Identification of the Solution of the Burgers. Equation on a Finite Interval via the Solution of an. Appropriate Stochastic Control Problem

Transcription

1 Ad. heor. Al. Mech. Vol. 3 no Idenificaion of he oluion of he Burger Equaion on a Finie Ineral ia he oluion of an Aroriae ochaic Conrol roblem Arjuna I. Ranainghe Dearmen of Mahemaic Alabama A & M Unieri.O.Bo 36 Normal A 3576.UA arjuna_ranainghe@ahoo.com huli Wickramainghe Dearmen of hic he College of New Jere Ewing NJ 868. UA wick@cnj.edu Abrac oluion of he inhomogeneou Burger equaion wih gien homogeneou Iniial/Dirichle boundar condiion are idenified b a ochaic conrol roblem. Our aroach coni of idenifing he oluion of Burger equaion [ - ] wih he oimal conrol of an aroriae ochaic conrol roblem. echnique decribed in hi aer ma hae far reaching alicaion in aronomical arial differenial equaion. Mahemaic ubjec Claificaion: 35Q35 35Q Keword: ochaic Conrol heor agrangain Brownian moion Duali and Wiener Meaure

2 38 A. I. Ranainghe and. Wickramainghe. Vicou and Inicid Burger Equaion wih mall Vicoi he Burger equaion wih mall icoi can be wrien a follow. e є > be gien and i i he icoi of he fluid. Conider η f. on a b ubjec o he boundar daa b. and iniial daa a b.3 Aume ha f i a funcion of he cla C [ ] [ a b] aifing f f b..4 When we hae he correonding inicid equaion. i.e. f..5 On a b ubjec o he boundar daa b.6 And iniial daa a b..7 Where f aifie.4. he conergence of he oluion of he icou Burger equaion o ha of he inicid equaion a he icoi become ufficienl mall ha been hown b eeral reearcher [3] and i ubjec o he ame iniial and boundar condiion. In hi aer we will how he connecion beween he oluion of he Burger equaion and an oimal conrol roblem.. ochaic Oimal Conrol and he oluion of he Burger Equaion e define a funcion f d c є a b a b є []. c Noe ha for each fied є [ ] є C [a b] a a funcion of ince f i C [ ] [a b]. Conider he comleel oberable ochaic conrol roblem goerned b he ae equaion

3 Idenificaion of he oluion of he Burger equaion 39 dξ u d dw. Where u i he conrol w i he one-dimenional Brownian moion defined on a comlee robabili ace Ω ζ w. he weak oluion roce{ ξ } aifie ξ and ω i uch ha w. w i he robabili induced b he Weiner meaure. e u be he conrol roce. When we mu ecif he informaion aailable o he conroller. We furher aume ha he ae ξ i comleel oberable a each ime. hi aumion i eenial o ha we can coner he ochaic conrol roblem o a arial differenial equaion ia dnamic rogramming. Furher he conrol a he end oin i. e. a b i. We can rewrie he equaion. a follow: ξ u τ dτ w r.3 Define he co funcion a follow. I E d.4 We need o find an admiible feedback conrol law which minimize.4 where E i he eecaion oeraor wih reec o he robabili meaure induced b ξ є haing he roer ξ є. Now for each є [ ] define he following oeraor. A.5 A.6 l..7 e є be a oluion of he dnamic rogramming equaion defined b [ A l ] min єϑ..8 [ ] for ϑ.9 where ϑ [ ] {a b} U { { } a b. ν ince A є є ν є he dnamic rogramming equaion ha he form

4 4 A. I. Ranainghe and. Wickramainghe min ν ν ν ϑ g [.]... he oimal conrol law i found b minimizing he ereion. a a funcion ofν. e m ν ν ν. hen m ν ν. and m m >. herefore ν i a local minimum. I i alo a global minimum on [.]. hi imlie ha..3 Now le u eek a oluion of.8.9 wih <. hen >. If hen and. hu for є > he minimal co є of oer he cla of feedback conrol U aifie.4 wih..5 i.e..6.7 Now we conider he deerminiic conrol roblem i.e.. no icoi ξ u.8 ξ..9 where u i he conrol roce u ξ and { ξ } i he oluion roce. he co funcion i gien b I u E d.

5 Idenificaion of he oluion of he Burger equaion 4 d. ξ. e be he oimal conrol law which minimize.. A before for є [ ] we hae A. l..3 Again le be a oluion of he dnamic rogramming equaion [ A l ] min [] єϑ.4 for є ϑ..5 ince A he dnamic rogramming equaion ake he form min [] єϑ.6..7 e be he oimal conrol law found b minimizing. a a funcion of. A before hu for є he minimal co oer he cla of feedback conrol U aifie g.8..9 Obere ha he oimal co є and aif he following Cauch roblem: F.3.3 where F.3 and F where F.35 Alo oimal co are reeciel gien b

6 4 A. I. Ranainghe and. Wickramainghe d E ξ.36. d ξ.37 he agrangian aociaed wih can be gien b Furher we can how ha є and reeciel aif { } ma F.4 { } ma F.4 Where. For eamle conider є in which { } F ma ma.4 I i ea o ee ha maimum occur a -. herefore { }.. ma Y F.43 imilarl we can low ha i he duali of F. I can be alo hown ha є and are cone in. For eamle є where є F. herefore for all є R є. I i clear ha. for When. We ge d f ξ ξ.44 imlie ha. f.45 Recall ha and hence we obain d f

7 Idenificaion of he oluion of he Burger equaion 43 f d d f d..46 Uing he erminal boundar condiion on we ge b.47 When є > imlie ha E f d.48 Where again u for є > our aumion on he conrol ee age 3 imlie ha he conrol anihe a a and b. herefore b. Hence aifie he ame boundar condiion a є for є. I i ea o ee from.6 and.8 ha є a b..49 imilarl.5 hi eabliher he connecion beween he oluion of Burger equaion and he oimal conrol roblem. Reference [] A. Ranainghe and F. Majid Mahemaical Model for Oimum Waer Diribuion in a Remoe Ci wih Waer ower Alabama Journal of Mahemaic Vol. 3 No. & [] A. Ranainghe and F. Majid oluion of Blaiu Equaion b Decomoiion Journal of Alied Mahemaical cience Vol. 3 9 no

8 44 A. I. Ranainghe and. Wickramainghe [3] F. Majid and A. Ranainghe oluion of he Burger Equaion Uing an Imlici inearizing ranformaion Journal of Communicaion in Nonlinear cience and Numerical imulaion Vol Receied: Augu 9