A DELAY-DEPENDENT APPROACH TO ROBUST STABILITY FOR UNCERTAIN STOCHASTIC NEURAL NETWORKS WITH TIME-VARYING DELAY

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1 Journal of Marine Science and echnology Vol. 8 No. pp (00) 77 A DELAY-DEPENDEN APPROACH O ROBUS SABILIY FOR UNCERAIN SOCHASIC NEURAL NEWORKS WIH IME-VARYING DELAY Chien-Yu Lu* Chin-Wen Liao* Koan-Yuh Chang** and Wen-Jer Chang*** Key words: ime-varying delay linear marix inequaliy neural neworks uncerainy robus sabiliy. ABSRAC his paper invesigaes he global delay-dependen robus sabiliy in he mean square for uncerain sochasic neural neworks wih ime-varying delay. he acivaion funcions are assumed o be globally Lipschiz coninuous. Based on a linear marix inequaliy approach globally delay-dependen robus sabiliy crierion is derived by inroducing some relaxaion marices which when chosen properly lead o a less conservaive resul. wo numerical examples are given o illusrae he effeciveness of he mehod. I. INRODUCION Recenly i has been well-known ha ime delays are frequenly encounered in various elecronic implemenaion of neural neworks wih ime delay such as Hopfield neural neworks cellular neural neworks and bi-direcional associaive memory neworks and is exisence is a source of oscillaion and insabiliy of neural neworks. herefore he research of he dynamical characerisics (include sable unsable oscillaory and chaoic behavior) of neural neworks wih ime delays is an imporan opic in he neural neworks heory. In paricular globally sabiliy is one of he mos desirable dynamic properies of neural neworks here have been growing research ineres on he sabiliy analysis problem for delayed neural neworks. Recenly considerable effors have been exensively applied o he analysis of he sabiliy in Paper submied 08/9/08; revised 0//09; acceped 0/6/09. Auhor for correspondence: Wen-Jer Chang ( wjchang@mail.nou.edu.w). *Deparmen of Indusrial Educaion and echnology Naional Changhua Universiy of Educaion Changhua aiwan 500 R.O.C. **Deparmen of Elecronic Engineering Chienkuo echnology Universiy Changhua 500 aiwan R.O.C. ***Deparmen of Marine Engineering Naional aiwan Ocean Universiy Keelung 0 aiwan R.O.C. signal and image processing arificial inelligence indusrial auomaion and oher fields [ 6 7 9]. I is noed ha so far mos works on delayed neural neworks have deal wih he sabiliy analysis problem for neural neworks wih ime delays. Much of he curren ineres in neural neworks sems no only from heir richness as a heoreical model of collecive dynamics bu also from he promise hey have shown as a pracical ool for performing parallel compuaion. In carrying ou he compuaion here are various sochasic perurbaions o he neworks and i is imporan o undersand how hese perurbaions affec he neworks. Consequenly i is very urgen o know wheher he neworks are sable or no under he perurbaions. According he sabiliy crieria for sochasic neural neworks become an aracive research problem of prime imporance. Laely some iniial resuls have jus appeared for example he sabiliy problem for he sochasic neural neworks wihou delays has been sudied in [3] and sochasic delayed neural neworks has been discussed in [3 6 7 ]. he sabiliy analysis of sochasic delayed cellular neural neworks in erms of linear