Novel robust stability criteria of neutral-type bidirectional associative memory neural networks

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1 0 Novel robus sabiliy crieria of neural-ype bidirecional associaive memory neural neworks Shu-Lian Zhang and Yu-Li Zhang School of Science Dalian Jiaoong Universiy Dalian 608 P. R. China Absrac he exisence uniqueness and global robus exponenial sabiliy is analyzed for a class of uncerain neural-ype bidirecional associaive memory BA neural neworks wih ime-varying delays. Wihou assuming he boundedness of he acivaion funcions by consrucing a novel class of augmened Lyapunov- Krasovskii funcional new relaxed delay-dependen sabiliy crieria of he unique equilibrium poin are presened in erms of linear marix inequaliies LIs. Following he idea of convex combinaion and free-weighing marices mehod less conservaive resuls are obained. wo examples are given o illusrae he effeciveness of our proposed condiions. Keywords: Global robus exponenial sabiliy globally exponenial sabiliy linear marix inequaliyli neural-ype bidirecional associaive memory BA neural nework.. Inroducion Bidirecional associaive memory BA neural neworks which were proposed by Kosko in [0] generalized he single-layer auoassociaive Hebbian correlaor o a wolayer paern-mached heeroassociaive circuis. here are many applicaions for BA neural neworks such as paern recogniion arificial inelligence solving opimizaion problem and auomaic conrol engineering. herefore he BA neural neworks have been one of he mos ineresing research opics and have araced he aenion of many researchers. Up o now many imporan resuls on he sabiliy of BA neural neworks have been repored in he lieraure see e.g. [] [4] [7] [8] [] [5] [9] and references herein. On he oher hand he sabiliy of a neural nework may ofen be desroyed by is unavoidable uncerainies due o he exisence of modeling errors exernal disurbance and parameer flucuaion in he applicaions and designs of neural neworks. herefore he robus sabiliy analysis of neural neworks has gained much research aenion [5] [7] [9] [4] [7]. However he exisence of ime delays in hese DNN models indicaes ha ime delays are dependen on he pas sae. In fac many pracical delay sysems can be modeled as differenial sysems of neural ype whose differenial expression concludes no only he derivaive erm of he curren sae bu also concludes he derivaive erm of he pas sae such as parial elemen equivalen circuis and ransmission lines in elecrical engineering conrolled consrained manipulaors in mechanical engineering populaion dynamics sysem and so on see []. I is naural and imporan ha sysems should conain some informaion abou he derivaive of he pas sae o furher describe and model he dynamics for such complex neural reacions. here have been many resuls abou neural-ype cellular neural neworks bu o he bes of our knowledge few researchers sudied he sabiliy for BA neural neworks which is described by nonlinear delay differenial equaions of he neural ype. By using Jensen inegral inequaliy Liu e al. [] recenly proposed sabiliy condiions of neural-ype BA neural neworks wih consan delays which are expressed in erms of LIs. Based on an inequaliy Park e al. [4] obained a condiion of globally asympoic sabiliy for such sysems also wih consan ime delays. Up o now here are no resuls abou such sysems wih ime varying delays. oivaed by he preceding discussions he aim of his paper is o relax he consrain on he boundedness of he acivaion funcion and sudy he exisence uniqueness and global robus exponenial sabiliy for uncerain neural-ype BA neural neworks wih ime-varying delays. he auhors inroduce a novel form of augmened Lyapunov-Krasovskii funcional ha akes ino accoun k new erms e u Pu and k e v Q v whose derivaives are direcly coupled wih boh neural and rearded sysems. he proposed Lyapunov funcional also includes he erms of cross producs k e u Pu k e v Qv and some inegral erms of cross producs such as ks e u s R g u sd s ks e v s S f v s s d which are no considered in previous resuls. Following he idea of convex combinaion and free-weighing marices mehod [9] we derive several new sufficien Copyrigh c 04 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

