Exponential synchronization of inertial neural networks with mixed time-varying delays via periodically intermittent control

Neurocomputing 338 (2019) 181–190

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

Exponential synchronization of inertial neural networks with mixed time-varying delays via periodically intermittent control Qian Tang a, Jigui Jian a,b,∗ a b

College of Science, China Three Gorges University, Yichang, Hubei 443002, China Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, Hubei 443002, China

a r t i c l e

i n f o

Article history: Received 31 August 2018 Revised 10 December 2018 Accepted 29 January 2019 Available online 7 February 2019 Communicated by Choon Ki Ahn Keywords: Inertial neural network Exponential synchronization Mixed time-varying delay Periodically intermittent control Inequality technique

a b s t r a c t This paper is concerned with the problem on the exponential synchronization of inertial neural networks with discrete and finite distributed time-varying delays using intermittent control. Two kinds of time varying delays are considered: one is whose derivatives are strictly smaller than one and the other is without any restriction on the delay derivatives. Based on Lyapunov–Krasovskii functional method and applying inequality techniques, some new delay-dependent criteria are obtained to ensure the global exponential synchronization for the discussed networks, which are very simple to implement in practice and reduce the computational burden. Moreover, the exponential synchronization convergence rates depend on the norm, the transformation parameters, the control parameters and the width index of the control. Finally, some numerical examples are presented to demonstrate the validity of our results.

1. Introduction The dynamic property analysis of neural networks have gained the widespread attention in various domains including image processing, pattern recognition and engineer optimization. Meanwhile, time delays frequently appear in the signal transfer among the neurons, which can result in some poor performance such as oscillations, instability, chaos. Now, there have many interesting results on the dynamics of neural networks with delays [1–7]. Nevertheless, there exist many neural networks which are inherently unstable. As a result, some controllers need be added to the neural networks to assure the relevant asymptotic properties. Nowadays, various control strategies such as adaptive control [8], sliding mode control [9,10], pinning control [11], impulsive control [12,13] and intermittent control [14,15] are adopted to realize synchronization and stabilization of neural networks. On the other hand, an important subject in system analysis is to seek less control cost, simple and efficient methods for system control. Recently, discontinuous control methods such as impulsive control and intermittent control have attracted more and more attentions due to their preferable effectiveness and robustness in some engineering domains. Nevertheless, the intermittent control is different from the impulsive control. Intermittent control ∗ Corresponding author at: College of Science, China Three Gorges University, Yichang, Hubei 443002, China. E-mail addresses: [email protected] (Q. Tang), [email protected] (J. Jian).

https://doi.org/10.1016/j.neucom.2019.01.096 0925-2312/© 2019 Elsevier B.V. All rights reserved.

© 2019 Elsevier B.V. All rights reserved.

is activated during certain nonzero time intervals, while impulsive control is mobilized only at some isolated moments. In any period of intermittent control, the time consists of control time (or work time) and uncontrolled time (or rest time), the controller is active within control time and is out within the uncontrolled time. So, the system output is detected periodically instead of continuously. Generally, compared with the continuous control methods, intermittent control is a straightforward engineering approach to control and synchronize the systems. In communications, in order to compensate the lost signal and enable received signal at the terminal to achieve the desired requirement, the external control signal is added so long as the strength of the system signal is below the required level [16]. And then, the external control can be considering the lower cost [17]. Due to those virtues, intermittent control scheme has been well used in stabilization and synchronization for various neural networks [18–22]. Fig. 1 shows that the control time of intermittent control is periodic. Hence, the periodically intermittent control can become the continuous control for u = v and the impulsive control for u = 0, respectively. It is worth mentioning that dozens of previous literature mainly concerned with first derivative of the states in neural networks. The inertial term is taken as a critical tool to generate bifurcation and chaos. Hence, it is also of great significance to introduce an inertial term into the artificial neural networks. In 1986, Babcock and Westervelt [23] showed that the dynamics could be complex when the neuron couplings contain an inertial nature. Wheeler and Schieve [24] first presented a second-order inertial neural

182

Q. Tang and J. Jian / Neurocomputing 338 (2019) 181–190

Notation: In this article, Rn and Rn×m show the set of ndimensional real vectors and n × m real matrices, respectively. AT represents the transposition of matrix A. λmin (A) (λmax (A)) refers to minimal (maximal) eigenvalue of matrix A. A < 0 represents that A is negative definite. Let N+ = {1, 2, . . .} and  = {1, 2, . . . , n}. 2. Preliminaries Consider the following INNs Fig. 1. Sketch map of periodically intermittent control.

 d2 xi (t ) dxi (t ) = −ai − bi xi (t ) + ci j f j (x j (t )) dt dt 2 n

j=1

network (INN) and investigated its dynamic characteristic. Later, the Hopf bifurcation and chaos in a single inertial neuron with delay were considered in [25,26]. The authors [27] studied the stability of INNs by using the comparison principle. Wan and Jian [28] discussed the global convergence of impulsive INNs with time-varying delays. By utilizing inequality techniques and analytical method, the authors [29] investigated the ultimate boundedness and global exponential attractive set of INNs with timevarying delay. Wang and Tian [30] discussed the global boundedness for INNs with discrete and distributed time varying delays. Zhang et al. [31] analyzed the global exponential dissipativity of memristive INNs with discrete and distributed time-varying delays. In [32], the matrix measure method was employed to study the exponential stabilization for complex-valued INNs with delays by applying impulsive control. Li et al. [33] researched the synchronization and asymptotic stability for INNs with delay via non-reduced order method. In [34], the finite-time synchronization of INNs without delay is investigated by designing continuous and discontinuous controllers. The authors [35,36] studied the finite-time synchronization for memristive INNs with time-varying delay. Dharani et al. [37] explored pinning sampled-data synchronization of coupled delayed INNs with reaction–diffusion terms. Rakkiyappan et al. [38] analyzed the stability and synchronization of memristive INNs. The authors [39] analyzed the delaydependent passivity for a class of memristive impulsive INNs with time-varying delays. Zhang et al. [40] studied the exponential stability of INNs via periodically intermittent control. Nevertheless, there are few literatures focusing on designing periodically intermittent controllers to synchronize two INNs with mixed timevarying delays. Inspired by the above discussions, the objective of this article is to explore the exponential synchronization of INNs with mixed time-varying delays via periodically intermittent control. The main contributions of this paper are the following aspects: (i) Under two classes of activation functions, two kinds of time varying delays are considered: one is whose derivatives are strictly smaller than one and the other is without any restriction on the delay derivatives. (ii) By creating suitable Lyapunov–Krasoviskii functionals and applying inequality techniques, several delay-dependent criteria in terms of algebraic inequality and linear matrix inequality are obtained to insure the exponential synchronization. (iii) Our results indicate that the exponential synchronization convergence rate relies on the norm, the transformation parameters, the control parameters and the ratio of the control width to the control period, but does not depends on control width or control period. In some sense, it is easy to point out that the discussed models in this paper contain those in [27–29,33,40] as special cases. The rest of this paper is organized as follows. In Section 2, model description, definitions and lemmas are given. Section 3 mainly covers the derivation of synchronization criteria. Some illustrative examples with simulations are presented to clarify the main conclusions in Section 4. Finally, conclusions are presented in Section 5.

