Existence and Uniqueness of the Optimal Control in Hilbert Spaces for a Class of Linear Systems

Transcription

1 Iellge Iformao aageme 34-4 do:436/m6 Pblshed Ole Febrar (hp://wwwscrporg/joral/m Esece ad eess of he Opmal Corol lber Spaces for a Class of Lear Ssems Absrac ha Popesc Ise of ahemacal Sascs ad Appled ahemacs of he Romaa Academ chares Romaa Emal: ma popesc@ahoocom We aalze he esece ad eess of he opmal corol for a class of eacl corollable lear ssems We are eresed he mmzao of me eerg ad fal mafold rasfer problems The sae varables space X ad respecvel he corol varables space are cosdered o be lber spaces The lear operaor T( whch defes he solo of he lear corol ssem s a srog semgrop Or aalss s based o some resls from he heor of lear operaors ad fcoal aalss The resls obaed hs paper are based o he properes of lear operaors ad o some heorems from fcoal aalss Kewords: Esece ad eess Opmal Corol Corollable Lear Ssems Lear Operaor Irodco A parclar mporace shold be assged o he aalss of he corol of lear ssems sce he represe he mahemacal model for varos damc pheomea Oe of he fdameal problems s he fcoal opmzao ha defes he performace de of he damc prodc Ths der dffereal ad algebrac resrcos oe deermes he corol correspodg o fcoal eremsao der cosderao [] Varaoal calclao offers mehods ha are dffcl o se order o vesgae he esece ad eess of opmal corol The mehod of deermg he feld of eremals (sweep mehod ha aalzes he esece of cojgaed pos across he opmal rasfer rajecor (a sffce opmm codo proves o be a ver effce oe hs coe [3 5] Throgh her reslg applcaos me ad eerg mmzao problems represe a mpora goal ssem damcs [6 ] Rece resls for corollable ssems epress he mmal eerg hrogh he corollabl operaor [ 3] Also sabl codos for ssems whose eerg eds o zero fe me are obaed he lerare [34] sg lear operaors lber spaces hs sd we shall aalze he esece ad eess of opmal corol rasfer problems The goal of hs paper s o propose ew mehods for sdg he opmal corol for eacl corollable lear ssem The mmzao of me ad eerg rasfer problem s cosdered The mmzao of he eerg ca be see as beg a parclar case of he lear reglaor problem aomacs sg he adjo ssem a ecessar ad sffce codo for eac corollabl s esablshed wh applcao o he opmzao of a broad class of aer-pe fcoals mm Tme Corol Esece Problem Formlao We cosder he lear ssem : d d A A ( ( ( where s a lber space wh er prodc ad orm A: D( A s a boded operaor o whch geeraes a srog sem grop of operaors o ( A S e : s a boded lear operaor o aoher lber space for eample L : [] s a sare egrable fco represeg he ssem corol ( For a corol fco here ess a solo of ( gve b Coprgh ScRes

2 POPESC 35 ( e A e s d s ( s A ( The corol problem s he followg oe: Gve I z X ad he cosa ( le s deerme ( ( sch ha ( z f ( ad ere X ad are lber spaces Theorem The opmal me corol ess for he above formlaed problem Proof Le be a decreasg moooos seece sch ha Also le s cosder he seece ( wh We have ( whch gves ( ( (3 ( ( d (4 ( d ( d (5 S ( beg a coos lear operaor s herefore boded I follows ha we have: S ( ( (6 ( ( d (7 If we cosder he se of all he admssble corols (8 he s a weakl compac closed cove se As s weakl compac a seece ( possesses a weakl coverge sbseece o a eleme Ths for ad a (he dal of we have Sce lm (9 follows ha Also we have ( So For ever wrg : X S ( ( S ( ( X F he Epresso (3 becomes or le s evalae he dfferece ( d k ( d k (3 d L k (4 L L L L (5 F Therefore F ( L L L L ( ( L L L d d (6 (7 sg he properes of he operaor oe obas F ( ( ( ecase he seece S ( d d (8 coverges weakl o he frs erm (8 eds o zero for (see(9 O he oher had s a srogl coos semgrop ad hece from s bodedess follows ha here ess a cosa K sch ha Coprgh ScRes

