Robust Exponential Stability Analysis of Switched Neural Networks with Interval Parameter Uncertainties and Time Delays

Xu, Xiaohui; Xue, Huanbin; Peng, Yiqiang; Zhang, Jiye

doi:https://doi.org/10.1155/2018/6318986

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

Research Article | Open Access

Volume 2018 | Article ID 6318986 | https://doi.org/10.1155/2018/6318986

Robust Exponential Stability Analysis of Switched Neural Networks with Interval Parameter Uncertainties and Time Delays

Xiaohui Xu,¹Huanbin Xue,²Yiqiang Peng,³and Jiye Zhang⁴

Academic Editor: Yongping Pan

Received11 May 2018

Revised22 Jul 2018

Accepted05 Aug 2018

Published23 Sept 2018

Abstract

In this paper, the stability of switched neural networks (SNNs) with interval parameter uncertainties and time delays is investigated. First, the conditions for the existence and uniqueness of the equilibrium point of the system are discussed. Second, the average dwell time approach and M-matrix property are employed to obtain conditions to ensure the globally exponential stability of the delayed SNNs under constrained switching. Third, by resorting to inequality technique and the idea of vector Lyapunov function, sufficient condition to ensure the robust exponential stability of the delayed SNNs under arbitrary switching is derived. The form of the constructed Lyapunov functions is simple, which has certain commonality in studying delayed SNNs, and the proposed results not only are explicit but also reveal the relationship between the constrained switching and the arbitrary switching of the SNNs. Finally, two numerical examples are presented to illustrate the effectiveness and less conservativeness of the main results compared with the existing literature.

1. Introduction

In the past years, neural networks have been widely studied and successfully applied to various realms such as dynamic optimization, associative memory, and pattern recognition and to solve nonlinear algebraic equations and so on [1–5]. In the real world, the connections among different nodes of the networks are not always fixed or consistent, which frequently result in link failure and new link creation. Therefore, the abrupt changes in the structures and parameters of the neural networks often occur, which bring about switchings among certain different topologies and the instability of the networks [6]. In application’s point of view, a fundamental problem of applying neural networks is stability. This is a prerequisite for ensuring that the developed networks can work normally [7–10]. Thus, a popular topic about the stability analysis and stabilization of SNNs has been considered in [11–24].

A switched neural network is a hybrid system, which is essentially composed of a family of subnetworks and a switching signal which defines a specially designated subnetwork being activated at each instant of time. SNNs have attracted significant attention and have been successfully applied to many fields such as artificial intelligence, high-speed signal processing, and gene selection in DNA microarray analysis [25–28]. Generally, a switching system can be described by the following differential equation: where is a family of functions parameterized by some index set and switching signal is a piecewise constant and right continuous function of time mapping from to . The original motivation for the study on switched systems comes partly from that switching among different systems may cause many nonlinear system behaviors such as chaos and multiple limit cycles [29]. In recent years, switched systems have gained increasing attention because many practical systems (for example, constrained robotics, computer-controlled systems, and automated highway systems) can be modeled as switched systems. Furthermore, from the point of view of control, multicontroller switching is an effective way to deal with complex systems. It is well-known that time delays are inevitable in a practical control design which usually leads to unsatisfactory performances and the stability of the dynamic systems may even be destroyed with the increase of delays [30–35]. Attributing to the interaction among the discrete dynamics, continuous dynamics, and time delays, the behaviors of delayed SNNs are very complicated. Besides, due to many inevitable factors such as modelling errors and external perturbations, the models certainly contain uncertainties which can have a serious effect on the dynamical behavior of the systems. To analyze the robustness of the SNNs, one feasible method is to assume that the parameters are included in certain intervals [36]. Therefore, the robust stability analysis of SNNs with interval parameter uncertainties and time delays is of practical and theoretical importance.

