POSITIVITY AND REACHABILITY OF FRACTIONAL ELECTRICAL CIRCUITS

Transcription

1 asz Kaczo Posy a achably o Facoal Elccal cs POSIIVIY ND EHIIY OF FION EEI IUIS asz KZOEK* *Facly o Elccal Egg ałyso Usy o chology l Wsa D - ałyso aczo@sppwpl bsac: oos o h posy o acoal la lccal ccs copos o ssos cols cosaos a olag c socs a sablsh I s show ha: h acoal lccal cc copos o ssos cols a olag soc s pos o ay als o h ssacs cacs a soc olags a oly h b o cols s lss o qal o h b o s laly p shs h acoal lccal cc s o pos o ay als o s ssacs capacacs a soc olags ach s bach coas sso capaco a olag soc I s also show ha h acoal pos lccal ccs o yp a achabl a oly h cocacs bw h os a zo a h acoal pos lccal ccs o yp a achabl a oly h ssacs blogg o wo shs a zo INODUION yacal sys s call pos s acoy sag o ay oga al sa as o h pos oha o all oga ps ow o sa o h a pos syss hoy s g h oogaphs Faa a al ; Kaczo Vay o ols hag pos bhao ca b o gg coocs socal sccs bology a c c h oo o coollably a obsably a h coposo o la syss ha b oc by Kala hs oos a h basc cocps o h o cool hoy sals Mchl 6; Kaczo 999 ; Kalah 98; osboc 97; Woloch 97 hy ha b also x o pos la syss Faa a al ; Kaczo h coposo o h pa a o h pos sc- la sys has b ass Kaczo b h achably o la syss s closly la o h coollably o h syss Spcally o pos la syss h coos o h coollably a ch sog ha o h achably Kaczo ss o h achably a coollably o saa a pos la syss a g Kaczo 8b ; Klaa 99 h sably obs sably a sablzao o pos la syss ha b sga słowcz 8a 8b 8c 9 ; Kaczo c alyss o acoal lccal ccs has b ass Kaczo a a b I hs pap h cssay a sc coos o h posy a achably o acoal la lccal ccs copos o ssos cols cosaos spcosaos a olag c socs wll b sablsh h pap s ogaz as ollows I sco h sa qaos o h acoal la lccal ccs a h solos a ps h pos acoal la lccal ccs copos o ssos cosaos cols a olag socs a aalyz sco h achably o h acoal pos lccal ccs s sga sco oclg as a g sco h ollowg oao wll b s: h s o al bs h s o al acs h s o acs wh oga s a h s o Mzl acs al acs wh oga o-agoal s h y ax FION SE EQUIONS ND HEI SOUIONS I hs pap h ollowg apo o o h a-gal o acoal o wll b s Kaczo 8a c Γ < < N { } wh x Γ x x > s h gaa co a s h classcal o a

2 aca chaca aoaca ol o h c a spcosao sholy cosao wh h capacy b h o a o s chag q Kaczo a c q < < Usg q w oba wh s h olag o h cosao Slaly l h olag o col co wh h cac b h o a o s agc lx Ψ Kaczo a c Ψ < < 6 ag o acco ha Ψ w oba 7 wh s h c h col os a lccal cc copos o ssos capacos a olag socs Usg h qao a h Kchho s laws w ay scb h as sas h lccal cc by h acoal al qao x x < < 8 wh h copos o h sa co a p co a h olags o h cosaos a soc olags spcly os a lccal cc copos o ssos cols a socs Slaly sg h qao 6 a h Kchho s laws w ay scb h as sas h lccal cc by h acoal al qao x x < < 9 wh I hs cas h copos o h sa co a h cs h cols ho Solo o h qao 8 sasyg h al coo has h o x Φ x Φ wh Φ Φ < Γ Γ < [ ] Poo s g Kaczo 8a c Now l s cos lccal cc copos o ssos capacos cols a olag c soc s h sa aabls h copos o h sa co w choos h olags o h capacos a h cs h cols Usg h qaos 7 a Kchho s laws w ay w o h acoal la ccs h as sas h sa qao x x x x < < a wh h copos a olags o h cosaos h copos a cs h cols a h copos o a h soc olags b So xapls o lccal ccs scb by h qao a g Kaczo c c ho h solo o h qao o < < < < wh al coos x x x a x has h o x Φ x [ Φ Φ ] a wh x x x x x x I o l o l l b o l l l o l > Φ l l Γ l l l Φ l Γ l l Φ l Γ l l [ l ] [ ] Poo s g Kaczo c c h xso o ho o syss cossg o sbsyss wh acoal o s g Kaczo b

