ON THE RELATION OF DELAY EQUATIONS TO FIRST- ORDER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

Transcription

1 O THE ELATIO OF DELAY EQUATIOS TO FIST- ODE HYPEBOLIC PATIAL DIFFEETIAL EQUATIOS Io Kfyi * d Mio Ki ** * Dee of Mei io Tei Uieiy of Ae Zogfou Cu 578 Ae Geee ei iok@e.u.g ** De. of Mei d Aeoe Eg. Uieiy of Cifoi S Diego L Jo CA U.S.A. ei ki@ud.edu Ab Ti e ebie e equiee bewee ye deibed by ige fi-ode yeboi i diffeei equio d ye deibed by ieg dey equio. Sye-eoei eu e oided fo bo e of ye og e oee Lyuo eu. Te ooed fewok ow e udy of dioiuou ouio fo oie ye deibed by ige fi-ode yeboi i diffeei equio ude e effe of eube iu ig o e boudy d/o o e diffeei equio. A iuie ee ow e oeio of ye deibed by ige fi-ode yeboi i diffeei equio o ieg dey ye iify oideby e ouio of e oeodig obu feedbk biiio obe. Keywod ieg dey equio fi-ode yeboi i diffeei equio oie ye.. Ioduio Te eio of fi-ode yeboi Pi Diffeei Equio PDE wi dey equio i we kow. I y e ye of Fi-Ode Hyeboi PDE FOH-PDE be foed o ye deibed by eded Fuio Diffeei Equio ee [893] O e oe d e ee wok [34] e ow ye of fi-ode yeboi PDE be uiied fo e biiio of dey ye. Howee ee wok o e oo of ye deibed by FOH-PDE ee [ ] e ed o e udy of o-dd ye of FOH-PDE wi e foowig feue i e boudy odiio e gie by fuio of e woe e ofie d ii e diffeei equio ioe fuio of e woe e ofie d o oy oi ue of e e. I oud be oed fo u ye ee i o ibe eiee-uiquee eoy ii o e eoy of dd FOH-PDE ee [54] e eee e uiied foio gue wi guee io yeeoei oeie. Moeoe e eiee of fuio of e woe e ofie i e ei deiio of u ye do o ow e we o e foowig queio

2 W of dey ye be ued fo e deiio of o-dd ye of FOH-PDE? I i wok we we e boe queio fo of ye deibed by ige FOH- PDE. Fi we ow e oie of dey ye i wi bee ey udied i.e. ye of e fo f w w W. wee W i o-ey oy o e wi U > L [ ; w L [ ]; W e defied by fo [ w w fo [ ] d f L [ ; L [ ]; W i ig. We e boe of dey ye ye deibed by Ieg Dey Equio IDE. Sye of e fo. e bee udied i [9678]. I oud be oed i oi ye deibed by IDE e bee uiied fo og ie i e biiio of fiie-dieio oo ye wi iu dey ee [34] eey edio feedbk i ye deibed by IDE woe iu i e e of e fiie-dieio oo ye. Ideed edio feedbk fo e fiiedieio ye y g y y k wee we ue o deoe e oo iu d eu i i feedbk w of e fo wee u L o f y u ; deoe e oo uo eo L o ; deoe e euee eo. Cey e boe oed-oo ye i e feedbk ieoeio of ie-ii ye deibed by ODE d ieg dey ye of e fo.. Teefoe e udy of ye deibed by IDE i io o i ow. Io ye-eoei oeie fo e of ye deibed by IDE e oided i Seio of e ee e. Seody we ow of ye deibed by o-dd ige FOH-PDE ey ye of e fo g K w fo > i... i i fo >. d....3 G w fo.4 wee > i iege w W W i oy o e > i o g g... g G W L ]; K i W L ]; i... e fuio g i i... e uffiiey egu fuio d i e e ofie i.e. fo d [] e equie o ye deibed by IDE.

3 Te equiee ow e deeoe of io ye-eoei eu fo e of ye deibed by..3 d.4 Seio 3 of e ee e. Te obied eu ow e udy of eube iu w W wi i io feue beue i y e boudy odiio of e fo.4 eu fo e ieeio of biiig feedbk w ee [345]. Howee e oo io oe ogee wi oo uo eo d euee eo wi e yiy odeed by eube iu. Fiy e obied eu ow e deeoe of eodoogy fo e ouio of oo obe fo ye deibed by o-dd ige FOH-PDE ey ye of e fo Te eodoogy i eeed i Seio 4 of e ee e by e of iuie ee. Te ee ow y io feue of e ooed eodoogy d oio i de wi oe eiig eodoogie. oio. Tougou i e we do e foowig oio [. Le I be ie d U be e. By L I; U we deoe e e of eube d eeiy bouded fuio u defied o I d kig ue i U. By L I; U wee [ we deoe e e of eube fuio u defied o I d kig ue i U fo wi u d < L o I; U we deoe e e of eube d I. By oy eeiy bouded fuio u defied o I d kig ue i U i.e. fo eey o ie J I i od u L J; U. Fo eey L I; wee I i ie we deoe by e eei ueu of i.e. u L I; wee Fo eo I i ie wi [ b I b. Fo I < we defie u [ b. < b we deoe by i oe d by i Euide o.. Sye Deibed by Ieg Dey Equio We oide ye. ude e foowig uio H Tee ei o-deeig fuio M u fo eey > d fo eey y L [ ; w L [ ]; W wi y w e foowig iequiie od f w f y w H Fo eey > fo [ u < L ;.e. d f w y M u < y fo. f w. [ w L [ ]; W e fuio [ defied by fo [.e. ifie L ; [. 3

