#### Abstract

This paper is concerned with the passivity and synchronization for multiple multi-delayed neural networks (MMDNNs) under impulsive control. To ensure the passivity, input strict passivity, and output strict passivity in MMDNNs, a suitable impulsive controller is designed. Moreover, an impulsive time-dependent Lyapunov functional is exploited to obtain the synchronization criterion of MMDNNs, where the criterion is formulated by linear matrix inequalities. Numerical examples are given to verify the validity of the theoretical results.

#### 1. Introduction

Neural networks (NNs) have received extensive attention due to their successful applications in vision system [1], associative memory [2], pattern recognition [3], and image compression [4]. NNs always require stability, which is a prerequisite for many applications. Therefore, the stability of NNs has become a hot issue in recent years [5–11]. Zhang et al. [7] considered some asymptotic stability criteria of NNs with distributed delays based on Lyapunov–Krasovskii functionals. By improving the auxiliary polynomial-based functions, Li et al. [9] solved the stability problem in delayed NNs. Since the main property of passivity is to keep the system internally stable, some researchers have focused on the passivity for NNs [12–17]. Lian et al. [14] proposed a kind of switched NNs with time-varying delays and stochastic disturbances, and the passivity of networks was analyzed by designing a state-dependent switching law and a hysteresis switching law. Cao et al. [17] addressed the robust passivity issue of uncertain NNs with additive time-varying delays and leakage delay, and a general activation function was utilized to ensure that the proposed network model was passive.

In addition, multi-weighted network models [18–21] can be used to describe many real-world networks including public transportation road networks, communication networks, social networks, and so forth. Recently, some researchers have investigated the dynamical behaviors of complex networks with multiple weights [20, 21]. Wang et al. [20] concentrated on two types of multi-weighted complex networks with several different weights between two nodes, and sufficient conditions ensuring the synchronization were developed by utilizing the pinning control method. Under the help of pinning adaptive control techniques, the passivity of multi-weighted complex networks with different dimensions of input and output was discussed in [21]. However, a few authors have considered the stability and passivity of NNs with multiple delays.

Compared with continuous control, there are many advantages of impulsive control strategies, which include low maintenance costs, high reliability, ease of installation, and high efficiency [22]. So far, a series of investigations in regard to the stability [23–28] and passivity [29–31] for impulsive NNs have been reported. Zhu and Cao [25] dealt with the stability problem of impulsive stochastic BAM NNs with mixed time delays and Markovian jump parameters by exploiting Itô’s formula and stochastic analysis theory. According to the comparison principle and compression mapping theorem, the global exponential stability of periodic solution was considered in an array of Cohen–Grossberg NNs with time-varying delays and periodic coefficients via impulsive control in [26]. Zhou [31] took into account the passivity of recurrent NNs with multiproportional delays and impulse. But very few authors have discussed the stability and passivity problems of multi-delayed NNs under impulsive control.

At present, most of the literatures with respect to identifying network structures from the observation and control of dynamical behavior are concentrated in a single neural network [5, 7, 12, 13]. In practical applications, some tasks are difficult to complete by a single neural network; even if the network can accomplish these tasks, it may result in high costs. But multiple NNs can solve some difficult problems through cooperation with each other so that the cost can be reduced. Recently, some cooperative control problems [32–35] involving passivity and synchronization [36–38] have been concerned in multiple NNs. Unfortunately, as far as we know, the passivity and synchronization of multiple multi-delayed NNs (MMDNNs) via impulsive control have never been considered. Inspired by the above discussion, this paper aims to further study the passivity and global exponential synchronization of MMDNNs by using impulsive control techniques. The contributions of this paper are as follows. First, compared with the traditional impulse-time-independent Lyapunov functional, the impulse-time-dependent feature of the Lyapunov functional in this paper can capture more dynamical behaviors of MMDNNs. Second, with the help of some piecewise linear functions and inequality techniques, the passivity problems of MMDNNs are addressed via impulsive control. Third, a newly designed impulsive controller is applied to synchronize the proposed networks.

#### 2. Preliminaries

##### 2.1. Notations

Let , . The fixed moments satisfy and . denotes the smallest (largest) eigenvalue of a matrix. For any , , .

The notation represents the set of impulse time sequences satisfying for all , in which . For a given impulse time sequence , some piecewise linear functions can be defined as follows:

Let . It is obvious to see that , , for , and , .

