Finite-Time Synchronization and Passivity of Multiple Delayed Coupled Neural Networks via Impulsive Control

This paper focuses on the finite-time passivity (FTP) and finite-time synchronization (FTS) for multiple delayed coupled neural networks (MDCNNs) with and without uncertain parameters. A novel FTP criterion for MDCNNs with different dimensions of input and output is derived by designing an impulsive controller and using the Jensen’s inequality. Then, by means of the Lyapunov functional and integral inequality techniques, a sufficient condition is established to ensure the FTS of MDCNNs via impulsive control. Furthermore, the robust FTP and FTS in MDCNNs are considered. Finally, a numerical example is given to demonstrate the validity of the presented results.


I. INTRODUCTION
During the past two decades, neural networks (NNs) have been successfully applied in various fields including medicine and health care [1], image processing [2], data mining [3], and signal processing [4]. Through coupling interaction, a large number of NNs can form coupled NNs (CNNs). CNNs have attracted increasing attention because of their many potential applications in secure communications and electronic circuits. Generally these applications of CNNs are strongly dependent on their dynamical behaviors, such as synchronization, dissipativity, stability, passivity, and chaos. Because synchronization is ubiquitous in natural and man-made systems and has important potential applications in different areas such as image encryption, encoded communications, and secure transmission of voice messages, it has received considerable attention in recent years. In particular, the synchronization in CNNs has been investigated and meaningful achievements have been obtained [5]- [8]. In [7], The associate editor coordinating the review of this manuscript and approving it for publication was Juntao Fei . synchronization stability was studied for arrays of linearly coupled delayed NNs with nonsymmetric coupling, and a separate synchronization criterion was established, which was the combination of stability condition of node networks and the constraint condition of coupling configuration matrix. More importantly, the authors of [7] discovered the relation between synchronization and stability of NNs in an analytical expression, which had important significance for the development of neural network theory. In [8], a detailed discussion on stability and synchronization of complex NNs was presented, and it showed that relative stability was the essential fundamental of stability and synchronization. This is a significant breakthrough in the qualitative theory of complex NNs. In this meaning, synchronization is also called synchronization stability in many references. On the other hand, passivity is closely related to circuit analysis and has played an important role in many areas involving stability, robotic systems, synchronization, power systems, etc. Consequently, investigations related to the passivity of CNNs have been carried out [9]- [13]. Ren et al. [10] presented two kinds of CNNs with and without time delays, and sufficient conditions were developed to ensure the passivity for these models by means of pinning control strategies.
However, in most of the existing literature [14]- [19], synchronization in CNNs can only be achieved when the time is infinite. In practice, due to the limited lifespan of machines and humans, CNNs are expected to realize synchronization over a finite-time interval. Based on this motivation, finite-time synchronization (FTS) has been presented, which means that synchronization can be obtained in a setting time. In addition to fast convergence speed, FTS also has better robustness and disturbance attenuation. Hence, some researchers have dealt with the FTS of CNNs [20]- [22]. In [22], the FTS of coupled hierarchical hybrid NNs was discussed. It should be pointed out that passivity can ensure synchronization for CNNs in some cases. Therefore, the finite Regretfully, to the best of our knowledge, the FTP and FTS of multiple delayed CNNs (MDCNNs) via impulsive control have never been considered.
Motivated by the above discussions, this paper presents a class of multiple delayed coupled neural network models with and without uncertain parameters. The main contributions are as follows: 1) There is a lack of research on multiple delayed coupled neural network models in which the state of each node is affected not only by the current states of other nodes but also by the past states of other nodes.
2) Based on some inequality techniques, the finite-time boundedness (FTB), robust FTB, FTP, and robust FTP criteria of MDCNNs under impulsive control are formulated for the first time.
3) This study is the first to achieve FTS and robust FTS for MDCNNs with the aid of impulsive control law and two Lyapunov functionals.

C. ASSUMPTIONS
Assumption 1: There exists R ϑ h > 0 (h = 1, 2, . . . , n) such that Assumption 2 [41]: For a given positive parameter η and a positive constant T p , the input vector u(t) satisfies D. DEFINITIONS Definition 1 [41]: Given a positive parameter η, a matrix F > 0, and three positive constants T p , c 1 , c 2 with Definition 2 [41]: For a given positive constant T p , the MDCNN (6) with input u(t) ∈ R nN and output y(t) ∈ R υN realizes FTP with respect to (c 1 , c 2 , T p , F, η), if the following conditions hold: where u(t) satisfies (8).
Remark 1: In the proof of Theorem 1, it is not required thaṫ V 1 (t) is a negative definite or seminegative definite function, which is different from the classical Lyapunov function for neural network models in the case of asymptotical stability.
Remark 2: It should be noted that (10)- (12) are related to the parameters c 1 , c 2 , T p , µ, η. In practical applications, the values of T p , η, c 1 are usually prescribed, and c 2 , µ can be taken as optimization variables. The optimization problem can be expressed as follows: The minimum values of c 2 , µ satisfy (10)- (12).
Remark 3: In [24], the adaptive controllers for ensuring the FTP of directed and undirected CNNs were devised. However, if the definition of FTP in [24] is used, it is difficult to achieve FTP of CNNs under an impulsive controller. Compared with [24], by exploiting the definition of FTP in [41] and the impulsive control strategy, a sufficient condition is given to ensure the FTP of MDCNNs in this paper.
Remark 4: This paper cannot prove that the MDCNNs are finite-time bounded and passive under the impulsive law only based on the proof method in [41]. Compared with [41], the two inequalities (21) and (33) play an important role in dealing with the FTP problem of MDCNNs with impulsive control.

