Stability analysis for stochastic BAM nonlinear neural network with delays

Transcription

1 Joural of Physcs: Coferece Seres Sably aalyss for sochasc BAM olear eural ework wh elays o ce hs arcle: Z W Lv e al 8 J Phys: Cof Ser 96 4 Vew he arcle ole for upaes a ehacemes Relae coe - Robus sably for sochasc brecoal assocave memory eural eworks wh me elays H S Shu Z W Lv a G L We - New expoeal sably crera for sochasc BAM eural eworks wh mpulses R Sakhvel R Samura a S M Aho - A lear marx equaly approach o global sychrosao of o-parameer perurbaos of mul-elay Hopfel eural ework Shao Ha-Ja Ca Guo-Lag a Wag Hao-Xag hs coe was owloae from IP aress 4858 o 7/8/8 a :54

2 7 Ieraoal Symposum o Nolear Dyamcs (7 ISND IOP Publshg Joural of Physcs: Coferece Seres 96 (8 4 o:88/ /96//4 Sably aalyss for sochasc BAM olear eural ework wh elays Z W Lv H S Shu G L We College of Apple Mahemacs Doghua Uversy Shagha 5 Cha School of Iformao Scece a echology Doghua Uversy Shagha 5 Cha E-mal: hsshu@hueuc Absrac I hs paper sochasc brecoal assocave memory eural eworks wh cosa or me-varyg elays s cosere Base o a Lyapuov-Krasovsk fucoal a he sochasc sably aalyss heory we erve several suffce coos orer o guaraee he global asympocally sable he mea square Our vesgao shows ha he sochasc brecoal assocave memory eural eworks are globally asympocally sable he mea square f here are soluos o some lear marx equales(lmis Hece he global asympoc sably of he sochasc brecoal assocave memory eural eworks ca be easly checke by he Malab LMI oolbox A umercal example s gve o emosrae he usefuless of he propose global asympoc sably crera Irouco I [5]-[6] Kosko propose a ew class of eworks calle brecoal assocave memory (BAM eural eworks hs class of eworks has bee successfully apple o paer recogo ue o s geeralzao of he sgle-layer auoassocave Hebba correlaor o a wo-layer paer-mache heeroassocave crcurecely he yamcs such as sably a perocy of BAM eural eworks have receve much aeo ue o her poeal applcao assocave memory parallel compuao a opmzao problems Some mpora resuls have bee obae Refs [][]- [7][]-[][5][] Mos eural ework moels propose a scusse he leraure are eermsc As s well kow a real sysem s usually affece by exeral perurbaos whch may cases are of grea uceray a hece may be reae as raom as poe ou by Hayk [] ha real ervous sysem syapc rasmsso s a osy process brough o by raom flucuao from he release of eurorasmers a oher probablsc causes herefore s of prme mporace a grea eres o coser sochasc effecs o he sably of eural eworks o ae some resuls o sably of sochasc cellular eural eworks a sochasc Cohe-Grossberg eural eworks have bee repore (see [8][]-[][7][9]-[][] However o he bes of our kowlege he sably of BAM eural eworks wh elays have bee sue ([9][][]bu few auhors suy he sably of sochasc BAM eural eworks wh elays Movae by he above scusso hs paper we aalyze he sochasc BAM eural ework moels wh cosa or me-varyg elays By ulzg a Lyapuov-Krasovsk fucoal a coucg he sochasc aalyss we erve several suffce coos orer o guaraee he o whom ay correspoece shoul be aresse c 8 IOP Publshg L

