A Consecutive Quasilinearization Method for the Optimal Boundary Control of Semilinear Parabolic Equations

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1 Appled Maheacs Publshed Ole March 4 ScRes hp://wwwscrporg/joural/a hp://dxdoorg/436/a45467 A Cosecuve Quaslearzao Mehod for he Opal Boudar Corol of Selear Parabolc Equaos Mohaad Dehgha aer * Al Vahda Kaad Appled Maheacs Depare Ferdows Uvers of Mashhad Mashhad Ira Eal: * u_de34@su-aluacr udeaer@galco kaad@ahuacr Receved 5 Deceber 3; revsed 5 Jauar 4; acceped 3 Jauar 4 Coprgh 4 b auhors ad Scefc Research Publshg Ic Ths work s lcesed uder he Creave Coos Arbuo Ieraoal Lcese (CC BY hp://creavecoosorg/lceses/b/4/ Absrac Opal boudar corol of selear parabolc equaos requres effce soluo ehods applcaos Soluo ehods bpass he olear dffere approaches Oe approach ca be quaslearzao (QL bu s applcabl s locall e oeheless cosecuve applcaos of ca for a ew ehod whch s applcable globall e Dvdg he corol proble equvalel o a fe cosecuve corol subprobles he ca be solved cosecuvel b a QL ehod The proposed QL ehod for each subproble cosrucs a fe sequece of lear-quadrac opal boudar corol probles These probles have soluos whch coverge o a opal soluos of he subproble Ths ples he uqueess of opal soluo o he subproble B ergg soluos o he subprobles he soluo of orgal corol proble s obaed ad s uqueess s cocluded Ths uqueess resul s ew The proposed cosecuve quaslearzao ehod s uercall sable wh covergece order a leas lear Is cosecuve feaure preves large scale copuaos ad creases ache applcabl Is applcabl for globalzao of locall coverge ehods akes aracve for desgg fas hbrd soluo ehods wh global covergece Kewords Quaslearzao; Opal Boudar Corol; SQP Mehods; Selear Parabolc PDE s Iroduco The soluo ehods for he opal corol of olear sses pass fro olear o lear dffere * Correspodg auhor ow o ce hs paper: aer MD ad Kaad AV (4 A Cosecuve Quaslearzao Mehod for he Opal Boudar Corol of Selear Parabolc Equaos Appled Maheacs hp://dxdoorg/436/a45467

2 approaches For exaple he grade ehods odf eravel he prevous approxae soluo b learl seekg a suable dreco horough solvg a lear proble [] The SQP ehods seek he opal soluo b learzg he opal sses usg soe verso of ewo s ehod [] [] Our approach hs respec s o learze he sae equao hrough a quaslearzao ehod Quaslearzao ehod for olear equaos has s org he heor of dac prograg ad has pora feaures coo wh ewo s ehod especall s for [3] For a foral explaao le Y ad Z be ordered Baach spaces L: Y Z be a bouded lear operaor ad : Y Z be a olear dffereable operaor Cosder he equao L = ( ( I he covex case Equao ( ca be wre as ( ax ( ( ( L = = v v v ( v Y where he rgh had sde s quaslear The sarg fro Y horough he quaslearzao ehod a sequece of lear equaos s defed L = (3 ( ( ( ( whch produces he sequece of approxae soluos { } (; see [3] [4] Ths ehod has he followg feaures { } Y covergg o he soluo of ( or s oooc Ths ses fro posv ad verse posv of L ad L he covergece s globall he sese ha ca be a lower soluo of ( e L ( 3 The rae of covergece s quadrac For deals o hese feaures refer o [3] [5] There are soe exesos refees ad geeralzaos o he quaslearzao ehod whch preserve he above feaures bu relax he covex assupo o ; for a coplee surve see [5]-[7] Quaslearzao ehod has ae coeco wh he heor of posve ad oooe operaors axu operao ad dffereal equales; cofer [8] Sec 433; [4] [9] I order o roduce he proposed cosecuve quaslearzao ehod for opal corol probles le U be a Baach space J : Y U be a fucoal ad B: Y U be a bouded lear boudar operaor Cosder he followg opal boudar corol proble: u L J( ( u( ( = ( ( ( ( B = u u U where belogs o a e erval [ T ] For v Y cosder he followg approxao o (4: J ( u( u ( ( = ( ( ( ( ( ( ( ( = ( L v v v B u u U Sarg fro u be he opal soluo of (5 wh v u coverges o he soluo of (4 wh he followg feaures: The covergece s occurred for T < T for soe T > The covergece s globall he sese ha ca be chose a elee a subspace of Y 3 The rae of covergece s a leas lear bu s o ecessarl super-lear or quadrac ere he sequece { } or { u } s o ecessarl oooc eve whe s covex or cocave For he case T > T he opal corol proble s decoposed o a fe opal corol subprobles each o a e erval wh legh less ha soe T ad he he above ehod be appled o each of he cosecuvel ere T < T s such ha he sabl s preserved The opal boudar corol proble whch s vesgaed has he sadard quadrac objecve of rackg pe ad a sae cosra coprsed of a selear parabolc equao wh xed boudar pe For such corol probles due o lack of covex of he soluo se here s o geeral uqueess resul based o he opal heor of opal corol probles [] [] [] owever a uqueess resul for such probles s le ( (4 (5 = The he sequece {( } 69