For , can be written aswhere

*Definition 1. *(see [21]). A system is said to be passive with output and input , if there is a storage function and a matrix satisfyingfor any and .

*Definition 2. *(see [21]). A system is said to be strictly passive with output and input , if there is a storage function , matrices , , , and eigenvalues satisfyingfor any and .

The system is input-strictly passive if and output-strictly passive if .

#### 3. Passivity of MMDNNs via Impulsive Control

##### 3.1. Network Model

The MMDNNs are considered as follows:in which ; is the state vector of node ; ; ; ; ; means the input vector of node ; and is the transmission delay and .

In this paper, there is such thatfor any . Let .

Suppose that is an equilibrium solution of an isolated node of the MMDNNs (6). Then, one gets

For the MMDNNs (6), construct the following impulsive controller:in which ; ; means the impulsive coupling matrix, where is described as follows: if there is a link from node to node , then ; otherwise, ; and

It is derived from (6) and (9) thatwhere , . Suppose .

Let . Then, by (8) and (11), we acquire

The output vector of the MMDNNs (12) is chosen asin which and are known matrices.

##### 3.2. Passivity Criteria

Theorem 1. *Under the impulsive controller (9), the MMDNNs (12) are passive over if there exist matrices , , and and a scalar such thatin which , , , and .*

*Proof. *Let .

Then, by (14), we haveThe impulse-time-dependent Lyapunov functional for the MMDNNs (12) is considered as follows:where .

Then,where .

Obviously,Substituting (19) into (18) yieldsFrom (13), (16), and (20), one obtainsin which .

Integrating (21) with respect to from to , we acquirein which and .

At the impulse time , according to the definition of , one getsIt is found from (12), (15), and (17) thatBy (21)–(24), we haveHence,for any and , in which . □

Theorem 2. *Under the impulsive controller (9), the MMDNNs (12) are input-strictly passive over if there exist matrices , , , and and a scalar such thatin which , , , .*

*Proof. *Let .

Then, from (27), one obtainsSelecting the same as (17) for the networks (12) and using (29), one hasin which .

Integrating (30) with respect to from to , we acquirein which and .

At the impulse time , according to the definition of , one acquiresOn the basis of (12), (17), and (28), we getConsidering (30)–(33), it is obtained thatThus,for any and , in which .

Theorem 3. *Under the impulsive controller (9), the MMDNNs (12) are output-strictly passive over if there exist matrices , , , and and a scalar such thatin which , , , and .*

*Proof. *Let .

Then, it is derived from (36) thatSelecting the same as (17) for the networks (12) and utilizing (38), we getin which .

Integrating (39) with respect to from to , one hasin which and .

At the impulse time , according to the definition of , we deriveIt is obtained from (12), (17), and (37) thatUsing (39)–(42), we can acquireTherefore,for any and , in which .

*Remark 1. *In recent years, as an effective method to study synchronization, the passivity of NNs has been investigated by some researchers [12–17]. Impulsive control is a popular control method among control methods due to its reliability, low cost, and flexibility [22]. Using impulsive control strategies, some authors have focused on the passivity and synchronization of NNs. However, the passivity and synchronization of MMDNNs have not been considered under impulsive control.

#### 4. Synchronization of MMDNNs via Impulsive Control

Setting in the MMDNNs (6), we obtain

The initial value of (45) is given bywhere is the set of continuous functions from to .

Suppose that is an equilibrium solution of an isolated node of the MMDNNs (45). Then, one has

For the MMDNNs (45), select the following impulsive controller:in which and have the same meanings as in the third section. Assume .

It is obtained from (45) and (48) thatwhere , . Suppose .

Let . Then, it is found from (47) and (49) that

*Definition 3. *(see [20]). The MMDNNs (45) achieve synchronization if

Theorem 4. *If there exist matrices and and scalars such thatwhere , , and , then under the impulsive controller (48), the MMDNNs (45) achieve global exponential synchronization over with convergence rate .*

*Proof. *Let , .

Then, by (52), we haveThe impulse-time-dependent Lyapunov functional for the MMDNNs (50) is considered as follows:in which .

For , , differentiating along the solution of the MMDNNs (50) and applying the fact that , for all and , one getsMoreover,It is obtained from (54), (56), and (57) thatTherefore,At the impulse time , on the basis of the definitions of and , one hasCombining (50), (53), and (55) together, we derive that