marix inequaliy [5] approach was proposed in []. By using he mehod of variaion parameer inequaliy echnique and sochasic analysis he sufficien condiions o guaranee he mean square exponenial sabiliy of an equilibrium soluion are given in [7]. I should be poined ou ha he aforemenioned resuls in [ ] were obained as delay-independen condiions ha is delay-independen condiions do no include any informaion relaing o he magniude of he delays. However i is known ha delay-dependen condiions are generally less conservaive han delay-independen condiions; paricularly when he magniude of he delay is small. Alhough delayindependen resuls on he robus sabiliy problem for sochasic neural neworks wih discree delay and mixed delays were presened in [] and [8] respecively no delaydependen robus sabiliy resuls on sochasic neural neworks wih ime-varying delay are available in he lieraure which moivaes he presen sudy. In his paper we are concerned wih he global delaydependen robus sabiliy condiions in he mean square for

2 F :)( R k 78 Journal of Marine Science and echnology Vol. 8 No. (00) sochasic uncerain neural neworks wih ime-varying delay. Some crieria on globally delay-dependen robus sabiliy condiions are presened. Based on Lyapunov-Krasovskii funcional combining wih linear marix inequaliy (LMI) echniques globally delay-dependen robus sabiliy condiions for sochasic uncerain neural neworks wih imevarying delay which are given in erms of quadraic forms of sae and LMI are derived. We will also inroduce some relaxaion marices which when chosen properly produce a less conservaive resul [0 0]. A numerical example will be finally given o deermine he effeciveness of he mehod. Noaion: hroughou his paper he noaion X Y (respecively X > Y ) for symmeric marices X and Y means ha he marix X Y is posiive semidefinie (respecively posiive definie) Z represens he ranspose of he marix Z. he vecor norm refers o he Euclidean vecor norm ha is W λ ( W W) where λ M M (W) (respecively λ m (W)) sands for he operaion of aking he maximum (respecively minimum) eigenvalue of W. II. SYSEM DESCRIPION AND PROBLEM FORMULAION he delayed neural nework model is defined by he following uncerain sae equaions wih ime-varying delay or equivalenly u () ( A+ A ()) u () + ( W+ W ()) gu ( ()) + ( W + W ( )) g( u( τ ( ))) () n i i + ai i + ij + ij j j j u () ( a ()) u () ( w w ()) g ( u ()) n ( w ij w ij ( )) g j ( uj ( τ ( ))) i... n () j + + where u() [u () u () u n ()] is he sae vecor of he neural nework A diag(a a a n ) is a diagonal marix wih posiive enries a i > 0 i n W [w ij ] n n and W [w ij ] n n represen he connecion weigh marix of he neurons and he delayed connecion weigh marix of he neurons respecively g(u()) [g (u () g (u () g n (u n ())] denoes he neuron acivaion funcion wih g(0) 0 τ() is he ransmission delay saisfying 0 τ( ) τ and τ () d. In pracical implemenaion of neural neworks he values of he consan a i and weigh coefficiens w ij and w ij depend on cerain resisance and capaciance values which are subjec o uncerainies. his may rigger some deviaions in he values of a i w ij and w ij. So A() W() and W () are unknown marices represening ime-varying parameer uncerainies and are assumed o be of he form [ A ( ) W( ) W( )] HF( )[ E E E] (3) 