Control of distributed parameter systems, 1982 : proceedings of the Third IFAC symposium, Toulouse, France, 29 June-2 July 1982

603 Pages·1983·23.839 MB·English

by  Jean-Pierre Babary, Laurent Le Letty, International Federation of Automatic Control

/FGA
Titlienst he ProceedinSgesr ies
AKASHCIo:n tSrcoile anncdTe e chnoflootrgh yPe r ogroefss JANSSEPNA,U STRASZADKy:n amMiocd elalnidnC go ntorfo l
SocieVtoyl,u mes NatioEncao&ln omics
7
ATHERTONM:u ltivaTreicahbnloel Soygsitceamls JOHANNSERNIJ NSDORAPn:a lyDseissi,ag nndE, v aluaotfi on
BABARYL EL ETTYC:o ntorfDo ils triPbaurtaemdeS tyesrt ems Man-Ma&c Shyisntee ms
& LAUBERS:a feotfCy o mputCeorn tSryoslt ems
(1982)
BANKS PRITCHARCDo:n torfoD li striPbaurtaemde ter LEININGCEoRm:p utAeird eDdes igonfM ultivariable
Sys&t ems TechnolSoysgtiecmasl
BEKEY SARIDIISd:e ntifaincdSa ytsitoPenam r ameter LEONHARCDo:n tirnoP lo weErl ectraonndEi lcesc tDrriicvaels
Est&i mation LESKIEWICZZ AREMBAP:n eumaatnidHc y draulic
CAMPBEL(CL1o:n9 t8Ar2so)ple ocftP sr osthaentdOi rctsh otics Compon&ea nntdIs ns trumiennA tust omaCtoinct rol
VanC AUWENBERGHIEns:t rumenatnadtA iuotno maitnit ohne MAHALANABTIhSe:o arnydA pplicoaftD iiogniC toanlt rol
PapeRru,b bePrl,a satnidPc osl ymeriIsnadtuisotnr ies MILLERD:i striCboumtpeudtC eornt Sryoslt ems
(1981)
CICHOCKIS TRASZASKy:s tAenmasl yAspipsl icattoi onsM UNDAYA:u tomaCtoinct irnoS lp ace
ComplP&er xo grams
NAJIMA BDEL-FATTSAyHs:t Aepmpsr ofoarcDh e velopment
CRONHJORTR:e aTli mPer ogramming &
(1978) (1980)
CUENODC:o mputAeird eDde siogfCn o ntSryoslt ems NIEMIA: L inBke tweSecni eanncdAe p plicaotfAi uotnosm atic
DeG IORGOR OVEDAC:r itfoerrS ieal ecAtpipnrgo priateC ontrol
Techno&l uongdieDeris ff erCeunltt ural, aTnedSc ohcniiaclaN lO VAK: SofofrtC woamrpeu ter Control
Conditions
O'SHEAP OLIS: AutionMm iantiinoMgni, n eraanldM etal
DUBUISSOINn:f ormaantdiS oyns tems Proc&e ssing
ELLICSo:n tPrroolb laenmdDse viicnMe asn ufactTuercihnngo lOoSgHyI MAI:n formatCioonnt Prroolb lienMm asn ufacturing
Technology
(1980) (1977)
FERRTAE PUENTES:o ftwfaorCreo mputCeorn trol RAUCH:C ontArpopll icaotfNi oonnlsi nPeraorg ramming
&
GELLIET AV ASTD:i striCboumtpeudCt oenrt Sryoslt ems REMBOLDI:n formatCioonnt Prroolb lienMm asn ufacturing
GHONA&I MSYy:s tAepmpsr oafocrDh e velopment (1982T)e chnology
(1977) (1979)
HAASER:e aTli mPer ogramming RIJNSDOCRaPs:Se t udiinAe ust omarteiloantt oHe udm anization
(1980) ofW ork
HAIMESK INDLEWRa:t earn dR elaLtaendRd es ourScyes tems
& SAWARAGIA KASHEIn:v ironmSeynsttaPellm asn nDiensgi,g n
HALMEM:o dellaindnC go ntorfoB li otechPnriocceasls es
andC ont&r ol
HARRISODNi:s triCboumtpeudtC eorn tSryoslt ems
SINGHT ITLCIo:n tarnodMla nagemoefIn ntt egrIantdeuds trial
HASEGAWRAe:a Tli mPer ogram(m1in9g8 1) Com&p lexes
HASEGAWAI NOUEU:r baRne,g ioannadNl a tioPnlaaln ning
SMEDEMAR:e aTli mPer ogramming
-Envir&o nmAesnpteaclt s (1977)
SUBRAMANYAM: CoAmpppultiecrai tnLi aorngSsec aPloew er
HERBSTA:u tomaCtoincti rnoP lo weGre neraDtiisotnr iabnudt ion
Systems
Protection
TITLIS INGHL:a rgSec aSlyes teTmhse:oa rnydA pplications
ISERMANINd:e ntifiacnadSt yisotnPe amr ameEstteirm ation
&
VanW OERKOMA:u tomaCtoinct irnoS lp ace
(1979) (1982)
ISERMANNK ALT ENECKEDRi:g iCtoamlp uAtpeprl icattoi ons
ProcCeos&n st rol
NOTICET O READERS
DeaRre ader
Ify oulri briansro yat l reaas dtya ndcionngt/i nuoartdcieuorsn t otmote hrsi esr imeasyw, e r ecommtehnadyt o pul aacs et anding/
continuoartdtieoorr ne ceiimvmee diuaptoepnlu yb licaalntlei wvo onl umSehso.uy lodfiu n dt hatth evseo lumnoel so ngseerry voeun re eds,
youorr dcearbn ec ancealtal neytd i mwei thnooutti ce.
