On exponential stability of bidirectional associative memory neural networks with time-varying delays Ju H. Park a, * , S.M. Lee b , O.M. Kwon c a Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea b Platform Verification Division, BcN Business Unit, KT Co. Ltd., Daejeon, Republic of Korea c School of Electrical and Computer Engineering, Chungbuk National University, Cheongju 361-763, Republic of Korea Accepted 19 April 2007 Abstract For bidirectional associate memory neural networks with time-varying delays, the problems of determining the expo- nential stability and estimating the exponential convergence rate are investigated by employing the Lyapunov func- tional method and linear matrix inequality (LMI) technique. A novel criterion for the stability, which give information on the delay-dependent property, is derived. A numerical example is given to demonstrate the effectiveness of the obtained results. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction As an extension of the unidirectional autoassociator of Hopfield [1], Kosko [2] has proposed a series of neural net- works related to bidirectional associative memory (BAM). This class of networks has good application in the area of pattern recognition and artificial intelligence. Therefore, the BAM neural networks has been one of the most interesting research topics and has attracted the attention of many researchers. For instance, refer to Refs. [3–10]. Also, time delay will inevitably occur in the communication and response of neurons owing to the unavoidable finite switching speed of amplifiers in the electronic implementation of analog neural networks, so it is more in accordance with this fact to study the BAM neural networks with time delays. The existence of time delay is frequently a source of oscillation and insta- bility [11–19]. Therefore, the study of the stability and convergent dynamics of BAM with delays has raised considerable interest in recent years, see for example [20–22] and the references cited therein. In this paper, the problem of exponential stability for BAM with time-varying delays is considered. When it comes to design a neural network, one concerns not only on the stability of the system but also on the convergence rate, that is to say, one usually desires a fast response in the network, so it is important to determine the exponential stability and to estimate the exponential convergence rate [23–28]. Based on the Lyapunov theory and linear matrix inequality frame- work, a novel less conservative criterion is given in terms of LMI. The advantage of the proposed approach is that 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.05.003 * Corresponding author. E-mail