2 condiions for he global exponenial sabiliy of BA neural neworks wih ime-varying delays. he derived condiions are expressed in erms of linear marix inequaliies LIs which can be checked numerically very efficienly via he LI oolbox. Some comparisons beween he obained resuls in his paper and previous resuls are made by wo illusraive examples. he res of his paper is organized as follows. In Secion II problem formulaion and preliminaries are given. In Secions IIIIV new delay-dependen condiions are esablished for he exisence uniqueness and exponenial sabiliy. In Secion V he new sabiliy condiions are exended o neural neworks wih normbounded uncer-ainies. Secion VI provides wo illusraive examples. Finally some conclusions are drawn in Secion VII.. Problem descripion Considering he following neural-ype neural neworks wih ime-varying delays: x Ax Bf y Cf y Dx J y Ay Bg x Cg x D y E x x x... x n and y y y... ym are he neural sae vecors JE are he consan exernal inpu vecors. A A A B B B C C C D D D A A A B B B C C C D D D. A diag a a... a n and A diag a a... a m are posiive diagonal marices B b i nm C c i nm D d i nn B bi mn C ci mn D di are mm known consan marices A B C D A B C D are parameric uncerainies 0 0 are he ime-varying delays are posiive consans. f x f y f y... f m ym g x g x... g n xn denoe he neural acivaion funcions. I is assumed ha here exis consans l i l i l l such ha 4 f i s f i s l i l i s s g s g s l l4 s s for any s s R s... ; s i m... n. oreover we assume ha he iniial condiion of neural neworks has he form x yi i [ max 0] i i... m;... n are coninuous funcions. hroughou his paper le y denoe he Euclidean norm n of a vecor yr W W W m W and W W W denoe he ranspose he inverse he larges eigenvalue he smalles eigenvalue and he specral norm of a square marix W respecively. Le W>0<0 denoe a posiive negaive definie symmeric marix I denoe an ideniy marix wih compaible dimension. he definiion of exponenial sabiliy is now given. Definiion [0] he neural nework is said o be globally exponenially sable if here exis consans k 0 and > such ha x y sup x x 0 k sup y y e 0 k is called he exponenial convergence rae. Clearly he equilibrium poin of neural nework is robus sable if and only if he zero soluion of neural nework 4 is robus sable. he ime-varying uncerain marices are defined by: A B C D H0E0 G G G G4 4 A B C D HE G5 G6 G7 G8 H0 H Gi i...8 are known real consan marices wih appropriae dimensions. E 0 E are unknown ime-varying marices saisfying E E I E E I In order o obain he main resuls we need he following lemmas. Lemma see [] Le XY and P be real marices of appropriae dimensions wih P>0. hen for any posiive scalar he following marix inequaliy holds: XYYX XP X YPY. n n Lemma see [7] Coninuous map z : R R is homeomorphic if z is inecive and lim z. z Copyrigh c 04 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

3 Exisence and uniqueness of he equilibrium poin Firsly we consider neural nework wihou uncerainies ha is x Ax Bf y Cf y Dx J 6 y Ay Bg x Cg x Dy E. In order o sudy he exisence and uniqueness of he equilibrium poin we define map H w as H w [ H w H w] [ w x y ] and H w Ax B C f y J H w A y B C g x E. heorem. Under assumpions and given a consan k 0 suppose ha here exis posiive definie symmeric marices P [ Pi ] Q [ Qi ] nonnegaive definie symmeric marices R [ Ri ] S [ Si ] Ui Zi i posiive diagonal marices real marices X Y 4 wih compaible dimensions such ha he following LIs hold i=: i k i e U 0 0 k 0 e U i 44 kp P R A A P R R X X LL4 ki A P X X 4 R AR L L4 6 P R D P 7 AZ P R B P R C k kp e R X X X X L4L4 X X 4 k 5 e R L L4 4 k 6 P D P e R P B P C X X R 46 R D 48 B Q S 49 B Q 4 R B B S 4 R C 44 B Z e R k 55 4 e R C Q S k C Q 5 C S 54 C Z k 66 e R 67 D Z 77 Z Z R U 7 ZB 7 ZC 88 kq Q S A A Q S S Y Y L L 89 ki A Q Y Y 8 S AS L L 8 Q S D Q 84 A Z k 99 kq e S Y Y Y Y L L 90 Y Y4 k 9 e S L L k 9 QD Q e S 00 Y4 Y4 S k S D e S k e S k e S 4 D Z 44 Z Z S U [ X X ] [ 0 X X ] [ Y Y ] [ Y Y ] Li diag li li... lin i 4 and oher parameers 7 i i are all equal o zero's hen neural nework 6 have a unique equilibrium poin. Proof. As done in [6] we will firsly prove ha nm H w H w for any w w w wr. Now suppose H w H w ha is A x x B C f y f y 0 A y y B C g x g x 0. I is easy o see ha [ x x P R P g x g x R] A xx BC f y f y 0 8 [ y y Q S Q f y f y S] A yy B C g x g x 0. 