3 obaed here as a b-produc of he covergece of proposed cosecuve quaslearzao ehod The orgazao of paper s as follows Seco roduces he sae equao ad soe esaes cocerg soluo of lear al-boudar value probles Seco 3 proves he exsece of a opal soluo Seco 4 roduces he quaslearzao ehod ad proves s covergece for T < T Seco 5 explas how o appl he quaslearzao ehod cosecuvel o he opal boudar corol proble whe T > T Also he uqueess of opal soluo s saed here I Seco 6 he error ad sabl aalss of cosecuve quaslearzao ehod s vesgaed Seco 7 preses a uercal exaple cocerg he obaed resuls The Sae Equao k Le Ω be a ope bouded doa k wh boudar Ω of class β Le T > Q = ( T Ω ad Σ = [ T ] Ω Cosder he corol sse descrbed b he selear parabolc al-boudar value proble: x Ax = f xx x Q C β for soe ( ] ( ( ( ( ( x ( cx ( = ux ( ( x Σ ( x = ( x x Ω where x ( ad ux ( are respecvel he sae ad he dsrbued corol of sse f( xx ( s he sse olear ad k ( ( ( ν ( ( x = a x x x x Σ j j= s he oral dervave of assocaed wh A where ν ( ν ν The followg assupos are posed o he sse ad daa: (A A s a secod order dffereal operaor dvergece for: k ( ( = ( ( j= k (6 = s he ouward u oral o Ω Ax : j aj x x (7 where a C β ( Ω ad A s uforl ellpc e for ever ξ ( ξ ξ j µ Also s cosdered ( k = k k aj ( x ξξ j µξ x Ω (8 j= for soe > c L Σ (A f : Q sasfes Caraheodor s codo e f s easurable o Q ad couous o ad he esk s operaor F : L ( Q L ( Q defed b F: = f xx x Q s bouded ad couous A suffce codo for ha s ( ( ( ( ( ( ( for soe k L ( Q Theore 3 [] f x ξ Cξ kx ae x Q ξ (9 (A3 f s wce coousl dffereable wh respec o ad ( ξ ( ( ξ ( f x M ae x Q ξ f x M ae x Q ξ for cosas M M > The sadard fuco spaces : = L ( Ω V : = ( Ω ad V : = ( Ω are used he paper Idefg wh s dual resuls he evoluo rples where he ebeddgs are dese couous ad copac The sadard soluo space of parabolc probles ad s or s defed as { ( ( } : = L TV ; L TV ; : = L ( TV ; L ( TV ; 693

4 The couous ebeddgs ad are well-kow ad he laer s copac For dealed defos ad properes of he above spaces refer o []-[4] The blear for assocaed wh Equao (6 s defed as follows: k [ φ] : = j jφdd φdd φ ( ; ( B a x c x L TV Q j= B Assupo (A he coeffces ( are bouded Ths resuls he boudedess of B[ ] Le deoes he dual parg bewee L( TV ; ad s dual L( TV ; ad ( ad ( L ( Σ deoe respecvel he er produc of L ( Q ad L ( Σ The Defo Le For u L ( Σ a fuco s called a weak soluo of (6 f ( = ad for all L ( TV ; Σ [ ] ( ( L ( φ B φ = F φ u φ Σ φ ex heore uder weaker assupos s proved Theore 3 of [] Theore Le L ( Ω The uder Assupos (A-(A3 for ever u L ( Σ proble (6 ads u I he followg secos he lear al-boudar value probles of he pe below are used: a uque weak soluo ( A = d h Q c = g o Σ ( = Ω where d L ( Q h L ( Q c L ( Σ g L ( B [ ; ] : V V ae [ T] b k Σ ad ( Defe he fal of blear fors [ ; ] : = d j j ( d ae [ ] ( Bwv a w v x c wv x wv V T Ω j= Ω B Assupo (A he coeffces ( are bouded for ae Thus B [ ;] s bouded o V for ae [ T] Le be he dual parg bewee V ad s dual V ad ( V V be he er produc of = L ( Ω ad L ( Ω Defo A fuco s called a weak soluo of ( f ( v B [ v] = ( dv ( hv ( gv V V L ( Ω = ad ; V ad ( L ( Ω for all v V T or esaes cocerg soluo of proble ( are coo he leraure of lear al-boudar value probles []-[4] ex heore saes soe of he clarfg he e depedec qual of her cosas Is proof has bee cluded due o lack of suable referece o he bes of our kowledge for he for saed here Theore The al-boudar value proble ( has a uque weak soluo wh or esaes ad ae [ ] ( ( ( ( ( ( C T h g L Σ (3 C T h g L Q L Q L Σ (4 ( ( ( ( ([ ] C ( T h C T L ( Q g L ( Σ (5 ; where C ( T ad C ( T are bouded whe T vares boudedl ad lt ( C T = If h L ( Q g L ( Σ ad L ( Ω he L ( Q Proof B Theore 53 Ch III of [5] for soe K > he followg esae exss: 694