4 Sye of e fo. eee ieg dey equio of e fo w q w i i i i wee > <... e o i W i... e oy Lii ig W i o-ey oy o e d q W i oy Lii ig. I be ow uio H H od fo i of ye. Te f uio H od fo i of ye i die oequee of Le o ge 4 i [6]. Te foowig eu ow u ye. be egded ie-ii ye wi iu woe e e i L [ ; d woe iu e i L [ ]; W. Sye. ifie e i eigou oey e Boudede-Iie-Coiuio BIC oey d e oey of Lii deedee o e iii odiio ee []. Teoe. Coide ye. ude uio H H. Te fo eey L [ ; w L o [ ; W ee ei w ] wi > 5 u f w fo w d uique ig L ; d o [ ifyig [.e. d fo [.e.. Moeoe if < e i u. Fiy ee ei o-deeig fuio P G u fo eey L [ ; y L [ ; w L o [ ; W i od y G e P y fo [.3 i u u y u w wee w y w o [ w y L o [ y w; L ; d e e uique ig ifyig [ w.e. y f y w fo [ y u.e. d [.e.. f w fo y y fo w L o [ ; W. Defie u u d ee wi. Defie fo [.e. d fo [ Poof Le biy L [ ; biy Cey L [ ; wi u. <. Wiou o of geeiy we y ue e o-deeig fuio ioed i. ifie fo. Le be u < fo 5.4 Uig iduio d uio H we guee e equee of fuio k [ fo k defied by 4

5 5 k fo [.e. d k k w f fo [.e. fo k ifie k L ; [ fo k. oie. i ojuio wi e f < u d u u iie u < d u <.5 We e i 5 u... k < fo k.6 Te oof of e i i de by iduio. We fi oie by iue of.5 e i i ue fo k d k. e uoe e i i ue fo ei k. oie by iue of. d e f 5 e foowig iequiy od fo k... d fo o [ [ u u u u M M w f w f < < < <.7 Sie.7 od fo o [ d ie fo o [ d fo we ge [ [ fo k....8 Iequiy.8 iie [ [ fo... k.9 Uig e ige iequiy eeedy d e f / oequee of.4 we ge fo k... [ [ [ [ [. I foow fo.5 d.

6 u < k k [ 5. d eefoe we e oed.6. Te eiee of L ; [ ifyig f w fo [.e. d fo [.e. foow oey e oof of B fied oi eoe o e oed e L [ ; B wee B deoe e oed b of diu 5. Sie.9 od fo e equee k L [ ; B i Cuy equee. Teefoe ee ei uique ii L [ ; B. Auio H iequiy. iie e ig d ide of e iequiy f w k f w f k w be de biiy fo uffiiey ge k d fo o [ d oequey we obi f w fo [.e.. I ode o oe uiquee we ue odiio gue. Suoe ee ei L [ ifyig y f y w fo [.e. d y fo [.e. wi y. Le be e e ue boud of [ wi u y. A y ; u > < < odiio gue ow <. Sie [ i e e ue boud of [ wi u y i foow u y > fo. A odiio < < gue ow u y. Defie u y u u w. I < < < foow fo. fo o d y f w f y w u y <. Iequiy. iie u y u y u y < < u y < u y < fo < < < we obi u y < i < odiio. Sie e uique fuio L ; fo ei > fo fo d ie < i i.e. [ ifyig f w fo [.e. d [.e. i bouded i foow by eeig e gue boe ee L [ ; ifyig f w fo [.e. d [.e.. Moe eifiy by iue of.4 > ifie d uique fuio fo 5 u u w <.3 6

7 Le w ] be e e ue boud of > fo wi ee ei fuio L [ ; ifyig f w fo [.e. d fo [.e.. If < i.e. i fiie e fo eey > ifyig f w fo [ < o [ ε ee ei ε.e. d d fuio L [ ; > fo [ ee ei fuio L ;.e. d fo [.e.. o.e.. I foow i y e [ ifyig f w fo Fo e e < we oie.3 iie e iequiy 5 u o < u w 5 u u ε w fo ε < ε ε > d > ε. Te iequiy 5 u u w o od fo < ε ε > if u <. Teefoe if < e i u. < Fiy we oeed o e ouie oof of eie.3. Le L [ ; L [ ; w L o [ ; W d e L [ w; y L [ y w; y e uique ig ifyig f w o fo w o be [.e. y f y w fo [.e.. Defie y w.e. d y y fo [ w y w. Fo we defie u u y u w d i wee d i i. Iequiy. iie fo o [ i deoe e iege of e e y f w f y w u q y q M y q<.4 oie ie / we ge fo.4 u q< q y q M y.5 Uig iduio d ii gue we oe e foowig iequiy od fo i... u q< i q y q M i y.6 Sie fo i we obi fo.6 i u q< q y q i M e M y.7 7

8 Iequiy.7 diey iie eie.3 wi G M d M P. Te oof i oee. Hig oed Teoe. we e juified fo biy L [ ; o e e uique ig L [ ; ifyig f w w L o [ ; W o fo [.e. fo [.e. fo ei w ] e ouio of. wi iii odiio fo [.e. oeodig o iu w L o [ ; W wiou eii efeee o e i eiee ie of e ouio w ]. A ube of ye-eoei oeie be oed by uig e eu of Teoe.. Moe eifiy we udy ye. ude uio H H d e foowig uio H3 w d u wee d D i o e u U i oy o e wi U d. Moeoe ee ei K L [ D u L [ ]; U e foowig iequiy od d ]; b u fo eey L ; [ f d u b u u.8 Auio H3 e e ye' ig-d ide i uifoy bouded wi ee o e diube d d e ig-d ide i eo we e e d iu u e eo ieeie of e diube d. Ti e i iu e oigi i equiibiu we e iu u i eo ieeie of e ue of e diube d wi eude fo ee ye e ddiie diube i e ode' ig-d ide. I Ee.7 we oide ie ye ue oe of e eee of uio H3 d e ye' deedee o d. We e edy o gie i of oeie fo ye. e deied fo e eu of Teoe.. Poey obue of e equiibiu oi. Auio H3 guee L [ ; i equiibiu oi fo ye. we u i eo d fo y d. Howee i ode o udy e obu biiy oeie of e equiibiu oi of. we eed oge eio ey L [ ; i obu equiibiu oi fo e ye. wi iu u U ee []. Teoe. Coide ye. ude uio H H H3. Te fuio L [ ; i obu equiibiu oi fo e ye. wi iu u U i.e. fo ε > d T ee ei ε T > u fo eey L [ ; d L [ ; D u L o [ ; U wi u u < ee ei d u T ] d uique ig L ; [.e. d < ε o [ ifyig f d u fo [ T ]. fo.e. fo [ 8

