Avalable ole a www.scecedrec.com Proceda Egeerg 5 (0) 86 80 Advaced Corol Egeergad Iformao Scece Sably Crero for BAM Neural Neworks of Neural- ype wh Ierval me-varyg Delays Guoqua Lu a* Smo X. Yag ab a College of AuomaoChogqg Uversy Chogqg 000Cha b School of Egeerg Uversy of Guelph Guelph Oaro Caada Absrac I hs paper he asympoc sably for bdrecoal assocave memory (BAM) eural eworks of eural-ype wh erval me-varyg delays s vesgaed. he dscree delay s assumed o be me-varyg ad belog o a gve erval whch meas ha he lower ad upper bouds of erval me-varyg delays are avalable. By employg he Lyapuov-Krasovsk fucoal mehod ad usg he lear marx equaly (LMI) echque a ew delay-rage-depede sably crero s esablshed erms of LMI. I addo he proposed LMI based resuls ca be easly checked by LMI corol oolbox Malab. 0 Publshed by Elsever Ld. Ope access uder CC BY-NC-ND lcese. Seleco ad/or peer-revew uder resposbly of [CEIS 0] Keywords:Asympoc sably; Bdrecoal assocave memory eural eworks; Neural-ype; Lear marx equaly; Ierval me-varyg delays;. Iroduco I s well kow ha bdrecoal assocave memory (BAM) eural ework s a ype of recurre eural ework. BAM eural ework was roduced by []-[]. Durg he pas years he dyamcs of BAM eural eworks have bee wdely suded due o her exesve applcaos may areas such as assocave memory paer recogo opmzao ad auomac corol. I pracce me delays are lkely o be prese due o he fe swchg speed of amplfers ad occur sgal rasmsso amog * Correspodg auhor. Address: College of Auomao Chogqg Uversy Chogqg 000 Cha. E-mal address: guoqualu98@homal.com. 877-7058 0 Publshed by Elsever Ld. do:0.06/j.proeg.0.08.5 Ope access uder CC BY-NC-ND lcese.
Guoqua Lu ad Smo X. Yag / Proceda Egeerg 5 (0) 86 80 87 euros he elecroc mplemeao of eural eworks I addo me delay s ofe a source of oscllaos chaos ad sably varous ypes of eural eworks. hus he sudy of he sably problem of BAM eural eworks wh me delays has receved grea aeo rece years ad a umber of resuls have bee repored []-[7]. O he oher had may dyamcal eural eworks are descrbed wh eural fucoal dffereal equaos ha clude eural delay dffereal equaos. hese eural eworks are called eural eural eworks or eural eworks of eural-ype. Recely a few resuls abou he global asympoc or expoeal sably for BAM eural eworks of eural-ype have bee derved he leraures [8]- [0]. I [8] a delay-depede global asympoc sably crero s preseed for BAM eural eworks of eural-ype by usg he Lyapuov mehod. I [9] Lu ad zhag furhermore vesgaed asympoc sably for BAM eural eworks of eural-ype a ovel delay-depede sably codos were esablshed. I [0] by ulzg he Lyapuov-krasovsk fucoal ad combg wh he LMI approach; hree suffce codos were gve esurg he global expoeal sably for BAM eural eworks of eural-ype wh me-varyg delays. Up o ow he asympoc sably problem has o bee ouched for BAM eural eworks of eural-ype wh erval me-varyg delays whch s sll ope problem. Based he aforemeoed dscussos a class of BAM eural eworks of eural-ype wh erval me-varyg delays s cosdered hs paper. Based o he Lyapuov-krasovsk sably mehod ad he LMI echque a ew asympoc sably crero s preseed erms of LMI. Noaos: he oaos are que sadard. R ad R deoe he -dmesoal Eucldea space ad he se of all real marces respecvely; For a real symmerc marx X he oao X 0 (respecvely X > 0 ) meas ha X s posve sem-defe (respecvely posve defe); he superscrps " " ad "-" sad for marx rasposo ad marx verse respecvely; he mahemacal expecao