Effects of infinite occurrence of hybrid impulses with quasi-synchronization of parameter mismatched neural networks

Neural Networks 122 (2020) 106–116

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Effects of infinite occurrence of hybrid impulses with quasi-synchronization of parameter mismatched neural networks Rakesh Kumar a , Subir Das a , Yang Cao b , a b

∗

Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005, India Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China

article

info

Article history: Received 1 June 2019 Received in revised form 6 September 2019 Accepted 8 October 2019 Available online 25 October 2019 Keywords: Neural networks Quasi synchronization Hybrid impulses Parameter mismatch Mixed time-varying delays

a b s t r a c t This article is deeply concerned with the effects of hybrid impulses on quasi-synchronization of neural networks with mixed time-varying delays and parameter mismatches. Hybrid impulses allow synchronizing as well as desynchronizing impulses in one impulsive sequence, so their infinite time occurrence with the system may destroy the synchronization process. Therefore, the effective hybrid impulsive controller has been designed to deal with the difficulties in achieving the quasisynchronization under the effects of hybrid impulses, which occur all the time, but their density of occurrence gradually decrease. In addition, the new concepts of average impulsive interval and average impulsive gain have been applied to cope with the simultaneous existence of synchronizing and desynchronizing impulses. Based on the Lyapunov method together with the extended comparison principle and the formula of variation of parameters for mixed time-varying delayed impulsive system, the delay-dependent sufficient criteria of quasi-synchronization have been derived for two separate cases, viz., Ta < ∞ and Ta = ∞. Finally, the efficiency of the theoretical results has been illustrated by providing two numerical examples. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Over the past three to four decades, the dynamical behaviors of neural networks, such as stability, periodic attractors, bifurcation, and chaotic attractors have been extensively investigated due to their potential applications in pattern recognition, optimization, signal processing, and associative memories. Moreover, chaos synchronization of chaotic neural networks has been a fascinating problem since the pioneering work of Pecora and Carroll (1990). Till date, various types of synchronization of neural networks have been studied, such as projective synchronization (Chen & Cao, 2012), quasi-synchronization (Tang, Park, & Feng, 2018), lag-synchronization (Hu, Yu, Jiang and Teng, 2010), lag quasi-synchronization (Huang, Li, Huang, & Han, 2013) etc., due to their theoretical importance and practical applications in secure communication (Lakshmanan et al., 2018), tracking control (Yang, Feng, Feng and Cao, 2017), and image encryption (Wen, Zeng, Huang, Meng, & Yao, 2015). In most of the synchronization schemes, master and slave systems are identical. But in the real world, it is common to have parameter mismatches ∗ Corresponding author. E-mail address: [email protected] (Y. Cao). https://doi.org/10.1016/j.neunet.2019.10.007 0893-6080/© 2019 Elsevier Ltd. All rights reserved.

between the systems and their existence may slow the speed of convergence or even destroy the synchronization. Therefore, many researchers have paid their attention towards finding the implications of parameter mismatches to the synchronization process, see the references He et al. (2015), Huang et al. (2013) and Tang et al. (2018). One more thing is worth mentioning that time-delay in neural network is inevitable due to the finite-time speed of signal propagation, and finite time required for information processing, etc. Presence of time-delay in the system may result in instability or stability of the system’s trajectory depending on the value of delay. Therefore, many significant results have been found regarding the implications of time-delay on systems’ dynamics (Balasubramaniam & Vembarasan, 2012; Chen & Cao, 2012; Rahman, Blyuss, & Kyrychko, 2015; Yang, Huang and Li, 2017; Zhou & Cai, 2018; Zhu & Cao, 2010). Mainly, there are two types of time-delay, viz., discrete and distributed time-delays, which play a significant role to change the dynamical behaviors of the systems. Basically, the structure of a neural network is a spatial nature because of having so many parallel pathways with a variety of axon sizes and lengths, so it is logically meaningful to consider distributed delay in its modeling. In order to achieve synchronization, most of the systems’ networks need some external forces, namely, controller. Only

R. Kumar, S. Das and Y. Cao / Neural Networks 122 (2020) 106–116

a few networks exist in nature and the artificial world; those could achieve synchronization by adjusting their systems’ parameters (Arenas, Diaz-Guilera, Kurths, Moreno, & Zhou, 2008; Matheny et al., 2014). Many effective control functions have been developed for investigating synchronization and stability problems of neural networks, such as adaptive control (Cao, Wang, & Sun, 2007), intermittent control (Zhang, Huang, & Wei, 2011), quantized intermittent pinning control (Xu et al., 2018), integral sliding mode control (Shi, Zhu, & Qin, 2014), and impulsive control (Chen, Shi, & Lim, 2018; He, Qian, Cao, & Han, 2011; He et al., 2015; Li, Ho, & Lu, 2017; Li, Lou, Wang, & Alsaadi, 2018; Lu, Ho, & Cao, 2010; Tang et al., 2018; Yang, Huang, & Zhu, 2011; Yi, Feng, Wang, Xu, & Zhao, 2017), etc. In the viewpoint of engineering applications, the intermittent and impulsive controllers are very efficient in reducing the cost of control and the amount of transmitted information due to their discontinuity in nature. The working mechanism of both controllers could be described in two points: (i) intermittent control is activated interval-wise, i.e., in some intervals it works but in others it does not work, for example in Hu, Yu et al. (2010), Cheng et al. had applied periodically intermittent control to achieve the lag-synchronization of neural networks with mixed time-varying delays, where controller works only for periodically partitioned intervals. (ii) impulsive controller is activated only at discrete points which can be found in He et al. (2011), where the controller works only at the impulsive moments of the impulsive sequence {tk }∞ k=1 to control the states of slave system so that they get synchronized with the states of master system within a synchronization error bound. It is clear from the working mechanism of both controllers that impulsive controller is more efficient than an intermittent controller to reduce control cost and amount of transmitted information. In recent years, very progressive efforts have been reported in Chen and Cao (2012), Chen et al. (2018), He, Qian, and Cao (2017), He et al. (2011), He et al. (2015), Lu et al. (2010), Tang et al. (2018), Yang et al. (2011), Yi et al. (2017), Zhang, Li, and Huang (2017) and Zhang, Tang, Fang, and Wu (2012) to make impulsive controller more effective in investigating stability and synchronization of less conservative non-linear dynamical systems. Generally, impulses are characterized in to three categories: synchronizing impulses (|µk | < 1), desynchronizing impulses (|µk | > 1), and inactive impulses (|µk | = 1). Most of the works using impulsive controller have been done for synchronizing impulses and desynchronizing impulses, separately. There are a few articles in which synchronization or stability problems have been investigated by using synchronizing and desynchronizing impulses, simultaneously. For example, in Zhang et al. (2012), the authors have studied the stability of delayed neural networks with the effects of stabilizing (|µk | < 1) and destabilizing impulses (|µk | > 1), simultaneously. It is shown in the article that the simultaneous effects of stabilizing and destabilizing impulses could not affect the stability of delayed neural networks, adversely, if stabilizing impulses can prevail over the influence of destabilizing impulsive effects. Lu et al. (2010) have developed a new concept of average impulsive interval and studied a unified synchronization of impulsive dynamical systems with simultaneous effects of synchronizing and desynchronizing impulses. The main motivation behind the concept was the idea of average dwell time (Hespanha & Morse, 1999) and the intuition to enhance the results like (Guan, 2018; Hu, Jiang and Teng, 2010; Li & Rakkiyappan, 2013; Li & Song, 2013; Pu, Liu, Jiang, & Hu, 2015; Sheng & Zeng, 2018), which are based on supremum and infimum of impulsive intervals. Later on, many results on the synchronization of impulsive dynamical systems using average impulsive interval are published, see Chen et al. (2018), He et al. (2011), He et al. (2015), Li, Shi, and Liang (2019), Lu et al. (2010),