ROBERMTA XWELL
Publishera tP ergamoPnr ess
CONTROL OF DISTRIBTUED
PARAMETER SYSTEMS
1982
Proceedionfgt sh eT hir!dF A Symposium,
C
ToulousFer,a nc2e9,J une-J2u l1y9 82
Editebdy
JEAN-PIERREB ABARY
Laboratdo'iAruet omateitdq 'uAena lydseesS ysti!dmueC se ntNraet iodneal laR echerScchiee ntifique,
ToulouFsrea,n ce
and
LAURENT LE LETTY
Centdr'eE tuedted seR echercdheTe osu louTsoeu,l ouFsrea,n ce
Publishedt he
for
INTERNATIONAL FEDERATION OF AUTOMATIC CONTROL
by
PERGAMON PRESS
OXFORD NEW YORK TORONTO SYDNEY PARIS FRANKFURT
· · · · ·
U.K. PergamPorneL stsd H.e,a dinHgitlHolan l l,
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U.S.A. PergamPorneIs nsc M.a,x weHlolu sFea,i rvPiaerwk ,
ElmsfoNredw,Y or1k0 52U!1.,S .A.
CANADA PergamPorneC sasn aLdtad S.u,i 1t0e4 ,
15C0o nsumReorsa dW,i llowOdnatrlaieM,o2 J1 P9C,a nada
AUSTRALIA PergamPorne( sAsu sPtt.yL).t dP..,O . 5B4o4x,
PotPtosi nNt.,S .2W0.1 A1u,s tralia
FRANCE PergamPorneS sAsR L2,4r udee Esc oles,
7524P0a rCiesd,e 0x5F ,r ance
FEDERARLE PUBLIC PergamPorneG smsb HH,a mmerw6e,g
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Copyrig1h9t8I©3F AC
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Arellp roducsetdo,r ienda retriesvayls teomrt ransmititnae ndy
formo rb ya nym eanse:l ectroenliecc,t rosmtaagtniect,ti acp e,
mechanicpahlo,t ocopyinrge,c ordionrgo therwiwsiet,h out
permissiinow nr itifnrgo mt hec opyright holders.
Firesdti t1i9o8n!1
LibroafrC yo ngrCeasst aloignPi unlbgi caDtaitoan
Maienn turnyd etri tle:
Controofdl i stripbaurtaemde styesrt e1m9s8,2 .
(IFApCr oceedsienrgise s)
Proceedoiftn hg3esr dI FACS ymposoinu" mC ontorfo l
Distributed SPyasrtaemmest"e.r
Incluidnedse x.
Automactoinct- rCooln gress2e.Cs o.n trol
tIh.e ory-Cong3r.De issstersi.b puatreadm estyesrt ems
-CongresIs.Be asb.a rJye,a n-PiIeIrL.re eL .e tty.
LaurenItI.II .F ACS ymposoinut mh Ceo ntroofl
DistriPbaurtaemde Styesrt emIsV.I. n ternational
FederaotfAi uotno maCtoinct roVl.S. e ries.
TJ212.2.1C96863 629.8'38132- 2273
Data
BritLiisbhr aCrayt alogiuniP nugb lication
Controofd li stripbaurtaemde styesrt 1e9m8s2 .
-­
(IFApCr oceedsienrgise s)
Contrtohle ory-Congresses
II..T itlIeI B.a barJye,a n-Pierre
IIIL.eL ettLya,u reInVt.I nternational
FederaotfAi uotno maCtoinct rVo.lS eries
629.8Q A402.3
ISBN0 -08-029361-1
Thesep roceediwnegrse r eproducbeyd m eanso ft he
photo-ofpfrsoecteu sss intgh em anuscripstusp plibeydt he
authoorfst hed ifferpeanpte rsT.h em anuscrihpatvse
beent ypeuds indgz fefr entty pewriatnedrt sy pefacTehse.
lay-out, figures oafsn do metp aabpleerdssi dn ota gree
completweiltyth h es tandarredq uiremecnotnss;e quently
ther eproductdiooensn otd isplcaoym pleutnei formity.
