LMI conditions for global asymptotic stability results for neutral-type neural networks with distributed time delays☆
Introduction
The key features of neural networks are asynchronous parallel processing, continuous time dynamics and global interaction of network elements. Neural networks and their various generalizations have attracted the attention of the scientific community due to their promising potential for tasks of classification, associative memory, and parallel computation and their ability to solve difficult optimization problems [1], [7], [8]. In such applications, the stability of the network is prerequisite. Many researchers have a lot of contributions to these subjects. Stability is a basic knowledge for dynamical systems and is useful in the application to the real systems. Because of this, the stability of neural networks has been deeply investigated in the literature. Delay systems have been largely used to describe propagation and transport phenomena or population dynamics. In economic systems, delays appear in a natural way since decisions and effect (investment policy, commodity market evolution, etc.) are separated by some time interval. The existence of time delays may degrade system performance and cause oscillation in a network, leading to instability. Therefore, it is important to investigate the stability of delayed neural networks. The global asymptotic stability results for different classes of delayed neural networks were proposed in [2], [3], [4], [9].
Most works on delayed neural networks have dealt with the stability analysis problems for neural networks with discrete delays. Neural network usually has a spatial nature due to the presence of various parallel pathways with a variety of axon sizes and lengths, so it is desirable to model them by introducing unbounded delays. In these circumstances the signal propagation is not instantaneous and cannot be modelled with discrete delays. A more appropriate way is to incorporate continuously distributed delays in neural network models. In recent years there has been a growing research interest in the study of neural networks with distributed delays [15], [16], [17], [18], [20], [21]. Another type of time-delays, namely, neutral-type time-delays, has recently drawn much research attention. So far there are only a few papers that have taken neutral-type phenomenon into account in delayed neural networks, see for example [5], [6], [10], [11], [12], [13], [19]. Practically, such phenomenon always appears in the study of automatic control, population dynamics and vibrating masses attached to an elastic bar, etc. Recently, Rakkiyappan and Balasubramaniam [14] studied the global exponential stability results for neutral-type neural networks with distributed delays by using LMI approach.
Based on the above discussions, we consider a class of neutral-type neural networks with unbounded distributed time delays described by a neutral integro differential equations. The main purpose of this paper is to study the global asymptotic stability for neutral-type neural networks with unbounded distributed delays. To the best of the authors knowledge, there were no results for global asymptotic stability analysis problem for neutral-type neural networks with distributed delays. It is therefore, our intension in this paper is to investigate the global asymptotic stability analysis problem for a class of neutral-type neural networks with distributed delays. By using the Lyapunov–Krasovskii functional technique, global asymptotic stability conditions for the considered delayed neural networks are given in terms of LMIs, which can be easily calculated by MATLAB LMI toolbox. We also provide a numerical example to demonstrate the effectiveness of the proposed stability results.
Section snippets
Global stability results
Throughout the manuscript we will use the notation to denote that A is a symmetric and positive definite (or negative definite) matrix. The notation and means that the transpose of A and the inverse of a square matrix, respectively. If A, B are symmetric matrices , then is positive definite (positive semi-definite). denotes the Euclidean norm of a vector z and denotes the induced norm of the matrix A, that is .
In this paper, we consider a class
Main results
We establish the following stability criteria by using the Lyapunov method in terms of LMIs: Theorem 3.1 The equilibrium point of system(4) is globally asymptotically stable if there exist positive definite matrices and a positive diagonal matrices such thatwhere
Example
Consider the following coefficient matrices of two-neuron neural networks of neutral-type with distributed delays satisfying (2)where . Solving the LMI in Theorem 3.1 using MATLAB LMI toolbox, we obtain the solution asThe above results shows that all the
Conclusion
A new sufficient condition is derived to guarantee the global asymptotic stability of the equilibrium point for neutral-type neural networks with unbounded distributed delays. To the best of our knowledge, the results presented here have been not appeared in the related literature. The stability criteria is expressed in terms of LMIs, which are less conservative and less restrictive and can be easily verified by using MATLAB LMI toolbox.
References (21)
- et al.
A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach
Chaos Soliton Fract.
(2005) - et al.
Stability in delayed Cohen–Grossberg neural networks: LMI optimization approach
Physica D
(2005) - et al.
Stability analysis of delayed cellular neural networks
Neural Networks
(1998) - et al.
Global stability analysis for a class of neural networks with time varying delays and control input
Appl. Math. Comput.
(2007) Global stability of almost periodic solutions of Hopfield neural networks with neutral time-varying delays
Appl. Math. Comput.
(2008)- et al.
Global asymptotic stability for cellular neural networks with discrete and distributed time-varying delays
Chaos Soliton Fract.
(2007) - et al.
LMI optimization approach on stability for delayed neural networks of neutral-type
Appl. Math. Comput.
(2008) - et al.
A new stability criterion for bidirectional associative memory neural networks of neutral-type
Appl. Math. Comput.
(2008) - et al.
New global exponential stability results for neutral type neural networks with distributed time delays
Neurocomputing
(2008) - et al.
Stability analysis of Cohen–Grossberg neural networks with both time-varying and continuously distributed delays
J. Comput. Appl. Math.
(2006)
Cited by (57)
A new Lyapunov functional for stability analysis of neutral-type Hopfield neural networks with multiple delays
2020, Neural NetworksCitation Excerpt :Therefore, in the past literature, most of the existing research results have dealt with the stability issues of various classes of neutral-type neural systems including single or discrete delays. In Dong, Guo, and Hao (2020), Lee, Kwon, and Park (2010), Mahmoud and Ismail (2010), Orman (2012), Park, Kwon, and Lee (2008), Rakkiyappan and Balasubramaniam (2008a, 2008b), Weera and Niamsup (2016), Xu, Lam, Ho, and Zou (2005) and Zheng et al. (2017), the stability of neutral neural-type networks defined by (3) and (4) has been studied, and by constructing different suitable Lyapunov functionals together with employing some lemmas and new mathematical techniques, different sets of novel stability criteria for the considered neutral-type neural networks in the forms of linear matrix inequalities have been presented. In Liu and Du (2015), Manivannan, Samidurai, Cao, Alsaedi, and Alsaadi (2018a), Samidurai et al. (2017), Shi, Zhong, Zhu, Liu and Zen (2015) and Shi, Zhu, Zhong, Zeng and Zhang (2015), the stability problem for delayed neutral-type neural networks possessing discrete delays has been tackled, and by employing various proper Lyapunov functionals with the triple or four integral terms, new global stability criteria have been proposed.
LMI-based global exponential stability of equilibrium point for neutral delayed BAM neural networks with delays in leakage terms via new inequality technique
2016, NeurocomputingCitation Excerpt :So far, some authors have studied the stability of neural networks involving time-delay in the leakage term, for example, see [13,14,24–31,42]. Up to now, there are some papers that have taken neutral-type phenomenon into account in delayed neural networks [8,18,32–39,44–46]. Recently, there are only a few papers [31,43,45] considered the periodic solution or almost periodic solution for neutral-type delayed neural networks with delays in the leakage terms.
Almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales
2014, Applied Mathematics and ComputationStability of stochastic neural networks of neutral type with Markovian jumping parameters: A delay-fractioning approach
2014, Journal of the Franklin Institute
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The work of the authors was supported by UGC-SAP(DRS), New Delhi India under the sanctioned No. F510/6/DRS/2004 (SAP-1).