107

Tang et al. (2018), Yang et al. (2011), Yang, Lu, Ho, and Song (2018) and Yi et al. (2017). Unfortunately, in Chen et al. (2018), He et al. (2011), He et al. (2015), Li et al. (2019), Lu et al. (2010), Tang et al. (2018), Yang et al. (2011) and Yi et al. (2017), authors could not discuss the effects of hybrid impulses over synchronization scheme when impulses occur infinitely, but sparsely, i.e., Ta = ∞. In the real world, there exists an impulsive sequence in which impulses occur all the time, but the length of their impulsive intervals increases with time. Such type of impulses does not essentially adversely affect the synchronization of the coupled neural networks. Inspired from this fact, authors in Wang, Li, Lu, and Alsaadi (2018) proposed two new concepts of average s and average impulsive impulsive interval Ta = limt →∞ N t − (t ,s) |µ1 |+|µ2 |+···+|µN

ζ

(t ,s) |

ζ , and derived the unified gain µ = limt →∞ Nζ (t ,s) synchronization criteria for an array of coupled neural networks with hybrid impulses under the influence of Ta = ∞. Motivated from all the discussions mentioned above, in this article, the quasi-synchronization of different neural networks with mixed time-varying delays has been investigated for Ta = ∞ using the hybrid impulsive controller. The new concepts proposed in Wang et al. (2018) are adopted in this article to deal with the situation of Ta = ∞ and the simultaneous effects of synchronizing and desynchronizing impulses. The main contributions of this article can be described in the following points. (1) Different from Tang et al. (2018), we have designed a hybrid impulsive controller for time-varying impulses. Further, for dealing with the simultaneous effects of different types of impulses (|µk | < 1 and |µk | > 1), the concept of average impulsive gain has been adopted from Wang et al. (2018). (2) We have successfully applied the modified version of the average impulsive interval to study the case of infinite but sparse occurrence of impulses, i.e., the situation when impulsive interval Ta = ∞, on quasi-synchronization of neural networks with parameter mismatches and mixed time-varying delays. (3) Using some mathematical techniques and the extended comparison principle for time-varying delayed impulsive differential equation combined with the formula for the variation of parameters, the delay-dependent criteria for quasi synchronization of the neural networks with mixed time-varying delays and parameter mismatches have been derived for Ta < ∞ and Ta = ∞. Also, the small domains of convergence containing the origin have been obtained into which the trajectories of the controlled neural networks are converging globally exponentially at the respective rate of convergence. The remaining portion of this article is organized as follows. In Section 2, the problem formulation for the models of neural networks is done and then some important preliminaries, such as definitions, assumptions, and lemmas which are needed to prove the main results of this article, are listed. For two different cases, one for Ta < ∞ and another for Ta = ∞, the sufficient criteria of quasi-synchronization between different neural networks with mixed time-varying delays are derived in Section 3. Two examples are considered in Section 4 to validate the theoretical results proposed in this article. Finally, the overall conclusions of this work have been drawn in Section 5. Notations: The following standard notations will be used in this article. R is a set of real numbers. Rn denotes the Euclidean space of column vectors of dimension n. λmax (.) presents the largest eigenvalue of a square matrix. Rn×n denotes the Euclidean space of square matrices of order n. The notation ‘‘T ’’ means transpose√ of a matrix or a vector. ∥.∥ is 2-norm which is defined

∑n

√

2 as ∥y∥ = λmax (AT A) 1=1 yi for a column vector y and ∥A∥ = for a square matrix A. [t ] indicates the least integer of number less than or equal to t.

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R. Kumar, S. Das and Y. Cao / Neural Networks 122 (2020) 106–116

Assumption 2. The trajectory of Eq. (2) is bounded, i.e., there exists M > 0 such that ∥x(t)∥ ≤ M , ∀t ∈ [−σ , +∞].

2. Problem formulation and some preliminaries Consider a neural network with mixed time-varying delays whose state equation is described as follows: x˙ i (t) = − a¯ i xi (t) +

n ∑

b¯ ij f1j (xj (t)) +

j=1

+

n ∑

d¯ ij

n ∑

Lemma 2 (Gu, 2000). For any positive constant matrix P ∈ Rn×n , P = P T , a scalar value α > 0, and a vector valued function F : [0, α] → Rn , the following integral inequality is well-defined

c¯ij f2j (xj (t − σ1 (t)))

j=1

t

∫

f3j (xj (s)) ds + Ii ,

(∫

t −σ2 (t)

j=1

(1)

¯ ¯ 1 (x(t)) + C¯ f2 (x(t − σ1 (t))) x˙ (t) = − Ax(t) + Bf t

f3 (x(s))ds + I ,

x(t) =φ (t) ∈ C ([−σ , 0], R ),

(2)

where n denotes the number of neurons in the network; x(t) = [x1 (t), x2 (t), . . . , xn (t)]T ∈ Rn is the state vector associated with the neurons at time t; A¯ = diag(a¯ 1 , a¯ 2 , . . . , a¯ n ) > 0; B¯ = (b¯ ij )n×n ∈ Rn×n is the connection weights matrix of the neurons at time t; ¯ = (d¯ ij )n×n ∈ Rn×n are the matrices of C¯ = (c¯ij )n×n ∈ Rn×n and D weights of connections among the neurons with and without delay, respectively; f1 (x(t)) = [f11 (x1 (t)), f12 (x2 (t)), . . . , f1n (xn (t))]T ∈ Rn , f2 (x(t − σ1 (t))) = [f21 (x1 (t − σ1 (t))), . . . , f2n (xn (t − σ1 (t)))]T ∈ Rn and f3 (x(t)) = [f31 (x1 (t)), f32 (x3 (t)), . . . , f3n (xn (t))]T ∈ Rn are activation functions of the neurons at time t, t − σ1 (t) and distributed time delay t − σ2 (t) respectively, such that f1 (0) = 0, f2 (0) = 0 and f3 (0) = 0; σ = max{σ1 , σ2 }, where σ1 and σ2 are upper bounds of σ1 (t) and σ2 (t) respectively; the initial condition of Eq. (1) is φ (s) = [φ1 (s), φ2 (s), . . . , φn (s)]T belongs to the set of continuous functions C ([−σ , 0], Rn ); the external input value to the network is denoted by I = [I1 , I2 , . . . , In ]T . Now, we will consider some assumptions, lemmas, and definitions which will be used throughout the article to achieve quasi-synchronization of neural networks under hybrid impulses. Assumption 1. For any u1 , u2 ∈ R and i = 1, 2, . . . , n, the continuous activation functions f1 (.), f2 (.) and f3 (.) satisfy the following conditions: 0≤ 0≤ 0≤

f1i (u1 ) − f1i (u2 ) u1 − u2 f2i (u1 ) − f2i (u2 ) u1 − u2 f3i (u1 ) − f3i (u2 ) u1 − u2

P

)

0

0

α

∫

≤α

F (s)ds

F T (s)PF (s)ds.

(3)

0

˜ ˜ 1 (y(t)) + C˜ f2 (y(t − σ1 (t))) y˙ (t) = − Ay(t) + Bf

t −σ2 (t) n

α

To formulate the problem of this article, we need another state equation of neural network whose connection weights matrices are different from Eq. (2). Consider the state equation of the slave system as

In a compact form it can be re-written as

∫

)T (∫

α

F (s)ds

xi (t) =φi (t) ∈ C ([−σ , 0], R), i = 1, 2, . . . , n.