9 By inequaliies and we ge y y L L y y f y f y f y f y y y LL f y f y 0 xx L 4 L4 xx gx gx 4 gx gx xx L L g x g x Copyrigh c 04 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

4 hese ogeher wih Eqs.89 give G 0 0 G G i 44 [ x x g x g x y y f y f y ] G P R P A A P R P L 4 L4 G A R 4 L L4 G4 P R P B C G 4 G B C F4 Q G4 R B C B C S G Q S Q A A Q S Q L L G A S L L \ G On he oher hand one can infer from inequaliy 7 i== ha 0. Le I I I I I B I I I I I 0 0 muliplying by B and B on is lef and righ side respecively we obain I I 0P 0I 0 G k 0 0 Q 0 I I k k e R e S diag 0 I I. 0 0 Noe ha k0 P0 Q0 R0 S00 i i 0 0 hus G<0. Obviously his conradics wih 0. he conradicion implies ha H w H w. Hence map H is inecive. Nex we show ha H w as w. o prove his i suffices o show ha H w H 0 as w. Similar o above proof from Lemma and assumpions we obain [ x P R P g x g0 R] H w H0 [ y Q S Q f y f0 S ] H w H 0 x x g x g0 g x g0 G y y f y f0 f y f0 x x g x g0 g x g0 G y y f y f0 f y f 0 G x x g x g0 g x g0 y y f y f0 f y f0 G x x x x y y y y G x y G w diag... n i max l i li diag ñ... ñn ñi max li l4i i... n. From assumpions and above discussion we have 0 0 G 0. By Schwarz inequaliy and assumpions we have G w [ x P R P g x g0 R] H w H0 [ y Q S Q f y f0 S] H w H0 x P R P g x g0 R H w H0 y Q S Q f y f0 S H w H0 P R P R x H w H0 Q S Q S y H w H0 H w H0 P R P R Q S Q S. ha is w H w H 0 G i i 4. Exponenial sabiliy resul of he equilibrium poin In order o prove he robus sabiliy of he equilibrium * * poin x y of neural nework we will firs simplify * neural nework as follows. Le u x x v * y y hen we have u Au Bf v Cf v Du v Av Bgu Cgu Dv Copyrigh c 04 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

5 4 * * f v f v y f y g u * * i i i i i i i g u x g x wih f 0 g 0 0 i... m;... n. By assumpions and we can see ha fi s fi s l i li s s l i g s g s l. 4 s s Nex we consider neural nework wihou uncerainies ha is u Au B f v C f v Du 4 v Av Bgu Cgu Dv. heorem. he unique equilibrium poin of neural nework 4 is sable wih exponenial convergence rae k if he condiions of heorem are saisfied. Proof. Consider he following Lyapunov-Krasovskii funcional: V u V u 5 i wih k k V u e P e Q ks d V u e s R s s ks e s S sd s ks ks V u s e u s U u sds s e v s U v sd s [ u u ] [ v v ] s [ u s g u s u s] s [ v s f v s v s]. For convenience we denoe u u v v. he ime derivaive of funcional 5 along he raecories of neural nework 4 is obained as follows: k V u e k P P k Q Q k V u e R S k e R k e S i V u e u U u v U v k k s d k s e u s U u s s e v s U v sd s. Based on Leibniz-Newon formula for any real marix X i i...4 wih compaible dimensions we ge k 0 e u X u X u u u sd s 6 k 0 e u X u X 4 k 0 e v Y v Y u u u sd s 7 v v v sd s 8 k 0 e v Y v Y4 v v v s d s. 9 I is easy o ge he following inequaliies by using Lemma : u X u X u sds k e u s U u sds k e U 0 u X u X u sds 4 k e u s Uu sds k e U v Y v Y v sds k e v s U v sds k e U 4 vy v Y vs ds k e v s U v sds k e U [ u u u g u g u u u v v v f v f v v v ]. On he oher hand one can infer from inequaliies ha he following marix inequaliies hold for any Copyrigh c 04 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

