A New Approach to the Numerical Solution of Fractional Order Optimal Control Problems
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- Godfrey Summers
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1 Avilble hp://pvu.edu/ Appl. Appl. Mh. ISSN: Vol. 8 Issue Decebe 3 pp Applicions nd Applied Mheics: An Inenionl ounl AAM A Ne Appoch o he Nueicl Soluion o Fcionl Ode Opil Conol Pobles T. Abin nd M. Keynpou Depen o Applied Mheics Fculy o Mheicl Sciences Univesiy o Guiln Rsh In Abin.@gil.co;.eynpou@gil.co Received: uly 4 ; Acceped: Sepebe 9 3 Absc In his icle ne nueicl ehod is poposed o solving clss o cionl ode opil conol pobles. The cionl deivive is consideed in he Cpuo sense. This ppoch is bsed on cobinion o he peubion hooopy nd peeizion ehods. The conol uncion u is ppoied by polynoil uncions ih unnon coeiciens. This ehod conves he cionl ode opil conol poble o n opiizion poble. Nueicl esuls e included o deonse he vlidiy nd pplicbiliy o he ehod. Keyods: Fcionl ode opil conol Hooopy peubion ehod Cpuo cionl deivive. AMS-MSC No.: 49 6A33. Inoducion Fcionl opil conol pobles FOCPs e opil conol pobles ssocied ih cionl dynic syses. The cionl opil conol heoy is vey ne opic in heics. FOCPs y be deined in es o dieen ypes o cionl deivives. Bu he os ipon ypes o cionl deivives e he Rienn-Liouville nd he Cpuo cionl deivives. In Agl 4 Agl nd Blenu 7 he uhos obined 53
2 54 T. Abin nd M. Keynpou necessy condiions o FOCPs ih he Rienn-Liouville deivive nd ee ble o solve he poble nueiclly. Agl 8 pesened qudic nueicl schee o clss o cionl opil conol pobles FOCPs. In Agl 8c he FOCPs e ouled o clss o disibued syses hee he cionl deivive is deined in he Cpuo sense nd nueicl echnique o FOCPs pesened. Blenu e l. 9 used diec nueicl schee o ind soluion o he FOCPs. In Biss nd Sen FOCPs ih ied inl ie e consideed nd nsvesely condiion is obined. FOCPs ih dynic consin involving inege nd cionl deivives e lso consideed Biss nd Sen b. Bsed on he epnsion oul o cionl deivives ne soluion schee s poposed in elicic nd Peovci 9. Loi e l. used Legende ohonol polynoil bsis o solve he FOCPs. A diec ehod using Eigen uncions o solve he FOCPs o -diensionl syse s pesened in zdei e l. 9 hee he G nld-leniov ppoiion s used o ppoie he cionl deivives. Siil eps hve been de by sevel eseches o solving he FOCP o disibued syses Hsn e l. Rpic nd elicic. Ticud nd Chen pesened nueicl schee o FOCPs bsed on inege ode opil conols poble. In Youse e l. he usge o Legende ulivele bsis nd collocion ehod s poposed o obining he ppoie soluion o FOCPs. Ticud nd Chen b poposed ionl ppoiion bsed on he Hnel d i o he ipulse esponse o obin soluion o he genel ie-opil poble. The ineesed ede is eeed o Evigen nd zdei Wng nd Zhou d zdei 9 Agl 8b nd Fedeico e l. 8 o uhe inoion. The hooopy peubion ehod HPM s pplied o solve ODE nd PDE equions in He 3 He 5 Biz nd Ghzvini 9. The nueicl ehod o FOCPs pesened in his ppe ollos he ppoch pesened in Keynpou nd Azizse Keynpou nd Abin Bozbdi e l. Bozbdi nd Azizse. O couse in his ppe e develop hybid o peizion nd odiied hooopy peubion ehod o solve FOCPs. This ppe is ognized s ollos: In Secion e pesen soe bsic deiniions. In secion 3 e descibe ou ehod. In secion 4 e epo ou nueicl esuls. Finlly secion 5 e conclude he ppe.. Fcionl Opil Conol Poble Seen.. Bsic Deiniions We pesen soe bsic deiniions eled o cionl deivives. The Le Rienn-Liouville deivive o cionl ode o uncion is deined by:
3 AAM: Inen.. Vol. 8 Issue Decebe 3 55 d d d D hile he Righ Rienn-Liouville cionl deivive is given s:. d d d D b b Anohe cionl deivive he le Cpuo cionl deivive is deined s: * d d d D hile he igh Cpuo cionl deivive given by: * d d d D b b hee... Fcionl Opil Conol Poble Foulion The FOCPs in he sense o Cpuo e ouled s ollos: b d u u Miniize subjec o: * u G D he iniil condiions o he pobles e = -. s b c hee
4 56 T. Abin nd M. Keynpou G g g l u u C l b R R l u in hich nd u e he se nd conol vibles especively b snds o he ie nd nd G e given nonline uncions. Hee e ssue h FOCPs hve unique soluion. The bsic eisence nd uniqueness ollo o he Lipschiz condiion by using concion pping heoe nd eighed no ih Mig-Lele in Y e l. Lshinh nd Mohp Podlubny 999 So e l. 993 Sh nd Y Descipion o Mehod In his secion he poposed ehod is descibed nd n ssocied lgoih is pesened. The coninuous conol uncion u is ppoied ih inie cobinion o eleens o bsis Rudin 976 s ollo: u j j q j. 8 Since he FOCPs e solved by hooopy peubion ehod e consuc conve hooopy s ollos: d d d p d G u D* p 9 nd suppose he soluion o Equion6 hs he olloing o: p p hee j j e unnon uncions. Subsiuing Equion ino Equion 9 o nd equing he coeiciens o he es ih idenicl poes o p e deive: d p : c d d d p : G u D* d d d d p : G u D* d d.
