Novel delay-dependent exponential stability criteria for neutral-type neural networks with non-differentiable time-varying discrete and neutral delays
Introduction
During the last few decades, there have been extensively investigations on the stability analysis of neural networks due to their extensively applications such as signal processing, pattern recognition, associative memories, and combinatorial optimization, see [3], [7], [8], [9], [10], [11], [12]. An important type of neural networks is the recurrent neural networks with time delays which has broad applications in modeling dynamic behavior of many biological and cognitive activities such as heartbeat, respiration, mastication, locomotion, and memorization, see [4].
In performing a periodicity or stability analysis of a neural network, the conditions to be imposed on the neural network are determined by the characteristics of various activation functions and network parameters. When neural networks are created for problem solving, it is desirable for their activation functions are not too restrictive. As a result, there have been considerable research works on the stability of neural networks with various types of activation functions under more general conditions [4], [14], [19]. On the other hand, during the implementation of neural networks, the occurrence of time delays is unavoidable during the processing and transmission of the signals because of the finite switching speed of amplifiers in electronic networks or finite speed for signal propagation in biological networks, and the delays are often the source of instability, hidden oscillations, divergence, chaos or other poor performance behavior. Therefore, several researchers have focused on the study of stability analysis of delayed neural networks (DNNs) [13], [15], [20], [27], [28], [31], [35], [36], [37].
In order to derive delay-dependent stability criteria for time-delay systems, several authors assumed that the time-varying delay varies from 0 to a given upper bound. In practical, time delay may vary in a range for which the lower bound is not restricted to be zero; such time-delay is called interval time-varying delay. A typical example of dynamic systems with interval time-varying delays is networked control systems [32]. Nonetheless, in lots of works on stability analysis, some restrictions are imposed on derivatives of time-varying delays, [1], [21], [26], [30], [36], [37]. For example, in order to derive stability criteria for neural networks, it is assumed that the derivative of time-varying delay is less than 1. As a result, the stability criteria may neither be used for fast time-varying delay nor non smooth time-varying delay.
Furthermore, it is common that the time delay occurs not only in system states but also in the derivatives of system states. Systems containing the information of past state derivatives are called neutral-type delay systems. Accordingly, the stability analysis of neutral-type neural networks has also been received considerable attention in recent years. In many practical systems, the phenomena on neutral-type delay often appears such as in heat exchanges, distributed networks containing lossless transmission lines, partial element equivalent circuits and population ecology are examples of neutral systems, see [2], [5], [18], [25], [29] where some recent results on DNNs with neutral-type pertaining to the scope of this paper were reported. Furthermore, in neural networks, it might occur that there are connections between past state derivatives in the systems. As a result, it is more natural to consider neural networks with activation functions of past state derivatives.
In the study of stability of neural networks, exponential stability is more desired property than asymptotic stability since it gives faster convergence rate to the equilibrium point and provide information about the decay rates of the networks. Accordingly, it is particularly important, when the exponential stability property guarantees that, whatever transformation occurs, the network stability to store rapidly the activity pattern is left invariant by self-organization. Therefore, exponential stability analysis of neural networks with interval time-varying delays is worth investigating, see [10], [28].
In order to obtain less conservative stability criteria, numerous important and interesting methods have been proposed. For examples, free-weighting matrix technique, see [10], [37] and a new convex combination technique is developed based on the inequality , see [34]. A piecewise delay method is introduced in the analysis by using DCP method, see [36]. Delay-fraction technique is investigated in [22]. However, from these existing methods, the discrete delay is required to be differentiable and the information on derivative of neutral delay such as boundedness of the derivative is required.
Motivated by above discussions, the main objective of this paper is to further investigate the exponential stability problem for neutral-type neural networks with time-varying delays both in system states and state derivatives under more general activation functions. Two main contributions of our study are the following. Firstly, by constructing a novel augmented Lyapunov–Krasovskii functional, a novel exponential stability criterion will be derived in terms of linear matrix inequalities (LMIs). An advantage of our development is that the discrete delay is not necessarily differentiable and the information on derivative of neutral delay is not required. To the best of our knowledge, this is the first study under this conditions on discrete and neutral delays. Meanwhile, this restriction is required in some existing results, see [17], [22], [24], [33]. Secondly, we consider the case when there are interconnections between past state derivatives, namely, neural networks contains activation function of past state derivatives, . Moreover, the obtained exponential stability in this work is more applicable in the sense that it may still be applied to the situation that there are no interconnections between past state derivatives.
