Matrix Measure Approach for Stability and Synchronization of Complex-Valued Neural Networks with Deviating Argument

Zhou, Wenbo; Li, Biwen; Zhang, Jin-E

doi:https://doi.org/10.1155/2020/8877129

Mathematical Problems in Engineering

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

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Synchronization, Estimation, Observation and Applications of Dynamical Network Systems

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Research Article | Open Access

Volume 2020 | Article ID 8877129 | https://doi.org/10.1155/2020/8877129

Matrix Measure Approach for Stability and Synchronization of Complex-Valued Neural Networks with Deviating Argument

Wenbo Zhou,¹Biwen Li,¹and Jin-E Zhang¹

Academic Editor: Sanbo Ding

Received21 Sept 2020

Revised17 Oct 2020

Accepted27 Oct 2020

Published16 Nov 2020

Abstract

This paper concentrates on global exponential stability and synchronization for complex-valued neural networks (CVNNs) with deviating argument by matrix measure approach. The Lyapunov function is no longer required, and some sufficient conditions are firstly obtained to ascertain the addressed system to be exponentially stable under different activation functions. Moreover, after designing a suitable controller, the synchronization of two complex-valued coupled neural networks is realized, and the derived condition is easy to be confirmed. Finally, some numerical examples are given to demonstrate the superiority and feasibility of the presented theoretical analysis and results.

1. Introduction

During the recent years, recurrent neural networks have always been an object with immense attention due to their specific self-learning ability, which makes neural networks have been extensively applied in combinatorial optimization, signal processing, parallel computing, and some other fields [1–8]. These widespread applications of neural networks are practically based on their affluent dynamic properties in theory, including stability, synchronization, dissipativity, and chaos. Therefore, some researchers have become more interested in the theoretical study about the dynamical behaviors of neural networks and achieved lots of excellent works [9–18].

In a large number of theoretical and applied results, complex signals frequently occur and complex-valued neural networks (CVNNs) are preferable. Actually, CVNNs could be regarded as an expansion about the real-valued neural networks (RVNNs) in a certain sense, in which the state variable, activation functions, and synaptic strength matrices are all complex-valued. Therefore, such networks have much more sophisticated features than the RVNNs in many aspects and accordingly make it capable to settle a lot of matters that cannot be settled by the real-valued ones [19]. As an example, when some real-world applications contain complex signals, complex-valued model can reduce and simplify the complexity of information processing because the phase and amplitude information of complex signal can be encoded simultaneously [20]. In consequence, it is very significant to research the dynamical behaviors of CVNNs, especially the stability and synchronization. In [21], master-slave CVNNs with time-varying delay have been researched, by the concepts of average impulsive gain and interval, and a suitable hybrid impulsive controller has been designed to get a condition for insuring the exponential synchronization of the coupled CVNNs. In [22], a class of memristor-based delayed CVNNs has been discussed, and some sufficient criteria have been obtained to achieve the exponential synchronization of such networks by the Lyapunov function method and the inequality technique. In [23], complex-valued BAM neural networks with time delays have been concerned, and based on the inequality techniques, upper right-hand Dini derivative concepts, and Halanay inequality, some novel conditions for the dissipativity are presented. In [24], the global asymptotic stability of fractional-order memristive delayed CVNNs has been investigated, and based on the Filippov framework and differential inclusion method, some sufficient conditions for the stability are proposed. Moreover, ultimate Mittag–Leffler synchronization between two fractional-order systems has been also achieved by establishing fractional differential inequality. For more interesting works on the complex-valued neural networks, see [25–27] and the references therein.

