Passivity and Synchronization of Coupled Different Dimensional Delayed Reaction-Diffusion Neural Networks with Dirichlet Boundary Conditions

Lin, Shanrong; Huang, Yanli; Yang, Erfu

doi:https://doi.org/10.1155/2020/4987962

Complexity

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 4987962 | https://doi.org/10.1155/2020/4987962

Passivity and Synchronization of Coupled Different Dimensional Delayed Reaction-Diffusion Neural Networks with Dirichlet Boundary Conditions

Shanrong Lin,¹Yanli Huang,¹and Erfu Yang²

Academic Editor: Lingzhong Guo

Received05 Jul 2019

Accepted21 Nov 2019

Published08 Jan 2020

Abstract

Two types of coupled different dimensional delayed reaction-diffusion neural network (CDDDRDNN) models without and with parametric uncertainties are analyzed in this paper. On the one hand, passivity and synchronization of the raised network model with certain parameters are studied through exploiting some inequality techniques and Lyapunov stability theory, and some adequate conditions are established. On the other hand, the problems of robust passivity and robust synchronization of CDDDRDNNs with parameter uncertainties are solved. Finally, two numerical examples are given to testify the effectiveness of the derived passivity and synchronization conditions.

1. Introduction

In recent years, complex networks (CNs) have attracted much attention since they are ubiquitous under the circumstance of our daily life, for instance, communication networks, food webs, and social networks. Coupled neural networks (CNNs), as a particular type of CNs, have been put into use successfully in various fields, e.g., pattern recognition, chaos generators design, and brain science [1–3]. Strictly speaking, these applications in a large extent are depending on some properties of dynamics in CNNs (e.g., synchronization). Thus, the problem of synchronization for CNNs has attracted comprehensive attention and been developed into a hot research topic. So far, many important and interesting results have been derived on this topic recently [4–10]. Through exploiting the properties of random variables, Yang et al. [4] acquired some synchronization conditions for randomly delayed CNNs. In [7], several synchronization criteria were gained for delayed CNNs in accordance with the Lyapunov functional strategy.

Nevertheless, most of the synchronization results in aforesaid literatures [4–10] neglected reaction-diffusion phenomena. In fact, the reaction-diffusion phenomena are unavoidable for CNs such as neural networks and cellular networks when they are implemented by means of electric circuits in practical situations [11]. Hence, it is necessary to research coupled reaction-diffusion neural networks (CRDNNs). Recently, some significant synchronization results for CRDNNs have been acquired [12–17]. In [12], the authors researched synchronization of CRDNNs and presented some synchronization criteria. By designing appropriate pinning controllers, several synchronization criteria were established for CRDNNs in [17]. Furthermore, owing to the existence of external interferences, the noises of environment, and equipment restrictions, it is very difficult to ensure network models containing the certain parameter values in some practice situations. Consequently, some authors studied parametric uncertainties of neural networks [18–22], and a few interesting results have been derived regarding robust synchronization of CNNs with uncertain parameters [23–25]. In [24], the scholars presented the memristive CNNs model with parametric uncertainties and obtained some adequate conditions for guaranteeing the robust synchronization for the considered network. By utilizing impulsive functional strategy combined with the stability theory, Li et al. [25] presented some robust synchronization criteria for the CNNs with uncertainties.

Actually, passivity is also one of the most important behaviors of dynamics for CNs which can guarantee a system’s internal stabilization in system theory. Due to the potential applications in plenty of fields, e.g., fuzzy control and sliding mode control, the passivity problem of CNNs has been investigated extensively in the past years [26–33]. In [27], the authors proposed several passivity criteria for delayed CNNs. Ren et al. [28] analyzed the model of CRDNNs and established some passivity and pinning passivity conditions for the considered network. Unfortunately, the passivity results in aforementioned studies [26–33] are based on the case that the dimension of input is identical with output. As far as we know, only few scholars have addressed the problem of passivity for the network with nonidentical dimensional output and input [34–36]. The authors in [34] established some conditions for ensuring that the CRDNNs with the input and output in different dimensions achieve passivity. Ren et al. [36] discussed the (pinning) passivity problems of CNNs with nonidentical dimensional output and input and acquired some corresponding passivity criteria.

