Finite-time lag synchronization of inertial neural networks with mixed infinite time-varying delays and state-dependent switching

doi:10.1016/j.neucom.2020.12.059

Neurocomputing

Volume 433, 14 April 2021, Pages 50-58

https://doi.org/10.1016/j.neucom.2020.12.059 Get rights and content

Abstract

In this article, we design a new control scheme to investigate the finite-time lag synchronization (FTLS) of inertial neural networks (INNs) with mixed infinite time-varying delays and state-dependent switching. Several novel and easily verified conditions are gained guaranteeing the FTLS of INNs via the finite-time stability theory and nonsmooth analysis. Moreover, it is worth emphasizing that here we directly analyze INNs without using variable substitution, which is different from the reduced-order approach used in correspondingly previous works. At last, a numerical example and an application in image encryption are given to verify the correctness and practicability of the obtained results.

Introduction

Nonlinear system is a hot research topic and deserve further study [1], [2], [3], [4]. Especially, neural networks has gained widespread attention of researchers in recent decades for its applications in many areas including optimization problems, associative memory and so on [3], [5], [6], [7]. As stated in [7], all such practical applications are heavily related to the dynamical characteristics of neural networks. It is known that synchronization is part of the dynamic behaviors of neural networks. And lag synchronization means that the state of the master system is synchronized with the past state of the slave system at a constant time lag [8]. This is also a kind of synchronization and has been extensively applied to information security [9], [10]. In addition, it has been found that lag synchronization can clearly indicate the vulnerability of the nervous system compared to complete synchronization [8], [9]. For this purpose, the lag synchronization of neural networks has received extensive attention and some pretty results have been achieved [8], [9], [10], [11], [12], [13].

The above-mentioned lag synchronization works mainly focused on the analysis of first-order neural networks, and yet, it is also vital to explore the inertial/second-order neural networks, because neural networks with inertial terms contribute to generating complex dynamic behaviors including chaos and bifurcation, and it also has a strong biological and engineering backgrounds [14], [15]. On the other hand, time delay is inevitable in neural networks and has a negative impact on the system [16]. Furthermore, since neural networks have many parallel pathways with different axon sizes and lengths, another type of time delay referred to distributed delay also exists [17], [18]. So, it is reasonable to introduce inertial terms and mixed time delays into nervous system base on the theoretical and practical application backgrounds. Up to now, some new results have been obtained regarding the dynamic behavior of delayed inertial neural networks (INN), see [5], [6], [7], [9], [18], [19], [20], [21], [22], [23], [24]. It should be pointed out that discrete delays or distributed delays therein are all assumed to be bounded, which means that the present states of a neuron only relate to its local history. Actually, the whole history of a neuron affects its current state [25]. Thus, both discrete and distributed delays in neural networks should be regarded as infinite, so that the behavior of neurons in the human brain can be displayed more realistically.

Switched systems have always been an important research topic in engineering and physics applications. It consists of many subsystems depicted by differential or difference equations and a switching mode orchestrating switching between these subsystems [26]. As we know, switched systems can be divided into two categories under the different switching modes. One is time dependent switching, and the other is state-dependent switching. Compared with the time-dependent switching systems, state-dependent switching systems with different initial values may take on different equations at the same instant, which leads to more complex to analyze the dynamical behaviors of state-dependent switching system. Based on the characteristics of memristors and differential inclusion theory, the authors of previous work [27] constructed and analyzed the global stability of a class of state-dependent switching neural networks (SDSNNs). After this, many outcomes about such SDSNNs including stability and synchronization have been reported, see [4], [17], [18], [19], [28], [29], [30], [31], [32].

