Stability Analysis for Uncertain Complex-Valued Recurrent Neural Networks
This paper addresses the global robust stability problem for the complex-valued recurrent neural networks with time delay, where the system matrix parameters are time-varying within given intervals. Based on the Lyapunov functional method and some propert
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Stability Analysis for Uncertain Complex-Valued Recurrent Neural Networks Weiqiang Gong, Jinling Liang and Jinde Cao
Abstract This paper addresses the global robust stability problem for the complex-valued recurrent neural networks with time delay, where the system matrix parameters are time-varying within given intervals. Based on the Lyapunov functional method and some properties of the norm of matrices, sufficient criteria are presented to ascertain the global stability of the uncertain neural networks. One numerical example is also provided to illustrate the effectiveness of the obtained results. Keywords Complex-valued recurrent neural networks Parameter uncertainties Time delay
96.1
Robust stability
Introduction
In the last few decades, due to the extensive applications such as classification of pattern recognition, signal processing and associative memory and so on, the recurrent neural networks have been widely studied, see [1, 2] for example. It is well known that time delays often occur in the processing of information storage and transmission, which may bring oscillation, instability, and bifurcation to neural networks, see [3, 4] for more details. Therefore, it is necessary to investigate the behaviors of neural networks with time delays. A large amount of significant results can be seen in literatures such as [5, 6]. Recently, the neural networks have been used in electronics for real-time applications. Some parameters of neural networks may be influenced by the tolerances of electronic components. Hence, it is essential to consider the dynamical variations of neural networks subject to parameter uncertainties. For real-valued neural networks, a great deal of results on the robust stability has been presented, see references [7, 8] for example. To the best knowledge of the authors, there are W. Gong J. Liang (&) J. Cao Department of Mathematics, Southeast University, Nanjing 210096, China e-mail: [email protected] © Springer Science+Business Media Singapore 2016 R. Wang and X. Pan (eds.), Advances in Cognitive Neurodynamics (V), Advances in Cognitive Neurodynamics, DOI 10.1007/978-981-10-0207-6_96
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not enough papers which are concerned about the robust stability of uncertain complex-valued recurrent neural networks with delays. Motivated by the above discussions, the aim of this paper is to study the robust stability of complex-valued recurrent neural networks with parameter uncertainties and time delay. pffiffiffiffiffiffiffi Notations: Let i be the imaginary unit, i.e., i ¼ 1. The superscript ‘′’ represents the conjugation and transposition of a complex matrix. jAj ¼ ðjakj jÞmn for A ¼ ðakj Þmn 2 Cmn . The notation X [ Y means that X Y is positive definite. PR and PI denote, respectively, the real and the imaginary parts of matrix P 2 Cmn .
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Problem Formulation and Some Preliminaries
Consider the following complex-valued recurrent neural networks with time delays described by the nonlinear delay differential equations with form as follows: :
uðtÞ ¼ CuðtÞ
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