On Control Problem Described by Infinite System of First-Order Differential Equations

Transcription

1 Ausalian Jounal of Basic and Applied Sciences 5(): ISS On Conol Poblem Descibed by Infinie Sysem of Fis-Ode Diffeenial Equaions Gafujan Ibagimov and Abbas Badaaya J'afau Insiue fo Mahemaical Reseach and Depamen of Mahemaics Faculy of Science Univesii Pua Malaysia 434 UPM Sedang Selango Malaysia Absac: We sudy conol poblem descibed by infinie sysem of fis ode diffeenial equaions in Hilbe Space Conol paamee is subjeced o inegal consain Ou goal is o ansfe he sae of he sysem fom a given iniial posiion o he oigin fo a finie ime In his pape we obained necessay and sufficien condiion fo which he goal o be achieved Conol funcion is consuced in an explici fom Fuhemoe equaion fo calculaing opimal ime fo he ansfe pocess is obained Key wods: Infinie sysem Conol Inegal consain Hilbe space IRODUCIO he impoance of conol poblems moivaed he developmen of many boos on fundamenal esuls (see fo example Avdonin and Ivanov 989: Buovsiy 975: Egoov 4: Ki 998: Pinch 993: and Ponyagin e al 969) Conol poblems descibed by paial diffeenial ae of significan impoance in solving eal life poblems and in ohe eseach aeas such as engineeing and economics his foms a moivaion fo he exensive sudy of such ype of poblems by many eseaches using diffeen appoaches (eg Avdonin and Ivanov 989: Buovsiy 975: Chenous o 99: Ibagimov 3: Ilin : Osipov 977: uhasinov and Mamaov 9: and Saimov and uhasinov 6) Among he appoaches found in lieaue is he use of decomposiion mehod o educe he poblem o he one descibed by an infinie sysem of diffeenial equaions (see fo example Buovsiy 975: Chenous o 99: Ibagimov 3: uhasinov and Mamaov 9: and Saimov and uhasinov 6) In Chenous o 99 a conol sysem descibed by he following paial diffeenial equaion u Au w () was educed o he one descibed by he sysem of odinay diffeenial equaions z () z () w () () and invesigaed whee in () uu( x) is a scala funcion of x n R and ime ; w is a conol paamee; n u Au aij ( x) i j x i x j is a linea diffeenial opeao whose coefficiens do no depend on In () w ae conol paamees and consans saisfy he condiion Fuhemoe in uhasinov and Mamaov 9; and Saimov and uhasinov 6 conol poblems descibed by () consising of wo conol sysem wih conflicing goals unde diffeen foms of esicions wee invesigaed he decomposiion mehods wee also employed his appoach hins he significan elaionship beween conol poblems descibed by paial diffeenial equaions and hose descibed by infinie sysem of diffeenial equaions heefoe he lae can be sudied in a sepaae heoeical famewo Fo insance in Ibagimov and Hasim pusui and evasion diffeenial Coesponding Auho: Abbas Badaaya Insiue fo Mahemaical Reseach and Depamen of Mahemaics Faculy of Science Univesii Pua Malaysia 434 UPM Sedang Selango Malaysia s: abbas@mahupmedumy 736

2 Aus J Basic & Appl Sci 5(): game descibed by () was sudied in a famewo independen of ha descibed by paial diffeenial equaions In his pape we invesigae conol poblem descibed by he sysem () in he case of negaive coefficiens in a famewo diffeen fom he one descibed by paial diffeenial equaions Saemen of he poblem: Le space be a bounded sequence of negaive numbes and be a eal numbe We inoduce he l ( ) : wih inne poduc and nom / l Le L( ; l ) w( ) ( w w) : w ( ) d w( ) L( ) () ( ) L ( ; l ) w w d whee is a given numbe We examine a conol poblem descibed by he following sysem of diffeenial equaions z z w z z (3) ( ) ( ) ( ) () whee z w R z ( z z) l w w ae conol paamees Definiion : A funcion w() w :[ ] l wih measuable coodinaes subjec o d w ( ) whee is a given posiive numbe is efeed o as he admissible conol We denoe he se of all admissible conols by S( ) Definiion : A funcion z ( ) ( z( ) z( )) is called he soluion of he sysem (3) if each coodinaes z () of ha ) is coninuously diffeeniable on ( ) and saisfies he iniial condiions z() z; ) has he fis deivaive z () almos eveywhee on ( ) and saisfies he equaion z () z() w () almos eveywhee on ( ) Definiion 3: he numbe is an opimal conol ime if 737