address: jessie@ynu.ac.kr (J.H. Park). Chaos, Solitons and Fractals xxx (2007) xxx–xxx www.elsevier.com/locate/chaos ARTICLE IN PRESS Please cite this article in press as: Park JH et al., On exponential stability of bidirectional associative memory , Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003 resulting stability criterion can be performed efficiently via existing numerical convex optimization algorithms such as the interior-point algorithms for solving the linear matrix inequality inequalities [30]. The rest of this paper is organized as follows: in Section 2, we formulate the problem and state the well-known facts and lemmas which would be used later; in Section 3, a new stability criterion for exponential stability of BAM with time-varying delays will be established; in Section 4, some conclusions are drawn. Notations: Throughout the paper, R n denotes the n dimensional Euclidean space, and R nÂm is the set of all n  m real matrices. I denotes the identity matrix with appropriate dimensions. q denotes the elements below the main diagonal of a symmetric block matrix. diagfÁÁÁg denotes the diagonal matrix. For symmetric matrices X and Y, the notation X > Y(respectively, X P Y ) means that the matrix X–Y is positive definite, (respectively, nonnegative). k M ðÁÞ and k m ðÁÞ denote the largest and smallest eigenvalue of given square matrix, respectively. 2. Problem statement Consider the following BAM neural networks with time-varying delays: _ u i ðtÞ¼Àa i u i ðtÞþ X m j¼1 w ji g j ðv j ðt ÀsðtÞÞÞ þ I i ; i ¼ 1; 2; ; n; _ v j ðtÞ¼Àb j v j ðtÞþ X n i¼1 v ij  g i ðu i ðt À hðtÞÞÞ þ J j ; j ¼ 1; 2; ; m; ð1Þ in which u ¼ðu 1 ; u 2 ; ; u n Þ T 2 R n and v ¼ðv 1 ; v 2 ; ; v m Þ T 2 R m are the activations of the ith neurons and the jth neu- rons, respectively, w ji and v ij are the connection weights at the time t, I i and J j denote the external inputs, sðtÞ > 0 and hðtÞ > 0 are positive time-varying delays which correspond to the finite speed of axonal signal transmission satisfying sðtÞ <  s, _ sðtÞ 6 s d < 1 and hðtÞ <  h, _ hðtÞ 6 h d < 1, respectively, sðtÞ Ã ¼ maxf  h;  sg, and a i > 0; b j > 0. In this paper, it is assumed that the activate functions g i and  g i possess the following properties: (A1) g i and  g i are nondecreasing and bounded on R; i ¼ 1; 2; ; maxfm; ng. (A2) There exist real numbers k 1i > 0 and k 2i > 0 such that 0 6 g i ðn 1 ÞÀg i ðn 2 Þ n 1 À n 2 6 k 1i ; i ¼ 1; 2; ; m; 0 6  g i ðn 1 ÞÀ  g i ðn 2 Þ n 1 À n 2 6 k 2i ; i ¼ 1; 2; ; n: ð2Þ It is clear that under the assumptions (A1) and (A2), system (1) has at least one equilibrium. Assume that u à ¼ðu à 1 ; u à 2 ; ; u à n Þ T and v à ¼ðv à 1 ; v à 2 ; ; v à m Þ T are the equilibrium