Nonlinear system may be seriously affected by deviating arguments in the running process [28]. As a hybrid dynamical system, the nonlinear control system with deviating argument integrates the peculiarities of continuous systems and discrete systems. For many problems in ecology, demography, and economics, the present decisions or current behaviors are often influenced by the past and future information. By modeling the deviating arguments, the deviating function can be used to describe the past and future information. Akhmet et al. [29] propose the neural networks with deviating argument and establish stability conditions by the second Lyapunov method. Then, the neural network model with deviating argument has become a new and important research direction. In [30], a class of neural networks with impulsive and piecewise constant arguments has been studied, and stability criterion is derived for the existence and uniqueness of the periodic solution by using the Razumikhin-type technique and the Banach fixed-point theorem. In [31], the fractional-order neural networks with generalized piecewise constant arguments have been investigated, and the existence for the unique equilibrium point has been proved; besides that, some Mittag–Leffler stability conditions have also been presented. In [32], a kind of stochastic neural networks has been discussed, which can be affected by advanced and retarded argument. Several criteria have been provided to ascertain the uniqueness and existence of solution in the form of algebraic inequalities; in addition, the global mean square exponential stabilization for such system has also been implemented by the stability theory of stochastic differential equation.

Matrix measures can trade off the influences of positive and negative values of matrices, while noticing that many previous studies about stability and synchronization are expressed in norm or algebraic form, which make the range of nonnegative constants to be limited, and thus matrix measure approach is more effective when the synaptic strengths include both positive value and negative value. Compared with the Lyapunov function approach, the matrix measure method allows us to express the Lyapunov function directly by matrix measures of variables or error vector. It is well known that there are many papers for analyzing the dynamic characteristics of RVNNs by matrix measure approach [33–35]. In the recent few years, the matrix measure approach has been used in CVNNs [36–39]. In [36], the matrix measure approach is used to study the stability behaviors of CVNNs with time-varying delays, and several stability conditions are obtained based on the Halanay inequality. And in [38], the matrix measure method is also employed to investigate the synchronization problem of CVNNs with time-varying delays, by separating CVNNs to the real and imaginary parts, and some criteria are presented to ensure exponentially synchronization for the addressed CVNNs. However, as far as we know, the stability and synchronization of CVNNs with deviating argument have not been investigated by using the matrix measure approach.

Motivated by the above discussions, we try to use matrix measure approach to analyze global exponential stability and synchronization for CVNNs with deviating argument. The main results about this paper have been organized as follows: (1) Matrix measure approach is used firstly to analyze the dynamic behaviors of CVNNs with deviating argument, which is susceptive about the sign of the system matrix entries, hence more efficient and convenient to dispose the network. (2) By introducing the deviating argument, our model is more practical. Without structuring any Lyapunov auxiliary function, global exponential stability conditions for CVNNs with deviating argument under two activation functions are derived. (3) A proper complex-valued feedback controller is designed, and a sufficient condition is obtained to realize global exponential synchronization between drive and response CVNNs with deviating argument.

The remaining contents about this paper are constituted by the following four parts. In Section 2, model description is introduced, two assumptions on activation functions are given, and some lemmas and definitions are proposed. In Section 3, two stability conditions and a synchronous condition for CVNNs with deviating argument are made by matrix measure approach. Then, Section 4 gives numerical calculations to prove the effectiveness of the derived main results. At last, the conclusion of this paper is summarized in Section 5.

Notations: denote and as the sets of nonnegative real numbers and natural numbers, i.e., , , respectively. , , , and represent the sets of real vectors with -dimensional, real matrices, complex vectors with -dimensional, and complex matrices. Let be the imaginary unit, i.e., , and is denoted as the identity matrix of corresponding dimension. Superscript “” represents the transpose of a vector or matrix. Fix two sequences with real-valued , , such that for all , , with as . represents the real (imaginary) part of matrix .

2. Preliminaries

We are centered on the CVNNs with deviating argument described by the differential equations as follows:in which means the state vector. , for , , if . with is the self-inhibition matrix. and denote the synaptic strength matrices. And the complex-valued output functions are represented as and . denotes the input vector. is defined as the initial state.

Remark 1. For all , , function . System (1) is said to be an advanced system if . On the other hand, system (1) is said to be a retarded system if . Therefore, the deviating function causes (1) to become a mixed system. For the derivation of the main results, some assumptions about the activation functions as follows are necessary.