3 36 K POPESC (9 So follows ha he secod erm (8 eds o zero as The seece ( X s weakl coverge o ( z X f for a X X we have whch gves lm ( ( ( lber spaces p p Spaces l L p 3 If he aach spaces are rod he he prodc space s rod oo 4 form cove spaces are rod b he coverse mplcao s o re eess We assme ha ad are opmal Therefore z ( ( d ( From ( oe ges ( d z ( ad hece s he opmal corol A mpora resl whch s gog o be sed for provg he eess belogs o A Fredma: Theorem (bag-bag Assmg ha he se s cove a eghborhood of he org ad s he corol of opmal me he problem formlaed ( ([ ] follows ha for almos all ] [ eess of Tme-Opmal Corol Rod Space Le be he sphere he aach space ad le be s bodar [] The space s sad o be rod f he followg evale codos are sasfed: a f follows ha here ess a scalar sch ha ; b each cove sbse K has a leas oe eleme ha sasfes z K z K ; c for a boded lear fcoal f o here ess a leas a eleme sch ha f f ( f d each s a po of ereme of Eamples of rod spaces: ; ( d z (3 ( d z (4 addg Eaos (3 ad (4 oe obas ( ( ( d z I follows ha ( ( ( are opmal sg Theorem we have ( k (5 ( ( ( Sce he codo ( s sasfed space ad Ths ad herefore s a rod (6 (7 whch eds he proof of he eess 3 mm Eerg Problem 3 Problem Formlao Le A (see( be a corollable ssem wh fe dmesoal sae space X [6 4] Le I ] ( a X b X be gve Le be [ he lber space L ( I The mmm orm corol Coprgh ScRes

4 POPESC 37 problem ca be formlaed as follows: deerme sch ha for some I ( b s mmzed o ] where represes [ he orm o L ( I Le s f Cosder he lear operaor defed b We have Theorem 3 Le ( L : L ; L : s s d s (9 ( a L b (8 (3 L : be a lear mappg bewee a lber space ad a fe dmesoal space [] The here ess a fe dmesoal space sch ha he resrco L of L o s a jecve mappg Proof Le e be a bass for he rage of Gve a ca be wre as where each The L L L e (3 ca be epressed b f (3 f L f e (33 f are learl depede ad geerae a dmesoal sbspace From he properes of lber spaces (he projeco heorem follows ha Le Le The f Ths ker L L ker L We ge (34 L f e (35 Sce e are depede ad so ker L L maps f bjecvel o he rage of ad hece s a jecve mappg L o Le L (36 ecase L Sce we have L L (37 (38 The a e mmm orm L L ess ad s he mmm orm eleme sasfg L Remark The e solo of he eao L wh mmm orm corol s he projeco of oo he closed sbspace f f ece from (3 here ess a corol L ( ; rasferrg a o b me f ad ol f b a The corol whch Im L acheves hs ad mmzes he fcoal E s d s called he eerg fcoal s Defe he lear operaor L b S a (39 ( Q r r d r (4 We have he followg resls (see [ ]: Proposo The fco s he e solo of he Eao [3] where d d D( A Q Q C Q D( A Q C R D( A A I (4 (4 Coprgh ScRes

5 38 POPESC If A geeraes a sable semgrop he Q lm Q (43 ess ad s he e solo of he eao Q A D( A (44 The proof of Proposo s gve [] The followg heorem gves geeral resls for he fcoals E ( a b he mmal eerg for rasferrg a o b me ad E ( b where a b (see [] Theorem 4 For arbrar ad a b Q a b E ( a b (45 If ( s sable ad he ssem S corollable me he Q b b E ( b s ll A 3 oreover here ess C sch ha C Q b E ( b Q b b 4 Nmercal ehods for mal Norm Corol 4 Preseao of he Nmercal ehods Le be a damc ssem wh R A R m (46 (47 X R I [93] Gve b deerme sch ha ( b p s mmzed for p [ Now e b ( d d (48 I order o make Eao (48 re chose o he erval ] Cosder he -h compoe of he vecor e : e ms be [ e ( d f ( (49 where s he row of he mar ad f s he e fcoal correspodg o he er prodc (Reszrepreseao heorem Le be a arbrar scalar The Sce e f f ( ( R s a lber space he er prodc e f ( e (5 (5 beg a vecor wh arbrar compoes If Eao (5 s re for a leas dffere learl depede he Eao (48 s also re We have f ölder s eal (5 From Eaos (5 ad (53 p f e p p e (53 (54 Ever corol drvg he ssem o he po b ms sasf Eao (54 whle for he opmal (mmm orm corol Eao (54 ms be sasfed wh eal (Theorem Nmercall oe ms search for a vecors sch ha he rgh-had sde of Ea- Coprgh ScRes

