GLOBAL EXPONENTIAL STABILIZATION OF FREEWAY MODELS

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GOA EXPONENTA STAZATON OF FEEWAY MODES o Kyll * M Koto ** d Mo Pgeogo ** * Det. o Mtemt Ntol Tel Uvety o Ate Zogo m 5780 Ate Geee eml o@etl.t.g ** Sool o Podto Egeeg d Mgemet Tel Uvety o ete 7300 Geee eml mo@dl.t.g moto@dl.t.g Att T wo devoted to te otto o eed lw w gtee te ot glol eoetl tlty o te ogeted eqlm ot o geel dete-tme eewy model. Te eed otto ed o otol yov to o d elot et mott oete o eewy model. Te develoed eed lw e teted mlto d detled omo mde wt etg eed lw te ltete. Te ote oete o te oeodg loed-loo ytem wt eet to meemet d modelg eo e lo tded. Keywod ole ytem dete-tme ytem eewy model glol eoetl tlty.. todto Feewy t ogeto dg e eod d det eome gt olem o mode oete w led to eeve dely eded t ety d eed el omto d evometl ollto. Te m t otol mee emloyed let ot lwy ote wy to tle t ogeto e m meteg M d vle eed lmt S. M mlemeted y e o t lgt otoed t o-m to otol te eteg t low [6. S e ed o eed mozto t eet tde ve demotted tt t my e ed mtem meteg deve well [4. To eve te gol tee otol mee mt e dve y ote otol ttege. A o elted ee odeed ole otml otol d MP Model Pedtve otol etwo-wde eewy t otol o ee e.g. [ 3 0. Howeve oly de to te volved otol ttegy omlety oe o te ooed metod dved to eld-oetol tool. Aote gt o eewy t otol ee odeed elt eed otol oe to tle ogeto olem. A oeeg develomet t deto w te -tye lol eed m meteg eglto ANEA [4 w ee ed ded o el eld mlemetto od te wold ee e.g. [5 7. ANEA otol te t eteg om o-m d tget tl dety te mtem megg egmet o to mmze te eewy togt. Ote ooed lol eed otol lgotm o m meteg lde [3 5 9 30 to meto t ew. o eteo d modto o ANEA wee ooed d eld-mlemeted ove te ye to dde e emegg eed. Mot elevt te eet otet te eteo to P-tye eglto o to eetly dde ottlee w e loted dowtem o te mege e [3; d te llel deloymet o P-tye eglto to dde mltle otetl ottlee dowtem o te meteed o-m [3. O te ote d eed otol oe o mtem t otol y e o S ve ee te e ee [5; ee lo [4 o eet eteo to te mltle ottlee e. To deqtely dde te eg eewy t ogeto olem t eetl to vetgte develo d deloy te otetlly mot eet metod; d eet otol teoy dve old e otely eloted to t ed. t wo we ovde goo metodology o te otto o elt eed lw tt gtee te ot glol eoetl tlty o te ogeted eqlm ot o geel ole detetme eewy model. We o o dete-tme eewy model w e geelzed veo o te ow t-

ode dete Godov omto to te emt-wve tl deetl eqto o te W-model ee [3 8 wt ole [9 o eewe le ell Tmo Model - TM [7 otlow to. Te otted eewy model llow ll ole e o te eltve ote o te low to e te to ot d eve llow tme-vyg d ow oty le. Te otto o te ot glol eoetl eed tlze ed o te otol yov Fto F o ee [6 well o et mott oete o eewy model. mmy te otto o te eet wo teeold F otted o te eewy model; te oml o te yov to e elt d e ed tgtowd wy o vo oe mott oete o geel ole d et dete-tme eewy model e oved metezed mly o glol eoetl eed tlze o te ogeted eqlm ot o eewy model otted. Te eved tlzto ot wt eet to ll oty le tt e ed o te low. A detled omo mde y me o mlto wt etg eed lw ooed te ltete d emloyed te. Moe elly we o o te dom oted ottlee P-tye eglto w w ooed [3 d te mot otted o te vey ew omle eed eglto tt ve ee emloyed eld oeto [7. Te mlto eeted Seto 4 o te eet wo tdy te ote oete o te oeodg loed-loo ytem de te eet o meemet d modellg eo well eome e. mot e t w od tt te eome d te ote oete gteed y te mlemetto o te ooed eed lw wee good d omle to te eome d te ote oete ded y te P eglto. te ee o meemet d modellg eo te eome o te ooed eed lw w ette t te eome o te P eglto. Notto. Togot t mt we dot te ollowg otto [0. Fo evey et S S S S tme ts deote te teo o 0 S. o evey otve tege. y A ; Ω we deote te l o oto to o. Fo et A w te vle S m Ω. y A ; Ω wee tege we deote te l o to o A wt oto devtve o m ode w te vle Ω. et. Te toe o deoted y. y we deote te Elde om o.. Feewy model d te oete We ode eewy w ot o 3 omoet o ell; tyl ell legt my e 00-500 m. E ell my ve etel otollle low o-m low loted e te ell tem ody; d etel otlow o-m low loted e te ell dowtem ody Fge. Te me o vele t tme t 0 omoet {... } deoted y t. Te totl otlow d te totl low o vele o te omoet {... } t tme t 0 e deoted y q t 0 d F t 0 eetvely. All low dg tme tevl e meed [ve. oeqetly te le o vele oevto eqto o e omoet {... } gve t t q t F t... t 0.. E omoet o te etwo toge ty 0.... O t mto tte tt te et low om e ell e ott eetge o te totl et low.e. tee et ott [0... tt low o vele q t om omoet to omoet o....