Furthermore, among the synchronization results of existing state-dependent switching INNs (SDSINNs), most of the results herein are infinite time stable, i.e., asymptotical synchronization [33], [34] and exponential synchronization [7], [35]. However, synchronization should be realized with a finite time for practical control applications. Recently, the finite-time synchronization issue of delayed SDSINNs was discussed in [36], [37] via a designed delay-dependent controller. Subsequently, the authors in [38] focused on the finite-time synchronization of SDSINNs with hybrid bounded time delays based on the finite-time stability theorem, feedback control and adaptive feedback control. Lately, with the inequality techniques and feedback control, several novel sufficient criteria ensuring the finite-time synchronization of SDSINNs with mixed bounded time delays are obtained in [39]. In order to make the settling time estimated independent on the initial value in the finite-time analysis, then fixed-time synchronization of delayed SDSINNs was studied in [40]. But on the finite-time lag synchronization (FTLS) of SDSINNs with mixed infinite time-varying delays (MITVDs) remains unsolved, it thus motivates us to fill this gap.

Note that the finite-time synchronization results of SDSINNs obtained in [36], [37], [38], [39], [40] are all through the order-reduction method, that is, change the second-order SDSINNs into the first-order system through variable substitution. Such order-reduction approach not only increase the dimension of SDSINNs, but also expands the difficulty of theoretical analysis of SDSINNs. Moreover, for the results in [36], [38], [39], [40], in order to achieve finite-time synchronization of SDSINNs, two designed controllers were introduced into the converted first-order system respectively. In fact, the more controllers, the more difficult it is to implement in practice. Thus, employing few controllers and considering the master and slave system directly from the SDSINNs themselves are more practically and correctly.

Inspired by the aforementioned discussions, this article explores the FTLS of SDSINNs with MITVDs by adaptive hybrid control and non-reduced order approaches. The main contributions are listed as follows.

(1) Discrete and distributed delays in the latest works [7], [19], [24], [38], [39] are assumed to be bounded, but in this paper, they are regarded as infinite, thus relaxing the application range of neural networks.
(2) Different from these previous works [36], [38], [39], [40], in order to get the synchronization results of SDSINNs, two controllers were introduced into the conversion first-order systems. In this article, using only one controller, some FTLS criteria for SDSINNs are obtained with a non-reduced order approach. Moreover, these results are also applied to image encryption.
(3) Unlike previous control methods [8], [9], [10], [11], [12], [13] used to realize lag synchronization of delayed neural networks, this article employs adaptive hybrid control to achieve FTLS for SDSINNs with MITVDs.
(4) The inertial term, uncertain connection weights, discrete and distributed delays are all taken into account in the dynamic model of neural networks, which makes the discussed model more versatile.

The remaining part of this article is outlined as follows. Some basic preliminaries are introduced in Section 2. In Section 3 gives the main theoretical results. Two examples are given in Section 4. In Section 5, conclusions are presented.

Section snippets

Preliminaries

Let $Ω = {1, 2, \dots, n}, R^{n}$ represents the n-dimensional Euclidean space. $C ((- \infty, 0], R^{n})$ stand for the set of all continuous functions mapping from $(- \infty, 0]$ to $R^{n}$ . For any $s = (s_{1}, s_{2}, \cdot \cdot \cdot, s_{n}) \in R^{n}$ , the norm $‖ s ‖ = \sum_{p = 1}^{n} | s_{p} |$ . $a_{pq} = \max {| a_{pq}^{*} |, | a_{pq}^{* *} |}, b_{pq} = \max {| b_{pq}^{*} |$ , $| b_{pq}^{* *} |}, c_{pq} = \max {| c_{pq}^{*} |, | c_{pq}^{* *} |}, p, q \in Ω$ .