+ +

n  j=1 n 

di j f j (x j (t − ν (t )))  wi j

j=1

t

t −μ(t )

f j (x j (ϑ ))dϑ + Ii (t ), i ∈  ,

(1)

where t ≥ 0, xi (t) is the ith neuron state, the second-order derivative is referred to as an inertial term of (1). ai and bi are positive constants, cij , dij and wij are constants and represent the connection strengths, fi ( · ) shows the activation function, Ii (t) is the external input, the time-varying delays ν (t) and μ(t) satisfy ν (t) ≤ ν , μ(t) ≤ μ, ν˙ (t ) ≤ ν1 , μ˙ (t ) ≤ μ1 . The initial values of (1) are

xi ( s ) = ψi ( s ),

dxi (s ) = ϕi (s ), s ∈ [−h, 0], i ∈  , ds

where h = max{ν, μ}, ψ i (s) and ϕ i (s) are continuous and bounded functions. Remark 1. It is obvious that network (1) is more general than the ones in [27–29,33,40]. For example, when wi j = 0, system (1) becomes into the INNs in [28,29,33]. When wi j = 0 and Ii (t ) = I, system (1) reduces to the INNs in [27,40]. For some chosen scalar ξi ∈ R, using the following variable transformation

yi (t ) =

dxi (t ) + ξi xi (t ), i ∈  , dt

then (1) can be changed into

⎧ dxi (t ) ⎪ = −ξi xi (t ) + yi (t ), ⎪ ⎪ dt ⎪ ⎪ n  ⎪ dy ( t ) i ⎪ ⎪ ⎨ dt = −αi xi (t ) − βi yi (t ) + ci j f j (x j (t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+ +

n  j=1 n 

j=1

di j f j (x j (t − ν (t ))) wi j

j=1

t

t −μ(t )

(2)

f j (x j (ϑ ))dϑ + Ii (t ),

where αi = bi + ξi (ξi − ai ) and βi = ai − ξi , the initial conditions can be expressed as

xi (s ) = ψi (s ), yi (s ) = ψi + ξi ϕi (s )  φi (s ), s ∈ [−h, 0], i ∈  . Let system (2) be the driver INN, and the response INN is shown by

⎧ du (t ) i ⎪ ⎪ ⎪ dt = −ξi ui (t ) + vi (t ) + Ki (t ), ⎪ ⎪ n ⎪ ⎪ dvi (t ) = −α u (t ) − β v (t ) +  ⎪ ci j f j (u j (t )) ⎨ dt i i i i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+ +

n  j=1 n  j=1

j=1

di j f j (u j (t − ν (t ))) wi j

t

t −μ(t )

(3)

f j (u j (ϑ ))dϑ + Ii (t ) + ϒi (t ),

where ui (t), vi (t) denote the state of the response INN, the remainder notations are the same as (2). Ki (t), ϒ i (t) are the periodically intermittent controllers described by

Q. Tang and J. Jian / Neurocomputing 338 (2019) 181–190

⎧ K (t ) = −ki (ui (t ) − xi (t )), ϒi (t ) = −γi (vi (t ) − yi (t )), ⎪ ⎨ i lT ≤ t < (l + θ )T , ⎪ ⎩ Ki (t ) = ϒi (t ) = 0, ( l + θ )T ≤ t < ( l + 1 )T ,

ϑi = pξi − ( p − 1 ) − |αi | −

j=1

υi = pβi − 1 − ( p − 1 )|αi | − ( p − 1 )

|d ji |li |w ji |li μ p−1 εμ ≥ 0, q ji = e ≥ 0, 1 − ν1 1 − μ1 ε > 0, r ji ≥ 0, i, j ∈  ,

ε, rji are some properly chosen numbers.

<

l+ , j

ˇ (0 )ek(1−ϑ )t , t ≥ 0. ˇ (t ) ≤ W W

then (A2) describes the



Let L1 = diag l1− l1+ , l2− l2+ , . . . , ln− ln+ , L2 = diag



Lemma 2 [30]. If numbers aˆ and bˆ meet bˆ > aˆ, m × m constant matrix W > 0 and vector function ω (t ) : [aˆ, bˆ ] → Rm , then the following inequality holds



bˆ aˆ

ω ( ς )d ς

l1− +l1+ l2− +l2+ 2 , 2 ,...,

e(t ) = (e1 (t ), e2 (t ), . . . , en (t ),

e1 (t ),

e2 (t ), . . . ,

en (t )) , T

(s ) = ( x1 ( s ), x2 ( s ), . . . , xn ( s ), y1 ( s ), y2 ( s ), . . . , yn (s )) , T

their p-norm for p > 1 are defined as

 |||| p =

j=1 n 

sup | x j ( s )| p +

j=1 −h≤s≤0

bˆ

W aˆ

    bˆ ω (ς )dς ≤ bˆ − aˆ ωT (ς )W ω (ς )dς . aˆ

Theorem 1. Under Assumptions (A1) and (A3), systems (2) and (3) with intermittent controller (4) are exponentially synchronized if there exists ε > 0 such that

=

ε

p

− λ ( 1 − θ ) > 0,

p





where λ = max max{ki }, max{γi } . i∈

1 p

V (t ) =

n  

eεt |ei (t )| p + eεt |

ei (t )| p

i=1

+eεν



n 

+ eεν

,

pi j

sup | y j ( s )| p

j=1 −h≤s≤0

where xi ( s ) = ψi ( s ) − ψ i ( s ), yi (s ) = φi (s ) − φ i (s ).

n 

1 p

,

+

n  j=1

 qi j

t

t −ν (t )

 ri j

j=1



i∈

Proof. Construct the following Lyapunov–Krasovskii functional

j=1



j=1 n 

T 

In this segment, we will select the appropriate T, γ i and ki in (4) to realize exponential synchronization between (2) and (3).