For switched dynamical systems, the unpredictable change of system dynamics, such as abrupt perturbation of external environment or sudden change of the system structure due to the failure of a component, may cause the sudden change of the switching signal. In these cases, in order to keep the system working, the system should be stable under arbitrary switching. A typical approach for the stability analysis of switched dynamical systems with arbitrary switching signal is to search for a suitable common Lyapunov function (CLF) such that the rate of the decrease of along the trajectories of systems is not affected by switching (see, e.g., [37–40] and the references therein). If the CLF for the systems does not exist or is not known, in this case, we can study the stability of the system by using multiple Lyapunov functions (MLFs) , (see [37, 41, 42]). However, it is worth noting that to apply this MLF method, one needs to know some information of the state at each switching time. This is to be contrasted with the Lyapunov second method, which do not need to know the knowledge of the solutions. For example, Wu et al. studied the exponential stability of delayed SNNs by using a linear matrix inequality approach and an average dwell time method [12]; based on the piecewise Lyapunov function technique and average dwell time approach, the problem of the exponential stability of SNNs with constant and time-varying delays was investigated, respectively, in [43] and in [44]; by resorting to a novel delay division method, the stability analysis for uncertain SNNs with mixed time-varying delays was addressed. A common feature in these articles is that they all resort to scalar Lyapunov function (or functional). In this paper, the stability of the delayed SNNs with switching signal will be studied by using the idea of vector Lyapunov function with simple forms, which have certain commonality in studying SNNs, and this is the main reason why the obtained results in this paper have less conservativeness. By using the M-matrix property and average dwell time approach, the differential inequalities with time delays will be constructed. By the stability analysis of the differential inequalities, the sufficient conditions to ensure the robust exponential stability of the SNNs under arbitrary switching and constrained switching will be obtained.

Compared with the existing results on SNNs, the contributions of this paper are listed as follows: (a) the forms of the constructed Lyapunov functions are simple, which have certain commonality in studying delayed SNNs under arbitrary switching; (b) unlike asymptotic stability, we analyze the exponential stability of SNNs which include uncertainty and time delays, and the exponential convergence rate can also be obtained; (c) the obtained results not only have less conservativeness but also reveal the relationship between the constrained switching and the arbitrary switching of the delayed SNNs; and (d) comparing with most of the previous results obtained by linear matrix inequalities approach (to apply LMIs approach, one has to determine too many unknown parameters), the proposed criteria are straightforward, which are conducive to practical applications.

Notation. Let denote a column vector of (the symbol “” denotes transpose), denote , and denote a vector norm defined by . For , means that each pair of the corresponding elements of and satisfies the inequality “>.” For matrix , denote . denotes the set of continuous functions mapping from to .

2. Preliminaries

The model of a delayed SNNs can be described by the delayed differential equations as follows: where , is the number of neurons, is the state of neuron at time , is the switching signal, which is a piecewise constant and right continuous function of time, and means that the th subnetwork is activated. denotes the neuron self-feedback coefficient matrix of the subnetwork, and represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the networks and external inputs; is the activation functions of neurons at time ; and are the connection weight matrices of the th subnetwork, and and denote the connection strengths of the th neuron on the th neuron at time and , respectively; the delay is the bounded function with and ; is the constant external input vector of the th subnetwork. is the initial condition of the system, where .

We assume that the switching signal is unknown a priori. Corresponding to the switching signal , we have a switching sequence , which means that the th subsystem is activated when . We also assume that there is only finite switching in any finite interval and satisfy the following conditions.

Assumption 1. Each activation function in the delayed SNNs (2) is assumed to satisfy for any , where and are known constant scalars and .

Remark 1. Assumption 1 was first proposed in [45]. The constants and in this assumption are allowed to be any real number (positive, negative, or zero). Therefore, the activation functions can be nonmonotonic, which are more general than commonly used Lipschitz conditions and sigmoid functions. Such assumption is very useful to obtain less conservative results.
To facilitate the following analysis, let with . In order to study the stability of SNNs under parameter uncertainties, for , the matrices are intervalized as follows: Define

Definition 1. For the delayed SNNs (2), the equilibrium point is said to be robustly exponentially stable if for each , , and , there exist constants and such that where .

3. Existence and Uniqueness of the Equilibrium Point

The purpose of the present section is to give a sufficient condition which ensures that the equilibrium point of each subsystem satisfies the existence and uniqueness, which implies that for any initial condition , system (2) admits a solution which exists in a maximal interval , where .

Definition 2. A real matrix is said to be an M-matrix if , and all successive principal minors of are positive.

Lemma 1 ([46]).
Let be an matrix with nonpositive off-diagonal elements. Then the following statements are equivalent: (i) is an M-matrix(ii)There exists a vector such that .