point of the system, then we will shift the equilibrium points to the origin by the transformation x i ðtÞ¼u i ðtÞÀu à i , y j ðtÞ¼v j ðtÞÀv à j ,  f i ðx i ðtÞÞ ¼  g i ðu i ðtÞÞ À  g i ðu à i Þ, and f j ðy j ðtÞÞ ¼ g j ðv j ðtÞÞ Àg j ðv à j Þ. Then, the transformed system is as follows: _ x i ðtÞ¼Àa i x i ðtÞþ X m j¼1 w ji f j ðy j ðt ÀsðtÞÞÞ; i ¼ 1; 2; ; n; _ y j ðtÞ¼Àb j y j ðtÞþ X n i¼1 v ij  f i ðx i ðt ÀhðtÞÞÞ; j ¼ 1; 2; ; m; x i ðsÞ¼/ i ðsÞ; y j ðsÞ¼w j ðsÞ; s 2½Às à ; 0; i ¼ 1; 2; ; n; j ¼ 1; 2; ; m; ð3Þ where the activate functions f i and  f i satisfy the following properties: (H1) f i and  f i are bounded on R; i ¼ 1; 2; ; maxfm; ng, (H2) There exist real numbers k 1i > 0 and k 2i > 0 such that 0 6 f i ðn 1 ÞÀf i ðn 2 Þ n 1 À n 2 6 k 1i ; i ¼ 1; 2; ; m; 0 6  f i ðn 1 ÞÀ  f i ðn 2 Þ n 1 À n 2 6 k 2i ; i ¼ 1; 2; ; n; ð4Þ (H3) f i ð0Þ¼0;  f i ð0Þ¼0; 8i. For convenience, we can rewrite Eq. (3) in the form 2 J.H. Park et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx ARTICLE IN PRESS Please cite this article in press as: Park JH et al., On exponential stability of bidirectional associative memory , Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003 _ xðtÞ¼ÀAxðtÞþWf ðyðt ÀsðtÞÞÞ; _ yðtÞ¼ÀByðtÞþV  f ðxðt ÀhðtÞÞÞ; ð5Þ where xðtÞ¼ðx 1 ðtÞ; x 2 ðtÞ; ; x n ðtÞÞ T ; yðtÞ¼ðy 1 ðtÞ; y 2 ðtÞ; ; y m ðtÞÞ T , A ¼ diagða 1 ; a 2 ; ; a n Þ, B ¼ diagðb 1 ; b 2 ; ; b m Þ, W ¼ðw ij Þ mÂn , V ¼ðv ij Þ nÂm , f ¼ðf 1 ; f 2 ; ; f m Þ T , and  f ¼ðf 1 ; f 2 ; ; f n Þ T . The following facts, definition, and lemmas will be used for deriving main result. Fact 1 (Schur complement). Given constant symmetric matrices R 1 ; R 2 ; R 3 where R 1 ¼ R T 1 and 0 < R 2 ¼ R T 2 , then R 1 þ R T 3 R À1 2 R 3 < 0 if and only if R 1 R T 3 R 3 ÀR 2 "# < 0; or ÀR 2 R 3 R T 3 R 1  < 0: Fact 2. For any z; y 2 R nÂm , and any positive definite matrix X 2 R nÂn , the following inequality: 2z T y 6 z T X À1 z þ y T Xy holds. Definition 1. For system defined by (5), if there exist the positive constants k and l > 1 such that kxðtÞk þ kyðtÞk 6 le Àkt sup Às à 6h60 kxðhÞk þ sup Às à 6h60 kyðhÞk  8t > 0; then, the trivial solution of the system (5) is exponentially stable where k is called the convergence rate (or degree) of exponential stability. Lemma 1. [29] Suppose that (2) holds, then Z u v ½g i ðsÞÀg i ðvÞds 6 ½u À v½g i ðuÞÀg i ðvÞ; i ¼ 1; 2; ; n: Lemma 2 [32]. For any constant matrix R 2 R nÂn , R ¼ R T > 0, scalar c > 0, vector function x : ½0; c!R n such that the integrations concerned are well defined, then Z c 0 xðsÞds  T R Z c 0 xðsÞds  6 c Z c 0 x T ðsÞRxðsÞds: ð6Þ 3. Main result In this section, we present a stability criterion for exponential stability of system (1) using