Consider the following a class of SDSINNs with MITVDs $\begin{matrix} {\ddot{s}}_{p} (t) = & - α_{p} s_{p} (t) - d_{p} {\dot{s}}_{k} (t) + \sum_{q = 1}^{n} a_{pq} (s_{p} (t)) F_{q} (s_{q} (t)) \\ + \sum_{q = 1}^{n} b_{pq} (s_{p} (t)) ϖ_{q} (s_{q} (t - h_{q} (t))) \\ + \sum_{q = 1}^{n} c_{pq} (s_{p} (t)) \int_{- \infty}^{t} K_{q} (t - θ) ω_{q} (s_{q} (θ)) d θ, \end{matrix}$ where $t ⩾ 0, p \in Ω, α_{p} > 0, d_{p} > 0$ and $s_{p} (t)$ is the state of the p

Main results

Here, different from the order-reduction approach used in the existing work [37], and also unlike these previous works [36], [38], [39], [40], two controllers are added into the converted first-order systems. We derive the FTLS results of SDSINNs by directly constructing Lyapunov functional and using only one controller. In addition, some relevant corollaries are also given.

Theorem 1

Suppose that Assumption 1, Assumption 2, Assumption 3 hold. Then the FTLS between SDSINNs (1) and SDSINNs (6) are achieved

Numerical examples

In this part, a numerical example and an application in image encryption are given to illustrate the validity and practicability of the derived results.

Example 1

Consider the 2-D SDSINNs with MITVDs as follows

\begin{matrix} {\ddot{s}}_{p} (t) = & - α_{p} s_{p} (t) - d_{p} {\dot{s}}_{k} (t) + \sum_{q = 1}^{2} a_{pq} (s_{p} (t)) F_{q} (s_{q} (t)) \\ + \sum_{q = 1}^{2} b_{pq} (s_{p} (t)) ϖ_{q} (s_{q} (t - h_{q} (t))) \\ + \sum_{q = 1}^{2} c_{pq} (s_{p} (t)) \int_{0}^{+ \infty} K_{q} (θ) ω_{q} (s_{q} (t - θ)) d θ, \\ t ⩾ 0, p = 1, 2, \end{matrix}

where

h_{q} (t) = 0.5 \ln (1 + t), α_{1} = α_{2} = 1, d_{1} = d_{2} = 1, K_{q} (θ) = e^{- θ}, F_{q} (\cdot) = ω_{q} (\cdot) = ϖ_{q} (\cdot) = \tanh (\cdot), q = 1, 2

Γ_{1} = Γ_{2} = 1

a_{11} (s_{1} (t)) = 3, a_{12}^{*} = 0.4, a_{12}^{* *} = 0.5

a_{21}^{*} = 4, a_{21}^{* *} = 5, a_{22} (s_{2} (t)) = 5

b_{11}^{*} = - 1.5, b_{11}^{* *} = - 1.2, b_{12}^{*} = 0.3, b

Conclusions

This article has explored the FTLS problem of SDSINNs with MITVDs. Under the newly designed adaptive hybrid control, several novel criteria have been given to guarantee the FTLS of SDSINNs via the finite-time stability theory and nonsmooth analysis. It is worth emphasizing that the results obtained herein are very concise, easy to verify, and without computing the complex algebraic criteria or LMIs [20], [35], [36], [37], [38], [39], [40]. Different from the order-reduction approach used in the

CRediT authorship contribution statement

Changqing Long: Writing - original draft, Software. Guodong Zhang: Methodology, Software, Writing - review & editing. Zhigang Zeng: Conceptualization. Junhao Hu: Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by the National Science Foundation of China Nos. 61976228, 61673188 and 61876192, the Natural Science Foundation of Hubei Province of China No. 2019CFB618, and the Fundamental Research Funds for the Central University of South-Cental University for Nationalities Nos. KTZ20051, CTZ20020 and CTZ20022.