Definition 3. A mapping is a homeomorphism of onto itself if , is one to one, is onto, and the inverse mapping , where denotes the set of continuous functions.

Lemma 2 ([47]).
If satisfies the following conditions: (i) is injective on (ii) as Then is a homeomorphism of .

Theorem 1. Under Assumption 1, if for all , are nonsingular M-matrices, then for each specified switching signal , system (2) has a unique equilibrium point.

Proof 1. Because the equilibrium point of subsystems, satisfies the following equation: for and . Let where for . In the following, we will give a proof that are homeomorphisms of onto itself.
First, we prove that are injective mappings on . Actually, if there exist vectors , , and such that ; then for and . From Assumption 1, it can be derived that for . That is, Let . Obviously, have nonpositive off-diagonal entries and which implies that are nonsingular M-matrices. From Theorem 2.3 of [48], we can get . That is, which is a contradiction. As a result, are injective mappings on .
Next, we prove that as .
Because are nonsingular M-matrices, we know that there exist positive diagonal matrices , which make matrices positively definite. Let where for . Calculate Using Schwartz inequality, we have When , we get which implies as .
Since implies , by Lemma 2, we know that are homeomorphisms of . So each subnetwork has a unique equilibrium point. Therefore, for each specified switching signal , system (2) has a unique equilibrium point. The proof is completed.

4. Exponential Stability of the Delayed SNNs

4.1. Exponential Stability under Constrained Switching

In this section, we will give a sufficient condition ensuring the global exponential stability of delayed SNNs (2) by using the average dwell time method. Let be an M-matrix; we denote

Definition 4 (see [37]).
Let denote the number of discontinuities of a switching signal on an interval . is called the average dwell time, if for any and , hold.

Theorem 2. Under Assumption 1, if for all , are nonsingular M-matrices, then for all , , and any external input , the delayed SNN (2) is robustly exponentially stable for any switching signal with the average dwell time satisfying where is determined by inequalities for some given and with

Proof 2. According to Theorem 1, we know that if are M-matrices, then the system has a unique equilibrium point for each specified switching signal. Let be an equilibrium point of system (2) and be any solution of system (2). Denote , , and ; then system (2) can be rewritten as with .
Due to being M-matrices, by Lemma 1 (ii), we know that there exist and such that Consider a Lyapunov functional candidate Calculating the upper right derivative of along the solutions of (25), we get Defining functions, Obviously, . Since are continuous functions, there exist , such that . Let ; we can get . Combining it with inequality (28), we get So for , For convenience, we denote when . That is, the th subnetwork is activated for ; then where Let ; we can get Combining (31) and (34) yields When , the subnetwork is activated; then where Combining (27), (35), and (36) yields Let ; (38) becomes Let and ; we have when and . According to Definition 1, equilibrium point system (2) is robustly exponentially stable. The proof is completed.

Remark 2. For all , are M-matrices which mean that delayed SNN (2) is globally exponentially stable under constrained switching. From the definitions of functions, we know that the value of relies on vector . So, for obtaining the maximum convergence rate or the minimum average dwell time , one can solve the optimization problem under constraint conditions .

4.2. Exponential Stability under Arbitrary Switching

Define the indicator function where with . Therefore, delayed SNN system (25) can be described as follows:

For any switching signal, only one subnetwork is activated at any time, so it follows that .

Theorem 3. Under Assumption 1, the equilibrium point of delayed SNNs (2) is robustly exponentially stable for all , , in , and any switching signal if the following conditions are satisfied: (i), are nonsingular M-matrices(ii) is nonempty.Moreover, the exponential convergence rate of system (2) is equal to , which is determined by for a given vector .

Proof 3. Consider Lyapunov function candidates . Calculating the upper right derivative of along the solutions of (43), we get Substituting into the above inequality, we can get for .
Since are nonsingular M-matrices and is nonempty, from Lemma 1, we know that there exists at least one vector such that for .
Consider functions with and .
By inequality (47) and the definition of functions , it is clear that and . Because , there are constants such that Let ; then for and .
Let , where . So For , we claim that . If this is not true, there exist some and corresponding , which make , , and for , . However, applying (44) and (46) leads to This is a contradiction. So , for . That is, for , Let ; then we can get for . From Definition 1, the equilibrium point of system (2) is robustly exponentially stable. Moreover, the exponential convergence rate is . The proof is completed.