low o vele om omoet to ego ot o te q t eewy o.....3 Te ott e ow et te.e. oto o q tt e od o te o-m o te -t ell. Se te -t ell te lt dowtem ell o te odeed eewy we my me tt. We lo me tt < o... d tt ll et to ego ot o te etwo.e. ll o-m well te m et ommodte te eetve et low. O eod mto delg wt te ttemted otlow.e. te low tt wll et te ell tee 0 et e te dowtem ell. We me tt tee et to [0 ; wt 0 < < o ll 0 d vle t [0... o tt q t t... t 0 d q t t..4 t Te vle t [0 o e... dte te eetge o te ttemted otlow om ell tt eome tl otlow om te me ell. Te to [0 wt te elzed ltete o T Egeeg ee e.g. [9 lled te demd-t o te dmetl dgm o te -t ell.e. te low tt wll et te ell tee et e te dowtem ell. Note tt eqto.4 o q t ollow om o mto tt ll et to ego ot o te etwo.e. ll o-m well te m et ommodte te eetve et low. et 0... deote te ttemted etel low to omoet {... } om te ego ot o te eewy. Tylly... oeod to etel o-m low w my e detemed y m meteg otol ttegy. Fo te vey t ell we me o oveee tt tee t oe etel low ; t deed te e we e ddeg mtem meteg olem [5; o te ote d we e ddeg m meteg olem [6 ome ot te o-m low o ell d te mtem demd low vg om tem. et te vle w t [0... dte te eetge o te ttemted etel low to omoet {... } tt eome tl low. Te we ot om. d.4 F t w t d F t w t t t.....5 t t O et mto te yl eqemet tt te low o vele t te ell {... } t tme t 0 deoted y t 0 ot eeed ed eetge o te me o ee oto o vele o ell F {... } t tme t 0.e. F t t... t 0.6 wee 0... e te ed eetge o te me o ee oto o vele. Followg [8 we me tt we t t t o ome {... } te we ve t w t. Smlly we t t te we ve w t. We t t t o ome {... } o we t t te we ve F t t o F t. Teeoe we get t t t F t m t t 0.7 t t d t m m 0 d m t t t t t t... t 0.8 F t m t... t 0.9 3

Fge Te eewy model emtlly. wee d t [0... t 0..0 e tme-vyg mete. Altog te mete d t [0 e etmted y e o eml o tte-elted ee e.g. [ dt we tey e ott o we tey e lowly vyg we wll tet tem ow tme-vyg mete dte. Te ede old ote tt y todg te mete d t [0 d y llowg tem to e tme-vyg we ve te to ot ll ole e o te eltve ote o te low d we lo llow te oty le to e tme-vyg; ee [6 7 8 o eewy model wt e oty le w e el e o o geel o. All te ove e lltted Fge. omg eqto...3.4.7 d.9 we ot te ollowg dete-tme dyml ytem m w. m w o... m w...3 wee [0... e gve y.8 d w [0.... Dee S 0 0 0. Se te to [0 ty 0 < < o ll 0 t ollow tt...3 et otol ytem o S 0 0 0.e.... S wt t d... d [0... 0 d dte d. Note tt te etty d d... d [0 e te eqto.. d.3 oly we ogeto eome e eet te te t ell.e. oly we t t t o ome {... }. We et me te ollowg mto o te to [0... H Fo e... tee et δ 0 tt eg o [ 0 δ d o-eg o [ δ. Moeove tee et ott 0 δ 0 δ tt [0 o 0 δ d o ll 0 δ d o ll 0 δ. H Fo ll... te eqlty δ < old. Amto H elet te oete o te o-lled demd to [9; weey δ te tl dety wee eve mmm vle. Note tt Amto H lde te olty o eded low o ovetl dete.e. we δ to elet te ty do eomeo ooed [0. t 4

Amto H tel mto w ted o lmot ll e. Howeve mto H H ve o-tvl oeqee. A lt o te mot mott oeqee o mto H H gve elow. oeqee o mto H H Te mg [ 0 0 e o-deeg o.... Poety det oeqee o te t tt o ll 0 δ d te t tt o-eg o δ. [ Fo e... tee et ott λ 0 G [0 tt λ d G o ll [0 δ.4 Poety det oeqee o te t tt tee et ott 0 δ 0 δ tt [0 o 0 δ d o ll 0 δ d o ll 0 δ. We olde tt.4 old wt λ 0 d G. 3 Tee et ott 0... tt o ll 0 d.... Poety 3 det oeqee o te t tt t tt o ll 0 δ [ o ll 0 δ det oeqee o te. [ δ d te deto m m 0 Te ollowg oeqee ovde el le lowe od o wegted m o et te. t oo ovded t te Aed. 4 Fo evey... t d o evey... 0 wt δ < o... tee et ott 0 tt te ollowg eqlty old o ll S 0 0 U 0 [0... [0 d d... [0 wt d..5 em. Te oo o oety 4 mle tt te ott 0 e omted tgtowd wy we.... We dee te otve ott ϕ δ o... d Y 0... g te eve oml Y m Y ϕ... wt Y wee 0... e te ott volved Poety 3. Te te ott 0 e eleted Y. Flly te lt oeqee ovde el eqlte d eqlte o wegted m o ll vele dete o te eewy. t oo ovded t te Aed. 5 Fo evey... t te ollowg eqlty old o ll S 0 0 d... d [0... 0 d wt 5

