 (t ), ex2 (t ), . . . , exn (t ))T , ey (t ) = (e1 (t ), e2 (t ), . . . , em (t ))T . The initial conditions ex (s ) = ψ1 (s ) = φ1 (s ) − ϕ1 (s ) ∈ C ([−τ , 0], Rn ) and ey (s ) = ψ2 (s ) = φ2 (s ) − ϕ2 (s ) ∈ C ([−τ , 0], Rm ). To derive the main results, we need two assumptions.





A1. There exist constants Wj > 0 such that  f j (· ) ≤ W j , j = 1, 2, . . . , m. A2. There exist constants Ni > 0 such that |gi (· )| ≤ Ni , i = 1, 2, . . . , n.

(7) of

system

(5)

are

C. Chen et al. / Applied Mathematics and Computation 322 (2018) 100–110

Lemma 1.







103



sign(exi (t ))(−σi ( xi (t )) xi (t ) + σi (xi (t ))xi (t )) ≤ −σ i exi (t ) + Ti σi − σi , for i = 1, 2, . . . , n. Proof. Four cases are considered, respectively. (1) When |xi (t)| < Ti and | xi (t )| < Ti ,

sign(exi (t ))(−σi ( xi (t )) xi (t ) + σi (xi (t ))xi (t ))









= −sign(exi (t ))(σi xi (t ) − σi xi (t )) = −σi exi (t ) ≤ −σ i exi (t )

(8)

(2) When |xi (t)| > Ti and | xi (t )| > Ti ,

sign(exi (t ))(−σi ( xi (t )) xi (t ) + σi (xi (t ))xi (t ))









= −sign(exi (t ))(σi xi (t ) − σi xi (t )) = −σi exi (t ) ≤ −σ i exi (t )

(9)

(3) When |xi (t)| ≤ Ti and | xi (t )| ≥ Ti ,

sign(exi (t ))(−σi ( xi (t )) xi (t ) + σi (xi (t ))xi (t )) = −sign(exi (t ))[σi ( xi (t ))exi (t ) + (σi ( xi (t )) − σi (xi (t )))xi (t )]









≤ −σ i exi (t ) + Ti σi − σi 

(10)

(4) When |xi (t)| ≥ Ti and | xi (t )| ≤ Ti ,

sign(exi (t ))(−σi ( xi (t )) xi (t ) + σi (xi (t ))xi (t )) x

= −sign(ei (t ))[(σi (xi (t )) − σi (xi (t ))) xi (t ) + σi (xi (t ))exi (t )]









≤ −σ i exi (t ) + Ti σi − σi 

The proof is completed.

(11)



Lemma 2.









sign(eyj (t ))(−ξ j ( y j (t )) y j (t ) + ξ j (y j (t ))y j (t )) ≤ −ξ j eyj (t ) +  T j ξ j − ξ j , for j = 1, 2, . . . , m. Proof. The proof is similar to that of Lemma 1.   + + + Lemma 3. |Fi (t)| ≤ i , where i = m j=1 2W j (a ji + b ji + τ2 p ji ), i = 1, 2, . . . , n. Proof. By means of Assumption A1, it can be easily proved from the definition of Fi (t).  ), j = 1, 2, . . . , m. Lemma 4. |Gj (t)| ≤  j , where  j = ni=1 2Ni (ci+j + di+j + τ2 q+ ij



Proof. By means of Assumption A2, it can be easily proved from the definition of Gj (t).