+ D¯

For any X , Y ∈ Rn , we have 2X T Y ≤ X T X + Y T Y .

Lemma 1.

+ D˜

t

∫

f3 (y(s))ds + U(t) + J , t −σ2 (t)

y(t) =ϕ (t) ∈ C ([−σ , 0], Rn ),

(4)

where y(t) = [y1 (t), y2 (t), . . . , yn (t)] is the state vector of the networks’ neurons with initial condition y(t) = ϕ (t) = [ϕ1 (t), ϕ2 (t), . . . , ϕn (t)]T ∈ C ([−σ , 0], Rn ); A˜ = diag(a˜ 1 , a˜ 2 , . . . , a˜ n ) > ˜ = (d˜ij )n×n ∈ 0; B˜ = (b˜ij )n×n ∈ Rn×n , C˜ = (c˜ij )n×n ∈ Rn×n and D Rn×n are the connection weights matrices among the neurons of the network without and with delays; J = [J1 , J2 , . . . , Jn ]T is column vector of external input values to the network; U(t) is the hybrid impulsive control function which will be discussed later. In order to investigate the quasi-synchronization between the systems’ network, the error neural network which is defined as e(t) = y(t) − x(t) has been constructed by using Eqs. (4) and (2). Derivative of the error neural network can be written as T

˜ e˙ (t) = −Ae(t) + B˜ fˆ1 (e(t)) + C˜ fˆ2 (e(t − σ1 (t))) + D˜

∫

t

fˆ3 (e(s))ds + R(x(t), σ1 (t), σ2 (t)) + U(t),

(5)

t −σ2 (t)

where fˆ1 (e(t)) = f1 (y(t)) − f1 (x(t)), fˆ2 (e(t − σ1 (t))) = f2 (y(t − σ1 (t))) − f2 (x(t − σ1 (t))), fˆ3 (e(s)) = f3 (y(s)) − f3 (x(s)), and

¯ ¯ 1 (x(t)) R(x(t), σ1 (t), σ2 (t)) =(−A˜ + A)x(t) + (B˜ − B)f + (C˜ − C¯ )f2 (x(t − σ1 (t))) ∫ t ¯ + (D˜ − D) f3 (x(s))ds + (J − I).

≤ lf1i , ∀u1 , u2 ∈ R , ≤ lf2i , ∀u1 , u2 ∈ R ,

(6)

t −σ2 (t)

From Assumption 1, we have

≤ lf3i , ∀u1 , u2 ∈ R ,

where Lf1 = diag(lf11 , lf12 , . . . , lf1n ) > 0, Lf2 = diag(lf21 , lf22 , . . . , lf2n ) > 0, and Lf3 = diag(lf31 , lf32 , . . . , lf3n ) > 0 are positive diagonal matrices. Remark 1. The projective synchronization between the systems (4) and (2) was investigated using the linear feedback controller in Chen and Cao (2012) but the Lipschitz condition imposed on activation functions is conservative as compare to the condition given in Assumption 1. Therefore, the results of this article are more general than the existing results in Chen and Cao (2012). Remark 2. Throughout the article it is assumed that both delays satisfy 0 ≤ σ1 (t) ≤ σ1 and 0 ≤ σ2 (t) ≤ σ2 . There is no need to impose differentiability condition on σ1 (t) and σ2 (t). Thus, restrictions imposed on both type of delays are less conservative.

√ λmax (LTf1 Lf1 )M , √ ∥f2 (x(t − σ1 (t)))∥ ≤ λmax (LTf2 Lf2 )M , √ ∥f3 (x(s))∥ ≤ λmax (LTf3 Lf3 )M .

∥f1 (x(t))∥ ≤

(7) (8) (9)

Now, using Assumption 2 and taking norm ∥.∥ on both sides of the equality (6), we get

∥R(x(t), σ1 (t), σ2 (t))∥ ≤∥A¯ − A˜ ∥∥x(t)∥ + ∥B˜ − B¯ ∥∥f1 (x(t))∥ + ∥C˜ − C¯ ∥∥f2 (x(t − σ2 (t)))∥ + ∥D˜ − D¯ ∥ ∫ t × ∥f3 (x(s))∥ds + ∥J − I ∥ t −σ2 (t)

≤(∥A¯ ∥ + ∥A˜ ∥)M + (∥B˜ ∥ + ∥B¯ ∥) √ × λmax (LTf1 Lf1 )M + (∥C˜ ∥ + ∥C¯ ∥)

R. Kumar, S. Das and Y. Cao / Neural Networks 122 (2020) 106–116

√ × λmax (LTf2 Lf2 )M + (∥D˜ ∥ + ∥D¯ ∥) √ × λmax (LTf3 Lf3 )M σ2 + ∥J ∥ + ∥I ∥.

Definition 3 (Wang et al., 2018, 2019 Average Impulsive Gain).The average impulsive gain of all impulses {µ1 , µ2 , . . .} is designed as (10)

It is clear from the inequality (10) that ∥R(x(t), σ1 (t), σ2 (t))∥ is bounded ∀t ≥ −σ . Suppose that Ξ = supt ≥−σ ∥R(x(t), σ1 (t), σ2 (t))∥, where Ξ < ∞. For achieving the quasi-synchronization between the systems (4) and (2), a hybrid impulsive controller is designed as Ui (t) = −γi ei (t) +

∞ ∑ (

) µk ei (t) − ei (t) δ (t − tk ), i = 1, 2, . . . , n,

k=1

(11) where Γ = diag(γ1 , γ2 , . . . , γn ) ≥ 0 and µk is impulsive strength at point t = tk of impulsive sequence ζ = {t1 , t2 , . . . , tk }. The impulsive sequence is strictly increasing and unbounded above, i.e., t1 < t2 < t3 < · · · < tk−1 < tk < · · · and limk→∞ tk = +∞. The function δ (.) is the Dirac delta function. Substituting the control function (11) into the error neural network (5), we obtain

⎧ ⎪ ⎨e˙ (t) =

−(A˜ + Γ )e(t) + B˜ fˆ1 (e(t)) + C˜ fˆ2 (e(t − σ1 (t))) ∫t + D˜ t −σ (t) fˆ3 (e(s))ds + R(x(t), σ1 (t), σ2 (t)), t ̸= tk , ⎪ ⎩e(t + ) = µ e(t − 2), t = t , k = 1, 2, . . . , n. k k k k (12) The solution of impulsive delayed system (12) is assumed to be right-hand continuous and left-hand discontinuous at t = tk (k = 1, 2, . . . , n), i.e., e(tk+ ) = e(t) ̸ = e(tk− ). This means that the solution of (12) exhibits jump kind of discontinuities from the left side of t = tk . For simplicity, throughout the article we will write R(x(t), σ1 (t), σ2 (t)) as R(.). Due to the presence of parameter mismatches between the neural networks (4) and (2), the equilibrium point of the impulsive differential equation (12) cannot be zero. Therefore, the complete synchronization is not possible, but it is found that the states of systems can be synchronized up to a small synchronization error bound by using effective controller. This type of synchronization is called quasi-synchronization. Definition 1. The synchronization between the systems (4) and (2) is said to be quasi-synchronization with a small error bound e¯ > 0 if there exists a compact set ∆ = {e ∈ Rn : ∥e(t)∥ ≤ e¯ } in to which the trajectory of the error system (12) converges globally exponentially as time t → ∞. Definition 2 (Wang et al., 2018; Wang, Lu, Liang, Cao, & Perc, 2019 Average Impulsive Interval).The average impulsive interval Ta of the impulsive sequence ζ = {t1 , t2 , t3 , . . .} is defined as t −s

