ISS IN DIFFERENT NORMS FOR 1-D PARABOLIC PDES WITH BOUNDARY DISTURBANCES
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- Octavia Sherman
- 6 years ago
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1 ISS I DIFFERE ORMS FOR -D PARABOLIC PDES WIH BOUDARY DISURBACES Io Kfyll * Mol K ** * De. of Mem ol el Uey of Ae Zofo Cm 578 Ae Geee eml ok@el.. ** De. of Mel Aeoe E. Uey of Clfo S Deo L Joll CA 99- U.S.A. eml k@.e A Fo -D ol PDE e o oe e e e oe ISS eme o om. De o e lk of ISS Lyo fol fo oy e e oof meooloy e eefo eo of e olo fe-ffeee eme. e ISS eme fo e -om le o efeme of e ell-ko mmm le fo e e eqo. Flly e oe el e le o q- emoely moel ole olol oy oo. Smll- oo ee e lol eoel ly of e eo olo fo moel e ee. Keyo ISS ol PDE oy e emoely.. Ioo e eeo of e I-o-Se Sly ISS oey f eeloe y E. D. So [7] o yem e ee y Pl Dffeel Eqo PDE eqe oel meml ool oel oe. Eeo of e ISS oey o PDE yem e ee e my ee le ee fo emle [56789]. I l fo PDE yem ee e o qlely loo ee e e e om e e e e PDE e oy e e e oy oo. Mo of e e el e lee el e e. Boy e ee mjo llee ee e fomo of e oy e o e e le o e ISS oey ee o e oy e ome of me ee ee fo emle []. ele y e e of oe oeo fo e eeo of e effe of e oy e ee e ele o [7] fo fe-meol yem e eee y me of oe le oeo. Moeoe lo e oo of Lyo fol fo PDE oee fly e l ye ee fo emle [68] oe of e ooe Lyo fol e e fo e eo of e ISS oey... oy e. Iee ee o ee ISS Lyo fol e fo PDE oy e e ll oee Lyo el eqe e o-lle oeo oey fo e lloe e ee []. Ufoely e oeo oey oe o ol fo oy e. omley oe [] ee e e e me-y loely ele o e oy e elf. e ee le [] ee meooloy fo o ISS eme... oy e fo -D ol PDE. e m e [] e eefo eo of e
2 olo e oe of eq e e o ke le o e e e eolo of e e ee o ke le meee oe e. e oe ISS eme ee eee ee L om. e foe o -D ol PDE e o o oy e e e. e oe e elme of e ISS oey o om. Wle ISS eme e L om ee eoem. y efom ly ml o ooe [] o e e ffee fo om. o e e eelo ee fe-ffeee eme llo e eo of ISS eme eee ee L L om eoem. eoem.. Fe-ffeee eme e ee e e e fo o PDE ee fo emle []. e oe eme of e of e oy e e e e ee o e le o eqlm ofle fo e om e ely eql o e eme. eefoe e lme e ee e eelo oel meml ool e e fo e eo of e ISS oey fo oy e e lo of feffeee eme. Aoe mo fee of e ee ok e f e ISS eme L om fo e e eqo Coolly.5 le o efeme of e ell-ko mmm le fo e e eqo ee fo emle [] o e 5. e oe ISS eme e eloe y me of mll- me fo e ly ly of ol PDE olol oy oo. o Seo of o e ee e y -D q- emoely moel. e moel ole olol oy oo ooe f e [78]. Sly oo fo -D q emoely moel e ee e [78] [95] fo mel ly of eo eme. We elo e ISS eme fo -D ol PDE oe mll oo fo e lol eoel ly of e eo olo o om eoem. eoem. eoem.. e oe ly oo e eke e oo ooe e lee. e e of e e follo. Seo eoe o e eeo of e olem e eme of e m el llo e eo of e ISS eme o om eoem. eoem. eoem.. Sef el e oe fo e e eqo Dle oy oo Coolly.5. e m el o e lol eoel ly o om of e eo olo fo q- emoely moel e e Seo eoem. eoem. eoem.. Seo of e ee ok o e oof of e m el. e ol emk of e e e oe Seo 5. Flly e Ae o e oof of ome ly el el ee e Seo. oo. oo e e o e follo oo. Le U e e o-emy eo le e e. By C U ; e eoe e l of oo m o U ke le. By C k U ; ee k e eoe e l of oo fo o U e oo ee of oe k o U ke le. We o elly e.e. e e e C k U e me. Fo C [ ] eoe e ee ee o [ ]. eoe e e of o-ee el me. Fo el me [] eoe e ee of.e. e ee ee le o eql o. Le C [ ] e e. We e e oo [] o eoe e ofle e.e. [ ] fo ll [ ]. H eoe e Soole e of ooly ffeele fo o [ ] mele qe ele eo ee. L [ ] eoe e eqlee l of mele fo / f [] fo f.