Toe nsurrea pipdu blicattihoisdn is crepanccoyu lndo tb e
changendo;r c oultdh eE nglibsehc heckecdo mpletely.
Therefotrheer, e adearrseas kedt oe xcusea nyd eficiencies
oft his publiwchaitcimhoa ny b ed uet ot hea bovmee n­
tionrede asons.
TheE ditors
Printeidn G reaBtr itabiyn Wheaton Co.L td.E,x eter
A. (I
THIRD IFAC SYMPOSIUM ON
CONTROL OF DISTRIBUTEDP ARAMETER SYSTEMS
Organbiys ed
AssociatFiroann c;aidseeC ybernetiqEuceo nomiqueetT echniqu(eA .F.C.E.T.)
Centrde' Etudeetsd eR echerchdeesT ouluose( CE.. R .T .)
InstiNtautti ondaelR echercehneI nformatiqeutAe u tomatiq(uIe. N.R.I.A.)
Laboratodi'rAetu omatiqeuted 'AnalydseesS ysetmes( L.A.A.S.)
Sponsboyr ed
TheI nternatioFneadle ratoifoA nu tomatCiocn troaln di tTse chnicCaolm mittees:
Theory
Mathemtaicosf C ontrol
Applicoantsi
InternaPrtoigornaaml CommitteeO rganiCzoimnmigt tee
J.LL.I ONS,F ranc(eC hairman) J.PB.A BARY (Chairman)
H.T.B ANKS,U .S.A(.V ice-Chairman) L.L E LETTY (Vice-Chairman)
M. AMOUROUX, France A. LE POURHIET
J.PB.A BARY,F rance D. VOLPERT
A.G.B UTKOVSKIY,U .S.S.R. M.AMOUROUX
R.F.C URTAIN,N etherlands A.T ITLI
M.C.D ELFOUR,C anada J.PY.V ON
H. GORECKI,P oland C.S AGUEZ
L.L E LETTY,F rance
L.L JUNG,S weden
S.K.M ITTER,U .S.A.
U.M OSCO, Italy
M.P.P OLIS,C anada
A.]. PRITCHARUD.,K .
W.H.R AY,U .S.A.
C.S AGUEZ,F rance
Y.S AWARAGI, Japan
S. GT.Z AFESTA S,G reece
P.K.C. WANG, U.S.A.
J.PY.V ON, France
J.Z ABZCYK,P oland
FOREWORD
Tihsv olucmeo ntapianpsea rcsc epbtyte hdIe n ternaPtrirooagnmmaCelo mtmetief or
presenattat tii31orenId F ASCy mposoinu" mC ontorfDo itlsr ibuPtraeatmdee Sry tse"m,sh eld
inT oulo(ucEsloeeN atioSnuapleer iaeelu 'erAreo tniaeque utd el 'Es)p,aF craenocveef ror u
dayfsr,o mJ unteo Ju,l y
29 2 1982.
Itf olwleotdh fei trI sFASCy ompsmih uelidnB naffC,a n,a idna antdh see cond
heladtt hCeno trTohle ory aCtte hnUetn reiervi syto fW arwiicnkC ov1e9n7E1t,nr gynl,dai , n
1977.