Note that the networks are composed of identical nodes in the aforementioned works [4–36]. Unfortunately, this case is very rare in the real-world networks. Consequently, the CNs consisting of nonidentical nodes in the same dimension [37–41] have been discussed firstly by researchers. Zhao et al. [39] dealt with the synchronization problem of CNs with nonidentical nodes. As a matter of fact, the networks constructed by nonidentical nodes of different dimensions can reflect more real networks in many circumstances. Note that the networks constructed by nonidentical nodes in different dimensions can exhibit different and even more sophisticated dynamical behaviors, which makes the passivity and synchronization methods for the networks with the same dimensional nonidentical nodes or identical nodes in the abovementioned works [4–41] invalid. Hence, it is necessary and meaningful to develop some new stabilization and synchronization strategies for the network with nonidentical nodes of different dimensions [42–45]. In [43], by designing appropriate decentralized controllers, the authors devoted to establishing stabilization and synchronization criteria for CNs consisting of nonidentical nodes. Up to now, only a few researchers considered CNNs constructed by the nonidentical nodes of different dimensions [46–48]. In [47], the authors investigated generalized synchronization of delayed CNNs with different dimensional nodes by making use of the Lyapunov functional method. To the best of knowledge, the problems of synchronization and passivity for CDDDRDNNs have not yet been investigated. Consequently, it is essential to put forth some efforts to study passivity and synchronization of CDDDRDNNs.

In terms of the above introduction, the main aim in this paper is to address passivity and synchronization of CDDDRDNNs without and with parametric uncertainties. On the one hand, we discuss the CDDDRDNNs without parametric uncertainties, and several conditions are derived to guarantee the considered network to achieve passivity and synchronization. On the other hand, the problems of robust synchronization and robust passivity for CDDDRDNNs with parametric uncertainties are also studied.

2. Preliminaries

Let the matrix , the notation signifies G is symmetric and negative (positive, seminegative, and semipositive) definite. represents the minimum (maximum) eigenvalue of the corresponding matrix. An open-bounded domain in with smooth boundary is defined by . For any , we have

Definition 1 (see [49]). Let and denote the input and output of a system. Assume that there exists a storage function which satisfiesfor any and , then the system with supply rate is dissipative. Moreover, a system is passive if the system is dissipative withwhere the matrix . Furthermore, assume that a system is dissipative within which , , and , then the system is strictly passive. Especially, if , then the system is called to be output-strictly passive; if , then the system is called to be input-strictly passive.

Lemma 2.1 (see [50]). Let be a cube and real-valued function satisfy . Then,where .

3. Passivity and Synchronization of CDDDRDNNs

3.1. Network Model

The CDDDRDNNs considered in this section is stated as follows:in which , denotes the state vector of ith neuron; ; ; ; , , and denote the activation functions for the lth neuron in neural network i; ; denotes the control input; the inner coupling matrix is defined by ; is a known matrix; is the coupling configuration matrix denoting coupling weight, which satisfies if node i and node j are connected, or else . In addition, the time-varying delay satisfies and .

Remark 1. As a special type of CNs, CNNs have attracted much attention due to their extensive applications on plenty of fields. So far, a large number of scholars have acquired some interesting research results about synchronization and passivity of CNNs [4–36]. Nevertheless, the considered CNs in the aforementioned literature are made up of identical nodes. As a matter of fact, it is utterly impractical that the networks have totally identical nodes in many practical situations. For instance, due to the differences of the parameters, it is impossible that the neurons in the nervous system of neural networks are entirely the same as each other. Consequently, it is significant to study CNNs composed of nonidentical nodes. To our knowledge, a few investigators have discussed CNs consisting of the same dimensional nonidentical nodes in recent years [37–41]. However, the networks with nonidentical nodes of different dimensions can describe more practical networks. In addition, the considered networks in these existing works did not take the reaction-diffusion terms into account. Therefore, we pay our attention on the CRDNNs with different dimensional nodes. As far as we know, this is our first step toward addressing the passivity and synchronization problems of CDDDRDNNs.
For network (6), the initial value condition and boundary value condition are described bywhere is bounded and continuous on and .
Suppose that the functions and satisfy the following global Lipschitz condition; there exist positive constants and such thathold for all .
Let the constant vector be an equilibrium point of an isolated node of network (6). Then,where .
For the error vector , we havewhere and .
The output vector of the network (11) is given as follows:where and are the known matrices.
In the whole paper, we denote