3 7 Ieraoal Symposum o Nolear Dyamcs (7 ISND IOP Publshg Joural of Physcs: Coferece Seres 96 (8 4 o:88/ /96//4 global asympocally sable he mea square Dffere from he commoly use marx orm heores (such as he M marx meho a ufe lear marx equaly(lmi approach s evelope o esablsh suffce coos for he eural eworks o be global asympocally sable Noe ha LMIs ca be easly solve by usg he Malab LMI oolbox a o ug of parameers s requre [8] A umercal example s prove o show he usefuless of he propose global asympoc sably coo m Noaos: hroughou hs paper a eoe respecvely he -mesoal Euclea space a he se of all mreal marces For symmerc marces X ay he oao X > Y (res- -pecvely X Y meas ha X Y s posve efe(respecvely o-egave I eoe he comp -pable meso ey marx Deoe by LF [ h ]; R he famly of all -measurable ( { θ : h } C( [ h ]; R -value raom varables ξ = ξ( θ such ha sup ( p h θ E ξ θ < where E{} sas for he mahemacal expecao operaor wh respec o he gve probably mea- P he shorha ag { M M M } eoes a block agoal marx wh agoal blocks -sure beg he marces M M M N N Somemes he argumes of a fuco or a marx wll be ome he aalyss whe o cofuso ca arse he BAM eworks wh cosa elays escrbe by he followg ffereal equaos ([][4][7]: u ( = Au( + W g( v( τ + I ( v ( = Bv( + V g ( u( δ + J whch m u= u u R v = v v R A= ag ( a a > g = ( g g B= ag ( ( m m g = ( ( ( ( ( m I = J = m = j = m j m ( b b > g g I I J J W w V v τ δ are cosas hroughou hs paper we assume ha he acvae fucos g possess he followg properes: (A g are boue o R = max{ m} (Ahere exs real umbers F > such ha g ( x g ( y M x y for ay x y R M = max{ m} I s clear ha uer he assumpo (A a (A sysem ( has a leas oe equlbrum * * u = u u v = ( v * m m of sysem ( o he org hs rasformao x = u u * y = v( v * f ( x ( = g u ( g u * ( * f y = g v g v pu sysem ( o sysem ( * * * I orer o smplfy our proof we shf he equlbrum po ( v ( ( ( ( ( ( ( ( ( x ( = Ax( + W f ( y( τ y ( = By( + V f ( x( δ where x( = ( x( x ( y( = ( y( ym( f = ( f fm f = ( f f M = ag M M M = ag M M ( ( m (

4 7 Ieraoal Symposum o Nolear Dyamcs (7 ISND IOP Publshg Joural of Physcs: Coferece Seres 96 (8 4 o:88/ /96//4 Obvously he acvae fucos ( H f are boue o max{ m} j f j sasfy he followg properes: R = (H here exs real umbers such ha = max{ m} ( H ( { } M > ( ( f x f y M x y for ay x y R f = = max m I s ofe he case pracce ha he ework s surbe by evromeal oses ha affec he sably of he equlbrum I hs paper he sochasc BAM eworks wh cosa elays escrbe by he followg ffereal equaos: x = Ax + W f y τ + σ x y τ ω ( ( ( ( ( ( ( ( y( = B ( V ( ( δ y + f x + σ ( y( x( δ ω ( p q ( ω ωp ω ( ( ω( ωq( where ω( ( ( = = are Browa moo efe + m p m m q o Ω F { F } Ρ a assume ha σ : R σ : + are ( locally Lpschz couous a sasfes he lear growh coo Moreoverσ σ sasfes ( x( y( ( x( y( x( + y( ( y( x( ( y( x( y( + x( race σ τ σ τ τ race σ δ σ δ δ (5 Respecvely where Σ a Σ Σ are kow cosa agoal marces wh approprae m- -esos x φ y ϕ o Now accorg o [4] s obvous ha sysem ( has a uque global soluo ( ( b for ay al value φ [ τ ]; F ( σ = ~ σ = are requre such ha sysem (4 has a rval soluo Furhermore ( ( x ( y ( Defo a Lemmas I hs par we wll focus our aeo o suyg he sably of sysem (o oba our resuls we ee rouce he followg efo a lemmas m L τ ; η L ( σ ; ( Defo For he eural ework ( a every ξ F [ ] [ ] he rval soluo (equlbrum po s globally asympocally sable he mea square f he followg hols: ( ξ y( η lm( E x ; + E ; = m LemmaLe x y a ε > he we have x y+ y x εx x+ ε y y Lemma For ay posve efe marx M > scalar > γ vecor fuco :γ [ ] ha he egraos cocere are well efe he followg equaly hols: ( ( ( ( γ ( ( ( γ γ γ ω s s M ω s s ω s Mω s s F ( (4 (6 ω such