9 Poof Sie D i o we e i e oiio o defie Q d. Wiou o of geeiy we y ue e fuio b K ioed i.8 ifie b fo. Le biy ε > T be gie d defie ε T κ ε / g g... T κ g ρ wee g 5b fo T ρ ie d dd ρ 5b Q ε.9 T i e iege of d i e fuio ρ ioed i.. Le biy L [ ; d L [ ; u u < d oide e uique ig L o ; D u L o [ ; U wi f [ ifyig d u fo [.e. fo [.e. fo ei d u ]. Uig.8 e f u u < e f ε die oequee of defiiio.9 e f Q d dd d oeedig ey i e oof of Teoe. we ow Fo wi ε [ i od g fo ]. [ ρ i We e i iρ g fo i... iρ i e o fo iρ. Te i i die oequee of oey. d defiiio.9. Teefoe we ge κ fo [ ρ]. Sie T ε T ρ die oequee of e f ; e.9 d ρ κ / e.9 we obi < ε fo [ T ]. Te oof i oee. Poey obu Gob Ayoi Sbiiy d Iu-o-Se Sbiiy Uig e eu oied i [] we e i oiio o defie e oio of Uifo obu Gob Ayoi Sbiiy d Iu-o-Se Sbiiy ISS fo ye. ude uio H H H3. Te oio of ISS fo ye deibed by IDE i oeey ogou o e oeodig oio iodued by E. D. Sog fo fiie-dieio ye i [5]. Defiiio.3 Coide ye. ude uio H H H3 d ue fo eey L [ ; d L [ ; D u L o [ ; U ee ei L o [ ; ifyig f d u fo [.e. fo [.e.. We y ye. i Iu-o-Se Sbe ISS fo e iu u U uifoy i d D if ee ei oiuou o-deeig fuio γ u e foowig oeie od obu Lgge Sbiiy Fo eey ε > i od u ε d L [ ; D u L [ U < u u γ o ;. obu Lyuo Sbiiy Fo eey ε > ee ei ε > u u d L [ ; D u L [ U ε u u γ o ; < 9.

10 Uifo obu Aiiy Fo eey ε > d ee ei ε > u If {} GAS fo.. u d L [ ; D u L [ U ε u u γ o ; < U e we y L [ ; i Uifoy obuy Goby Ayoiy Sbe Le. Le 4. i [] d Teoe. i [] gie u e foowig eu. Teoe.4 Coide ye. ude uio H H H3. Te e foowig ee e equie Sye. i ISS fo e iu u U uifoy i d D. b Tee ei σ KL d oiuou o-deeig fuio γ u fo eey L [ ; d L [ ; D u L o [ ; U ee ei L o [ ; ifyig f d u fo [.e. fo [ σ γ u fo..e. d. u Fueoe if U {} e e foowig ee i equie o ee b Sye. i obuy Fowd Coee i.e. fo eey T d i od u{ [ T ] d L [ ; D } < wee deoe e ouio of. wi fo [.e. oeodig o iu d L [ ; D d e oey of Uifo obu Aiiy of Defiiio.3 od. Poey 3 Lyuo Ceiio of GAS Teoe 3.4 i [] d e eu of Teoe.. ow u o obi oee Lyuo eiio fo e GAS oey fo ye.. Teoe.5 Coide ye. ude uio H H H3 d ue {} L [ ; i GAS fo. if d oy if ee ei U. Te equiibiu oi fuio V L [ ; o-deeig fuio Q d fuio K u e foowig iequiie od V V fo L ; Moeoe fo eey L [ ; V y Q y y fo y L [ ; [.. d L [ ; D e ouio of. wi iii odiio fo [.e. oeodig o iu d L [ ; D ifie e V fo.3 V

11 Iequiy. guee e fuio V L [ ; i Lii o bouded e of e e e L [ ;. Howee iequiy. doe o guee Fee diffeeibiiy of e fuio V L [ ; o e ii i V V ei fo e ouio of. wi iii odiio fo [.e. oeodig o iu d L [ ; D. oie iequiy.3 guee fo eey L [ ; d L [ ; D e ouio of. wi iii odiio fo [.e. oeodig o iu d L [ ; D ifie iu V V V Poey 4 Suffiie Codiio fo Sbiiy Poeie. Teoe.5 i o e o oeie wy of oig GAS o ISS fo.. Fo i uoe we ue e foowig eu wi i eeio of e i uiki eoe fo ie dey ye ee [8]. Teoe.6 Coide ye. ude uio H H H3. Aue ee ei oiuou oiie defiie d diy ubouded fuio W oiuou o-deeig fuio γ d o λ u e foowig iequiy L [ ; d L [ ]; D u L [ ]; U od fo W f d u λ u W γ u < Te ye. i ISS fo e iu u U uifoy i d D..4 Poof Defie e fuio V u e σ W fo L [ ;.5 < wee σ i o. Sie W i oiuou oiie defiie d λ diy ubouded fuio ee ei fuio K u W fo.6 Uig.5 d.6 we oude e foowig iequiy od V fo L ; wee σ fo e Le biy L [ ;. [.7 d L [ ; D u L [ ; U d oide e uique ouio of. wi iii odiio fo [.e. oeodig o iu d L [ ; D u L o [ ; U. Le biy d u wee d u > i e i eiee ie of e ouio d e d u. Uig defiiio.5 we ge

12 V < e e < u u e e σ W σ W u e σ σ u e σ W u e σ < σ V u W < Uig.4 we ge fo o [ W q W q < < W W f d u λ u W u W γ u q q q q< < q.8.9 wee u u u. Defiiio.5 d iequiy.9 gie u W q λ e q< < wi diey iie u W q λ e σ V γ u λ q< σ V u W γ u.3 λ Sie λ e σ e σ fo σ oie λ.3 fo d u wi σ σ λ we ge fo.8 σ V γ u V e.3 λ Uig e Boudede-Iie-Coiuio oey iequiie.7.3 d dd odiio gue we oude d u. Teefoe we oude.3 od fo d wi. σ λ Uig iduio d.3 we ow V e iσ V γ u jσ iege i d σ λ ow V e σ q i V γ u d fo i i i j e λ fo. Coequey e eiou iequiy i ojuio wi.3 e σ q fo iege i λ e σ σ λ q. Sie fo eey ee ei iege i d σ λ q wi i q we obi σ λ λ e σ V e σ V γ u fo.3 λ λ e σ Iequiy.3 i ojuio wi.7 gie fo ε >