3.2. Passivity Analysis

It follows from (11) and (12) that the following system with output and input can be obtained as follows:where and .

Theorem 1. System (14) reaches output-strict passivity if there are matrices , and satisfyingwhere .

Proof. Construct the Lyapunov functional for system (14) as follows:Then,By means of the boundary condition and Green’s formula, one can obtain the following:where . Let , one hasThen, we obtainObviously, there is a real matrix satisfyingThus,Let , for , it follows from boundary condition (8) that . In terms of Lemma 2.1, we can getTherefore,In addition, we can derive thatSimilarly, we haveBy (25)–(27), one hasThus,where . On the basis of (16), one getsThen,for any and . In other words,where .
The following results can be deduced by using the similar method.

Corollary 1. System (14) realizes passivity if there are matrices and satisfyingwhere .

Corollary 2. System (14) realizes input-strict passivity if there exist matrices , and satisfyingwhere

Remark 2. To the best of our knowledge, the concept of passivity is proposed for the first time in circuit analysis and has discovered comprehensive potential applications in lots of areas after that. Over the past few decades, some scholars have investigated the passivity of CNNs and CRDNNs and derived many meaningful results [26–33]. Note that the passivity problem is solved in the abovementioned works based on the case that the input has identical dimension as the output. As far as we know, only a few passivity results have been obtained for the CRDNNs with different dimensional input and output until now [34–36]. However, the networks considered in these works are coupled by identical nodes. Therefore, we devote ourselves to studying the passivity of CRDNNs consisting of different dimensional nodes (i.e., CDDDRDNNs), in which the input has different dimensions as output. Hence, the acquired passivity results are more general and less conservative in this paper.

3.3. Synchronization Analysis

Definition 2. Network (6) achieves synchronization if for all ,under the condition .

Theorem 2. Network (6) is synchronized if there are matrices satisfyingwhere .

Proof. Choose the same Lyapunov functional as (17) for system (11), then,where .
From (40), we haveFrom (17) and (40), one knows that all terms of are bounded and is nonincreasing. Hence, converges to a finite nonnegative real number. According to (41), we can infer that exists and is finite. Furthermore, since is bounded, it is easily derived thatHence, is not only existing but also is a nonnegative real number. Next, we will prove that . If not, one can getObviously, there is a real constant such thatHence,where . According to (40) and (45), we obtainIt follows from (46) thatThis is unreasonable. Consequently,Then, we have . Hence, network (6) is synchronized.

4. Robust Passivity and Robust Synchronization of CDDDRDNNs

4.1. Network Model

Taking the parameter uncertainties into account, a CDDDRDNN with uncertain parameters is stated as follows:in which represent the same senses as in model (6). The parameters can be changed within a certain parameter range of precisions as follows:

In addition, for convenience, we denotewhere .

According to (10), we can get the error system of network (49) as follows:where the ranges of , and are described by using (50).

4.2. Robust Passivity Analysis

The output vector of system (52) is defined similarly as (12). In terms of (12) and (52), the error system with output and input can be described bywhere and .

Theorem 3. If there are matrices , and satisfyingwhere , then, system (53) under the given ranges of parameters (50) is output-robustly passive.

Proof. Choose the same Lyapunov functional as (17) for system (52). Then,Obviously,Similar to the deduction of (25), one can getIn addition,According to (56)–(59), one getsHence,From (54), one can getThen,for any and . In other words,in which .
The following results can be deduced by using the similar method.