5 7 Ieraoal Symposum o Nolear Dyamcs (7 ISND IOP Publshg Joural of Physcs: Coferece Seres 96 (8 4 o:88/ /96//4 Lemma he LMI Q( x S( x > S ( x R( x where Q( x = Q ( x R( x = R ( x a S ( x epes affely o x s equvale o ( Q( x ( ( > R x S x Q( x S( x > ( R( x > Q( x S( x R( x S ( x > Ma resuls heoremif here exs posve scalars ρ > ρ > ε > ( = Q a posve agoal marces P Q posve efe marces P wh approprae mesos sasfyg P < ρi (7 Q < ρi PA + APεMM ρσσ δmq M ρw PW > WP ε (8 QB + BQεMM ρσσ τmpm ρσ Σ QV > VQ ε hols he yamcs of he eural ework ( s globally asympocally sable he mea square Proof: Coser he followg Lyapuov-Krasovsk fucoal caae V x y = x P x + y Q y + f y ξ P f y ξ ξ + f x η Q f x η η ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( τ δ ( ( ξ P ( ( ξ ξξ ( ( η Q τ f y f y + ξ δ f x + η ( ( η ηη + + f x (9 where P Q P Q are he posve soluo o (8 a P s efe by P = ε I + M ρ M ( Q s efe by Q = εi + M ρ M ( Employg I oˆ ' s ffereal rule oe ca euce ha V( x( y( = x (( PA AP x( + y ( ( QB BQ y( + x ( PW f ( y ( τ + y QV f x δ + race( σ x y τ Pσ x y τ + ( ( ( ( ( ( ( ( ( ( σ ( y( x( δ Q σ( y( x( δ + f ( y( ( P + τp f ( y ( ( ( τ P ( ( τ ( ( ( Q δq ( ( ( ( δ ( ( δ ( ( ξ P ( ( ξ ξ ( ( η Q ( ( η η race f y f y + f x + f x f x Q f x f y f y f x f x τ ex follows from (4 a (7 ha δ ( 4

6 7 Ieraoal Symposum o Nolear Dyamcs (7 ISND IOP Publshg Joural of Physcs: Coferece Seres 96 (8 4 o:88/ /96//4 ( σ ( x( y( τ Pσ( x( y( τ λmax( P ( σ σ race race For he posve scalars ε > follows from Lemma ha ( P x( y( τ λmax + [ x ( x( y ( τ ( τ ] ρ + y ( ( ( τ ε ( ( τ ( ( τ + ε ( x PW f y f y If y x PW WPx (4 Furhermore ca be see from Lemma ha τ ( ( ( ( ( ( ( ( ( ( τ τ Smlarly we ca oba f y ξ P f y ξ ξ τ f y ξ ξ P f y ξ ξ ( σ ( y( x( δ Q ( ( ( ( ( σ y x δ ρ y y + x ( δ x( δ race y QV f x δ ε f x δ If x δ + ε y QV VQ y where ε > δ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( δ δ f x η Q f x η η δ f x η η Q f x η η Usg ( o (8 a( H we oba from ( ha ( ( ( ( ε ρ ( δ ε ρ ( τ ( (5 (6 ( ( + ( ( QB ( BQ QV VQ M P P M y x ( δ ( ρ ( ( ( ( M Q ε I M x δ y τ ρ M ( P εi M y ( τ V x y x PA AP PW WP M Q Q M x y ( ( ( ( ( ( ( ( η η δ ( ( ( ( f y P Q τ f y τ f x δ τ ξ ξ ξ ξ δ η η (7 (8 f x (9 Usg ( o ( a by some mapulaos we oba from (9 ha V x y α π α + β π β ( ( ( ( ( ( ( where x ( y ( α ( = β ( = ( ( ξ ξ f y τ ( ( η η f x δ PA AP+ εmm + ρ + δmq M + ε PW WP+ ρ π = τ P QB BQ+ εmm + ρ + τmpm + ε QV VQ+ ρ π = δ Q I lgh of (8 a ( we kow ha ( 5