Ta = lim

t →∞

Nζ (t , s)

,

(13)

where Nζ (t , s) is the number of impulsive instants of impulsive sequence ζ on the interval (t , s). Remark 3. The concept of average impulsive interval was first introduced in Lu et al. (2010) to deal with the occurrence of different types of impulses. Unfortunately, most of the results published in the literature using tT−s − N0 ≤ Nζ (s, t) ≤ tT−s + N0 a a are restricted to Ta < ∞. More precisely infinite occurrence of impulses has not been considered in the previous results. Practically, there may be the situation when Ta = ∞. Keeping this situation in mind, the authors in Wang et al. (2018) had developed the new concept of average impulsive interval Ta = s limt →∞ N t − to use it for more generalized impulsive sequence. (t ,s) ζ

109

µ = lim

|µ1 | + |µ2 | + · · · + |µNζ (t ,s) | Nζ (t , s)

t →∞

> 0,

(14)

where µk for all k = 1, 2, . . . , n denotes impulsive gain at t = tk . Remark 4. There are three types of impulses classified on their behavior towards synchronization of the networks. (i) Desynchronizing impulses (|µk | > 1) which often potentially destroy the synchronization because they enlarge the difference between states of the systems. (ii) Synchronizing impulses (|µk | < 1) which can enhance synchronization because they reduce the difference between states of the systems. (iii) Inactive impulses (|µk | = 1) which neither affect synchronization negatively nor positively. Most of the existing results on effects of impulses to synchronization are either based on |µk | > 1 or |µk | < 1. Inspired from Lu et al. (2010), the authors in Wang et al. (2018) have developed the new concepts of average impulsive gain µ = limt →∞

|µ1 |+|µ2 |+···+|µNζ (t ,s) |

Nζ (t ,s) s limt →∞ N t − ζ (t ,s)

> 0 and average impulsive interval

to deal with infinite occurrence of hybrid Ta = impulses which occur infinitely but sparsely. To the best of our knowledge, there is not any result concerned with effects of the infinite occurrence of hybrid impulses to quasi-synchronization. In the present article, it is found that using the hybrid impulsive controller (11) we can allow Ta = ∞ for quasi-synchronization of neural networks with mixed time-varying delays. This enhances the novelty of this article. Lemma 3 (He et al., 2017). If there exists a positive constant c such that

{

D+ (u(t)) ≤ G(t , u(t), u(t − σ1 (t))) + c u(tk ) ≤ Ik (u(tk )), k ∈ N −

∫t t −σ2 (t)

u(s)ds, t ̸ = tk ,

(15) and

{

D+ (v (t)) > G(t , v (t), v (t − σ1 (t))) + c

v (tk ) ≥ Ik (v (tk− )), k ∈ N,

∫t t −σ2 (t)

v (s)ds, t ̸= tk , (16)

where G(t , u, u¯ 1 ) : R × R × R → R is non-decreasing in u¯ 1 for any fixed (t , u) and Ik (u) : R → R is non-decreasing in u. Then u(t) ≤ v (t), ∀t ∈ [−σ , 0] implies that u(t) ≤ v (t) for t > 0, where u(t +h)−u(t) D+ (u(t)) = limh→0+ sup . h +

3. Main results In this section, we have studied the influence of hybrid impulses when Ta < ∞ on quasi-synchronization of neural networks (4) and (2) under hybrid impulsive controller (11). Furthermore, we have considered the case of Ta = ∞ and derived sufficient criteria for quasi-synchronization. 3.1. Quasi-synchronization criteria for Ta < ∞ Theorem 1. Suppose that the activation functions f1 (.), f2 (.) and f3 (.) satisfy Assumption 1. The controlled error neural network (12) converges globally exponentially at the convergence rate λ2 > 0 into ¯ 1 containing the origin if a small compact domain ∆

ξ1 + 2

ln µ Ta

+ δ + ξ2 + ξ3 σ2 < 0,

(17)

110

R. Kumar, S. Das and Y. Cao / Neural Networks 122 (2020) 106–116

{

} { × V (e(t)) + λmax LTf2 Lf2 V (e(t − σ1 (t))) { }∫ t + λmax σ2 LTf3 Lf3 V (e(s))ds + ∥R(.)∥2 .

}

˜ T D˜ + LT Lf , ξ2 = where ξ1 = λmax 2(−A˜ − Γ ) + B˜ T B˜ + C˜ T C˜ + D f 1 1

{ } { } λmax LTf2 Lf2 , ξ3 = λmax σ2 LTf3 Lf3 and λ is a unique solution of λσ2 (t) −1

ln µ

equation λ + 2 T + ξ1 + δ + ξ2 eλσ1 (t) + ξ3 e a domain of convergence is

λ

= 0. The compact

{

}

{ ¯ 1 = e(t) ∈ Rn : ∥e(t)∥ ≤ √ ∆ ( − ξ1 +

Ξ . ) 2 ln µ − ξ − δ − ξ σ 2 3 2 Ta

˜ T D˜ + Suppose that ξ1 = λmax 2(−A˜ − Γ ) + B˜ T B˜ + C˜ T C˜ + D

}

{

LTf Lf1 , ξ2 = λmax LTf Lf2 1

t ̸ = tk ,

(18)

where e(t) = [e1 (t), e2 (t), . . . , en (t)] is the state vector of impulsive system (12). The Dini derivative of Eq. (18) with respect to t along trajectories of the controlled error neural network (12) could be written as

}

2

{

}

and ξ3 = λmax σ2 LTf Lf3 . Then, we have 3

D (V (e(t))) ≤ ξ1 V (e(t)) + ξ2 V (e(t − σ1 (t))) +

Proof. Suppose the Lyapunov function candidate is constructed as follows: V (e(t)) = eT (t)e(t),

(27)

t −σ2 (t)

+ ξ3

t

∫

V (e(s))ds + ∥R(.)∥2 .

(28)

t −σ2 (t)

For t = tk , the Lyapunov function candidate will be

T

D+ (V (e(t))) = 2eT (t)e˙ (t).

(19)

=µ2k eT (tk− )e(tk− ) V (e(tk+ )) =µ2k V (e(tk− )).

(29)

Based on (28) and (29), the following impulsive system with distributed delay can be derived by using the comparison principle as ⎧ ∫t 2 ⎪ ⎪ ⎨z˙ (t) = ξ1 z(t) + ξ2 z(t − σ1 (t)) + ξ3 t −σ2 (t) z(s)ds + ∥R(.)∥ + ϵ, t ̸= tk ,

From system (12), we have D+ (V (e(t))) =2eT (t)(−A˜ − Γ )e(t) + 2eT (t)B˜ fˆ1 (e(t))

+ 2eT (t)C˜ fˆ2 (e(t − σ1 (t))) + 2eT (t)D˜ ∫ t × fˆ3 (e(s))ds + 2eT (t)R(.).

