A simple algorithm for Optimal Control problems governed by Non-linear Hammerstein Integral Equations
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- Ashlee Holmes
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1 A ple lgor for Opl Corol proble govered b o-ler Here Iegrl Eqo Rez Mrbqer Mood Sool Od Sol Frd Abr I d ovel opol frewor for olvg l of opl orol proble for e govered b o-ler Here egrl eqo preeed A eore for e overgee e vld of e ppro lo gve del erl epere opro w e olo ofr e effe e r of e propoed eqe Ide Ter Opl Corol proble o-ler Here egrl eqo Ierve eod Approe-ll olo I ITRODUCTIO Ierve ee opzo reled o egrl eqo re wo proe feld of reer ppled ee egeerg Te or prpoe of opzo o deere proedre of ow opll ge or flee rel e o eve dered rel T reqre o relze lrge-le opzo rege w reg ople w r ove e develope of erl eqe for opzo prpoe O e oer el forlo of pl peoe egrl eqo re lw eoered ve red eo I f egrl eqo re por dfferel eqo pper vre of pplo feld ldg o e poel eor geop eler ge e eor of ge eredr peoe bolog q e el eoo poplo gee ede fld e ed e e odo rdve e rfer proble [ ] I w ere re dre dre erl olo for egrl eqo [ ] Frerore opl orol of e govered b egrl eqo re oeo pplo e opl orol proble reled o e Ore-Ulebe proe w re fro l oo eor [8] I pper we fo o e Mrp reeved Febrr 8 5 -T wor bee ppored b reer or of e Il Azd Uver Irr br Rez Mrbqer J Depre of Me Irr Br Il Azd Uver Irr Ir Mood Sool PD Sde P-e-oor Uver Md Ir Od Sol Frd Sool of Me Coper See Dg Uver of B See Dg Ir forlo of l of opl orol proble govered b o-ler Here egrl eqo follow Mze J f d d d e ow fo f C[] R R re gve fo w re defed o e ervl be eped o e Tlor ere bo Bede C [] re e reor orol fo repevel Here ed e proble - qe olo everele ere o reer o olve e opl orol proble - T e prpoe of d o or erve ee o ob e pproe olo lo e ll olo of e proble e for of polol ere olo II TAYLOR SOLUTIO I eo followg e wor of Mood [] ed for gve e o-ler egrl eqo for w re fo ve ble dervve e ervl e olo e for og Eq Tlor polol of degree re oeffe o be deered Ug Eq we lo oder Eq epreed er of Tlor polol 8 wwwerpbloorg
2 A ple lgor for Opl Corol proble govered b o-ler Here Iegrl Eqo 84 wwwerpbloorg 4 5 W re Tlor polol of degree Te oeffe re o-ler obo of follow: 7 A Mr repreeo of e opoe To ob e olo of e for of epreo we fr dfferee e w repe o : d d d d 8 e le r repreeo Aordg o [5] Eq8 be wre F 9 d d d d F Here we ve d d d
3 85 wwwerpbloorg d e ] [ d Ug Lebz' rle we ob ] [ d oe Eq If we pproe Eq 4 5 we ob d d F Sbg Eq Eq9 gve d d 4 or brefl K H 5 d d H > 6 d K 7 If we e relo 5 rede o e of o-ler eqo for ow oeffe follow: K H Y X 8 X Y Ad H K re re defed 6 7 Te o-ler e of eqo obed 8 be olved g rd e oolboe MATLAB III THE ITERATIE METHOD AD ITS COERGECE Le Q be e be of e prod pe [] [] C C o ll pr w f Eq Alo le Q be e be of Q og of ll pr preerzed orol fo e followg polol 9 e ered olo of e egrl eqo w odered polol of degree o e Here R R e : re oo fo ow we oder e zg of J o Q
4 A ple lgor for Opl Corol proble govered b o-ler Here Iegrl Eqo w { } ow T obvol opzo proble deol pe { R : } J be odered fo J Sppoe be e olo of Q zg J o ; polol for of e e 9 g llow o ppl e propoed eod Seo for erg polol olo of w rel { } obg eqee of reor fo Tlor ere fll o eve zg eqee { } Le If { } f Q J for overge eqee e Proof Proof [6 9] ow be olded } { overge bee odereg boded fro below eqee Teore If e J l Proof Proof [6 9] f Te bove do rel be rzed erl lgor for obg e pproe olo for e opl orol Eq Algor Cooe > > for e r of e olo Sep Le J e e e e e e Sep Le f Q J < Q fd Sep If e go o Sep 4 oerwe go o Sep Sep 4 Le fd J go o Sep 5 f Q < Sep 5 If e op oerwe go o Sep 4 I UMERICAL EXPERIMETS ow we ow e effe of e eod derbed g e followg eple I ll eple e pproe olo wll be opred w e e olo Eple Le fr oder e opl orol proble govered b Fredol-Here egrl eqo follow Mze J d d Te e opl olo of - re w e opl rero J J we ve Y K I e fr ero e for ge X I e e ero for X Y 8 / / /4 K we So b g e ero Algor e erl rel re llred Tble Fgre -4 Tble : Te Approe-All rel for Eple Ier J 86 wwwerpbloorg
