A VECTOR SMALL-GAIN THEOREM FOR GENERAL NONLINEAR CONTROL SYSTEMS

Transcription

1 A VECTOR SMALL-GAIN THEOREM FOR GENERAL NONLINEAR CONTROL SYSTEMS Iao Kaafyll ad Zhog-Pg Jag Depame of Evomeal Egeeg Techcal Uvey of Cee 73 Chaa Geece emal: Depame of Eleccal ad Compe Egeeg Polyechc Ie of New Yo Uvey Sx Meoech Cee Booly NY USA emal: Abac A ew Small-Ga Theoem peeed fo geeal olea cool yem The ovely of h eeach wo ha veco Lyapov fco ad fcoal ae lzed o deve vao p-o-op ably ad p-o-ae ably el I how ha he popoed appoach ecove eveal ece el a pecal ace ad exedble o eveal mpoa clae of cool yem ch a lagecale complex yem olea ampled-daa yem ad olea medelay yem A applcao o a bochemcal cc model llae he geealy ad powe of he popoed veco mall-ga heoem Keywod: Ip-o-Op Sably Feedbac Syem Small-Ga Theoem Veco Lyapov Fco ad Fcoal Iodco The mall-ga heoem ha bee wdely ecogzed a a mpoa ool fo obe aaly ad ob coolle deg wh he cool yem commy Fo ace clacal mall-ga heoem [443] have played a ccal ole fo lea ob cool of cea yem bec o dyamc ceae [44] A odced he famewo of clacal mall-ga a eeal codo fo p-op ably of a feedbac yem ha he loop ga le ha oe Th codo elyg po o he cocep of lea fe-ga wa f elaxed by Hll [7] ad he Maeel ad Hll [6] g he oo of moooe ga ad olea opeao Some olea veo of he clacal mall-ga heoem ae deved fo p-op ably of olea feedbac yem Qcly afe he bh of he oo of p-o-ae ably ISS ogally odced by Soag [33] a olea geealzed mall-ga heoem wa developed [9] Th olea ISS mall-ga heoem dffe fom clacal mall-ga heoem ad he olea mall-ga heoem of [7] ad [6] eveal apec Oe of hem ha boh eal ad exeal ably popee ae dced a gle famewo whle oly p-op ably addeed pevo mall-ga heoem A demoaed [9] ad he beqe wo of may ohe olea mall-ga ha led o ew olo o eveal challegg poblem ob olea cool ch a ablzao by paal-ae ad op feedbac ob adapve acg ad olea obeve Moe eegly h pepecve of olea mall-ga ca fd efl applcao moooe yem a mpoa cla of yem mahemacal bology ee [5] Fhe exeo of h ool o he cae of o-fom me ably dcee-me yem ad Lyapov chaacezao ae ped by eveal aho depedely; ee fo ace [ ] Th wo ha bee ppoed pa by he NSF ga DMS-5446

2 Th pape ae a ep fhe o boade he applcably ad geealy of olea mall-ga by emovg wo eeal eco pevo mall-ga heoem A commo feae of he eale olea mall-ga heoem ha he em-gop popey eqed mplcly o explcly fo he olo of he feedbac yem qeo whehe he feedbac yem decbed by oday dffeeal eqao o ae he fom of hybd ad wched yem We wll adop a wea em-gop popey whch mch moe elaxed ha he em-gop popey ee [89] A how o ece wo [] he wea em-gop popey allow dyg a wde cla of olea feedbac yem ch a hybd ad wched yem Whle he mall-ga heoem o be odced h pape ea he above-meoed feae of o eale olea ISS mall-ga heoem wll eoy a addoal ew feae ha boh fe-dmeoal ad fe-dmeoal yem ca be addeed Hee we maly foc o me-delay yem decbed by eaded fcoal dffeeal eqao We wll acheve h omehow ambo a by mag e of veco Lyapov fco ad fcoal Becae of he lae we co o ew mall-ga heoem peeed h pape veco mall ga The advaage of veco Lyapov fco ve gle Lyapov fco olea ably aaly ha bee well docmeed pa leae [58] Rece wo [ 4] povde fhe evdece o he efle of veco Lyapov fco o he cae of p-o-ae ably I h pape we wll how ha he veco mall-ga heoem ca ecove eveal ewly odced mallga heoem fo lage-cale complex yem [354] I addo how ha wh veco mall-ga fom ad o-fom p-o-op ad p-o-ae ably popee ca be ded fo vao mpoa clae of olea dyamcal coolled yem Example of hee yem clde hoe epeeed by Oday Dffeeal Eqao ODE Readed Fcoal Dffeeal Eqao RFDE ad ampled-daa yem A eeg applcao o a bochemcal cool cc model gve o demoae he effecvee of h veco mall-ga mehodology The e of he pape ogazed a follow I Seco we povde cea efl el o moooe dceeme yem The el coaed h eco ae ed exevely beqe eco Seco 3 of he pape povde a bef evew of he yem-heoec famewo odced [89] ad he ma el aed Theoem 3 I Seco 4 ffce Lyapov-le codo fo he vefcao of he hypohee of Theoem 3 ae peeed fo hee ype of yem: Syem decbed by ODE Syem decbed by RFDE ad Sampled-Daa yem The el coaed Seco 4 ae exploed Seco 5 whee example ad applcao of he veco mall-ga mehodology ae gve The coclo of he pape ae povded Seco 6 The poof of mo of he el of he pape ae gve he Appedx Noao Thogho h pape we adop he followg oao: We deoe by K he cla of pove coo fco defed o R : { x R: x } We ay ha a fco ρ : R R pove defe f ρ ad ρ > fo all > By K we deoe he e of pove defe ceag ad coo fco We ay ha a pove defe ceag ad coo fco ρ : R R of cla K f lm ρ By KL we deoe he e of all coo fco R R σ σ : R wh he popee: fo each he mappg σ of cla K ; fo each he mappg σ o-ceag wh lm σ R { x R : x x } R : x { e} deoe he adad ba of R Z deoe he e of o-egave ege Le x y R We ay ha x y f ad oly f R We ay ha a fco ρ : R of cla y N f ρ coo wh ρ ad ch ha ρ ρ y fo all Fo le [ V ] [ : p p V τ V τ τ [ τ [ e p [ a b] [ V τ V τ V τ x y R wh x y R τ R be a boded map We defe deoe he eeal pemm of x m Fo a meaable ad eeally boded fco We ay ha Γ : R R o-deceag f Γ Γ y fo all we defe Γ Γ o443 Γ o o Γ whe m me We defe R If v he v x y R x :[ a b] R wh x y Fo a ege

3 By we deoe he om of he omed lea pace By we deoe he Ecldea om of R Le U wh U [ ]: U ; we deoe he eeco of U wh he cloed By { } B U phee of ad ceeed a U If U R he U deoe he eo of he e U R Le U be a be of a omed lea pace U wh U By M U we deoe he e of all locally boded fco : R U By we deoe he decally zeo p e he p ha afe U fo all If U R he M U deoe he pace of meaable locally boded fco : R U Global Aympoc Sably fo Moooe Dcee-Tme Syem Code he dcee-me yem x Γ x x R whee Γ : R R a o-deceag map wh Γ Fo he dy of he above yem we adop he adad ably oo fo dcee-me yem ee fo ace [4] ad efeece hee Moe pecfcally we ay ha R a Globally Aympocally Sable GAS eqlbm po fo f lm Γ fo all R ad fo evey ε > hee ex δ > ch ha x δ R mple x x Γ ε fo all Nex a eceay codo fo he Global Aympoc Sably popey ad a echcal el ha gaaee covegece o zeo ae povded Popoo : If R GAS he he followg mplcao hold: Γ x x x Poof: Sppoe ha hee ex x R x wh Γ x By dco follow ha Γ x Leg lead o a coadco Th hold < fo all Lemma : Le Γ : R R be a coo o-deceag map afyg wh Γ If he eqaly Γ x hold fo ome R he lm Γ y fo all R wh y x x y Poof: The eqece Γ o-ceag he ee ha Γ Γ Γ x ad heefoe he lm lm Γ ex By ve of coy of he mappg Γ : R we have Γ Codo gaaee ha Γ Γ y Defo 3: Le z ad coeqely lm Γ fo all ad coeqely lm Γ y < x x R x 3 R Moeove f y x he by dco hold ha y y y R We defe z MA{ x y} whee z z R afe z { x y } fo Smlaly fo z MA{ m } wh z { m } Rema 4: m R we have a The MA opeao dffee fom he opeao whch mplcly defed o evey e a he mal eleme of a e Fo example f x R y 3 R he he opeao { x y} o defed whle MA { x y} 3 Of coe f x y he y { x y} MA{ x y} 3