. Denote

|

e j (t )|

mT ≤ t < (m + ϑ )T ,

ˇ (t ) ≤ 0, D+ W

3. Main results



|e j (t )| +

r ji > 0.

j=1

where m ∈ N+ , T > 0, 0 < ϑ < 1 and k > 0, then

monotone nondecreasing activations. When l − < l+ ≤ 0, then (A2) j j shows the monotone nonincreasing activations. So, (A2) here can possess better application.

|| e(t )|| p =

n 

ˇ (t ) ≤ kW ˇ (t ), (m + ϑ )T ≤ t < (m + 1 )T , D W

j = .

n 

υi + pγi − ε − eεν

+

or positive. If l − = −l + , then (A2) is equivalent to (A1) such as j j

p

p ji > 0,

|| e(t )|| p ≤ η|||| p e−σ t , t ≥ 0. 

Remark 2. In fact, functions satisfying (A2) are the generalization of functions meeting (A1). In (A2), l − and l + can be negative, zero j j

n 

n 

ˇ (t ) ≥ 0 meets that Lemma 1 [21]. If function W

(A2). For ∀ζ , ϑ ∈ R and ζ = ϑ, there are constants l + , l− meeting j j



( |ci j | + |di j | + |w i j | )l j ,

Definition 1. Drive-response systems (2) and (3) with (4) are said to be exponentially synchronized, if there are constants η > 0, σ > 0 such that

| f j (ζ ) − f j (ϑ )| ≤ l j |ζ − ϑ|, f j (0 ) = 0, j =  .

those in Refs. [33–35,38,40]. If 0 ≤

n 

j=1

In this article, the following assumptions are necessary. (A1). For ∀ζ , ϑ ∈ R, there exists constant lj > 0 meeting

l− j

p ji =

ϑi + pki − ε − eεν

j=1



(|c ji |li + μq ji ),

Remark 3. Under (A3), there must be a number ε > 0 such that

(5)

if (l + θ )T ≤ t < (l + 1 )T , ⎪ ⎪ ⎪ dei (t ) ⎪ ⎪ = −ξi ei (t ) +

ei (t ), ⎪ ⎪ dt ⎪ ⎪ n

 d e ( t ) ⎪ i ⎪ = −αi ei (t ) − βi

ei (t ) + ci j g j (e j (t )) ⎪ ⎪ dt ⎪ j=1 ⎪ ⎪ n ⎪  ⎪ ⎪ + di j g j (e j (t −ν (t ))) ⎪ ⎪ j=1 ⎪ ⎪ ⎪ n t  ⎪ ⎪ + wi j t −μ(t ) g j (e j (ϑ ))dϑ . ⎩

f j (ζ ) − f j (ϑ ) ≤ l +j , ζ −ϑ

r ji > 0, where

j=1

j=1

vi (s ) = φ i (s ), s ∈ [−h, 0], i ∈  ,

⎧ if lT ≤ t < (l + θ )T , ⎪ ⎪ ⎪ dei (t ) ⎪ ⎪ = −(ki + ξi )ei (t ) +

ei (t ), ⎪ ⎪ dt ⎪ ⎪ n

 dei (t ) ⎪ ⎪ = −αi ei (t ) − (βi + γi )

ei (t ) + ci j g j (e j (t )) ⎪ ⎪ dt ⎪ j=1 ⎪ ⎪ n ⎪  ⎪ ⎪ + di j g j (e j (t −ν (t ))) ⎪ ⎪ j=1 ⎪ ⎪ n ⎪ t  ⎪ ⎪ + wi j t −μ(t ) g j (e j (ϑ ))dϑ , ⎨

ln− +ln+ 2

n 

n 

j=1

where ψ i (s ) and φ i (s ) are continuous and bounded functions. Let e j (t ) = u j (t ) − x j (t ),

e j (t ) = v j (t ) − y j (t ), g j (e j (t )) = f j (u j (t )) − f j (x j (t )), g j (e j (t − ν (t ))) = f j (u j (t − ν (t ))) − f j (x j (t − ν (t ))), then we have the following error system

l −j ≤

υi + pγi −

p ji > 0,

j=1

(4)

where i ∈  , ki , γ i are the control gains, l ∈ N+ , T > 0 represents the control period, 0 < θ < 1 is called the width index of the control. And the initial values of (3) are

ui ( s ) = ψ i ( s ),

n 

(A3). ϑi + pki −

183

t

0

0 −μ(t )

eεϑ |e j (ϑ )| p dϑ

eεϑ |

e j ( ϑ )| p d ϑ 

t

t+ϑ



eεη |e j (η )| p dηdϑ .