the Lyapunov stability theory and linear matrix inequality approach. Now the following theorem gives a new criterion for the stability of system (1). Theorem 1. For given positive matrices K 1 ¼ diagfk 11 ; k 12 ; ; k 1n g, K 2 ¼ diagfk 21 ; k 22 ; ; k 2n g, positive scalars  h and  s, the equilibrium point of system (1) is globally exponentially stable with convergence rate k if there exist two positive diagonal matrices D ¼ diagfd 1 ; d 2 ; ; d n g and E ¼ diagfe 1 ; e 2 ; ; e n g, positive definite matrices P, Q, R 1 ,R 2 ,Z 1 ,Z 2 , L 1 , L 2 and any matrices N i ði ¼ 1; 2; ; 10Þ satisfying the following LMI: J.H. Park et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx 3 ARTICLE IN PRESS Please cite this article in press as: Park JH et al., On exponential stability of bidirectional associative memory , Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003 P ¼ U 1 þ N 1 A þ AN T 1 2kD þ A T N T 2 A T N T 3 N 1 þ A T N T 4 A T N T 5 I R 1 À 2DAK À1 2 0 N 2 0 IIU 2 N 3 0 IIIN 4 þ N T 4 þ  hL 1 N T 5 IIIIÀ  h À1 e À2k  h L 1 IIIII IIIII IIIII IIIII IIIII 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 PW ÀN 1 W 00 00ÀN 2 W þ DW 00 V T Q À V T N T 6 V T E ÀV T N T 7 ÀN 3 W À V T N T 8 ÀV T N T 9 ÀV T N T 10 00ÀN 4 W 00 00ÀN 5 W 00 U 3 þ N 6 B þ BN T 6 2kE þ B T N T 7 B T N T 8 N 6 þ B T N T 9 B T N T 10 I R 2 À 2EBK À1 1 0 N 7 0 IIU 4 N 8 0 IIIN 9 þ N T 9 þ  sL 2 N T 10 III IÀs À1 e À2k  s L 2 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 < 0; ð7Þ where U 1 ¼ 2kP À 2PA þZ 1 ; U 2 ¼Àe À2k  h ð1 À h d ÞR 1 À e À2k  h ð1 À h d ÞK À1 2 Z 1 K À1 2 ; U 3 ¼ 2kQ À 2QB þ Z 2 ; U 4 ¼Àe À2ks ð1 À s d ÞR 2 À e À2ks ð1 À s d ÞK À1 1 Z 2 K À1 1 : Proof. Consider a Lyapunov function candidate as V ¼ V 1 þ V 2 þ V 3 þ V 4 þ V 5 ; ð8Þ where V 1 ¼ e 2kt x T ðtÞPxðtÞþe 2kt y T ðtÞQyðtÞ; V 2 ¼ 2 X n i¼1 d i e 2kt Z x i ðtÞ 0  f i ðsÞds þ 2 X m i¼1 e i e 2kt Z y i ðtÞ 0 f i ðsÞds; V 3 ¼ Z t tÀhðtÞ e 2ks  f T ðxðsÞÞR 1  f ðxðsÞÞds þ Z t tÀsðtÞ e 2ks f T ðyðsÞÞR 2 f ðyðsÞÞds; V 4 ¼ Z t tÀhðtÞ e 2ks x T ðsÞZ 1 xðsÞds þ Z t tÀsðtÞ e 2ks y T ðsÞZ 2 yðsÞds; V 5 ¼ Z t tÀ  h Z t s e 2ku _ x T ðuÞL 1 _ xðuÞdu ds þ Z t tÀ  s Z t s e 2ku _ y T ðuÞL 2 _ yðuÞdu ds: Now, let us calculate the time derivative of V i along the trajectory of (5). First the derivative of V 1 is _ V 1 ¼ e 2kt f2kx T ðtÞPxðtÞþ2x T ðtÞP _ xðtÞþ2ky T ðtÞQyðtÞþ2y T ðtÞQ _ yðtÞg ¼ e 2kt f2kx T ðtÞPxðtÞþ2x T ðtÞP ðÀAxðtÞþWf ðyðt ÀsðtÞÞÞÞ þ2ky T ðtÞQyðtÞ þ 2y T ðtÞQðÀByðtÞþV  f ðxðt À hðtÞÞÞÞg: ð9Þ Second, we get the bound of _ V 2 as 4 J.H. Park et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx ARTICLE IN PRESS Please cite this article in press as: Park JH et al., On exponential stability of