4 b If z MA{ x y} ad y MA{ w v} he z MA{ x w v} Alo MA { x y w} MA{ x MA{ y w}} If x z ad y z he MA{ x y} z c Le x y R I geeal we have MA{ x y} MA{ Γ Γ y} Γ fo ay o-deceag map R Γ : R Defo 5: We ay ha x y R he followg eqaly hold: R Γ : R MA-peevg f R Γ : R o-deceag ad fo evey MA{ x y} MA{ Γ Γ y} Γ 3 The above defed MA-peevg map eoy he followg mpoa popey Popoo 6: Γ : R R wh Γ Γ Γ MA-peevg f ad oly f hee ex odeceag fco γ : R R wh Γ γ x fo all x R Poof: Defe γ : Γ e fo all Le ha MA{ x e x e } Γ x R e x x e x e wh x Noce x ad coeqely MA{ Γ xe Γ x e } Γ x e Γ x e } γ x { defo Γ γ x < Γ Theefoe The covee aeme a dec coeqece of he Nex eceay ad ffce codo ae povded fo GAS of fo he cae of a coo MA-peevg map The poof of he followg popoo povded he Appedx Popoo 7: Sppoe ha Γ : R R wh Γ Γ Γ MA-peevg ad hee ex fco γ N wh Γ γ x The followg aeme ae eqvale: R GAS fo I hold ha γ < fo all > Fhemoe f > he he followg e of mall-ga codo hold fo each : fo all { } f γ γ o o < o > γ 3 The followg mplcao hold: Γ x x x v hold ad fo each ad hold ha Q MA { x Γ Γ Γ } x R Γ Rema 8: Noce ha a R Q R R Moeove Γ Q Q fo all : a coo MA-peevg map wh Q ad Q a a fo all x R The ex popoo a efl echcal el whch wll be ed he followg eco Popoo 9: Sppoe ha R Γ γ Γ : R wh Γ Γ MA-peevg ad hee ex fco γ N wh Γ x Moeove ppoe ha mplcao hold ad ha x MA{ a Γ } { a Γ a Γ a Γ a } Q a MA fo cea 4 x a R The x Qa whee

5 Poof: Sppoe ha x MA{ a Γ } The Γ MA { Γ a Γ } ad x MA { a Γ a Γ } dco agme x MA { a Γ a Γ a Γ } fo all Popoo 7 ha x MA { Q a Γ } fo all Sce lm Γ By a I follow fom aeme v of we oba x Qa < 3 A Veco Small-Ga Theoem fo a Wde Cla of Syem 3A Revew of he Syem-Theoec Famewo I h wo we mae e of he yem heoec famewo peeed [89] Fo eao of compleee he bac oo ae ecalled hee The oo of a Cool Syem-Defo []: A cool yem Σ : Y M U M D φ π H wh op co of a e U cool e whch a be of a omed lea pace U wh U ad a e M U MU allowable cool p whch coa a lea he decally zeo p a e D dbace e ad a e M D M D whch called he e of allowable dbace a pa of omed lea pace Y called he ae pace ad he op pace epecvely v a coo map H : R U Y ha map boded e of R U o boded e of Y called he op map v a e-valed map R M U M D x π x [ wh π x fo all x R M U M D called he e of amplg me v ad he map φ : A φ whee A φ R R M U M D called he ao map whch ha he followg popee: Exece: Fo each x R M U M D hee ex > ch ha x Aφ [ Idey Popey: Fo each x R M U M D hold ha φ x x ~ 3 Caaly: Fo each x Aφ wh > ad fo each ~ d M U M D wh ~ ~ τ d τ τ d τ fo all τ [ hold ha x ~ d ~ Aφ wh ~ φ x φ x ~ d 4 Wea Semgop Popey: Thee ex a coa > ch ha fo each wh x Aφ a τ x d Aφ fo all [ τ b φ τ φ τ x φ x fo all τ π x d x Aφ π x d τ x [ d x [ ] A π τ φ τ x π x : c f he hold ha π fo all τ wh φ we have [ τ The BIC ad RFC popee-defo []: Code a cool yem Σ : Y M U M D φ π H wh op We ay ha yem Σ ha he Bodede-Imple-Coao BIC popey f fo each x R M U M D hee ex a mal exece me e hee ex 5

6 x ] ch ha A { x } I : φ [ x d R M U M D [ x addo f < he fo evey M > hee ex wh φ > M obly fowad complee RFC fom he p M U f ha he BIC popey ad fo evey T hold ha { φ x ; M B [ ] M [ T ] x [ T ] d M } < U U D p The oo of a ob eqlbm po-defo 3 []: Code a cool yem Σ : Y M U M D φ π H ad ppoe ha H fo all We ay ha a ob eqlbm po fom he p M U fo Σ f fo evey R R M D wh hold ha φ fo evey ε > T h R hee ex δ : δ ε T h > ch ha fo all x [ T ] M U τ ] wh x p < δ hold ha τ x A fo all d M D ad [ h U { φ τ x ; d M τ [ h] [ ] } < ε p D T Nex we pee he Ip-o-Op Sably popey fo he cla of yem decbed pevoly ee alo [937] The oo of IOS UIOS ISS ad UISS-Defo 5 []: Code a cool yem Σ : Y M U M D φ π H wh op ad he BIC popey ad fo whch a ob eqlbm po fom he p M U Sppoe ha Σ RFC fom he p M U If hee ex fco σ KL β K γ N ch ha he followg emae hold fo all M U x R M D ad : H φ β x p γ x σ Y τ τ φ he we ay ha Σ afe he Ip-o-Op Sably IOS popey fom he p M U wh ga γ N Moeove f β he we ay ha Σ afe he Ufom Ip-o-Op Sably UIOS popey fom he p wh ga γ N M U Fo he pecal cae of he dey op mappg e H x : x he Ufom Ip-o-Op Sably popey fom he p M U called Ufom Ip-o-Sae Sably U ISS popey fom he p M U The eade hold oce ha ohe eqvale defo of he ISS popey ae avalable he leae ee [63] U 3B A New Small-Ga Theoem We code a abac cool yem Σ : Y M U M D φ π H wh he BIC popey fo whch a ob eqlbm po fom he p M U Sppoe ha hee ex map : R U wh V R R V fo all ad a MA-peevg coo map Γ : R wh Γ ch ha he followg hypohee hold: H Thee ex fco σ KL ν c K ζ a p N p N L : R R wh L fo all ch ha fo evey x R M U M D he mappg 6