3 . ISS o om Le C []; C []; q C [ ] e e fo oe e Sm- Lolle oeo A D [ ] efe y L q Af f f fo ll f D. ee D e e of ll fo f H fo f f f f. ee e el o fy e follo mo H Ee o. H Ee o. FAC ee Ce [] e [] All eele of e Sm-Lolle oeo A D L [ ] efe y.. e H H e el. ey fom fe e eqee lm. o e eele... oeo ely oe eefo D C [ ] fe A. e eefo fom oooml of L [ ]. I e [] e lo mke e follo mo fo e Sm-Lolle oeo A D [] efe y... L H e Sm-Lolle oeo A D L [ ] efe y.. ee e el o fy H H fe. Coe e meee oe e m. C [] X.5 mee. Ge C C [ ] X e y e olo C []; C []; fo [ ] X fo ll fo ll [ ] q fo ll.6 I oe o e oe e olo of e eolo eqo.6 fe fo ll e oy oo
4 .7 e C e oy e e oly e oy oo le e C [ ] e e e ffeel eqo. Ue ffe ely oee fo e e C e e oo o ee ll olo e fo e eolo eqo.6.7. ee y e follo el. eoem. Soe mo H H H ol. e fo eey C C [] X e eolo eqo.6.7 l oo X qe olo C [] C [ ] fo ] X fo ll fo ll [ ]. [ eoem. ee fo y e C [ ] C [ ] ee e o-emy e C ; e follo mlo ol If e e eolo eqo.6.7 l oo C [] qe olo C [] C [ ] fo [ ] C [] fo ll fo ll [ ]. Iee eoem. ee C ; e follo el e f m el of e eo. I oe ISS eme of e olo of.6.7 ee L om e e follo mo. H ee e fo C []; o q fo ll [ ]. Moeoe e eqle ol. eoem. Soe mo H H H H ol. e fo eey C [ ] C [] e qe olo C [] C [ ] of e eolo eqo.6.7 l oo C [ ] fe e follo eme fo ll ee [ ] m m m e [] m m.8 [ ] m..9 e eo m el of e ee ok oe ISS eme of e olo of.6.7 ee L om.
5 eoem. Soe mo H H H ol. e fo eey C [ ] C [] e qe olo C [] C [ ] of e eolo eqo.6.7 l oo C [ ] fe e follo eme fo ll e [ ] [] C m e. ee C m C ~ C m m ~. C. C ~. /. [ ] ~ ~ C [] e qe olo of e oy le olem q ~ fo [] ~ ~ ~ ~ C [ ] e qe olo of e oy le olem q. fo [ ] Flly e m el of e eo oe ISS eme ee L om fo el e of e ol PDE.6 Dle oy oo. eoem. Le k e e o k. Fo eey C [ ] C [] le C ; e e o-emy e fo e eolo eqo k fo.5 fo.6 l oo C [ ] qe olo C [] C [] [ ] C [ ] fo ll fo ll [ ]. e fo eey C [ ] C [ ] e qe olo C [] C [] of e eolo eqo.5.6 l oo [] fe e follo eme fo ll C 5
6 [ ] e k [] m m e k m k e m.7 ee [ ] e..8 Remk o e ee m el e ISS eme.8 ffee fom e ISS eme..7 eme.8 eee y me of m-oeo e m-fomlo of e ISS oey; ee [] le e ISS eme..7 e eee y me of mmo. e oof of eoem. eoem. e ml ey e o elo feffeee eme fo e omo of e olo of e PDE. O e oe e oof of eoem. ele o eefo eo lo le []. eoem. oly lle o e PDE.5 Dle oy oo.6 le eoem. eoem. e lle o m moe eel ol PDE o oy oo Dle em o Ro. Amo H H e o ee. o ee oe e oy oo ole Amo H ly e e ~ ee ~ Amo H ol. Sml mlo my e oe fo e fo of Amo H ell f eee. Amo H eqe e eee of oe eefo of e Sm-Lolle oeo A D L [ ] efe y.. e olem of e eee of oe eefo fo ell oeo ee e e lee ee [5] o e efeee ee. Hoee m e oe ee eee of feeom e eleo of e oy oo o ef oy oo e me o ol fo e fo C []; e fo oly eqe o fy e oy eqle. e Se e eqlm o oeo o e o e o e e fo ~ C [] C [ ] eeely follo e of e oy e ole o e ISS eme ] M e [] m m fo ll fo e o [ M m fy e eqle C C ee C C e e y... O e oe eoem. ee C C fo ll Coeqely e ee e emo of e me y eoem.. Moeoe foml.. ee e of e oy e e ome o e kolee of e eele e eefo of e Sm-Lolle oeo A D L [ ] efe y... e oly e ee o ko o e eele e eefo of e Sm- Lolle oeo A D L [ ] efe y.. Amo H H H ol. 6
7 I oe o e o ell e of e e eme y e eqle.8..7 e e ele e el o e e eqo Dle oy oo. Coolly.5 Le e e o. Fo eey C [ ] le C ; e e o-emy e fo e eolo eqo fo.9.6 l oo C [ ] qe olo C [] C [] [ ] C [ ] fo ll fo ll [ ]. e fo eey C [ ] e qe olo C [] C [ ] of e eolo eqo.9.6 l oo C [ ] fe e follo eme fo ll m m e. m e m e m m e m m. m fo ll. Remk.6 I le e emo fom... of e of e e e o e o le o eqlm ofle fo e oeo ee om e ely eql o e eme. By ele e e fom. e follo eme ol fo ll / m me m m m. Eme. efeme of e ell-ko mmm le fo e e eqo ee [] o e 5 e ll mmm le oe eme. fo /. k / e o fom. e follo mle eme m ol fo ll. m e m m m 7.
8 . Alo o emoely Moel Smll-G Ame Q- moel of emoely oe l meo el e PDE.9 oy oo. ee C [ ] e e fo ee [78]. e le e meole eoy oo [ ] meole l le me meole me le of l of omoeeo oo mel. Coo fo e fo [] e moe ee [] o ee e o-emy e C C [] ee L fo eey e eolo eqo.9. qe olo C [] C [ ] [ ] C [ ] fo ll fo ll [ ]. Hoee e ll o el ee e eee/qee olem fo e olo of.9.. e ee eo eoe o e ly ly of e eo olo fo e yml yem ee y.9.. e follo el e e ISS eme of e eo eo ojo mll- me. eoem. Soe ee e o. e ee e o M fo eey e qe olo C [] C [] of e eolo eqo.9. [ ] C [ ] fo ll fo ll [ ] fe e follo eme fo ll ee [ ] m. eoem. Soe [ ] M e []. / /. e ee e o M fo eey e qe olo C [] C [] of e eolo eqo.9. [ ] C [ ] fo ll fo ll [ ] fe e follo eme fo ll 8
9 M e [].5 [ ] ee / [ ]. eoem. Soe.6 e ee e o M fo eey e qe olo C [] C [] of e eolo eqo.9. [ ] C [ ] fo ll fo ll [ ] fe e follo eme fo ll M e [].7 [ ] ee [ ]. Remk. e oof of eoem. eoem. eoem. o oo..6 e of oo. e mll- oo fo e e eqo.9 of oy feek e.. Sml ly oo e ee e [78] [95] fo mel ly of eo eme. I ol e oe oo. oe eelo of e ly oo e [78].8 e ee f oo.8 ol e ee e o eqle. ol. e ee y oy of e fo P P e f eqle.8 mly e eqle P P. ee ffeee eee o el e el [78]. e el [78] ee e m [] ee e oo.6 le o el ee e eoel oeee of [] o [] o [ ] o eo e oo.. o.6 eeely. Hoee o el o o ele e oly [] o [] o [ ] m ee fo e eo. 9