Definition 1. MBAMNNs (1) and (4) are said to be exponentially synchronized, if there exist constants ϖ > 0 and γ > 0 such that

ex (t )2 + ey (t )2 ≤ sup (ψ1 (s )2 + ψ2 (s )2 )e−γ t , t ≥ 0. −τ ≤s≤0

3. Main results Theorem 1. If Assumptions A1 and A2 hold, MBAMNN (4) will be asymptotically synchronized with MBAMNN (1) under the adaptive feedback controllers:

ui (t ) = −ζi (t )exi (t ) − ηi (t )sign(exi (t )),

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

(12)

and

v j (t ) = −α j (t )eyj (t ) − β j (t )sign(eyj (t )), with



2 ζ˙i (t ) = ki(exi (t ))  , ζi (0 ) = 0, x   η˙ i (t ) = li ei (t ) , ηi (0 ) = 0,

j = 1, 2, . . . , m,

(13)

(14)

104

C. Chen et al. / Applied Mathematics and Computation 322 (2018) 100–110

and

α˙ j (t ) = r j (eyj (t )) 2 , α j (0 ) = 0, β˙ j (t ) = s j eyj (t ), β j (0 ) = 0,

(15)

where ki > 0, li > 0, rj > 0 and sj > 0 are constants. Proof. We design the following Lyapunov function:

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

V1 (t ) =

(16)

 n n 

1 x 1 1 (ei (t ))2 + (ζi (t ) − ζi )2 + (ηi (t ) − ηi )2 , 2 2ki 2li i=1

i=1





m m

1 y 1 1 V2 (t ) = (e j (t ))2 + (α j (t ) − α j )2 + (β j (t ) − β j )2 , 2 2r j 2s j j=1

(17)

j=1

ζ i , ηi , α j and β j are constants to be determined, i = 1, 2, . . . , n, j = 1, 2, . . . , m. The derivative of V1 (t) along system (5) can be calculated as:

V˙ 1 (t ) =

n  

exi (t )sign(exi (t ))[−σi ( xi (t )) xi (t ) + σi (xi (t ))xi (t ) + Fi (t ) + ui (t )] i=1

+

n

(ζi (t ) − ζi )(exi (t ))2 +

n

i=1

  (ηi (t ) − ηi )exi (t )

i=1

n  n     



exi (t ) −σ i exi (t ) + Ti σi − σi  + i − ≤ ζi (t )(exi (t ))2 i=1

−

i=1

n

i=1

=

n

n n  

 

ηi (t )exi (t ) + (ζi (t ) − ζi )(exi (t ))2 + (ηi (t ) − ηi )exi (t ) i=1

(−σ i − ζi )(exi (t ))2 +

i=1

i=1

n

  (Ti |σi − σi | + i − ηi )exi (t ),

(18)

i=1

where Lemmas 1 and 3 have been used.   Choose ζi > −σ i and ηi ≥ Ti σi − σi  + i , i = 1, 2, . . . , n. Then we have V˙ 1 (t ) < 0. Similarly, the derivative of V2 (t) along system (5) can be calculated as:

V˙ 2 (t ) ≤

m

j=1

(−ξ j − α j )(eyj (t ))2 +

m

    ( T j ξ j − ξ j  +  j − β j )eyj (t ).

(19)

j=1

Choose α j > −ξ j and β j ≥  T j |ξ j − ξ j | +  j , j = 1, 2, . . . , m. Then, we have V˙ 2 (t ) < 0.

Since V˙ (t ) = V˙ 1 (t ) + V˙ 2 (t ) < 0, we say MBAMNNs (1) and (4) can achieve asymptotic synchronization. This completes the proof.  Theorem 2. Suppose Assumptions A1 and A2 hold. For given constant μ, if μ satisfies 0 < μ ≤   2 min σ 1 , σ 2 , . . . , σ n , ξ 1 , ξ 2 , . . . , ξ m , MBAMNN (4) will be exponentially synchronized with MBAMNN (1) under the adaptive feedback controllers:

ui (t ) = −ζi (t )sign(exi (t )) − ηi (t )xi (t )sign(exi (t )xi (t )),

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

(20)

and

v j (t ) = −α j (t )sign(eyj (t )) − β j (t )y j (t )sign(eyj (t )y j (t )), with

and

  ζ˙i (t ) = kiexi (t )eμt , ζi (0 ) = 0, η˙ i (t ) = li exi (t )xi (t )eμt , ηi (0 ) = 0,    α˙ j (t ) = r j eyj (t )eμt , α j ( 0 ) = 0,  y  β˙ j (t ) = s j e j (t )y j (t )eμt , β j (0 ) = 0,

j = 1, 2, . . . , m,

(21)