7 V φ x V x ad L L φ x V φ d ae locally boded o ad he followg emae hold fo all : [ L Γ [ V ] [ [ V MA σ [ ] ζ 3 U [ x p [ V ] p [ L ν c a [ ] 3 U [ whee he mal exece me of he ao map of Σ H R GAS fo he dcee-me yem H3 Thee ex fco b N g N x R U : L g V x κ μ x b whee V x V x V x Fo fe efeece V x V x V x Σ : Y M M φ π H U D The ma el of he pee wo aed ex μ β κ K ch ha he followg eqale hold fo all ad L b β x 33 called he veco Lyapov fco fo he yem Theoem 3: Code yem Σ : Y M U M D φ π H wh he BIC popey fo whch a ob eqlbm po fom he p ad ppoe ha hee ex map : R U wh M U fo all ad a MA-peevg coo map H-3 hold V R R Γ : R wh Γ ch ha hypohee V The hee ex fco ~ σ KL emae hold fo all : ad ~ β K ch ha fo evey x R M U M D he followg whee ad Q MA { x Γ Γ Γ } ~ β x G [ V MA ~ σ 34 U [ ] { Q σ p Q σ p Q ζ Q ζ } G MA 35 Moeove f Fally f addo o H-3 he followg hypohe hold: β c K ae boded he ~ β K boded H4 Thee ex q N ch ha he followg eqaly hold fo all x R U : whee V x V x V x V x H x q Y 36 7

8 he yem Σ afe he IOS popey fom he p M U wh ga γ : q G whee G G G he mappg defed by 35 Moeove f β c K ae boded he yem Σ afe he UIOS popey fom he p M U wh ga γ : q G Rema 3: Noce ha fo he cool p-fee cae e Theoem 3 mple Ufom Rob Global Aympoc Op Sably RGAOS fo he coepodg yem Moeove f hee ex M ch ha σ M fo all he he fco G N wh G G G ae gve by: whee ϕ { Mp Mp ϕ ζ ϕ ζ ζ } G : ϕ { γ γ o : { } } γ o ad γ N ae he fco wh Γ γ x ad Γ Γ Γ The eade hold oce ha f hypohe of Popoo 7 hold fo he fco γ N he fo each we have ehe ϕ o hee ex a dex e { wh o epeaed dex ch ha ϕ γ γ oγ o } 4 Veco Lyapov Fco ad Fcoal I h eco we povde ffce Lyapov-le codo fo he vefcao of Theoem 3 fo hee ype of yem: Syem decbed by Oday Dffeeal Eqao ODE Syem decbed by Readed Fcoal Dffeeal Eqao RFDE ad Sampled-Daa yem Noce ha ce famle of Lyapov fco o fcoal ae employed he obaed el coe codo fo veco Lyapov fco o fcoal fo he UIOS popey 4A Syem of ODE We code yem decbed by Oday Dffeeal Eqao ODE of he fom: x& f x Y H x R Y R N U d D 4 l m whee D R U R wh U ad f : R R U D R H : R R R ae coo mappg wh H f fo all R D ha afy he followg hypohee: A Thee ex a ymmec pove defe max P R ch ha fo evey boded I R ad fo evey boded S R U hee ex a coa L afyg he followg eqaly: A Thee ex a K x y P f x f y L x y I x y S S d D γ K ch ha f x a x γ fo all x R R U D A3 Thee ex fco V C R R ; R W C R R ; R a a a K N a 3 4 μ β κ K ζ N g N γ N p N wh γ fo a famly of pove 8

9 defe fco ρ C R ; R ad a coa λ ch ha he followg eqale hold fo all x R R U : a H V a x 4 β μ x g V V κ W a x a β V W W p f d x : d D W λ ζ p 44 x ad fo evey ad x R R U he followg mplcao hold: If γ V V ζ he p f d x ρ V V d D V x 45 O ma el coceg yem of he fom 4 he followg el whch povde ffce codo fo Theoem 3 o hold I poof povded he Appedx Theoem 4 Veco Lyapov Fco Chaacezao of he IOS popey: Code yem 4 de hypohee A-3 If he followg e of mall-ga codo hold fo each : γ γ o o < o > 46 γ 3 fo all { } l f l he yem 4 afe he IOS popey wh ga γ a oθ N fom he p MU whee p γ ϕ ζ ζ θ : ϕ p ζ γ ϕ ζ 47 ad ϕ : l { γ γ o : { } } γ l l l o 48 l Moeove f o N β K boded he yem 4 wh op Y H afe he UIOS popey wh ga γ a θ fom he p MU Comme o Theoem 4: The poof of Theoem 4 how ha eqale 4 45 ae ed fo he devao of eqale 3 ad 36 whle eqale ae ed fo he devao of eqale 3 ad 33 Hypohee A ad A ae mmal eglay hypohee ha gaaee qee of olo ad coy of he olo wh epec o al daa fo yem 4 Moeove clea fom mplcao 45 ha h cae he dagoal ga fco γ play o ole mplcao 45; ha why hey ae amed o be zeo Fo he ISS cae whee H x oe ca e W Theoem 4 ad oba he followg coollay Coollay 4 Veco Lyapov Fco Chaacezao of he ISS popey: Code yem 4 de hypohee A- ad ppoe ha hee ex a famly of fco V C R R ; fco R a a K β K ζ N γ N wh γ fo ad a famly of pove defe fco ρ C R ; R ch ha: a x V a x β R R 49 ad mplcao 45 hold fo evey ad x R R U If addoally he mall-ga codo 46 hold fo each ad fo all { } l f l he yem 4 afe he ISS 9

10 popey wh ga γ a oθ N fom he p MU whee θ N defed by Moeove f β K boded he yem 4 afe he UISS popey wh ga γ a oθ N fom he p MU whee θ N defed by Compao of Theoem 4 ad Coollay 4 wh exg el: The eade hold compae he el of Coollay 4 wh Theoem 34 [] I clea ha Theoem 34 [] a pecal cae of Coollay 4 wh γ a fo all whee a N wh a < fo > O he ohe had veco Lyapov chaacezao baed o he ma el [3] eg Theoem 36 [] eqe Lpchz eglay of he ghhad de of he ODE ce he ma el [3] baed o cea qalave chaacezao of he ISS popey povded [3435] Th exacly he eao why he ma el [3] doe o povde a fomla fo he ga fco of he oveall yem coa wh he ma el he pee wo The ame comme hold fo Theoem [5] I pacla hold be meoed ha he mall-ga codo Coollay 4 ae exacly he ame a he cyclc mall-ga codo [4 5] I ode o demoae he applcably of o el o lage-cale ecoeced yem code he cae x& f d x x x x R N d D U whee x R N N l D R a o-empy compac e U R a oempy e wh U f : D R U R ae locally Lpchz mappg wh f d fo all d D We ame ha he UISS popey hold fo each byem x & f d x wh p x x x x Le V C R R ; R be ISS-Lyapov fco fo each oe of he byem e pove defe ad adally boded fco fo whch he followg eqale hold fo : γ V x V x N p V x f d x : d D U x x x R ζ < x d D fo cea fco ζ N γ N wh γ fo Wog wh he Lyapov fco V C R R ; R ad explog Coollay 4 we ca gaaee ha he UISS popey hold fo he above yem f he mall-ga codo 46 hold fo each ad fo all { } l f l I hold be clea ha he fco γ N ae he acal ga fco e he followg N eqale hold fo all R ad M U : V x σ V x ζ p τ γ p V x τ τ m τ fo cea σ KL whch ae ohg ele b he eqale of he -fomlao of he UISS popey fo each byem x & f d x wh p x x x x Aga ca be ee ha h cae he dagoal ga fco γ play o ole; ha why hey ae amed o be zeo 4B Syem Decbed by RFDE l m Le D R be a o-empy e U R a o-empy e wh U ad Y a omed lea pace We deoe by x he qe olo of he al-vale poblem: x& f T x d Y H T R Y Y d D U 4