10 . Poof of M Rel We e oof of eoem.. Poof of eoem. Le y C C [ ] X e e. Fo e eee/qee of e eolo eqo.6.7 l oo X e mly ly e fomo y ee y C []; C []; e qe fo fe y[ ] X ell.5 fo ll y fo ll [ ] y Ay[ ] A A fo ll. ee C C C C oe Amo H H ee. e eee/qee of y C []; C []; fe y[ ] X fo ll y fo ll [ ]. ee y eoem. []. e oof omlee. e oof of eoem. le e follo el lemm. I oof oe e Ae. Lemm. Le y j j j j e eqee fy e follo oee P P j {... m} y j K j j j j {... m} y j K j j y j fo e el o K fo e oe ee m. e e follo eqly ol fo ll j {... m} j m m K m y... j j. Poof of eoem. Le C []; e e fo ole Amo H. oe e my me C [ ] e e y eeo of e fo o of e el [ ]. I e follo eoo e ll e e oo q fo [ ]. I ffe o o ee e o fo eey me fo eey e follo eme ol fo ffely le
11 m m e m m. ee e o ole Amo H m m m m m....5 fo.6 fo.7 e fo ] [ [] e y e foml fo ll ] [.8 fo ll ] [].9 Iee e m m. e o fom. m m m e m m m M ee m M. Se e fo ] [ [] e y.8.9 oo fo fomly oo o e om e ] [ ] [ e e lm lm lm e ee efo.6.7 y le e e.8 fo ll m e m m m
12 Se e oe eqly ol fo ll ] e lm m m e oe eqly mle.8 ol fo ll. Ieqly.8 ol fo ell. eefoe e e o oe ee e o fo eey me fo eey eme. ol fo ffely le. Le e y me. Fo eey [ ] e o... Le e o o e follo eqle ol fo ll m. m oe e eee of o. ol fo ll e oeqee of Amo H H. e eee of o.7 ol fo ll e oeqee of Amo H e f. Flly.5.6 e oeqee of e oy of e fo e f fo ll [ ]. We e ele ee efe....8 ee.9 C o o e elee
13 m m C. oe.9 ojo efo ee e follo eqle fo ll fo I follo fom e f fo eey C ] [ e follo eqly ol P. ee P.5 Femoe follo fom. efo.8 fo eey e follo eqle ol P fo....6 U.7 e follo eqle.7 e o e eqle
14 .8.9 Coeqely e o fom.6 fo fo eey / e follo eqle ol Q. Q. ee P Q. P Q. We e oe e fo ] [ [] e y.8.9 oo fo y e of e f fo ll ] [ ell Amo H efo. fe fo ].... Le y e e le e elee y fo ll [] ] [.5 fo ll ] [ ] [.6 / fo ll ] [.7
15 5 ee ] [ [] e fo efe y.8.9. e eee of fe.5.6 follo fom fom oy of e fo o e om e ] [ ] [. Iee oe oy of e m fo [] ojo e oy of e m fo [] e f.6 ol fo ll mle.6 ol fo ll ] [ e m oo fo ] [. e eee of fe.7 follo fom fom oy of e fo ] [ [] efe y.8.9 o e om e ] [ ] [. Le e ee ffely le m efe y me of.8. e le C m efe m. oe.9 ol fo eleo m ee efe y.8. Moeoe oe e efo of ee ee e o S S Q Q P m... fo ll ] [.8 oe y e of. e o S my e elee o e eee of e me. U e f ell mo H e le eqly... e e fo ll ] [ S.9 S fo.... S.. Coeqely eqle. efo.8.5 e o fom.9.. fo ll ] [
16 ee m S m m. I me of fo mlo o o eme... ojo efo.8.5 eqle.7 mly e follo mlo ol fo ll ] [ Q If m e m. Q If m e m m. ee m m. By e of Q Q e f m follo oee P P of Lemm. ol fo e eqee m j y j j fo... j K m. oe.7 ojo.9 e f m mle. eefoe e o fom Lemm. m m m m m.5 U.5 e f m e m e f m e e.. e oof omlee. We e oee e oof of eoem..