bidirectional associative memory , Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003 _ V 2 ¼ 2 X n i¼1 d i e 2kt ð2k Z x i ðtÞ 0  f i ðsÞds þ  f i ðx i ðtÞÞ _ x i ðtÞÞ þ2 X m i¼1 e i e 2kt ð2k Z y i ðtÞ 0 f i ðsÞds þ f i ðy i ðtÞÞ _ y i ðtÞÞ ¼ X n i¼1 4kd i e 2kt Z x i ðtÞ 0  f i ðsÞds þ 2e kt  f T ðxðtÞÞD _ xðtÞþ X m i¼1 4ke i e 2kt Z y i ðtÞ 0 f i ðsÞds þ 2e kt f T ðyðtÞÞE _ yðtÞ 6 e 2kt f4k  f T ðxðtÞÞDxðtÞÀ2  f T ðxðtÞÞDAxðtÞþ2  f T ðxðtÞÞDWf ðyðt ÀsðtÞÞÞg þ e 2kt f4kf T ðyðtÞÞEyðtÞ À 2f T ðyðtÞÞEByðtÞþ2f T ðyðtÞÞEV  f ðxðt ÀhðtÞÞÞg ð10Þ where Lemma 1 is utilized. Third, the bound of _ V 3 is as follows: _ V 3 6 e 2kt  f T ðxðtÞÞR 1  f ðxðtÞÞ À e 2kðtÀ  hÞ ð1 À h d Þ  f T ðxðt ÀhðtÞÞÞR 1  f ðxðt ÀhðtÞÞÞ þ e 2kt f T ðyðtÞÞR 2 f ðyðtÞÞ À e 2kðtÀsÞ ð1 À s d Þf T ðyðt ÀsðtÞÞÞR 2 f ðyðt ÀsðtÞÞÞ: ð11Þ Next, we obtain the followings: _ V 4 6e 2kt x T ðtÞZ 1 xðtÞÀe 2kðtÀ  hÞ ð1 À h d Þx T ðt ÀhðtÞÞZ 1 xðt À hðtÞÞ þ e 2kt y T ðtÞZ 2 yðtÞÀe 2kðtÀsÞ ð1 À s d Þy T Âðt ÀsðtÞÞZ 2 yðt ÀsðtÞÞ: ð12Þ Finally, we have _ V 5 ¼ e 2kt  h_x T ðtÞL 1 _xðtÞÀ Z t tÀ  h e 2ks _x T ðsÞL 1 _xðsÞds þ e 2kt s_y T ðtÞL 2 _yðtÞÀ Z t tÀs e 2ks _y T ðsÞL 2 _yðsÞds 6 e 2kt  h _ x T ðtÞL 1 _ xðtÞÀe 2kðtÀ  hÞ Z t tÀ  h _ x T ðsÞL 1 _ xðsÞds þ e 2kt  s _ y T ðtÞL 2 _ yðtÞÀe 2kðtÀ  sÞ Z t tÀs _ y T ðsÞL 2 _ yðsÞds 6 e 2kt  h _ x T ðtÞL 1 _ xðtÞÀe À2k  h  h À1 Z t tÀ  h _ xðsÞds  Þ T L 1 Z t tÀ  h _ xðsÞds  þ  s _ y T ðtÞL 2 _ yðtÞ  Àe À2k  s  s À1 Z t tÀ  s _ yðsÞds  T L 2 Z t tÀ  s _ yðsÞds  ) 6 e 2kt  h _ x T ðtÞL 1 _ xðtÞÀe À2k  h  h À1 Z t tÀhðtÞ _ xðsÞds ! T L 1 Z t tÀhðtÞ _ xðsÞds ! þ  s _ y T ðtÞL 2 _ yðtÞ 8 < : Àe À2k  s  s À1 Z t tÀsðtÞ _ yðsÞds ! T L 2 Z t tÀsðtÞ _ yðsÞds ! 9 = ; ; ð13Þ where Lemma 2 is used in the second inequality. Thus, it follows that: _ V 6 e 2kt ( 2kx T ðtÞPxðtÞþ2x T ðtÞP ðÀAxðtÞþWf ðyðt ÀsðtÞÞÞÞþ2ky T ðtÞQyðtÞþ2y T ðtÞQðÀByðtÞþV  f ðxðt ÀhðtÞÞÞÞ þ4k  f T ðxðtÞÞDxðtÞÀ2  f T ðxðtÞÞDAxðtÞþ2  f T ðxðtÞÞDWf ðyðt ÀsðtÞÞþ4kf T ðyðtÞÞEyðtÞÀ2f T ðyðtÞÞEByðtÞ þ2f T ðyðtÞÞEV  f ðxðt ÀhðtÞÞþ  f T ðxðtÞÞR 1  f ðxðtÞÞÀe À2k  h ð1Àh d Þ  f T ðxðt ÀhðtÞÞÞR 1  f ðxðt ÀhðtÞÞÞ þf T ðyðtÞÞR 2 f ðyðtÞÞÀe À2k  s ð1Às d Þf T ðyðt ÀsðtÞÞÞR 2 f ðyðt ÀsðtÞÞÞþx T ðtÞZ 1 xðtÞ Àe À2k  h ð1Àh d Þx T ðt ÀhðtÞÞZ 1 xðt ÀhðtÞÞþy T ðtÞZ 2 yðtÞÀe À2k  s ð1Às d Þy T ðt ÀsðtÞÞZ 2 yðt ÀsðtÞÞþ  h _ x T ðtÞL 1 _ xðtÞ Àe À2k  h  h À1 Z t tÀhðtÞ _ xðsÞds !! T L 1 Z t tÀhðtÞ _ xðsÞds !! þ  s _ y T ðtÞL 2 _ yðtÞÀe À2k  s  s À1 Z t tÀsðtÞ _ yðsÞds ! T L 2 Z t tÀsðtÞ _ yðsÞds ! 