3 Aus J Basic & Appl Sci 5(): ) hee exiss an admissible conol o ensue ha z( ) ; ) hee exiss no admissible conol o ensue z( ) fo any ( ) he poblem is o find a conol w() ( w() w()) such ha he sae of he sysem (3) can be ansfeed o he oigin fo a finie ime Main esul: I is no difficul o veify ha he h equaion in (3) has a unique soluion ( ) z ( ) z e w ( ) e d (4) whee Le C( ; l ) be he space of coninuous funcions z () z() z() wih values in l he following asseion can be poved simila o (Avdonin and Ivanov 989) Asseion : If is a bounded below sequence hen he funcion z () z() z() belongs o he space C( ; l ) defined by (4) Le e F () () z () (5) Lemma : Le z l he seies in (5) is convegen a any fixed if and only if z l Poof Suppose ha () z is convegen Le ( ) he inequaliy e ( ) is ue since ( ) is inceasing and appoaches as appoaches zeo heefoe we have z z z e Hence () z and so l z Convesely le z l We have z e F () (6) z z e e 738

4 Aus J Basic & Appl Sci 5(): heefoe he seies (6) is convegen if he seies (7) z e And z e (8) ae convegen As he funcion ( ) is inceasing hen eplacing by (since < ) in (7) we obain z z e e heefoe he seies (7) is convegen since z l Convegence of he seies (8) follows fom he elaions z z z e e e his is because ( ) e is deceasing funcion and z l Convegence of (7) and (8) imply he convegence of he seies (6) he poof of he lemma is complee Moeove if zl l hen he funcion F ( ) has he following popeies: (i) F () is a deceasing funcion of (ii) F () appoaches as (iii) F ( ) appoaches z as (iv) F () is coninuous Indeed popey (i) follows fom he fac ha each em of he seies is deceasing funcion of Since each em of F () appoaches as heefoe popey (ii) is ue We now pove popey (iii) If zl l hen F () is convegen fo any We fix As F () is convegen hen fo any hee exiss a numbe such ha z ( ) (9) And z () Inequaliy (9) eeps o hold if we incease since ( ) ae deceasing funcions hee exiss such ha ( ) z z 739

5 Aus J Basic & Appl Sci 5(): wheneve since ( ) appoaches as appoaches and he sum consiss of a finie numbe of summands We now have ( ) z z ( ) z z ( ) z z Hence popey (iii) is poved Finally we pove popey (iv) Coninuiy of F () follows fom he fac ha i is convegen and ( ) ae coninuous e We now have fom he popey (i) and (iii) ha F ( ) z () ow conside he equaion F () () If equaion () has a soluion hen accoding o () z (3) Convesely le (3) be ue Since F () appoaches infiniy as appoaches zeo; F () appoaches z as appoaches infiniy and F () is coninuous and deceasing hee exiss a unique such ha F( ) heefoe he following asseion is ue Asseion : he equaion defined by () has a oo if and only if (3) is ue and his oo is unique heoem : hee exiss an admissible conol o see he sae of he sysem (3) ino he oigin if and only if (3) holds In he poof of his heoem we use he following lemma Le s { w( ) ( w( ) w( )) w ( s) e ds w( ) S ( ) Lemma : Among all he conols w() he conol w() such ha w ( ) ( ) e minimizes he funcional Q ( ) w ( s) ds 74