ThSey pmosmif uixtehdfe o llowionbgj ectives
-tou niete tsbalsihesdp eacliissa tnmde bmerosft eawmhsho a vweo rkientd h e field for
seveyreaalr s,
-tom akken onwenmw e thoofda sp proiamcemhd,ti eap roebmlsa,g reemeanndtdfsi f erences
concertnhviean irgo us moefta hpopdrsot aotc hhte h esmo eft hceo nferaesan w choel, e
-tor epoorntt h per rosegso fa n ubmeorf s utdieasn sdb ujesco tfa ctiyv oin titsth heme
whictho hu cona wisdpete rcmuo ft poicisnp yhsi;ci snf atct or epoorntt h ger owaicntgi ­
viyt int hfei ledb oht tehoretiacnaadl sfl ayar s m anayp lpciatiaornceso nceed.r n
-tom ake known :
practiincdtaurlasi la plpiciaotns
•
appraoitxoimnmet houdsse idnt hdei ffetrheenmectso ndseirewdhi,c ahr nee ecsasry
•
foarn cyo ncraepltpcieai toonf t hselo utitooni sn findiitmee nspirooenbmalsl
reasrecihn tsobu -tomipasl otloiunasn ad ggregpartoeebmdls ifmolpren mteatoino n
•
microcomputers
-top ormoctoen taancddti csusisonbsew teerne seawrocrhk eernsg,i naeneidrn sdt urasiliss t
ind iffesrhpeenrt:e s
resea(rmhcaehtm iac,tsa plpeidma thaetmsi,ca umtaoticco ntrdoaltp,ar eoscisn,g
pyhsics),
indtursy,
unisvieirteasn "dg ranedceosl" e (siHghIenrts tiuotniosf L ear)ni.n g
-toe ncoupreaorgnseac lo ntbaecwtte erne srecawho rketresa,c hienrdtsur,asi lissa ntd
sutdenbtohstn ationaanidl nltye rnaalytl.i on
ORGIASNATION
ThSey pmsoimuw aosr ngziaesdic entilfyai ncmdaa letirayl (l.i.eC alflo Pra pers,
PrgorammPer,e nptr)sib yt hCee ntre d'dEetR uedcehse rdceeTh toe usl oauntsd heLe ba oratoire
d'Atuomateitqd 'uAeln yasdee Sssy tmeeis nc ollbaorantw iihot thIen sttuiNta tiodneRa elc her­
cheen I nfotrimeqa eutA utaoitmqu(eP reosfrsJ o.LL.ISO .)N
Ini tcspa atcyi aansI F ..AS.yCpm.o smi tuhceno ferebnneceef iftreodtm h seop nsorship
oft hfeo lwligonI F.A...Cc omtmtise e:
Theyo r
MatahteimocfsC n otrol
Aplpciations
andw apsa lceudn dtehrfe i nncaiaanlad d msirntaitei rvepsonisbliiyt oft hAes sotcinio a
Fr�aanidseeC b yeerntqiuEec onomeitTq ecuhen iwqhuierc ehp sreentthsIe F .A..Ci.nF rnac.e
xv
xvi Foreword
commnuiactioonvse 1r2 p0r oepodps apearnsdn ear1l4y0s b umitetda bsrtac,tw sere
selectb9e1y td h eI ntaetrnioPnraglor ammCeo mmitte,e asnedev np lnearsye sosniwse rper esebnyt ed:
JL..L ION(rSFa nec),A .JP.RT ICHA(RUDn iKtiendgm d),oR .FC.U RITNA( Hlonald),H .TB.AN K(SnU ited
Sattse)A,. GB.U TKOV(SUK..ISYS. .)RM,. PP.O LI(SCn aadaan)d C S..K UBRUaSnLHdY . M ALEBRANCHE
(rBaz)i.Tlw op apehrasvb ee ecna nlceledb yt hea uthoarfst tehreS ypmosiTuhmep. er sevnot­l u
met hecno nitnas9 8p paer.s Therxetepr aap enrostp rieovulsys bumittheadv bee epnr eseantt ed
thec onfer;te hnecayer ie ncluadtet dhee dn otfh ivso mleu.
MAINT HESM DEEBATED
The9 1c ommunicawteirodeni tssr biuteovde 2r4 s essioonnst hef olwliontgh emes
I.M oedllin(g2s eisnoss, c5o mmuincantsi)o
2.I defnitctiiaoann dE stimat(i4os nes sio,n 1s5c ommnuicaotn)is
3.S atbilitziao(n1 s essio,n c3om munciaitosn)
4. abSitlizaanbdilC iotnytl rlboaili(tIys esosni,c o4mmu nicaotn)is
OptimlaC ontlr (o2s esosn,is6 c ommnuicaiton)s
S.
6.O pitmziatni (o3s essio,n1 s1c ommunicast)i on
7.O pmtailD esi(g2sn esosn,is cmomnuicat)i ons
MethoadnsdT echnioqfuC eo7 snr tol( 5s esosn,is c21mmo unicantsi)o
8. SothcatsiPc rosceess sessio4n c,mmo unicast)i on
I9O.. F elxilbeS rtuctu(r(1se1 sses io,n c4o mmcuanotini)s