13 ε λ e σ u ε e σ γ fo.3 λ λ e σ Iequiy.3 ow ee ei σ KL d oiuou o-deeig fuio γ u fo eey L [ ; d L [ ; D u L [ ; U ee ei L o [ ; ifyig f d u fo [.e. fo [.e. d σ γ u fo. A dd uiy gue ow e eie σ u γ u od fo d u L o [ ; U. Teefoe Teoe.4 iie ye. i ISS fo e iu u U uifoy i d D. Te oof i oee. Ee.7 Coide e ie ye deibed by e ige IDE d q d u d [ ] u.33 wee q [ ] i oiuou fuio. Uig Teoe.6 wi W we oude ye.33 i ISS fo e iu u ude e uio ee ei λ u of e iu q d λ. I f foowig e oof of Teoe.6 we ow e gi fuio u i ie. 3. Sye Deibed by Sige Fi-Ode Hyeboi Pi Diffeei Equio Coide e ye deibed by ige FOH-PDE of e fo..3.4 ude e foowig uio A Tee ei o-deeig fuio L σ u e fuio G L ]; K L ]; i... ify e foowig iequiie fo i y L u y G w G w y Ki w Ki w y L u i < < fo w W wi w d y L ];[ ] 3. fo G w i K w σ i w W wi w d L ];[ ] 3. 3

14 We ow e ow o fo ye..3.4 ude uio A o equie ye deibed by IDE. Coide i ouio of..3.4 wi uffiiey egu iii odiio fo ] oeodig o uffiiey egu iu w W. We ue e ouio ei fo J wee J i ie wi o-ey ieio d J. Fi we defie fo J G w 3.3 If we defie q e q q d q q e q q d g q d fo o q ] 3.4 e iegio o e eii ie ow u o oude e ouio of..3.4 u ify e foowig equio fo J d [] e d e dg d Le L [ ; be e ig defied by fo J [ e L [ ; be e ig defied by fo J [ equio 3.5 ow u o defie fo eey L [ ; A L [ ; L ]; d ig B L [ ; L ]; foowig equio od fo wee J A B 3.5. Siiy. Cey bouded ie oeo o e 3.6 A e q q dq B e q q dq g d I foow fo 3.3 d 3.6 e foowig equio od fo F w fo i... i i wee e ig i W L [ ; w W L [ ; by e equio F w K w A B i i fo ] 3.6b J d F w 3.7 F... i e defied fo eey fo i... d F w G w A B 3.8 4

15 I oe wod eey i ouio of..3.4 oeod o ouio L [ ; of e ye deibed by e ieg dey equio Te oedue deibed boe i igoou d e ye deibed by e IDE i equie o e ye deibed by e ige FOH-PDE Ti i ow by e foowig eoe. Teoe 3. Aue C []; i oy Lii d g i C []; i.... Moeoe ue ee ei o-deeig fuio L σ u e fuio G W L ]; K i W L ]; i... ify iequiie fo. Fiy ue e foowig uio od fo e ig F W L [ ; wee F w F w... F w d F W L [ ; i... e defied by 3.8 wi e id of 3.6b. i A Fo eey > L [ ; w L [ ]; W eey fuio ξ [ defied by ξ fo [.e. d ξ F w fo [.e. ifie ξ L [ ;. Te fo eey L [ ]; w L o [ ; W ee ei w ] d uique oy bouded ig [ L []; ifyig fo [.e. L o [ d wi ; e e dg d d fo fo > fo [ d o []. Te ig L []; i obied fo 3.5 d e ouio of e ye 3.7 wi biy iii odiio L [ ; ifie 3.4. Moeoe e ig L [ ]; i oiuou o [ fo eey [ d if < e i u u. Fiy ee ei o-deeig fuio P Q u fo eey L [ ]; y L [ ]; w L o [ ; W i od 3.9 u y Q e P u y fo [ 3. i u u u u y u w d e < < L []; y L []; e e uique ig ifyig.3.4 wee w y w ig 3.3 fo [.e. wi L o [ ; 3.9 fo [ d o [] we e ogou equio wi y eig. 5

16 Te ede oud oie o eguiy eu i oided fo e ig L [];. I i e e uique ig L []; ifyig fo [.e. wi L o [ ; 3.9 fo [ d o [] i o i ouio of e iii ue obe..3.4 wi iii odiio fo [].e. wee L [ ];. Te ig L []; ifyig fo [.e. wi L o [ ; 3.9 fo [ d o [] i oe o e oio of e id ouio fo fi-ode ye of yeboi PDE ee [7] d efeee eei. Poof Te oof of e eoe i og iooig ee dii eu o we diide i i fou e. Se We ow ye 3.7 ifie uio H H. Teefoe by iue of Teoe. fo eey iu w L o [ ; W d fo eey iii odiio L [ ; ifie 3.4 fo ei L [ ]; ye 3.7 di uique ouio. Te obied ouio of ye 3.7 gie u diey by e of 3.4 d 3.5 oy bouded ig [ L []; ifyig fo [.e. L o [ d 3.9 fo d o []. wi ; [ Se We ow fo eey w L o [ ; W L [ ]; ee ei o oe oy bouded ig [ L []; ifyig fo [.e. wi L o [ ; d 3.9 fo [ d o []. Teefoe e ig obied i Se i e uique ig ifie fo [.e. wi [ ; L o d 3.9 fo [ d o []. Ti e fo eey iii odiio L [ ; ifie 3.4 fo e gie L [ ]; e ouio of ye 3.7 wi be e e. Se 3 Mkig ue of e eu of Teoe. we ow if < e i u u d ee ei o-deeig fuio P Q u fo eey [ ]; y [ ]; w L o [ ; W eie 3. od. L L Se 4 We ow fo eey w L o [ ; W L [ ]; e oy bouded ig [ L []; ifyig fo [.e. wi L o [ ; d 3.9 fo [ d o [] gie u ig L [ ]; wi i oiuou o fo eey [. [ Se Auio A guee uio H od fo ye 3.7. Likewie e idiy of uio H fo ye 3.7 i die oequee of uio A. To ee i e L [ ; L [ ; wi be gie d e L ]; be defied by A B A B wee A L [ ; L ]; d B L [ ; L ]; e e ig defied by 3.6b i.e. 6