Assumption 1. Denote with . and satisfy . Moreover, they can be expressed as follows:where , and , , , and satisfy the conditions as follows:in which are arbitrary numbers in , and , , , and are known positive constants.
Furthermore, the nonlinear functions , and are usually to be limited with greater condition.

Assumption 2. Let with . and satisfy . Moreover, they can be expressed as follows:where , and , , , and satisfy that with :in which , , , and are known positive constants.

Remark 2. In many existing literature studies, the complex-valued activation function cannot be explicitly decomposed into real part and imaginary part. Under this condition, it is generally assumed that the activation function satisfies the Lipschitz condition on the complex domain. Then, the dynamic behaviors of CVNNs can be analyzed in the similar approach as RVNNs. However, if activation functions do not satisfy the Lipschitz condition, we have to separate the CVNNs into its real and imaginary parts. Moreover, activation functions are mostly restricted by certain conditions such as smooth, bounded, and nonconstant in RVNNs. But such restrictions for the activation functions of CVNNs cannot be allowed, the functions might become a scalar over the all complex field on the basis of Liouville’s theorem. Therefore, choosing the right activation function for the CVNNs is one of the main challenges.
With the above assumptions, let , we can rewrite system (1) asor in a more simplified form with the initial statein whichFor further discussion, several important definitions need to be presented and two useful lemmas are introduced as follows, which are employed for the later theoretical results.

Definition 1. (see [33]). Give a matrix with and the following is the matrix measure of :The norms of matrix are defined as follows:and then the matrix measure can be given as follows:

Definition 2. For any , if there exist constants and and the following inequality holdsthen the equilibrium point of system (7) is called globally exponentially stable.

Lemma 1 (see [36]). The following are the characteristics of matrix measure mentioned in Definition 1 for all :(1)(2)(3)

Lemma 2 (see [36]). The inequality holds, ifwhere , , , , and with when , otherwise, .

3. Main Results

After the previous introduction and preparations, in this section, the dynamic performance of the CVNNs (1) with deviating argument is explored. Some conditions about the global exponential stability of our model are derived by the matrix measure approach and several lemmas. Moreover, by reasonably designing the feedback controller, the exponential synchronism of the master-slave CVNNs with deviating argument is achieved.

3.1. Global Exponential Stability

Suppose system (7) has an equilibrium point ; then

By settingthus we can convert system (7) into the following differential equation:where

Lemma 3. Assumed be a solution of system (16), and there exists a scalar such that . If , thenfor all , where with , , , , and .

Proof. For any , , it follows thatUnder Assumption 1, we haveThe following inequalities can be obtained by using the same method as (20):By substituting inequalities (20)–(23) into (19), we haveBy Gronwall’s inequality, we haveFurthermore, for , it followsTogether with (25) and (26),thenfor . For all , (18) holds.

Theorem 1. Let Assumption 1 hold, if the following condition satisfieswhere , , , , and , then the system (7) is globally exponentially stable.

Proof. Differentiating , we can derive thatin whichSubstitute inequalities (20)–(23) into (31), and it follows from (30) thatBy definition of upper right-hand Dini derivative and Lemma 3, one obtainswhere , and thenwhich can be claimed that system (6) achieves global exponential stability, that is, model (1) is also globally exponentially stable.

Remark 3. In a finite-dimensional space, diverse norms can be considered equivalent and satisfy the relationship , whereas the corresponding matrix measure could not satisfy the same condition. Therefore, if we can find the matrix norm and measure that meet the inequality (29), then Theorems 1 holds.

Remark 4. As everyone knew that there are some differences between matrix norms and matrix measures. The matrix norm has to be nonnegative, but matrix measures can be assigned positive or negative values. Moreover, , whereas matrix measures are generally sign-sensitive in that . If the connection weights include positive and negative values, the measure method will be more accurate and extensive than the norm form previously. Therefore, the results obtained by matrix measure in this paper are more excellent.