(6) 

184

Q. Tang and J. Jian / Neurocomputing 338 (2019) 181–190



When lT ≤ t < (l + θ )T , computing the Dini-derivative of V(t) with respect to time along the solutions of (5), one has

D V (t )|(5) ≤ +

n  

peεt |ei (t )| p−1



− (ξi + ki )|ei (t )| + |

ei (t )|

t −μ(t )



+

|ci j ||g j (e j (t ))| +

n 

j=1

+

n 

|wi j |

+ eεν

j=1

−ν (t ))|

p

|di j ||g j (e j (t − ν (t )))|

≤



≤

n 

≤

n 

q i j ( 1 − μ1 )



t −μ(t )

j=1

+

n 

eεη |e j (η )| p dη

(7)

≤

|eε t



|ei (t )|

|ci j ||g j (e j (t ))| ≤

j=1 n 

 |

ei (t )| p + ( p − 1 )|ei (t )| p ,

|αi

t −μ(t )

n 

+

p

+

t −μ(t )





t

t −μ(t )

|g j (e j (ϑ ))|dϑ

t

t −μ(t )

+ ( p − 1 )|

ei (t )| p ,

t

t −μ(t )

n 

 |e j (ϑ )| p dϑ + ( p − 1 )|

ei (t )| p . (12)



  |c ji |li + eεν p ji + μq ji + ε |ei (t )| p

n 



eεt − p(βi + γi ) + 1 + ( p − 1 )|αi | + ( p − 1 )

(9)

n  n   i=1

|ci j | peεt |

ei (t )| p−1 l j |e j (t )|

n  

≤



n 

|di j ||g j (e j (t − ν (t )))|

j=1

 |di j |l j − pi j (1 − ν1 ) |e j (t − ν (t ))| p

j=1



n 



n 

≤

n 

|di j |l j eε t



D+V (t ) ≤ p

≤ =

j=1

(13)

n 



eεt − pξi + ( p − 1 ) + |αi | +

(11)

|e j ( ϑ )| · 1d ϑ



t

t −μ(t )



that is

t

t −μ(t )

|e j ( ϑ )| p d ϑ |e j ( ϑ )| p d ϑ

1  p

·

1  p

·

t

t −μ(t )

μ(t )

1q · 1d ϑ

1 q

1

≤ μq

q

t

t −μ(t )

|e j ( ϑ )| p d ϑ

1 p

|c ji |li + p ji eεν

j=1

+|di j | + |wi j | )l j + eεν

1



n  

 +μq ji + ε |ei (t )| p  n n   + eεt − pβi + 1 + ( p − 1 )|αi | + ( p − 1 ) ( |ci j | i=1

t −μ(t )







On the basis of the Hölder inequality, one has t

t −μ(t )

|e j ( ϑ )| p d ϑ

r ji + ε |

ei (t )| p

i=1



|e j (t − ν (t ))| + ( p − 1 )|

ei (t )| . p

j=1



t

 p ji + ε |ei (t )| p

j=1

eεt − υi − pγi + eεν



Similarly, if (l + θ )T ≤ t < (l + 1 )T , one can get

peεt |

ei (t )| p−1 |di j |l j |e j (t − ν (t ))|

j=1



≤ 0.

j=1

≤

|ci j |

r ji + ε |

ei (t )| p

eεt − ϑi − pki + eεν

i=1

(10)

n   j=1

|wi j |l j μ p−1 − qi j (1 − μ1 )e−εμ

j=1 n 

n 

i=1

n 

 + ( p − 1 )|

ei (t )| p

n  

+ eε t

+

i

p

eεt − p(ξi + ki ) + ( p − 1 ) + |αi |

+|di j | + |wi j | l j + eεν

+

|ci j |l j eεt |e j (t )| p + ( p − 1 )|

ei (t )| p ,

n 

l j |e j ( ϑ )|d ϑ

|e j ( ϑ )|d ϑ



(8)

j=1

peεt |

e (t )| p−1

|e j ( ϑ )| p d ϑ

i=1



j=1



t

  |wi j |l j eεt μ p−1

j=1

peεt |ei (t )|αi ||

ei (t )| p−1 | ≤

t

i=1

On account of the fact that pab p−1 ≤ a p + ( p − 1 )b p for a > 0 and b > 0, one can get

n 

n 

|wi j |l j eε t

D+V (t ) ≤

j=1

peεt |

ei (t )| p−1

n 







From (7) to (12) and combining Assumptions (A3), there exists

qi j μeεt |e j (t )| p .

peεt |

ei (t )||ei (t )| p−1 ≤ eεt



peεt |

ei (t )| p−1 |wi j |

ε > 0 such that

t

|wi j |

j=1

ri j eεt |

e j (t )| p

j=1

−

n 

j=1

pi j eεt |e j (t )| p − (1 − ν1 )eε (t−ν ) |e j (t + eεν

p

≤ μq

|e j ( ϑ )| p d ϑ .

n 

j=1





t −μ(t )

peεt |

ei (t )| p−1

|g j (e j (ϑ ))|dϑ + ε eεt |

ei (t )| p

t −μ(t )

j=1 n 

t

t

p

j=1

j=1







Then, combining with the above inequality, one can obtain



+ peεt |

ei (t )| p−1 − (βi + γi )|

ei (t )| + |αi ||ei (t )| n 

|e j ( ϑ )|d ϑ

= μ p−1

i=1

+ε eεt |ei (t )| p

t

+

n  i=1

, +

n  j=1

eε t



 n 

n 

j=1

 r ji + ε |

ei (t )| p

j=1

 |di j |l j − pi j (1 − ν1 ) |e j (t − ν (t ))| p

j=1

|wi j |l j μ p−1 − qi j (1 − μ1 )e−εμ



t

t −μ(t )

|e j ( ϑ )| p d ϑ



Q. Tang and J. Jian / Neurocomputing 338 (2019) 181–190

≤

n 



eεt − ϑi − pki + eεν

i=1

+

n 

 |ei (t )| p

g(e(t − ν (t ))) = (g1 (e1 (t − ν (t ))), g2 (e2 (t − ν (t ))), . . . , gn (en (t − ν (t )))T , then (5) can be turned into

eεt pki |ei (t )| p



n 

eεt − υi − pγi + eεν

i=1

+

p ji + ε

j=1

i=1

+

n 

n 

⎧ if lT ≤ t < (l + θ )T , ⎪ ⎪ ⎪ ⎪ de(t ) ⎪ ⎪

)e(t ) +

= −( + K e(t ), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ d

e(t ) ⎪

)

⎪ = −α e(t ) − (β + ϒ e(t ) + C g(e(t )) ⎪ ⎪ dt ⎪ t ⎪ ⎨ + Dg(e(t − ν (t ))) + W t −μ(t ) g(e(ϑ ))dϑ ,



r ji + ε |

ei (t )| p

j=1

n 

eεt pγi |

ei (t )| p

i=1

≤

n 

eεt p(ki |ei (t )| p + γi |

ei (t )| p )

i=1

≤ λ pV (t ).