ThermParlo cCeosnrsotl (Is esosni,c o3mm nuicaotn)is
1112.. EneSrytgseym (s1 s essio,n c5o mmunaitcio)n s
ContorfoT lh ermParleo cssess esns,i ocoummniactions)
13. (1 3
Twop arlaelsle sosniwse roer ganiinzceddli unogn wei tshi umlnteaouisn tertpirneg
(Frhe/nEcnsghlE,in gliesnhhc/).FT rhes essiohnasv bee etne nttiavleyo rganiinzo erdd er to
presseinumtlt aneloyuc sonbturtipeadp eorfst heotrieciantle rte asndme thodyo alnodtg hose
of mporraec taiicmaoslr i ndtursiaapllp icat.i ons
Thet hmees (eitohiregri noarls ruvepypa er)s daetbeidnt hec ourosfet h pe elnary
sessiownesr teh ef olwligon
J.LL.IO NS
"Sulrec ontrooptliem adle ss ytsemedsi tsriubeisn stbaels"
A.JP.RT ICHARD
"Nonliinnefairnd iitemen siosnyatsle mtsh eory"
R.FC.U RTAIN
"Cpoemnsadteosrif gondr i srtibutperadam teesry tsem"s
H.TB.AN KS
"Ditsrbiuetds ytsemop timla contarnodpl ra maeteers mtaitiocno mpuotnalatt iechniques
usinsgpl inaep porxitmiaon"
A.GB.U TKOVSKIY
"oSmen ewr esultisn d itsirbutpeadmr eatesry tsemc notorl("a r eview)
M.PP.O LIS
"Thdei tsirbutesdys tepma rameitdeern titfiiopcnra oblae smru veoyf r ecernets su"l t
C..SK UBRUaSnLdHY . M AELBRANCHE
"As uvreyo no pmtails ensaonrdcs o nltrlroes altoicionnd itsribuptaemrdeat esry tsem"s
Ther egistpearretpdina ctinsu bmere12d,5 i nlcduin5g0 F rnecha nd7 5v iistorfsro m
23c ounets,ra ilthouangoht hiempro arnttI FASCy pmosi(unom I deinfticataindPo anr eaetmrE stima­
tiont)o opklc aei nW ashgitnotnw ow eekbseo fr.e
A permanntee xhiitbiwoint ah s imulaatpipoanrs a,t ut"Dhleef Pta arllPerlo screos",
waso rganbiyzPe rdeo sfsDoerc ketre'asfm r otmh eUn irvseiotyfD elftH,ol lnad.
Thretece hnailcv iists weroer ganizteotd h ef ollowing
AirB usI ndtursy,
Centdr'teEd uese td eR echerdcehT eosu sleo,u
Laboratdo'iAurteo matdi'qAulneya sdeeet sS ytseems.
Fowroerd xvii
A certnaumibne orf papers shelaevcet feobdpre oe ssni lbep builctaioinn A utomcaat i
byt heI ntaetrinnoaPlr omgmreCa omimtteaen tdh ec harimeonf s esosni.s
TheN atioOnranglia ziCnogmm itteteh eSo ypfmo isumi sp articuilnadrelbtyto et dh e
followibnogd ifeostr h eifri nianacslp upor:t CenNtartoeinl ad el aR echerScicheen tifique
(CN..R..)SD,i rectdieosRn e cherecthE etsud Teecsh niq(uD.e.RsE ..)TE,elc tricdietF ern ace
(.ED..)FI,nt situNta tiodneal laR ecehcrhee n IrnmfaotieqtuA eu tomat(i.IqNu.e.AR .).I,
anda slos peciianldleyb ttoea dl melmb erso ft heI nteartoinnaPlr omgmreCa oimmttefeo trh eir
acticvoeo pieor.na t
Toulo,uN osvebeemr
1982
J.PB.AB A-RLY. L EL ETTY
Copyrti gh©I FAC3 rdS ymposium PLENARLYE CTURES
ControofDl i stribPuatreadm eStyesrt ems
ToulouFsrea,n c1e9,8 2
ON OPTIMAL CONTROL OF UNSTABLE
DISTRIBUTEDS YSTEMS
J-.LL.i ons
Colleed geF ranc1e1,p lcaeM arlcieBne rth7e5l2oP3ta1r, i Cse dexF ra0n5c,e
and
JNRI,AD omaidneeV olucPe.0a. uB ,o x1 50, 7L8e1C 5h3e snCaeyd eFxra,n ce
Abstt.rI anctrodpeurescnttoarttyoti hooenp m taiclo ntorfuo nlsl teoa rb
nownle ls esty emsst.
INTCRTOIDOUN
Leutsc onsiinadv eergr ye nearnadl forTmhapilors b mlp eermiaut nsi sqoutleiu on
manans eytrse tmh set aotfwe ih chha bse en marykv(e.)d
0
epxredsi snte heeq iuoa:tn
Leutsr eutrtnot hgee ne(raafnloda rm l)
()I (y)$ v(=) cas()eI. I ti sas sumtehda� th eirsae
Ji. BansapcahYc seot hat
whreei sap artdiifafelr eone trtiaoa rl
(6_p) rlomeb( )Ip ermiaut nsi sqlouuet ion
(oJip ewriattho rp artfiucantli odnesr ived
yv()i nY.