5 L ( Ω v K v K v v V Usg he Gardg equal (Proposo 45 [4] or he ellpc eerg esaes (Sec 6 Theore [] s obaed for ae [ T] [ ] ( β v Bvv ; dvv γ v v V (6 V where β > ad γ The he exsece of a uque weak soluo of ( whch sasfes he esae (3 s deduced usg a Galerk procedure (Proposo 33 [4] To oba esaes (4 ad (5 le be he weak soluo of ( The Defo wh v = ( elds ( ( ( ( = ( ( ( ( ( ( ( [ ] ( B ; d f g ae T (7 V V L Ω Furherore d ( ( = ( ae [ T] V V d h h ae T ( ( ( ( ( ( [ ] ( [ ] ( ( ( ( ( L ( Ω L ( Ω L ( Ω g ε g ε ae T ε > (8 where he equao s proved Ch III Proposo [] ad he equales are obaed b Cauch s equal The couous ebeddg elds v L ( CV v v V for soe C > Ω V V Ch II Theore 33 [] Cosequel (6-(8 wh v = ( eld d d ( ( ( β εcv ( V ( ( ( B ; ( ( ( d ( ( γ ( V V ( ( γ ( ε ( L ( Ω h g for ae [ T] ow le ( ( η = ad ξ( h ( ε g ( L ( Ω = The (9 ples ( ( ( ( [ T] η γ η ξ ae ad he dffereal for of Growall s equal (Appedx B [] elds Sce ( ( ( ( γ η = = s obaed ( Iegrag ( fro o T resuls ( ( ( [ ] η e η ξ s d s T ( ( ( ( [ ] Σ ( γ ε L (9 e h g T ( ( ( γ T e L ( Q h g ε ( ( L ( ( Σ γ B eplog he equal ad C( T lt = he esae (4 wh C( T a b a b = ( γ T e ε ( γ s cocluded 695

6 The esae (5 s a cosequece of ( wh C ( T ( γ T ε e = The las assero of Theore s proved Proposo 33 of [] e also ee he backward for of proble ( e he lear fal-boudar value proble p Ap = dp h Q p cp = g o Σ ( pt wh d L ( Q h L ( Q c L ( Σ g L ( = p Ω f ( Σ ad pf All resuls of Theore are vald for ( Theore 3 The al-boudar value proble ( has a uque weak soluo p wh or esaes ( ( f ( f ( f ( ( p C T p h g L Σ ( p C T p h g (3 Σ ( ( ( ( L Q L Q L p ([ ] C ( T p h C T L ( Q g L ( Σ (4 ; where C ( T ad C ( T are bouded whe T vares boudedl ad lt C( T = If h L ( Q g L ( Σ ad pf L ( Ω he p L ( Q Proof The subsuo w ( pt ( ( he forward for = ( elds he followg equvale proble o he proble ( ( ( ( ( w Aw= d T w h T Q w c T w= g T o Σ w = p Ω f (5 Proble (5 sasfes all he assupos whch proble ( sasfes Therefore he asseros of Theore ad he esaes (3-(5 are vald for w Sce w( = p( T = p( whe X s oe of X X X spaces L ( Q C( [ T] ; or L ( Q he asseros of heore ad he esaes (-(4 are verfed 3 The Opal Sse Le U = u L ( Σ a u b wh ab L ( Σ L ( Σ ad where > { } α J u u u Uad sasfes (6 Cosder he followg corol proble ( : = d L ( Q L ( Σ (6 α ad d L ( Q Theore 4 Uder Assupos (A-(A3 he opal corol proble (6 has a opal soluo ( u Uad Proof 3 B Theore he opal corol proble (6 s feasble Also he esk operaor F= f( xx ( ( x Q as a operaor fro L( TV ; o L ( TV ; s copleel couous Because whe weakl L ( TV ; he copac ebeddg elds srogl L ( Q Cosequel he cou of F : L ( Q L ( Q ad he couous ebeddg resuls F F srogl L ( TV ; (cofer Assupo (A Thus he exsece of a opal soluo ( u U for he proble (6 ca be deduced fro ad Theore 45 of [] Theore 5 A ecessar codo for ( u U be a local soluo (or a opal soluo of ad proble (6 s ha here exss p such ha 696