operaor wh respec o he gve probably measure P s deoed by E{}. dag{ } deoe s a block dagoal marx; deoes he elemes below he ma dagoal of a symmerc block marx.. Problem Formulao Cosder he followg BAM eural eworks of eural-ype wh erval me-varyg delays: u &() = Au () Wf ( v ()) Wf( v ( τ ())) Wv & ( h()) I () v& () = Bv () Vg ( u( )) Vg( u( σ ())) Vu& ( h ( )) J... u = [ u u... u ] ad v = [ v v v ] are he euro sae vecors. A= dag { a a... a } > 0 B = d ag { b b... b m } > 0 W W W V Vad V are kow cosa marces wh approprae dmeso f ad g deoe he euro acvaos I ad J deoe he cosa exeral pus. τ () adσ () represe he dscree rasmsso delays wh 0 τ & τ() τ & τ( ) τd < 0 σ σ( ) σ τ( ) σd < () τ τ τ d σ σ adσ d are cosas. h() ad h () represe he eural delays wh 0 h() h h& () h d < 0 h() h h& () hd <. () Assume ha he euro acvao f ucos f ad g sasfy he followg hypoheses (A) f ad g are bouded fucos. (A) f ad g are Lpschz couous.e. here exs real scalars l > 0 ad k > 0 such ha j
88 Guoqua Lu ad Smo X. Yag / Proceda Egeerg 5 (0) 86 80 f ( ς ) f ( ς ) lj ς ς g( ς) g( ς) k ς ς for all ς ς R ad ς for ay =... j =... m. I s clear ha uder he assumpos (A) ad (A) sysem () has a leas oe equpme po. Suppose ha u ( u u... u = m ) ad v = ( v v... v ) be oe equlbrum po of sysem (). For coveece we shf u v o he org by ake he followg rasformao: x() = u() u f() = f ( u()) f ( u ()) y() = v() v g( v()) = g( v()) g( u ()). () he sysem () ca be wre as x& ( ) = Ax ( ) Wf ( y ( )) Wf ( y ( τ ( ))) Wy & ( h( )) (5) y &() = By () Vgx ( ()) Vgx ( ( τ ())) Vx &( h()). From (A) ad (A) we ca derve ha acvao f ad g sasfy (G) f ad g are bouded fucos. (G) f ad g are Lpschz couous.e. here exs real scalars l > 0 ad k > 0 such ha f ( ς ) f ( ς ) lj ς ς g( ς) g( ς) k ς ς for all ς ς R ad ς ς for ay =... j =... m wh f(0)) = 0 g(0)) = 0. I order o oba he ma resul a basc lemma s always made hroughou hs paper. Lemma. For ay cosa marx M > 0 ay scalars a ad b wh a< b ad a vecor fuco x ( ):[ ab ] R such ha he egrals cocered as well defed he followg holds: b b b x() sds M xsds () ( b a) x() smxsds (). a a a (6) j. Sably Aalyss I hs seco we propose a ew sably crero for BAM eural eworks of eural-ype wh erval me-varyg delays descrbed (5). heorem. Uder assumpos (G) ad (G) hold. he equlbrum po of sysem (5) s asympocally sable f here exs posve defe marces P P Q =...6 R = Z Z ad dagoal posve defe marces M M such ha he followg LMI holds: wh Ω 0 Ω 0 0 Ω Ω 0 Ω5 Ω= < 0 Ξ 0 Ξ Ξ Ξ0 0 Ξ Ξ Ξ Ξ 0 Ξ 0 Ξ Ω = Ω = Ω = dag Ξ Ξ { Z Z } { R R R R } 5 6 Ξ7 Ξ 8 Ξ Ξ5 Ξ6 Ξ7 Ξ Ξ Ξ Ξ Ξ 9 8 9 0 Ξ 0 Ξ5 Ξ 0 Ξ Ω = Ω = dag 6 7 5 Ξ8 0 Ξ 9 (7)
Guoqua Lu ad Smo X. Yag / Proceda Egeerg 5 (0) 86 80 89 ( τ τ ) PA AP Q A Q6A R R Z PW A Q6W AM Ξ = Ξ = Ξ = Ξ = PW A Q6W Ξ 5 = PW A Q6W Ξ 6 = Q PB BP B Q5B R R ( σ σ) Z Ξ = BM Ξ = PV BM B QV Ξ = PV B QV Ξ = PV B QV Ξ = M W QW 6 MV W M W QW 6 MV 5 W QW 6 Ξ = Ξ = Ξ = Ξ = Ξ = Ξ = Ξ = Ξ = Ξ = ) W QW 6 7 8 5 9 5 0 5 Q MW W Ξ = Ξ = Ξ = Ξ = 6 MV W QW 6 7 Q MV V M V QV 5 8 MW V QV 5 9 V QV 5 0 MW V QV 5 V QV 5 ( τd) Q ( σd) Q ( τd Q Ξ = W QW Ξ = ( σ ) Q V QV Ξ = V QV Ξ = ( h ) Q W QW Ξ = ( h ) Q V QV. 5 6 6 d 6 7 5 8 d 5 6 9 d 6 5 Proof. Cosruc a lyapuov-krasoskll fucoal for sysem (5) as follows V() = V() V() V() V() (8) yj() x() = j 0 j 0 j= = V () x () Px () y () P y () m f () s ds m g () s ds V ( ) = x( sqxs ) ( ) f ( ys ( )) Qf ( ys ( )) ds y () sqys () g (()) xs Qgxs (()) ds τ() σ() y& () s Q5y& () s ds x () s