(22)

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C. Chen et al. / Applied Mathematics and Computation 322 (2018) 100–110

105

where ki > 0, li > 0, rj > 0 and sj > 0 are constants, i = 1, 2, . . . , n, j = 1, 2, . . . , m. Proof. We design the following Lyapunov function:

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

V1 (t ) =

(24)

 n n 

1 μt x 1 1 e (ei (t ))2 + (ζi (t ) − ζi )2 + (ηi (t ) − ηi )2 , 2 2ki 2li i=1

i=1





m m

1 μt y 1 1 V2 (t ) = e (e j (t ))2 + (α j (t ) − α j )2 + (β j (t ) − β j )2 , 2 2r j 2s j j=1

(25)

j=1

ζ i , ηi , α j and β j are constants to be determined, i = 1, 2, . . . , n, j = 1, 2, . . . , m. The derivative of V1 (t) along system (5) is calculated as:

V˙ 1 (t ) = eμt

n 

exi (t )e˙ xi (t ) +

μ 2

i=1

= eμt

n 

    (exi (t ))2 + (ζi (t ) − ζi )exi (t ) + (ηi (t ) − ηi )exi (t )xi (t )

−σi ( xi (t ))(exi (t ))2 − (σi ( xi (t )) − σi (xi (t )))exi (t )xi (t )

i=1









+exi (t )Fi (t ) − ζi (t )exi (t ) − ηi (t )exi (t )xi (t ) +









+(ζi (t ) − ζi )exi (t ) + (ηi (t ) − ηi )exi (t )xi (t ) ≤ eμt

n 



 

μ 2

(exi (t ))2







−σ i (exi (t ))2 + σi − σi  · exi (t )xi (t ) + i exi (t )

i=1

+

μ

= eμt

    (exi (t ))2 − ζi exi (t ) − ηi exi (t )xi (t )

2 n 

−σ i +

μ 2

i=1

 (exi (t ))2 + ( i − ζi )|exi (t )| + (|σi − σi | − ηi )|exi (t )xi (t )| ,

(26)

where Lemma 3 has been used. Similarly, the derivative of V2 (t) along system (5) is calculated as:

V˙ 2 (t ) ≤ eμt

m 

−ξ j +

j=1

μ 2

 (eyj (t ))2 + ( j − α j )|eyj (t )| + (|ξ j − ξ j | − β j )|eyj (t )y j (t )| .

      ηi ≥ σi − σi , α j ≥  j , β j ≥ ξ j − ξ j , i = 1, 2, . . . , n,   2 min σ 1 , σ 2 , . . . , σ n , ξ 1 , ξ 2 , . . . , ξ m , then V˙ (t ) ≤ 0, it follows that V(t) ≤ V(0), t ≥ 0. ζ i ≥ i ,

Choose

On the other hand,



V (0 ) =

n ζ 2 η2 1

(exi (0 ))2 + i + i 2 ki li i=1

1+δ ≤ sup 2 −τ ≤s≤0

n



i=1

ζi2 ki

+

ηi2



li

+

m



j=1

m

α 2j rj

+

β 2j



j=1

i=1

( (t )) + 2

m

j=1



(28)





n

≤ δ sup

−τ ≤s≤0

(exi (s ))2 +

i=1

m

 (eyj (s ))2 .

(29)

j=1





n m



1+δ ≤ V (t ) ≤ V (0 ) ≤ sup (exi (s ))2 + (eyj (s ))2 , t ≥ 0. 2 −τ ≤s≤0 i=1

( (t )) ≤ (1 + δ ) sup eyj

0 0 satisfies n



(27)

2

−τ ≤s≤0

n

i=1

( (s )) + exi

2

m

j=1

(30)

j=1

 ( (s )) eyj

2

e−μt , t ≥ 0.

(31)

106

C. Chen et al. / Applied Mathematics and Computation 322 (2018) 100–110

Fig. 1. The state trajectories of the drive system (33).