7 ( ( ( ( x Ax = f xx Q (7 A x ( cx ( ux ( ( = ( Ω ( ( ( ( ( ( ( d ( = Σ ν x x p x Apx = f xx px x x Q px ( cpx ( = Σ ptx ( = Ω ( α ( ( ( ( L ( Σ ux px ux vx uv U Proof 4 Corollar 3 [] (or Theore 48 [] pu of he opal sse (7-(9 belogs o Theore 6 A soluo ( ( ( ( L Σ Proof 5 As U ad s bouded ulzg Theore (or Theore 3 [] s deduced L ( Q Theore [] elds p L ( Q Lea The opal codo (9 ca be wre he equvale for bellow: for ae ( ( ad ( ux ( ( ( ( ( bx ( ax = αux ( px ( = ax < ux < bx ux = x Σ Proof 6 Refer o Lea [] u p U = sasf (3 The Corollar Le ( u s ad ad ( α ( ( ( ( (8 (9 The (3 αu x u x p x p x ae x Σ (3 p s do o ecessarl sasf a opal sse x Σ ad ( u p x Proof 7 Le ( occurs for ( u p a ( = sasf (3 a ( ( = α ( ( ( = ( ( α ( = α ( αu x ax p x αu x p x p x bx u x x The oe of he hree cases below Slarl oe of he hree such cases occurs for ( u p a ( x Le he frs case be occurred for ( u p a ( x The oe of he hree cases below us be cosdered for αu αu a ( x = αax ( αax ( = αu( x αu( x p( x p( x αax ( αu( x = αu( x αu( x p( x p( x αax ( αbx ( = αu( x αu( x p( x p( x As ou see each of he hree cases above sasfes (3 a ( oher cases of ( u p a ( (3 a ( x 4 The Quaslearzao Mehod x I a slar argue for each of he wo x hree relaos as he above ca be wre provg ha each of he sasf Cosder proble (6 uder Assupos (A-(A3 e vesgae sead of he opal sse (7- (9 he followg oe where he opal codo (9 has bee replaced b s equvale for (3 cofer Lea : 697

8 ( ( ( ( x Ax = f xx Q (3 A x ( cx ( ux ( o ( = ( Ω ( ( ( ( ( ( ( d ( = Σ ν x x p x Apx = f xx px x x Q px ( cpx A ( ( = Ω = Σ ν ptx ( ux ( ( ( ( ( bx ( ax = αux ( px ( = ax < ux < bx o Σ ux = (33 (34 B Theore 5 ad Theore 4 opal sse (3-(34 has a leas oe soluo Theore 7 Le ( p u U be a soluo of opal sse (3-(34 The here exss a ad { } sequece ( p u = Uad whose elees are he uque soluo of he followg lear opal sses ad here exss > whe T T As a cosequece whe T T opal sse (3-(34 has a uque soluo ( ( ( x A x T such ha hs sequece coverges a leas learl o ( pu ( ( ( ( ( ( = f x x f x x x x Q ( x c ( x u ( x ( = ( Ω = o Σ x x ( ( ( p x Ap x ( ( ( ( d ( = f x x p x x x Q p( x cp( x ( T x = Ω = o Σ p ( ( ( ( ( ( ( ax = u x αu ( x p( x = ax < u x < bx o Σ u x = bx { } Proof 8 Abou he exsece of sequece ( p u opal sse of followg lear-quadrac opal corol proble = (35 (36 (37 Uad oe ha (35-(37 s he α J( u : = d u sasfes ( 35 (38 L ( Q L U ( Σ u ad whch has a uque opal soluo (Theore 43 [] The he opal heor for lear-quadrac opal corol probles elds he exsece of a uque soluo ( p u Uad of he sse (35- (37 whe L ( Q L ( Q cofer Secos 5-7 [] Referrg o Theores ad 3 s deduced ad p L ( Q ow le ( p u U be a soluos of he opal sses (3-(34 Defe ad Y : = U : = u u P : = p p (39 The (3 ad (35 ad he ea value heore eld 698