17 7 e e d g d d fo ] [.e. 3. e e d g d d fo ] [.e. 3. Defie [] M d [] g C 3.3 d oie 3. d defiiio 3.3 iy M C e u u 3.4 Uig 3. defiiio 3.8 d 3.4 we obi M C w F e σ fo ; [ L W w wi d w 3.5 Iequiy 3.5 ow iequiy. od fo e o-deeig fuio M C e σ fo oie defiiio 3.3 iie M i o-deeig. Uig 3. defiiio 3.8 d 3.4 we obi u u K L K L w F w F < < fo W w wi w 3.6 wee M C K e. Defie ] [ u L d oie < L fo ie []; C i ued o be oy Lii. Uig d defiiio 3.3 we obi fo ] [.e. d B B 3.7 wee e M B e 3 M L C C L B. Cobiig 3.6 d 3.7 we obi fo W w wi w

18 F w F w L K B L K B L K B L K B u < u u < < u < 3.8 Iequiy 3.8 ow iequiy. od fo oie o-deeig fuio M. Le biy L [ ]; d w L o [ ; W. Coide ow e ouio of ye 3.7 oeodig o iu w L o [ ; W wi iii odiio L [ ; wi d 3.4 i.e. e d fo o [ 3.9 Teoe. guee ee ei w ] o ye 3.7 di uique ouio Lo [ ;. Uig 3.5 d 3.9 we obi oy bouded ig [ L []; wi ifie fo [.e. d 3.9 fo d o []. [ Se We ow by odiio fo eey w L o [ ; W L [ ]; ee ei o oe oy bouded ig [ L []; ifyig fo.e. wi [ L o [ d 3.9 fo d o [] ; [. Suoe ee ei w L o [ ; W L [ ]; o > d wo oy bouded ig [ L []; [ L []; o 3.9 od fo [ d o [] equio e e d g d fo d fo > 3. fo [ d o [] od fo [.e. equio i Ki w fo i... d od fo [.e. d u G w 3. 8

19 u [ [] u > 3.3 Le [ be e gee owe boud of ie [ fo wi u >. oie by iue of 3.3 [ i we defied d u u fo e e >. [] We ow e ee ei > o u u [] [ [ [] wi odi e f [ i e gee owe boud of ie [ fo wi u >. [] A io oey i ued i oi i e o-ed i eigou oey ee [] i.e. e f eey oy bouded ig [ L []; ifyig fo [.e. wi L o [ ; d 3.9 fo [ d o [] ifie e foowig ideiy fo d fo [ ] e e [ d g d fo d fo > fo [].e. 3.4 Teefoe ideiy 3.4 od fo [ d fo [ ]. Siiy e foowig ideiy e d g d fo e d fo > fo [].e. 3.5 od fo [ d fo [ ]. Uig 3.4 d 3.5 wi fo e e > d fo e e i ojuio wi wi iie fo [ ] we obi e 3 e M C C L L M fo fo > u 3.6 9

20 fo ] [.e. d fo wee u u u w ] [ u L Uig d 3. we obi u L fo o 3.7 u u L L < fo o i d < 3.8 We defie fo i Δ < < < u u u u E 3.9 Uig d defiiio 3.9 we obi fo i E Δ Γ Δ 3.3 E E E Δ Γ Γ Δ Γ 3.3 wee e 3 L L C L C M Γ e L M Γ. Uig 3.3 d uig Γ Γ we ge E Δ Γ. I i e iequiy 3.3 iie Δ Γ Γ Δ wi gie Δ E fo Γ Γ Γ i 3.3 Iiio 3.3 defiiio 3.9 d e f u u [] ] [ iy u u [] ] [. Sie ] i biiy oe o we ge u u [] [. Tu e oy bouded ig []; [ L wi ifie fo [.e. d 3.9 fo [ d o ] [ wi w foud i Se i uique.

21 Se 3 I i e we ow if < e i u u d ee ei odeeig fuio P Q u fo eey L [ ]; y L [ ]; w L o [ ; W eie 3. od. Ideed if < Lo [ ; of 3.7 oeodig o iu w L o [ ; W wi iii odiio L [ ; ifyig d 3.9 e Teoe. guee iu. By iue of iequiy 3. d equio fo e uique ouio iu o e ue i u u. Teoe. guee ee ei o-deeig fuio P G u fo eey L [ ; L [ ; w L o [ ; W i od G e P fo [ 3.33 wee i w w u u u w d Lo [ w; Lo [ w; e e uique ouio of 3.7 oeodig o iu w L o [ ; W wi iii odiio L [ ; d L [ ; eeiey. Le L [ ]; L [ ]; w L o [ ; W d oide e oy bouded ig [ w L []; [ w L []; wi ify fo [.e. d 3.9 fo [ d o [] we e ogou equio wi eig d eig. Te ig [ w L []; [ w L []; e foud by obiig e ouio Lo [ w; Lo [ w; of 3.7 wi iu w L o [ ; W d iii odiio L [ ; L [ ; eeiey d by uig d e d fo o [ 3.34 Iequiy 3. fo oie o-deeig fuio P Q i die oequee of 3.6 wi d 3.. Se 4 I ode o ow e ig L [ ]; i oiuou o [ fo eey [ i uffie o ue e i eigou oey ideiy 3.4 d o ow fo eey [ ε > ee ei > u d ε fo [ ] 3.35

22 Sie C []; i dee i L []; fo eey [ fo eey ε > d [ ]; ee ei [ ]; wi d ε / 3 d u. Moeoe ee C ei u ε u u 3 u. Uig e iequiy L d / d / d / d / we obi d ε fo [ ]. Coequey fo eey [ fo eey ε > d [ ]; ee ei u u d ε L. Uig e ideiy d d d fo [ ] we ge wee d u / u w ge fo i / / ] d o ] [ d fo i / / ] 3.36 [. Moeoe uig 3.9 d defiiio 3.3 we M C M e 3.37 Cobiig 3.36 d 3.37 we obi fo i / / ] Le [ M C M d e d 3.38 be u u d ε. I foow fo e eiou iequiy i ojuio wi od wi / ε e M ε i. 3 C M 3 3 Te oof i oee.

