Spectral Galerkin Method for Optimal Control Problems Governed by Integral and Integro- Differential Equations

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1 Mh. Sci. Le. Vol. o Mheicl Sciences Leers An Inernionl SP url Sciences Publishing Cor. Specrl Glerin Mehod or Opil Conrol Probles Governed by Inegrl nd Inegro- Dierenil Equions Mos A. El-Kheb Mnl. E. Al-Hohly nd Hussien S. Hussien 3 D Universiy College o Science nd Ars yriy dep. o Mhes. Sudi Arbi D Universiy Fculy o girls El D dep. o Mhes. Sudi Arbi 3 Dep. o Mh. Fculy o Science King Fisl Universiy Al-Ahs398 P.O. Bo38 Sudi Arbi originl ddress: Dep. o Mh. Fculy o Science Souh Vlley Universiy en Egyp Absrc::In his pper Legendre inegrl ehod is proposed o solve inegrl nd inegro- dierenil probles nd opil conrol probles governed by inegrl nd inegro- dierenil equions. Glerin ehod is used o reorule he proble s consrined opiizion proble. The resuling consrined opiizion proble is solved by Hybrid penly pril qudric inerpolion echnique. uericl resuls re included o conir he eiciency nd ccurcy o he ehod. Keywords:Specrl ehods- Legendre polynoils Hybrid penly pril qudric inerpolion echnique- Inegrl equions Inegro dierenil equions - Opil conrol Probles.. Inroducion Specrl ehods using epnsion in orhogonl polynoils such s Chebyshev or ulrsphericl polynoils is successul in he nuericl pproiion o vrious boundry vlue probles see or insnce Ahues e l [] Cnuo e l [3]nd Kopriv [5]. I hese polynoils re used s bsis uncions hen he re o decy o he epnsion coeiciens is deerined by he soohness properies o he uncion being epnded. This choice o ril uncions is responsible or he superior pproiion properies o specrl ehods copred wih inie dierence nd inie eleen ehods. El-Hwry e l. [4] derived soe useul properies o he ulrshpericl polynoils. They inroduced n ulrshpericl pproiion or ny coninuous uncion nd is inie inegrls. They derived error esies or his pproiion. They inroduced n lgorih h gives n opil pproiion o he inegrls. Opil conrol probles governed by inegrl nd inegro- dierenil equions rise in vriey o probles wih eory eecs including econoics populion dynics epideiology. Belbs[] presened nd nlyzed novel ehod or solving opil conrol probles or Volerr inegrl equions bsed on pproiing he conrolled Volerr inegrl equions by sequence o syses o conrolled ordinry dierenil equions. Peng e l.[6] considered clss o second-order nonliner ipulsive inegro-dierenil equions o ied ype whose principle is ie-vrying genering operors wih unbounded perurbion on Bnch spces. They discussed he perurbion o ie-vrying operor ri nd consruced corresponding he evoluion syse genered by operor ri. ingning[7] considered he nuericl siulion or clss o consrined opil conrol probles governed by inegrl equions o Fredhol ype. He provided superconvergence nlysis or he Glerin pproiion o hese conrol probles. Bsed on he resuls o he superconvergence nlysis he esblished recovery ype poseriori error esior which cn be used or dpive esh reineen The purpose o his pper is o solve inegrl nd inegro- dierenil probles nd opil conrol probles governed by inegrl nd inegro- dierenil equions. We shll cobine Legendre inegrl pproiion wih Glerinehodo reorule he inegrl nd inegro- dierenil proble o unconsrined opiizion proble. Opil conrol probles governed by inegrl nd inegro- dierenil equions shll be reoruled o be consrined opiizion proble. The resuling unconsrined opiizion proble is solved by penly pril qudric inerpolion echnique wheres he consrined opiizion proble is reed by Hybrid penly pril qudric inerpolion echnique.

2 34 The error bound or he used pproiion is discussed o ensure he eiciency o he proposed ehod. We include enough nuericl eples nd coprisons o conir ccurcy o he ehod. This pper is orgnized s ollows. In Secion we will inroduce he Legendre specrl pproiion. In Secion 3 we will discuss error bound. In Secion 4 we will se bsis o Glerin ehod. Seing o he proble is ound In Secion 5. In Secion 6 we will describe he proposed ehod o soluion which uses Glerin ehod o reorule he proble s consrined opiizion proble. In Secion 7 we will se Hybrid penly pril qudric inerpolion echnique o solve he resuling consrined opiizion proble. In Secion 8 we will give nuericl resuls. Finlly in Secion 9 we will conclude his pper.. The Legendre Specrl Approiion The Legendre polynoils re sequence o polynoils... ech respecively sisying he ollowing wo recurrence relions. D D D!!.b The recurrence orul. cn be used o genere he Legendre polynoils sring wih nd see Szegö [8]. The order derivive o he Legendre polynoils cn be genered by he recurrence orul.b. where is he order o diereniion. We hve ro. where X... Polynoil o lower order The Legendre polynoils sisy he orhogonl propery d where.3 nd i i We presen here he Legendre pproiion o he inegrion o ny uncion C [ ] eqully spced poins S.... Le be pproied Sby Legendre polynoils nely.