9 ( = ( ( ( ( f ( x η Y f ( x ( Y Y Q Y AY f x f x f x Y Y = Y cy = U o Σ Y ( = Ω where η ( x les bewee ( x ad x ( ( x Thus cosderg (4 as he lear proble ( wh ( ( ( η ( ( Also (33 ad (36 eld (4 Q B Assupo (A3 f ( x L( Q d= f x s cocluded b Theore ( ( ( Σ Y CT f x f x Y U L ( ( ( ( ( ( ( P AP = f x p f x p Y = f x P f x f x p Y Q P cp = o Σ P ( T = Ω Sce f ( x L( Q cosderg (4 as he lear proble ( wh d f ( x cocluded b Theore 3 ( ( ( P C ( T f ( x f ( x p Y Referrg o Theore 4 p belogs o L ( Q value heore s obaed (4 (4 = s (43 Cosequel eplog Assupo (A3 ad he ea ( f ( x f ( x p C Y ( L ( Q (44 where C: f ( x p ( M p = γ whch γ L ( Q ( x les bewee ( x ( x Q Therefore (43 elds L Owg o Corollar ad he couous ebeddgs ow cobg (4 (45 ad (46 resuls Cosequel s obaed where Q L ( ( ( Q ( ad x ( P C T C Y Y (45 α U P C P (46 L ( Σ L ( Σ Σ ( Y CT M Y α CC T C Y Y L Q L Q Σ L Q L Q Referrg o Theore C( T ( ( ( ( ( ( C ( T = Y C T Y (47 ( ( ( ( M α CC Σ ( T C CT ( α CC ( T CT ( whe T Σ ad C ( (48 T s bouded Cosequel here exss 699

10 T > such ha for <T T < C T Ths elds he covergece of Y o zero ( L ( T; for T T hereb he covergece of P o zero ( T = for T T va (45 ad he covergece of U o zero L ( Σ L ( T; L ( Ω for T T va (46 The esae (3 Theore for he al boudar value proble (4 elds he deueraor (48 be posve ad ( Y C T f x Y f x Y Y U ( ( η ( ( ( L ( Σ ( ( L ( Q L ( Q L ( Σ C T M Y M Y U (49 where he secod equal s obaed usg he ea value heore Cosequel he covergece of Y o zero for T T s obaed Referrg o (47 he covergece of Y L ( Q s a leas lear whereb he covergece of P ad U L ( Σ wll be a leas lear for T T cofer (45 ad (46 The s cocluded fro he esaes (49 ha he covergece rae of Y o zero s a leas lear for T T { } = The sequece ( p u produced b (35-(37 s depede fro ( pu ad coverges o Uad pu ca be a soluo of opal sse (3-(34 hs s possble excep opal sse (3-(34 has ol oe soluo The ex wo corollares are used he error aalss Seco 6 Corollar Uder assupos of Theore 7 here exss T <T T such ha for T T he followg esae s vald As ( Y C ( T ad ( C T 3 Y C T C T Y (5 ( ( ( ; ( where C3 T are as (39 (48 ad (53 respecvel Proof 9 The proof follows he les of proof of Theore 7 As Y sasfes (4 he esae (5 Theore elds ex eplog he esaes (45 ad (46 resul As where Y ( ( ( ( Y C T M C( [ T] ; Y U (5 L Σ ( ( ( Y C T M ([ ]; Y α CC C T T C Y Y L ( Q Σ L ( Q L ( Q TY s deduced fro he above equal ( ([ ]; C T C ( T ( ( Y C3 T Y (5 C( [ T] ; = ( M α CC Σ ( T C C( T α CC ( T C( T T 3 Referrg o Theore C ( T ad ( Σ (53 C T are bouded whe T whereb here exss T > such ha he deueraor (53 s posve for <T T Se T = { T T} wh T beg deered Theore 7 The (5 s obaed fro (5 b repeaedl eplog he esae (47 Corollar 3 Suppose he quaslearzao ehod Theore 7 sead of he accurae al value he approxae al value s used Le T be as Corollar The for T T he followg esae s vald where C ( T C ( T ad C5 ( ( ( ( ( Y C3 T C T Y C 5 T (54 C( [ T] ; 3 T are as (48 (53 ad (59 respecvel Proof The proof follows he les of proof of Theore 7 As Y sasfes (4 Q wh 7

11 ( ( ( Y = = = he esae (5 Theore elds ex eplog he esae (45 ad (46 resul As Y ( ( ( ( Y C( [ T] ; C T M Y U (55 L Σ ( Y ( ( ([ ]; C T M Y C T α CC T C Y Y L ( Q Σ L ( Q L ( Q TY choosg ( ([ ]; T > as Corollar (55 for T T elds C T ( ( ( Y C3 T Y C 4 T (56 C( [ T] ; C ( T where C4 ( T = α CC Σ C ( T T ow order o coclude (54 we eed a esae lke (47 (47 s for he case ( Y ( = Such a esae s obaed followg he les whch (47 obaed As Y ( = he esae (4 Theore elds ( Y C T M Y U L Q L Q L ( ( ( ( Σ The eplog he esaes (45 ad (46 resul ( C T Y C T Y ( L ( Q L ( Q α CC Σ CT ( Y = here Y sasfes (4 wh where T T T beg deered afer (48 Referrg o (48 whou loss of geeral s cosdered C ( T C T > CC α Σ ( ( T CT ( Eplog repeaedl (57 elds whereb s obaed ( Y C T Y (57 L Q ( ( ( ( ( ( ( ( ( ( Y C T Y C T C T C T ( L ( Q ( T ( T ( T ( T C = C( T Y C L T ( Q C C T Y C C where he las equal s obaed fro ( (54 wh ( (58 < C T < for T T ow ulzg (58 (56 resuls ( T ( T C C5( T = C3( T C4 T C 5 Applcao o he Opal Boudar Corol Probles ad he Uqueess The proposed quaslearzao ehod Theore 7 s coverge o he e ervals [ ] ( (59 T for T T T beg deered Theore 7 I order o appl he quaslearzao ehod o he opal corol proble (6 up o a arbrar fal e T s possble o decopose he proble o a fe opal corol probles each o a erval wh legh less ha T I order o follow such a approach le T T I order o preserve he sabl T s chose as Corollar (also cofer Seco 6 7