Theorem 2. Under Assumption 2, the global exponential stability for system (7) can be guaranteed if the following condition satisfieswhere , , , , , , , and

Proof. DefineBy rewriting by its definition, thenwhereFrom Assumption 2, the following inequalities can be holdSimilarly, we can also havewhereBy substituting (38) and (41) into (31), and also considering (22) and (23), one hasConsequently, from Lemma 3, we can getWe have the following inequality by Lemmas 1 and 2:Thus, one concludes thatwhere , and thenThe condition (47) guarantees that network (7) and network (1) are globally exponentially stable by Definition 2.

3.2. Global Exponential Synchronization

Consider the corresponding response system of drive system (1) as following differential equations:where with , and denotes the controller.

Obviously, system (48) can also be divided into the following forms:

Then subtract the master network (6) from the slave network (49), define synchronization errors , and then the error system can be obtained as follows:in which

The control input is designed as follows:where matrices , , , and are positive definite and diagonal.

Definition 3. For all , if there exist constants and and the following inequality holdsthen the error system (50) achieves global exponential stability under controller (52), that is, the master systems (6) and the slave system (49) under the controller (52) are globally exponentially synchronizable.

Theorem 3. Suppose there exists a scalar such that . If the following conditions hold(1)(2), where , , , ,then global exponential synchronization can be achieved for the master network (6) and slave network (49).

Proof. Assume and are random solutions of network (1) and network (48), respectively. By differentiating , we can easily getwherein whichFrom Assumption 1, we can getand the following inequality can be also obtained:where , , and .
By substituting inequalities (56)–(59) into (55), we havein which .

Fix , for any , from inequalities (58) and (59), it follows thatin which .
Then the same as Lemma 3, finally we obtainfor , and .
Further, from (60) and (62), we can getwhere , thenFrom the above discussions, the master network (6) and slave network (49) achieve global exponential synchronization by the feedback controller (52).

Remark 5. Synchronization is different from the stabilization of single network, which mainly happens in multiple networks. Actually, some alterable coupling between networks must exist for synchronism achieved, and over the course of time, the errors between the coupled networks get more smaller until zero. We employ drive-response systems (1) and (48) to achieve exponential synchronization in the process of describing our model. Theoretically, the error system (50) can stabilize after a long enough time by several proper feedback control designs, and thus the coupled systems (1) and (48) can be said exponentially synchronized.

Remark 6. Based on the Lyapunov functions method, many papers on the dynamic characteristics of neural networks have been well done. In fact, choosing a desired Lyapunov function is considerably difficult, and the procedure of proofing the stability/synchronization for neural networks is very complicated and cumbersome. Matrix measure approach allows us to express the Lyapunov function directly in terms of matrix measures of variables or error vectors. Therefore, the criteria had been obtained more concisely.

Remark 7. It should be noted that CVNN (1) with deviating argument has more complex dynamic behaviours by the accession of deviating function . Compared with traditional neural networks, such system can not only rely on the present state, but also the previous and subsequent states, whereas many published works can only consider the present one. As a matter of fact, the deviating argument can be used to denote the past and future messages, that is to say, our theoretical analysis and results are more complex and comprehensive.

Remark 8. By decomposing the considered CVNNs into real and imaginary parts and then combining them in matrix form, we obtain an equivalent -dimensional RVNNs. Based on the definitions of matrix norm and measure, we use the matrix measure method to analyze the dynamic changes of such RVNNs, which increases the computation but greatly reduces the difficulty of analysis.

4. Illustrative Examples

In order to verify the credibility of the derived main results, based on computer emulation, two examples are given in this section.

Example 1. We focus on a two-dimensional CVNN model described as follows:with two real-valued sequences and , the deviating function , if , , and for with , , and the activation functions are as follows:Then, we can easily get ,It is obvious that Assumption 1 holds with . Here we take , then , , , , and . Then, , , and .
Hence, we have . According to Theorem 1, system (65) is globally exponentially stable.
The evolutionary behaviors of the solutions for network (65) under the following eight initial states are described in Figures 1 and 2. Case 1: ; Case 2: ; Case 3: ; Case 4: ; Case 5: ; Case 6: ; Case 7: ; and Case 8: .