Corollary 3. System (53) under the given ranges of parameters (50) is robustly passive if there are matrices satisfyingwhere .

Corollary 4. System (53) under the given ranges of parameters (50) is input-robustly passive if there are matrices , and satisfyingwhere .

4.3. Robust Synchronization Analysis

Definition 3. Network (49) under the given ranges of parameter (50) realizes robust synchronization if for all , ,under the condition .

Theorem 4. Network (49) under the given ranges of parameters (50) achieves robust synchronization if there are matrices satisfyingwhere .

Proof. Select the same Lyapunov functional as (17) for system (52); then,where .
Then, can be proved similarly as in Theorem 2. Consequently, network (49) under the given ranges of parameters (50) realizes robust synchronization.

Remark 3. In Section 3, we deal with the passivity and synchronization problems of CDDDRDNNs and establish some adequate conditions to achieve the passivity and synchronization of network (6). Note that the parameters in matrices of network model (6) are fixed. However, it is utterly unrealistic for the networks with some certain parameters due to the noises of environment and equipment restrictions in some practical situations [18–25]. Thus, it is necessary to consider the case that the parameters in matrices of network model (6) belong to some given ranges and investigate robust dynamical properties of the considered network. As far as we know, the robust synchronization and robust passivity of CDDDRDNNs with parametric uncertainties have never been studied. In this section, we present several robust synchronization and robust passivity criteria of CDDDRDNNs with uncertain parameters in Theorems 3 and 4 and Corollaries 3 and 4, respectively.

5. Numerical Examples

Example 1. Consider the following CDDDRDNNs:where , , and . The matrices are chosen as follows:

Case 1. Obviously, the equilibrium point of an isolated node of network (70) is and . It is easy to calculate the following matrices , and satisfying (15) and (16):and , whereOn the basis of Theorem 1, the network (70) with the input and output as described in (12) realizes output-strictly passivity. Figure 1 displays the simulation results.

Case 2. , and are similar as those in Case 1. By exploiting the MATLAB Toolbox, the matrix P satisfying (38) and (39) can be computed as follows:whereAccording to Theorem 2, the network (70) realizes synchronization. Figures 2–4 show the simulation results.

Example 2. Consider the following CDDDRNNs:in which , , , and . The matrices are chosen as follows: The parameters in network (76) can be changed in the precisions given as follows:

Case 1. Obviously, and the equilibrium point of an isolated node of network (76) is . It is not difficult to calculate the matrices , and satisfying (54) as follows:and , whereIn terms of Theorem 3, network (76) with the input and output as described in (12) under the given parameters defined in (78) realizes output-strictly passivity. Figure 5 displays the simulation results.

Case 2. , and are the same as those in Case 1. By exploiting the MATLAB Toolbox, the matrices P satisfying (68) can be computed as follows:whereAccording to Theorem 4, network (76) under the given ranges of parameters defined in (78) achieves robust synchronization. Figures 6–8 show the simulation results.