wee w.6 o.... Moeove o evey... 0 wt δ < o... te ollowg eqlty old o ll d S U [0.7 wee U 0 [0... [0 d 0 te ott volved.5. olo te model.-.3 geelzed veo o te ow t-ode dete Godov omto to te emt-wve tl deetl eqto o te W-model ee [3 8 wt ole [9 o eewe le ell Tmo Model - TM [7 otlow to. Howeve te eeted mewo lo ommodte eet modto o te W-model [0 to elet te o-lled ty do eomeo. 3. ot Glol Eoetl Stlzto o Feewy ode te eewy model...3 de mto H H. We oe tt tee et 0 0... d veto... 0 δ 0 δ wt... d <... <...3. t eto we tegte mto H d we me tt H3 Te ollowg eqlty old o ll... t te ogeted eqlm ot o te eewy model Note tt H3 H. Dee * δ. 3. m δ δ o... d m δ. 3. De to te eqlte 3. d te t tt < t ollow tt o.... Moeove t ollow om...3 d 3. tt te ollowg eqto old we Ω 0... 0 d [0 o... * w o... d o... 3.3 o.... 3.4 Dee te veto eld F [0 d... 0 S 0 S o ll S 0 0 d d... [0 d 6

F d F d... F d wt F d m F d m o... F d m d d m m 0 d m o.... 3.5 Te ollowg elt o m elt eed deg. Te elt ow tt oto ot glol eoetl tlze et o evey eewy model de mto H H3. Te oml o te eed lw elt. Teoem 3. ode ytem...3 wt 3 de mto H H3. et λ 0 G 0... o... e te ott volved Poety. et... 0 e ty ott d let 0 e ott o tt m λ m λ G <.... Te tee et et {... } o te et o ll de {... } wt 0 ott 0 o d ott 0 tt o evey 0 te eed lw S deed y... wt m γ o ll S d o ll S 3.6 wee γ d 0 m o ll S 3.7 eve ot glol eoetl tlzto o te ogeted eqlm ot... 0 δ 0 δ o ytem...3.e. tee et ott M 0 0 tt o evey 0 S d o evey eqee { } d t [0 t 0 te olto t S o t F d t t t wt F [0 S 0 S deed y 3.5 0 0 te te etmte t M e t 0 o ll tege t 0. Moeove o evey 0 tee et ott A K 0 o tt te to S deed y A K m 0 P o ll S 3.8 wee o... d P m 3.9 yov to o te loed-loo ytem...3 wt.e. tee et ott ott λ [0 K K 0 tt te to S te te eqlte K F d λ o ll d [0 S. K 7

8 Poo wt ollow we wll ve. et * e el ott wt < δ o... d let 0 e te ott volved.5. et }... { e et o te et o ll de }... { o w 0 d tt < m... d <... m 3.0 wee * o... e te ott deed y 3.. S et }... { lwy et o emle }... { e te et o ll de }... { o w 0. eqlte 3.9 mly tt tee et ott 0 ε d 0 o tt m... d ε... m. 3. We et eom te ollowg oede Dee... m d Ω ote tt wee 0 te ott volved.5. Dee...... m m m d ε... m. Dee m d let 0. Dee γ o d d 0 A o tt A γ. Fd 0 K o tt A A K ε... m m. We et ove te mlto 0... 0 Ω [0 d d [0 o... te. 3. wee m m... G λ λ d d F. deed g 3.3 3.4 d deto 3.7 we get o ll 0... 0 Ω [0 d d [0...

m m 0 m 0 0 m 0 3.3 Ug.4 te t tt δ o... oeqee o 3. d te t tt eg o [ 0 δ o... oeqee o H we get 0 G m 0 m o ll 0.... 3.4 [ Ug oete te t tt δ o... oeqee o 3. we get 0 m 0 m λ o ll 0.... 3.5 omg 3.3 3.4 3.5 we ot mlto 3.. Net we ow te mlto S d [0 d [0 o... te P P. 3.6 wee F d. deed 3.6 det oeqee o 3. d deto 3.9 we Ω 0... 0. O te ote d we S \ Ω tee et t let oe {... } o w m d oeqetly deto 3.9 gve P. Teeoe deto 3.7 mle... oeqee o te t tt... oeqee o 3.9 we get P P we S \ Ω. wt ollow we ve F d old [ m. Se P o ll S. Net we ow tt o ll S d [0 te ollowg eqlty We dtg two e.. 3.7 e Ω 0... 0 d [0. Deto 3.8 d eqto.6 3.3 3.4 wt gve K m 0 P A 3.8 wt. Ug 3. oety te t tt δ o... oeqee o 3. d deto mλ m λ G we get om 3.8... 9

0 0 m P K A 3.9 t ollow om te omto o 3.6 d eqlty 3.9 tt te ollowg eqlty old o ll Ω 0 m m P K A γ 3.0 Deto Ω mle tt. 3. Ug 3.9 d 3. we get 0 m m P K A γ 3. We et dtg two e e. t e we ve γ o ll. Se oeqee o < we get om 3. d deto 3.6 tt γ o ll d m m. Se o... m P... m γ oeqee o m 0 m... o... oeqee o deto 3.7 we get