(14)

Combining (13), (14) and Lemma 1, one has for t ≥ 0

V (t ) ≤V (0 )eλ p(1−θ )t .

(15)

Then, from (15), we obtain

|| e(t )|| pp =

n  i=1

+

n 

j=1

 qi j

j=1

 ≤ ×



0

−μ ( 0 )

0

ϑ

1 + ν eεν max

1≤ j≤n

n 

⎛

0 −ν ( 0 )

11 ⎜ T12 ⎜ 0 ⎜ T = ⎜ ⎜ C T P2 ⎜ D P2 ⎝W T P

eεϑ |e j (ϑ )| p dϑ

eεη |e j (η )| p dηdϑ ]e−(ε−λ p(1−θ ))t



n 

 pr j

r=1

 + μ2 max

1≤ j≤n

n 



+

qr j

|ψi (κ ) − ψ i (κ )| p e−(ε−λ p(1−θ ))t

= p −(ε −λ p(1−θ ))t

| φi ( κ ) − φ i ( κ ) | e

i=1

M = 1 + ν eεν max1≤ j≤n {

where κ∈ [−h, 0], μ2 max1≤ j≤n { nr=1 qr j }. Therefore

(18)

|| e(t )|| p ≤M p |||| p e−t , 1

n

0 R1 L2 0 0

33 0 R2 L2 0

P2C R1 L2 0 μ2 P4 − R1 ∗ 0

P2 D 0 R2 L2 0 −R2 0

⎞

P2W 0 ⎟ ⎟ 0 ⎟ ⎟ ≤ 0, 0 ⎟ ⎟ 0 ⎠ −e−εμ P4

r=1

1 max q



λmax (P1 ) + νλmax (P3 )  + μ2 λmax (MP4 M ), λmax (P2 ) > 1,

(16) pr j } +

(17)

by Definition 1, the inequality (17) shows that systems (2) and (3) with the intermittent controller (4) are exponentially synchronized. The proof is completed. If denote

e(t ) = (

e1 (t ),

e2 (t ), . . . ,

en (t )) , e(t ) = (e1 (t ), e2 (t ), . . . , en (t )) , T

0 0

) + ε P ,  = −P α + P ,  = −2P ( where 11 = −2P2 (β + ϒ 2 12 2 1 22 1

) + P − R L + ε P ,  = −(1 − ν )e−εν P − R L . +K 3 1 1 1 33 1 3 2 1 (ii)

r=1

≤ M|||| pp e−(ε−λ p(1−θ ))t ,

12 22

2

i=1 n 

⎪ if (l + θ )T ≤ t < (l + 1 )T , ⎪ ⎪ ⎪ ⎪ ⎪ de(t ) ⎪ ⎪ = −e(t ) +

e(t ), ⎪ ⎪ dt ⎪ ⎪ ⎪ d

e(t ) ⎪ ⎪ = −α e(t ) − β

e(t ) + C g(e(t )) + Dg(e(t − ν (t ))) ⎪ ⎪ dt ⎪ ⎩ t + W t −μ(t ) g(e(ϑ ))dϑ .

Theorem 2. Under Assumption (A2), systems (2) and (3) with intermittent controller (4) are exponentially synchronized if there exist scalar ε > 0, four positive definite matrices P1 , P2 , P3 , P4 and two positive diagonal matrices R1 , R2 such that the following conditions hold (i)

(|ei (t )| p + |

ei (t )| p )

≤ e−εt V (t ) ≤ e−εt V (0 )eλ p(1−θ )t  n n   ≤ [|ei (0 )| p +|

ei (0 )| p +eεν pi j i=1

185

T

σ=

ε 2

−

λ q

( 1 − θ ) > 0,



), λmax (P ϒ where q= min{λmin (P1 ), λmin (P2 )}, λ= max{λmax (P1 K 2 )}. Proof. Constructing the Lyapunov functional as follows

V (t ) = eεt eT (t )P1 e(t )+eεt

eT (t )P2

e(t )+ +μ



t

t−μ



t

ϑ



t

t −ν (t )

eεϑ eT (ϑ )P3 e(ϑ )dϑ

eες gT (e(ς ))P4 g(e(ς ))dς dϑ .

(19)

 = diag{ξ1 , ξ2 , . . . , ξn },

= diag{k1 , k2 , . . . , kn }, K

= diag{γ1 , γ2 , . . . , γn }, ϒ



α = diag{α1 , α2 , . . . , αn },

β = diag{β1 , β2 , . . . , βn }, mi = max{| |, | |}, M = diag{m1 , m2 , . . . , mn },

li+

li−

D = (di j )n×n , C = (ci j )n×n , W = (wi j )n×n , g(e(t )) = (g1 (e1 (t )), g2 (e2 (t )), . . . , gn (en (t )))T ,

For lT ≤ t < (l + θ )T , computing the derivative of V(t) along the solutions of (18), we get

V˙ (t )|(18) = 2eεt eT (t )P1 e˙ (t ) + ε eεt eT (t )P1 e(t ) + 2eεt

eT (t )P2

e˙ (t ) + ε eε t

eT (t )P2

e(t )+ eεt eT (t )P3 e(t ) −(1 − ν˙ (t ))eε (t−ν (t )) eT (t − τ (t ))P3 e(t − ν (t ))  t +μ2 eεt gT (e(t ))P4 g(e(t )) − μ eεϑ gT (e(ϑ ))P4 g(e(ϑ ))dϑ



t−μ

)

)e(t ) − 2

≤ eεt − 2eT (t )P1 ( + K eT (t )P2 (β + ϒ e(t )

186

Q. Tang and J. Jian / Neurocomputing 338 (2019) 181–190

From (26), (27) and there exist κ ∈ [−h, 0], we can get

+ 2

eT (t )(−P2 α + P1 )e(t ) + 2

eT (t )P2C g(e(t )) + 2

eT (t )P2 Dg(e(t − ν (t ))  t  + 2

eT (t )P2W g(e(ϑ ))dϑ

|| e(t )||22 =

n 

|ei (t )|2 +

i=1

t −μ(t )

+μ2 eεt gT (e(t ))P4 g(e(t )) −(1 − ν1 )eε (t−ν ) eT (t − ν (t ))P3 e(t − ν (t ))  t −μ eεϑ gT (e(ϑ ))P4 g(e(ϑ ))dϑ .