(DtiirsbuStyesdta emni)dnw i hcvhi st he
conlt( rocoro mmavnadr)i (aobfrlu en c;t ioTnh)ceo rnlotp rloembcsa tnh -eant l eafsotr
iti sas sumtehda t al ragneu emrob ft hsoew hiacrohe fi nter­
esfto arp ipclat-iboefn osrma utleidnt he
(2v)E. !f/115/1t=po loogvieccatlso praaacsle flolowwian:yog n ceo nstiwdfoeu trniscao lns
assumbeyBd a n*;ai cn(h )Io nmeu satdt dh e
bouanrcdyo ndiatn,idi o,fnJi' sia sne volut(i7n)g
opeor,rta htien itciodanilnt ;sii on is (8) y + lR if +
ano pertahtaloter af drsot mo a s (pla)ic.ciJn e E. +lR , '!'(A)+ "" + A +"" •
wihcthh per oblhema asOd IJ i re.c tionanfdo arl vl Eoo/,tn,se e ts
(I) CJ
Toc alirftyih sl,eu tst akase mi lpee xpal:me (9) 'J(v=)<l> ( (yv))+ 'l'(lvllaz}l
letb ean no pesneo tflR n w itrhe gubloaurn ­(wh1ev1rIl ay e i nditchaneto erosmf i6/.1)nv.
daryo nceo indsetrheseq uation Ift herset li ils
r;
(10)U/ /=c oncvleeoxdss e(tn eomntp yo)f tl/I
()3 -Y�+ y o=i nn ad
onteh elno ofkosr
with
(I in.[ff v()v,E 15/l a•d
I) [J
(4) ��= vnor. Prlomebso ft ihst ypaen td-hi erv ery numer­
2 ouvsai ran-thsa gvievr eients oag reat
Onaess umtehsva tE L()r= l/O.P orbml( e3)
nuemrob fd eovpemleunptwoshn i wcesh ah ll
(4i) se quivtaofl ienndytHiE1 (nn*g*) s o noetn therer.e Ouorbe jctiintv hei s
that arltieic st os hwohwao tnc ea dno- o rt ry
tod o-!w hehny psoit(sh6e i) sn oatpl cpi­
I
(5) ay(.P,)= rfv ;odVrE'P H(n,) able0.
or
Onceos nidtehressint l alne quation
a( )y fn= [ � y' + y\O dx. buitnc aswehset rhese o tliucoann( eIn=) xo its t
'\0 v..p l orr atwhheertr heei rsae ni nf f
inrtY"Q
sloutiooran gsaw ihnet rhesel o utiiso n
*wea ssumhee trheaa tls lp accoenss idered
copmletuenlbsytli aenr etliatoonv arinas tio
arree .a Bluittm abyei netsenrtg(i ni
ofv .
relaptreomdb,s pl aerticiuntl heaeqr ulay­
tioonfSs c hroetdoci onngsetirhdc)eeo mrp ­
Leutst akane elxera imgaphwt:aO ynceo­ n
lecxa sewle .la s
siders
3
1 (1) 2 �y-y= ivnn
**H ist hneo rSmoablo slpeavc e:
1 (Q) 2 with
H (n) ={ <P,la� \Odipx E.L (n,) =il ,. • .n,}. (1) 3y 0 on an
]. r
2 J.L-.L ions
2
tehrerfeoi,f v E ()nL,i ti sh ighplryo b­ (j)jQ uestoifomn osr phosgielsnde e tot he
ablteh at thsetrieal nlii nsf inoift y controfom lul itplseat tsey smtse( owfhi ch
soliuotn-sa ndi na nyc ase itsht irsuf eo r (12an)d ( 13a)ree x apmle.s R)efereinsc e
a densseei tn L2 (nf)u cntionosfv .C f.o n maed tot hweo rdko nbey J,K erPn.e vez
thissu ebcjAt .B ahri A.B aharnid M.C .D uban Joly andt ot hbe(o Io)k,
H.B erteyscki It(t Ih)e,r ebfeocroem es undeprr ep(aI)rGb,a.y tJ i.o( nKIP e).re nve,x
(I).
difcfuilttoc onsiydv(e)r( t a leatsotf ol­ J,L .L ioannsdD .T homas
()I.
lotwh be ifurcbartaensdc). h eOnei st here­
(j)jT jhper esaepnptr omaacybh eu seffuolr
froel etd oa c hanogfeo ne'pso intv ioefw .
thteh eocraelstu tidyo fc ernt uanistable
prombslee,. gf.o trh es tiuodfty h es turcteu r
Onet hecno nsid()e1u rnsed rt hfeo llowgi n
ofd atfao wrh icah g enelyrl naonwe lls et
ascpte:o nec onsiad perriso trhies eotf
probpleemrm tisa soloun.t iCf.F uriskov
paivrs,s zo t hat
()2.
(14)v E z YE (I)
1..(, Thep lna ftohrea rtlieci s faosl wls:o
with UnstaSbylsmets e
1)
2)M ultpilSet aStyets ems
(1)5 (z=) fiJ(v;)
.A 3) Nwoenls le ptr ombsl e
onee tsabilshtehshe y ptoehsis Bbilgiropah.y
(61) theirsea se{tv,zJ veirfnygi
0 0 Preimlinarreypt osor nt hre estusol uitnled
(1)(415s)o th0Ea°4itd , v hereahfatvebere epnr ovibdyJe d.L .L ions
I (2())34()(5a)n di n tchoeu rastets h e
ando nce onsid(ecromsp wairte(h )9 ): Collgeed eF ranicne1 8908/1a nd1 89/182.