3 where H E E and E 3 R are known real consan marices and F() is an unknown real-valued ime-varying marix saisfying F () F() I. () I is assumed ha all he elemens of F() are Lebesgue measurable. A() W() and W () are said o be admissible if boh (3) and () hold. Suppose here exiss a sochasic perurbaion o he neural nework and he sochasically perurbed nework is described by a sochasic sysem wih ime-varying delay dx () [ ( A+ A ()) x () + ( W+ W()) g( x ()) + ( W + W ( )) g( x( τ ( )))] d + σ( x ( ) x ( τ( ))) db ( ) (5) wih iniial daa x () ϕ() [ τ 0]. (6) Here B() is an n-dimensional Brownian moion defined on he given complee probabiliy space. We assume ha σ (): n n n m R R R is locally Lipschiz coninuous and saisfies linear growh condiion. σ( x x ) σ( y y ) G ( x y ) + G ( x y ) (7) for all x x y and y R n where G and G are known real consan marices. Equaion (5) has a unique global soluion on 0 and we show he soluion by x (; ϕ ). Also we assume ha σ (0 0) 0 for he sabiliy purpose and hence (5) permis a rivial soluion x (; 0) 0 [8]. Assumpion (Lipschiz condiion): he acivaion funcion g(x) is nondecreasing bounded ha is g j( ξ) g j( ξ) 0 α j ξ ξ ξ ξ j n. (8) he following definiion is necessary in he heory of sochasic differenial equaions [5]. Definiion : If here exis β > 0 and γ > 0 such ha β E x() γ sup E ϕ( s) e > 0. (9) τ s 0 hen he sysem (5) is said o be exponenially mean square sable

3 C.-Y. Lu e al.: A Delay-Dependen Approach o Robus Sabiliy for Uncerain Sochasic Neural Neworks 79 he main resuls of his paper hinge on he following fac. Fac [9]: he rivial soluion of a sochasic delayed differenial equaion dx ( ) F( x ( ) x ( τ( )) d ) + Gx ( ( ) x ( τ( )) dw ) ( ) x() ϕ () [ τ 0] (0) on [ 0 ] wih iniial daa f ( ϕ ) F( ϕ(0) ϕ( τ ) ) and g ( ϕ ) G( ϕ(0) ϕ( τ ) ) () n n n where F: R R [ 0 ] R and G: R n R n [ 0 ] R n m n for ( ϕ ) C([ τ 0]; R ) [ 0 ] is globally asympoically sable in probabiliy if here exiss a funcion V( x) C (U + R n ) which is posiive definie in he Lyapunov sense and saisfies he generaor LV V LV + ( gradv ) F + (r G G ) HessV < 0 () for x 0 and Vx ( ) + as x. he marix HessV is he Hessian marix of he second-order parial derivaives. his fac is analogous o he well-known heorem of Lyapunov for deerminisic sysems. III. MAIN RESULS In his secion he exponenially mean-square sabiliy for uncerain sochasic sysem (5) wih ime-varying delays is explored. An LMI approach is developed o solve he robus sochasic sabiliy if he sysem associaed o (5) is meansquare asympoically sable for all admissible uncerainies A() W() and W (). he analysis firs gives some resuls which are essenial o inroduce he following Lemma for he developmen of our main heorem. Lemma []: For any vecors or marices z and y wih appropriae dimensions and any posiive consan ε he following inequaliy is saisfied: Define a new sae variable z y εz z+ ε y y. (3) z() ( A+ A()) x() + ( W + W()) g( x()) + ( W + W ( )) g( x( τ ( ))). () Equaion (5) can be rewrien as dx( ) z( ) d + σ( x( ) x( τ( ))) db( ). (5) For any marices N i and S i (i 3 5) of appropriae dimensions i can be shown ha [ x () N + x ( τ()) N + g ( x()) N + g ( x( τ()) N 3 + z () N5] [ x() x( τ ()) dx()] s 0 (6) τ () [ x () S + x ( τ()) S + g ( x ()) S + g ( x ( τ()) S 3 + z () S ] {[ ( A+ A()) x() + ( W + W()) g( x()) 5 + ( W + W ( )) g( x( τ ( )))] z( )}. (7) heorem : For any given τ > 0 and 0 d < saisfying 0 τ( ) τ and τ () d if here exis marices P > 0 R > 0 Q > 0 Q > 0 S 5 < 0 X 0 X 0 X 33 0 X 0 X 55 0 X 0 X 3 0 X 0 X 5 0 X 3 0 X 0 X 5 0 X 3 0 X 35 0 X 5 0 and diagonal marix > 0 and any marices N i (i 3 5) S S S 3 and S of appropriae dimensions and posiive scalars ε ε and ρ such ha and ( ) N H S X N N SH ε 0 < 0 0 ε N 0 R (8) (9) P ρi. (0) hen he sysem described by (5) is exponenially sable in he mean square where ( ) Π+ τ X Π Π Π3 Π Π5 Π Π Π3 Π Π5 Π Π3 Π3 Π33 Π3 Π 35 Π Π Π3 Π Π 5 Π5 Π5 Π35 Π5 Π55

4 80 Journal of Marine Science and echnology Vol. 8 No. (00) Π Q + ( τε + δ ) G G + N + N + ε E E + S A+ A S + N N A S Γ SW A S 3 ε E E N3 Π + + N SW A S ε E E3 Π N5 P S A S5 Π ( dq ) + ( τε + δ ) GG N N Π Π SW N Π 5 S N5 3 SW N3 Π + 33 Q S 3W W S3 ε E E Π + Π 35 S3 W S5 3 N S3W W S ε E E3 Π ( dq ) + ε EESW W S 3 3 Π Π 55 S5 + S5 + τ R 5 S W S5 X X X X X X X X X X X X X X X X X X X X X X5 X5 X35 X5 X ε + τ τ s V ( x( ) ) G x( v) G x( v ( v)) dvds (5) ( ( ) ) V ( ) ( ). 5 x z vrzv dvds (6) τ s Along rajecories of (5) and making use of he I ô - differenial rule [9] hen he generaor LV ( x( ) ) for he evoluion of V ( x ( ) ) is given by LV ( x( ) ) x ( ) Pz( ) + race σ ( x ( ) x ( τ( ))) Pσ( x ( ) x ( τ( ))) + x ( Qx ) ( ) τ ( d) x ( τ()) Qx ( ()) + g ( x ()) Qg( x ()) ( dg ) ( x ( τ( ))) Qgx ( ( τ( ))) + τε Gx ( ) + Gx ( τ( )) τ ε G xs () + Gxs ( τ()) s ds+ τz () Rz () z () srzds () τ τ 3 τ 5 + [ x () N + x ( ()) N + g ( x()) N + g ( x( ()) N + z () N ] [ x ( ) x ( τ ( )) dxs ( )] τ () N [ N N N N N ] S [ S S S S S ] Proof: Consider he following Lyapunov-Krasovskii funcional for he sysem (5) V( x( ) ) V ( x( ) ) + V ( x( ) ) + V ( x( ) ) + V ( x( ) ) where 3 + V ( x( ) ) () 5 τ 3 τ 5 [ x () S+ x( ()) S + g (()) x S + g (( x ()) S + z () S] {[ ( A + A ( )) x ( ) + ( W+ W( )) g( x ( )) + ( W + W ( )) g( x( τ( )))] z( )} + τ q ( ) Xq( ) τ () q () Xqd () α g ( x ()) g( x ()) + g ( x ()) g( x ()) (7) V ( x( ) ) x ( ) Px( ) () V ( x( ) ) x ( s) Q x( s) ds (3) τ () where q( ) [ x ( ) x ( τ( )) g ( x( )) g ( x( τ( ))) z ( ) ] V (()) x g (()) x s Q g(()) x s ds () 3 τ () X 0 and diagonal marix > 0. I follows from Lemma ha for any ε > 0 and ε > 0

5 C.-Y. Lu e al.: A Delay-Dependen Approach o Robus Sabiliy for Uncerain Sochasic Neural Neworks 8 [ x ( ) N + x ( τ ( )) N + g ( x( )) N 3 + τ + 5 τ () g ( x( ( )) N g ( ) N ] dx( s) ε τ () q ( ) N z( s) ds+ q ( ) NN q( ) + ε σ( xs ( ) xs ( τ( s)) dbs ( ) (8) τ () q ( ) SHF( ) Eq( ) q ε SHH S q( ) + ε q ( ) E Eq ( ) wih E E 0 E E3 0. Nex i follows from he condiion (7) and (0) ha race σ ( x ( ) x ( τ( ))) Pσ( x ( ) x ( τ( ))) (9) x ( ) ρg G x( ) + x ( τ( )) ρg G x( τ( )) (30) where ρ λ max (P). Using Assumpion yields g ( x()) Hg( x()) g ( x()) H Γ x() (3) where Γ diag( α α α n ). Moreover τ() τ() E σ( xs ( ) xs ( τ( s)) dbs ( ) E Gxs ( ) Gxs ( τ( s)) + ds Combining now (7)-(3) yields { } { Π } { Π } τ () (3) E LV(()) x E q () q() E q ( s) q( s) ds wih q ( s) [ q ( ) z ( s)]. Nex from (8) and (9) we can show LV < 0. Define a new funcion as (33) β Y( x( ) ) e V( x( ) ) (3) is infiniesimal operaor L is given by β β LY( x( ) ) βe