9 = ; : ð14Þ Here note that J.H. Park et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx 5 ARTICLE IN PRESS Please cite this article in press as: Park JH et al., On exponential stability of bidirectional associative memory , Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003 À 2  f T ðxðtÞÞDAxðtÞ 6 À2  f T ðxðtÞÞDAK À1 2  f ðxðtÞÞ; À 2f T ðyðtÞÞEByðtÞ 6 À2f T ðyðtÞÞEBK À1 1 f ðyðtÞÞ; À e À2k  h x T ðt ÀhðtÞÞZ 1 xðt À hðtÞÞ 6 Àe À2k  h  f T ðxðt ÀhðtÞÞÞK À1 2 Z 1 K À1 2  f ðxðt ÀhðtÞÞÞ; À e À2k  s y T ðt ÀsðtÞÞZ 2 yðt ÀsðtÞÞ 6 Àe À2k  s f T ðyðt ÀsðtÞÞÞK À1 1 Z 2 K À1 1 f ðyðt ÀsðtÞÞÞ; ð15Þ where the property H2 and Fact 2 are used to derive the inequalities. For any appropriate dimensional matrices N i ði ¼ 1; 2; ; 10Þ, the following equations hold: 2 " x T ðtÞN 1 þ  f T ðxðtÞÞN 2 þ  f T ðxðt À hðtÞÞÞN 3 þ _ x T ðtÞN 4 : þ Z t tÀhðtÞ _ xðsÞdsÞ ! T N 5 3 5 ½ _ xðtÞþAxðtÞÀWf ðyðt ÀsðtÞÞÞ ¼ 0; 2½y T ðtÞN 6 þ f T ðyðtÞÞN 7 þ f T ðyðt ÀsðtÞÞÞN 8 þ _ y T ðtÞN 9 þ Z t tÀsðtÞ _ yðsÞdsÞ ! T N 10 # ½ _ yðtÞþByðtÞÀV  f ðxðt ÀhðtÞÞÞ ¼ 0: ð16Þ Substituting Eq. (15) into Eq. (14) and utilizing the relationship (16) gives that _ V 6 e 2kt z T ðtÞPzðtÞ; ð17Þ where zðtÞ¼ x T ðtÞ  f T ðxðtÞÞ  f T ðxðt ÀhðtÞÞÞ _x T ðtÞ Z t tÀhðtÞ _xðsÞdsÞ ! T y T ðtÞ f T ðyðtÞÞ f T ðyðt ÀsðtÞÞÞ _y T ðtÞ Z t tÀsðtÞ _yðsÞdsÞ ! T 2 4 3 5 T : Since the matrix P given in Theorem 1 is the negative definite matrix, we have _ V 6 0, it follows that V 6 V ð0Þ. Then we have the followings: V ð0Þ¼x T ð0ÞPxð0Þþ2 X n i¼1 d i Z x i ð0Þ 0  f i ðsÞds þ Z 0 Àhð0Þ e 2ks  f T ðxðsÞÞR 1  f ðxðsÞÞds þ Z 0 Àhð0Þ e 2ks x T ðsÞZ 1 xðsÞds þ y T ð0ÞQyð0Þþ2 X m i¼1 e i Z y i ð0Þ 0 f i ðsÞds þ Z 0 Àsð0Þ e 2ks f T ðyðsÞÞR 2 f ðyðsÞÞds þ Z 0 Àsð0Þ e 2ks y T ðsÞZ 2 yðsÞds þ Z 0 À  h Z 0 s e 2ku _ x T ðuÞL 1 _ xðuÞdu ds þ Z 0 Às Z 0 s e 2ku _ y T ðuÞL 2 _ yðuÞdu ds: ð18Þ Also, we further get the bound of V ð0Þ as follows: V ð0Þ 6 k M ðP Þk/k 2 þ 2d M k 1M k/k 2 þðk M ðR 1 Þk 2 1M þ k M ðZ 1 ÞÞ Z 0 À  h e 2ks x T ðsÞxðsÞds þ k M ðQÞkwk 2 þ 2e M k 2M kwk 2 þðk M ðR 2 Þk 2 2M þ k M ðZ 2 ÞÞ Z 0 À  s e 2ks y T ðsÞyðsÞd s þk M ðL 1 Þ Z 0 À  h Z 0 s _ x T ðuÞ _ xðuÞdu ds þ k M ðL 2 Þ Z 0 À  s  Z 0 s _ y T ðuÞ _ yðuÞdu ds; ð19Þ where d M ¼ maxðd i Þ, e M ¼ maxðe i Þ, k 1M ¼ maxðk 1i Þ, k 2M ¼ maxðk 2i Þ, k/k¼sup À  h6h60 kxðhÞk, and kwk¼sup Às6h60 kyðhÞk. It follows from Fact 2 that: _ x T ðsÞ _ xðsÞ 6 2x T ðsÞA T AxðsÞþ2f T ðyðs ÀsðsÞÞÞW T Wf ðyðs ÀsðsÞÞÞ 6 2k M ðA T AÞk/k 2 þ 2k M ðW T W Þk M ðK 2 1 Þkwk 2 _ y T ðsÞ _ yðsÞ 6 2y T ðsÞB T ByðsÞþ2  f T ðxðs À hðsÞÞÞV T V  f ðxðs À hðsÞÞÞ 6 2k M ðB T BÞkwk 2 þ 2k M ðV T V Þk M ðK 2 2 Þk/k 2 : ð20Þ 6 J.H. Park et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx ARTICLE IN PRESS Please cite this article in press as: Park JH et al., On exponential stability of bidirectional associative memory , Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003 From the relationship (20) and simple calculation, we