5 Fgre : E pproe reor fo for Eple Eple We oder e opl orol proble govered b olerr-here egrl eqo follow J d Mze d Te e opl olo of e proble -4 re w e opl rero J J Ug e ero Algor e erl rel re llred Tble Fgre 5- Tble : Te Approe-All rel for Eple Ier J Fgre : E pproe orol fo for Eple Fgre : E pproe reor fo for Eple Fgre 5: E pproe reor fo for Eple Fgre 4: E pproe orol fo for Eple Fgre 6: E pproe orol fo for Eple 87 wwwerpbloorg
6 A ple lgor for Opl Corol proble govered b o-ler Here Iegrl Eqo Fgre 7: E pproe reor fo for Eple Eple Fll Le oder e opl orol proble govered b olerr-fredol-here egrl eqo follow Mze J d 5 ep d 6 ep d 5e e Te e opl olo of 5-6 re w e opl rero J J Tble Fgre -4 gve e erl rel for eple Tble : Te Approe-All rel for Eple Ier J Fgre 8: E pproe orol fo for Eple E 6 55 Fgre 9: E pproe reor fo for Eple Fgre : E pproe reor fo for Eple Fgre : E pproe orol fo for Eple Fgre : E pproe orol fo for Eple 88 wwwerpbloorg
7 Fgre : E pproe reor fo for Eple Fgre 4: E pproe orol fo for Eple COCLUSIO Opl orol proble govered b o-ler Here egrl eqo re ll dffl o olve lll o eer o ob e pproe olo Te pree eod effeve for e e ow fo ve ffe dervve e gve ervl Oe of e dvge of eod e opl olo e e reor orol fo re epreed Tlor ere red Terefore el be evled for rbrr vle of w low opo effor T eod wll o wor for e e gve fo do o ve eog dervve A ereg fere of eod we ob ll olo e ow e eple REFERECES [] MA Abdo O e olo of ler oler egrl eqo Appl M Cop 46 pp [] S Adere M Hdzde Te Copol Effe Of Wl Approo For Two-Deol olerr egrl eqo A-Erope Jorl of Me 4 pp 6-7 [] E Bbol J Bzr AR d Te deopoo eod ppled o e of Fredol egrl eqo of e eod d Appl M Cop 48 pp [4] C Hoe IM vo Opl orol of flow w doo J of opol p 87 pp [5] M I Gl Pov of Gree' Fo o olerr Iegrl Hger Order Iegro-Dfferel Eqo Al Applo 7 pp [6] SC Hg RP Sw Te Treffz eod egrl eqo Adv Eg Sofwre 4 pp 57-6 [7] S Jg Rol Seod d egrl eqo for e ll poel eor o ope rfe II J Cop P 95 pp -6 [8] T Kl Soe egrl eqo w "orol" erel IEEE Tr If T IT pp [9] PK Ke P Pr Copol Meod of Ler Iegrl Eqo Brer Boe /o Sprger-erlg ew Yor [] D Lg B Zg erl l of grded e eod for l of eod d egrl eqo o rel le J M Al Appl 94 pp 48-5 [] Y Mod Tlor polol olo of o-ler olerr-fredol egrl eqo I J of Coper Me 8 pp [] K Mleed M Kr Ug e WPG eod for olvg egrl eqo of e eod d Appl M Cop 66 pp - [] K Mleed M Kr erl olo of o-ler Fredol egrl eqo b g l-wvele e Perov-Gler eod Appl M Cop 68 pp - [4] K Mleed Y Mod erl olo of ler Fredol egrl eqo b g brd Tlor Blo-Ple fo Appl M Cop 49 pp [5] K Mleed Y Mod Tlor polol olo of g-order o-lerolerr-fredol egro-dfferel eqo Appl M Cop 45 pp [6] O S Frd M Sool Tlor olver for Fredol opl orol proble J of Adv Reer Dfferel Eqo pp - [7] J Rd A Pr Se-Orogol Sple Sllg Fo For Solvg Here Iegrl Eqo I J of Wvele Mlreolo Iforo Proeg IJWMIP 9 pp [8] J Sber-df M Tgr Modfed Hooop Perrbo Meod for Solvg Iegrl Eqo I J of Moder P B IJMPB 4 pp [9] M Sool O S Frd erl See for Fredol Iegrl Eqo Opl Corol Proble v Bere Polol Arl J of B Appled See 4 pp [] I YM Se KJE er JFM Ipe Opl orol eor: geer ool for defo orol of Bo-el reor Al rewev of orol 6 pp 57-7 [] W Wg A ew el lgor for olvg e eod d of Fredol egrl eqo Appl M Cop 7 pp [] S Ylb K Erde Approe olo of oloer olerr Iegrl Eqo Se I J of Moder P B IJMPB 4 pp [] SC Yg A vego o egrl eqo eod volvg erl glr erel for o erg J Sod b 4 pp 5-9 [4] T Zdee T oegrd C Kele Ke S J Gler C ele Opl orol MR peroop: erl pleeo SIMPSO Jorl of Mge Reoe 97 pp -4 [5] S Zg Iegrl Eqo Cogqg pre Cogqg C 987 Rez Mrbqer J reeved e MS degree Me ArKbr Uver of Ter H reer ere re e re of ppled e He e fl of Il Azd Uver Irr Ir Mood Sool PD de of Opl Corol P or Uver of Md Ir H reer ere re: Opl Corol Operg Reer Fzz Od Sol Frd oe profeor Appled Me e fl of Dq Uver Ir H reer ere re: Opl Corol Operg Reer Fzz 89 wwwerpbloorg
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