23 Hig oed Teoe 3. we e juified fo biy L [ ]; w L o [ ; W o e e uique ig [ L []; ifyig fo [.e. wi [ ; L o d 3.9 fo [ d o [] e ouio of..3.4 wi iii odiio L [ ]; oeodig o iu w L o [ ; W wiou eii efeee o e i eiee ie of e ouio w ]. A ube of ye-eoei oeie be oed by uig e eu of Teoe 3.. Moe eifiy we udy ye..3.4 ude uio A A d e foowig uio A3 w d u wee d D i o e u U i oy o e wi U d. Moeoe ee ei b K u fo eey L [ ]; d D u U e foowig iequiy od G d u K d u b u u i i 3.39 We e edy o gie i of oeie fo ye..3.4 e deied fo e eu of Teoe 3.. Poey obue of e equiibiu oi. Auio A3 guee L [ ]; i equiibiu oi fo ye..3.4 we u U i eo fo y d D. Howee i ode o udy e obu biiy oeie of e equiibiu oi of..3.4 we eed oge eio ey L [ ]; i obu equiibiu oi fo e ye..3.4 wi iu u U ee []. Teoe 3. Coide ye..3.4 ude uio A A A3. Te fuio L [ ]; i obu equiibiu oi fo e ye..3.4 wi iu u U i.e. fo ε > d T ee ei ε T > u fo eey L [ ]; d L [ ; D u L o [ ; U wi u u u < e uique ouio L []; of..3.4 wi iii odiio L [ ]; oeodig o iu d L [ ; D u L o [ ; U ifie u < ε fo [ T ]. Poof Te oof i die oequee of Teoe. ude uio A3 ye 3.7 ifie uio H3. Dei e ef o e ede. Poey obu Gob Ayoi Sbiiy d Iu-o-Se Sbiiy Uig e eu oied i [] we e i oiio o defie e oio of obu Gob Ayoi Sbiiy d Iu-o-Se Sbiiy fo ye..3.4 ude uio A A A3. 3

24 Defiiio 3.3 Coide ye..3.4 ude uio A A A3 d ue fo eey L [ ]; d L [ ; D u L o [ ; U e ouio L []; of..3.4 wi iii odiio L [ ]; oeodig o iu d L [ ; D u L o [ ; U ei fo. We y ye..3.4 i Iu-o-Se Sbe ISS fo e iu u U uifoy i d D if ee ei oiuou odeeig fuio γ u e foowig oeie od obu Lgge Sbiiy Fo eey ε > i od u u u γ u u ε d L [ ; D u L [ ; U < obu Lyuo Sbiiy Fo eey ε > ee ei ε > u u u d L [ ; D u L [ ; U ε u u u γ o < Uifo obu Aiiy Fo eey ε > d ee ei ε > u u u d L [ ; D u L [ ; U ε. u u u γ o < If U {} e we y L [ ]; i Uifoy obuy Goby Ayoiy Sbe GAS fo o.. Le. Le 4. Teoe. i [] d Teoe gie u e foowig eu. Teoe 3.4 Coide ye..3.4 ude uio A A A3. Te e foowig ee e equie Sye..3.4 i ISS fo e iu u U uifoy i d D 4. b Tee ei σ KL d oiuou o-deeig fuio γ u fo eey L [ ]; d L [ ; D u L o [ ; U e ouio L []; of..3.4 wi iii odiio [ ]; oeodig o iu d L [ ; D L L [ U ifie u σ u u γ u fo u o ; Sye 3.7 i ISS fo e iu u U uifoy i d D.. Fueoe if U {} e e foowig ee i equie o ee b d Sye..3.4 i obuy Fowd Coee i.e. fo eey T d i od u [ T ] u d L [ ; D < wee L [ ]; u deoe e ouio of..3.4 wi iii odiio L [ ]; oeodig o iu d L [ ; D d e oey of Uifo obu Aiiy of Defiiio 3.3 od.

25 Poey 3 Lyuo Ceiio of GAS Teoe 3.4 i [] d e eu of Teoe ow u o obi oee Lyuo eiio of e GAS oey fo ye Teoe 3.5 Coide ye..3.4 ude uio A A A3 d ue U {}. Te equiibiu oi L [ ]; i GAS fo..3.4 if d oy if ee ei fuio V L []; o-deeig fuio Q d fuio K u e foowig iequiie od u V u V V y Q u u y u y fo L [ ]; 3.4 fo y L [ ]; 3.4 Moeoe fo eey L [ ]; d L [ ; D e ouio L [ ]; of..3.4 wi iii odiio [ ]; oeodig o iu d L [ ; D ifie L e V fo 3.4 V Agi i oud be oied ou iequiy 3.4 guee e fuio V L []; i Lii o bouded e of e e e L [];. Howee iequiy 3.4 doe o guee Fee diffeeibiiy of e fuio V L []; o e ii i V V ei fo e ouio of..3.4 wi iii odiio L [ ]; oeodig o iu d L [ ; D. oie iequiy 3.4 guee fo eey L [ ]; d L [ ; D e ouio of..3.4 wi iii odiio L [ ]; oeodig o iu d L [ ; D ifie u V V V i. Poey 4 Suffiie Codiio fo Sbiiy Poeie Teoe 3.5 i o e o oeie wy of oig GAS o ISS fo Fo i uoe we ue e foowig eu wi i die oequee of Teoe.6 d e equie deiio of ye..3.4 by e of ye 3.7. I oof i oied. Teoe 3.6 Coide ye..3.4 ude uio H H H3. Aue ee ei oiuou oiie defiie d diy ubouded fuio W oiuou o-deeig fuio γ d o λ u e foowig d D u U iequiy od fo L [ ; W F d u λ u W γ u <

