The Penalty Cost Functional for the Two-Dimensional Energized Wave Equation

Transcription

1 Lonardo Jornal of Scncs ISSN Iss 9, Jly-Dcmbr 006 p Th Pnalty Cost Fnctonal for th Two-Dmnsonal Enrgd Wav Eqaton Vctor Onoma WAZIRI, Snday Agsts REJU Mathmatcs/Comptr Scnc dpartmnt, Fdral nvrsy of Tchnolog Mnna 90003, Ngr Stat, Ngra; Natonal Opn Unvrsy of Ngra, Vctora Island, Lagos, Ngra; Abstract Ths papr constrcts th pnalty cost fnctonal for optmng th two-dmnsonal control oprator of th nrgd wav qaton. In som mltplr mthods sch as th Lagrang mltplrs and Pontrygan maxmm prncpl, th cost of mrgng th constrant qaton to th ntgral qadratc objctv fnctonal to obtan an nconstrant qaton s normally gssd or obtand from th frst partal drvatvs of th nconstrand qaton. Th Extndd Conjgat Gradnt Mthod ECGM) ncssats that th pnalty cost b sqntally obtand algbracally. Th ECGM problm contans a fnctonal whch s compltly gvn n trms of stat and tm spatal dpndnt varabls. Kywords Pnalty fnctonal, nrgd wavs, Control oprator, Optmal control, optmal control and ECGM Introdcton W now dfn th statmnt of th two-dmnsonal problm n a charactrstc dxtry as n []: Problm P): mn J, ), = mn [,, ), x y t sbjct to th nrgd wav qaton: ] dxdydt. 45

2 Th Pnalty Cost Fnctonal for th Two-Dmnsonal Enrgd Wav Eqaton Vctor Onoma WAZIRI, Snday Agsts REJU =. wh nal and bondary condons: 0 0) = t 0) = t tt 0) = tt x 0) = x 0) = y 0) = y 0) = y) 0, = 0, = 0, = 0 x 0, = y 0, = 0, = 0, = x, =, = 0, = y, =, = 0 0 t ; 0 x ; 0 y Th nconstrant problm of th two-dmnsonal nrgd wav qaton s dfnd as: Problm P) mn, J,, µ ) y x t = mn { [ ] dtdxdy µ, y x t dtdxdy}.3 In solvng qaton.), w wll nd th nmrcal val of th pnalty cost constant µ 0. To achv ths, w mst drv a pnalty fnctonal n stat varabls sch that µ 0. Thraftr, w shall obtan th program cods for som optmal nmrcal vals at dffrnt stat profl strata. Th control and stat partal drvatvs ar qvalntly drvd by th Hamltonan form akn to [] and [] as follows: thn: = = = 0) sn t x sn y t In a frthr dvlopmnt, snc th rat of chang of th optmal control s th stat,.4 46

3 Lonardo Jornal of Scncs ISSN Iss 9, Jly-Dcmbr 006 p = 0)snxsny) t 0) sn x sn y t 0)sn xsn y).5 = Dffrntatng qaton.5) frthr wh rspct to tm t: = t 3 0)snxsny) t 3 3 t 0)snxsny.6 Also qaton.6) frthr drvatv ylds: = tt 3 4 0)sn x sn y 3 4 0)snxsny) 0)snxsny t 4 t 0)sn x sn y).7 Dffrntatng th stat qaton wh rspct to x twc: = )[ 0)snxsny) t 0)snxsny.8 47

4 Th Pnalty Cost Fnctonal for th Two-Dmnsonal Enrgd Wav Eqaton Vctor Onoma WAZIRI, Snday Agsts REJU Also dffrntatng th stat qaton wh rspct to y twc rslts n: = )[ 0)snxsny) t.9 0)snxsny Th drvd qatons.3) throgh to.9) ar sfl tools n th mplmntaton of th ECGM algorhm onc th control oprator s rady. Th constrcton of th pnalty cost fnctonal In th follow-p and n lght of th drvaton n [ 3 ] control oprator, w constrct th pnalty cost fnctonal for th two-dmnsonal nrgd wav qaton. By dfnon, w assmd that th optmal control oprator B and B obtand for th two-dmnsonal cas ar symmtrc posv dfn; hnc comparably qvalnc. In th control oprator, w obtand ths: B =µ ). B = µ ) ). t It s not dffclt to s that qatons 0) and ) ar qvalntly xprssd as: = µ ) µ.3 Basd on Rj [], th acton of th analytcal stat on th spac varabls y) can b convnntly b xprssd as: = µ ) µ.4 It s asy to xprss qaton.4) n ths ordr: µ =.5 Q whr 48