17 7 Poof of eoem. I ffe o oe e follo eme m m m e m e.6 fo e olo of fo.7 e fo.8 ee o. oe f y olo of.5.6 e e fo y e efe fo ] [ olo of.7.8 k e. I oe o o eme.6 ffe o o fo eey me fo eey e follo eme ol fo ffely le W W e e.9 ee m m m m ] o f f.5 W..5 oe [] efe y.5 oo fo fo ll ] [. Iee.9 ffe fo e oof of eme.6 e m ee e o m W.5 eefoe e o fom.9 e e M W W
18 8 ee m m M. Le e e.5 fo ll W W e e Se e oe eqly ol fo ll ] e lm e oe eqly mle.6 ol fo ll. Ieqly.6 ol fo ell. eefoe e e o oe fo eey me fo eey eme.9 ol fo ffely le. Le e e y me. Fo eey ] [ e o... We e ele ee efe....5 ee.5 o o e elee. oe.5 ee e follo eqle fo ll.55 I follo fom.7.. e f fo eey ] [ e follo eqly ol P.56 ee P.57 Femoe follo fom.56 efo.5 fo eey e follo eqle ol P We e oe e fo [] e y.5 oo fo fe o fo
19 Le y e e le e elee y eqle.5.6 ol / fo ll.6 e eee of fe.5.6 follo fom fom oy of e fo o e om e [ ] [ ]. Iee oe oy of e m fo [ ] ojo e oy of e m fo [ ] e f.7 ol fo ll mle.7 ol fo ll [ ] e m oo fo [ ]. e eee of fe.6 follo fom fom oy of e fo [] efe y.5 o e om e [ ]. Le e ee ffely le m / efe y.5. e le m efe. oe.5 ol fo eleo m m ee efe y.5. Moeoe oe e efo of ee ee e o S m P fo ll [ ].6... U e le eqly.55 e e fo ll [ ] e.6 Coeqely efo.5.5 e o fom.6.59 fo ll [ ] W W m ee m m e..6. Aly.6 ely fo e eqee W j fo j... m e f m e f e e m W W e 9
20 ee m m. e oe eqly ojo e f e m mle.9. e oof omlee. e e e e oof of eoem.. Poof of eoem. Se e eefo of e Sm-Lolle oeo [] L D A efe y.. e Amo H H fom oooml of [] L follo Pel ey ol.e. ] [ fo ll.6 ee fo By e of.6 follo fom eee eo y e follo eqle ol fo ll A q q q e e fo ll A.66 I follo fom.66 e f A efo.65 e follo eqo ol fo ll.67
21 e e o fo ll.68 Iee e eqo ee.7 e fom e o Eqo.68 follo ely fom e oe eqo e f. Moeoe e eqo ee.7 e fom e o e f ojo e oe eqo mle.69 U e o fo ll.7 Ie e ffeel eqo.7 e o fo ll... e e e e.7 Coy of e m.7 mle e follo eqo fo ll...
22 e e e e.7 Eqo.7 ojo e eo eqly mly e follo eme fo ll... e m e m e e.7 ee m m. We e e e eqly ol fo eey ojo.7. U.7 fo e e e e ll... m m e e.7 A e e e e eqly ol fo eey ojo.7. We o fo ll... m m e e.75 We oe.75 ol fo ell. Se fo ll... e o fom.75 e follo eme fo ll... m m e e.76 eefoe y e of eme.76.6 e follo eme ol fo ll
23 m m ~ m m [] e e ] [ C C C.77 ee C.78 C.79 ~ C.8 We oe e fo... e e m oeqely m fo.... eefoe eqly. mle. U e Cy-S eqly e f fo... e e fo.... e e ~ C U Lemm. [] follo e oy le olem ~ ~ q fo ll ] [.8 ~ ~ ~ ~.8 qe olo ] [ ~ C fe ~.8 C y Lemm. [] follo e oy le olem q fo ll ] [.8
24 .85 qe olo C [ ] fe.86 e oof omlee. Poof of Coolly.5 Ieqle.. e fo lo of eoem. eoem.. Ieqly. oeqee of ml ly o eee e oof of eoem. eee e. Se e fo fom oooml of L [ ] follo Pel ey ol.e. ee fo ll.87 fo By e of.6.9 follo fom eee eo y e follo eqle ol fo ll Ie e oe ffeel eqo e o fo ll... e e.89 Coy of e m.89 mle e follo eqo fo ll... e e.9 Eqo.9 mly e follo eme fo ll... e m m e.9 eefoe e o fom.9 e follo eme fo ll e Sele e e e m m fo e o fom.9 fo ll