(a)

(b)

(a)

(b)

Example 2. We focus on the following two-dimensional complex-valued master and slave networks with deviating argument.with two real-valued sequences and , the deviating function , if , , , and control input is taken as , with .
For with , the activation functions are as follows:It is clear that Assumption 1 holds with .
Here we take , and it is obvious that , , , , and . Then, , , and .
Hence, we have . On the basis of Theorem 3, the exponential synchronism of master network (68) and slave network (69) can be derived. The trajectories of the solutions are demonstrated in Figures 3 and 4 depicts synchronization errors between and .

(a)

(b)

(c)

(d)

5. Conclusion

The investigation of the dynamic behavior of CVNNs has attracted widespread considerations. However, based on the matrix measure approach, the dynamic analysis about CVNNs with deviating argument has scarcely been researched. We explore the stability/synchronization problem for CVNNs with deviating argument via matrix measure method. As a hybrid dynamical system, the CVNNs with deviating argument integrate the peculiarity of successive and discrete networks, and it has the prospective to represent the property of its inward principle. In addition, most of the research studies about the dynamic performance of neural networks are achieved by Lyapunov functions method. Under the characteristics of the matrix measure method, we do not need to structure any auxiliary functions, and under two nonlinear activation functions, some criteria for ascertaining the stability of CVNNs with deviating argument are obtained. Moreover, we achieved the global exponential synchronism between the master network and the slave network. Furthermore, we are considering extending this method to fractional-order networks with deviating argument to solve more practical problems.

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 work of the first author for master student Wenbo Zhou was done under the direction of Assistant Professor Jin-E Zhang. This work was supported by the Natural Science Foundation of China under Grants 61976084 and 61773152.