i=1

−ν ( 0 )

(20)

t−μ

1 q

|

ei (t )|2 ≤ e−εt V (t )

2λ 1 ≤ V (0 )e−(ε− q (1−θ ))t q 1 ≤ [eT (0 )P1 e(0 ) +

eT (0 )P2

e (0 ) q  0 + eεϑ eT (ϑ )P3 e(ϑ )dϑ

+ ε eεt eT (t )P1 e(t ) + ε eεt

eT (t )P2

e(t ) + eεt eT (t )P3 e(t )

+μ

By Lemma 2, one can obtain



n 



0



−μ

0

ϑ

2λ eες gT (e(ς ))P4 g(e(ς ))dς dϑ ]e−(ε− q (1−θ ))t

(21)

1 ≤ [λmax (P1 ) + νλmax (P3 ) + μ2 λmax (MP4 M )] q n  2λ × |ψi (κ ) − ψ i (κ )|2 e−(ε− q (1−θ ))t

As g(e(t)) meets with (A2), there exist positive diagonal matrices R1 , R2 such that

 2λ 1 + λmax (P2 ) |φi (κ ) − φ i (κ )|2 e−(ε− q (1−θ ))t q

−μ

t

t−μ

eεϑ gT (e(ϑ ))P4 g(e(ϑ ))dϑ ≤

−eε (t−μ)





t

t −μ(t )

T 

e(t ) g(e(t ))

g(e(ϑ ))dϑ

−R1 L1 R1 L2



R1 L2 −R1

T

 P4

−R2 L1 R2 L2



t

t −μ(t )



T 

e(t − ν (t )) g(e(t − ν (t )))



g(e(ϑ ))dϑ .



e(t ) g(e(t ))

R2 L2 −R2



≥ 0,

e(t − ν (t )) g(e(t − ν (t )))



(23)





+ 2

eT (t ) − P2 α + P1 e(t ) + 2

eT (t )P2C g(e(t )) + 2

eT (t )P2 Dg(e(t − ν (t ))  t + 2

eT (t )P2W g(e(ϑ ))dϑ t −μ(t )

  +g (e(t )) μ2 P4 − R1 g(e(t )) + 2eT (t )R1 L2 g(e(t ))   + eT (t − ν (t )) − (1 − ν1 )e−εν P3 − R2 L1 e(t − ν (t )) + 2eT (t − ν (t ))R2 L2 g(e(t − ν (t ))

4. Illustrative examples t

t −μ(t )

g(e(ϑ ))dϑ



≤ eεt zT (t )z(t ) ≤ 0,

(24)

where = (

eT (t ), eT (t ), eT (t − ν (t )), gT (e(t )), gT (e(t − ν (t ))), ( tt−μ(t ) g(e(ϑ ))dϑ )T )T . Similarly, if (l + θ )T ≤ t < (l + 1 )T , one has zT (t )



e(t ) + 2eεt

V˙ (t )|(18) ≤ eεt zT (t )z(t ) + 2eεt eT (t )P1 K eT (t )P2 ϒ e(t ) ε t T ε t T



≤ 2λmax (P1 K )e e (t )e(t ) + 2λmax (P2 ϒ )e e (t )e(t ) ≤

2λ V (t ). q

(25)

2λ V (t ) ≤ V (0 )e q (1−θ )t , t ≥ 0.

(26)

Under (A2), one has

e (ς )MP4 Me(ς ) − gT (e(ς ))P4 g(e(ς )) ≥ 0.

Example 1. Consider the inertial system of model (1) as follows for n = 2:

 d2 xi (t ) dxi (t ) = −ai − bi xi (t ) + ci j f j (x j (t )) 2 dt dt 2

j=1

+

2 

(27)

di j f j (x j (t − ν (t )))

j=1

+

2  j=1

Combining (24) and (25) and using Lemma 1, one derives

T

(28)

Remark 5. Up to now, there are lots of results on the dynamical characteristics of various INNs such as [27–40] and references therein. However, to our knowledge, there are no results of exponential synchronization for INNs with mixed time-varying delays via utilizing periodically intermittent control. So, this indicates that the results presented here are novel.

T

t −μ(t )

|| e(t )||2 ≤ 2 ||||2 e−σ t .

Remark 4. From Assumption (A3), the conditions of Theorem 1 need the bounds of time-varying delays satisfying ν 1 < 1 and μ1 < 1, but Theorem 2 satisfies Assumption (A2) and do not meet ν 1 < 1 and μ1 < 1. So Theorem 1 and Theorem 2 do not embody each other.

) + ε P2

+

eT (t ) − 2P2 (β + ϒ e(t )

− gT (e(t − ν (t ))R2 g(e(t − ν (t ))  t   −e−εμ g(e(ϑ ))dϑ T P4

≤ ||||

Based on (28) and Definition 1, one knows that systems (2) and (3) with intermittent controller (4) are exponentially synchronized.





i=1

2 −(ε − 2qλ (1−θ ))t . 2e

1

≥ 0.

) + ε P1 + P3 − R1 L1 e(t ) V˙ (t )|(18) ≤ eεt eT (t ) − 2P1 ( + K



n

Therefore



Combining (20)–(23), one has



(22)

i=1

 wi j

t

t −μ(t )

f j (x j (ϑ ))dϑ + Ii (t ),

(29)

where a1 = a2 = 5, b1 = b2 = 11, c11 = d22 = w11 = −w22 = 0.3, c12 = −c21 = d11 = d21 = w12 = w21 = 0.1, c22 = − d12 = 0.2, fi (xi ) = 0.3 tanh(xi ), I1 (t ) = 2 cos(t ), I2 (t ) = 2 sin(t ), μ(t ) = 0.5 cos(t ) + 0.5 and ν (t ) = 0.4 sin2 (t ). Selecting ξ1 = ξ2 = 3, we can get α1 = α2 = 5, β1 = β2 = 2. Then (29) can be changed into

Q. Tang and J. Jian / Neurocomputing 338 (2019) 181–190

187

Fig. 2. Synchronization curves of x1 , u1 , x2 , u2 .

Fig. 3. Synchronization curves of y1 , v1 , y2 , v2 .

