DELAY-DEPENDENT ASYMPTOTICAL STABILIZATION CRITERION OF RECURRENT NEURAL NETWORKS Grienggrai Rajchakit Division of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand griengkrai@yahoo.com Keywords: Neural networks; Time-varying Delay; Stability; Quadratic Lyapunov functional approach. Abstract. This paper deals with the problem of delay-dependent stability criterion of discrete-time recurrent neural networks with time-varying delays. Based on quadratic Lyapunov functional approach and free-weighting matrix approach, some linear matrix inequality criteria are found to guarantee delay-dependent asymptotical stability of these systems. And one example illustrates the exactness of the proposed criteria. Introduction A recurrent neural network (RNNs) is a very important tool for many application areas such as associative memory, pattern recognition, signal processing, model identication and combinatorial optimization. With the development of research on RNNs in theory and application, the model is more and more complex. Parameter uncertainties and nonautonomous phenomena often exist in real systems due to modeling inaccuracies [1, 2]. Particularly when we consider a longterm dynamical behavior of the system and consider seasonality of the changing environment, the parameters of the system usually will change with time [3, 4]. Simultaneously, in implementations of artificial neural networks, time delay may occur due to finite switching speeds of the amplifiers and communication time [5, 6]. In order to model those systems with neural networks, the neural networks with time- varying delay appear in many papers [7, 8]. So in this paper we consider the stability of the following discrete-time recurrent neural networks: In this paper, we consider control discrete-time system of neural networks of the form ( 1) () ( ( )) (( ( ))) () + = + + − + + vk Cv k AS v k BS v k hk Du k f , (1) where () n vk ∈Ω⊆ R is the neuron state vector, 2 2 0 () , 0,1, 2, , < ≤ ≤ ∀ = … h hk h k 1 { , , } = … n C diag c c , 0 ≥ i c , 1, 2,..., i n = is the n n × constant relaxation matrix, , AB are the n n × constant weight matrix, D is n m × constant matrix, () m uk ∈ R is the control vector, 1 ( , , ) n n f f f = ∈ … R is the constant external input vector and 1 1 () [ ( ), , ( )] T n n Sz s z s z = … with [ ] 1 , ( 1,1) i s C ∈ − R where i s is the neuron activations and monotonically increasing for each 1, 2,..., i n = . The asymptotic stability of the zero solution of the delay-differential system of Hopfield neural networks has been developed during the past several years. Much less is known regarding the asymptotic stability of the zero solution of the control discrete-time system of neural networks. Therefore, the purpose of this paper is to establish sufficient condition for the asymptotic stability of the zero solution of (1) in terms of certain matrix inequalities. Preliminaries The following notations will be used throughout the paper. + R denotes the set of all non-negative real numbers; + Z denotes the set of all non-negative integers; n R denotes the n-finite-dimensional Euclidean space with the Euclidean norm . and the scalar product between x and y is defined by ; T xy nm × R denotes the set of all ( ) n m × -matrices; and T A denotes the transpose of the matrix A ; Applied Mechanics and Materials Vol. 330 (2013) pp 1045-1048 Online available since 2013/Jun/27 at www.scientific.net © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.330.1045 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 202.28.38.174-26/08/13,07:12:13)