Remark 3. The existence of exponential convergence rate has been proved, and from the definitions of functions and (44), we know that the value of relies on vector . So, for obtaining maximum convergence rate , one can solve the optimization problem under constraint conditions .
By virtue of Theorem 3, it is easy to get the following result.

Corollary 1. Under Assumption 1, if for , are nonsingular M-matrices, then for all , , and , the equilibrium point of system (2) is robustly exponentially stable for any switching signal with the average dwell time satisfying where and is determined by inequality for some given .

Proof 4. Consider Lyapunov function candidates . For convenience, we assume that the subsystem is activated when . Calculating the upper right derivative of along the solutions of (25), we get It can be known from Theorem 3 that there exists such that Let ; then for .
By the proof of Theorem 3, we know that for . Since the system state is continuous, it follows from (60) that where and . Let and yield When , . According to Definition 1, system (2) is robustly exponentially stable, and the exponential convergence rate is . The proof is completed.

Remark 4. Stability conditions in Theorems 2 and 3 and Corollary 1 are explicit for SNNs, which are convenient to verify in practice. However, they have the disadvantage of neglecting the signs of entries in the connection weight matrices and , and thus, differences between excitatory and inhibitory effects might be ignored.

Remark 5. Theorems 2 and 3 and Corollary 1 reflect the relationship between arbitrary switching and constrained switching of system (2). If are M-matrices for all , then system (2) would be exponentially stable at least under constrained switching. If is nonempty, then system (2) is exponentially stable for any switching signal.

5. Numerical Examples

We present two examples to illustrate the main results.

Example 1. Consider a delayed SNNs with two subnetworks, and the relevant parameters of system (25) are given as follows [18, 21]: Take the activation functions as and .
Obviously, satisfy Assumption 1 and .

Step 1. Determine whether are M-matrices. are both M-matrices, which imply that the considered SNN is at least exponentially stable under constrained switching. If is nonempty, then the system is exponentially stable under arbitrary switching.

Step 2. Determine whether is nonempty.
Let ; then we can get and . That is, . Therefore, the considered SNN is globally exponentially stable for any switching signal.

Step 3. Calculate the maximum exponential convergence rate .
By using LINGO solver, we can get the maximum convergence rate under the constraint conditions , and the corresponding vector .

The numerical simulations are given in Figures 1–5. We can see that the state trajectories converge to the equilibrium point of the system, which is consistent with the conclusion of Theorem 3. On the other hand, from [18, 21], we know that when the average dwell time of switched signal is greater than or equal to 9.1936 and 0.8396; then the considered neural network is exponentially stable. Table 1 shows that the stability criteria obtained in this paper are less conservative than those in [18, 21].

Example 2. Consider the second-order delayed SNNs in system (25) described by [49]: , , , , and Obviously, and satisfy Assumption 1 and ,

Step 1. Determine whether , are M-matrices. are both M-matrices, which mean that the considered system is at least globally exponentially stable under constrained switching.

Step 2. Determine whether , is nonempty.
As shown in Figure 6, . Therefore, we can not claim that the considered system is stable under arbitrary switching.

Step 3. Calculate the average dwell time .
By using LINGO solver, we can get the maximum convergence rate under the constraint conditions , , , and the corresponding vectors , , and . So we can get the average dwell time .

For numerical simulation, let , , and , where , and choose the initial value . Figures 7–9 display the state responses and state norm responses of these two subnetworks. Figures 10–13 display the state responses and state norm responses of the delayed SNNs under two different switching signals. From Figures 10 and 11, we can see that with the dwell time that is less than , the trajectories can not converge to the equilibrium point of the system; Figures 12 and 13 show that with the dwell time that is larger than , the trajectories converge to the equilibrium point of the system. This is consistent with the conclusion of Corollary 1.

These two examples indicate the correctness and effectiveness of the results proposed in this paper.