V (e(tk+ )) =eT (tk+ )e(tk+ )

(20)

t −σ2 (t)

z(tk+ ) = µ2k z(tk− ), ⎪ ⎪ ⎩z(s) = ∥ϕ (s) − φ (s)∥2 ,

t = tk ,

− σ ≤ s ≤ 0,

Using Lemma 1, we could find these inequalities as

(30)

˜ 2e (t)B˜ fˆ1 (e(t)) ≤ eT (t)B˜ T Be(t) + f1ˆT (e(t))fˆ1 (e(t)), T T 2e (t)C˜ fˆ2 (e(t − σ1 (t))) ≤ e (t)C˜ T C˜ e(t)

(21)

+ f2ˆT (e(t − σ1 (t))) × fˆ2 (e(t − σ1 (t))), ∫ t T ˜ ˜ T De(t) ˜ 2e (t)D fˆ3 (e(s))ds ≤ eT (t)D

(22)

T

z(t) = W (t , 0)z(0) +

t −σ2 (t)

(∫

t

fˆ3 (e(s))ds

+

)T (∫

t −σ2 (t)

fˆ3 (e(s))ds ,

T

(23)

+ ξ3

+ f1ˆT (e(t))fˆ1 (e(t)) + eT (t)C˜ T C˜ e(t) ˜ + f2ˆT (e(t − σ1 (t)))fˆ2 (e(t − σ1 (t))) + eT (t)D˜ T De(t) (∫ )T t

fˆ3 (e(s))ds

t

) fˆ3 (e(s))ds

z(r)dr + ∥R(.)∥ + ϵ

ds, t ≥ 0,

(31)

where W (t , s) is the Cauchy matrix of the following linear impulsive system z˙ (t) = ξ1 z(t), t ̸ = tk , z(tk+ ) = µ2k z(tk− ), t = tk , k = 1, 2, . . . , n. From Definitions 2 and 3, the Cauchy matrix can be calculated as W (t , s) =µ21 µ22 ...µ2Nζ (t ,s) eξ1 (t −s)

( |µ1 | + |µ2 | + · · · + |µN (t ,s) | )2Nζ (t ,s) ζ ≤ eξ1 (t −s) Nζ (t , s)

t −σ2 (t)

×

] 2

s−σ2 (s)

{

˜ D+ (V (e(t))) ≤2eT (t)(−A˜ − Γ )e(t) + eT (t)B˜ T Be(t)

(∫

ξ2 z(s − σ1 (s))

s

∫

(24)

Substituting the inequalities (21), (22), (23), and (24) in Eq. (20), we get

+

W (t , s) 0

t −σ2 (t)

T

[

t

∫

)

t

2e (t)R(.) ≤ e (t)e(t) + R (.)R(.). T

where z(t) is a unique solution for all ϵ > 0. From Lemma 3, it is concluded that V (t) ≤ z(t) for all t ≥ 0. By employing the extended formula for variation of parameters, z(t) could be written as

+ eT (t)e(t) + RT (.)R(.).

t −σ2 (t)

=e

2Nζ (t ,s) ln

(25) 2 ln

From Assumption 1 and Lemma 2, we have

˜ D+ (V (e(t))) ≤eT (t)2(−A˜ − Γ )e(t) + eT (t)B˜ T Be(t) + eT (t)LTf1 Lf1 e(t) + eT (t)C˜ T C˜ e(t) ˜ + eT (t − σ1 (t))LTf2 Lf2 e(t − σ1 (t)) + eT (t)D˜ T De(t) ∫ t + σ2 LTf3 Lf3 eT (s)e(s)ds + ∥R(.)∥2 (26) t −σ2 (t) { } ≤λmax 2(−A˜ − Γ ) + B˜ T B˜ + C˜ T C˜ + D˜ T D˜ + LTf1 Lf1

=e

|µ1 |+|µ2 |+···+|µN (t ,s) | ζ Nζ (t ,s)

eξ1 (t −s)

|µ1 |+|µ2 |+···+|µN (t ,s) | ζ Nζ (t ,s) (t −s) t −s Nζ (t ,s) eξ1 (t −s)

2 ln

From (13) and (14), we have limt →∞

.

|µ1 |+|µ2 |+···+|µN (t ,s) | ζ Nζ (t ,s) t −s Nζ (t ,s)

=

2 ln µ . Ta

That is, for any µ > 1 or µ = 1 or µ < 1, there exists a sufficiently large T > 0 for given δ > 0 such that (

W (t , s) ≤ e

2 ln µ +δ+ξ1 Ta

) (t −s)

, t > T.

(32)

R. Kumar, S. Das and Y. Cao / Neural Networks 122 (2020) 106–116

( ) ϵ + ∥R(.)∥2 eλσ2 − 1 α¯ t ∗ σ1 ≤e η+ + ξ2 η e + ξ3 η Ω λ ( ) ∗ ¯ )t ) 1 e(α+λ ϵ + ∥R(.)∥2 ( × − + ξ 2 + ξ 3 σ2 + Ω α¯ + λ α¯ + λ Ω ] ( ∗)

Substituting inequality (32) into the integral equation (31), we get (

z(t) ≤ ∥ϕ (0) − φ (0)∥2 e

2 ln µ +δ+ξ1 Ta

)

(

t

∫

t

e

+

2 ln µ +δ+ξ1 Ta

) (t −s)

0

[ × ξ2 z(s − σ1 (s)) + ξ3

]

s

∫

z(r)dr + ∥R(.)∥ + ϵ ds 2

s−σ2 (s)

(

≤ ηe [

2 ln µ +δ+ξ1 Ta

)

(

t

∫

t

e

+

× ξ2 z(s − σ1 (s)) + ξ3

∗

z(r)dr + ∥R(.)∥ + ϵ ds, 2

where η = sup−σ ≤s≤0 ∥ϕ (s) −φ (s)∥ . Define ψ (λ) = λ+ λσ2 −1

ξ1 + ξ2 eλσ1 + ξ3 e

a

∗

2 ln µ Ta

+δ+

eλσ2 (λσ2 −1)+1

0, ψ (+∞) > 0 and ψ ′ (λ) = 1 + ξ2 σ1 eλσ1 + ξ3 > 0, λ2 this implies that ψ (λ) = 0 has a unique solution λ > 0. ln µ Since λ > 0, ϵ > 0 and −ξ1 − 2 T − δ − ξ2 − ξ3 σ2 > 0, then a we have

−σ ≤t ≤0

ϵ + ∥R(.)∥2 −ξ1 − 2 lnTaµ − δ − ξ2 − ξ3 σ2

, − σ ≤ t ≤ 0. (34)

Using the mathematical principle: proof by contradiction, we will proceed to show the inequality (34) is true for all t > 0. That is,

ϵ + ∥R(.)∥2 −ξ1 − 2 lnTaµ − δ − ξ2 − ξ3 σ2

,

t > 0.

(35)

ϵ + ∥R(.)∥ −ξ1 −

ln µ 2 T a

2

− δ − ξ 2 − ξ 3 σ2

ϵ + ∥R(.)∥2 −ξ1 − 2 lnTaµ − δ − ξ2 − ξ3 σ2

For the sake of simplicity assume α¯ =

.

(36)

, t < t ∗.

2 ln µ Ta

(37)

+ δ + ξ1 and Ω = − δ − ξ2 − ξ3 σ2 . Further, considering the inequalities (33), (37) and equation ψ (λ) = 0, we have [ ∫ t∗ ( ϵ + ∥R(.)∥2 ) ∗ ∗ z(t ∗ ) <ηeα¯ t + eα¯ (t −s) ξ2 ηe−λ(s−σ1 (s)) + Ω 0 ] ∫ s ( ϵ + ∥R(.)∥2 ) −λr 2 + ξ3 ηe + dr + ∥R(.)∥ + ϵ ds Ω s−σ2 (s) [ ∫ t∗ ϵ + ∥R(.)∥2 α¯ t ∗ σ1 ¯ )s
ln µ 2 T a

t

× 0

e−α¯ s

s

t

drds + (∥R(.)∥2 + ϵ )

s−σ2 (s)

− δ − ξ2 − ξ3 σ2

.

(38)

V (e(t)) = eT (t)e(t) ≤ z(t) < ηe−λt +

∥R(.)∥2 −ξ1 −

2 ln µ Ta

− δ − ξ2 − ξ3 σ2

.