11 wh al codo T x x C [ ]; R whee > a coa T x : θ ; θ [ ] ad he mappg f : R C [ ]; R U D R H : R C [ ]; R Y afy f H fo all R D The followg hypohee wll be mpoed o yem of he fom 4: S The mappg x f x coo fo each fxed ad hee ex a ymmec pove defe max P R wh he popey ha fo evey boded I R ad fo evey boded S C [ ]; R U S Thee ex a K hee ex a coa L ch ha: y P f x f y I x y S S d D x R C [ ]; R U D L τ y τ τ [ ] L x y γ K ch ha f x d γ a x fo all S3 Thee ex a coable e A R whch ehe fe o A { ; } wh > > fo all ad lm ch ha he mappg x R \ A C [ ]; R U D f x coo Moeove fo each fxed x R C [ ]; R U D we have lm f x f x S4 The mappg H : R C [ ]; R Y coo Fo x C [ ]; R we defe x : θ We wll e he coveo x C []; R R we have x x θ [ ] C [];R R ad f The cla of fcoal whch ae almo Lpchz o boded e wa odced [3] ad ed exevely he pee wo Fo he ae of compleee we ecall hee he defo [3] Defo 43: We ay ha a coo fcoal V :[ a C [ ]; R R > a almo Lpchz o boded e f hee ex o-deceag fco M : R R P : R R G : R [ ch ha fo all R he followg popee hold: P Fo evey x y { x C [ ]; R ; x R} hold ha: V y V M R y x [ a R] P Fo evey abolely coo fco x :[ ] R wh x R ad eeally boded devave hold ha: V h V hp R e p x& τ fo all [ a R] ad h τ G R e p x& τ τ Fo he cae we ay ha a coo fcoal V :[ a C [ ]; R R almo Lpchz o boded e f V :[ a R R locally Lpchz oce ha h cae C [ ]; R R e fo evey compac S [ a R hee ex L ch ha V V τ y L τ L x y fo all S τ y S

12 If he coo fcoal V :[ a C [ ]; R R almo Lpchz o boded e he we ca defe he devave V x; v he followg way ee alo [3] fo x v R C [ ]; R R V h Eh x; v V V x; v : lm p h h whee E h x; v wh h < deoe he followg opeao: E h θ h v fo h < θ x; v : 4a θ h fo θ h : Paclaly fo he cae we defe E h x; v : hv 4b The followg heoem povde ffce Lyapov-le codo fo he UIOS popey The ga fco of he IOS popey ca be deemed explcly em of he fco volved he ampo of he heoem I poof povded he Appedx Theoem 44: Code yem 4 de hypohee S-4 ad ppoe ha hee ex almo Lpchz o boded e fcoal Q :[ C [ ]; R R wh Q :[ C [ ]; R R wh fco a a a3 a4 K μ β κ K ζ N g N γ N p N pove defe fco ρ C R ; R ad a coa λ ch ha fo all x R C R [ ]; U he followg eqale hold: H V a x a β Y μ x g V V κ W a x 3 4 β 4 a 43 whee p Q T d D V : x; f x Q T p θ [ ] Q θ T V λ ζ p 44 θ W : p Q θ T θ θ [ ] 45 ad fo evey ad x R C R [ ]; U he followg mplcao hold: If ζ γ V Q T he p Q T x; f x ρ Q T d D Fhe ppoe ha he followg e of mall-ga codo hold fo 46 γ > 47 < ad fo each : γ γ o o < o > 48 γ 3 fo all { } l f l The yem 4 afe he IOS popey wh ga γ a oθ N fom he p MU whee θ N defed by Moeove f β K boded he yem 4 afe he UIOS popey wh ga o N γ a θ fom he p MU whee θ N defed by 47 48

13 Whe H x eg Q Theoem 44 we oba he followg coollay o he ISS of yem 4 Coollay 45: Code yem 4 de hypohee S-4 ad ppoe ha hee ex a famly of almo Lpchz o boded e fcoal Q :[ C [ ]; R R wh fco a a K β K ζ N γ N ad a famly of pove defe fco ρ C R ; R ch ha fo all x R C R [ ]; U he followg eqaly hold: whee x V a x a β V : p θ [ ] Q θ T 49 θ 4 ad mplcao 46 hold fo evey ad x R C R [ ]; U If addoally he e of mall-ga codo hold he yem 4 afe he ISS popey wh ga γ a oθ N fom he p MU whee θ N defed by Moeove f β K boded he yem 4 afe he UISS popey wh ga γ a oθ N fom he p MU whee θ N defed by Rema 46: I of ee o oe ha ome of he fcoal Q :[ C [ ]; R R Theoem 44 ad Coollay 45 ae allowed o be fco cae of Th emd he cae of Razmh fco whch ae ed feqely fo he poof of ably popee of yem decbed by RFDE ee [3734] Coeqely Theoem 44 ad Coollay 45 allow he flexbly of g Lyapov fcoal wh Razmh fco ode o pove deed ably popee Rema 47: I hold be clea ha he coveo C [];R R allow Theoem 44 ad Coollay 45 o be ed he cae of yem decbed by ODE cae of I h cae Theoem 44 ad Coollay 45 ae geealzao of Theoem 4 ad Coollay 4: he e of locally Lpchz fco allowed ad dcoe of he gh-had de of he dffeeal eqao wh epec o me ae allowed I h cae he dagoal ga fco γ fo play o ole whaoeve ad coeqely ca be e eqal o zeo Howeve he medelay cae cae of > he dagoal ga fco γ fo play a gfca ole ee Example 5 below 4C Sampled-daa Syem We code wched yem decbed he followg way: gve a pa of e D R U R wh U a pove fco h : R U ] whch boded by a cea coa > ad a pa of veco feld f : R R D U U R H : R R we code he wched yem ha podce fo each ~ x R R ad fo each ple of meaable ad locally boded p d : R D d : R R : R U he pecewe abolely coo fco R va he followg algohm: Sep : ~ Gve τ ad τ calclae τ g he eqao τ τ exp d τ h τ τ Compe he ae aecoy x [ τ τ a he olo of he dffeeal eqao x& f τ d τ 3 Calclae x τ g he eqao τ lm τ Fo we ae τ ad x τ x al codo Schemacally we we l m 3

14 x& f τ d τ [ τ τ ~ τ τ τ exp d τ h τ τ Y H 4 wh al codo x x Swched yem of he fom 4 ae called ampled-daa yem ee alo [9] ad [39] fo he cae of ae-depede amplg peo I he pee wo we dy yem of he fom 4 de he followg hypohee: R f x x d coo wh epec o x d R D U ad ch ha fo evey boded S R R U U hee ex coa L ch ha R Thee ex a fco x y f x x d f y x d L x y x x S D y x S D a K ch ha d x x f x x a d x x U U D R R R3 H : R R a coo map wh H R4 The fco h : R U ] a pove coo ad boded fco The followg heoem povde ffce Lyapov-le codo fo he UIOS popey The ga fco of he IOS popey ca be deemed explcly em of he fco volved he ampo of he heoem I poof povded he Appedx Theoem 48 Veco Lyapov Fco Chaacezao of UIOS: Code yem 4 de hypohee R-4 ad ppoe ha hee ex oegave fco V C R ; Q C R ; a a a3 a4 K ζ N g N γ N p N R R coa μ κ λ ad pove defe fco ρ C R ; R ch ha he followg eqale hold fo all x x R R U U : a H V x a x x g V V Q x a x a3 4 4 κ 43 V p V p Q f x x d μ Q λ ζ ζ p x 44 d D ad fo evey ad x R U U he followg mplcao hold: ζ x ad x A h x he p V f x x d ρ V 45 If ζ γ V V d D whee he famly of e-valed map R R T A T R defed by 4