3 asz Kaczo Posy a achably o Facoal Elccal cs POSIIVE FION EEI IUIS Do h acoal lccal cc 8 o 9 s call h ally pos acoal sys h sa co o ay al coos a all Do sqa al ax [ ] s call h Mzl ax s o-agoal s a oga o Kaczo c ho h acoal lccal cc 8 s ally pos a oly M wh s h s o Mzl acs Poo s g Kaczo c Fo ho appl o h acoal cc ollows ha h lccal cc s pos a oly M ; Facoal yp lccal ccs ho h acoal lccal cc s o pos ach s bach coas ssos cosao a olag soc h poo s sla o h poo o ho Kaczo a os h acoal lccal cc show o Fg wh g cocacs ; capacacs a soc olags Fg Facoal lccal cc Usg a h Kchho s laws w ay w h qaos a Fo w ha Sbso o o yls 6 wh 7 Fo 7 ollows ha a ho h ollowg ho has b po ho h acoal lccal cc show o Fg s pos o ay als o h cocacs ; capacacs a soc olag I gal cas l s cos h acoal lccal cc copos o q cocacs ; capacacs a soc olags b h b o laly p os o h lccal cc a > Usg h Kchho s laws w ay w h qao 8 wh s h olag o h -h capaco s h olag o h -h o s h agoal Mzl ax a Usg h o ho w oba H F 9 wh s a Mzl ax a ag o acco ha o 9 w oba H F

4 aca chaca aoaca ol o Sbso o o 8 yls wh F H h lccal cc scb by h qao s pos a oly h ax s a Mzl ax a h ax has oga s ho h ollowg ho has b po ho h la lccal cc copos o q ssos capacos a soc olags s pos a oly < a M F H Facoal yp lccal ccs os h lccal cc show o Fg wh g ssacs cacs a soc olags Fg Facoal lccal cc Usg 7 a h sh ho o h lccal cc w oba h ollowg qaos a wh b No ha h s ax has pos s Fo w ha 6 wh 7 Fo 7 ollows ha a oly a 8 ho h acoal lccal cc s pos a oly h coo 8 s I gal cas l s cos h acoal -sh lccal cc wh g ssacs cacs o a sh soc olags Do by h sh cs I a sla way as o h lccal cc show o Fg sg h sh ho w oba h qao 9a wh 9b No ha a Plplyg 9a by w oba a wh b h acoal lccal cc s pos a oly h ax s a Mzl ax M ho h ollowg ho has b po ho h acoal la lccal cc copos o ssos cols a olag socs s pos o > s ssacs a cacs sasy h coo

5 asz Kaczo Posy a achably o Facoal Elccal cs a I h cas s possbl o choos h laly p shs so ha o ach sh blogs oly o col h h ax [ ] a h coo s o ay als o h ssacs a cacs o h lccal cc a No ha s possbl o choos h laly p shs so ha o ach sh blogs oly o col all bachs blogg o h sa o coa h cols I hs cas w ca la o o h bach cs sg h ac ha h s o h cs h cols s qal o zo Fo ho a a w ha h ollowg poa ho ho h acoal la lccal cc copos o ssos cols a olag socs s pos o alos all als o h ssacs cacs a soc olags a oly h b o cols s lss o qal o h b o s laly p shs a h cos o h sh cs a coss wh h cos o h sh soc olags Facoal yp lccal ccs os h acoal lccal cc show o Fg wh g ssac cac capacac a soc olag Fg Facoal lccal cc Usg h Kchho s laws w ca w h qaos whch ca b w h o wh a b h ax has ga o-agoal y -/ a s o a Mzl ax o ay als o ho h acoal lccal cc s o pos o o ay als o h ssacs cac capacac I gal cas w ha h ollowg ho ho 6 h acoal lccal ccs o yp s o pos o alos all als o s ssacs cacs capacacs a soc olags a las o s bach coas cac a capacac Poo I s wll-ow ha h la p shs o h lccal ccs ca b chos so ha h bach coag h cac a capacac blogs o h s o h qao o h s sh coas h ollowg wh a a h soc olag a c o h s sh a s h olag o h capacac Fo a / ollows ha h ax o h lccal cc has a las o ga oagoal y ho h ax s o a Mzl ax a h lccal cc s o pos o os h lccal cc show o Fg wh g ssacs cacs capacacs a soc olags Fg Facoal lccal cc 6