further have V ð0Þ 6 k M ðP Þk/k 2 þ 2d M k 1M k/k 2 þðk M ðR 1 Þk 2 1M þ k M ðZ 1 ÞÞk/k 2 1 À e À2k  h 2k þ k M ðQÞkwk 2 þ 2e M k 2M kwk 2 þðk M ðR 2 Þk 2 2M þ k M ðZ 2 ÞÞkwk 2 1 À e À2k  s 2k þ  h 2 k M ðL 1 Þð2k M ðA T AÞk/k 2 þ 2k M ðW T W Þk M ðK 2 1 Þkwk 2 Þ þ  s 2 k M ðL 2 Þð2k M ðB T BÞkwk 2 þ 2k M ðV T V Þk M ðK 2 Þ 2 Þk/k 2 Þ ¼fk M ðPÞþ2d M k 1M þðk M ðR 1 Þk 2 1M þ k M ðZ 1 ÞÞ 1 À e À2k  h 2k þ 2  h 2 k M ðL 1 Þk M ðA T AÞ þ 2  s 2 k M ðL 2 Þk M ðV T V Þk M ðK 2 Þ 2 Þgk/k 2 þ k M ðQÞþ2e M k 2M þðk M ðR 2 Þk 2 2M þ k M ðZ 2 ÞÞ 1 À e À2k  s 2k  þ2  h 2 k M ðL 1 Þk M ðW T W Þk M ðK 2 1 Þþ2  s 2 k M ðL 2 Þk M ðB T BÞ  kwk 2  c 1 k/k 2 þ c 2 kwk 2 : ð21Þ Furthermore, we have V P e 2kt ðk m ðPÞkxðtÞk 2 þ k m ðQÞkyðtÞk 2 Þ: Then we easily obtain kxðtÞk þ kyðtÞk 6 ffiffiffi 2 p ðkxðtÞk 2 þkyðtÞk 2 Þ 1=2 6 lðk/k 2 þkwk 2 Þ 1=2 e Àkt 6 lðk  /kþk  wkÞe Àkt for all t P 0, where l P 1 is a constant and k  /k¼ sup Às à 6h60 kxðhÞk; k  wk¼ sup Às à 6h60 kyðhÞk: Thus by Definition 1, system (5) is exponentially stable and has the exponential convergence rate k. This completes the proof. h Remark 1. The criterion given in Theorem 1 is delay-dependent. It is well known that the delay-dependent criteria are generally less conservative than delay-independent criteria when the delay is small. Remark 2. The solutions of Theorem 1 can be obtained by solving the eigenvalue problem with respect to solution variables, which is a convex optimization problem [30]. In this paper, we utilize Matlab’s LMI Toolbox [31] which implements interior-point algorithm. This algorithm is significantly faster than classical convex optimization algo- rithms [30]. Example 1. Consider the following BAM neural networks (5) with  f i ðxÞ¼ 1 2 ðjx i þ 1jÀjx i À 1jÞ, f j ðyÞ¼ 1 2 ðjy j þ 1jÀjy j À 1jÞ, s =1,h ¼ 0:5 and A ¼ I; B ¼ 2I; W ¼ 0:05 0:25 0:05 0:10:05 0:15 0:15 0:15 0:05 2 6 4 3 7 5 ; V ¼ 0:75 0:75 0:95 00:50:15 0:15 0:15 0:05 2 6 4 3 7 5 : ð22Þ From the functions  f i ðxÞ and f j ðyÞ, we can easily obtain K 1 ¼ K 2 ¼ I. When the exponential convergence rate is taken as k = 0.4, the criteria given in [27,28,26] cannot determine that system (22) is exponentially stable. However when our criterion given in Theorem 1 is applied to the system (22), our maximum allowable convergence rate for guaranteeing exponential stability of the system (22) is k = 0.57. Thus our result is less conservative than those of the existing works [26–28]. When the time-varying delays are considered for the system (22) with hðtÞ 6 1 and sðtÞ 6 0:5, the maximum allowable convergence rate is summarized in Table 1. Table 1 Convergence rate k h d ð¼ s d Þ 0.3 0.5 0.7 0.9 Maximum allowable convergence rate k 0.52 0.47 