Global robust asymptotic stability analysis of BAM neural networks with time delay and impulse: An LMI approach
Introduction
Bi-directional associative memory (BAM) neural networks model is a two-layer nonlinear feedback neural networks model introduced by Kosko in [1], [2]. Because BAM neural networks model provides a flexible nonlinear mapping from input space to output space, it possesses an important attribute in its simple architecture and has good application perspective in pattern recognition. Time delay will inevitably occur in electronic neural networks owing to the unavoidable finite switching speed of amplifiers, so it is more in accordance with this fact to study the BAM neural networks with constant or time-varying delays. In recent years, some sufficient conditions for the existence of the equilibrium points, bounds of trajectories, and stability of delayed BAM neural networks have been investigated by many researchers [6], [8], [16], [17]. Moreover, parameter fluctuation in neural network implementation on very large integration chips is also unavoidable. This fact implies that a good neural network should have certain robustness which paves the way of introducing the theory of interval matrices and interval dynamics to investigate on robust stability [5], [7], [9].
However, besides delay effect, impulsive effects are also likely to exist in neural networks [4], [10], [11], [12], [13], [15], [18], [19]. For instance, in implementation of electronic networks, the state of the networks is subject to instantaneous perturbations and experiences abrupt change at certain instants, which may be caused by switching phenomenon, frequency change or other sudden noise, that is, it exhibits impulsive effects [12]. Therefore, it is necessary to consider both impulsive effect and delay effect on dynamical behaviors of neural networks. Some results on impulsive effect have been gained for delayed neural networks, see [4], [10], [11], [12], [13], [15], [18], [19] and the references therein.
Recently, LMI-based techniques have been successfully used to tackle various stability problems for neural networks. The main advantage of the LMI-based approaches is that the LMI stability conditions can be solved numerically using the effective interior-point algorithm [3].
To the best of our knowledge, the stability analysis for impulsive BAM neural network with time-varying delays has seldom been investigated and remains important and challenging. It is known that BAM neural network model is one of the most popular and typical neural network models. So it is not only theoretically interesting but also practically important to determine the global robust stability (GRAS) for impulsive BAM neural network with time-varying delays. Motivated by the above discussions, the objective of this paper is to study the GRAS of impulsive BAM neural network with time-varying delays. Applying the idea of the Lyapunov–Krasovskii functional with the linear matrix inequality (LMI), several new criteria on the GRAS are presented.
Section snippets
Model description and assumptions
Consider the following impulsive Cohen–Grossberg-type BAM neural networks model with delays:where and are the state of the ith neuron from the neural field and the jth neuron from the neural field at time t, respectively; denote the activation functions of the jth neuron from and
Robust stability for BAM neural networks with constant delays and impulses
In this section, some sufficient conditions of the GRAS for system (2.2), (2.3), which describe the bounds of the interconnection matrix and activation functions, are obtained. Theorem 3.1 Suppose that there exist diagonal matrices , and , such that the following LMIs hold:and The impulsive operators and satisfy
Robust stability for BAM neural networks with time-varying delays and impulses
In this section, we will consider the BAM neural networks with time-varying delays and impulses, which is described bywhere and are two functions, . Furthermore, we assume that and are differentiable and . Theorem 4.1 Suppose the system (4.1) is GRAS if it satisfies hypotheses
Examples
In this section, we give examples to illustrate our results. Example 1 Consider the following system (2.3) with constant delays: for , with the initial values . Let . To take , we have The system parameters areSince that
Acknowledgements
This research was partly supported by the National Natural Science Foundation of China (No: 10926128 and 10801109), the Natural Science Foundation of Zhaoqing University (0825) and the Natural Science Foundation of Wuhan University of Science and Engineering (2008Z25).
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2015, NeurocomputingCitation Excerpt :In the design of neural networks, it is important to ensure that the system be stable with respect to these uncertainties. Here, there exist several related results on robust stability of the delayed BAM neural networks; see, e.g. [18–30]. However, most of the existing stability results derived for the BAM neural networks can be applicable when only a pure delayed neural network model is considered.
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2014, Neural NetworksCitation Excerpt :Moreover, in practical implementation of neural networks, the stability of networks can often be destroyed by its compulsory uncertainty issuing from the existence of modeling errors, external disturbance and parameter fluctuations. In addition, several studies with interesting results examining robust stability analysis of BAM neural networks were published in Arık (2014), Cao, Ho, and Huang (2007), Li, Cao, and Wang (2007), Li and Jia (2013), Liu, Yi, Guo, and Wu (2008), Senan, Arık, and Liu (2012), Sheng and Yang (2009) and Zhou and Wan (2010). Hence, robustness of the designed network should be considered.