⎧ dxi (t ) ⎪ ⎪ ⎪ dt = −ξi xi (t ) + yi (t ), ⎪ ⎪ 2 ⎪ ⎪ dyi (t ) = −αi xi (t ) − βi yi (t ) +  ⎪ ci j f j (x j (t )) ⎨ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+ +

2 

j=1 2 

j=1

di j f j (x j (t − ν (t ))) wi j

j=1

t

t −μ(t )

that the synchronization of (30) and (31) is delay-dependent. So, Figs. 2–5 testify the validity of the results for Theorem 1. Example 2. Consider the inertial system of model (1) with n = 2

(30)

f j (x j (ϑ ))dϑ + Ii (t ),

and the response system is shown as

⎧ dui (t ) ⎪ ⎪ = −ξi ui (t ) + vi (t ) + Ki (t ), ⎪ ⎪ dt ⎪ 2 ⎪  dvi (t ) ⎪ ⎪ ⎨ dt = −αi ui (t ) − βi vi (t ) + ci j f j (u j (t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+ +

2 

j=1 2  j=1

j=1

di j f j (u j (t − ν (t ))) wi j

t

t −μ(t )

⎧ 2 d x1 (t ) dx1 (t ) ⎪ ⎪ ⎪ dt 2 = −3.5 dt − 5x1 (t ) + 0.1 f1 (x1 (t )) ⎪ ⎪ ⎪ ⎪ − 0.05 f2 (x2 (t )) − 0.05 f1 (x1 (t − ν (t ))) ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ + 0.1 f2 (x2 (t − ν (t ))) + 0.15 t −μ(t ) f1 (x1 (ϑ ))dϑ ⎪ ⎪ ⎪ t ⎪ ⎨ − 0.1 f2 (x2 (ϑ ))dϑ + I1 (t ), t −μ(t )

(31)

f j (u j (ϑ ))dϑ + Ii (t ) + ϒi (t ),

when lT ≤ t < (l + θ )T , Ki (t ) = −ki (ui (t ) − xi (t )) and ϒi (t ) = −γi (vi (t ) − yi (t )), otherwise, Ki (t ) = ϒi (t ) = 0, the parameters are the same as system (29). For p = 2, we select positive constants r11 = r12 = r21 = r22 = 0.05 and the controller parameters k1 = k2 = 2, γ1 = γ2 = 2.5. By simple calculation, we obtain l1 = l2 = 0.3, p11 = p21 = 0.05, p12 = 0.1, p22 = 0.15, q11 = q22 = 0.18, q12 = q21 = 0.06. Choose ε = 2.4 such that θ > 0.52. So the conditions of Theorem 1 here are all satisfied, then the drive-response systems are exponentially synchronized via the intermittent control. By taking the control period T = 3, the width index of the control θ = 0.6, and the error initial conditions are

x1 ( s ) = y2 (s ) = −0.6, x2 ( s ) = y1 (s ) = 0.6, s ∈ [−1, 0]. Under the intermittent control, Figs. 2 and 3 show that synchronization curves of states x1 , x2 , y1 , y2 of (30) and states u1 , u2 , v1 , v2 of (31), respectively. Fig. 4 shows time responses of the error variables e1 (t ), e2 (t ),

e1 (t ),

e2 (t ) with time-varying delays μ(t ) = 0.5 cos(t ) + 0.5 and ν (t ) = 0.4 sin2 (t ). In addition, let the time-varying delays μ(t ) = 0.5 sin2 (t ) + 0.5, .8et ν (t ) = 01+ , all the other parameters and the error initial condiet tions are the same as the above for systems (30) and (31), then it is easy to verify that the conditions of Theorem 1 hold. Under the intermittent control, Fig. 5 shows time responses of the error variables e1 (t ), e2 (t ),

e1 (t ),

e2 (t ). From Figs. 4 and 5, the synchronization time in Fig. 4 is shorter than that in Fig. 5, which indicates

(32)

⎪ d x2 (t ) dx2 (t ) ⎪ ⎪ = −3.5 − 5x2 (t ) + 0.15 f1 (x1 (t )) ⎪ ⎪ dt 2 dt ⎪ ⎪ ⎪ − 0.1 f2 (x2 (t )) − 0.1 f1 (x1 (t − ν (t ))) ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ + 0.05 f2 (x2 (t − ν (t ))) + 0.05 t −μ(t ) f1 (x1 (ϑ ))dϑ ⎪ ⎪ ⎪ t ⎩ − 0.15 t −μ(t ) f2 (x2 (ϑ ))dϑ + I2 (t ), 2

where f j (x j ) = 0.1(|x j + 1| − |x j − 1| ), I j (t ) = 1.5 sin(t ), j = 1, 2, ν (t ) = 0.3 sin(t ) + 0.8 and μ(t ) = 1.01 cos(t ) + 1.09. Choosing ξ1 = ξ2 = 2 and writing system (32) in the matrix form, we can get the corresponding matrices

 = diag{2, 2}, α = diag{2, 2}, β = diag{1.5, 1.5},

 C=

 W =











0.1 0.15

−0.05 −0.05 , D= −0.1 −0.1

0.1 , 0.05

0.15 0.05

−0.1 1.5 sin(t ) , I= . −0.15 1.5 sin(t )



The corresponding response system is shown as

⎧ du(t ) ⎪ = −u(t ) + v(t ) + K (t ), ⎪ ⎪ ⎨ dt dv(t ) = −α u(t ) − βv(t ) + C f (u(t )) + D f (u(t − ν (t ))) ⎪ ⎪ ⎪ dt t ⎩ + W t −μ(t ) f (u(ϑ ))dϑ + I (t ) + ϒ (t ),

(33)

(u(t ) − x(t )) and ϒ (t ) = where lT ≤ t < (l + θ )T , K (t ) = −K

(v(t ) − y(t )), otherwise, K (t ) = ϒ (t ) = 0, the parameters are −ϒ the same as system (32). We select the control parameters k1 = k2 = 2, γ1 = γ2 = 1.5. Using simple calculation, we obtain l1+ = l2+ = 0.2, l1− = l2− = −0.2,

188

Q. Tang and J. Jian / Neurocomputing 338 (2019) 181–190

Fig. 4. Time responses of states e1 , e2 ,

e1 ,

e2 with μ(t ) = 0.5 cos(t ) + 0.5, ν (t ) = 0.4 sin (t ). 2

Fig. 5. Time responses of states e1 , e2 ,

e1 ,

e2 with μ(t ) = 0.5 sin (t ) + 0.5, ν (t ) = 2

0.8et 1+et

.