6 POPESC 39 o (54 akes s mamm 4 A Smple Eample We cosder a sgle op lear damc ssem where A ecase Therefore I e b R X R I R fed he S ( e ad from ölder s eal where we have assmed ha ( b ( d (55 ( d ( d (56 ( d (57 ( L ( I The mmm orm corol f ess wll sasf From (56 ad from (57 we have ad respecvel e (58 sg( sg( ( [ ] (59 ( h (6 [ ] h arbrar cosa The corol sasfes he relao or p p p p ( h ( k ( (6 From codo (59 p sg ( k ( (6 p k sg ( ( ( (63 sbsg Eao (63 o Eao (55 he cosa k ca be deermed 4 The Parclar Case p I hs case he Eao (55 represes he er prodc L ; ( The relao (63 becomes The Ths b (64 k sg ( ( ( (65 b k ( d k (66 ad he opmal corol s gve b 5 Eac Corollabl 5 Adjo Ssem b k (67 b( ( (68 Le [ ] be he fdameal solo of a homogeeos ssem assocaed o he lear corol ssem A (see( [453] Ths we have From oe obas whch mples ha Sce d A I [ ] (69 d d d I S ( S ( d d d S d (7 ( S ( A (7 ( A S ( S (7 ( S ( S (73 he ssem becomes d d S A S ( [ ] ( ( (74 I follows ha S [ ] s he fdameal solo for he adjo Ssem (69 5 A Class of Lear Corollable Ssems Coprgh ScRes

7 4 POPESC We cosder he class of lear corol ssems vecoral form [34835] d A C ( (75 d Le be a eral po of he closed boded cove corol doma Ths doma coas he space E m of varables We ake ( m (76 hs rasformao ssem (75 becomes d A C (77 d Ths oe rasfers he org of he coordaes of space E m A he same me he org of he coordaes of space E m s a eral po sde he doma Le s deoe b ( he solo of ssem (75 whch correspods o he corol Ths corol s admssble as he org of he coordaes belogs o he doma ad sasfes he al codo ( As a resl d ( A ( C d (78 Le be a corol ad ( be he rajecor correspodg o ssem (75 We have We ake d ( A ( C ( (79 d ( ( ( (8 The from (78 ad (79 oe obas ece for d ( A ( ( d ( he ssem (75 becomes A A : ( R (8 Ths he lear corol ssem belogs o he class defed b (8 A 53 Opmal Corol Problems For a gve le s deerme he opmal corol ~ ha eremses he fcoal of fal vales J F c ( ( (8 ad sasfes he dffereal cosras represeed b he ssem (8 The adjo varable R sasfes eao (83 where s he amloa assocaed o he opmal problem (84 Sce he fal codos ( are free ad he fal me has bee dcaed from he codo of rasversal oe obas ( I follows ha he adjc ssem becomes wh solo A A : ( (85 ( S ( (86 Proposo For he class of opmm problems der cosderao he followg de holds re: d (87 Proof Assmg ha C ad ( D( A follows ha C Iegrag b pars sce ( : : we have A d A d d d A d d A d ( (88 Coprgh ScRes

8 POPESC 4 Oe obas d (89 The de has bee demosraed Ths resl ca be eeded for arbrar L ( ; ad ( A mpora resl referrg o he eac corollabl of he lear ssem (8 s saed Theorem 5 The ssem A s eacl corollable f ol f he followg codo s sasfed S ( d c (9 Proof We assme ha s eacl corollable A Le L ( ; ad ( We cosder ha he applcao L : ( T (9 s well defed Le : L ( ; be he verse of L From Theorem 3 follows ha here ess a fe dmesoal sbspace ker L sch ha he resrco s jecve L L (9 ker L Sce L ( L ( ( follows ha rasfers ( for he ssem A We choose ( a d ( ( ( I follows ha ( d (93 For ( L L sg ölders s eal oe obas ( d ( ( d d d (94 From (93 ad (94 for or ( d we have d d (95 (96 chagg becomes ad S ( s rasformed o S ( Therefore we ge oe evale relao of corollable ssem whch sbses S ( c ( d S d The drec mplcao has bee demosraed We assme ha codo (9 s flflled The For a d c (97 (98 we ake he se ( ( ( s solo of A A We cosder he solo ( for whch correspods o he above meoed corol defe he boded operaor Sce ( ( Le s : ( L (99 he de (89 becomes ( Therefore here ess a cosa ( s sasfed I follows ha d c c ( for whch s posvel defed Ths resmes he coclso ha he ssem corollable A s Coprgh ScRes