6. Conclusion

The existence, uniqueness, and robust exponential stability of the equilibrium point of SNNs with time delays were investigated in this paper. For each specified switching signal , conditions for guaranteeing the existence and uniqueness of the delayed SNNs were obtained by resorting to the homomorphism mapping theorem and M-matrix theory. By using average dwell time approach, sufficient conditions to ensure the robust exponential stability of the delayed SNNs under constrained switching were derived, and inequality technique and the idea of vector Lyapunov function were employed to obtain conditions for ensuring the globally exponential stability of the delayed SNNs under arbitrary switching. The obtained results not only have less conservativeness but also reveal the relationship between the constrained switching and the arbitrary switching of the delayed SNNs. Finally, two numerical examples were presented to demonstrate the effectiveness and less conservativeness of the main results over existing literature.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the Scientific Research Foundation of the Education Department of Sichuan Province under Grant 17ZA0364, in part by the National Natural Science Foundation of China under Grant 11572264 and 11402214, in part by the Foundation for Distinguished Young Talents in Higher Education of Guangdong under Grant 2016KQNCX103, and in part by the Open Research Fund of Key Laboratory of Automobile Measurement and Control & Safty, Xihua University, Sichuan Province under Grant szjj2017-074. The authors deeply appreciate Ling Zhao and Yan Yan from Xihua University for their helpful constructive suggestions in the revision of the language of this article.