Similar as the handling method in Theorem 1, it is clear that
On the basis of (42) and (50), one gets Hence, the MDCNN (38) reaches FTS with respect to c 3 , c 4 , T p 1 , F 1 under the impulsive controller (4).
Remark 5: Compared with [40], the main difficulties in studying the FTP and FTS of MDCNNs with impulsive control come from the design of the impulsive controller and the proofs of these theorems. By designing the impulsive controller and the Lyapunov functional and using inequality techniques, this paper considers the FTP and FTS of MDCNNs (see Theorems 1-3).
Proof: The Lyapunov functional for the MDCNN (6) with parameter ranges defined in (51) is constructed as follows: Then, we acquirė It can be derived from Assumption 1 that Obviously, According to Lemma 2 in [18], one gets Considering (52) and (56)-(61), we can derivė (z)dz. Similar to the proof of Theorem 1, we arrive at Consequently, the MDCNN (6) with parameter ranges defined in (51) is robustly finite-time bounded with respect to c 1 , c 2 , T p , F, η by the impulsive controller (4).
Proof: By Theorem 4, the MDCNN (6) with parameter ranges defined in (51) is robustly finite-time bounded with respect to c 1 , c 2 , T p , F, η by the impulsive controller (4).
Next, we prove the robust FTP of the MDCNN (6). Consider the same Lyapunov functional as Theorem 4 and define the following functional: (z)dz. It is found from (63) and (64) thaṫ Based on the proof of Theorem 4 and the condition V 2 (0) = 0, we have Therefore, the MDCNN (6) with parameter ranges defined in (51) attains robust FTP with respect to c 1 , c 2 , T p , F, η under the impulsive controller (4).
Proof: For the MDCNN (40), select the same Lyapunov functional as Theorem 4 and define the following functional: In view of (68) and (71), we havė Similar to the proof of Theorem 3, one derives Thus, the MDCNN (38) with parameter ranges defined in (51) achieves robust FTS with respect to c 3 , c 4 , T p 1 , F 1 under the impulsive controller (4).
Remark 6: It should be noted that parameter uncertainty may occur frequently in neural network models due to modeling errors, external disturbances, and parameter fluctuations [42], [43]. As far as we know, there are no reports on the robust FTP and FTS of MDCNNs via impulsive control.
Remark 7: As we all know, the dimensions of input and output vectors for many systems are different [10], [30]. But it is usually assumed that the dimensions of input and output vectors are the same in a large number of studies on the FTP [23]. The FTP in MDCNNs with diverse dimensions of input and output is investigated under impulsive control in this paper.
Remark 8: To date, some research results regarding the FTP of CNNs mainly assume that the controller is continuous [24].  This paper introduces an impulsive controller to address the FTP problem because of the great advantages of this control scheme in designing finite-time control strategies [41].   (38) is displayed in Fig. 3, which clearly demonstrates that the FTS can be realized.
Remark 9: Recently, some researchers have discussed the passivity and synchronization of CNNs [5]- [7], [40]. In many cases, CNNs need to obtain passivity and synchronization in finite time. Unfortunately, the FTP and FTS for MDCNNs via impulsive control have not yet been considered. This paper investigates the issues of FTP and FTS in MDCNNs by designing a suitable impulsive controller.

VI. CONCLUSION
In this paper, a multiple delayed coupled neural network model with and without uncertain parameters has been proposed. The main contribution is that the FTP and FTS of MDCNNs via impulsive control have been studied for the first time. Unlike the continuous control method, impulsive control is a discontinuous control method, which can reduce communication costs and save bandwidth more effectively. Moreover, based on the impulsive control strategy, the problem of the robust FTP in MDCNNs has been addressed. By utilizing the Lyapunov functional, the criterion for ensuring the robust FTS of MDCNNs has been expressed in terms of linear matrix inequalities. A numerical example has been designed to examine the correctness of the theoretical criteria.
The impulsive control method is a useful control law for FTP and FTS in CNNs. In addition, NNs can be implemented through electronic circuits, and reaction-diffusion phenomenon is unavoidable in electronic circuits. Therefore, in the future, we will investigate the FTP and FTS for multiple weighted coupled reaction-diffusion NNs under impulsive control.