15 A T x R T : d M ζ M V γ V φ x ; d fo all D U [ ] ad wh φ x ; d x V 46 ad φ x ; d deoe he olo of x & f x d wh al codo x x coepodg o d M D M U Fhemoe f he e of mall-ga codo hold he yem 4 afe he UIOS popey wh ga γ a oθ N fom he p MU ad zeo ga fom he p d ~ M R whee θ N defed by Fo he ISS cae whee H x oe ca e Q Theoem 48 ad oba he followg coollay Coollay 49 Veco Lyapov Fco Chaacezao of UISS: Code yem 4 de hypohee R-4 ad ppoe ha hee ex a famly of fco V C R ; R fco a a K ζ N γ N ad a famly of pove defe fco C R ; R ρ ch ha he followg eqaly hold fo all x R : ad mplcao 45 hold fo evey a x V x a x 47 ad x R U U whee he famly of e-valed map R R T A T R defed by 46 ad φ x ; d deoe he olo of x & f x d wh al codo x x coepodg o d M D M U Ude he e of mall-ga codo yem 4 afe he UISS popey wh ga γ a oθ fom he p M U ad zeo ga fom he p d ~ M R whee θ N defed by N Rema 4: I woh og ha Theoem 3 ad Coollay 33 [4] ae pecal cae of Theoem 48 ad Coollay 49 wh γ a fo all ad Q whee a N wh a < fo > Moeove amed [4] ha hee ex a coa R ad a fco p K ch ha x R p H I he pee wo ch a hypohe o eeded The epeao of he famly of e-valed map R R T A T R defed 46 he followg he ame wh [4]: each A T R he e of all ae x R o ha he olo of x & f x d wh al codo x x ca be coolled o x R me le o eqal ha T by mea of appopae p d M D M U ha afy ζ p V ad ch ha he [ ] p γ V V x I geeal vey dffcl o aecoy of he olo afe he coa [ ] oba a accae decpo of he e-valed map Howeve fo evey g C R ; R we have: g g { x R : g x g T b } 5 R R T A T R defed by 46 A T B T x T R whee g b : g ξ f ξ x d : d D ζ ad V C R ; ae he fco volved hypohee of Theoem 48 R { } V γ { V ξ V x } V < x R

16 5 Example ad Applcao Example 5: Code he me-delay yem: x& a x g d T 5 whee wh m d D R a > ad g : D C [ ]; R R ae coo mappg p g d c d D x 5 fo cea coa c We ex how ha C [ ]; R RGAS fo 5 f c < a fo all ad he followg mall-ga codo hold fo each : fo all { } f c c c < a a a 53 3 F we oce ha hypohee S-4 hold fo yem 5 de hypohe 5 wh op H : x C [ ]; R Defe he famly of fco Q x ad V : p Q θ x fo x C [ ]; R Thee mappg afy eqaly 49 ad θ [ ] defo 4 wh a : a : β ad : Le λ ad oce ha c mplcao 46 hold wh γ : ad ρ : λ a Codo 53 ad he fac ha c < a λ a fo all mple ha he mall-ga codo hold fo λ ffcely cloe o We coclde fom Coollay 45 ha C [ ]; R RGAS fo 5 I mpoa o oce ha he codo o he dagoal em cao be avoded geeal Sch a ao occ fo example whe x& a x c x c x wh c < Example 5: Code he followg bochemcal cool cc model: & g & τ τ a a R 54 whee a > τ ae coa ad g C R ; R a fco wh g > fo all > Th model ha bee ded [3] ee page 58-6 ad I h boo fhe amed ha p g C R ; R boded ad cly ceag a ypcal choce fo g C R ; R g wh p p beg a pove ege I how ha f hee oe eqlbm po fo 54 he aac all olo If hee ae wo eqlbm po he all olo ae aaced o hee po Hee we dy 54 de he followg ampo: > H Thee ex K > ad λ wh a g ad ch ha whee a a K 6 a g K λ fo all 55

17 Ug Small-Ga Aaly we ae a poo o pove: Code yem 54 de hypohe H ad le : τ The fo evey C [ ]; R he olo of 54 wh al codo T afe lm whee R wh a g fo I hold be clea ha coa o he aaly pefomed [3] fo 54 baed o he moooe dyamcal yem heoy we do o ame ha g C R ; R boded o cly ceag Moeove eve f hee ae wo eqlbm po oce ha 55 allow g ad heefoe R ca be a eqlbm po we pove almo global covegece o he o-val eqlbm A ypcal aaly of he eqlbm po of 54 de hypohe H how ha hee ex a eqlbm po R afyg: a g 56 I ode o be able o dy olo of 54 evolvg we code he followg afomao: R exp x 57 Theefoe yem 54 de afomao 57 expeed by he followg e of dffeeal eqao: g exp x τ x& exp a x 58a g x& exp x x a τ x x R 58b F we oce ha hypohee S-4 hold fo yem 58 de hypohe H wh op H : x C [ ]; R ad ha C [ ]; R a eqlbm po fo 58 Defe he famly of fco Q x ad V : p Q θ x fo x C [ ]; R Thee [ ] θ mappg afy eqaly 49 ad defo 4 wh a : a : β ad : We defe γ fo ad : [ log θ exp ] beg he coa volved hypohe H ad b : Q x g ; a γ whee θ λ b K Noce ha exp x τ g exp a x g exp x τ exp x g We code he followg cae: b λ 7

18 x I h cae he lef had de eqaly 55 mple ha g < exp x g τ b exp x τ b b exp x τ b exp x τ γ Q mple θ exp x τ x eqaly eqale gve: V x l exp x τ whee b : K The whch combed wh he pevo ; g Q x a exp x τ exp ax g b bθ exp b bθ exp x 59 x I h cae he gh had de eqaly 55 mple ha g exp x τ λ exp x g mple θ exp x τ x l τ λ exp x τ The eqaly γ Q whch combed wh he pevo eqale gve: V x λθ exp g exp x Q x ; a τ exp x ax x g 5 Combg he wo cae we oba fom 59 ad 5 ha he followg mplcao hold: γ V g exp x τ Q Q x ; a exp ρ Q 5 g x wh b bθ exp ρ : a m λθ exp b bθ exp ad : [ log μ exp ] We ex defe fo γ fo γ whee μ > o be eleced Wog a mla way a above we oba fo : γ V Q x Q x ; a exp x τ x ρ Q x 5 wh exp : μ a μ ρ fo Theefoe we coclde fom 5 ad 5 ha mplcao 46 hold Fally we chec he mall-ga codo Explog he pevo defo of he fco γ we coclde ha he mall-ga codo hold f ad oly f γ γ > Sce γ o γ o oγ [ log μ ] θ exp he mall-ga codo hold wh μ θ 8 o o o fo all γ < Th Coollay 45 mple ha C [ ]; R Ufomly Globally Aympocally Sable fo yem 58 Tag o acco afomao 57 h mple ha fo evey C [ ]; R he olo of 54

19 wh al codo T afe a g fo < lm whee R wh 6 Coclo A ovel Small-Ga Theoem peeed whch lead o veco Lyapov chaacezao of he fom ad o-fom IOS popey fo vao mpoa clae of olea cool yem The el peeed h wo geealze may ece mall-ga el he leae ad allow he explc compao of he ga fco of he oveall yem Moeove ce he ga map Γ : R R allowed o coa dagoal em he obaed el have dec applcao o me-delay yem Example have demoaed he effecvee of he veco mall-ga mehodology o lage-cale me-delay yem ch a hoe ecoeed boechology O fe wo wll be deced a applcao of he veco mall-ga heoem o he olea feedbac deg e fo vao clae of olea cool yem Aohe eeg opc fo fe eeach o dy he eal ad exeal ably popee fo copled yem volvg egal p-o-ae able ISS a weae oo ha ISS [36] byem fom a vewpo of veco mall-ga Some pelmay el ae epoed po [8] fo ecoeced yem cog of wo ISS ad/o ISS byem Appedx-Poof Poof of Popoo 7: We pove mplcao v ad v ce he mplcao a coeqece of Popoo v : Clealy ce Γ Q fo all x R we have Γ Q fo all Coy of he mappg { } x R oce ha Γ x R Q MA x Γ Γ Γ whch a dec coeqece of coy of he mappg Γ mple ha fo evey ε > hee ex δ > ch ha x δ x R mple Q ε oce ha Q Th mple ably Γ : MA-peevg we have { } Sce R R Γ Q MA Γ Γ Γ fo all x R Moeove ce Γ Q fo all x R follow ha Γ Q Q fo all x R Lemma coco wh he fac ha hold ad x Q fo all R mple ha lm Γ fo all R x x : If hee ex > ad ome ege ch ha γ he he o-zeo veco wh x ad x fo wll volae Coeqely γ fo all > < x R Nex ppoe ha > Sppoe ha hee ex ome > { } dce { } wh f o Who lo of geealy we may ame ha fo ch ha γ γ o o γ 3 ad coeqely γ o γ 3 o oγ The o-zeo veco x R wh x x γ o γ o oγ fo ad x fo > afe Γ x ad coeqely hypohe volaed Theefoe m hold v The poof of h mplcao a dec coeqece of he fac ha Γ { γ o γ oγ x : { } } 9