12 ad T = T for soe Le : T = Q : = ( Ω ad : [ ] be a Baach space The L ( T; X s orsoorphc o ( L ; X = ( ( ( L T; X L ; X ( ( ( : = : = = L ( T; X L ( ; X = Replacg X b elds ha L ( Q L ( T; L ( Q L ( wh he or de = = ; Σ = Ω : = Le X hrough he soorphs = be orsoorphc o elds ha be orsoorphc o he closed subspace = = where { ( ; ( ; } { ( ( } = L TV L TV = L ; V L ; V = L ( Q ad replacg X b V = L ( Q c of c = ( ( = ( = = = wh he or de Thus f sasfes he al-boudar value proble (6 he : = sasf cosecuvel he followg al-boudar value probles ad vce versa: where : ( ( ( ( ( ( ( A( x c( x u( x ( x ( = ( Ω x A x = f x x x Q = Σ ν x x x (6 = = Cosequel he opal corol proble (6 s equvale o he cosecuve opal corol subprobles α J u : = u sasfes 6 (6 u Uad where Uad { u u U Σ ad } ( ( ( ( d = Therefore solvg he opal corol proble (6 s equvale o cosecuvel solvg he opal corol subprobles (6 Furherore he proposed quaslearzao ehod Theore 7 s applcable o each opal corol subproble (6 I fac he subsuo he -h subproble (6 rasfors o a equvale proble o he e erval [ ] = [ T] whereb he quaslearzao ehod wll be applcable o Moreover as a cosequece of Theore 7 he soluo of opal sse of -h subproble (6 s uque Thus vew of Theore s 4 ad 5 each subproble (6 has a uque opal soluo Cosequel b he equvalece bewee proble (6 ad cosecuve subprobles (6 ca be saed Theore 8 Opal boudar corol proble (6 uder Assupos (A-(A3 has uque opal boudar corol soluo ad opal sae soluo oe ha he uqueess could o be esablshed horough he opal heor of opal corol probles whch was used for sag he exsece Seco 3 Ths s due o lack of covex of he soluo se of proble (6 A ssue cocerg he above cosecuve process s he relao bewee ( opal sse of proble (6 ad ( (6 ( pu sasfes (7-(9 o Q ad ( p u sasfes pu he soluo of p u he soluo of opal sse of -h subproble 7

13 ( ( ( ( x A x = f x x Q (6 ν A( x c( x u( x o ( = ( Ω ( ( ( ( ( ( ( = Σ x x ( p x Ap x = f x x p x x x Q ν d Ap( x cp( x o ( = Ω ( ( ( ( L ( Σ = Σ p T x αu x px u x vx u v U ad (63 (64 I vew of Theore s 8 4 ad 5 opal sse of proble (6 has a uque soluo Coseque coparg (6-(64 wh (6-(9 s cocluded ha J( u = J( u = ad ( ( ( ad u ( = u( ( p s sce p sasfes (63 ad p ( = bu ( possble geeral o cosruc p fro (8 6 Error Aalss = Bu here s o a slar relao bewee he cosaes p ad p s o ecessarl zero; cofer (8 Also s o p s; however afer obag s p ca be copued fro Le ( pu be he soluo of opal sse (3-(34 ( B he cosecuve quaslearzao ehod Seco 5 he opal corol proble (6 s solved hrough cosecuve opal corol subprobles (6 Each subproble s solved b he quaslearzao ehod Theore 7 whch s a erave ehod wh fe eraos I applcaos s pleeed up o a fe eraos hereb producg error Cosequel durg solvg each subproble here exss a error produco ad a error propagao p u be he soluo of -h opal sse (6-(64 ad ( p u be he soluo provded b he quaslearzao ehod a erao for he -h opal sse e oe whch sasfes (35-(37 o Q For he frs subproble he quas = = = ad s eraed afer learzao ehod sars wh he accurae al value ( ( ( erao wh he fal value ( The error equals o Y ( ( ( value s accurae Corollar wh T = elds = As T ad he al ( ( ( C( T; L ( Q Y Y C T C T Y (65 3 = he quaslearzao ehod sars wh he approxae - For he -h subproble o [ ] : al value ( = ( Y ( ad s eraed afer erao wh fal value ( ror of fal value equals o Y ( = ( ( ex we esae hs error The er- The subsuo he -h subproble (6 rasfors o a equvale proble o he = T Seg T = T ad ulzg Corollar 3 for he equvale proble elds e erval [ ] [ ] he esae (54 wh = ( Y ( he esae The ulzg he reverse subsuo resuls ( [ ]; Y Y C ( ( ( ( ( ( C T C T Y C T Y 3 5 (66 ow begg fro = dow o = repeaedl eplog (66 resuls 73