0 m m m...... P γ γ omg 3. wt te ove eqlty we ot A γ. 3.3 t ollow om 3.3 d te t tt A γ tt 3.7 old we. e. t e γ o ll. Deto 3.6 mle tt o ll. Moeove t e tee et t let oe }... { o w *. Se eg o 0 [ o... oeqee o H d te t tt δ we olde tt tee et t let oe }... { o w. oeqetly we get om te t tt m... d o ll... m omg 3.6 3.0 wt te ove eqlty d g te t A γ we olde tt 3.7 old we. e Ω \ S [0 d. t e tee et t let oe }... { o w. Teeoe deto 3.7 mle... m d oeqetly deto 3.9 gve P. Moeove deto 3.6 gve o ll det oeqee o γ. omg we get om deto 3.8 d 3.6

0 m 0 m A K P K A 3.4 Ug.7 te t tt o ll ε... m d ε... m w ot mly tt we get 0 m 0 m. 3.5 omg 3.4 d 3.5 we get 0 m A K. 3.6 Deto 3.7 oto wt 3.6 mle tt te ollowg eqlty old A K 0 m m. 3.7 Te t tt tee et t let oe }... { o w d te t tt ε... m mly tt ε...... m m. 3.8 Ug 3.7 3.8 we ot K A K ε... m 0 m m 3.9 Se A A K ε... m m we olde om 3.9 d deto 3.8 tt 3.7 old.

3 Se 0 we get o ll S. 3.30 Smlly g deto 3.7 we get o ll S 0. 3.3 Ug 3.3 te t tt o... deto 3.9 d te t tt oeqee o.7 3. d te t tt we get o ll S P P m 0 m m 0 m 0 m 0 m 0 m 0 m 3.3 t ollow om deto 3.8 d 3.30 3.3 3.3 tt tee et ott 0 K K tt te eqlty K K o ll S 3.33 old. Ug 3.7 3.30 d 3.33 we get o ll S [0 d K. Te ove eqlty mle tt te eqlty d F λ o ll S [0 d 3.34 old wt K λ. Note tt 0 λ. Glol Eoetl Stlty o te ogeted eqlm ot 0 0... δ δ o ytem...3 wt det oeqee o eqlte 3.33 3.34. deed te tte e wee d ot 0 0 S d o dte wee eet te we wold e le to e Teoem 3. o ge 765-766 [. Howeve e te et dyml ytem...3 wt deed o 0 0 S wt dte [0 d we ot e Teoem 3. o ge 765-766 [. Howeve we e 3.34 dtvely d ot te etmte 0 t t λ o evey olto o te

4 dyml ytem...3 wt o evey eqee { } 0 [0 t t d d o evey tege 0 t. Te eqed etmte o te olto oted y omg te evo etmte wt 3.33. Te oo omlete. em 3. Te oo o Teoem 3. ovde metodology o otg etmto o te et }... { d te tl ott 0. et 0... e ty ott. et * e el ott wt < δ o... d let 0 e te ott volved.5. Selet }... { to e et o te et o ll de }... { o w 0 d o w tee et 0 tt m... d <... m wee * o... e te ott deed y 3.. et 0 ε e ott w te ε... m. Te etmto o te tl ott 0 my e doe te ollowg wy Dee... m d Ω wee 0... 0 Ω. Dee...... m m m d ε... m. Te etmted vle o 0 gve y m. Howeve te etmted vle o 0 w oted y lyg te ove metodology my e oevtve gtly mlle t te tl vle. Te e o te eleto o e low tt mt e otolled o te tlzto o te ogeted eqlm ot o eewy l. Te ollowg emle lltte ow Teoem 3. e ed o eleto. Emle 3.3 ode eewy tet w ot o 4 ell wee te t tee ell ve oe o-m d te lt oe oe o-m. E ell mmm dety 0 d we lo ve 4... 0 o 3. Te otlow to e gve y [ 50 9 / 54 05 0 9 o o 4....