9 4 POPESC Ths sce s versable for a sae ( bee demosraed s sch ha here ess a corol whch rasfers 6 Coclsos The heorem has The ma corbo hs paper cosss provg esece ad eess resls for he opmal corol me ad eerg mmzao problem for corollable lear ssems A ecessar ad sffce codo of eac corollabl opmal rasfer problem s obaed Also a mercal mehod for evalag he mmal eerg s preseed I Sbseco 5 he ohomogeeos lear corol ssems are rasformed o lear homogeeos oes wh a ll al codo For he corol of sch a class of ssems oe eeds o cosder he adjo ssem correspodg o he assocaed opmal rasfer problem (Sbsecos 5 ad 53 The above heor ca be sed for solvg varos problems spaal damcs (redez-vosspaal saelle damcs space prsg [3489] Also hs heor ca be sccessfl appled o aomacs robocs ad arfcal ellgece problems [6 8] modelled b lear corol ssems 7 Refereces [] Q W Olbro ad L Padolf Nll corollabl of a class of fcoal dffereal ssems Ieraoal Joral of Corol Vol 47 pp [] J Sssma Nolear corolabl ad opmal corol arcel Dekker New York 99 [3] Popesc Varaoal rasor processes ad olear aalss opmal corol Techcal Edcao chares 7 [4] Popesc Sweep mehod aalss opmal corol for redezvos problems Joral of Appled ahemacs ad Compg Vol 3 pp [5] Popesc Opmal corol lber space appled orbal redezvos problems Advaces ahemacal Problems Egeerg Aerospace ad Scece; (ed Svasdaram Cambrdge Scefc Pblshers pp [6] S Che ad I Lasecka Feedback eac ll corollabl for boded corol problems lber space Joral of Opmzao Theor ad Applcao Vol 74 pp [7] Popesc O mmm adrac fcoal corol of affe olear corol Nolear Aalss Vol 56 pp [8] Popesc Lear ad olear aalss for opmal prs space Advaces ahemacal Problems Egeerg Aerospace ad Scece; (ed Svasdaram Cambrdge Scefc Pblshers pp [9] Popesc Opmal corol lber space appled orbal redezvos problems Advaces ahemacal Problems Egeerg Aerospace ad Scece; (ed Svasdaram Cambrdge Scefc Pblshers pp [] Popesc mm eerg for corollable olear ad lear ssems Semerre Theore ad Corol verse Savoe Frace 9 [] Popesc Sabl ad sablzao damcal ssems Techcal Edcao chares 9 [] G Da Prao A J Prchard ad J Zabczk O mmm eerg problems SIA J Corol ad Opmzao Vol 9 pp 9 99 [3] E Prola ad J Zabczk Nll corollabl wh vashg eerg SIA Joral o Corol ad Opmzao Vol 4 pp [4] F Gozz ad P Lore Reglar of he mmm me fco ad mmm eerg problems: The lear case SIA Joral o Corol ad Opmzao Vol 37 pp [5] Popesc Fdameal solo for lear wo-po bodar vale problem Joral of Appled ahemacs ad Compg Vol 3 pp [6] L Frsol A orell ad oager e al Arm rehablao wh a roboc eoskeleleo Vral Real Proceedgs of IEEE ICORR 7 Ieraoal Coferece o Rehablao Robocs 7 [7] P Garrec Ssemes mecaes : Coffe P e Kheddar A Teleoperao e eleroboe Ch ermes Pars Frace [8] P Garrec F Geffard Y Perro (CEA Ls G Pola ad A G Frederech (AREVA/NC La age Evalao ess of he eleroboc ssem T-TAO AREVANC/ La age ho-cells ENC 7 rssels elgm 7 Coprgh ScRes