26 wee F d u F d u... F d u d F i D U L [ ; i... e defied by 3.8. Te ye..3.4 i ISS fo e iu u U uifoy i d D. Te ue of Teoe 3.6 i iued by e foowig ee. Ee 3.7 Coide ye..3.4 wee K d u d d G d u g g ye Fo i ee ye 3.7 i gie by U {} g i o d D [ ] ey e gd d d 3.44 d d gd d 3.45 d e ouio of 3.44 i eed o e ouio of 3.45 by e of e equio g w dw d 3.46 Cey uio H H H3 e ified fo ye 3.45 d uio A A A3 e ified fo ye I i ee we ow e equiibiu oi L [ ]; i GAS fo ye 3.44 oided g < 3.47 Ideed we y Teoe 3.6 wi F d u d d gd d d W k wee k > g u i o. oie W F d u d d g u g g k d d g u k g k u W d < < < < Coequey 3.43 od wi λ g k < wi γ. Te odiio 3.47 i fo g d d ye 3.44 di e fiy of o ouio Q fo [] wee Q d oequey e equiibiu oi L [ ]; i o GAS. 6

27 4. A Iuie Ee Coide ye..3.4 wee U K u g g g i o d d D i iee. Ti obe w udied i [] d boudy feedbk oo of e fo u k d G 4. w deiged wee e kee k C [ ]; w eiiy gie d gueed fiie ie biiy fo e oeodig oed-oo ye. Hee we wi oide eie eodoogy fo e deig of obu biie fo ye..3.4 wi G u u ey e ye g u 4. Fo i ee ye 3.7 i gie by g d u 4.3 d e ouio of 4. i eed o e ouio of 4.3 by e of e equio g w dw 4.4 Cey uio H H H3 e ified fo ye 4.3 d uio A A A3 e ified fo ye 4.. Equio 4.3 iy e foowig diffeei equio od fo o fo wi e ouio of 4.3 ei d d g g g 4.5 Coequey e foowig equio od fo fo wi e ouio of 4.3 ei e g e g g d g e g d 4.6 Uig 4.6 wi we ow fo eey iii odiio d fo eey u L o ; e ouio of 4.3 i bouded fo [ eefoe Teoe. iie e ouio ei fo [. Uig iduio we oude fo eey iii odiio d fo eey u L o ; e oeodig ouio of 4.3 ei fo. Moeoe uig 4.3 d 4.6 wi we oude ye 4.3 ifie e foowig equio fo.e. 7

28 g e g e g d g u e g d 4.7 I i e e feedbk w e g e g d g u g e g d 4.8 guee e oed-oo ye 4.3 wi 4.8 ifie fo.e. g e gw w e g e gw w dw 4.9 Uig 4.3 d e equio fo. g d we oude fo 4.9 A i oi we oud eie we e ieed fiie-ie biiio of e ye 4.3 by e of e feedbk w 4.8. Howee we e o ieed biiio of ye 4. by e of feedbk w of e fo 4.. Ti ei o be ow. Ideed uig 4.4 we ge fo 4.8 u g g g Iegig by we ge g g e g e g e g e g d g d g e g e g d g e g dd g e g d d g e g e g d oequey we obi dd g d g d d dd g e g d e g d e g e g d g e g d d dd e g e g dd e g d d u g e g d wi i boudy feedbk w of e fo 4. wi k g e g. Coequey uig 4.4 we oude e feedbk w 4. wi k g e g iee fiie-ie biiio of ye 4.. Ideed we oe e eiee of o M o e foowig eie od fo e ouio of e oed-oo ye 4.3 wi 4.8 8

29 u u < < Mσ u u fo 4. < < if [ wee σ. Uig 4.4 we ge u u g u. Te eiou if < < < iequiy eie 4. d e f 4.4 u od ow u by eeig fo [ wi by iue of 4.4 gie q q fo q ] o obi e eie g M u u σ fo 4. < Te yi eeed o f owed u o obi oe diffee feue fo e yi i [] Te ieeio of e feedbk w I foow fo 4.4 e feedbk w 4.8 be ieeed by uig e foowig diibued dey feedbk w e g e g d g 9 < u g e g d 4. Te feedbk w 4. ow we do o eed o eue e woe e ofie we oy eed o eue wo diffee e oi d. Te iue of e e e I oud be eied e feedbk w u g e g d i ey e e wi obied i []. Howee we e oed eie 4. od fo iii odiio fo [] wi [ ];. Moeoe o oibiiy odiio e.g. i g e g d L i equied o od. 3 Coo uo eo Te ieeio of e feedbk w u g e g d y eu o e equio u g e g d w fo 4.3 wee w L ; i e oo uo eo. Uig e fewok deibed i e ee wok we e i oiio o udy e oed-oo ye 4. wi 4.3. I i e we ge e ieg dey ye g w g e g e g d g d wi w L ; iu. Uig 4.7 we ge fo e g d 4.4 w w g w d 4.5