20 fo all x R Q MA { x Γ Γ Γ } have Q MA { Γ Γ Γ } ad Ug may be how ha Γ fo all x R Γ fo all x R x R Sce By dco follow ha Γ Q Q fo all Γ Q fo all x R R Γ : R MA-peevg we A a el we oba Γ Q Q fo all x R Sce x Q we oba The fac ha mplcao hold how by coadco Sppoe ha hee ex a o-zeo x R wh Γ x Coeqely fo evey { } hee ex p { } wh γ p x p x Wh hee eqale md hee a lea oe { } wh x > ad a cloed cycle ch ha γ o γ o o x x γ The poof h compleed < whch coadc Theefoe he mplcao > v hold Poof of Theoem 3: The poof co of wo ep: Sep : We how ha Σ RFC fom he p followg eqaly hold fo all : M U ad ha fo evey x R M U M D σ L Q ζ [ he V MA Q A U [ Theefoe by ve of A 33 popee P ad P of Lemma 7 [] hold fo yem Σ wh V V ad γ G Moeove f β K boded he 33 mple ha popee P ad P of Lemma 8 [] hold fo yem Σ wh V V ad γ G Sep : We pove he followg clam Clam: Fo evey ε > Z R T hee ex τ ε R T ch ha fo evey x R M U M D wh [ T ] ad x R he followg eqaly hold: Q σ L G [ V MA Q Pε Γ fo all U [ τ A Moeove f β c K ae boded he fo evey ε > Z R hee ex τ ε R ch ha fo evey x R M U M D wh x R eqaly A hold Noce ha hypohe H ad eqaly 33 gaaee he exece of ε T R Z ch ha l Q Pε Γ Q σ b R β fo all l If β K boded he depede of T Theefoe T by ve of A popey P3 of Lemma 7 [] hold fo yem Σ wh V V ad γ G Moeove f β c K ae boded he A mple ha popey P3 of Lemma 8 [] hold fo yem Σ wh V γ G V ad The poof of Theoem 3 h compleed wh he help of Lemma 7 o Lemma 8 [] Now we e o eablh RFC a clamed Sep

21 Sep : Le x R M U M Ieqaly 3 mple fo all [ D [ V ] MA L Γ [ V ] [ [ Popoo 9 coco wh A3 mple A fo all σ [ ζ A3 U [ [ We how ex ha Σ RFC fom he p M U by coadco Sppoe ha < The by ve of he BIC popey fo evey M > hee ex [ wh φ x > M O he ohe had emae A coco wh he hypohe < how ha hee ex M ch ha p V τ M The fac ha V boded coco wh emae 3 mple ha hee ex τ < coa M ch ha p L τ M τ < μ φ x b L g V κ [ x > M I follow fom 33 ad eqaly ha he ao map of Σ e φ x boded o d ad h coadc he eqeme ha fo evey M > hee ex wh φ Hece we m have Theefoe we coclde ha Σ RFC fom he p ad ha A hold fo all M U [ Sep : Poof of he Clam The poof of he clam wll be made by dco o Z F we how eqaly A fo Le abay ε > R T x R M U M D wh [ T ] ad x R Ieqaly 3 coco wh eqaly A gve fo : V MA σ L Γ Q σ L A4 U [ U [ ΓQ [ ] γ τ ζ [ Sce Γ Q Q ad Q x fo all x R eqaly A4 mple fo all : V MA σ L Γ Q σ L A5 U [ Q ζ [ Smlaly eqaly 3 coco wh eqaly A gve fo : [ ] x p Q σ L p Q ζ τ p [ L ν c a A6 U [ U [ Noce ha 33 mple L b β x b R β Ug he popee of he KL fco we ca T gaaee ha hee ex τ ε R T ch ha σ β b R τ ε Noce ha f β K he τ T depede of T The follow fom A5 ha we have

22 V MA Pε Γ fo all U [ τ Sce G Q ζ fo all a coeqece of 35 ad Q ε ε we coclde ha eqaly A hold fo Q σ L Q ζ [ Nex ppoe ha fo evey ε > R T hee ex τ ε R T ch ha fo evey x R M U M D wh [ T ] ad x R A hold fo ome Z Le abay ε > [ T R T x R M U M D wh ] ad x R Noce ha he wea emgop popey mple ha π x [ τ τ ] Le π x [ τ τ ] The 3 mple: Moeove eqaly A gve: Ieqaly A6 alo mple: L L Γ [ V ] [ τ ] V MA σ [ γ fo all U [ A7 [ ] MA Q ε Γ Q σ L G [ fo all [ A8 V [ U ν c a R p Q σ L A9 U [ ] U [ ] [ ] p Q ζ τ p [ Ug A8 ad he fac ha Γ G G fo all a dec coeqece of defo 35 ad he fac ha Γ Q Q fo all Γ x R we oba: [ ] MA Q ε Γ Q σ L fo all [ A G [ V [ U Ieqaly A coco wh eqaly A7 he fac ha G Q ζ ζ ha τ mple: fo all ad he fac V MA σ L Γ τ Q ε Q σ L U [ ] fo all τ A G [ Ieqaly A9 coco wh he fac ha σ p G σ p Q ζ G he fac ha whee τ [ T ] ad x R mple ha L τ MA σ f ε T R τ G [ fo all ad σ U [ fo all τ A f ε T R : τ ε R T ν c a T R p Q σ b R β A3 T The eade hold oce ha f β c K ae boded ad τ depede of T he f ca be choe o be depede of T a well Noce ha by combg A ad A we ge:

23 V MA σ f ε T R Γ τ Q ε Q σ L U [ fo all τ A4 Clealy hee ex ε R T τ ch ha σ f ε T R τ ε G [ Defe: τ ε R T τ ε R T τ ε R T A5 Aga he eade hold oce ha f f ad τ ae depede of T he τ depede of T a well Sce Q ε ε we oba fom A4: V MA Q ε Γ σ fo all U [ τ A6 Q L G [ whch how ha A hold fo The poof complee < Poof of Theoem 4: We wa o how ha all hypohee of Theoem 3 hold wh L : W V A7 Noce ha hypohe H3 of Theoem 3 a dec coeqece of eqale 4 43 ad defo A7 Moeove hypohe H4 of Theoem 3 a dec coeqece of eqaly 4 wh q : a x fo all R x Code a olo x of 4 coepodg o abay M U M D wh abay al codo x R Clealy hee ex a mal exece me fo he olo deoed by Le V V W W abolely coo fco o [ ad le L L Moeove le I [ be he zeo Lebege meae e whee x o dffeeable o x& f d By ve of 45 follow ha he followg mplcao hold fo \ I ad : V γ V [ ζ V & ρ V A8 ad by ve of 44 we ge fo [ \ I : W & W λ ζ p V A9 Lemma 35 [4] coco wh A8 mple ha hee ex a famly of coo fco of cla KL wh σ fo all ch ha fo all ad we have: σ V p σ ζ p τ [ V ; p σ p γ V τ τ τ τ Moeove eqaly A9 decly mple ha fo all we have: [ τ σ A 3