3 where.4 S d We cn wrie rpezoidl rule or pproiing his inegrl b '' b h b g d h g g h so equion * will be: '' 3 * 35.4b where he double prie on he suion denoes h he irs nd ls er hlved. Fro Eqs we cn deine he eleens o he rices D n n d i n... i i.5 where d n n i i D d S n i wih o. w D n=. s ollows: n S.6 We pproie he -ies inegrl o uncion i i i... dd... d qi ies S i i [ ] qi where i i.7! q i i i S by he ollowing relion:.5b.7b nd q i i r r r d r

4 36 3. Error Bound Le ] [ C be pproied by.4. We deine he uncion: -P - u nd choose such h U hs roo sisying hen P P 3. Since C [ ] [ ] P C [ ] C i ollows h u hs ore hn + roos. This ensure h u hs les one roo hen here eiss in [-] which u P 3. Since P nd by.!! Then ro 3. nd 3. we hve:! P hence! P. This error is bounded since! R X X P X Hence! R X 3.3 ow Le be pproied.4 hen by using.4b nd J so 3 J

5 37 by diereniion - ies we ge D D i di E i Where E D 3! D i i i E i is deined by i D i D The error bound o he inegrl is obined ro i i i... dd... d q R S i i where i i i R... dd... d ies i i i ies dd d 3 nd... in []. nd re he Legendre polynoils Glerinehod Consider he specrl pproiion u is used o solve he inegro- dierenil proble L u orhogonl bse uncion. The Glerin ide is o require he residul [] R... L u To be orhogonl o ech o he bse uncions. Th is he inner produc R using he inner produc deiniion [7] w R d Wih w is he weigh uncion corresponds o he orhogonl bse uncions.... wih 5. Seing o he proble We re deling wih he opil conrol proble governed by inegrl or inegro- dierenil equions. The probles cn be sed s: Find he conrol u which iniizes he uncionl I u d 5. subec o n inegro- dierenil equion 4.

6 38 L u g where L u r K s s. u s ds 5.b Ar r 5.c Where r denoes he rh- ordinry derivive o nd he coeiciens r r... y be uncions in nd [ ]. 6. Descripion o he soluion We pproie he derivives in equion 5.b vi diereniion specrl ri.5 i. e. n n i d i n... We use inegrion ri.7 o pproie he inegrl in 5. nd 5.b. Furherore We shll use pproiing or he conrol vrible M i u d Wih M re unnown coeiciens. The obecive 5. cn be pproied by: I qi u 6.3 The consrins 5. in view o Glerin ehod becoe s ollows b L L u d g d Eq 6.4 in view o 5.b becoe b L K s s u s ds d Which cn be rewrien ing use o.5 nd.7 s ollows b g d... b L ql l dl dl l q K l s s u s q g... Also 5.c becoe 6.5 r d A r... r 6.6 Hence he proble cn be pproied by: Find he conrol u which iniizes he uncionl6.3 subec o

7 39 We cn reorule his opil conrol proble s he consrined opiizion proble Miniize I 6.7- subec o F L 6.7-b The proble 6.7 hs +M+ unnown vribles. I will be solved vi he ollowing echnique. 7. Hybrid penly pril qudric inerpolion echnique The proble o solving opil conrol probles by ens o our Legendre pproiion ehod reduced o consrined opiizion proble. We pply he Hybrid penly pril qudric inerpolion echnique HPPI see Sli e l. [9] or he consrined opiizion proble. The consrined iniizion proble cn be sed s: Miniize T T {... } 7. subec o g T or ll i M 7. where he uncions nd g i i M re ssued o be wice coninuously dierenible uncion o T. The Hybrid penly pril qudric inerpolion echnique HPPI o solve he consrined opiizion proble 7. by ens o sequenil iniizion o he Penly uncion T T g T 7.3 Le T is n iniil esible or inesible vlue nd. We solve 7.3 s unconsrined opiizion proble by ens o HPPI lgorih [see Sli e l. [9]. 8. uericl resuls: ow we consider wo eples o show he eeciveness o our echnique in cse o opil conrol probles governed by Volerr equion or inegro- dierenil equion. Eple 8.: Miniize u I d 8. Subec o s s usds. 8. Approiing nd u using 6. nd 6. respecively ing use o.7 or pproiion inegrls nd 4.3 or reing he condiion8. he given proble is rnsored o: J q u L q q u s F L The opil cos J nd iu bsolue error in he consrins Fobined or his proble is presened in Tble. These resuls re copred o hose obined by Sli [] nd El-Kdy[]. nd conrol vribles obined ro presen ehod. In igure we plo he se Eple 8. Miniize u I d 8.3

