Passivity and Synchronization of Multiple Multi-Delayed Neural Networks via Impulsive Control

School of Computers, Guangdong University of Technology, Guangzhou 510006, China Chongqing Key Lab on IFBDA, School of Mathematical Sciences, Chongqing Normal University, Chongqing 400047, China School of Mathematical Sciences, Chongqing Normal University, Chongqing 400047, China School of Computer Science and Technology, Tiangong University, Tianjin 300387, China School of Artificial Intelligence, Chongqing University of Technology, Chongqing 401135, China


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. erefore, the stability of NNs has become a hot issue in recent years [5][6][7][8][9][10][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][13][14][15][16][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][19][20][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].
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][33][34][35] involving passivity and synchronization [36][37][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. e 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. ird, a newly designed impulsive controller is applied to synchronize the proposed networks.
Definition 2 (see [21]). A system is said to be strictly passive with output for any t ϱ , t 0 ∈ [0, +∞) and t 0 ≤ t ϱ . e system is input-strictly passive if 0 < A 1 and outputstrictly passive if 0 < A 2 .

Passivity of MMDNNs via Impulsive Control
e MMDNNs are considered as follows: .. , J n ) T ∈ R n ; v i (t) ∈ R n means the input vector of node i; and τ ι is the transmission delay and 0 ≤τ ι ≤ τ.

Discrete Dynamics in Nature and Society
Suppose that z * � (z * 1 , z * 2 , . . . , z * n ) T ∈ R n is an equilibrium solution of an isolated node of the MMDNNs (6). en, one gets For the MMDNNs (6), construct the following impulsive controller: N×N means the impulsive coupling matrix, where B ij is described as follows: if there is a link from node i to node j, then ; otherwise, B ij � 0(i ≠ j); and It is derived from (6) and (9) that (8) and (11), we acquire e output vector y i (t) ∈ R ϵ of the MMDNNs (12) is chosen as in which C 1 ∈ R ϵ×n and C 2 ∈ R ϵ×n are known matrices.
en, one has For the MMDNNs (45), select the following impulsive controller: in which B ij and Φ have the same meanings as in the third section. Assume

Conclusion
In this paper, a class of multiple multi-delayed neural network models has been introduced, and the model can describe the dynamics of neurons more accurately. On the one hand, several passivity criteria for MMDNNs have been established by means of inequality techniques. On the other hand, the synchronization of the proposed model has also been researched based on stability theory. Finally, the correctness of the synchronization and passivity criteria has been verified by two numerical examples. In future work, we will design a suitable impulsive controller to further study the passivity and synchronization of multiple reaction-diffusion neural networks.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.