6 5 posiive diagonal marices i i 4 wih compaible dimensions k 0 e v L L v f v f v v L L f v 4 k 0 e v LLv f v f v v L L f v 5 k 0 e u L4Lu g u g u u L L g u 6 4 k 0 e u LLu g u g u u L L g u o ge less conservaive crierion we inroduce he following equaliies for any real marices Z Z wih compaible dimensions 0 u Z u Au B f v C f v D u 8 0 v Z v A v B g u C g u D v. 9 From 5-9 we obain k k V u e e U k e U k e U k e U. Noe ha 0 0 so k k e U e U k e U k e U 0 holds if and only if he following four inequaliies k k e U e U 0 0 k k e U e U 0 k k e U e U 0 k k e U e U 0 are rue. From he well-known Schur complemen inequaliies 0- are equivalen o 7 wih i= respecively hus Vu 0 holds if 7 i= are rue. Furhermore following he similar line in [7] from Lemma we have V u0 x sup u * 0 * 4 0 y sup v 4 P R R U 4 Q S 4 S U max l l max l l. and i n i 4 i i n i i eanwhile k V u e * P x Q m m * y min k e m P m Q x y * * by Lyapunov sabiliy heory he proof of heorem is compleed. Remark. One can noice ha he augmened Lyapunov funcional approach of his paper is quie differen from k previous ones. New erms e u Pu and k e v Qv are used o augmen he Lyapunov funcional whose derivaives are direcly coupled wih boh neural and rearded sysems. herefore he augmened Lyapunov funcional can lead o an reduce in he conservaiveness of he resuls which will be illusraed by hree examples. Remark. he proposed Lyapunov funcional also includes he erms of cross producs k k e u Pu e v Qv and some inegral erms of cross producs such as ks ks e u s R gd u s s e v s S fd v s s which are no considered in previous resuls. his approach provides a quasi-full-size Lyapunov funcional hrough augmenaion. oreover he se of Lyapunov- Krasovskii funcional inroduced in Park e al. [4] is us reduced form of he one proposed in his paper. I is well known ha as he Lyapunov funcional is reduced he corresponding resuls become more conservaive. herefore he proposed novel augmened Lyapunov funcional can yield less conservaive resuls han he exising mehods. Remark. I should be poined ou ha he condiion of heorem in [4] needs o be revised. In he proof one Copyrigh c 04 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

7 6 key sep is he following proposiion see inequaliy 4 in Page 79 of [4]: Proposiion. Le L L 4. he following inequaliy holds for any given L4 0 posiive definie marices Z: g y Zg y y L ZL y Unforunaely he above Proposiion is no valid in general his fac can be illusraed by he following example: Example.. Obviously g s g s s s saisfy condiions wih L 4 I. Se y [ ] hen g y [ ]. Furher se Z 5 we have g y Zg y y LZLy i.e. inequaliy 4 is false. 4 4 Based on above analysis he LIs of heorem in [4] may no be sufficien condiions assumed ha Z is a posiive definie marix. In fac if Z is revised o be a posiive scalar marix hen he above Proposiion and heorem in [4] are sill valid. Remark 4. If any of are unknown or any of are no differeniable by seing Pi Qi 0 i R S 0 in funcional 5 and adding he following zero equaions o he ime derivaive along he raecories of neural nework 4: 0 u K u Au B f v C f v D u 0 v K v A v B g u Cg u Dv Ki i are any real marices wih compaible dimensions. Following he similar line in heorem we can obain a sabiliy crierion similar o LIs 7. Remark 5. In erms of LIs heorem and Remark 4 provide a sufficien condiion for he global exponenial sabiliy of he delayed neural-ype neural nework in 4. One of he advanages of he LI approach is ha he LI condiion can be checked numerically very efficienly by using he inerior-poin algorihms which have been developed recenly in solving LIs []. Nex consider sysem 4 wih D D 0 ha is u Au Bf v Cf v 5 v Av Bgu Cgu. I is easy o see he following resul holds from heorem. Corollary. Under assumpions and given a consan k 0 suppose ha here exis posiive definie symmeric marices PQ nonnegaive definie symmeric marices RSZ i Ui i posiive diagonal marices real marices X Y 4 wih compaible dimensions such ha he LIs 7 i= wih D D 0 hold hen he equilibrium poin of neural nework 5 is exponenial sable. Remark 6. In Corollary by seing Pi Qi 0 i R S 0 we can employ his crierion o analyze he sabiliy of neural nework 5 when any of are unknown or any of are no differeniable. 