lely mto H H old wt δ δ 5 / 0... 4. We et llte te vle o ott 0 tt te.5 o... 4 d 3. 6 3 0. 0 d 4 0. Ug em. we get 0.... 4 d 0. 00895. * 3 Net we ode low 3. 6 0 d 0. Te ogeted eqlm ot et o ll < 9 / 0 d te mto H3. Te ott to e 5 o... 4 d o ll < 9 / 0. 4 * o... 4 tyg 3. e omted y 3. We et ode te qeto Fo wt vle o 09 / 0 te ogeted eqlm ot e glolly eoetlly tlzed y otollg oly te low.e. o wt vle o 09 / 0 we ve { }? eqlte 3.0 old o { } ovded tt <. Teeoe we olde tt te ogeted eqlm ot e glolly eoetlly tlzed y otollg oly te low o 3 < 0. 00895. Te we my e oevtve e te etmto o te ott 0 tt te.5 y me o em. oevtve d e te eleto... 4 ot eely otml. 4. Smlto t eto we ode eewy model o te om...3 wt 5 [ 06 / 5 6 / 50 o 5 0... 5... 4 5 5 / 85 o [ 06 / 5 5 o 0 5 5 5 / 85 o 6 / 50 * t o 34 5 0 o... 4. 4. Fo ede wo e tomed to te tdtol t o ve/ o low d ve/m o dete te emle model my e vewed to elet eewy tet wt 3-le ell wt eql legt o 500 m; wt tme te o 5. Alo e tem o...3 meed omlzed t o 6 ve. Wt tee ettg te tl dety o 6/5 4. oeod to 34 ve/m/le; wle te m dety o 0 4. oeod to 07 ve/m/le; d te ell low te o 3/5 o ell 5 d.6 o ell... 4 4. oeod to 638 ve//le d 048 ve//le eetvely. Fom 4. ell 5 0% lowe low ty e.g. de to gde o vte o tel o dge et. t te t o ell d teeoe otetl ottlee o te eewy. We lo me t emle tt tee e o temedte o-m d o-m te oly otol olty eg te low o te t ell. Amto H old wt δ δ 6 / 5... 5 /... 4 5 3/ 5. Te ogeted * * eqlm ot... 4 5 5 / et o < 3 / 5. Smlto eveled tt te ogeted eqlm ot glolly eoetlly tle o te oe-loo ytem d m low <. 007. Fo ge vle o te m low te ogeted eqlm ot ot glolly eoetlly tle de to te etee o ddtol ogeted eql. Amto H3 old o ll < 3 / 5.e. o ll vle o te m low o w te ogeted eqlm ot et. Moeove oety old wt λ G /... 4 λ 5 3/ 5 d G 5 / 5. Teeoe we e oto to eve glol eoetl tlzto o te ogeted eqlm ot o model 4. y g Teoem 3.. deed Teoem 3. gtee tt o evey 0 tee et ott 0 d ott γ 0 tt o evey γ γ te eed lw 00 deed y 0 5 m γ m 4. 5

eve ot glol eoetl tlzto o te ogeted eqlm ot... 06 / 5 06 / 5 o ytem 4.. 5 We eleted. 79 w vey loe to 3/5.8 te ty low o ell 5; e. 007 t oeod to ogeted eqlm w ot oe-loo glolly eoetlly tle. Te oeodg eqlm vle e *. 558 o... 4 d 5 3. 975. Te vle o te ott 0 w oe to e 0.0; t te low mmm low vle te t llow ee to tdy te dym oete o te eglto ode ele otol e. o vle o te ott 0 d γ 0 wee teted y eomg mlto tdy wt eet to vo tl odto. ode to evlte te eome o te otolle o vo vle o te mete 0 d γ 0 we ed eome teo te totl me o ele Etg te Feewy EF o te tevl [ 0 T.e. * EF T T t 0 5 5 t. 4.3 Note tt te eewy eom et d totl dely e mmed EF mmzed; te mmm teoetl vle o EF.8 T w eved ell 5 oetg t ty.8 t ll tme. Te elt o T 00 e ow te Tle ote tt te mmm teoetl vle o EF o T 00 57.8. t le om Tle tt low vle o 0 eqe lge vle o γ 0 ode to ve glol eoetl tlty o te loed-loo ytem. γ 0.95 0.9 0.85 0.75 0.5 0.3 0. 0.5 4.653 39.770 3.7489 No St. No St. No St. No St. 0.5 4.687 4.7043 39.63 No St. No St. No St. No St. 0. 39.839 4.4873 4.779 No St. No St. No St. No St. 0.35 36.9 38.03 39.7896 4.9896 No St. No St. No St..5 34.6698 34.804 34.9068 35.4509 43.875 No St. No St. 8 33.9565 33.9565 33.9565 33.9565 34.974 43.358 No St. 4400 33.9565 33.9565 33.9565 33.9565 33.9565 33.9565 43.7883 Tle le o EF 00 o te loed-loo ytem 4. wt 4.. 79 o vo vle o 0 d γ 0. Te tl odto w te ot 00...0.e. ll ell t m dety. Te No St. ety me tt o te e vle o 0 d γ 0 te ogeted eqlm ot ot glolly eoetlly tle. Te eoe o te dete o evey ell o te loed-loo ytem 4. wt te ooed eed eglto 4. wt 0. 95 γ 0. 5 d tl odto 0 [.4.4.4.43.5 e ow Fge. Note tt te tl odto vle e lgtly elow ty o te t o ell wle te dety o te 5 t ell ovetl lgtly ogeted. Fo t e we d EF 00 56. 67. Te eed eglto ee to eod vey ttoly t tet d eve odgly g eome. All ollowg tet o te ooed eglto 4. wee odted wt te me vle 0. 95 d γ 0. 5. A detled omo o te ooed eed eglto 4. w mde wt te dom oted ottlee P eglto w w ooed [3 d oe o te vey ew omle eed eglto tt ee emloyed eld oeto [7. Te P eglto o ytem 4. mlemeted ollow m 0 t t m v t K t t K t v t m m m m 6 o... 5 4.4 m v t ev t e v t o... 5 4.5