(39) Further, the inequality (39) could be written as

∥R(.)∥2 −ξ1 −

2 ln µ Ta

− δ − ξ2 − ξ3 σ2

,

(40)

which implies

∥e(t)∥ ≤

√

Ξ

λ

η e− 2 t + √

−ξ1 −

2 ln µ Ta

− δ − ξ2 − ξ3 σ2

.

(41)

It is clear from (41) that trajectory of the impulsive system (12) converges globally exponentially { at the convergence rate 2

=

¯1 = into a small compact set ∆ Ξ

√

−ξ1 − 2 Tln µ −δ−ξ2 −ξ3 σ2

}

e(t) ∈ Rn : ∥e(t)∥ ≤ e¯

as t → ∞. That is, the quasi-synchroni-

a

But the inequality (35) still holds as z(t) < ηe−λt +

2 ln µ Ta

It is obvious that the inequality (38) is contradicting the assumption (36). Thus, the inequality (38) is true for all t > 0. Let ϵ → 0, then we have

λ

Let the inequality (35) does not hold ∀t > 0, then there exist t ∗ > 0 such that ∗

−ξ1 −

∥e(t)∥2 ≤ ηe−λt +

−σ ≤t ≤0

z(t) ≥ ηe−λt +

ϵ + ∥R(.)∥2 ∗ − eα¯ t Ω ϵ + ∥R(.)∥2

∗

<ηe−λt +

. It is easy to verify that ψ (λ) is continuous λ ln µ function. Furthermore, ψ (0) = ξ1 + 2 T + δ + ξ2 + ξ3 σ2 <

z(t) < ηe−λt +

α¯

]

s

2

<ηe−λt +

e−α¯ t

≤ηeα¯ t + ηe−λt +

(33)

z(t) ≤ sup ∥ϕ (t) − φ (t)∥ ,

−

(t −s)

s−σ2 (s)

2

α¯

)

0

∫

1

×

2 ln µ +δ+ξ1 Ta

111

[

0

e−α¯ s ds

zation between the systems (2) and (4) is achieved with a small synchronization error bound e¯ under the impulsive control (11) with the average impulsive interval Ta < ∞. This completes the proof. □ Remark 5. The time delays in neural network affect the impulsive synchronization negatively. If we do not consider distributed time-varying delay in neural networks (2) and (4), then for any ln µ fixed δ > 0, the inequality (17) becomes ξ1 + 2 T + δ + ξ2 < 0. a Furthermore, in the case of delay free systems, the inequality (17) ln µ could be written as ξ1 + 2 T + δ < 0. Compared the delay free a

case with the delay case (17), it is seen that the less value of coupling strength γi in ξ1 and large value of average impulsive interval Ta are required, that means the cost of control in delay free case is lesser than the delayed case. 3.2. Quasi-synchronization criteria for Ta = ∞ In this subsection, we will study the case Ta = ∞ on quasisynchronization of neural √ networks. In fact, this situation will arise when Nζ (t , s) = [ t − s] in (13) which implies that the impulses occur all the time but their density of occurrence decreases as time goes to infinity. The study of this case on quasisynchronization is the first time in literature. Theorem 2.

Suppose that Assumption 1 holds and Ta = ∞. { ˜ T D˜ For the constants ξ1 = λmax 2(−A˜ − Γ ) + B˜ T B˜ + C˜ T C˜ + D

112

R. Kumar, S. Das and Y. Cao / Neural Networks 122 (2020) 106–116

}

{

}

{ } + LTf1 Lf1 , ξ2 = λmax LTf2 Lf2 , and ξ3 = λmax σ2 LTf3 Lf3 , we have two cases as follows: Case 1: For any δ ′ > 0 and µ > 1, the controlled error neural network (12) globally exponentially converges at the convergence ′ rate λ2{ > 0 into a small compact domain }containing the origin

¯ 2 = e(t) ∈ Rn : ∥e(t)∥ ≤ ∆ δ′ 2

Ξ

√

′

− δ2 −ξ1 −ξ2 −ξ3 σ2

if

+ ξ1 + ξ2 + ξ3 σ2 < 0,

(42)

where λ is a unique root of the equation λ + ′

λ′ σ2 − 1 λ′

δ′ 2

λ′ σ

+ ξ1 + ξ2 e

1

+

= 0.

Case 2: For 0 < µ ≤ 1, the solution of the controlled impulsive system (12) globally exponentially converges at the convergence rate λ′′ ¯3 = > 0 into a small compact domain containing the origin ∆ 2

{

e(t) ∈ Rn : ∥e(t)∥ ≤

√

Ξ −ξ1 −ξ2 −ξ3 σ2

}

if

ξ1 + ξ2 + ξ3 σ2 < 0,

(43)

where λ′′ is a unique root of the equation λ′′ + ξ1 + ξ2 e ′′ eλ σ2 −1 3 λ′′

ξ

λ′′ σ

1

W (t , s) =µ21 µ22 ...µ2Nζ (t ,s) eξ1 (t −s)

=e

|µ1 |+|µ2 |+···+|µN (t ,s) | ζ Nζ (t ,s)

=e

lim

t →∞

.

(44)

= 0,

it means that for any δ ′ > 0 there exists a sufficiently large T > 0 such that |µ1 |+|µ2 |+···+|µNζ (t ,s) | Nζ (t ,s) t −s Nζ (t −s)

≤

δ′ 2

,

( ′ δ 2

) +ξ1 (t −s)

t > T.

(45)

z(t) ≤ e

2

) +ξ1 t

η+

+ ξ3

.

(46)

z(t) ≤ e

t

∫

e

( ′ δ 2

) +ξ1 (t −s)

λ′ 2

> 0 as

(49)

ξ1 t

η+

[

t

∫

e

ξ1 (t −s)

ξ2 z(s − σ1 (s))

∫

]

s

z(r)dr + ∥R(.)∥ + ϵ 2

ds.

(50)

s−σ2 (s)

˜ (λ′′ ) = λ′′ + ξ1 + Similar to Case 1, define a continuous function ψ λ′′ σ

ξ2 eλ σ1 + ξ3 e λ2′′ −1 . From (43), ψ˜ (0) < 0. It is easy to verify that ψ˜ (+∞) > 0 and its derivative ψ˜ ′ (λ′′ ) > 0 for all λ′′ > 0. This ˜ (λ′′ ) = 0 has a unique positive real implies that the equation ψ root λ′′ . ′′

√

η e−

λ′′ t 2

Ξ , t ≥ 0. +√ −ξ1 − ξ2 − ξ3 σ2

(51)

As t → ∞, it is clear from (51) that the trajectory of error neural network (12) is converging globally exponentially into a {

¯2 = small compact domain containing the origin ∆ ∥e(t)∥ ≤

√

Ξ −ξ1 −ξ2 −ξ3 σ2

}

at the convergence rate

e(t) ∈ Rn :

λ′′

> 0. This

2

completes the proof of the second case. □ Corollary 1. Suppose that the neural networks (2) and (4) are ¯ = D˜ and I = J, then identical, i.e., A¯ = A˜ , B¯ = B˜ , C¯ = C˜ , D from Eq. (6), we get R(.) = 0 implies Ξ = 0. Under Assumption 1 and Ta < ∞, the trajectory of the error neural network (12) converges globally exponentially to zero at the rate of convergence λ > 0 if 2 ln µ Ta

+ δ + ξ2 + ξ3 σ2 < 0,

(52)

˜ T D˜ + where δ > 0, ξ1 = λmax 2(−A˜ − Γ ) + B˜ T B˜ + C˜ T C˜ + D

}

{

}

ξ2 z(s − σ1 (s))

{

LTf Lf1 , ξ2 = λmax LTf Lf2 , ξ3 = λmax σ2 LTf Lf3 2

solution of the equation λ+

[

3

ln µ 2 T a

}

and λ is a unique

λσ1 (t)

+ξ1 +δ+ξ2 e

λσ2 (t) −1

+ξ3 e

λ

= 0.