5 AAM: Inen.. Vol. 8 Issue Decebe 3 57 As p Equion 9 ends o Equion 6 nd Equion in os cses conveges o n ppoie soluion o Equion 6 i.e.. By subsiuion o Equion 8 nd Equion ino Equion 5 nd Equion 7 e obin n ppoie soluion o FOCPs s ollos: Le in s. : b b c s. j q d * be he opil vlue o Equion 3. A sopping cieion is poposed s ollos: j j 3 * * 4 hee he sll posiive nube is chosen ccoding o he ccucy desied. We popose he olloing lgoih hich is pesened in o sges. Algoih: Iniilizion sep: Choose o he ccucy desied nd se nd go o he in sep. Min sep Sep. Se u by Equion 8 nd go o Sep. Sep. Copue by Equion nd go o Sep 3. Sep 3. Then copue Sep 4. * in by Equion 3. I go o sep 5. Oheise go o Sep 4. I he sopping cieion Equion 4 holds hen sop; else go o Sep 5. Sep 5. nd go sep. 3. Nueicl Resuls In his secion e pply he ehod pesened in Secion 3 o solve he olloing o es eples. All copuions cied ou by he pcge MAPLE 3.
6 58 T. Abin nd M. Keynpou Eple. Conside he olloing ie invin poble u d 5 subjec o: D* u ih he iniil condiion. The ec soluion o is hee sinh cosh cosh sinh u cosh sinh sinh cosh.98 * ih objecive vlue u.999. In Figue he se vible nd he conol vible u e ploed o. I is obvious h by pplying he lgoih pesened in secion 3 he ppoie vlues o nd u convege o he ec soluions. Figue shos he se nd he conol inpu u s uncions o ie o dieen vlues o. Choosing 5 he esuls o he pplying he given lgoih e pesened in Tble. Tble. Nueicl esuls in Eple n *
7 AAM: Inen.. Vol. 8 Issue Decebe 3 59 Figue. Appoie soluions nd ec soluion o u o n 7 b Appoie soluions nd ec soluion o o n 7 o Eple b Figue.: Se s uncion o o dieen vlues o b Conol u s uncion o o dieen vlues o o Eple Eple. In his eple ie vying FOCP is consideed o ind he conol u hich iniizes he peonce inde
8 53 T. Abin nd M. Keynpou u d 6 subjec o: D* u ih ee einl condiion nd he iniil condiion. Figue 3 deonses he ppoiion o nd u o dieen vlues o. The esuls o pplying he lgoih e pesened in Tble. Figue 4 shos he se nd he conol vibles especively s uncion o ie o. 8 o dieen vlues o n. I is obvious h he ppoie vlues nd u convege o he ec soluions by incesing he vlues o n. Tble. Nueicl esuls in Eple n * b Figue 3. Se s uncion o o dieen vlues o u s uncion o o dieen vlues o o Eple b Conol
9 AAM: Inen.. Vol. 8 Issue Decebe 3 53 b Figue : Convegence o he se vible o he ie-vying syse o. 8 b Convegence o he conol vible o he ie- vying syse o.8. Tes pobles nd ee solved in Agl 8 in dieen y. Ou esuls shon in Figues -4 e in good geeen ih he esuls deonsed in Agl 8. Bu e chieved siscoy nueicl esuls in only 5 ieions hile in Agl 8 he nube o ppoiions ss in nd inceses up o 3. So i is signiicn h e chieved ou nueicl esuls ih vey sll ode o ppoiions. Also e ind he ppoie opil vlue o he objecive uncion o ech. 4. Conclusion In his ppe e hve developed he hooopy peubion nd peeizion ehods o solving clss o cionl opil conol pobles. By he poposed ehod e e ble o educe he in poble o n opiizion poble. The nueicl esuls hve deonsed he high ccucy o he poposed ehod. Acnoledgens We e vey geul o o nonyous eeees o hei ceul eding nd vluble coens hich led o he ipoveen o his ppe.
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