References

L. Chua and L. Yang, “Cellular neural networks: theory,” IEEE Transactions on Circuits and Systems, vol. 35, no. 10, pp. 1257–1272, 1988.
View at: Publisher Site | Google Scholar
L. O. Chua and L. Yang, “Cellular neural networks: applications,” IEEE Transactions on Circuits and Systems, vol. 35, no. 10, pp. 1273–1290, 1988.
View at: Publisher Site | Google Scholar
L. Chua, CNN: A Paradigm for Complexity, World Scientific, Singapore, 1998.
View at: Publisher Site
M. Gupta, L. Jin, and N. Homma, Static and Dynamic Neural Networks: From Fundamentals to Advanced Theory, Wiley-Interscience, New York, NY, USA, 2004.
J. Park, O. Kwon, and S. Lee, “LMI optimization approach on stability for delayed neural networks of neutral-type,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 236–244, 2008.
View at: Publisher Site | Google Scholar
J. Lian and J. Wang, “Passivity of switched recurrent neural networks with time-varying delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 2, pp. 357–366, 2015.
View at: Publisher Site | Google Scholar
J. Cao, K. Yuan, and H. X. Li, “Global asymptotical stability of recurrent neural networks with multiple discrete delays and distributed delays,” IEEE Transactions on Neural Networks, vol. 17, no. 6, pp. 1646–1651, 2006.
View at: Publisher Site | Google Scholar
J. Zhang, Y. Suda, and T. Iwasa, “Absolutely exponential stability of a class of neural networks with unbounded delay,” Neural Networks, vol. 17, no. 3, pp. 391–397, 2004.
View at: Publisher Site | Google Scholar
J. Zhang, “Global exponential stability of interval neural networks with variable delays,” Applied Mathematics Letters, vol. 19, no. 11, pp. 1222–1227, 2006.
View at: Publisher Site | Google Scholar
X. Xu, J. Zhang, Q. Xu, Z. Chen, and W. Zheng, “Impulsive disturbances on the dynamical behavior of complex-valued Cohen-Grossberg neural networks with both time-varying delays and continuously distributed delays,” Complexity, vol. 2017, Article ID 3826729, 12 pages, 2017.
View at: Publisher Site | Google Scholar
J. Lian and K. Zhang, “Exponential stability for switched Cohen-Grossberg neural networks with average dwell time,” Nonlinear Dynamics, vol. 63, no. 3, pp. 331–343, 2011.
View at: Publisher Site | Google Scholar
Z. Wu, P. Shi, H. Su, and J. Chu, “Delay-dependent stability analysis for switched neural networks with time-varying delay,” IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 41, no. 6, pp. 1522–1530, 2011.
View at: Publisher Site | Google Scholar
Y. Wu, J. Cao, Q. Li, A. Alsaedi, and F. E. Alsaadi, “Finite-time synchronization of uncertain coupled switched neural networks under asynchronous switching,” Neural Networks, vol. 85, pp. 128–139, 2017.
View at: Publisher Site | Google Scholar
C. D. Zheng, Y. Gu, W. Liang, and Z. Wang, “Novel delay-dependent stability criteria for switched hopfield neural networks of neutral type,” Neurocomputing, vol. 158, pp. 117–126, 2015.
View at: Publisher Site | Google Scholar
S. Wang, T. Shi, M. Zeng, L. Zhang, F. E. Alsaadi, and T. Hayat, “New results on robust finite-time boundedness of uncertain switched neural networks with time-varying delays,” Neurocomputing, vol. 151, pp. 522–530, 2015.
View at: Publisher Site | Google Scholar
J. Zhang, Z. Han, and H. Wu, “Robust finite-time stability and stabilisation of switched positive systems,” IET Control Theory & Applications, vol. 8, no. 1, pp. 67–75, 2014.
View at: Publisher Site | Google Scholar
X. Wu, Y. Tang, and W. Zhang, “Stability analysis of switched stochastic neural networks with time-varying delays,” Neural Networks, vol. 51, pp. 39–49, 2014.
View at: Publisher Site | Google Scholar
Y. Wu, J. Cao, A. Alofi, A. Abdullah, and A. Elaiw, “Finite-time boundedness and stabilization of uncertain switched neural networks with time-varying delay,” Neural Networks, vol. 69, pp. 135–143, 2015.
View at: Publisher Site | Google Scholar
S. Dharani, R. Rakkiyappan, and J. Cao, “New delay-dependent stability criteria for switched hopfield neural networks of neutral type with additive time-varying delay components,” Neurocomputing, vol. 151, pp. 827–834, 2015.
View at: Publisher Site | Google Scholar
L. Liu, J. Cao, and C. Qian, “pth moment exponential input-to-state stability of delayed recurrent neural networks with Markovian switching via vector Lyapunov function,” IEEE transactions on neural networks and learning systems, vol. 29, no. 7, pp. 3152–3163, 2017.
View at: Publisher Site | Google Scholar
W. Shen, Z. Zeng, and L. Wang, “Stability analysis for uncertain switched neural networks with time-varying delay,” Neural Networks, vol. 83, pp. 32–41, 2016.
View at: Publisher Site | Google Scholar
C. Huang, J. Cao, and J. Cao, “Stability analysis of switched cellular neural networks: a mode-dependent average dwell time approach,” Neural Networks, vol. 82, pp. 84–99, 2016.
View at: Publisher Site | Google Scholar
M. Ali and S. Saravanan, “Finite-time stability for memristor based switched neural networks with time-varying delays via average dwell time approach,” Neurocomputing, vol. 275, pp. 1637–1649, 2018.
View at: Publisher Site | Google Scholar
G. Bao and Z. Zeng, “Region stability analysis for switched discrete-time recurrent neural network with multiple equilibria,” Neurocomputing, vol. 249, pp. 182–190, 2017.
View at: Publisher Site | Google Scholar
T. X. Brown, “Neural networks for switching,” IEEE Communications Magazine, vol. 27, no. 11, pp. 72–81, 1989.