It means that

ex (t )2 + ey (t )2 ≤ (1 + δ ) sup (ψ1 (s )2 + ψ2 (s )2 )e−μt , t ≥ 0. −τ ≤s≤0

The proof is completed.

(32)



Remark 1. Although several synchronization criteria about MBAMNNs with discrete delay have been obtained [32,33], all of them need to solve linear matrix inequalities. In this paper, since the controllers that we utilize are adaptive feedback controllers, computing algebraic conditions and solving linear matrix inequalities are not needed to determine suitable control gains. What is more, compared with the existing works, the proofs of this paper are much simpler. Remark 2. In most references about the synchronization control of MBAMNNs, the considered MBAMNN models didn’t contain time delays or only contained discrete delay, and the rates of the neuron self-inhibition were fixed constants. However, the MBAMNN model considered in this paper has both time-varying discrete delay and time-varying distributed delay, what is more, the rates of the neuron self-inhibition are state-dependent. So the MBAMNN model in this paper is more general. 4. Numerical simulations In this section, we give some numerical simulations to verify the effectiveness of the theoretical results. Example 1. Consider the following MBAMNN with mixed time-varying delays:

⎧  t ) = −σi (xi (t ))xi (t ) + 2j=1 a ji (xi (t )) f j (y j (t )) ⎪ ⎪x˙ i ( t  ⎨ + 2j=1 b ji (xi (t )) f j (y j (t − τ (t ))) + 2j=1 p ji (xi (t )) t −ρ (t ) f j (y j (s ))ds, 2 y˙ (t ) = −ξ (y (t ))y (t ) + ci j (y j (t ))gi (xi (t )) ⎪ ⎪   ⎩+j 2 d (yj (tj ))g (xj (t − τ (ti=1 ))) + 2i=1 qi j (y j (t )) tt−ρ (t ) gi (xi (s ))ds, j i i i=1 i j

(33)

i = 1, 2, j = 1, 2, where T1 = T2 = 1,  T1 =  T2 = 2, σ1 = 1, σ1 = 1.5, σ2 = 0.8, σ2 = 1, a11 = 3.4, a11 = 2.9, a12 = −0.4, a12 = −0.22, a21 = 4.2, a21 = 3.9, a22 = 5.2, a22 = 5, b11 = −1.4, b11 = −1.2, b12 = 0.2, b12 = −0.1, b21 = 0.5, b21 = −0.2, b22 = −9.2, b22 = −6, p11 = −1.3, p11 = −1.18, p12 = 0.12, p12 = 0.05, p21 = −0.3, p21 = −0.2, p22 = −1.2, p22 = −0.6, ξ1 = 0.75,  = 1.81, c = 2.2, c = −0.14, c = 0.12, c = −1.9, c = −2.2, c = 5, c = 5.2, d  = ξ1 = 1, ξ2 = 1, ξ2 = 1.1, c11 11 12 12 21 21 22 22 11    = 0.15, d  = −0.2, d  = −0.18, d  = −2.5, d  = −2.3, q = 0.6, q = 0.65, q = 0.12, −0.95, d11 = −1.3, d12 = 0.08, d12 21 21 22 22 11 11 12 q12 = −0.12, q21 = −0.2, q21 = −0.18, q22 = −0.1, q22 = −0.12. ν +1|−|ν −1| Let f1 (ν ) = f2 (ν ) = g1 (ν ) = g2 (ν ) = | , τ (t ) = 2

et 1+et

and ρ (t ) = 1 + sint, we can get that W1 = W2 = N1 = N2 = 1, τ1 = 1, τ2 = 2, τ = 2. The initial conditions of MBAMNN (33) are ϕ1 (s ) = (2, 1 )T , ϕ2 (s ) = (−1, 0.2 )T , s ∈ [−2, 0]. Fig. 1 describes the state trajectories of MBAMNN (33). This is the corresponding response system:

⎧ 

x˙ i (t ) = −σi ( xi (t )) xi (t ) + 2j=1 a ji ( xi (t )) f j ( y j (t )) + ui (t ) ⎪ ⎪ t 2 ⎨+ 2 b (

x ( t )) f ( y ( t − τ ( t ))) + p j j j=1 ji i j=1 ji (xi (t )) t −ρ (t ) f j (y j (s ))ds, 2 ˙ j (t ) = −ξ j (



y y ( t )) y ( t ) + c ( y ( t )) g ( x ( t )) + v ( t ) ⎪ j j j i i i=1 i j ⎪ t j  ⎩ 2 + i=1 di j ( y j (t ))gi ( xi (t − τ (t ))) + 2i=1 qi j ( y j (t )) t −ρ (t ) gi ( xi (s ))ds,

(34)

i = 1, 2, j = 1, 2, where ui (t) and vi (t ) are the controllers. The initial conditions of MBAMNN (34) are φ1 (s ) = (1, 2 )T , φ2 (s ) = (1, −1 )T , s ∈ [−2, 0]. The evolutions of the synchronization errors between MBAMNNs (33) and (34) without control inputs are shown in Figs. 2 and 3.

C. Chen et al. / Applied Mathematics and Computation 322 (2018) 100–110

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Fig. 2. The synchronization errors ex1 (t ) and ex2 (t ) without control inputs.

Fig. 3. The synchronization errors ey1 (t ) and ey2 (t ) without control inputs.

Fig. 4. The synchronization errors ex1 (t ) and ex2 (t ) under the controllers.

Choose k1 = k2 = 1, l1 = 2, l2 = 1, r1 = r2 = 1, s1 = s2 = 1. According to Theorem 1, MBAMNN (34) can be asymptotically synchronized with MBAMNN (33) under the adaptive controllers (12) and (13). The evolutions of the synchronization errors under the controllers (12) and (13) are shown in Figs. 4 and 5 The evolutions of the control gains are presented in Figs. 6–9.

Remark 3. As far as we know, adaptive control has not been used to study the synchronization control of MBAMNNs until now. In the previous references about the synchronization control of MBAMNNs, to determine the suitable control gains, it is necessary to compute algebraic conditions or solve linear matrix inequalities based on the system parameters of the considered MBAMNNs, including memristive connection weights, activation functions, and time delays. Since we adopt adaptive control in this paper, extra calculations are not required to determine the suitable control gains, and we even needn’t know the system parameters of the considered MBAMNNs. Moreover, one can see from Figs. 6–9 that the control gains of the adaptive controller are very small.

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Fig. 5. The synchronization errors ey1 (t ) and ey2 (t ) under the controllers.

Fig. 6. The evolutions of the control gains ζ 1 (t) and ζ 2 (t).

Fig. 7. The evolutions of the control gains η1 (t) and η2 (t).

Fig. 8. The evolutions of the control gains α 1 (t) and α 2 (t).

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Fig. 9. The evolutions of the control gains β 1 (t) and β 2 (t).