7 7 Ieraoal Symposum o Nolear Dyamcs (7 ISND IOP Publshg Joural of Physcs: Coferece Seres 96 (8 4 o:88/ /96//4 PA AP+ εmm + ρ + δmq M + ε PW WP+ ρ < QB BQ+ εmm + ρ + τmpm + ε QV VQ+ ρ < so we oba π < π < he here mus exs scalar γ > γ > such ha γi γi π+ < π + < Followg from (oe ca oba ha Le γ = m{ γ γ } ( ( ( γ ( γ ( V x y α β he we oba from ( ha ( ( ( ( γ ( + ( V x y α β I mples ha V < for all α β Accorg o I oˆ ' s formula sysem ( s globally asympocally sable he mea square I he followg we wll suy he sably for sochasc BAM eworks wh me-varyg elays he moel s escrbe by he followg ffereal equao: x( = Ax( + W f ( y( τ ( + σ ( x( y( τ ( ω( (4 y( = By( + V f ( x( δ ( + σ ( y( x( δ ( ω ( I hs seco we always assume ha τ( δ( are ffereable oegave a boue τ ( τ δ τ ς < δ ς < ( ( ( δ a he ervave of τ( δ ( are less ha oe e ( ( heorem If here exs posve scalars ρ > ρ > ε > ( = P Q wh approprae mesos sasfyg P < ρi Q < ~ ρi ( ( ( posve efe marces PA + APκ εmm κ ρ ρ PW > WP ε (6 QB + BQκ εmmκ ρσσ ρσ Σ QV > VQ ε hols he yamcs of he eural ework (4 s globally asympocally sable he mea square Proof Coser he followg Lyapuov-Krasovsk fucoal caae ( ( ( ( ( ( ( ( ( ( ( ( V x y x P x y Q y f y ξ P f y ξ ξ = δ τ ( Q ( ( ( ( f x η f x η η where P Q are he posve soluo o (6 a P s efe by P = κ ε I + κ ρm M (8 Q s efe by Q M (9 = κ εi + κ ρm (5 (7 6

8 7 Ieraoal Symposum o Nolear Dyamcs (7 ISND IOP Publshg Joural of Physcs: Coferece Seres 96 (8 4 o:88/ /96//4 = ( ( ( we eoe κ f τ κ = f δ a employ Io's ˆ ffereal rule oe ca x R x R ( euce ha V ( x( y( = x (( PA AP x( + y (( QB BQ y( + x ( PW f( y ( τ ( + ( ( ( ( ( ( ( ( y ( x ( Q ( y ( x ( f y P f y y ( QV f ( x( δ( + race σ x y τ Pσ x y τ( ( σ δ σ δ + race ( + ( ( ( ( ( ( τ( P ( ( τ(( τ( + ( ( Q ( ( f y f y f x f x f ( x( δ ( Q f ( x ( δ( ( δ( x (( PA AP + MQ M + ε PW WP + ρ x( + y (( QB BQ + MPM y( y ( ( f ( x R ( ( ( ( ( ( ( ( δ ( + ε QV VQ + ρ + τ τ MPM + ρ + ε MM y τ + x δ [ f δ MQ M + ρ + ε MM] x x R (( ε ρ ( (( = x PA AP+ MQM+ PW WP+ x + y QB BQ + MPM + ε ( ( QV VQ+ ρ y( + y τ κmpm + ρ + εmm y( τ( + x ( δ ( [ κmq M + ρ + εmm ] x ( δ ( ( I lgh of (8 a (9 we oba ha V x y x ˆ π x y ˆ π y where ( ( ( ( ( ( ( ˆ π P A AP ε PW WP κ ρ κ ε MM ς δmq M ρ ˆ π = QB+ BQ ε QV VQ κ ρ κ ε MMςτMPM ρ = + I lgh of (6 we ca see ha ˆ π ˆ > π > he we have V < for all x y Accorg o Iô's formula sysem (4 s ( ( globally asympocally sable he mea square 4 A llusrave example Coser a wo-euro sochasc BAM eworks wh cosa elays (where A = B = W = V = = = = = 7 4 We have M = M j = ( = j= we ca easly oba M = M = I he cosa elays ca be ake as δ = 5 τ = 5 hus by usg he Malab LMI oolbox we solve he LMIs (7(8 for ( ε > = P > P Q Q ρ > ρ > > > > a oba P = Q= P= Q= ρ= 49 ρ=57 ε = 959 ε =