m v t m t m l {345} vl t m 4.6...5 t t t v. 4.7 Dety 3.6 3.4 3. 3.8 t ell d ell 3 d ell 4 t ell 5 t ell.6.4 5 0 5 0 5 30 35 40 45 50 t Fge Te eoe o te dete o evey ell o te loed-loo ytem 4. wt te ooed eed eglto 4. wt 0. 95 γ 0. 5. 79 d tl odto 0 [.4.4.4.43.5. Eetlly 4.4 elet te llel oeto o ve oded P-tye eglto oe o e ell; wle 4.5 eom eoetl mootg o te oted low ee [3 o te god d eog o t o. Te mete o te P eglto wee eleted te ome mel eemet to e K 0.05 K 0. 05 0. 5 e 0.5 m 0.0 d m.6. Note tt ll P eglto wee gve te me g vle o mlty d oveee ggeted [3. ll eoted tet te tl odto o te m P eglto w v 0 v 0 0. 79 o... 5. We led to te me tl odto 0 [.4.4.4.43.5 te P eglto led to vey ml ovegee te ooed eglto 4. Fge d eved detl eome EF 00 56. 67. Howeve evel ote eemet wt vo level o tl ogeto te ooed eglto 4. eted te eome t te P eglto. Fo emle Fge 3 ow te evolto o te Elde om t o te loed-loo ytem 4. wt te ooed eed eglto 4. le ve d o te loedloo ytem 4. wt te P eglto 4.4 4.5 4.6 4.7 ed ve. Te tl odto w 00...0 eletg lly ogeted ogl tte. t le tt te ooed eed eglto 4. eve te ovegee. T lo eleted te omted vle o EF 00 4. 6 o te ooed eed eglto 4. d EF 00 3. 88 o te P eglto. We et vetgted te ote o te ooed eed eglto wt eet to meemet eo. Te led oml o te meemet w t Ae t t P 4.8 wee P te oeto oeto o te loe o S e t omlzed veto d A 0 te mgtde o te meemet eo. t e te eed lw 4. w mlemeted ed o te tte meemet t gve y 4.8.e. 5 t m γ m 0 t. 4.9 Fo omo oe we lo eet te eome o te P eglto o ytem 4. de te me meemet eo. t e eqto 4.4 eled y te eqto 7

m 0 t t m v t K t t K t v t m m m o... 5 4.0 wee te tte meemet t gve y 4.8. 5 t- * 0 5 0 0 40 60 80 00 0 40 60 t Fge 3 Te evolto o t o te loed-loo ytem 4. wt te ooed eed eglto 4. le ve; d o te loed-loo ytem 4. wt te P eglto 4.4 4.5 4.6 4.7 ed ve. Te tl odto w 00...0. Fge 4 ow te eoe o te dete o evey ell o two e o te loed-loo ytem 4. wt te ooed eed eglto 4.9; d o te loed-loo ytem 4. wt te P eglto 4.0 o ω t 4.5 4.6 4.7; wee te tte meemet ot e gve y 4.8 wt A 0. 4 e t... 5 ω π. Te tl odto w te ogeted eqlm ot. Dety 3.4 3. 3.8.6 t ell d ell 3 d ell 4 t ell 5 t ell.4. Dety 3.4 3. 3.8.6 t ell d ell 3 d ell 4 t ell 5 t ell.4. 0 40 60 80 00 0 40 60 t Fge 4 Te eoe o te dete o evey ell o te loed-loo ytem 4. wt te ooed eed eglto 4.9; d te loed-loo ytem 4. wt te P eglto 4.0 4.5 4.6 4.7. o ω t ot e te tte meemet gve y 4.8 wt A 0. 4 e t... ω π ; tl 5 odto w te ogeted eqlm ot. t tet te P eglto le etve to meemet eo t te ooed eed eglto 4.9 te ltte odg vle oet Fge 4 tl lo o ell 5 w ede odgly te ttoy otlow. T lo eleted te omted vle o EF 00 5. 34 o te ooed eed eglto 4.9 w.3% le t te mmm vle o EF 00 d EF 00 56. 7 o te P eglto w 0.43% le t te mmm vle o EF 00 de to te meemet eo. Te ltmte me vle o te tte 8

e m loe to te eqlm vle o te P eglto t o te ooed eed eglto 4.9 dtg tt te P eglto eve m mlle me oet t e. t old e oted t t ot tt vo eqee ω wee teted o meemet eo. Wle Fge 4 tyl o medm d g eqee te P eglto eve mlle me oet t te ooed eed eglto 4.9 te elt dte ge etvty o te P eglto wt eet to meemet eo t low eqee. Fo low eqey meemet eo te ooed eed eglto 4.9 eve mlle me oet t te P eglto ow Fge 5. Fge 5 ow te eoe o te dete o evey ell o two e o te loed-loo ytem 4. wt te ooed eed eglto 4.9 d o te loed-loo ytem 4. wt te P eglto 4.0 4.5 4.6 4.7 wee te tte meemet ot e gve y 4.8 wt 0. 4 eqlm ot. o ω t ω 0.. Te tl odto w te ogeted 5 A e t... Dety Dety 5 4.5 4 3.5 3.5.5 5 4.5 4 3.5 3.5 t ell d ell 3 d ell 4 t ell 5 t ell t ell d ell 3 d ell 4 t ell 5 t ell.5 50 00 50 00 50 300 t Fge 5 Te eoe o te dete o evey ell o two e o te loed-loo ytem 4. wt te ooed eed eglto 4.9; d o te loed-loo ytem 4. wt te P eglto 4.0 4.5 4.6 4.7. ot e te tte meemet gve y 4.8 wt A 0. 4 o ω t e t... ω 0. ; tl odto w te ogeted eqlm ot. 5 Flly we vetgted te ote o te ooed eed eglto wt eet to modelg eo. We odeed te eee o ott otollle d ow low 0 w et to te 3 d ell o te eewy. Moeove we me eql ote o te low o te 3 d ell.e. we me tt 3 t w3 t. t e te model 4. eome 3 0 m 0 m m 0 3 3 m 0 3 0 m 0 3 3 m 4 3 3 3 0 m 0 4 4 m 5 4 4 4 3 3 3 4. 4. 4.3 4.4 9