5. Robus exponenial sabiliy resuls of uncerain delayed neural nework Now based on Lemma we invesigae he robus exponenial sabiliy condiion for neural nework wih uncerainies saisfying 4 and 5. Firsly by using he same funcional as in heorem we can easily obain he following resul. heorem. Under assumpions -5 and given a consan k 0 suppose ha here exis posiive scalars 0 posiive definie symmeric marices PQ nonnegaive definie symmeric marices RSZ i Ui i posiive diagonal marices real marices X Y 4 wih compaible dimensions such ha he following LIs hold i=: 0H0 H i H00 0I H 0 I k i 0 0 e U 0 k e U [ G 0000 \ G 0000G G 00] 0 4 [ G 0000 \ G 0000G G 00] 4 Copyrigh c 04 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

8 7 [0 0 0 G6 G7 0 0 G5 \ 0 0 \ 0 0 G8 0] [0 0 0 G G 0 0 G \ G 0] PR P R \ Z QS \ Q S Z [ ] [ ] and oher parameers are all defined in heorem hen he equilibrium poin of neural nework is robus exponenial sable Remark 7. Similar o Remark 4 if any of are no differeniable or any of are unknown by seing Pi Qi 0 i RS 0 in funcional 5 and adding he following zero equaions 0 u K u Au B f v C f v D u 0 v K v A v B g u C g u D v o he ime derivaive along he raecories of neural nework following he similar line in heorem we can obain a sabiliy crierion similar o LIs 6. Now consider sysem wih D D 0 ha is u Au Bf v Cf v v Av Bgu Cgu. I is easy o see he following resul holds from heorem. Corollary. Under assumpions and given a consan k 0 suppose ha here exis posiive scalars 0 posiive definie symmeric marices PQ nonnegaive definie symmeric marices RSU i i posiive diagonal marices real marices X Y 4 wih compaible dimensions such ha he LIs 6 wih D D 0 hold hen he equilibrium poin of neural nework 7 is robus exponenial sable. Remark 8. In Corollary by seing Pi Qi 0 i R S 0 we can employ his crierion o analyze he sabiliy of neural nework 7 when any of are unknown or any of are no differeniable. 6. Comparison and Illusraive Examples Nex we provide wo numerical examples o demonsrae he effeciveness and less conservaiveness of our delaydependen sabiliy crieria over some recen resuls in he lieraure. Example 5.. Consider neural nework 5 wih A I A 4 I B B C C L L 0 L L4 I. his model was sudied in [7] [8]. For his neural nework i is verified in [8] ha he resuls given in [] [] fail o ascerain he sabiliy for any ime delay. Furhermore if we se exponenial convergence rae k be fixed as 0.5 none of he crieria of [4] [9] can guaranee he sabiliy for any ime delay wih 0or 0. Se 0.50 all of he crieria given in [8] [7] fail o verify he sabiliy for any ime delay he allowable ime delay upper bound obained by Gau e al. [7] is 0.50 while our mehod shows ha he equilibrium poin of his neural nework is exponenially sable for any ime delay wih.06. his is much larger han he one of [7] which shows he less conservaiveness of our developed mehod. Example 5.. Consider neural nework 6 wih A diag.5 A diag B B c 0 C C c c c D D E J 6 4 f s f s g s g s 0.5 s s. Obviously he acivaion funcions saisfy assumpions and wih L L 0 L L4 I. Obviously none of he resuls in [] [4] [7] [8] [] [5] and [9] can be applied o verify he sabiliy of his model. However if we se 0.5 from heorems and we can conclude ha his neural nework has a unique equilibrium poin Copyrigh c 04 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

9 which is exponenial sable for any c wih 0 c Conclusion In his paper we have invesigaed he global robus sabiliy problem of uncerain BA neural neworks of neural-ype. By employing new Lyapunov-Krasovskii funcional we proposed several novel sabiliy crieria for he considered neural neworks. he obained resuls are all in he form of LI which can be easily opimized. Finally hree examples are given o show he superioriy of our proposed sabiliy condiions o some exising ones. References [] S. Boyd L. E.I. Ghaoui E. Feron V. Balakrishnan Linear arix Inequaliy in Sysem and Conrol heory SIA Philadelphia PA 994. [] J. Cao L. Wang Exponenial sabiliy and periodic