8 u 4 Tble : Resuls o eple 8. =M=4 Mehod J F Presen ehod E-9 Sli [] E-7 El-Kdy[] E-8 se vrible conrol vrible Subec o Figures : Se nd conrol vribles s u s ds Cos 8.4 wih nd. Approiing nd u using 6. nd 6. respecively ing use o.7 or pproiion inegrls nd 4.3 or reing he condiion8. he given proble is rnsored o: J q u The opil cos J nd iu bsolue error in he consrins Fobinedor his proble is presened in Tble. These resuls re copred o hose obined by El-Kdy[]. In igure we plo he se nd conrol vribles obined ro presen ehod.

9 u 4 Tble : Resuls o eple 8. =M=4 Mehod J F Presen ehod e-9 El-Kdy[] E -8 se vrible conrol vrible Figures : Se nd conrol vribles 9. Conclusion The bsic ide o our presen ehod is o cobine Legendre inegrl pproiion wih Glerinehodo reorule he inegrl nd inegro- dierenil proble o unconsrined opiizion proble. Opil conrol probles governed by inegrl nd inegro- dierenil equions shll be reoruled o be consrined opiizion proble. Solving he resuling consrined opiizion proble is esier hn solving he originl proble. The convergence o he proposed ehod depends on he Legendre pproiion ehod El-Hwry e l [4] nd he Hybrid penly pril qudric inerpolion echnique [9] Sli. The resuls given previously show h he suggesed echnique is quie relible. I cn be successully pplied o boh liner nd nonliner inegrl nd inegro- dierenil probles nd opil conrol probles governed by inegrl nd inegro- dierenil equions. The ehod produces n ccure soluion sll nuber

10 4 o nodes. The coprison o he iu bsolue error resuling ro he proposed ehod nd hose obined by Sli[] El-Kdy[] show vorble greeen nd is ore ccure. Acnowledgen The uhors re greul o Denship o Scieniic Reserch o Universiy o D Proec o. 33. Reerences [] Ahues M. Lrgillier A. nd Liye B.V. Specrl Copuions or Bounded Operors Chpn & Hll/CRC. [] Belbs S.A.A reducion ehod or opil conrol o Volerr inegrl equions Applied Mheics nd Copuion [3] Cnuo C. A. ureroni M. Y. Hussini nd Zng T. A. Specrl Mehods Evoluion o Cople Geoeries nd Applicions o Fluid Dynics Springer-Verlg Berlin Heidelberg 7 [4] El-Hwry H. M. Sli M. S nd Hussien H. S. An Opil Ulrsphericl Approiion o Inegrls Inern. J. o Copuer Mh [5]Kopriv D. A. Ipleening Specrl Mehods or Pril Dierenil Equions Algorihs or Scieniss nd Engineers Springer Science + Business Medi B.V. 9. [6] Peng Y. Xing X. nd Wei W. Second-order nonliner ipulsive inegro-dierenil equions o ied ype wih ie- vrying genering operors nd opil conrols on Bnch spces Copuers nd Mheics wih Applicions [7] ingning Y. B. Superconvergence Anlysis And A Poseriori Error Esiion O A Finie Eleen Mehod For An Opil Conrol Proble Governed By Inegrl Equions Applicions O Mheics o [8] Szegö G. Orhogonl Polynoils A. Mh. Soc. Colloq. Pub [9] Sli M. S. El-Kdy M.M. nd El-Sgheer A. M. Hybrid penly pril qudric inerpolion echnique in he consrined opiizion probles Journl o he Insiuion o Mheics nd copuer Science copuer science series [] El-Kdy M. M. uericl Sudies or Opil Conrol Probles Ph.D. Thesis Assui Universiy 994. [] Sli M. S uericl Sudied o Opil Conrol Probles nd is Applicions Ph.D. Thesis Assiu Universiy Egyp 99. [] Mlened K. Kni M. T. Solving liner inegro-dierenil equion syse by Glerin ehods wih hybrid uncions Applied Mheics nd Copuion