25 e m e e U e eqly e e e e Eme. ol fo ell. m fo ll....9 e f e e. fo y om e oof omlee. Fo e oof of eoem. eoem. e ee e follo el lemm. I oof oe e Ae. Lemm. Fo eey M ee e o e follo oey f y e lolly oe fo fo ee e o e follo eqly ol fo ll e e follo eqly ol fo ll M e y.9 M e y e.95 Se e oof of eoem. eoem. e ml ey e eee oee. Poof of eoem. eoem. Le y oe e qe olo C [] C [] of e eolo eqo.9. [ ] C [ ] fo ll fo ll [ ]. By e of Coolly.5 e olo fe eme.. fo ll fo ll.96 U Höle' eqly efo.96 e o fo ll [ ] [ ] [ ] [ ].97 Coeqely e e fom..97 fo ll [ ] e [] m [ ] 5.98
26 [ ] e [] m [ ].99 ee. By e of. o.6 e e ee e of eoem. e of eoem.. Coeqely ee e. By e of Lemm. fo e elee fo ee e o e follo oey f y e lolly oe fo fo eqly.9 ol fo ll M e eqly.95 ol. Aly oey o e fo y [ ] e of eoem. e fo y [ ] e of eoem. o eqly.9 ol fo ee fo y e of eme e emo oey fo e olo of e eolo eqo.9. e e fo eoem. [ ] fo eoem. [ ] e [] m [ ] e fo ll e [] m [ ] e fo ll.. Se eme.5 e of eoem. o.7 e of eoem. follo fom eme. o eme. eeely M. e oof omlee. Fo e oof of eoem. e ee e follo el lemm. I oof oe e Ae. Lemm. Fo eey M ee e o ] e follo oey f q y e lolly oe fo fo ee e o e follo eqly ol fo ll q m M e e e follo eqly ol fo ll q m M e Flly e ee e oof of eoem.. q y. q y e. 6
27 Poof of eoem. Le y oe e qe olo C [] C [] of e eolo eqo.9. [ ] C [ ] fo ll fo ll [ ]. By e of Coolly.5 e olo fe eme. fo ll e e o ole. e y.96. U Höle' eqly efo.96 e o fo ll [ ] [ ]. ee [ ] m. Coeqely e e fom.. fo ll [ ] m e [] m [ ].5 ee m. By e of. e e. Coeqely ee e. By e of Lemm. fo e elee fo ee e o ] e follo oey f q y e lolly oe fo fo eqly. ol fo ll M e eqly. ol. Aly oey o e fo q y [ ] o eqly. ol fo ee fo y e of eme.5 e emo oey fo e olo of e eolo eqo.9. e e [ ] e m [] m [ ] e fo ll.6 Se e ee e o K K K [ ] [ ] K [ ] eme. follo fom eme.6. e oof omlee. 5. Col Remk We e e -D ol PDE e o oe e e. O m el oe ISS eme ee L L L om. De o e lk of ISS Lyo fol fo oy e e e le fe-ffeee eme fo e oof of e ISS eme ee L L om. e ISS eme fo e -om le o efeme of e ell-ko mmm le fo e e eqo. We lo le e oe el o q- emoely moel ole olol oy oo. By me of mll- me e ee le o ee oo ee e lol eoel ly of e eo olo fo moel. Fe ok my ole e y of e oe of e loe-loo yem fo -D ol PDE e oy feek ool. I ol e oe e lo oy feek le o loe-loo yem ee y olol oy oo lke e yem ee e Seo of e ee ok; ee [6]. Moeoe fe ok my lo ole e eo of ISS eme eel L om ee. 7
28 Refeee [] Aomeo F. B. E. W C. Pe D -I-o-Se Sly of me-y oomoeeo Dffe Eqo Sje o Boy De Poee of e Ame Cool Cofeee Moel QC [] Aomeo F. B. C. Pe E. W S. Bemo A S Cool Lyo Fo fo Dffo Eqo W me-y De Coeffe IEEE o o Aom Cool [] Boye W. E. R. C. Dm Elemey Dffeel Eqo Boy le Polem 6 Eo Wley 997. [] Dkoky S. A. Moeko O e Ufom I-o-Se Sly of Reo- Dffo Syem Poee of e 9 Cofeee o Deo Cool Al GA USA [5] Dkoky S. A. Moeko Lol ISS of Reo-Dffo Syem Poee of e 8 IFAC Wol Coe Mlo Ily 8-. [6] Dkoky S. A. Moeko I-o-Se Sly of Ife-Dmeol Cool Syem Mem of Cool Sl Syem [7] Dy W. A. Eeo of Poey of e He Eqo o Le emoely Oe eoe Qely of Ale Mem [8] Dy W. A. A Dee Poey of Solo of Pol Eqo Alo o emoely Qely of Ale Mem [9] Ekol G. Fe Dffeee Meo fo olol Boy le Polem fo e He Eqo BI [] Fee G. J. C. Loe-Mo Glek Meo fo Semle Pol Polem olol Boy Coo Ae Comol Mem [] Jo F. Pl Dffeel Eqo E. Se-el e Yok 98. [] Kfyll I. Z.