6 aca chaca aoaca ol o 7 Usg h Kchho s laws w ca w h qaos o a o b whch ca b w h o 6a wh 6b a ] ag[ M 6c h lccal cc scb by h qao 6 s pos o all al o h ssacs cacs capacacs ho h ollowg ho has b po ho 7 h acoal la lccal cc o h sc show o Fg s pos o ay als o s ssacs cacs a capacacs EHIIY OF FION POSIIVE INE EEI IUIS os h acoal pos la lccal cc scb by h qaos 8 9 a Do h acoal pos lccal cc 8 s call achabl o ay g al sa h xss a p o [ ] ha ss h sa o h cc o zo al sa o h al sa x I y sa s achabl h h cc s call achabl h acoal pos lccal cc s call achabl o y h xs a p o [ ] whch ss h sa o h cc o o x al sqa ax s call ooal ach s ow a ach s col coas oly o pos y a h ag s a zo ho h acoal pos lccal cc 8 s achabl h ax Φ Φ > s ooal h p ha ss h sa o h lccal cc o o h sa x s g by h ola x Φ o ] [ h poo s g Kaczo a ho I h ax [ ] a o a ooal acs h h acoal pos lccal cc 8 s achabl Poo Fo ollows ha s agoal h h ax Φ a Φ a also ooal o ooal ax Fo w h o Φ Φ ] [ ollows ha h ax s ooal ho by ho h acoal sys s achabl Exapl os h acoal lccal cc show o Fg wh g cocacs capacac a soc olags Fg Facoal lccal cc Usg h Kchho s laws w ca w h qaos a

7 asz Kaczo Posy a achably o Facoal Elccal cs 8 wh 6 s a Mzl ax a Fo w oba 7 Sbso o 7 o 8 w oba 9 wh Fo ollows ha s a Mzl ax a h ax has oga s ho h acoal lccal cc s pos o all als o h cocacs a capacacs W shall show ha h acoal pos lccal cc show o Fg s achabl a oly No ha h ax 6 s agoal a oly I hs cas o ollows ha s a agoal Mzl ax a s a agoal ax wh pos agoal s ho by ho h acoal pos lccal cc s achabl I gal cas l s cos h acoal lccal cc show o Fg wh cocacs ; capacacs a soc olags Fg Facoal lccal cc ho h acoal lccal cc show o Fg s pos o all als o h cocacs capacacs a soc olags Poo Usg h Kchho s laws a h o ho o h lccal cc w ay w h qaos a a b wh ] ag[ ] ag[ ] ag[ c s h s o cocacs o all bachs blogg o h -h o h ax a has oga s Sbsg b o a w oba

8 aca chaca aoaca ol o 9 a wh M I ] [ b a c sc h acs a ha oga s ho h lccal cc s pos ho h acoal pos lccal cc show o Fg s achabl a oly a o Poo h ax by c s agoal a oly h coo s I hs cas h acs a a also agoal a o w oba a wh b No ha h sbsys a s achabl ho h pos lccal cc s achabl a oly h coo s sas Exapl os h acoal lccal cc show o Fg wh g ssacs cacs a soc olags Fg Facoal lccal cc Usg h Kchho s laws w ca w h qaos whch ca b w h o 6a wh 6b h acoal lccal cc s pos sc h ax s Mzl ax a h ax has oga s W shall show ha h acoal pos cc s achabl I hs cas 7 a 8 a o w oba 9 h ax 9 s ooal a by ho h acoal pos lccal cc s achabl Now l s cos h acoal -sh lccal cc wh g ssacs cacs a -sh soc olags I s ass ha o ach laly p sh blogs oly o cac I hs cas h ax by 9b s agoal o a h coo s ho h acoal pos -shs lccal cc wh oly o cac ach laly p sh s achabl o wh a s o h ax by 9b Poo I h coo s h h Mzl ax s agoal h ax by 9b s also agoal sc by asspo oly o cac blogs o ach laly p sh I hs cas h ax s agoal Mzl ax a s also agoal Fo agoal Mzl ax a agoal h ax s also agoal a h ax by s ooal y ho h pos lccal cc s achabl