V( x( ) ) + e LV( x( ) ). (35) By inegraing his relaion boh sides beween 0 o and hen aking expecaion yield { ( ( ) )} { ( (0) 0) } E Y x E Y x βα { ( ( ) )} { ( ( ) )} βα βe EVxs s ds+ e ELVxs s ds 0 0 (36) By using he similar analysis mehod in [5] i can be seen from () (3) and (36) ha if β > 0 is chosen small enough a consan γ > 0 can be found such ha { } γ { ϕ } EVx ( ( ) ) sup E ( s) e β. (37) I follows from () ha τ s 0 V( x( ) ) δ x ( ) x( ) (38) where δ min{λ min (P)} i can furher imply from (37) ha { } γ { ϕ } E x () x() sup E () s e β (39) τ s 0 where γ δ γ. From his he resul follows. ½ Remark : his paper is concerned wih he global delaydependen mean square exponenial robus sabiliy for sochasic uncerain neural neworks wih ime-varying delay. he delayed neural neworks via LMI approach in [ ] can be regarded as he specialiy of he sochasic delayed neural neworks. herefore he global asympoic sabiliy presened in [ ] is he specializaion of heorem obviously. Le us now work ou wo numerical examples o show he usefulness of he proposed resuls. IV. NUMERICAL EXAMPLES Example : Consider a sochasic uncerain delayed neural nework dx() [ ( A+ A ()) x () + ( W+ W()) g( x ()) + ( W + W ( )) g( x( τ ( )))] d + σ( x ( ) x ( τ( ))) db ( ) (0)

6 8 Journal of Marine Science and echnology Vol. 8 No. (00) where A W W H E E E3 G G he acivaion funcion is assumed o saisfy α 0.5 and α 0.3 in Assumpion. hen i can be shown ha he global delay-dependen robus condiions in [3 5 ] canno be saisfied for any τ > 0. Accordingly hey canno provide any resuls on he maximum allowed delay τ. However uilizing heorem in his paper he maximum allowable value of τ for differen d can be go as follows. d Maximum allowable value of τ herefore he sochasic uncerain neural neworks wih ime-varying delay in heorem of his paper are less conservaive han hose resuls in [3 5 ]. Example : Consider a sochasic uncerain delayed neural nework in (5) wih parameers A 0. 0 W W H E E E G G he acivaion funcion is assumed o saisfy Assumpion wih α 0.3 α 0. and α Furhermore he ime delay is assumed o saisfy 0 < τ ( ) 0.5. For he sochasic uncerain neural neworks wih ime-varying delay i is found ha he condiions in [3 7 ] and [] are no saisfied for any τ > 0. Accordingly hey fail o conclude wheher his sochasic uncerain neural nework wih ime-varying delay is globally delay-dependen robus sabiliy. However by using Malab LMI Conrol oolbox i can be verified ha his sochasic uncerain neural nework wih ime-varying delay is globally delay-dependen robus sabiliy for all 0 < τ() 6.5. In his case he soluion can be go as P Q Q R C ε.739 ε.78. herefore by heorem in his paper his sochasic uncerain neural nework wih ime-varying delay is globally robus delay-dependen sabiliy which implies ha for his example heorem in his paper can be less conservaive ha he exising resuls in he lieraure. V. CONCLUSIONS In his paper he problem of globally delay-dependen robus sabiliy for a class of sochasic neural neworks wih ime-varying delay has been considered. A sufficien condiion for he solvabiliy of his problem which depends on he size of he ime delay has been presened by means of he Lyapunov-Krasovskii funcional and he LMI approach. wo

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