References

C. Zhou, X. Zeng, J. Yu, and H. Jiang, “A unified associative memory model based on external inputs of continuous recurrent neural networks,” Neurocomputing, vol. 186, pp. 44–53, 2016.
View at: Publisher Site | Google Scholar
H. Huang, G. Feng, and J. D. Cao, “Robust state estimation for uncertain neural networks with time-varying delay,” IEEE Transactions on Neural Networks, vol. 19, no. 8, pp. 1329–1339, 2008.
View at: Google Scholar
Y. Shen and J. Wang, “Robustness analysis of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances,” IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 1, pp. 87–96, 2012.
View at: Google Scholar
Z. Guo, J. Wang, and Z. Yan, “Attractivity analysis of memristor-Based cellular neural networks with time-varying delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 4, pp. 704–717, 2014.
View at: Publisher Site | Google Scholar
A. Wu and Z. Zeng, “Lagrange stability of neural networks with memristive synapses and multiple delays,” Information Sciences, vol. 280, pp. 135–151, 2014.
View at: Publisher Site | Google Scholar
Q. Luo, Z. Zeng, and X. Liao, “Global exponential stability in Lagrange sense for neutral type recurrent neural networks,” Neurocomputing, vol. 74, no. 4, pp. 638–645, 2011.
View at: Publisher Site | Google Scholar
T. Huang, C. Li, S. Duan, and J. A. Starzyk, “Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects,” IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 6, pp. 866–875, 2012.
View at: Publisher Site | Google Scholar
S. Ding and Z. Wang, “Event-triggered synchronization of discrete-time neural networks: a switching approach,” Neural Networks, vol. 125, pp. 31–40, 2020.
View at: Publisher Site | Google Scholar
S. Ding, Z. Wang, and N. Rong, “Intermittent control for quasisynchronization of delayed discrete-time neural networks,” IEEE Transactions on Cybernetics, vol. 1, 2020, In press.
View at: Publisher Site | Google Scholar
A. Wu and Z. Zeng, “Global Mittag-Leffler stabilization of fractional-order memristive neural networks,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 1, pp. 206–217, 2017.
View at: Publisher Site | Google Scholar
A. Wu and Z. Zeng, “Output convergence of fuzzy neurodynamic system with piecewise constant argument of generalized type and time-varying input,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 46, no. 12, pp. 1689–1702, 2016.
View at: Publisher Site | Google Scholar
A. Wu and Z. Zeng, “Lagrange stability of memristive neural networks with discrete and distributed delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 4, pp. 690–703, 2014.
View at: Publisher Site | Google Scholar
A. Wu and Z. Zeng, “Exponential stabilization of memristive neural networks with time delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 12, pp. 1919–1929, 2012.
View at: Publisher Site | Google Scholar
A. Wu, H. Liu, and Z. Zeng, “Observer design and H∞ performance for discrete-time uncertain fuzzy-logic systems,” IEEE Transactions on Cybernetics, vol. 1, 2020, In press.
View at: Publisher Site | Google Scholar
J. Liu, H. Q. Wu, and J. D. Cao, “Event-triggered synchronization in fixed time for semi-Markov switching dynamical complex networks with multiple weights and discontinuous nonlinearity,” Communications in Nonlinear Science and Numerical Simulation, vol. 90, Article ID 105400, 2020.
View at: Google Scholar
M. Liu, H. Q. Wu, and W. Zhao, “Event-triggered stochastic synchronization in finite time for delayed semi-Markovian jump neural networks with discontinuous activations,” Computational & Applied Mathematics, vol. 39, no. 2, Article ID 118, 2020.
View at: Publisher Site | Google Scholar
W. Zhao and H. Q. Wu, “Fixed-time synchronization of semi-Markovian jumping neural networks with time-varying delays,” Advances in Difference Equations, vol. 2018, Article ID 213, 2018.
View at: Google Scholar
X. Peng, H. Wu, and J. Cao, “Global nonfragile synchronization in finite time for fractional-order discontinuous neural networks with nonlinear growth activations,” IEEE Transactions on Neural Networks and Learning Systems, vol. 30, no. 7, pp. 2123–2137, 2019.
View at: Publisher Site | Google Scholar
M. Bohner, V. Sree Hari Rao, and S. Sanyal, “Global stability of complex-valued neural networks on time scales,” Differential Equations and Dynamical Systems, vol. 19, no. 1-2, pp. 3–11, 2011.
View at: Publisher Site | Google Scholar
Z. Wang, X. H. Wang, Y. X. Li, and X. Huang, “Stability and Hopf bifurcation of fractional-order complex-valued single neuron model with time delay,” International Journal of Bifurcation and Chaos, vol. 27, no. 13, Article ID 1750209, 2017.
View at: Publisher Site | Google Scholar