Remark 4. Section 3 is devoting to investigating the synchronization and passivity of CDDDRDNNs. First, the network model of CDDDRDNNs is presented in Section 3.1. In Section 3.2, the passivity of CDDDRDNNs with certain parameters is studied. Moreover, we establish some adequate conditions to ensure the network being output-strictly passive in Theorem 1, passive in Corollary 1, and input-strictly passive in Corollary 2, respectively. Then, in Theorem 2 of Section 3.3, a synchronization criterion is obtained for the considered network. Because the precise values of parameters are difficult to acquire because of noises of environment and equipment limitations, we address the problems of robust synchronization and passivity for CDDDRDNNs with parametric uncertainties in Section 4. The main distinction on two kinds of CDDDRDNNs in Section 3 and Section 4 is whether in the network model of CDDDRDNNs is uncertain or not. More precisely, in Section 4.1, we give the CDDDRDNN model with parametric uncertainties firstly. After that, a robust output-strict passivity criterion is proposed for CDDDRDNNs with parameter uncertainties in Theorem 3 of Section 4.2. Meanwhile, we establish the related robust passivity condition in Corollary 3 and robust input-strict passivity condition in Corollary 4, respectively. Then, a robust synchronization criterion is obtained for the considered network in Theorem 4 of Section 4.3. In Section 5, we select the most intricate output-strict passivity condition for verifying the validity of the obtained passivity results. Actually, the related passivity or input-strict passivity condition can be illustrated similarly. In order to avoid repetition, we omit the simulation results for showing the effectiveness of the obtained passivity results in corollaries. Therefore, in Case 1 and Case 2 of Example 1, the correctness of the output-strict passivity and synchronization conditions in Theorems 1 and 2 for CDDDRDNNs (70) with parametric certainties are demonstrated, respectively. Similarly, in Case 1 and Case 2 of Example 2, the robust output-strict passivity criterion and robust synchronization criterion in Theorem 3 and 4 for CDDDRDNNs (76) with uncertain parameters described by (78) are demonstrated, respectively.

6. Conclusion

The synchronization and passivity of CDDDRDNNs with and without parameter uncertainties have been investigated in this paper. First, several new criteria for CDDDRDNNs with parametric certainties have been derived to guarantee the passivity and synchronization by taking advantage of the Lyapunov functional method. Second, we have also studied the problems of robust synchronization and robust passivity for CDDDRDNNs with uncertain parameters. Third, two numerical examples have been provided to display the effectiveness of the obtained passivity and synchronization results. In our future work, it would be very interesting to study the pinning adaptive passivity and synchronization of CDDDRDNNs.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors wish to express their sincere appreciation to Prof. Jinliang Wang for the fruitful discussions with him and good suggestions which helped to improve this paper. This work was supported in part by the Natural Science Foundation of Tianjin City under Grant 18JCQNJC74300, in part by the China Scholarship Council (No.: 201808120044), and in part by the National Natural Science Foundation of China under Grant 61773285. Dr. Yang is supported in part by the Advanced Forming Research Centre (AFRC) and Lightweight Manufacturing Centre (LMC) under the Route to Impact Programme 2019-2020 funded by Innovate UK High Value Manufacturing Catapult (Grant no.: AFRC_CATP_1469_R2I-Academy).