The rest of this paper is organized as follows. In Section 2, we give notations, definitions, propositions and lemma for using in the proof of the main results. Delay-dependent sufficient conditions for the exponential stability criteria for neutral-type neural networks with interval time-varying state and neutral-type delays are presented in Section 3. Numerical examples illustrated the effectiveness obtained results are given in Section 4. The paper ends with conclusions in Section 5.
Section snippets
Preliminaries
Consider the following neural networks with interval time-varying discrete and neutral delays:where is the neural state vector. is the state feedback coefficient matrix; is the activation of neurons. W0 is the connection weight matrix and are the delayed connection weight matrices; represents the external inputs. Assumption 1
Exponential stability analysis
Rewrite system (2.2) in the following descriptor system:For presentation convenience, in the following, we denote , , , . Theorem 3.1 Given . The system (3.1) is exponentially stable if there exist symmetric positive definite matrices positive diagonal matrices , ,
Numerical examples
In this section, we now provide examples to show the effectiveness of theoretical result. Example 4.1 Consider the following neutral-type neural networks with time-varying discrete and neutral delayswhere
, . It is worth noting that the delay functions is non-differentiable.
Conclusions
In this paper, we have investigated the exponential stability criteria for neutral-type neural networks with the discrete delay is not necessarily differentiable and the information on derivative of neutral delay is not required. The state delay function is not necessary to be differentiable which allows time-delay function to be a fast time-varying function. By constructing a set of improved Lyapunov–Krasovskii functional combined with Leibnitz–Newton׳s formula, the proposed stability criteria
Acknowledgments
Financial support from the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0355/2552) to Wajaree Weera and Piyapong Niamsup is acknowledged. The second author is also supported by Chiang Mai University, Chiang Mai, Thailand.
W. Weera is currently a Ph.D. student at the Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand under the supervision of Dr. Piyapong Niamsup. She has been supported from the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program Grant no. PHD/0355/2552. Her research interests include stability of time-delay systems and stability of neutral systems.
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2020, Neural NetworksCitation Excerpt :In Zhang, Liu, and Huang (2010), stability of system (3) has been investigated, and by utilizing an adequate Lyapunov functional and the series compensation technique, some global stability criteria have been derived. It is worth noting that the results of Balasubramaniam et al. (2010), Dong et al. (2020), Duan and Jian (2020), Lakshmanan et al. (2017), Lee et al. (2010), Lien et al. (2008), Liu and Du (2015), Ma et al. (2019), Mahmoud and Ismail (2010), Mai et al. (2009), Manivannan et al. (2018a, 2018b), Orman (2012), Park et al. (2008), Rakkiyappan and Balasubramaniam (2008a, 2008b), Samidurai et al. (2017), Shi, Zhong et al. (2015), Shi, Zhu, Zhong, Zeng and Zhang (2015), Shi, Zhu, Zhong, Zeng, Zhang, Wang (2015), Tu and Wang (2018), Weera and Niamsup (2016), Xu et al. (2005), Zhang et al. (2010), Zheng et al. (2017, 2018) and Zhu et al. (2009) use some various classes of linear matrix inequality techniques to obtain different sets of sufficient stability conditions for systems (3) and (4). However, as pointed out above, the stability of neutral-type neural networks having multiple delays is also an important issue to be addressed.
Robust stability of uncertain fractional order singular systems with neutral and time-varying delays
2020, NeurocomputingCitation Excerpt :In recent years, many scholars devote themselves to study the stability problem of neutral systems, for example see [41–47] and references therein. In [41], the exponential stability of neural networks with both non-differential time-varying delays and neutral delays was discussed. In [42], the robust stability of uncertain linear neutral systems with discrete and distributed delays was investigated.
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2019, Journal of the Franklin InstituteCitation Excerpt :In fact, in the past decade, mostly using the suitable Lyapunov functionals involving multiple integral term, many various sets of sufficient criteria for the stability of neutral systems (3) and (4) have been proposed [23–39]. The results of [23–39] have mainly established the conditions which are basically stated in the LMIs, (LMI stands for linear matrix inequality). On the other hand, different form the neutral system models (2) and (3), the neutral-type neural network stated by Eq. (1) cannot be expressed in the vector-matrix form since this network model involves multiple time delays τij and discrete neutral delays ζj.
W. Weera is currently a Ph.D. student at the Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand under the supervision of Dr. Piyapong Niamsup. She has been supported from the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program Grant no. PHD/0355/2552. Her research interests include stability of time-delay systems and stability of neutral systems.
P. Niamsup is currently an Associate Professor at the Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand. His research interests include stability of time-delay systems, stability of nonautonomous systems, stability of switched systems.