-P. J Sly Slo of ole Syem Se-el Loo See Commo Cool Eee. [] Kfyll I. M. K O e Relo of Dely Eqo o F-Oe Hyeol Pl Dffeel Eqo ESAIM Cool Omo Cll of o [] Kfyll I. M. K ISS W Ree o Boy De fo -D Pol PDE o e IEEE o o Aom Cool. [5] L Y. mel Solo of e He eqo olol Boy Coo Jol of Comol Ale Mem [6] Me F. C. Pe S Lyo Fo fo Semle Pol Pl Dffeel Eqo Meml Cool Rele Fel AIMS [7] Moeko A. H. Io Iel I-o-Se Sly of Ble Ife-Dmeol Syem Poee of e 5 IEEE Cofeee o Deo Cool Lo Aele Clfo USA [8] Moeko A. H. Io Coo of Lyo Fo fo Ieoee Pol Syem A ISS Ao X.58 [m.ds]. [9] Moeko A. Lol I-o-Se Sly Ceo Coeemle Syem Cool Lee [] Moeko A. F. W Reeme of I-o-Se Sly Ife Dmeo W Goe Wo me o e Ieol Symom o Meml eoy of Syem eok MS 6. [] Moeko A. F. W Glol Coee Lyo eoem fo Ife- Dmeol Syem me o e IFAC Symom o ole Cool Syem OLCOS 6. 8
29 [] ylo A. W. G. R. Sell Le Oeo eoy Eee See Se 98. [] Py A. Semo of Le Oeo Alo o Pl Dffeel Eqo e Yok Se 98. [] Pe C. F. Me ISS-Lyo Fo fo me-y Hyeol Syem of Ble L Mem of Cool Sl Syem - -. [5] Smolle J. Sok We Reo-Dffo Eqo Eo Se-el e Yok 99. [6] Smylye A. M. K Cloe-Fom Boy Se Feek fo Cl of -D Pl Ieo-Dffeel Eqo IEEE o o Aom Cool [7] So E.D. Smoo Slo Imle Come Foo IEEE o o Aom Cool Ae Poof of Lemm. Le y j {... m} e e. Defe e e Clely f A {... j} y K A. j A e e e j K y j K m y eme. ol e. eefoe e e me j A... j. We e follo e. Ce A. I e e e K y fo ll... j. Poey P mle fo ll... j. U o e e fo ll... j ely o eme. ol e. Ce A Defe m A oe e j A e o j. Defo A. mle K y K y fo ll... j. Poey P mle fo ll... j. U o e e fo ll... j. Moeoe Poey P mle y ome e eo j j eqly mle j y. e l eqly o eme. ol e ell. e oof omlee. Poof of Lemm. Le y e e le o o A. S o e y e of oy of e fo eooo of e f f. Defe f l M / l M l A. 9
30 oe efo A. ee M e e ]. U.9 e o fo ll Aly A. eeely e e y fo ll A. k k k k y fo ll A.5 U e f e e e fo ll e e e e e e e e e e e e e k e k y e y e y e e k k k e y e e k k k y e e k e k e k k k k Coeqely e follo eqly ol fo ll y e k e e e y e e A.6 Ieqly.9 mle M e y ome A.6 e fo ll M e e e e e M e e e e e e e e M e e e y e y e y e y y e y e
31 U e oe eqly ojo e f ] e o fo ll [ ] e M e y e A.7 e Se A.7 ol fo ll [ ] e ole e follo eme ol fo ll e M e y e A.8 e e Defo A. ee. Ieqly.95 e oeqee e of e eo eqly e f eqle A. A.8. e oof omlee. Poof of Lemm. Le y e e. Defe l M / l M m l A.9 oe efo A.9 ee M e e ] ]. U. e o fo ll q m Aly A. eeely e e q m q y fo ll A. k q m y fo ll A. k... k k U e f e ] e e fo ll q me me me me q e m e k y q e m e k y e k k... q e m e k y e k... q e m y e k... k... k k k k k k k k Coeqely e follo eqly ol fo ll q e y e A. q m e e f ] ] e fo ll Ieqly. mle q m M e q y ome A.
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