24 4 p p V p W W ζ λ A Le : σ σ whch a fco of cla KL ha afe σ fo all I follow fom A A ad defo A7 ha fo all [ ad we ge: p p V V V ζ γ A p p V L V ζ γ σ A3 p p V p W W ζ λ A4 Clealy eqale A3 how ha 3 hold wh R R Γ : Γ Γ Γ x x x wh x x γ Γ fo all ad x R Fhemoe ce γ fo ad aeme of Popoo 7 hold follow ha hypohe H of Theoem 3 hold a well Moeove eqale A ad A4 mply ha he followg emae hold fo all [ : p p p p V p V p W W ζ γ ζ λ A5 p p V V V ζ γ A6 Defe: : p p ζ ζ fo all A7 x p x x p : γ γ fo all x R A8 Combg emae A5 A6 ad explog defo A7 A7 ad A8 we ge fo all [ : p p p p V V p L L λ A9 Ieqaly 3 a dec coeqece of A9 eqale 4 43 ad Coollay [36] wh ν N p p N a defed by A7 A8 ad appopae N a ad K c The eade hold oce ha f K β boded he K c boded a well Coeqely all hypohee of Theoem 3 hold wh : σ σ whch a fco of cla KL ha afe σ fo all The e of poof a coeqece of Rema 3 coco wh defo A7 A8 The poof complee <

25 Poof of Theoem 44: We wa o how ha all hypohee of Theoem 3 hold wh L : W V A3 Noce ha hypohe H3 of Theoem 3 a dec coeqece of eqale 4 43 defo A3 45 ad Coollay [36] Moeove hypohe H4 of Theoem 3 a dec coeqece of eqaly 4 wh q : a x fo all x R We ex how ha hypohee H ad H of Theoem 3 hold a well The poof co of wo ep: Sep : We how ha hypohee H H of Theoem 3 hold fo abay T x x C [ ]; R Sep : We how ha hypohee H H of Theoem 3 hold fo abay T x x C [ ]; R R M U M D ad R M U M D ad Sep : Code he olo x of 4 coepodg o abay M U M D wh abay al codo T x x C [ ]; R Clealy hee ex a mal exece me fo he olo deoed by By ve of Lemma A [] we ca gaaee ha he fco Q Q T Q Q T ae abolely coo fco o Le x V V T p θ [ ] [ Q θ W W T p Q θ ad θ [ ] L L T be mappg defed o [ Moeove le I [ be he zeo Lebege meae e whee x o dffeeable o x& f T x d By ve of 46 follow ha he followg mplcao hold fo \ I ad : [ Q p ζ p γ V Q ρ Q A3 ad by ve of 44 we have Q Q λ ζ p V A3 Lemma 35 [4] coco wh A3 mple ha hee ex a famly of coo fco of cla KL wh ~σ fo all ch ha fo all ad we have: ~ σ Q p ~ σ ζ p τ [ Q ; p ~ σ p γ V τ τ τ τ Moeove eqaly A3 decly mple ha fo all we have: Q Q [ τ ~ σ A33 λ ζ p p p V A34 Ug he fac ha ~σ fo all we oba fom A33 fo all ad : [ Q ~ σ Q γ p V ζ p A35 5

26 Le σ be fco of cla KL defed by σ fo all [ ] ad σ ~ σ fo all > Ug he fac ha V V T p Q θ we oba fom A35 fo all ad : [ θ [ ] V σ V γ p V ζ p A36 Smlaly g A34 ad he fac ha W W T p Q θ we oba fo all [ : W W λ θ [ ] ζ p p p V A37 Le σ : σ whch a fco of cla KL ha afe σ fo all I follow fom A36 ad defo A3 ha fo all ad we ge: [ V V γ p V ζ p A38 V σ L γ p V ζ p A39 Clealy eqale A39 how ha 3 hold wh Γ : R Γ Γ Γ wh Γ γ x fo all ad x R R Moeove ce aeme of Popoo 7 hold follow ha hypohe H of Theoem 3 hold a well Fally eqale A37 ad A38 mply ha he followg emae hold fo all : [ W ζ p λ W p V p γ p V A4 p ζ p Defe: V V γ p V ζ p A4 p : ζ p ζ fo all A4 x p x : γ γ p fo all x R A43 Combg emae A4 A4 ad explog defo A3 A4 ad A43 we ge fo all : [ L L λ p p V p V p p A44 Ieqaly 3 a dec coeqece of A44 eqale 4 43 ad Coollay [36] wh ν p N p N a defed by A4 A43 ad appopae a N ad β K boded he c K boded a well c K The eade hold oce ha f 6

27 Sep : Le x R C [ ]; R M U M D Ieqale A39 mply fo he olo x of 4 coepodg o M U M wh al codo T x x C [ ]; R ad fo all : whee Q MA { x Γ Γ Γ } D σ L Q ζ [ [ V MA Q A45 U [ Ug 4 43 A3 A44 ad A45 we oba fco ρ K a K ch ha he olo x of 4 coepodg o M U M D wh al codo T x x C [ ]; R ad fo all : [ Lemma 6 [3] ad A46 mply ha yem 4 RFC fom he p We ex clam ha eqale 3 ad 3 hold fo all T x a ρ x p A46 M U x R C [ ]; R M U M D ad The poof wll be made by coadco Sppoe o he coay ha hee ex x R C [ ]; R M U M D ad x x > ch ha he olo x of 4 wh al codo T coepodg o p M U M D afe β x d > whee β x d L ν c a : V σ x p [ V ] p [ L Γ [ V ] [ U [ [ ] [ ζ τ U [ ] Ug coy of he mappg x L x V ad coy of he olo of 4 wh epec o he al codo we ca gaaee ha he mappg x β x d coo Ug dey of C [ ]; R C [ ]; R coy of he mappg x β x d we coclde ha hee ex ~ x C [ ]; R ch ha: β Th we oba a coadco x d β ~ x d x d β Coeqely all hypohee of Theoem 3 hold wh σ : σ whch a fco of cla KL ha afe σ fo all The e of poof a coeqece of Rema 3 coco wh defo A4 A43 The poof complee < Poof of Theoem 48: We wa o how ha all hypohee of Theoem 3 hold wh whee L : W V A47 Q W : exp μ x A48 Noce ha defo A48 coco wh eqale 43 mply he followg eqaly fo all R R : exp μ a x g V V κ W x a x 3 4 7

28 Ug Coollay [36] we ca fd fco a ~ K η K ch ha a ~ 3 exp μ a fo all η Coeqely we oba ~ exp μ a3 a η fo all Noce ha hypohe H3 of Theoem 3 wh β a dec coeqece of pevo eqale 4 ad defo A47 A48 Moeove hypohe H4 of Theoem 3 a dec coeqece of eqaly 4 wh q : a x fo all x R Code he olo x of 4 de hypohee R-4 coepodg o abay d d ~ M U M D M R wh abay al codo x R Noce ha ce yem 4 aoomo ee [8] ffce o code he cae By ve of Popoo 5 [8] hee ex a mal exece me fo he olo deoed by Le V V W W L L abolely coo fco o [ Moeove le π : { τ τ } be he e of amplg me whch may be fe f < ad p : { τ π : τ } q : m{ τ π : τ } Le I [ be he zeo Lebege meae e whee x o dffeeable o whee x& f τ d τ Clealy we have φ p p ; P d P fo all whee P p [ P d p Nex we how ha he followg mplcao hold fo \ I ad : [ V ζ p γ p V V & ρ V A49 p p I ode o pove mplcao A49 le [ \ I τ p ad ppoe ha V ζ p γ p V By ve of he emgop popey fo he pevo p p ζ τ ζ P V γ V φ τ ; P d P V fo all eqaly mple ha [ τ ] ad I h cae by ve of defo 46 ad he fac ha τ h τ τ follow ha τ A h τ τ Sce x & f τ d τ we coclde fom 45 ha V& V ρ Lemma 35 [4] mple ha hee ex a famly of coo fco σ fo all ch ha fo all ad we have: [ σ of cla KL wh V σ V ; p σ γ τ p V τ ; p σ ζ p τ p τ τ p τ τ τ A5 Le σ : σ whch a fco of cla KL ha afe σ fo all Ieqale 3 wh R Γ : R Γ Γ Γ wh Γ γ x fo all ad x R ae dec coeqece of he pevo defo emae A5 defo A47 ad he fac ha σ fo all ad Moeove ce aeme of Popoo 7 hold follow ha hypohe H of Theoem 3 hold a well Explog 44 ad defo A48 we ge fo \ I : [ W & W λ ζ p p p V p p A5 Ieqaly A5 decly mple ha fo all we have: [ 8