0.39 0.21 J.H. Park et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx 7 ARTICLE IN PRESS Please cite this article in press as: Park JH et al., On exponential stability of bidirectional associative memory , Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003 4. Concluding remarks A novel criterion for exponential stability of BAM neural networks with time-varying delays has been presented by combining the Lyapunov functional method with LMI framework. The criterion is delay-dependent and expressed by LMI. Throughout a numerical example, it is shown that our criterion is less conservative than those of existing results. References [1] Hopfield J. Neuron with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA 1984;81:3088–92. [2] Kosko B. Bidirectional associative memories. IEEE Trans Syst Man Cybernet 1988;18:49–60. [3] Gopalsamy K, He XZ. Delay-independent stability in bidirectional associative memory networks. IEEE Trans Neural Networks 1994;5:998–1002. [4] Cao J, Wang L. Periodic oscillatory solution of bidirectional associative memory networks. Phys Rev E 2000;61:1825–8. [5] Guo SJ, Huang LH, Dai BX, Zhang ZZ. 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Park et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx ARTICLE IN PRESS Please cite this article in press as: Park JH et al., On exponential stability of bidirectional associative memory , Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003 [30] Boyd B, Ghaoui LE, Feron E, Balakrishnan V. Linear matrix inequalities in systems and control theory. Philadelphia: SIAM; 1994. [31] Gahinet P, Nemirovski A, Laub A, Chilali M. LMI control toolbox user’s guide. Massachusetts: The Mathworks; 1995. [32] Gu K. An integral inequality in the stability problem of time-delay systems. In: Proceedings of IEEE conference decision control. Sydney, Australia: December 2000. p. 2805–10. J.H. Park et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx 9 ARTICLE IN PRESS Please cite this article in press as: Park JH et al., On exponential stability of bidirectional associative memory , Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003 . On exponential stability of bidirectional associative memory neural networks with time-varying delays Ju H. Park a, * , S.M. Lee b , O.M. Kwon c a Department of Electrical Engineering,. On exponential stability of bidirectional associative memory , Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003 4. Concluding remarks A novel criterion for exponential stability. used later; in Section 3, a new stability criterion for exponential stability of BAM with time-varying delays will be established; in Section 4, some conclusions are drawn. Notations: Throughout