(1)7 [ij(v,z=) + (l'vlll'l) UnstaSbyltseme s
�(z) 'lt 1.
thge eneprraolb olemft hceo ntoroful n sbtlae Supposne lR3; ocnoen siidneQ r =sn x0],[ ,T
disbturtiesdyts ems ist hetno f nid T>O givCae nny hotwh,es eotf p airs{,v }z so
that
(1)8 in.fiJv(,z,)v E 'l1td ,v{z,}verifying
(1)(41)5. (II.)
and
Ift heiresa pai{r,u }y sot hat
3
(91) (u,'!f)yf = nif. ('{fv,fz,) (I2.) �at -f',z- z= vu·i Q '
zx(,)O 0= in
ones ays {tuhy,a}t ainso pmtailp ai.r n,
z = 0 on = x] O T,
Thep rinicpaplr ombsla ere( aisn t hceal s­ l: r 2 [.
Note Fovr g vieni nL (Qt)h eiresn orm­
scialt heoorfoy p itmlac ontorfod li s­tri I.I.
alyl not otsaoll iuotn ti no(f.1 )2 (htings
butesdyt smes):
3 3
ardefi efrenitf- zisr epaclebdy + z in
(is)ut dyitnhge iseetxncoef o neo pmtail
paiarn ,di fp oss*i,bs lutedyitnhges turc- whiccha steh per omb lisew ells e.t )Wea re,
��- threefo,d reeailnhge rwei tahnu nstable
turoef tsheeot f o ptimpaarlis .
ditrsbiutesdyt sem.
(iip)r oviingdt hne ecess(naadr,y ifs ­spo
Not1e2. .T hec ondiiotnisn ( .1)2h avae
ibles,uf fiiecncto)dn iitonosfo pitamliyt
dicrteoin. Inf cat, ifc oonnsei zd aesr s
(potmialtiys ysmt);e
6
giveinnL (Qt)h etnh el ienapr romb le
(iipir)o viinadgp priomtxainagl gtohrmifso r
onoe ft hoept mialp aris-.W es halpl rovide
smoei nicdaitoncsoc nergn itnhessbuej cets (I• 3 ) -az;)t - f:,z= v +3 z2 E L (Q) ,
andr efteort he bboyo kJ .L ioLn(.)sI f or ('x 0)0 inn ' z= 0 on
all deatnadci olmlspmee nts. Iz = i:
2
pemritas unisqoulieuo tnin L (Q()c .f
Befogroien gi nttoh eex anmaiitoon fs ome
J.L .L ionansdE .M agenes andt his
eamxplelse uts g ivseo mhei nsto n the (I)),
soliuotni ss rton:g
motivatfioorn theo fps rtoumbdslyo e ft his
kin:d 2,1
4) z E H(Q)
(.I
()j froera saolnoslf w hichha ve more wohre re
lestsod ow ith energpyr omsbsal veionnge
2
ttouinr,,sn i sntt ohiac ecnno r naretsoinolgnf uu nbsmetroa fba lppelmc isa­,s yste(.1S) 2'H (1Q= )'{l''l,lax;-3'-'•1ax'.a' x1'.• a-tClaEL'!' 2( Q.) }
fori nsntcaei nc onnteoicnw itahi rfcrta. 1. 1. J
Onet hecno nsitdhecer oss ftu ntcoin( cmo­
A laragmeo uonftr easrecihs d one piornbt­o
pare (wi1t7ihn)t hIen trcotodinu:)
lmeso ft hitspy ee,s ecpiallbyys uibtlae
usagem iocfro ocepsrrsso.T his ldeiard­esc t 1 6 N 2
lyt ot eh sutdyo fh ypbeolrci unsbtlasey s­ (.1)6 (,v)z= 6z1-d1lzl6 +-zlvlll2
L (Q) L (Q)
tmes.
6
wherez isg iveinnL (Qa)n dwh erNe i s
d
*There laarreng uemae b ro f o(puernne svoe)ld giv>e no.