29 W W ζ p p p V λ A5 Moeove eqale A5 ad A5 mply ha he followg emae hold fo all : [ W ζ p λ W p V p γ p V p ζ p A53 V V γ p V ζ p A54 Defe: p : ζ p ζ fo all A55 x p x : γ γ p fo all x R A56 Combg emae A53 A54 ad explog defo A55 A56 ad A47 we ge fo all : [ L L p p V p V λ p p A57 Ieqaly 3 a dec coeqece of A57 eqale 4 43 wh ν c p N p N a defed by A55 A56 ad appopae a N Coeqely all hypohee of Theoem 3 hold wh σ : σ whch a fco of cla KL ha afe σ fo all The e of poof a coeqece of Rema 3 coco wh defo A55 A56 The poof complee < Refeece [] Agel D P De Leehee ad E D Soag A Small-Ga Theoem fo Almo Global Covegece of Moooe Syem Syem ad Cool Lee [] Agel D ad A Aolf A Tgh Small-Ga Theoem fo o ecealy ISS Syem Syem ad Cool Lee [3] Dahovy S B S Rffe ad F R Wh A ISS Small-Ga Theoem fo Geeal Newo Mahemac of Cool Sgal ad Syem [4] Deoe C ad M Vdyaaga Feedbac Syem: Ip-Op Popee Academc Pe New Yo 975 [5] Eco G A ad E D Soag Global Aacvy I/O Moooe Small-Ga Theoem ad Bologcal Delay Syem Dcee ad Coo Dyamcal Syem [6] Ge L Ip-o-Sae Dyamcal Sably ad Lyapov Fco Chaacezao IEEE Taaco o Aomac Cool [7] Hll D J A geealzao of he mall-ga heoem fo olea feedbac yem Aomaca [8] Io H ad Z-P Jag Neceay ad Sffce Small-Ga Codo fo Iegal Ip-o-Sae Sable Syem: A Lyapov Pepecve o appea IEEE Taaco o Aomac Cool [9] Jag ZP A Teel ad L Paly Small-Ga Theoem fo ISS Syem ad Applcao Mahemac of Cool Sgal ad Syem [] Jag ZP IMY Maeel ad Y Wag A Lyapov Fomlao of he Nolea Small-Ga Theoem fo Iecoeced Syem Aomaca

30 [] Jag ZP ad IMY Maeel A Small-Ga Cool Mehod fo Nolea Cacaded Syem wh Dyamc Uceae IEEE Taaco o Aomac Cool [] Jag ZP ad Y Wag A Covee Lyapov Theoem fo Dcee-Tme Syem wh Dbace Syem ad Cool Lee [3] Jag ZP Y L ad Y Wag Nolea Small-Ga Theoem fo Dcee-Tme Feedbac Syem ad Applcao Aomaca [4] Jag ZP ad Y Wag Ip-o-ae ably fo dcee-me olea yem Aomaca [5] Jag ZP ad Y Wag A Geealzao of he Nolea Small-Ga Theoem fo Lage-Scale Complex Syem Poceedg of he 7 h Wold Coge o Iellge Cool ad Aomao Chogqg Cha [6] Kaafyll I ad J Ta No-Ufom Tme ISS ad he Small-Ga Theoem IEEE Taaco o Aomac Cool [7] Kaafyll I The No-Ufom Tme Small-Ga Theoem fo a Wde Cla of Cool Syem wh Op Eopea Joal of Cool [8] Kaafyll I A Syem-Theoec Famewo fo a Wde Cla of Syem I: Applcao o Nmecal Aaly Joal of Mahemacal Aaly ad Applcao [9] Kaafyll I A Syem-Theoec Famewo fo a Wde Cla of Syem II: Ip-o-Op Sably Joal of Mahemacal Aaly ad Applcao [] Kaafyll I ad Z-P Jag A Small-Ga Theoem fo a Wde Cla of Feedbac Syem wh Cool Applcao SIAM Joal Cool ad Opmzao [] Kaafyll I C Kava L Syo ad G Lybeao A Veco Lyapov Fco Chaacezao of Ipo-Sae Sably wh Applcao o Rob Global Sablzao of he Chemoa Eopea Joal of Cool [] Kaafyll I P Pepe ad Z-P Jag Global Op Sably fo Syem Decbed by Readed Fcoal Dffeeal Eqao: Lyapov Chaacezao Eopea Joal of Cool [3] Kaafyll I P Pepe ad Z-P Jag Ip-o-Op Sably fo Syem Decbed by Readed Fcoal Dffeeal Eqao Eopea Joal of Cool [4] Kaafyll I ad C Kava Global Sably Rel fo Syem de Sampled-Daa Cool o appea he Ieaoal Joal of Rob ad Nolea Cool [5] Lahmaham V V M Maoov ad S Svadaam Veco Lyapov Fco ad Sably Aaly of Nolea Syem Klwe Academc Pblhe Dodech 99 [6] Maeel IMY ad DJ Hll Moooe Sably of Nolea Feedbac Syem Joal of Mahemacal Syem Emao ad Cool [7] Mazec F ad S-I Nclec Lyapov Sably Aaly fo Nolea Delay Syem Syem ad Cool Lee [8] Mchel AN ad RK Mlle Qalave Aaly of Lage-Scale Dyamcal Syem Academc Pe New Yo 977 [9] Nec D A R Teel ad D Caevale Explc Compao of he Samplg Peod Emlao of Coolle fo Nolea Sampled-Daa Syem o appea IEEE Taaco o Aomac Cool [3] Nclec SI Delay Effec o Sably A Rob Cool Appoach Hedelbeg Gemay Spge-Velag [3] Paly L ad Y Wag Sablzao pe of Mached Umodeled Dyamc ad a Eqvale Defo of Ip-o-Sae Sably Mahemac of Cool Sgal ad Syem [3] Smh H L Moooe Dyamcal Syem A Iodco o he Theoy of Compeve ad Coopeave Syem Mahemacal Svey ad Moogaph Volme 4 AMS Povdece Rhode Ilad 994 [33] Soag ED Smooh Sablzao Imple Copme Facozao IEEE Taaco o Aomac Cool [34] Soag ED ad Y Wag O Chaacezao of he Ip-o-Sae Sably Popey Syem ad Cool Lee [35] Soag ED ad Y Wag "New Chaacezao of he Ip-o-Sae Sably" IEEE Taaco o Aomac Cool [36] Soag ED Comme o Iegal Vaa of ISS Syem ad Cool Lee [37] Soag ED ad Y Wag Noo of Ip o Op Sably Syem ad Cool Lee [38] Soag ED ad B Igall A Small-Ga Theoem wh Applcao o Ip/Op Syem Icemeal Sably Deecably ad Iecoeco Joal of he Fal Ie [39] Tabada P Eve-Tggeed Real-Tme Schedlg of Sablzg Cool Ta IEEE Taaco o Aomac Cool [4] Teel A A Nolea Small Ga Theoem Fo he Aaly of Cool Syem Wh Saao IEEE Taaco o Aomac Cool

31 [4] Teel AR Coeco bewee Razmh-Type Theoem ad he ISS Nolea Small Ga Theoem IEEE Taaco o Aomac Cool [4] Teel AR Ip-o-Sae Sably ad he Nolea Small Ga Theoem Pep 5 [43] Zame G O he Ip-Op Sably of Tme-Vayg Nolea Feedbac Syem Pa I: Codo g cocep of Loop Ga Cocy ad Povy IEEE Taaco o Aomac Cool [44] Zho K J C Doyle ad K Glove Rob opmal cool New Jeey: Pece-Hall 996 3