References

B. A. Huberman and L. A. Adamic, “Internet: growth dynamics of the world wide web,” Nature, vol. 401, no. 16, pp. 23–25, 1999.
View at: Publisher Site | Google Scholar
S. H. Strogatz and I. Stewart, “Coupled oscillators and biological synchronization,” Scientific American, vol. 269, no. 6, pp. 102–109, 1993.
View at: Publisher Site | Google Scholar
Z. Zheng and L. Xie, “Finite-time path following control for a stratospheric airship with input saturation and error constraint,” International Journal of Control, vol. 92, no. 2, pp. 368–393, 2019.
View at: Publisher Site | Google Scholar
X. Yang, J. Cao, and J. Lu, “Synchronization of randomly coupled neural networks with Markovian jumping and time-delay,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 2, pp. 363–376, 2013.
View at: Publisher Site | Google Scholar
J. Wang, H. Zhang, Z. Wang, and Q. Shan, “Local synchronization criteria of markovian nonlinearly coupled neural networks with uncertain and partially unknown transition rates,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 47, no. 8, pp. 1953–1964, 2017.
View at: Publisher Site | Google Scholar
W. Wu and T. P. Chen, “Global synchronization criteria of linearly coupled neural network systems with time-varying coupling,” IEEE Transactions on Neural Networks, vol. 19, no. 2, pp. 319–332, 2008.
View at: Publisher Site | Google Scholar
J. D. Cao, G. R. Chen, and P. Li, “Global synchronization in an array of delayed neural networks with hybrid coupling,” IEEE Transactions on Systems, Man, and Cybernetics: Cybernetics, vol. 38, no. 2, pp. 488–498, 2008.
View at: Google Scholar
Z. Tang, J. H. Park, T. H. Lee, and J. Feng, “Mean square exponential synchronization for impulsive coupled neural networks with time-varying delays and stochastic disturbances,” Complexity, vol. 21, no. 5, pp. 190–202, 2016.
View at: Publisher Site | Google Scholar
H. Chen, P. Shi, and C.-C. Lim, “Exponential synchronization for markovian stochastic coupled neural networks of neutral-type via adaptive feedback control,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 7, pp. 1618–1632, 2017.
View at: Publisher Site | Google Scholar
H. Zhang, D. Gong, B. Chen, and Z. Liu, “Synchronization for coupled neural networks with interval delay: a novel augmented Lyapunov-Krasovskii functional method,” IEEE Transactions on Neural Networks and Learning Systems, vol. 24, no. 1, pp. 58–70, 2013.
View at: Google Scholar
A. Selivanov and E. Fridman, “Boundary observers for a reaction-diffusion system under time-delayed and sampled-data measurements,” IEEE Transactions on Automatic Control, vol. 64, no. 8, pp. 3385–3390, 2019.
View at: Publisher Site | Google Scholar
J.-L. Wang and H.-N. Wu, “Synchronization and adaptive control of an array of linearly coupled reaction-diffusion neural networks with hybrid coupling,” IEEE Transactions on Cybernetics, vol. 44, no. 8, pp. 1350–1361, 2014.
View at: Publisher Site | Google Scholar
J. L. Wang, H. N. Wu, and L. Guo, “Novel adaptive strategies for synchronization of linearly coupled neural networks with reaction-diffusion terms,” IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 2, pp. 429–440, 2014.
View at: Google Scholar
J.-L. Wang, H.-N. Wu, T. Huang, and S.-Y. Ren, “Pinning control strategies for synchronization of linearly coupled neural networks with reaction-diffusion terms,” IEEE Transactions on Neural Networks and Learning Systems, vol. 27, no. 4, pp. 749–761, 2016.
View at: Publisher Site | Google Scholar
S. Dharani, R. Rakkiyappan, and J. H. Park, “Pinning sampled-data synchronization of coupled inertial neural networks with reaction-diffusion terms and time-varying delays,” Neurocomputing, vol. 227, pp. 101–107, 2017.
View at: Publisher Site | Google Scholar
M. Xu, J.-L. Wang, and P.-C. Wei, “Synchronization for coupled reaction-diffusion neural networks with and without multiple time-varying delays via pinning control,” Neurocomputing, vol. 227, pp. 82–91, 2017.
View at: Publisher Site | Google Scholar
S.-X. Wang, Y.-L. Huang, and B.-B. Xu, “Pinning synchronization of spatial diffusion coupled reaction-diffusion neural networks with and without multiple time-varying delays,” Neurocomputing, vol. 227, pp. 92–100, 2017.