prbolemisn t hsir epscet. Spupo�sde w ith
OpitmaClo ntorfoU lnt sabDlies bturtieSdyts mes 3
(.I7 ) = clsoecdo pmlxe none mptsye t where
Onee s'bott(adflia 2 h s(e sLQ . ) thhypeo tehsi(sc ompwairteh (.I1 )8 w21,,. I s6 ( Q){,'1' 1'1',= ', ,�a x, i. ,' _a x_2i. tjiax _ J. a '
(1)6 in tIhnet urctoidotnh)a t ,at atji ELP( Q})
(.1)8 theerxie st{sv, v} with I)( .I2 )
0 0 (I. Theo pitmalyi styst(em.I 13-)( 1.71)c anb e
andv0 lhd· obtianebdy t hrdeefi ferenotd :sm eth
E
Onet hecno nsitdheepr rsol mbe offni ding (io)n ec anc nosidtehre aplieesndp romb le
(.19) li(n.1.�f2(a)v,n )zd Evl1t{,dv z·,} v eirfyi(n.IgI ) (.I91)J £ ( vz,)=ilzl-dz 1L616 ( Q+)
Oneh as +� 2 lvl1L212 (Q+)_!_2£ l l�at-l zl-3z-vlL22l (Q)
THEOREM Onea ssumetsh a(t.1 )8 takes
plcaeT.Ih .eIre.exi sttsh e{n,u }y with wher£e 0 isi ntentdole aded t woardOs;
2),u E4 and (1:1) onceno s>i ders
(I. ad 6
I)O r!f(u), yi=n .fif (v,z) (.I2 0) in.f1J£(v,z,)v EUad'z EL (Q,)
(I. az 2
{u,}y isa no pmtailp ai.r - llEzL (Q)
at
Thed eomnstartiuosne asc omcpntaesrseu slt zx(,)O 0= in z= 0 on
oft heR ellciht pye.O nec onsiad ers n, L;
miniimnigss qeuenocfe{ v, z} .I tf ollows theries(u E y,£1 soliuotonf • 2(0 ;)1t hrough
n n suiatblea p roifi esmtaitio(nessed etasi l
from. )6 that inJ .L .L ion(sI )o)n,ce n a gett het o
(I
{v, }z remaininsa set lmiit(£ + 0 )i nt hoep itmalisytsymt feor
(III. ) n n 2 6 .1)9;s etting
lmiite(dr ersitctbeydL )( Qx)L (Q.) (I 1 a£y 3
Butt hecno,n esriidn(g.1 3()f ovr, z) one PE= - -lly£- y£-u£)
n n £( (3�
cocnludfersot mh si that thissy smt ieswr itten
2•1
a(n.11ds)2 ni cnetzr heiemn aejiwcnitstio hoafHin2 ln' m1i (i Qto>-)2 fLH(- Qi()sQ ) (.I2 1)- ata PE- lip£- 32£y PE=( £y-dz)s inQ ,
compaictft o llowtsh aotn ce anex tcrtaf rom pE(,x)T 0= in p£ = 0 sur
ita sube-qsuenacgea mianr k{evd, z} so n, L,
2 2 1n n and
that, z+vu , yi nL (qH) ('Q w)e aakn d
3 n.3n 2 .2 2) (£p+ uN£) (uvf )d- xd t 0
z+y inL (wQe)a. k (I fQ �
n
Consnetquley V E'l{d ,U EE t(ad
V
ay lly y3 = u,i nQ , Onec ant hussh otwh ap£t rienmwsai thai n
at:- - cons.t roafWi2 n' 1 t6· '/( 5)Q ando nec ang ett o
yx(,)O = i0n y = o0n thelm iit.
n, L,
anda sl imi.n. f� (v, z) fi<u,)y,o nes ees
n n <! (iiA) s eocnmd eth-ocdl soetro the
tha{t,u }y iasno ptipmaail r. numericaaplp lica-tciooninsssti snd ecompo­
*
Aftethrsi o ne dcemaonnt srattheef ollowing snig thoeb lpaemrn dt hepne niaislngi t.
restuw lhicghi vtehsoe pmtailyi stsyt.em Onec onisdetrhse refore
ay 3
THEORE1M2. .T heries tar ip{lue, }y s,op (•I 2 3)at -lly = wv i+n Q ,
tha{tu, }y spuposaenos p itmapla iarn d-So
that yx(,)O = 0 inn ,y = o0nL ,
(.I1 )3 = u, wherve ( rpe.sw )i sg vieni nL2 (Q()o Lr6 ( Q)).
ap 2 5 Thep romb l(eI.32)p ermtisa uniqsuoel ount i
-at- lip- 3yp (=y d-)z inQ , markye(dv ),.w
(.141)y x(,)O p=x( ,)T = in0 It eivsi detnhtai tf
n,
(.142)y v(w,) w
(.II)S y = p on= 0 =
L,
and
(.I1) 6Q( Jp+N(uv))- xud dt�VOvEl.'°'a d'U El-'a d; *Them ethaocdat ulluys eidn t hneum erical
applicatiiostn hse oofLn aeg rangian
thefu cntoinp veirfies agmuente(daR ids)e. Cf.M .F ortni and
(.11) 7 pw 2•E1; 6/(5Q,) RR..GG lloowwiinn((1ss)1)kkaa niinddF J ..AF ng.Br oan,nnD da.(Gn) Ias.b ,a y