5. Conclusions This paper is concerned with the adaptive synchronization of MBAMNNs with mixed time-varying delays. We design two kinds of adaptive feedback controllers, under which the considered MBAMNNs can achieve asymptotic synchronization and exponential synchronization, respectively. It is well-known that adaptive control can avoid the high control gains effectively. Moreover, for adaptive control, computing algebraic conditions and solving linear matrix inequalities are not needed to determine suitable control gains. The effectiveness of the theoretical results is verified by numerical simulations, which show that the control gains of our adaptive feedback controllers are very small. In the future, we will consider the adaptive synchronization control of fractional-order memristor-based BAM neural networks. Acknowledgments The work is supported by the National Key Research and Development Program (Grant Nos. 2016YFB0800602 and 2016YFB0800604), the National Natural Science Foundation of China (Grant Nos. 61573067, 61771071 and 61472045), the Beijing City Board of Education Science and Technology Key Project (Grant No. KZ201510015015), and BUPT Excellent Ph.D. Students Foundation (Grant No. CX2017315). References [1] L.O. Chua, Memristor-the missing circuit element, IEEE Trans. Circuit Theory 18 (1971) 507–519. [2] D.B. Struko, G.S. Snider, D.R. Stewart, R.S. Williams, The missing memristor found, Nature 453 (2008) 80–83. [3] M. Sharifiy, Y. Banadaki, General spice models for memristor and application to circuit simulation of memristor-based synapses and memory cells, J Circuits Syst. Comput. 19 (2010) 407–424. [4] R. Zhang, D. Zeng, S. Zhong, Y. Yu, Event–triggered sampling control for stability and stabilization of memristive neural networks with communication delays, Appl. Math. Comput. 310 (2017) 57–74. [5] S.H. Jo, T. Chang, I. Ebong, B.B. Bhadviya, P. Mazumder, W. Lu, Nanoscale memristor device as synapse in neuromorphic systems, Nano Lett. 10 (2010) 1297–1301. [6] R. Zhang, D. Zeng, S. Zhong, Novel master-slave synchronization criteria of chaotic Lur’e systems with time delays using sampled-data control, J. Frankl. Inst. 354 (12) (2017) 4930–4954. [7] Y. Liu, B.Z. Guo, J.H. Park, S.M. Lee, Nonfragile exponential synchronization of delayed complex dynamical networks with memory sampled-data control, IEEE Trans. Neural Netw. Learn. Syst. (2016), doi:10.1109/TNNLS.2016.2614709. [8] Z. Tang, J.H. Park, H. Shen, Finite-time cluster synchronization of Lur’e networks: a nonsmooth approach, IEEE Trans. Syst. Man Cybern. Syst. (2017), doi:10.1109/TSMC.2017.2657779. [9] D. Zeng, R. Zhang, Y. Liu, S. Zhong, Sampled-data synchronization of chaotic Lur’e systems via input-delay-dependent-free-matrix zero equality approach, Appl. Math. Comput. 315 (2017) 34–46. [10] D. Zeng, R. Zhang, S. Zhong, J. Wang, K. Shi, Sampled-data synchronization control for Markovian delayed complex dynamical networks via a novel convex optimization method, Neurocomputing 266 (2017) 606–618. [11] R. Zhang, D. Zeng, S. Zhong, K. Shi, Memory feedback PID control for exponential synchronisation of chaotic Lur’e systems, Int. J. Syst. Sci. 48 (12) (2017) 2473–2484. [12] Z. Tan, M.K. Ali, Associative memory using synchronization in a chaotic neural network, Int. J. Mod. Phys. C 12 (1) (2001) 19–29. ´ M.E. Zaghloul, Synchronization of chaotic neural networks and applications to communications, Int. J. Bifurc. Chaos, 61996, 2571–2585. [13] V. Milanovic, [14] S. Wen, Z. Zeng, T. Huang, Q. Meng, W. Yao, Lag synchronization of switched neural networks via neural activation function and applications in image encryption, IEEE Trans. Neural Netw. Learn. Syst. 26 (7) (2015) 1493–1502. [15] C. Chen, L. Li, H. Peng, Y. Yang, T. Li, Finite-time synchronization of memristor-based neural networks with mixed delays, Neurocomputing 235 (2017) 83–89. [16] X. Han, H. Wu, B. Fang, Adaptive exponential synchronization of memristive neural networks with mixed time-varying delays, Neurocomputing 201 (2016) 40–50. [17] A. Abdurahman, H. Jiang, Z. Teng, Exponential lag synchronization for memristor-based neural networks with mixed time delays via hybrid switching control, J. Frankl. Inst. 353 (13) (2016) 2859–2880. [18] A. Ascoli, V. Lanza, F. Corinto, R. Tetzlaff, Synchronization conditions in simple memristor neural networks, J. Frankl. Inst. 352 (2015) 3196–3220. [19] X. Yang, J. Cao, J. Liang, Exponential synchronization of memristive neural networks with delays: Interval matrix method, IEEE Trans. Neural Netw. Learn. Syst. 28 (8) (2017) 1878–1888. [20] H. Bao, J.H. Park, J. Cao, Exponential synchronization of coupled stochastic memristor-based neural networks with time-varying probabilistic delay coupling and impulsive delay, IEEE Trans. Neural Netw. Learn. Syst. 27 (1) (2016) 190–201.