View at: Publisher Site | Google Scholar
Y. Du and R. Xu, “Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays,” ISA Transactions, vol. 53, no. 4, pp. 1015–1023, 2014.
View at: Publisher Site | Google Scholar
S. Li, X. Peng, Y. Tang, and Y. Shi, “Finite-time synchronization of time-delayed neural networks with unknown parameters via adaptive control,” Neurocomputing, vol. 308, pp. 65–74, 2018.
View at: Publisher Site | Google Scholar
Z. Cai, X. Pan, L. Huang, and J. Huang, “Finite-time robust synchronization for discontinuous neural networks with mixed-delays and uncertain external perturbations,” Neurocomputing, vol. 275, pp. 2624–2634, 2018.
View at: Publisher Site | Google Scholar
Y. Li, B. Luo, D. Liu, and Z. Yang, “Robust synchronization of memristive neural networks with strong mismatch characteristics via pinning control,” Neurocomputing, vol. 289, pp. 144–154, 2018.
View at: Publisher Site | Google Scholar
X. Ding, J. Cao, A. Alsaedi, F. E. Alsaadi, and T. Hayat, “Robust fixed-time synchronization for uncertain complex-valued neural networks with discontinuous activation functions,” Neural Networks, vol. 90, pp. 42–55, 2017.
View at: Publisher Site | Google Scholar
J. L. Liang, Z. D. Wang, Y. R. Liu, and X. H. Liu, “Robust synchronization of an array of coupled stochastic discrete-time delayed neural networks,” IEEE Transactions on Neural Networks, vol. 19, no. 11, pp. 1910–1921, 2008.
View at: Google Scholar
S. Yang, Z. Guo, and J. Wang, “Robust synchronization of multiple memristive neural networks with uncertain parameters via nonlinear coupling,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 45, no. 7, pp. 1077–1086, 2015.
View at: Publisher Site | Google Scholar
P. Li, J. Cao, and Z. Wang, “Robust impulsive synchronization of coupled delayed neural networks with uncertainties,” Physica A: Statistical Mechanics and its Applications, vol. 373, pp. 261–272, 2007.
View at: Publisher Site | Google Scholar
J.-L. Wang, M. Xu, H.-N. Wu, and T. Huang, “Finite-time passivity of coupled neural networks with multiple weights,” IEEE Transactions on Network Science and Engineering, vol. 5, no. 3, pp. 184–197, 2018.
View at: Publisher Site | Google Scholar
N. Li and J. Cao, “Passivity and robust synchronisation of switched interval coupled neural networks with time delay,” International Journal of Systems Science, vol. 47, no. 12, pp. 2827–2836, 2016.
View at: Publisher Site | Google Scholar
S.-Y. Ren, J. Wu, and P.-C. Wei, “Passivity and pinning passivity of coupled delayed reaction-diffusion neural networks with dirichlet boundary conditions,” Neural Processing Letters, vol. 45, no. 3, pp. 869–885, 2016.
View at: Publisher Site | Google Scholar
B.-B. Xu, Y.-L. Huang, J.-L. Wang, P.-C. Wei, and S.-Y. Ren, “Passivity of linearly coupled neural networks with reaction-diffusion terms and switching topology,” Journal of the Franklin Institute, vol. 353, no. 8, pp. 1882–1898, 2016.
View at: Publisher Site | Google Scholar
Y.-L. Huang, S.-Y. Ren, J. Wu, and B.-B. Xu, “Passivity and synchronization of switched coupled reaction-diffusion neural networks with non-delayed and delayed couplings,” International Journal of Computer Mathematics, vol. 96, no. 9, pp. 1702–1722, 2019.
View at: Publisher Site | Google Scholar
Y.-L. Huang, B.-B. Xu, and S.-Y. Ren, “Analysis and pinning control for passivity of coupled reaction-diffusion neural networks with nonlinear coupling,” Neurocomputing, vol. 272, pp. 334–342, 2018.
View at: Publisher Site | Google Scholar
Y.-L. Huang, S.-X. Wang, and S.-Y. Ren, “Pinning exponential synchronisation and passivity of coupled delayed reaction-diffusion neural networks with and without parametric uncertainties,” International Journal of Control, vol. 92, no. 5, pp. 1167–1182, 2019.
View at: Publisher Site | Google Scholar
Y.-L. Huang, S.-H. Qiu, and S.-Y. Ren, “Finite-time synchronisation and passivity of coupled memristive neural networks,” International Journal of Control, pp. 1–14, 2019.
View at: Publisher Site | Google Scholar
J.-L. Wang, H.-N. Wu, T. Huang, S.-Y. Ren, and J. Wu, “Passivity analysis of coupled reaction-diffusion neural networks with Dirichlet boundary conditions,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 47, no. 8, pp. 2148–2159, 2017.