Q6x() s ds h() & & h() = τ τ σ σ τ σ () = ( τ τ) () () δ ( σ σ) () () δ. τ δ σ δ V () x () s R x() s ds x () s R x() s ds y () s R y() s ds y () s R y() s ds V x s Z x s dsd y s Z y s dsd Calculag he dervave of V () alog he rajecory of sysem (6) s () = () () () () (9) () = x () P[ Ax () Wf ( y ()) Wf ( y ( τ ())) Wy & ( h()) ] y ( P ) [ By ( ) Vg ( x ( )) Vg ( x ( σ ( ))) Vx & ( h( )) ] Mf ( y ( )) [ By() Vgx ( ()) Vgx ( ( τ ())) Vx & ( h()) ] Mg ( x ())[ Ax () (0) Wf ( y ( )) Wf ( y ( τ ( ))) Wy & ( h ( )) ] () = x () Qx () ( & τ()) x ( τ()) Qx ( τ()) f ( y ()) Qf ( y ()) ( & τ()) f ( y ( τ())) Qf ( y ( τ( ))) y ( Qy ) ( ) ( & σ( )) y ( τ ( )) Qy ( τ ( )) g ( x ( )) Qgx ( ( )) ( & σ( )) g ( x ( σ( ))) Qg ( x ( σ( ))) [ By ( ) Vg ( x ( )) Vg ( x ( τ ( ))) Vx & ( h( )) ] Q5[ By ( ) Vg ( x ( )) Vg ( x ( τ ( ))) Vx & ( h( )) ] () ( h& ()) y& ( h()) Qy & 5 ( h()) [ Ax () Wf ( y ()) Wf ( y ( τ ())) Wy& ( h()) ] Q6[ Ax() Wf ( y()) Wf ( y ( τ ( ))) Wy & ( h ( )) ] ( h& ( )) x& ( h( )) Q6x& ( h( )) () = x () Rx () x ( τ) Rx ( τ) x () Rx () x ( τ) Rx ( τ) y () Ry () y ( σ) Ry ( σ) y () Ry () y ( σ) Ry ( σ) () τ () = ( τ τ) x () Zx () ( τ τ) x ( s) Zx() s ds ( σ σ) y () Zy() τ σ ( σ σ) y () s Z y() s ds. σ () By Lemma we kow ha τ τ ( τ τ) x () s Zx() s ds x() s ds Z x() s ds τ τ τ () τ
80 Guoqua Lu ad Smo X. Yag / Proceda Egeerg 5 (0) 86 80 σ σ σ σ σ σ σ ( σ ) y () s Z y() s ds y() s ds Z y() s ds. (5) By ulzg relaoshps (9)-(5) we have dv () ξ () Ω ξ() d (6) Ω s defed (9) ad ξ ( ) = x ( ) y ( ) f ( y( )) g ( x( )) x ( τ( )) y ( τ ( )) x ( τ) x ( τ ) y ( σ) τ σ ( τ ) ( σ ) y ( σ) f ( y( τ( ))) g ( x( σ( ))) y& ( h( )) x& ( h ( )) x( s) ds y( s) ds. I s obvous ha for Ω<0 whch dcaes from he Lyapuov sably heory ha he sysem (5) s asympoc sable. hs complees he proof.. Cocluso hs paper addresses he problem of he asympoc sably for BAM eural eworks of eural-ype wh erval me-varyg delays. A ew delay-rage-depede asympoc sably codo for he cosdered sysems s proposed erms of LMI based o he Lyapuov-Krasovsk fuco mehod ad he LMI echque. Ackowledgme. hs work was suppored by he Fudameal Research Fuds for he Ceral Uverses (No. CDJXS77). Refereces [] Kosko B Adapve bdrecoal assocave memores. Appled Opcs 987; 6: 97-960. [] Kosko B Bdrecoal assocave memores. IEEE ras. Sys. Ma Cyber 988; 8: 9-60. [] Lao X F Yu J B Che GR Novel sably crera for bdrecoal assocave memory eural eworks wh me delays. Ieraoal Joural of Crcu heory ad Applcaos 00; 0: 59-56. [] Che AP Cao J Huag LH. Expoeal sably of BAM eural eworks wh rasmsso delays. Neurocompug 00; 57: 5-5. [5] L Y Yag CB. Global expoeal sably aalyss o mpulsve BAM eural eworks wh dsrbued delays. Joural of Mahemacal Aalyss ad Applcaos 006;: 5-9. [6] Huag Z Luo XS Yag QG. Global asympoc sably aalyss of bdrecoal assocave memory eural eworks wh dsrbued delays ad mpulse. Chaos Solos & Fracals 007; : 878-885. [7] Hu L Lu H Zhao Y B. New sably crera for BAM eural eworks wh me-varyg delays. Neurocompug 009;7: 5-5. [8] Park J H Park CH Kwo O M Lee S M. A ew sably crero for bdrecoal assocave memory eural eworks of eural-ype. Appled Mahemacs ad Compuao 008;99: 76-7. [9] Lu J Zog G D. New delay-depede asympoc sably codos cocerg BAM eural eworks of eural ype. Neurocompug 009; 7: 59-555. [0] Balasubramaam P Rakkyappa R. Global expoeal sably for eural-ype BAM eural eworks wh me-varyg delays. Ieraoal Joural of Compuer Mahemacs 00; 87: 06-075.