That is, the complete synchronization between the neural networks (2) and (4) is achieved under the hybrid impulsive controller (11).

]

s

z(r)dr + ∥R(.)∥ + ϵ 2

ds.

¯ (λ′ ) = λ′ + Define a continuous function ψ ξ

with the convergence rate

0

1

(47)

s−σ2 (s)

′ eλ σ2 −1 . 3 λ′

2

−ξ2 −ξ3 σ2

{

0

∫

−ξ1 −

δ′

Substituting the inequality (49) in Eq. (31), we get

ξ1 + 2

Substituting inequality (46) into Eq. (31), we get ( ′ δ

(48)

W (t , s) ≤ eξ1 (t −s) .

Using inequality (45) in (44), we get W (t , s) ≤ e

}

Ξ

√

e

|µ1 |+|µ2 |+···+|µNζ (t ,s) | Nζ (t ,s) t −s Nζ (t −s)

, t ≥ 0.

t → ∞. The proof of the first case is completed. Case 2: For 0 < µ ≤ 1 and Ta = ∞, from inequality (44), we can obtain

∥e(t)∥ ≤

Case 1: From Definition 3 and Ta = ∞, when µ > 1, then we have 2 ln

2

− ξ 2 − ξ 3 σ2

get

ξ1 (t −s)

|µ1 |+|µ2 |+···+|µN (t ,s) | ζ Nζ (t ,s) (t −s) t −s Nζ (t −s) eξ1 (t −s)

δ′

Again, if we follow the similar proof as in Theorem 1, then we

( |µ1 | + |µ2 | + · · · + |µN (t ,s) | )2Nζ (t ,s) ζ ≤ eξ1 (t −s) Nζ (t , s) 2Nζ (t ,s) ln

−ξ1 −

¯ 2 = e(t) ∈ Rn : a small compact domain containing the origin ∆

+ ξ3

Proof. Following the similar proof as in Theorem 1, we get

Ξ

λ′

η e− 2 t + √

It is concluded from inequality (48) that the solution of the impulsive system (12) is converging globally exponentially into {

= 0.

2 ln

√

+

Thus, in both cases, the quasi-synchronization between the neural networks (2) and (4) is achieved with a small synchronization error bounds.

2 ln

∥e(t)∥ ≤

∥e(t)∥ ≤

′

ξ3 e

From now, following the similar proof as in Theorem 1, we obtain

δ′ 2

+ ξ1 + ξ2 eλ σ1 + ′

¯ (0) < 0. It can be easily verified From (42), we have ψ

¯ (+∞) > 0 and ψ¯ ′ (λ′ ) > 0 for all λ′ > 0. Thus, ψ¯ (λ′ ) = 0 that ψ must possess a unique positive real root λ′ .

Corollary 2. Suppose that Assumption 1, Ta = ∞ and R(.) = 0 hold. Case 1: For any δ ′ > 0 and µ > 1, the solution of the impulsive system (12) converges globally exponentially to zero at the rate of ′ convergence λ2 > 0 if

δ′ 2

+ ξ1 + ξ2 + ξ3 σ2 < 0,

(53)

R. Kumar, S. Das and Y. Cao / Neural Networks 122 (2020) 106–116

where λ′ is a unique root of the equation λ′ + ′ eλ σ2 −1 3 λ′

δ′ 2

+ ξ1 + ξ2 e

λ′ σ

1

+

ξ

= 0. Case 2: For 0 < µ ≤ 1, the solution of the impulsive system (12) converges globally exponentially to zero at the convergence rate λ′′ > 0 if 2 ξ1 + ξ2 + ξ3 σ2 < 0,

(54)

where λ′′ is a unique root of the equation λ′′ + ξ1 + ξ2 eλ ′′ eλ σ2 −1 3 λ′′

ξ

ξ2

{

′′ σ

1

+ }

= 0 and ξ1 = λmax 2(−A˜ −Γ )+B˜ B˜ +C˜ C˜ +D˜ D˜ + , { } { } = λmax LTf2 Lf2 , ξ3 = λmax σ2 LTf3 Lf3 are constants. In both T

T

T

LTf Lf1 1

cases, the complete synchronization between the systems (2) and (4) is achieved under the hybrid impulsive controller (11) with hybrid impulses which occur infinitely but sparsely. Remark 6. It is ′ worth ′′to observe that, in Theorem 2, the convergence rates λ2 and λ2 are independent of average impulsive interval Ta . Theoretically, we have shown that the infinite but sparse occurrence of impulses does not have a negative impact on the quasi-synchronization of different neural networks with mixed time-varying delays. Furthermore, since Ta = ∞, that is the length of the impulsive interval is very large, which means the control cost for the synchronization of neural networks is very less. Therefore, results obtained in Theorem 2 are effective for reducing the cost of control. Remark 7. In this article, different from Tang et al. (2018), the hybrid impulsive controller (11) has been designed for timevarying impulses. The impulsive controller designed in Tang et al. (2018) is restricted to invariant impulses, i.e., the impulses which do not change at different impulsive times. Thus, the hybrid impulsive controller (11) is more general than the existing one designed in Tang et al. (2018). Remark 8. In Theorem 2, the inequalities (42) and (43) are delay dependent sufficient criteria to achieve quasi-synchronization. If σ2 is very large, then we can adjust the value of coupling strength ′ γi > 0 such that the inequalities δ2 + ξ1 + ξ2 + ξ3 σ2 < 0 and ξ1 + ξ2 + ξ3 σ2 < 0 hold. That is, the linear feedback term −Γ e(t) in the controller (11) is playing a meaningful role for the case Ta = ∞. On the other hand, the impulsive control without linear feedback term as designed in the previous articles (Chen et al., 2018; He et al., 2017, 2011; Yang et al., 2011; Yi et al., 2017) will not work when time-delay is very large. Therefore, the controller (11) is perfectly designed to deal with the case Ta = ∞. Remark 9. In Theorems 1 and 2, one can observe that the synchronization error bounds can be reduced by increasing the coupling strength γi > 0 for i = 1, 2, . . . , n (provided other parameters are given) because ξ1 depends on the diagonal matrix Γ . Moreover, assume that γi and other parameters except µ > 0 are fixed then the synchronization error bound in Theorem 1 can be reduced by taking small value of the average impulsive gain µ. In contrast to Theorem 1, the radii of the compact spheres ¯ 2 and ∆ ¯ 3 in Theorem 2, i.e., the synchronization error bounds ∆ cannot be reduced if the coupling strength γi is fixed. One more thing is worth mentioning that the synchronization error bounds obtained in the theorems cannot be optimized to their minimum values, for the proof we refer to Kumar, Shreemoyee, Das, and Cao (2019). 4. Numerical simulation and discussions In this section, two examples have been considered to validate the theoretical results obtained in Theorems 1 and 2, where sufficient criteria of the quasi-synchronization are derived for Ta < ∞ and Ta = ∞.