30 wi gie u e eiee of o M γ u e foowig eie od u u Mσ u u γ u w fo 4.6 < < < < if [ wee σ. Eie 4.6 ow e ISS oey od fo ye 4.4. if Eie 4.6 ow uig ey e e gue boe e foowig eie od fo e ouio of 4. wi 4.3 u < g M u g γ u w σ fo 4.7 < wi i ey e ISS oey fo e oed-oo ye 4. wi 4.3. Hig eeed e yi of e ee we e i oiio o oi ou ei ooig of e ooed eodoogy of e oeio of ige FOH-PDE o ye deibed by IDE i Te jo ooig of e boe yi i e eee udie diffee ye ieg dey ye fo e oigi oe ye deibed by fi-ode yeboi de. Ti i io beue biiig feedbk w fo 4.3 y o be equie o e feedbk w fo 4. i y ioe dey e. Teefoe biiig feedbk w fo 4.3 we eeed i e oigi oodie y o gie u oed-oo ye of e fo 4. bu e ye deibed by ige fi-ode yeboi PDE wi dey. ii Aoe ooig of e boe yi i e eifi fo..3.4 fo wi e oeio o ye deibed by IDE be ued. o ye deibed by fi-ode yeboi de be eeed by..3.4 e equiee e fuio K i W L [ ]; i... e o owed o deed o [] i eiie. Tee e ye wi e udied i [] bu o be udied i e ooed fewok. Howee oe u be i id eiiy ieg foio be ied o e foed ye i of e fo Coudig ek Ti e ebie e equiee bewee ye deibed by ige fi-ode yeboi i diffeei equio d ye deibed by ieg dey equio. Sye-eoei eu e oided fo bo e of ye e Boudede-Iie-Coiuio oey d e oey of Lii deedee o iii odiio Teoe. d Teoe 3. e obue of equiibiu oi fo ye wi ee iu Teoe. d Teoe 3. 3 eiio of e Iu-o-Se Sbiiy d obu Gob Ayoi Sbiiy Teoe.4 d Teoe oee Lyuo eu Teoe.5 d Teoe uffiie odiio fo Iu-o-Se Sbiiy d obu Gob Ayoi Sbiiy Teoe.6 d Teoe

31 Te ooed fewok ow e udy of dioiuou ouio fo oie ye deibed by ige fi-ode yeboi i diffeei equio ude e effe of eube iu ig o e boudy d/o o e diffeei equio. Ti e i io fo oo-eoei oi of iew beue ye e ubje o diube odeig eo oo uo eo d euee eo wi e yiy odeed by eube iu. A iuie ee ow e oeio of ye deibed by ige fi-ode yeboi i diffeei equio o ieg dey ye iify oideby e ouio of e oeodig obu feedbk biiio obe. efeee [] Ao O. Diube ejeio i Lie Hyeboi Sye o e i IEEE Tio o Auoi Coo. [] Bi G. d J.-M. Coo Fue eu o Boudy Feedbk Sbiiio of Hyeboi Sye Oe Bouded Ie Poeedig of IFAC oo Boog Iy. [3] Bi G. d J.-M. Coo O Boudy Feedbk Sbiiio of o-uifo Lie Hyeboi Sye Oe Bouded Ie Sye d Coo Lee [4] Coo J.-M. G. Bi d B. d Ade-oe Diiie Boudy Codiio fo Oe- Dieio oie Hyeboi Sye SIAM Jou o Coo d Oiiio [5] Dige A. G. Bi d J.-M. Coo Lyuo Eoei Sbiiy of -d Lie Hyeboi Sye of Be Lw Auoi [6] Fiio A. V. Diffeei Equio wi Dioiuou ig-hd Side Kuwe Adei Pubie Dode 988. [7] H S.-Y. d A. T Lyuo Fuio d L -Sbiiy fo Diee Veoiy Bo Equio Couiio i Mei Pyi [8] He J.K. d S.M.V. Lue Ioduio o Fuio Diffeei Equio Sige- Veg ew Yok 993. [9] Kfyi I. P. Pee d Z.-P. Jig Sbiiy eu fo Sye Deibed by Coued eded Fuio Diffeei Equio d Fuio Diffeee Equio oie Ayi Teoy Meod d Aiio [] Kfyi I. d Zog-Pig Jig Sbiiy d Sbiiio of oie Sye Sige-Veg Lodo Seie Couiio d Coo Egieeig. [] Kfyi I. d M. Ki oie Sbiiio ude Sed d Deyed Meuee d wi Iu Subje o Dey d Zeo-Ode Hod IEEE Tio o Auoi Coo [] Ki M. d A. Syye Bkeig Boudy Coo fo Fi-Ode Hyeboi PDE d Aiio o Sye Wi Auo d Seo Dey Sye d Coo Lee [3] Ki M. Dey Coeio fo oie Adie d PDE Sye Bikäue Boo 9. [4] Ki M. Iu Dey Coeio fo Fowd Coee d Si-Feedfowd oie Sye IEEE Tio o Auoi Coo [5] Li T. T. Coobiiy d Obebiiy fo Quiie Hyeboi Sye oue 3. Hige Eduio Pe Beijig 9. [6] Meo-Agui D. V. Kioo. Loo Sbiiy Codiio fo Ieg Dey Sye Ieio Jou of obu d oie Coo -5. 3

32 [7] Meo-Agui D. O Sbiiy of Ieg Dey Sye Aied Mei d Couio [8] Meo-Agui D. Eoei Sbiiy of Soe Lie Coiuou Tie Diffeee Sye Sye d Coo Lee [9] Pe L. d L. Cg Lyuo-Bed Boudy Coo fo C of Hyeboi Lok-Voe Sye IEEE Tio o Auoi Coo [] Pee P. Te Lyuo Seod Meod fo Coiuou Tie Diffeee Equio Ieio Jou of obu d oie Coo [] Pieu C. d F. Me ISS-Lyuo Fuio fo Tie-Vyig Hyeboi Sye of Be Lw Mei of Coo Sig d Sye [] Pieu C. J. Wiki d G. Bi obu Boudy Coo of Sye of Coeio Lw Mei of Coo Sig d Sye [3] V. d S. I. iueu Oiio i Loe Pogio Mode Liuo- Kokii Ao IMA Jou of Mei Coo d Ifoio [4] u J. d M. Tyo Eoei Dey of Souio o Hyeboi Equio i Bouded Doi Idi Uieiy Mei Jou [5] Sog E.D. Soo Sbiiio Iie Coie Foiio IEEE Tio o Auoi Coo [6] Vque. M. Ki d J.-M. Coo Bkeig Boudy Sbiiio d Se Eiio of Lie Hyeboi Sye Poeedig of e 5 Cofeee o Deiio d Coo Odo. [7] Vque. M. Ki J.-M. Coo d G. Bi Lo Eoei H Sbiiio of Quiie Hyeboi Sye Uig Bkeig Poeedig of e 5 Cofeee o Deiio d Coo Odo. [8] Xu C.-Z. d G. Se Eoei Sbiiy d Tfe Fuio of Poee Goeed by Syei Hyeboi Sye ESAIM Coo Oiiio d Cuu of Viio