9 asz Kaczo Posy a achably o Facoal Elccal cs a h coo s h ssac o h bach blogg o wo laly p shs s zo hs sl s coss wh h o oba Exapl os h acoal lccal cc show o Fg wh g ssacs cacs capacac a soc olag Usg h Kchho s laws w ca w h qaos a a b Fo b w ha Sbsg o a w oba a wh b Fg Facoal lccal cc Fo b ollows ha h ax s o a Mzl ax I h coo s h h olag bw h pos a b s qal o zo a I hs cas h qao a as h o h acoal lccal cc scb by h qao s pos ho w ha h ollowg coollay oollay I h ssacs o h lccal cc sasy h coo h h acoal lccal cc s pos I gal cas w ha oollay Facoal opos lccal cc o so spcal choc o h paas ssacs ca b pos o Usg b s asy o chc ha ] a[ 6 a oly h coo s o sas ho w ha h ollowg coollay oollay h acoal saa opos lccal cc show o Fg s achabl a oly h coo s o sas Fo ollows ha h c acoal pos lccal cc s achabl hs cosaos ca b x o gal cas o yp lccal ccs ONUDIN EMKS h coos o h posy o acoal la lccal ccs copos o ssos cols cosaos a olag c socs ha b sablsh I has b show ha: h acoal lccal ccs copos o ssos cols a olag socs sholy call yp a pos o ay als o h ssacs c-

10 aca chaca aoaca ol o acs a soc olags a oly h b o cols s lss o qal o h b o s laly p shs ho h acoal lccal ccs copos o ssos cosaos a olag socs sholy call yp a o pos o ay als o s ssacs capacacs a olag socs ach h bach coas sso capaco a olag soc ho h acoal opos lccal ccs o h yp ca b pos o so spcal choc o h paas oollay h coos o h achably o h acoal pos lccal ccs ha b sablsh I has b show ha h acoal pos lccal cc o yp a achabl a oly h cocacs bw h os a zo ho a h acoal pos lccal ccs o yp a achabl a oly h ssacs blogg o wo shs a zo ho h acoal saa opos lccal ccs o yp a sally achabl a a achabl oly o so spcal choc o h paas h cosaos ha b llsa by xapls o la lccal ccs So o hs sls ca b also x o h coollably a obsably o h acoal la lccal cc Op pobl a xso o hs cosaos o h ollowg classs o h acoal syss: sb paas la syss; ola lccal ccs EFEENES sals PJ Mchl N 6 a Syss has oso słowcz M 8a Fqcy oa ho o sably aalyss o la coos- acoal syss I: Malows K a ows Es c acs ool a oao cac Pblshg Hos EXI Wasaw 8-9 słowcz M 8b Sably o la coos- acoal o syss wh lays o h a yp ll Pol ca Sc ch Sc Vol 6 No 9- słowcz M 8c obs sably o cox cobao o wo acoal g chaacsc polyoals ca Mchaca oaca Vol No - słowcz M 9 Sably aalyss o la coos acoal syss o cosa o Joal o oao Mobl obocs a Illg Syss Vol No -7 6 słowcz M obs sably o pos sc la syss o acoal o ll Pol ca Sc ch Vol 8 o Faa al S Pos a Syss; hoy a pplcaos J Wly Nw Yo 8 Kaczo 999 a ool Syss Vol J Wly Nw Yo 9 Kaczo Pos D a D Syss Spg Vlag oo Kaczo 8a Facoal pos coos- syss a h achably I J ppl Mah op Sc ol 8 o -8 Kaczo 8b achably a coollably o zo ss o saa a pos acoal sc- syss Joal Eopé s Sysès oasés JES Vol No Kaczo a alyss o acoal lccal ccs as sas OINS Szczy Pola a also Polsh Elccal w ol 86 o Kaczo b Dcoposo o h pa a o h pos sc- la syss chs o ool Sccs Vol No -7 Kaczo c Pos la syss wh acoal os ll Pol ca Sc ch Vol 8 o -8 Kaczo a Pos lccal ccs a h achably Poc o op pplcaos Elccal Egg pl - Poza Pola 6 Kaczo b Pos la syss cossg o sbsyss wh acoal os IEEE as a Sys ol 8 o7 7 Kaczo c Slc Pobls o Facoal Syss hoy Spg-Vlag l 8 Kalah 98 a Syss Pc-Hall Eglwoo ls Nw Yo 9 Kala E 96 O h al hoy o ool Syss Poc O h Fs I ogss o oac ool woh oo 8-9 Kala E 96 Mahacal Dscpos o a Syss SIM J ool Vol -9 Klaa J 99 oolably o Dyala Syss Klw cac Pbl Doh osboc HH 97 Sa-Spac a Mlaabl hoy J Wly Nw Yo Woloch W 97 a Mlaabl Syss Spg-Vlag Nw Yo hs wo was sppo by Msy o Scc a Hgh Ecao Pola wo No N N 689