5 5 5 5 m 0 5 4 4 4.5 Alyg 4. to 4.-4.5 lwy wt 0. 95 γ 0. 5 d. 79 wt 3 0.0 we oeved o vo tl odto te ovegee o te olto t to ot deet om te ogeted eqlm ot. Te P eglto 4.4 4.5 4.6 4.7 eve mlle oet. Te olto o 4.- 4.5 wt 4..e. te ooed eglto d te olto o 4.-4.5 wt 4.4 4.5 4.6 4.7.e. te P eglto e ow Fge 6. Te tl odto Fge 6 te ogeted eqlm o 4. wle te ow low 3 0.0. We d EF 00 55. 45 o te ooed eed eglto 4. dtg deee o 0.7% o te mmm EF 00 d we d EF 00 56. 6 o P eglto dtg deee o 0.44% o te mmm EF 00 de to te eee o ow low. Fo ge vle o 3 0 te etvty o te ooed eed lw 4. eome eve moe ooed; t evo de to te tlzto o te me vle. 79 4. w oweve ot led to ovegee to te deed eqlm eee o ow temedte ow - o otlow. Ogog wo med to move te evo o te eglto t eet. Dety 3.6 3.4 3. 3.8.6.4. t ell d ell 3 d ell 4 t ell 5 t ell Dety 3.6 t ell 3.4 3. 3.8.6.4. d ell 3 d ell 4 t ell 5 t ell 0 40 60 80 00 0 40 60 t Fge 6 Te eoe o te dete o evey ell o two e te olto o 4.-4.5 wt te ooed eglto 4.; d te olto o 4.-4.5 wt te P eglto 4.4 4.5 4.6 4.7. ot e ow low 3 0. 0 w led; te tl odto w te ogeted eqlm ot. Te olo o t mlto tdy e Te ooed eed eglto 4. eve te ovegee o te tte to te eqlm omed to te P eglto te ee o meemet d modelg eo. Te ooed eed eglto 4. qte ot to meemet d modellg eo. Howeve t m moe etve to modellg eo t te P eglto. teded te eteo e eeted to move te oete o te ooed eed eglto t eet. t lo moe etve to meemet eo o medm d g eqey t te P eglto t le etve to loweqey meemet eo t te P eglto. 0

5. olo T wo ovded goo metodology o te otto o metezed mly o elt eed lw tt gtee te ot glol eoetl tlty o te ogeted eqlm ot o geel ole d et dete-tme eewy model. Te otto o te glol eoetl eed tlze w ed o te F o well o et mott oete o eewy model. Detled mlto-ed omo wee mde wt etg eed lw w wee ooed te ltete d ve ee tl e. Moe elly we omed te eome d te ote oete o te loed-loo ytem de te eet o te ooed eed lw d de te eet o te dom oted ottlee P eglto [3. mot e t w od tt te eome d te ote oete gteed y te mlemetto o te ooed eed lw wee good d omle to te eome d te ote oete ded y te P eglto; meely eee o tog d ow - o otlow dowtem o te otollle low te ooed eglto my ode moe gt devto om te deed eqlm tte. Fte ee wll dde te ote e goo wy te owledge o yov to o te loed-loo ytem e eloted to t oe d elt oml o te g o vo t meemet o modellg eo e deved. Alo te etmto o te g o vo t llow te tdy d otol o teoeted eewy t etwo. Flly te eet o doe ot ode te mt o low otol o tem t low odto e.g. qee omg t o-m; te eteo wll dde tee e otely. Aowledgmet Te ee ledg to tee elt eeved dg om te Eoe ee ol de te Eoe Uo' Sevet Fmewo Pogmme FP/007-03 / E Gt Ageemet. [33 oet TAMAN. eeee [ -Ge H. d S. A Eml Moo Evlto o Feewy Mege-to Totto ee Pt 84 00 457-470. [ ge M. M. v de eg A. Hegy. De Stte d J. Helledoo odeto o Model-ed T otol Totto ee - Pt 35 03 9. [3 lo... Pml M. Pgeogo d A. Meme Otml Motowy T Flow otol volvg le Seed mt d m Meteg Totto See 44 00 38-53. [4 lo... Pml M. Pgeogo d A. Meme le Seed mt Mle Meteg Deve 89 t Al Meetg o te Totto ee od Wgto D.. Jy 0-4 00 e No. 0-59. [5 lo... Pml d M. Pgeogo ol Feed-ed Mtem T Flow otol o Motowy Ug le Seed mt EEE Tto o tellget Totto Sytem 0 6-76. [6 oog S. d M. A Dyml Poete o omtmetl Model o T Netwo Poeedg o te Ame otol oeee 04. [7 Dgzo. Te ell Tmo Model A Dym eeetto o Hgwy T otet Wt te Hydodym Teoy Totto ee Pt 84 994 69 87. [8 Dgzo.F. Te ell Tmo Model. Pt Netwo T Totto ee Pt 9 995 79 93. [9 Godov S. A Deee Metod o Nmel llto o Doto Solto o Hydodym Eqto Mtemte So 47 959 7 90. [0 Gome G. d. Hoowtz Otml Feewy m Meteg Ug te Aymmet ell Tmo Model Totto ee Pt 44 006 44-6. [ Hddd W. M. d. ello Nole Dyml Sytem d otol A yov-ed Ao Peto Uvety Pe 008 Peto. [ Hegy A.. De Stte d H. Helledoo Model Pedtve otol o Otml oodto o m Meteg d le Seed mt Totto ee Pt 3 005 85-09. [3 Ho Z. J.-X. X d J. Y A tetve eg Ao o Dety otol o Feewy T Flow v m Meteg Totto ee - Pt 6 008 7 97.