View at: Publisher Site | Google Scholar
Y. Tsividis, “Switched neural networks,” US Patent US4873661, 2003.
View at: Google Scholar
M. Muselli, “Gene selection through switched neural networks,” in NETTAB-2003, Workshop on Bioinformatics for Microarrays, Bologna, Italy, 2003.
View at: Google Scholar
H. Huang, Y. Qu, and H. X. Li, “Robust stability analysis of switched hopfield neural networks with time-varying delay under uncertainty,” Physics Letters A, vol. 345, no. 4-6, pp. 345–354, 2005.
View at: Publisher Site | Google Scholar
M. Kermani and A. Sakly, “On the stability analysis of switched nonlinear systems with time varying delay under arbitrary switching,” Journal of Systems Science and Complexity, vol. 30, no. 2, pp. 329–346, 2017.
View at: Publisher Site | Google Scholar
J. C. Principe, J. M. Kuo, and S. Celebi, “An analysis of the gamma memory in dynamic neural networks,” IEEE Transactions on Neural Networks, vol. 5, no. 2, pp. 331–337, 1994.
View at: Publisher Site | Google Scholar
Q. Song and J. Cao, “Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays,” Journal of Computational and Applied Mathematics, vol. 197, no. 1, pp. 188–203, 2006.
View at: Publisher Site | Google Scholar
H. Xue and J. Zhang, “Robust exponential stability of switched interval interconnected systems with unbounded delay,” Journal of Systems Science and Complexity, vol. 30, no. 6, pp. 1316–1331, 2017.
View at: Publisher Site | Google Scholar
H. Xue, J. Zhang, W. Zheng, and H. Wang, “Robust exponential stability of large-scale system with mixed input delays and impulsive effect,” Journal of Computational and Nonlinear Dynamics, vol. 11, no. 5, article 051019, 2016.
View at: Publisher Site | Google Scholar
X. Xu, J. Zhang, and J. Shi, “Dynamical behaviour analysis of delayed complex-valued neural networks with impulsive effect,” International Journal of Systems Science, vol. 48, no. 4, pp. 686–694, 2017.
View at: Publisher Site | Google Scholar
X. Xu, Q. Xu, Y. Peng, J. Zhang, and Y. Xu, “Stochastic exponential robust stability of delayed complex-valued neural networks with Markova jumping parameters,” IEEE Access, vol. 6, no. 1, pp. 839–849, 2018.
View at: Publisher Site | Google Scholar
N. Li and J. Cao, “Switched exponential state estimation and robust stability for interval neural networks with the average dwell time,” IMA Journal of Mathematical Control and Information, vol. 32, no. 2, pp. 257–276, 2015.
View at: Publisher Site | Google Scholar
D. Liberzon, Switching in Systems and Control, Birhäuser, Boston, 2003.
View at: Publisher Site
S. W. Lee and S. J. Yoo, “Robust fault-tolerant prescribed performance tracking for uncertain switched pure-feedback nonlinear systems under arbitrary switching,” International Journal of Systems Science, vol. 48, no. 3, pp. 578–586, 2017.
View at: Publisher Site | Google Scholar
M. A. Bagherzadeh, J. Ghaisari, and J. Askari, “Robust exponential stability and stabilisation of parametric uncertain switched linear systems under arbitrary switching,” IET Control Theory & Applications, vol. 10, no. 4, pp. 381–390, 2016.
View at: Publisher Site | Google Scholar
X. Lin, S. Huang, and C. Qian, “Smooth state feedback stabilization for a class of planar switched nonlinear systems under arbitrary switching,” in 2016 12th World Congress on Intelligent Control and Automation (WCICA), pp. 1454–1458, Guilin, China, June 2016.
View at: Publisher Site | Google Scholar
D. Tian and S. Liu, “Exponential stability of switched positive homogeneous systems,” Complexity, vol. 2017, Article ID 4326028, 8 pages, 2017.
View at: Publisher Site | Google Scholar
P. Li, X. Li, and J. Cao, “Input-to-state stability of nonlinear switched systems via lyapunov method involving indefinite derivative,” Complexity, vol. 2018, Article ID 8701219, 8 pages, 2018.
View at: Publisher Site | Google Scholar
L. Wu, Z. Feng, and W. X. Zheng, “Exponential stability analysis for delayed neural networks with switching parameters: average dwell time approach,” IEEE Transactions on Neural Networks, vol. 21, no. 9, pp. 1396–1407, 2010.
View at: Publisher Site | Google Scholar
W. Xie, H. Zhu, S. Zhong, H. Chen, and Y. Zhang, “New results for uncertain switched neural networks with mixed delays using hybrid division method,” Neurocomputing, vol. 307, pp. 38–53, 2018.
View at: Publisher Site | Google Scholar
Y. Liu, Z. Wang, and X. Liu, “Global exponential stability of generalized recurrent neural networks with discrete and distributed delays,” Neural Networks, vol. 19, no. 5, pp. 667–675, 2006.
View at: Publisher Site | Google Scholar
D. Šiljak, “Large-scale dynamic systems: stability and structure,” Tech. Rep., Elsevier North Holland, New York, NY, USA, 1978.
View at: Google Scholar
M. Forti and A. Tesi, “New conditions for global stability of neural networks with application to linear and quadratic programming problems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 42, no. 7, pp. 354–366, 1995.
View at: Publisher Site | Google Scholar
A. Berman and R. J. Plemmons, Nonnegative Matrices in Mathematical Sciences, Academic Press, New York, NY, USA, 1979.
H. Xue and Y. Wei, “Robust exponential stability of switched grey interconnected systems with delays,” Journal of Grey System, vol. 30, no. 2, pp. 14–27, 2018.
View at: Google Scholar

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