9 7 Ieraoal Symposum o Nolear Dyamcs (7 ISND IOP Publshg Joural of Physcs: Coferece Seres 96 (8 4 o:88/ /96//4 herefore follows from heorem ha he wo-euro sochasc BAM eworks wh cosa elays ( s globally asympocally sable he mea square 5 Coclusos I hs paper sochasc brecoal assocave memory eworks wh elays s sue By cosrucg a Lyapuov-Krasovsk fucoal a usg he sochasc sably aalyss heory we erve several suffce coos of globally asympocally sable he mea square A LMI approach has bee evelope o solve he problem aresse A smple example has bee use o emosrae he usefuless of he ma resuls Ackowlegme hs work s suppore by aural scece fu of Shagha (No 7ZR4 Refereces [] Che A P Huag L H a Cao J D Appl Mah Compu 4 [] Hayk S 994 Neural Neworks (NJ:Prece-Hal [] Huag X Cao J D a Huag D S 5 Chaos Solos a Fracals [4] Ju H Park 6 Chaos Solos a Fracals [5] Kosko B 987 Appl Op [6] Kosko B 988 IEEE ras Sys Ma Cybere 8 4 [7] L C D Lao X F a Zhag R 5 Chaos Solos a Fracals 4 9 [8] Lao X X a Mao X R 996 Sochas Aal Appl 4 65 [9] Lag J L a Cao J 4 Chaos Solos a Fracals 77 [] Lu Y R a Wag Z D 6 Chaos Solos a Fracals 8 79 [] Lu Y R Wag Z D a Lu X H 6 Neurocompug 7 4 [] Lu Y R Wag Z D Serrao A a Lu X H Physcs Leers A 6 48 [] Lou X Y a Cu B 7 Chaos Solos a Fracals 695 [4] Mao X R 997 Sochasc Dffereal Equaos a her Applcaos (UK:Horwoo Chcheser [5] Shu H S Lv Z W a We G L 6 J of Doghua Uversy 7 [6] Sog Q K a Cao J D 5 Chaos Solos a Fracals 4 [7] Su J a Wag L 5 I J Bfurca Chaos 5 [8] S Boy L EI Ghaou E Fero a V Balakrshma 994 Lear Marx Iequales Sysem a Corol heory (Phlaelpha PA: SIAM [9] Wag Z D Laura J a Lu X H 7 Chaos Solo Frac 6 [] Wag Z D Lu Y Fraser K a Lu X H 6 Physcs Leers A [] Wag Z D Lu Y L M a Lu X H 6 IEEE rasacos o Neural Nework 7 84 [] Wag Z D Shu H S Fag J A a Lu X H 6 Nolear Aalyss:Real Worl Applcaos 7 9 [] Zhag J Y a Yag Y R I J Crcu heory Appl