Y. Kan, J. Lu, J. Qiu, and J. Kurths, “Exponential synchronization of time-varying delayed complex-valued neural networks under hybrid impulsive controllers,” Neural Networks, vol. 114, pp. 157–163, 2019.
View at: Publisher Site | Google Scholar
D. Liu, S. Zhu, and E. Ye, “Synchronization stability of memristor-based complex-valued neural networks with time delays,” Neural Networks, vol. 96, pp. 115–127, 2017.
View at: Publisher Site | Google Scholar
C. Rajivganthi, F. A. Rihan, and S. Lakshmanan, “Dissipativity analysis of complex-valued BAM neural networks with time delay,” Neural Computing and Applications, vol. 31, no. 1, pp. 127–137, 2017.
View at: Publisher Site | Google Scholar
J. J. Chen, B. S. Chen, and Z. G. Zeng, “Global asymptotic stability and adaptive ultimate Mittag-Leffler synchronization for a fractional-order complex-valued memristive neural networks with delays,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 49, no. 12, pp. 2519–2535, 2018.
View at: Google Scholar
X. F. Chen, Q. K. Song, Z. J. Zhao, and Y. R. Liu, “Global -stability analysis of discrete-time complex-valued neural networks with leakage delay and mixed delays,” Neurocomputing, vol. 175, pp. 723–735, 2015.
View at: Google Scholar
R. Rakkiyappan, R. Sivaranjani, G. Velmurugan, and J. Cao, “Analysis of global O (t − α)stability and global asymptotical periodicity for a class of fractional-order complex-valued neural networks with time varying delays,” Neural Networks, vol. 77, pp. 51–69, 2016.
View at: Publisher Site | Google Scholar
X. Wang, Z. Wang, Q. Song, H. Shen, and X. Huang, “A waiting-time-based event-triggered scheme for stabilization of complex-valued neural networks,” Neural Networks, vol. 121, pp. 329–338, 2020.
View at: Publisher Site | Google Scholar
J.-E. Zhang, “Robustness analysis of global exponential stability of nonlinear systems with deviating argument and stochastic disturbance,” IEEE Access, vol. 5, pp. 13446–13454, 2017.
View at: Publisher Site | Google Scholar
M. U. Akhmet, D. Aruğaslan, and E. Yılmaz, “Stability analysis of recurrent neural networks with piecewise constant argument of generalized type,” Neural Networks, vol. 23, no. 7, pp. 805–811, 2010.
View at: Publisher Site | Google Scholar
T. Yu and D. Cao, “Stability analysis of impulsive neural networks with piecewise constant arguments,” Neural Processing Letters, vol. 47, no. 1, pp. 153–165, 2017.
View at: Publisher Site | Google Scholar
A. Wu, L. Liu, T. Huang, and Z. Zeng, “Mittag-Leffler stability of fractional-order neural networks in the presence of generalized piecewise constant arguments,” Neural Networks, vol. 85, pp. 118–127, 2017.
View at: Publisher Site | Google Scholar
L. Liu, A. Wu, Z. Zeng, and T. Huang, “Global mean square exponential stability of stochastic neural networks with retarded and advanced argument,” Neurocomputing, vol. 247, pp. 156–164, 2017.
View at: Publisher Site | Google Scholar
W. L. He and J. D. Cao, “Exponential synchronization of chaotic neural networks: a matrix measure approach,” Nonlinear Dynamics, vol. 55, no. 1-2, pp. 55–65, 2009.
View at: Publisher Site | Google Scholar
H. Bao, J. H. Park, and J. Cao, “Matrix measure strategies for exponential synchronization and anti-synchronization of memristor-based neural networks with time-varying delays,” Applied Mathematics and Computation, vol. 270, pp. 543–556, 2015.
View at: Publisher Site | Google Scholar
Y. Li and C. Li, “Matrix measure strategies for stabilization and synchronization of delayed BAM neural networks,” Nonlinear Dynamics, vol. 84, no. 3, pp. 1759–1770, 2016.
View at: Publisher Site | Google Scholar
W. Gong, J. Liang, and J. Cao, “Matrix measure method for global exponential stability of complex-valued recurrent neural networks with time-varying delays,” Neural Networks, vol. 70, pp. 81–89, 2015.
View at: Publisher Site | Google Scholar
Q. Tang and J. G. Jian, “Matrix measure based exponential stabilization for complex-valued inertial neural networks with time-varying delays using impulsive control,” Neurocomputing, vol. 273, pp. 251–259, 2017.
View at: Google Scholar
D. Xie, Y. Jiang, and M. Han, “Global exponential synchronization of complex-valued neural networks with time delays via matrix measure method,” Neural Processing Letters, vol. 49, no. 1, pp. 187–201, 2018.
View at: Publisher Site | Google Scholar
L. Li, X. Shi, and J. Liang, “Synchronization of impulsive coupled complex-valued neural networks with delay: the matrix measure method,” Neural Networks, vol. 117, pp. 285–294, 2019.
View at: Publisher Site | Google Scholar

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Copyright © 2020 Wenbo Zhou 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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