110

C. Chen et al. / Applied Mathematics and Computation 322 (2018) 100–110

[21] G. Velmurugan, R. Rakkiyappan, J. Cao, Finite-time synchronization of fractional-order memristor-based neural networks with time delays, Neural Netw. 73 (2015) 36–46. [22] J. Cao, R. Li, Fixed-time synchronization of delayed memristor-based recurrent neural networks, Sci. China Inf. Sci. 60 (032201) (2017) 1–15. [23] C. Chen, L. Li, H. Peng, Y. Yang, Fixed-time synchronization of memristor-based BAM neural networks with time-varying discrete delay, Neural Netw. 96 (2017) 47–54. [24] C. Chen, L. Li, H. Peng, Y. Yang, T. Li, Synchronization control of coupled memristor-based neural networks with mixed delays and stochastic perturbations, Neural Proc. Lett. (2017), doi:10.1007/s11063- 017- 9675- 6. [25] Q. Song, Synchronization analysis of coupled connected neural networks with mixed time delays, Neurocomputing 72 (2009) 3907–3914. [26] Z. Wang, H. Zhang, B. Jiang, LMI-based approach for global asymptotic stability analysis of recurrent neural networks with various delays and structures, IEEE Trans. Neural Netw. 22 (7) (2011) 1032–1045. [27] X. Yang, C. Huang, Q. Zhu, Synchronization of switched neural networks with mixed delays via impulsive control, Chaos Sol. Fractals 44 (10) (2011) 817–826. [28] B. Kosko, Bidirectional associative memories, IEEE Trans. Syst. Man Cybern. 18 (1988) 49–60. [29] J. Cao, Y. Wan, Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays, Neural Netw. 53 (5) (2014) 165–172. [30] J. Ge, J. Xu, Synchronization and synchronized periodic solution in a simplified five-neuron BAM neural network with delays, Neurocomputing 74 (2011) 993–999. [31] Y. Li, C. Li, Matrix measure strategies for stabilization and synchronization of delayed BAM neural networks, Nonlinear Dyn. 84 (3) (2016) 1759–1770. [32] K. Mathiyalagan, J.H. Park, R. Sakthivel, Synchronization for delayed memristive BAM neural networks using impulsive control with random nonlinearities, Appl. Math. Comput. 259 (C) (2015) 967–979. [33] R. Sakthivel, R. Anbuvithya, K. Mathiyalagan, Y.K. Ma, P. Prakash, Reliable anti-synchronization conditions for BAM memristive neural networks with different memductance functions, Appl. Math. Comput. 275 (C) (2016) 213–228. [34] J. Xiao, S. Zhong, Y. Li, F. Xu, Finite-time Mittag-Leffler synchronization of fractional-order memristive BAM neural networks with time delays, Neurocomputing 219 (2016) 431–439. [35] R. Anbuvithya, K. Mathiyalagan, R. Sakthivel, P. Prakash, Non-fragile synchronization of memristive BAM networks with random feedback gain fluctuations, Commun. Nonlinear Sci. Numer. Simul. 29 (2015) 427–440. [36] N. Li, J. Cao, New synchronization criteria for memristor-based networks: Adaptive control and feedback control schemes, Neural Netw. 61 (2015) 1–9. [37] H. Bao, J.H. Park, J. Cao, Adaptive synchronization of fractional-order memristor–based neural networks with time delay, Nonlinear Dyn. 82 (3) (2015) 1343–1354. [38] A. Wu, Z. Zeng, Anti-synchronization control of a class of memristivere current neural networks, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 373–385. [39] G. Zhang, J. Hu, Y. Shen, New results on synchronization control of delayed memristive neural networks, Nonlinear Dyn. 81 (3) (2015) 1167–1178.