113

Example 1. Consider the neural network (2) with the parameters given as

( ¯A = 1 0

)

0 , 1

( −1.5 C¯ = −0.2

B¯ =

) −0.1 , −4

¯ = D

(

(

2 −5

−0.3 0.1

) −0.1 , 4.5 ) 0.1 , −0.2

f1 (x(t)) = f2 (x(t)) = f3 (x(t)) = [tanh x1 (t), tanh x2 (t)]T with Lf1 = Lf2 = Lf3 = diag(1, 1); σ1 = 1, σ2 = 0.2, and the external input vector I = [0, 0]T . The phase portrait of chaotic attractors of the neural network (2) for its initial condition x(s) = [0.01, 0.1]T , −σ ≤ s ≤ 0 is shown in Fig. 2(a). Further, the parameters of the neural network (4) are given as A˜ = C˜ =

(

1 0

(

)

0 , 1

−1.7 −0.26

B˜ =

) −0.12 , −2.5

(

1.8 −5.2

( ˜D = 0.6 −2

) −0.15 , 3 .5 ) 0.15 . −0.12

The activation functions and the time delays are f1 (y(t)) = f2 (y(t)) = f3 (y(t)) = [tanh y1 (t), tanh y2 (t)]T and σ1 = 1, σ2 = 0.2, respectively. The chaotic attractors of the neural network (4) for its initial condition y(s) = [0.02, 0.01]T , −σ ≤ s ≤ 0 and the external input vector J = [0, 0]T can be observed from Fig. 2(b). First we will verify the results for the case Ta < ∞. For this purpose, set the coupling strength γi = 30, i = 1, 2. Using the given data of the parameters, we get ξ1 = −13.18, ξ2 = 1 and ξ3 = 0.2. Let the time-varying impulses are µ1 = 1.3 and µ2 = 0.9 with the average impulsive gain µ = 1.1 and the average impulsive interval Ta = 0.25. Hence, for fixed value δ = 0.5, the ln µ inequality ξ1 + 2 T + δ + ξ2 + ξ3 σ2 = −10.88 < 0 holds which a assured that the trajectories of the error neural network (12) λ converge globally exponentially at the rate of convergence { 2 = ¯ 1 = e(t) ∈ 1.135 into a small}compact domain of convergence ∆ Rn : ∥e(t)∥ ≤ 1.96 . The phase portrait of hybrid impulsive signal with average impulsive interval Ta = 0.25 is depicted in Fig. 3(a) and the corresponding error trajectory of system (12) is shown in Fig. 3(b) with the experimental error bound 0.0032 which is less than the theoretical error bound 1.96. This implies that the quasisynchronization between the neural networks (2) and (4) could be achieved with a given synchronization error bound under the hybrid impulsive controller (11). This verifies Theorem 1. Example 2. In this example, considering the same systems as in Example 1, we will verify the results obtained in Theorem 2. Let µ1 = 1.6 and µ2 = 0.7 are the impulsive strengths at different instants of the impulsive sequence ζ = {t1 , t2 , . . . , tk } with average impulsive gain µ = 1.15 > 1 and the average impulsive interval Ta = ∞. For fixed δ ′ = 1.5, the inequality δ′ + ξ1 + ξ2 + ξ3 σ2 = −11.39 < 0 holds which means that 2 the trajectory of the impulsive system (12) converges globally exponentially at the convergence rate λ2 { = 1.15 in to a small ¯ 2 = e(t) ∈ Rn : ∥e(t)∥ ≤ compact domain of convergence ∆ } 1.91 . Fig. 1 depicts the distribution of hybrid impulses with Ta = √ ∞ for Nζ (t , s) = [ 3 t − s] and the corresponding time evolution of the controlled error neural network (12) with experimental error bound 0.00276 is shown in Fig. 4(a). For the Case 2 of Theorem 2, set the strengths of the hybrid impulses as µ1 = 0.2 and µ2 = 0.5 with average impulsive gain µ = 0.35 < 1 and average impulsive interval Ta = ∞, then the inequality ξ1 + ξ2 + ξ3 σ2 = −12.14 < 0 holds which implies that the trajectory of error neural network (12) converges globally ′′ exponentially at the convergence rate λ2{ = 1.19 into a small ¯ 3 = e(t) ∈ Rn : ∥e(t)∥ ≤ compact domain of convergence ∆ } 1.86 . For the infinite but sparse occurrence of impulses as in ′

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√

Fig. 1. Phase portrait of hybrid impulses with Ta = ∞ for Nζ (t , s) = [ 3 t − s] in Example 2.

Fig. 2. Chaotic attractors of Eqs. (2) and (4) are shown in (a) and (b), respectively.

Fig. 3. Hybrid impulsive sequence with µ = 1.1 and the corresponding time evolution of controlled error neural network (12) for Ta < ∞ are shown in (a) and (b), respectively.

Fig. 1, the error trajectory of the system (12) is fluctuating within a small compact ball of radius 0.002672 which can be observed from the Fig. 4(b). Therefore, it is verified that in both cases of Theorem 2, the quasi-synchronization between neural networks (2) and (4) could be achieved within a given synchronization error bound under the controller (11) with hybrid impulses and infinite average impulsive interval Ta = ∞.

5. Conclusion In this article, the effects of the hybrid impulses have been deeply investigated for the quasi-synchronization of neural networks with mixed time-varying delays and parameters mismatch. A hybrid impulsive controller with feedback term has been designed so that the quasi-synchronization could easily be achieved, no matter how the simultaneous existence of synchronizing and

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Fig. 4. Time-evolution of error neural network (12) with µ = 1.15 and µ = 0.35 for Ta = ∞ are shown in (a) and (b), respectively.

desynchronizing impulses is affecting the networks’ synchronization. By employing the new concept of average impulsive interval, the average impulsive gain, some mathematical techniques, and the extended comparison principle for delayed impulsive system combined with the formula of variation of parameters, delaydependent quasi-synchronization criteria have been obtained for the neural networks with mixed time-varying delays when Ta < ∞ and Ta = ∞. Meanwhile, the compact sets containing the origin have been constructed for the cases Ta < ∞ and Ta = ∞ into which the impulsive error system converges globally exponentially with the exponential convergence rates. Finally, two examples are considered to validate the effectiveness of our theoretical results. The impulses considered in the present article vary with fixed time instants of the impulsive sequence. If the impulses are variable time impulses, then investigating their effects on dynamical behaviors of delayed neural networks and developing mathematical tools to study the effects are challenging issues in nonlinear dynamics which would be deeply investigated in our future research work. Acknowledgments The authors are extending their heartfelt thanks to the revered reviewers for their constructive suggestions towards improvement of the revised article. References Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., & Zhou, C. (2008). Synchronization in complex networks. Physics Reports, 469, 93–153. Balasubramaniam, P., & Vembarasan, V. (2012). Synchronization of recurrent neural networks with mixed time-delays via output coupling with delayed feedback. Nonlinear Dynamics, 70, 677–691. Cao, J., Wang, Z., & Sun, Y. (2007). Synchronization in an array of linearly stochastically coupled networks with time delays. Physica A. Statistical Mechanics and its Applications, 385, 718–728. Chen, S., & Cao, J. (2012). Projective synchronization of neural networks with mixed time-varying delays and parameter mismatch. Nonlinear Dynamics, 67, 1397–1406. Chen, H., Shi, P., & Lim, C. C. (2018). Pinning impulsive synchronization for stochastic reaction–diffusion dynamical networks with delay. Neural Networks, 106, 281–293. Gu, K. (2000). An integral inequality in the stability problem of time-delay systems. Proceedings of the 39th IEEE conference on decision and control (Cat. No.00CH37187), Vol. 3, (pp. 2805–2810). Guan, K. (2018). Global power-rate synchronization of chaotic neural networks with proportional delay via impulsive control. Neurocomputing, 283, 256–265. He, W., Qian, F., & Cao, J. (2017). Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control. Neural Networks, 85, 1–9. He, W., Qian, F., Cao, J., & Han, Q. L. (2011). Impulsive synchronization of two nonidentical chaotic systems with time-varying delay. Physics Letters. A, 375, 498–504.

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