14 ( [ ]; Y Y C ( ( ( ( ( ( C T C T Y C T Y 3 5 ( ( ( ( ( ( ( ( ( C T C T Y C T C T C T Y C T Y ( ( ( ( ( ( L ( Q L ( Q L ( Q L ( Q C T C T Y C T Y C T Y C T Y ( ( ( ( ( ( ( L ( Q C T Y C T C T C T C T Y 3 5 where he las equal s obaed b oe ha ( Y 5 Y L ( Q = L ( Q = ad he esae (65 Cosequel C5 T Y ( Y C3( T Y C C T ([ ] ; C T ( ( ( ( 5 (67 Y preses he accuulaed error cosss of he produco errors ad he propagao errors he cosecuve pleeao of quaslearzao ehod whe he pleeao s up o erao o each subproble I he esae (67 he er ( ad < ( < oal accuulaed error ( ehod Seco 5 s sable Furherore ( ( T ( T L ( Q C C3 T Y C 5 5 s depede fro C T ; cofer (48 ad hereafer Sce s fxed b creasg he uber of eraos he Y eds o zero Therefore he proposed cosecuve quaslearzao C5 T > for T > alhough C5 ( T ad C3 ( T decrease whe T decrease (or crease Cosequel a a rade off be ecessar bewee sze of (he uber of subprobles ad (he uber of requred eraos he pleeao of quaslearzao ehod order o have he desred oal error he cosecuve quaslearzao ehod 7 uercal Exaple A pcal exaple s preseed reflecg he obaed resuls he prevous secos applcaos Cosder he opal corol proble (6 wh he followg daa: Ω= ( ( Q = ( Ω α = a = a = a = a = f x x 5exp x x = 4x x x x = c = ( ( = ( ( ( a = b = Seg T = he cosecuve quaslearzao ehod s pleeed o he cosecuve subprobles (6 wh he opal sses (6-(64 The correspodg saes cosaes p ad corols u are approxaed b he elees ad boudar elees of couous lear fe elee spaces o Q [ = ] Ω wh h FEM = ad h = = T e whou dscrezao of e The lear opal sses (6-(64 are solved b he sesooh ewo s ehod [6] or Seco 5 [] ad he pleeao s doe wh MATLAB sofware Table preses he values of ad ( = 4 ( 4 4 ( 4 L ( Ω ( p = p4 ( 4 p4 ( 4 L ( Ω ( u = u4 ( 4 u4 ( 4 L ( Ω These values prese a leas a lear rae of covergece he quaslearzao ehod as was deduced fro (45-(47 Table preses he opal objecve values of proble whe he cosecuve quaslearzao ehod s pleeed wh dffere uber of subprobles bu fxed uber of eraos each quaslearzao d 74