Fig. 6. Synchronization curves of x1 , u1 , x2 , u2 .

Fig. 7. Synchronization curves of y1 , v1 , y2 , v2 .

Fig. 8. Time responses of error states e1 , e2 ,

e1 ,

e2 .

by using Matlab LMI control toolbox, the solutions can be derived as follows:



P1 =

 P3 =

1.2282 −0.0027 5.8920 0.0010



6.4249 R1 = 0





−0.0027 0.8391 , P2 = 1.2288 −0.0645





0.0010 1.0761 , P4 = 5.8874 −0.0022





0 2.5084 , R2 = 6.4415 0



−0.0645 , 0.8594



−0.0022 , 1.0813



0 . 2.5084

In view of Theorem 2, we can calculate that q = 0.7840,  = 12.2731 > 1 and acquire that the conditions of Theorem 2 hold. Therefore, the drive-response systems are exponentially synchronized via the intermittent control. By taking the control period T = 5, the width index of the control θ = 0.65, and the error initial conditions are

x1 (s ) = − x2 ( s ) = 0.7, y1 (s ) = −0.5, y2 (s ) = 0.4, s ∈ [−2.1, 0]. Under the intermittent control, Figs. 6 and 7 show that synchronization curves of states x1 , x2 , y1 , y2 of (32) and states u1 , u2 , v1 , v2 of (33), respectively. Fig. 8 implies that the curves of the time

Q. Tang and J. Jian / Neurocomputing 338 (2019) 181–190

response of the error variables e1 (t ), e2 (t ),

e1 (t ),

e2 (t ). Therefore, Figs. 6–8 indicate that the conclusion of Theorem 2 is correct. 5. Conclusions This paper have discussed the global exponential synchronization for INNs with mixed time-varying delays via periodically intermittent control. By using some inequality techniques and combining with the Lyapunov–Krasovakii functionals method, some novel delay-dependent criteria are obtained to guarantee the global exponential synchronization for the investigatory INNs. In addition, our results state that the exponential synchronization convergence rates depend on the norm, the transformation parameters, the control parameters and the width index of the control, but does not depends on the controlled width or the controlled period. Further, the method in this paper can also be extended to study the stabilization and the synchronization of discontinuous inertial neural networks, such as memristive INNs [31,35,36]. Finally, two examples are presented to expound the practicability and validity of our results. Acknowledgment The authors are grateful for the support of the National Natural Science Foundation of China (11601268). References [1] C.C. Hua, S.S. Wu, X.P. Guan, New robust stability condition for discrete-time recurrent neural networks with time-varying delays and nonlinear perturbations, Neurocomputing 219 (2017) 203–209. [2] G.D. Zhang, Z.G. Zeng, Exponential stability for a class of memristive neural networks with mixed time-varying delays, Appl. Math. Comput. 321 (2018) 544–554. [3] F.H. Zhang, Z.G. Zeng, Multistability and instability analysis of recurrent neural networks with time-varying delays, Neural Netw. 97 (2018) 116–126. [4] Q.k. Song, H.Q. Shu, Z.J. Zhao, Y.R. Liu, F.E. Alsaadi, Lagrange stability analysis for complex-valued neural networks with leakage delay and mixed time-varying delays, Neurocomputing 244 (2017) 33–41. [5] Y.C. Shi, J.D. Cao, G.R. Chen, Exponential stability of complex-valued memristor-based neural networks with time-varying delays, Appl. Math. Comput. 313 (2017) 222–234. [6] L. Zhang, Q.K. Song, Z.J. Zhao, Stability analysis of fractional-order complex-valued neural networks with both leakage and discrete delays, Appl. Math. Comput. 298 (2017) 296–309. [7] G. Velmurugan, R. Rakkiyappan, V. Vembarasan, J.D. Cao, A. Alsaedi, Dissipativity and stability analysis of fractional-order complex-valued neural networks with time delay, Neural Netw 86 (2017) 42–53. [8] S.Q. Jiang, X.B. Lu, C. Xie, S.M. Cai, Adaptive finite-time control for overlapping cluster synchronization in coupled complex networks, Neurocomputing 266 (2017) 188–195. [9] X.Y. Chen, J.H. Park, J.D. Cao, J.L. Qiu, Adaptive synchronization of multiple uncertain coupled chaotic systems via sliding mode control, Neurocomputing 273 (2018) 9–21. [10] H. Zhang, X.Y. Wang, X.H. Lin, Synchronization of complex-valued neural network with sliding mode control, J. Frankl. Inst. 353 (2016) 345–358. [11] X.W. Liu, Y. Xu, Cluster synchronization in complex networks of nonidentical dynamical systems via pinning control, Neurocomputing 168 (2015) 260–268. [12] H.L. Yang, X. Wang, S.M. Zhong, L. Shu, Synchronization of nonlinear complex dynamical systems via delayed impulsive distributed control, Appl. Math. Comput. 320 (2018) 75–85. [13] W.L. He, F. Qian, J.D. Cao, Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control, Neural Netw. 85 (2017) 1–9.

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Q. Tang and J. Jian / Neurocomputing 338 (2019) 181–190 Qian Tang received the B.S. Degree in College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, China, in 2015. She is currently a postgraduate student in College of Science, China Three Gorges University, Yichang, China. Her current research interests include behavior analysis of dynamic systems, complex-valued systems and neural networks.

Jigui Jian received the B.S. in Mathematics from China Southwest University, Chongqing, China, in 1987. His M.S. degree in Mathematics from Huazhong Normal University and his Ph.D. degree from Department of Control Science and Engineering from Huazhong University of Science and Technology, Wuhan, China, in 1994 and in 2005, respectively. He is a professor and M.S. advisor in China Three Gorges University. His present research interests are mainly in stability of dynamical systems, Fractional-order systems, nonlinear control systems and dynamics behavior of neural networks.