[4 oddo G.-.. ool. Pml d M. Pgeogo Feed-ed Mtem T Flow otol o Mltle ottlee o Motowy to e EEE Tto o tellget Totto Sytem. [5 Koo P. d K. Ozy Feed m Meteg tellget Totto Sytem Sge 003. [6 Kyll. d Z.-P. Jg Stlty d Stlzto o Nole Sytem Sge-elg odo See ommto d otol Egeeg 0. [7 Kyll. d M. Pgeogo Glol Stlty elt o T Netwo mtted to EEE Tto o otol o Netwo Sytem ee lo Xv40.0496 [mt.o. [8 Kzy A. A. d P. y Atve T Mgemet o od Netwo A Moo Ao Plool Tto o te oyl Soety A Mtemtl Pyl d Egeeg See 36898 00 4607 466. [9 eqe J. P. Te Godov Seme d Wt t Me o Ft Ode T Flow Model Poeedg o te 3t tetol Symom o Totto d T Teoy 996 Pegmo 647 677. [0 eqe J. P. Two-Pe oded Aeleto T Flow Model Alytl Solto d Alto Totto ee eod 85 Pe No. 03-4397 003 0-30. [ eqe J. P. d M. Koy Ft Ode Moo T Flow Model o Netwo te otet o Dym Agmet Totto Plg 004 9 40. [ eqe J. P. teeto Modelg Alto to Moo Netwo T Flow Model d T Mgemet T d Gl Flow 03 005 6 78. [3 gtll M.d G. Wtm O Kemt Wve Flow Movemet og ve. A Teoy o T Flow o og owded od Poeedg o te oyl Soety o odo Pt A 978 955 8 345. [4 Pgeogo M. H. Hd-Slem d J.-M. loevlle ANEA A ol Feed otol w o O- m Meteg Totto ee eod 30 99 58-64. [5 Pgeogo M. H. H-Slem d F. Mddelm ANEA ol m Meteg Smmy o Feld elt Totto ee eod 603 997 90-98. [6 Pgeogo M. d A. Kotlo Feewy m Meteg A Ovevew EEE Tto o tellget Totto Sytem 3 00 7-8. [7 Pml. M. Pgeogo. og d J. Gey Het m-meteg oodto Sttegy mlemeted t Mo Feewy Atl Totto ee eod 78 00 0-0. [8 d P. So Wve o te Hgwy Oeto ee 4 956 4 5. [9 Sly N. d P. Koo Feed m Meteg Ug Godov Metod ed Hyd Model Jol o Dym Sytem Meemet d otol 355 03. [30 S X. d. Hoowtz Set o New T-eove m-meteg Algotm d Moo Smlto elt Totto ee eod 959 007 9-8. [3 Wg Y. d M. Pgeogo ol m Meteg te e o Dtt Dowtem ottlee Poeedg o te EEE oeee o tellget Totto Sytem Tooto d Seteme 7-0 006 46-43. [3 Wg Y. M. Pgeogo J. Gey. Pml G. oe d W. Yog ol m Meteg dom-oto ottlee Dowtem o Meteed O-m Totto ee eod No. 78 00 90-00. Aed Poo o 4 Fo oveee we et o.... We et ove te ollowg lm lm Fo ll m... tee et ott 0 tt te ollowg eqlty old o ll S 0 0 U 0 [0... [0 d d... [0 m m m d A. m Poety 4 det oeqee o te ove lm.

3 Ft we ove te lm o m. deed g Poety 3 we get o ll 0 0 [0... [0 0 U [0... d d d wt m A. O te ote d o ll 0 0 [0... [0 0 U [0... d d d wt < o g.8 we get δ w omed wt te t tt mle δ d δ δ A.3 Note tt o te devto o A.3 we ve ed oety 3 d te t tt 0 δ. t ollow om A. d A.3 tt A. old wt m δ. Net we oe tt te lm old o }... { m d we ow tt te lm old o m. Ug A. o m d Poety 3 we ot o ll 0 0 S [0... [0 0 U [0... d d d wt m A.4 O te ote d we ve o ll 0 0 S [0... [0 0 U [0... d d d wt < o g.8 w omed wt te t tt mle δ d

4 δ δ A.5 Note tt o te devto o A.5 we ve ed A. o m d te t tt 0 δ. t ollow om A.4 d A.5 tt te lm old o m wt m δ. Poo o 5 Te ollowg eqto old o ll 0 0 S 0... [0... d d d wt d e det oeqee o...3 d deto o... w o... A.6 w A.7 Eqlty.6 oeqee o. A.6 A.7 d deto. omg A.7 d.5 we get w o ll [0 U S d A.8 Se t ollow om A.8 tt.7 old.

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