oscillaory soluion in BA neworks wih delays IEEE rans. Neural New. 00: [] J. Cao. Dong Exponenial sabiliy of delayed bidirecional associaive memory neworks Appl. ah. Compu.005:05-. [4] J. Cao D.W.C. Ho X. Huang LI-based crieria for global robus sabiliy of bidirecional associaive memory neworks wih ime delay Nonl. Anal : [5] A. Chen L. Huang J. Cao Exponenial sabiliy of delayed bidirecional associaive memory neural neworks wih reacion diffusion erms In. J. Sysems Science 00785: 4-4. [6] L. Chen H. Zhao New LI condiions for global exponenial sabiliy of cellular neural neworks wih delays Nonl. Anal.: Real World Appl. 0090: [7] Gau R.S. Hsieh J.G. Lien C.H. Global exponenial sabiliy for uncerain bidirecional associaive memory neural neworks wih muliple ime-varying delays via LI approach In. J. Circ. heor. Appl. 0086: [8] X. Huang J. Cao D.S. Huang LI-based approach for delay-dependen exponenial sabiliy analysis of BA neural neworks Chaos Solions Fracals 0054: [9] Y. He G.P. Liu D. Rees and. Wu Sabiliy analysis for neural neworks wih ime-varying inerval delay IEEE rans. Neural New : [0] B. Kosko Adapive bidirecional associaive memories Applied Opics 9876: [] B. Kosko Bidirecional associaive memories IEEE rans. Sysems an Cybernaics 9888: [] J. Liu G. Zong New delay-dependen asympoic sabiliy condiions concerning BA neural neworks of neural ype Neurocompuing 0097: [] H. ai X. Liao C. Li A semi-free weighing marices approach for neural-ype delayed neural neworks J. Compu. Appl. ah. 0095: [4] J. H. Park C. H. Park O.. Kwon S.. Lee A new sabiliy crierion for bidirecional associaive memory neural neworks of neural-ype Appl. ah. Compu : [5] J. H. Park Robus sabiliy of bidirecional associaive memory neural neworks wih delays Physics Leers A 00649: [6] S. Senan S. Arik Global robus sabiliy of bidirecional associaive memory neural neworks wih muliple ime delays IEEE rans. Sys. an Cybern. B 00705: [7] L. Sheng H. Yang Novel global robus exponenial sabiliy crierion for uncerain BA neural neworks wih ime-varying delays Chaos Solions Fracals 00940: 0-. [8] Z. Wang H. Zhang Sabiliy analysis of BA neural neworks wih ime-varying delays Progress in Naural Science 0077: 06-. [9] Z. Wang H. Zhang W. Yu Robus exponenial sabiliy analysis of neural neworks wih muliple ime delays Neurocompuing 00770: [0] S. Xu J. Lam D. W.C. Ho Y. Zou Delay-dependen exponenial sabiliy for a class of neural neworks wih ime delays" J. Compu. Appl. ah. 0058:6-8. [] Z. Zeng and J. Wang Improved condiions for global exponenial sabiliy of recurren neural neworks wih imevarying delays IEEE rans. Neural New. 0067:6-65. [] H. Zhang and Z. Wang Global asympoic sabiliy of delayed cellular neural neworks IEEE rans. Neural New. 0078: [] H. Zhang Z. Wang and D. Liu Global asympoic sabiliy of recurren neural neworks wih muliple imevarying delays IEEE rans. Neural New : [4] H. Zhang Z. Wang D. Liu Robus sabiliy analysis for inerval Cohen-Grossberg neural neworks wih unknown ime-varying delays IEEE rans. Neural New : [5] H. Zhang Z. Wang D. Liu Robus exponenial sabiliy of cellular neural neworks wih muliple ime varying delays IEEE rans. Circuis Sys. II : [6] H. Zhang Z. Wang and D. Liu Global asympoic sabiliy and robus sabiliy of a class of Cohen-Grossberg neural neworks wih mixed delays IEEE rans. Circuis Sys. I 00956: [7] C.D. Zheng H. Zhang and Z. Wang Novel delaydependen crieria for global robus exponenial sabiliy of delayed cellular neural neworks wih norm-bounded uncerainies Neurocompuing : Firs Auhor was born in Liaoning China in 96. She Graduaed from Dalian Universiy of echnology and oined he Deparmen of ahemaics Dalian Jiaoong Universiy China as Associae Professor in 00. Yu-Li Zhang was born in Jiangsu China in 978. She received he aser Degree in mahemaics from China Universiy of ining and echnology in 00 and hen oined he Deparmen of ahemaics Dalian Jiaoong Universiy China. Copyrigh c 04 Inernaional Journal of Compuer Science Issues. All Righs Reserved.