5 Lonardo Jornal of Scncs ISSN Iss 9, Jly-Dcmbr 006 p Q =.6 From qaton.6), w obsrv that th pnald paramtr s dpndnt on th stat and s partal drvatvs whch ar n thmslvs dpndnt on th spatal varabls x and y wh tm t. It s obvos that th paramtr µ s a fnctonal and gratr than ro. Ths by dfnon, th pnalty cost s posv dfn; hnc qaton.5) bcoms: µ,, ) =.7 Q) Now from War t al. 006b), w mst hav that: = t t.8 Also w hav from qaton.6) that: Q =.9 From [ 4 ] War t al. 006b), w can drv: = t 0) sn xsn y) t.0 = t t = t t = tt t. = { t t 49

6 Th Pnalty Cost Fnctonal for th Two-Dmnsonal Enrgd Wav Eqaton Vctor Onoma WAZIRI, Snday Agsts REJU = t tt tt t t. = ){ t 0) sn xsn y) = t = / 0)snxsny) } ){ t.3 0) snx sny t 0)snxsny) }.4 t t = ){ = t [ t ){ 0) snx sny 0) snx sny) t t } { t }] 50

7 Lonardo Jornal of Scncs ISSN Iss 9, Jly-Dcmbr 006 p t tt t t } t t t tt.5 Consqntl th sbstt of qaton.9) nto qaton.7) wll compltly dfn th acton of th stat trajctory on th pnalty. Programs cods wll gv s th ncssary dsrd nmrcal stat pnalty costs at varos spac profls; othrws, hrtofor rfr smply as to th nth-plan or n stratm. Th nmrcal otpt for th pnald optmal stat cost fnctonal Varos pnald strata for th stat fnctonal prodc th nmrcal otpts vals n tabl blow. Tabl. Otpts of pnalty cost constants s/n Nth-dmnsonal spac Pnalty cost constant n= n= n= n= n= n= n= n= Conclsons Th gnral analyss from both mprcsm and thortcal nvstgaton postlat that for a good constrant satsfacton, th pnalty cost constant sd n movng th dynamcal systm constrand problm nto th formlaton of an nconstrand control problm, mst 5

8 Th Pnalty Cost Fnctonal for th Two-Dmnsonal Enrgd Wav Eqaton Vctor Onoma WAZIRI, Snday Agsts REJU on on hand, b sffcntly larg; on th othr hand, ths paramtr mst not approach nfny as th convrgnc rat of most pnalty mthods dtrorat sharply at ths nstant, and many optmaton tchnqs, as a rslt, ar sbjctd to nmrcal nstably bcas of th drvatvs of th pnalty fnctons or pnalty fnctons thmslvs may ncras whot bonds n th vcny of th mnmm. Ths wold crat a paradox! An agl y obsrvaton of th pnalty cost constants n th abov tabl 3.) clarly agrs wh th assrtons n th paragraph bfor ths. As th nth-stratm spac ncrass, th pnalts cost constant nmrcal vals ncras posvly and mantan consstnt nmrcal stably whn th dmnsonal profl ncrass btwn th at all strata. Ths obvosly shows that th convrgnc s achvd for th pnalty cost fnctonal. Hnc, th optmal stat pnalty paramtrs gv xcllnt convrgnc rats. Ths wold gv an xcllnt ECGM otpts that wold compar favorably wh th analytcal manally f ndranc comptatonal procsss wll b sstand for q an laborat prod of tm. Rfrncs [] War V. O., Optmal Control of Enrgd Wav qatons sng th Extndd Conjgat Gradnt Mthod ECGM), Ph.D. Thss, Fdral Unvrsy of Tchnolog Mnna, Ngra, 004. [] Rj S. A., Comptatonal Optmaton n Mathmatcal physcs, Ph.D. Thss, Unvrsy of Ilorn, Ilorn, Ngra, 995. [3] War V. O., Rj S. A., Control Oprator for th Two-Dmnsonal Enrgd Wav Eqaton, Lonardo Jornal of Scnc, ISSN , Iss 9, p , 006. [4] War V. O., Rj S. A., Th Analyss of th Two-Dmnsonal Dffson Eqaton wh a Sorc, Lonardo Jornal Practcs and Tchnologs, Romana, ISSN , Iss 9, p.43-56,