GLOBAL EXPONENTIAL STABILIZATION OF FREEWAY MODELS

Transcription

1 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 [ 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-

2 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 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 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....

3 low o vele om omoet to ego ot o te q t eewy o 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 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

4 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 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 Se te to [0 ty 0 < < o ll 0 t ollow tt...3 et otol ytem o S e.... S wt t d... d [ 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

5 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 [ d wt 5

6 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 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 d [0 o... * w o... d o o Dee te veto eld F [0 d... 0 S 0 S o ll S 0 0 d d... [0 d 6

7 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 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 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 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 Ω [0 d d [0 o... te. 3. wee m m... G λ λ d d F. deed g d deto 3.7 we get o ll Ω [0 d d [0...

9 m m 0 m 0 0 m 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 [ Ug oete te t tt δ o... oeqee o 3. we get 0 m 0 m λ o ll omg 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 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 e Ω d [0. Deto 3.8 d eqto 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

10 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

11 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 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

12 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 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.

13 3 Se 0 we get o ll S Smlly g deto 3.7 we get o ll S 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 tt tee et ott 0 K K tt te eqlty K K o ll S 3.33 old. Ug 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 δ δ o ytem...3 wt det oeqee o eqlte 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 [. Howeve e te et dyml ytem...3 wt deed o 0 0 S wt dte [0 d we ot e Teoem 3. o ge [. Howeve we e 3.34 dtvely d ot te etmte 0 t t λ o evey olto o te

14 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 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 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 o 3. Te otlow to e gve y [ 50 9 / o o 4....

15 lely mto H H old wt δ δ 5 / We et llte te vle o ott 0 tt te.5 o... 4 d d 4 0. Ug em. we get d * 3 Net we ode low 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 < 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 / 85 o [ 06 / 5 5 o / 85 o 6 / 50 * t o o 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 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. Te ogeted * * eqlm ot / et o < 3 / 5. Smlto eveled tt te ogeted eqlm ot glolly eoetlly tle o te oe-loo ytem d m low < 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

16 eve ot glol eoetl tlzto o te ogeted eqlm ot / 5 06 / 5 o ytem 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 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 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 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 No St. No St. No St. No St No St. No St. No St. No St No St. No St. No St. No St No St. No St. No St No St. No St No St Tle le o EF 00 o te loed-loo ytem 4. wt o vo vle o 0 d γ 0. Te tl odto w te ot 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 γ 0. 5 d tl odto 0 [ 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 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 d γ 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 m v t ev t e v t o

17 m v t m t m l {345} vl t m t t t v. 4.7 Dety t ell d ell 3 d ell 4 t ell 5 t ell t Fge Te eoe o te dete o evey ell o te loed-loo ytem 4. wt te ooed eed eglto 4. wt γ d tl odto 0 [ 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 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 o We led to te me tl odto 0 [ te P eglto led to vey ml ovegee te ooed eglto 4. Fge d eved detl eome EF 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 ed ve. Te tl odto w eletg lly ogeted ogl tte. t le tt te ooed eed eglto 4. eve te ovegee. T lo eleted te omted vle o EF o te ooed eed eglto 4. d EF 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

18 m 0 t t m v t K t t K t v t m m m o wee te tte meemet t gve y t- * 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 ed ve. Te tl odto w 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 ; 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 t ell d ell 3 d ell 4 t ell 5 t ell.4. Dety t ell d ell 3 d ell 4 t ell 5 t ell 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 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 o te ooed eed eglto 4.9 w.3% le t te mmm vle o EF 00 d EF 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

19 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 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 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 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 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 m m m m m

20 m Alyg 4. to lwy wt γ 0. 5 d. 79 wt we oeved o vo tl odto te ovegee o te olto t to ot deet om te ogeted eqlm ot. Te P eglto eve mlle oet. Te olto o wt 4..e. te ooed eglto d te olto o wt e. te P eglto e ow Fge 6. Te tl odto Fge 6 te ogeted eqlm o 4. wle te ow low We d EF o te ooed eed eglto 4. dtg deee o 0.7% o te mmm EF 00 d we d EF 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 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 t ell d ell 3 d ell 4 t ell 5 t ell Dety 3.6 t ell d ell 3 d ell 4 t ell 5 t ell t Fge 6 Te eoe o te dete o evey ell o two e te olto o wt te ooed eglto 4.; d te olto o wt te P eglto ot e ow low 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

21 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/ / E Gt Ageemet. [33 oet TAMAN. eeee [ -Ge H. d S. A Eml Moo Evlto o Feewy Mege-to Totto ee Pt [ ge M. M. v de eg A. Hegy. De Stte d J. Helledoo odeto o Model-ed T otol Totto ee - Pt [3 lo... Pml M. Pgeogo d A. Meme Otml Motowy T Flow otol volvg le Seed mt d m Meteg Totto See [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 e No [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 [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 [8 Dgzo.F. Te ell Tmo Model. Pt Netwo T Totto ee Pt [9 Godov S. A Deee Metod o Nmel llto o Doto Solto o Hydodym Eqto Mtemte So [0 Gome G. d. Hoowtz Otml Feewy m Meteg Ug te Aymmet ell Tmo Model Totto ee Pt [ 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 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

22 [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 Xv [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 [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 [0 eqe J. P. Two-Pe oded Aeleto T Flow Model Alytl Solto d Alto Totto ee eod 85 Pe No [ eqe J. P. d M. Koy Ft Ode Moo T Flow Model o Netwo te otet o Dym Agmet Totto Plg [ eqe J. P. teeto Modelg Alto to Moo Netwo T Flow Model d T Mgemet T d Gl Flow [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 [4 Pgeogo M. H. Hd-Slem d J.-M. loevlle ANEA A ol Feed otol w o O- m Meteg Totto ee eod [5 Pgeogo M. H. H-Slem d F. Mddelm ANEA ol m Meteg Smmy o Feld elt Totto ee eod [6 Pgeogo M. d A. Kotlo Feewy m Meteg A Ovevew EEE Tto o tellget Totto Sytem [7 Pml. M. Pgeogo. og d J. Gey Het m-meteg oodto Sttegy mlemeted t Mo Feewy Atl Totto ee eod [8 d P. So Wve o te Hgwy Oeto ee [9 Sly N. d P. Koo Feed m Meteg Ug Godov Metod ed Hyd Model Jol o Dym Sytem Meemet d otol [30 S X. d. Hoowtz Set o New T-eove m-meteg Algotm d Moo Smlto elt Totto ee eod [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 [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 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.

23 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

24 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.