6 Aus J Basic & Appl Sci 5(): he poof of his lemma is simila o he poof of he lemma conained in (Ibagimov 3) wih Poof of heoem : Assume on he conay ha (3) is no ue z (4) and ha hee exiss an admissible conol w() such ha z( ) fo some ime hen (using (4)) w ( ) s s e ds z (5) By he lemma among all conols w () saisfying (5) he conol w (): w ( ) ( ) z e minimizes he funcional Q( ) ie ( ) Q ( ) z e ds z ( ) F ( ) Hence F() Q() (6) Using () (4) and (6) we obained z F ( ) Q ( ) meaning ha w ( ) ( w( ) w( ) ) is no admissible his is a conadicion Convesely suppose ha (3) holds Accoding o Asseion equaion () has a unique oo Consucion of he conol: z e w ( ) e (7) Admissibiliy of he consuced conol follows fom he elaions 4 z e w ( ) d d ( e ) 4 z e d ( e ) z e (hee we use (7) and definiion of in ()) Seeing he sysem o he oigin: Using (4) and conol (7) we have z e z ( ) e z d ( e ) e ( z z ) 74

7 Aus J Basic & Appl Sci 5(): his complees he poof of he heoem heoem : he numbe he oo of () is he opimal ime o ansfe he iniial posiion of he sysem (3) ino he oigin Poof: Suppose on he conay ha hee exiss [ ) such ha z( ) Consequenly we have inequaliy (6) holding (see Poof of heoem ) ow consideing he fac ha he conol w() saisfies he inegal consain; definiion of and ha he funcion F () is deceasing we have Q() F() F() heefoe Q( ) F( ) and his conadics (6) heefoe any iniial posiion z of he sysem (3) canno be ansfeed ino he oigin fo he ime [ ) Hence he numbe is opimal ansfe ime his complees he poof of he heoem Conclusion: A conol poblem descibed by sysem of fis ode diffeenial equaions in Hilbe space has been sudied he coefficiens of he sysem ae consideed o be bounded sequence of negaive numbes Conol funcion saisfies inegal consain ecessay and sufficien condiion fo which he conol pocess is possible has been obained his includes consucion of a conol funcion ha bings he sysem ino he oigin fo a finie ime Moeove his necessay and sufficien condiion depends on he seies F() As a esul of ha we sudied his seies and consequenly obained necessay and sufficien condiion fo he convegence of he seies F() Finally we succeeded in obaining opimal ime fo ansfeing he iniial posiion of he sysem (3) ino he oigin We have given an equaion o find his ime REFERECES Avdonin SA and SA Ivanov 989 he Conollabiliy of Sysems wih Disibued Paamees and Families of Exponenials UMKVO Kiev Buovsiy AG 975 Conol Mehods in Sysems wih Disibued Paamees aua Chenous'o FL 99 Bounded Conols in Sysems wih Disibued Paamees Pil Ma Meh 56(5): 8-86 Egoov AI 4 Pinciples of he Conol heoy aua Ibagimov GI and RM Hasim Pusui and Evasion Diffeenial games in Hilbe Space l Inenaional Game heoy Review (3): 39-5 Ibagimov GI 3 A Poblem of Opimal Pusui in Sysem wih Disibued Paamees Appl Mahs Mechs 66(5): Ilin VA Bounday Conol of a Sing Oscillaing a One End wih he Ohe End Fixed and Unde he Condiion of he Exisence of Finie Enegy Dol RA 378(6): Ki DE 998 Opimal Conol heoy An Inoducion Dove Publishes Lee EB and L Maus 967 Foundaions of Opimal Conol heoy John Wiley Osipov Yu S 977 Posiional Conol in Paabolic Sysems Pil Ma Meh 4(): 95- Pinch ER 993 Opimal Conol and Calculus of Vaiaions Oxfod Univesiy Pess Ponyagin LS VG Bolyansii RV Gamelidze and Ye E Mishcheno 969 Mahemaical heoy of Opimal Pocesses aua Moscow Saimov Yu and M uhasinov 6 Game Poblems on a Fixed Ineval in Conolled Fis-Ode Evoluion Equaions Mahemaical oes 8(4): uhasinov M and MSh Mamaov 9 On ansfe Poblems in Conol Sysems Diffeenial Equaions 45(3):