View at: Publisher Site | Google Scholar
J.-L. Wang, H.-N. Wu, T. Huang, S.-Y. Ren, and J. Wu, “Passivity of directed and undirected complex dynamical networks with adaptive coupling weights,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 8, pp. 1827–1839, 2017.
View at: Publisher Site | Google Scholar
S.-Y. Ren, J. Wu, S.-X. Wang, and Y.-L. Huang, “Passivity and pinning control of coupled neural networks with and without time-varying delay,” Transactions of the Institute of Measurement and Control, vol. 40, no. 9, pp. 2708–2717, 2017.
View at: Publisher Site | Google Scholar
C. J. Xu, Y. Zheng, H. S. Su, and X. G. Liu, “Adaptive synchronization for nonlinear coupled complex network with nonidentical nodes,” in Proceedings of the 33rd Chinese Control Conference, pp. 5750–5754, Nanjing, China, July 2014.
View at: Google Scholar
S. Wang, H. Yao, S. Zheng, and Y. Xie, “A novel criterion for cluster synchronization of complex dynamical networks with coupling time-varying delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 2997–3004, 2012.
View at: Publisher Site | Google Scholar
J. Zhao, D. J. Hill, and T. Liu, “Synchronization of dynamical networks with nonidentical nodes: criteria and control,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 58, no. 3, pp. 584–594, 2011.
View at: Publisher Site | Google Scholar
X. Yang, Z. Wu, and J. Cao, “Finite-time synchronization of complex networks with nonidentical discontinuous nodes,” Nonlinear Dynamics, vol. 73, no. 4, pp. 2313–2327, 2013.
View at: Publisher Site | Google Scholar
S. Lin, Y. Huang, and S. Ren, “Event-triggered passivity and synchronization of delayed multiple-weighted coupled reaction-diffusion neural networks with non-identical nodes,” Neural Networks, vol. 121, pp. 259–275, 2020.
View at: Publisher Site | Google Scholar
M. Tan and W. Tian, “Finite-time stabilization and synchronization of complex dynamical networks with nonidentical nodes of different dimensions,” Nonlinear Dynamics, vol. 79, no. 1, pp. 731–741, 2015.
View at: Publisher Site | Google Scholar
Y. Wang, Y. Fan, Q. Wang, and Y. Zhang, “Stabilization and synchronization of complex dynamical networks with different dynamics of nodes via decentralized controllers,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 59, no. 8, pp. 1786–1795, 2012.
View at: Publisher Site | Google Scholar
L. Zhang, Y. Wang, and Y. Huang, “Synchronization for non-dissipatively coupled time-varying complex dynamical networks with delayed coupling nodes,” Nonlinear Dynamics, vol. 82, no. 3, pp. 1581–1593, 2015.
View at: Publisher Site | Google Scholar
L. Zhang, Y. Wang, Y. Huang, and X. Chen, “Delay-dependent synchronization for non-diffusively coupled time-varying complex dynamical networks,” Applied Mathematics and Computation, vol. 259, pp. 510–522, 2015.
View at: Publisher Site | Google Scholar
M. Tan, “Stabilization of coupled time-delay neural networks with nodes of different dimensions,” Neural Processing Letters, vol. 43, no. 1, pp. 255–268, 2016.
View at: Publisher Site | Google Scholar
Y. Huang, W. Chen, S. Ren, and Z. Zheng, “Analysis and pinning control for generalized synchronization of delayed coupled neural networks with different dimensional nodes,” Journal of the Franklin Institute, vol. 355, no. 13, pp. 5968–5997, 2018.
View at: Publisher Site | Google Scholar
S. Lin, Y. Huang, and S. Ren, “Analysis and pinning control for passivity of coupled different dimensional neural networks,” Neurocomputing, vol. 321, pp. 187–200, 2018.
View at: Publisher Site | Google Scholar
J.-L. Wang, Z. Qin, H.-N. Wu, and T. Huang, “Passivity and synchronization of coupled uncertain reaction-diffusion neural networks with multiple time delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 30, no. 8, pp. 2434–2448, 2019.
View at: Publisher Site | Google Scholar
J. G. Lu, “Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions,” Chaos, Solitons & Fractals, vol. 35, no. 1, pp. 116–125, 2008.
View at: Publisher Site | Google Scholar

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Copyright © 2020 Shanrong Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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