Changqing Long received the B.S. degree in Applied Mathematics from the School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, China, in 2018. He is currently pursuing the M.S. degree in Operational Research and Cybernetics at the School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, China. His current research interests include Nonlinear differential-algebraic system, Discontinuous or continuous system, Neural Networks and

References (43)

L. Zou et al.
Moving horizon estimation with non-uniform sampling under component-based dynamic event-triggered transmission
Autom.
(2020)
Q.K. Song et al.
Stability criteria of quaternion-valued neutral-type delayed neural networks
Neurocomputing
(2020)
M. Li et al.
A bioeconomic differential algebraic predator-prey model with nonlinear prey harvesting
Appl. Math. Model.
(2017)
Q. Tang et al.
Exponential synchronization of inertial neural networks with mixed time-varying delays via periodically intermittent control
Neurocomputing
(2019)
G.D. Zhang et al.
Novel results on synchronization for a class of switched inertial neural networks with distributed delays
Inf. Sci.
(2020)
G.D. Zhang et al.
Exponential lag synchronization for delayed memristive recurrent neural networks
Neurocomputing
(2015)
J.C. Shi et al.
Global exponential stabilization and lag synchronization control of inertial neural networks with time delays
Neural Netw.
(2020)
Y.T. Cao et al.
New results on anti-synchronization of switched neural networks with time-varying delays and lag signals
Neural Netw.
(2016)
M.W. Zheng et al.
Finite-time generalized projective lag synchronization criteria for neutral-type neural networks with delay
Chaos Solitons Fract.
(2018)
J.J. Huang et al.
Finite-time lag synchronization of delayed neural networks
Neurocomputing
(2014)

D.S. Huang et al.

Finite-time synchronization of inertial memristive neural networks with time-varying delays via sampled-date control

Neurocomputing

(2017)

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Guodong Zhang received the Ph.D. degree in School of Automation from the Huazhong University of Science and Technology in 2014. He joined the South-Central University for Nationalities in 2014.07. Now, he serves as an Associate Professor and the supervisor in the School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, China. He has published over 20 international journal papers, he is also a Reviewer of several journals, such as IEEE Transactions on Systems, Man and Cybernetics, IEEE Transactions on Cybernetics, IEEE Transactions on Neural Networks and Learning Systems, Nonlinearity, Neural networks, Information Sciences, Chaos, Solitons and Fractals, Nonlinear Dynamics. His current research interests include Discontinuous system, Neural Networks and Intelligent control.

Zhigang Zeng received the Ph.D. degree in systems analysis and integration from the Huazhong University of Science and Technology, Wuhan, China, in 2003. He is currently a Professor with the School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, and also with the Key Laboratory of Image Processing and Intelligent Control, Education Ministry of China, Wuhan. He has authored over 140 international journal papers. His current research interests include theory of functional differential equations and differential equations with discontinuous right-hand sides, and their applications to dynamics of neural networks, memristive systems, and control systems. Dr. Zeng has been a member of the Editorial Board of Neural Networks since 2012, Cognitive Computation since 2010, and Applied Soft Computing since 2013. He was an Associate Editor of the IEEE TRANSACTIONS ON NEURAL NETWORKS from 2010 to 2011. He has been an Associate Editor of the IEEE TRANSACTIONS ON CYBERNETICS since 2014 and the IEEE TRANSACTIONS ON FUZZY SYSTEMS since 2016.

Junhao Hu received the B.Sc. degree from Mathematic Department of Central China Normal University, Wuhan,China, in 1998; M.Sc. degree from Computer Science Department of South-Central University for Nationalities, Wuhan, China, in 2003; Ph.D. degree from Control Science and Engineering Department of Huazhong University of Science and Technology, Wuhan, China, in 2007. He was a Post-Doctoral Fellow with the Huazhong University of Science and Technology in 2008–2010. He was a research associate of Department of the Mathematics and Statistics, University of Strathclyde in 2011-2012. He is now a Professor with the School of Mathematics and Statistics, South-Central University for Nationalities. He has published over 10 international journal papers. His current research interests include the areas of neural networks, nonlinear stochastic systems.

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Finite-time lag synchronization of inertial neural networks with mixed infinite time-varying delays and state-dependent switching

Abstract

Introduction

Section snippets

Preliminaries

Main results

Numerical examples

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

Autom.

Neurocomputing

Appl. Math. Model.

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Inf. Sci.

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Neural Netw.

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Chaos Solitons Fract.

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Phys. D

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J. Frankl. Inst.

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