15 Table The dfferece bewee eraos he quaslearzo ehod for he forh subproble a = 4 whe = 5 ad he uber of eraos s = ( e e- 64e-5 4e-8 79e- 8e-3 37e-6 e- e- e- ( p 8e- 33e-3 8e-6 e-9 8e- 49e-5 3e-8 e- e- e- ( u 39e- 95e-3 59e-6 3e-9 66e- 3e-4 4e-7 e- e- e- Table The opal objecve values wh dffere uber of subprobles bu fxed uber of eraos he quaslearzao ehod e = J = J = ehod e wh dffere s ad fxed As T = s soe sese he sep sze of e dscrezao s cree elds ore accurae approxao o he opal objecve value 8 Coclusos A cosecuve quaslearzao ehod was proposed for he opal boudar corol probles wh quadrac objecve of rackg pe ad a selear parabolc equao wh xed boudar as he sae cosra; cf (6 ad (3 The proposed ehod dvdes he corol proble equvalel o a fe cosecuve subprobles hrough parog he e erval o subervals; cf Seco 5 ad (6 The subprobles are solved cosecuvel b a quaslearzao ehod (hece he ae of proposed ehod Fall he opal soluo of corol proble s obaed b cosecuvel ergg opal soluos of subprobles The quaslearzao ehod for each subproble cosrucs a fe sequece of lear-quadrac opal boudar corol probles of for (38 The sequece of soluos o he opal sses of hese lear probles coverges o a soluos of he opal sse of subproble; cofer Theore 7 ad Seco 5 Ths ples he uqueess of soluo o he opal sse of a subproble hece he uqueess of opal soluo o he orgal corol proble; cofer Theore 8 Ths uqueess resul s ew o he bes of our kowledge he class of opal corol probles wh sae cosra of selear parabolc equao pe The covergece of quaslearzao ehod for each subproble depeds o he e erval legh of he subproble T ad here s a boud o T whch he covergece occurs T T T beg deered Theore 7 I coparso wh ehods whch requre he full dscrezao of orgal corol proble cf Chaper [] [] ad [7] T ca be cosdered as he e dscrezao sep legh I hs vew he cosecuve feaure of proposed ehod replaces he large scale copuaos full dscree ehods b he cosecuve sall scale copuaos he subprobles hece creasg he ache applcabl of ehod Specall quaslearzao ehod solvg he sequece of lear-quadrac corol probles he e dscrezao ca be avoded b choosg T eough sall cf Seco 7 I coparso wh superlear ehods whch are locall coverge as dffere versos of ewo s ehod ad/or Lagrage-SQP ehods (Chaper [] ad [] he cosecuve quaslearzao ehod s globall coverge ad s covergece order s a leas lear cf Theore 7 For exaple Table of Seco 7 preses a cubc covergece rae Thereb he cosecuve quaslearzao ehod s ver suable for he globalzao of locall coverge ehods b applg o fd a sarg soluo for hose ehods The quaslearzao ehod for subprobles has fe eraos bu applcaos s pleeed up o a fe erao Therefore s cosecuve applcao o he subprobles produces ad propagaes errors owever choosg T T guaraees he uercal sabl cf Seco 6 The posed boudedess assupos o he olear of proble ad he adssble corols are ecessar for he covergece proof cf Assupo (A3 Seco 3 ad proof of Theore 7 As he vesgaed corol proble here also has opal soluo wh uch weaker boudedess assupos cf [] applcao of cosecuve quaslearzao ehod hs case requres ew covergece proof 75

16 Refereces [] ze M Pau R Ulbrch M ad Ulbrch S (9 Opzao wh PDE Cosras Sprger Berl [] Trözsch F (999 O he Lagrage-ewo-SQP Mehod for he Opal Corol of Selear Parabolc Equaos SIAM Joural o Corol ad Opzao hp://dxdoorg/37/s [3] Bella R ad Kalaba R (965 Quaslearzao ad olear Boudar-Value Probles Aerca Elsever Publshg Copa ew York [4] Bella R (955 Fucoal Equaos he Theor of Dac Prograg V Posv ad Quas-Lear Proceedgs of he aoal Acade of Sceces of he Ued Saes of Aerca hp://dxdoorg/73/pas4743 [5] Buca A ad Precup R ( Absrac Geeralzed Quaslearzao Mehod for Cocdeces olear Sudes [6] Lakshkaha V ad Vasala AS (995 Geeralzed Quaslearzao for olear Probles Kluwer Acadec Publshers Dordrech [7] Carl S ad Lakshkaha V ( Geeralzed Quaslearzao ad Selear Parabolc Probles olear Aalss hp://dxdoorg/6/s36-546x(5-x [8] Beckebach EF ad Bella R (96 Iequales Sprger-Verlag Berl [9] Kalaba R (959 O olear Dffereal Equaos he Maxu Operao ad Moooe Covergece Joural of Maheacs ad Mechacs [] Raod JP ad Zda (999 aloa Porag s Prcples for Corol Probles Govered b Selear Parabolc Equaos Appled Maheacs ad Opzao hp://dxdoorg/7/s4599 [] Showaler RE (997 Moooe Operaors Baach Spaces ad olear Paral Dffereal Equaos Aerca Maheacal Soce Provdece [] Evas LC (998 Paral Dffereal Equaos Aerca Maheacal Soce Provdece [3] Ladzeskaja OA Solokov VA ad Ural ceva (968 Lear ad Quas-Lear Equaos of Parabolc Tpe Aerca Maheacal Soce Provdece [4] Zedler E (99 olear Fucoal Aalss ad Is Applcaos Vol II/A: Lear Moooe Operaors Sprger Berl [5] Showaler RE (994 lber Space Mehods for Paral Dffereal Equaos Elecroc Joural of Dffereal Equaos Moograph Dover Publcaos Meola [6] Ulbrch M ( Sesooh ewo Mehods for Varaoal Iequales ad Cosraed Opzao Probles Fuco Spaces SIAM hp://dxdoorg/37/ [7] eaaak P ad Tba D (994 Opal Corol of olear Parabolc Sses: Theor Algorhs ad Applcaos Marcel Dekker Ic ew York 76