Numerical Scheme for Fredholm Integral Equations Optimal Control Problems via Bernstein Polynomials

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1 Australia Joural of Basic ad Applied Scieces 4(): ISSN Nuerical Schee for Fredhol Itegral Equatios Optial Cotrol Proles via Berstei Polyoials Mahood Sachooli Oid Solayai Fard School of Matheatics ad Coputer Sciece Dagha Uiversity Dagha Ira. Meer of Youg Researchers Clu Islaic Azad Uiversity Aliaad Katul Brach Aliaad Katul Ira. Astract: I this paper we preset a ovel iterative ethod to approxiate the solutio to a class of optial cotrol proles govered y Fredhol itegral equatios. We are willig to costruct a direct schee ased o the Berstei polyoials ad paraeterizatio. The covergece of the ethod is also discussed i details ad at the ed soe uerical exaples illustrate the efficiecy ad accuracy of the ethod. Key words: Optial cotrol prole Fredhol itegral equatios Iterative ethods Approxiate- Aalytical solutio Nuerical schee. INTRODUCTION The advatages of optial cotrol theory ad calculus variatios are well estalished durig recet decades. The optial cotrol theory provides a systeatic ad direct approach to a large variety of cotrol desig proles icludig costraied optiizatio with iterrelated aipulated variales (Stes et al. ). Furtherore optial cotrol theory alog with optiizatio ethods are presetly eployed for various applicatios i differet fields e.g. aerodyaics eteorology cheistry uclear agetic resoace acoustics ecooical odels fiacial atheatics pharaceutical aufacturig coputatioal iology ad ioiforatics (Hoescu et al. 3; Zdeek et al. 9). I atheatical forulatio of physical pheoea itegral equatios are always ecoutered ad have attracted uch attetio. Itegral equatios are appeared i a variety of applicatios i ay fields icludig cotiuu echaics potetial theory geophysics electricity ad agetis kietic theory of gases hereditary pheoea i iology quatu echaics optiizatio optial cotrol systes atheatical ecooics populatio geetics edicie fluid echaics steady state heat coductio ad radiative heat trasfer proles (Adou et al. 3; Baolia et al. 4; Huag et al. 995; Jiag et al. 4; Kythe et al. ; Liag et al. 4; Malekejad et al. 4; Malekejad et al. 5; Malekejad et al. 5; Wag 6; Yag ; Zhag 987). I this paper we cosider the uerical solutio of a class of optial cotrol proles govered y itegral equatios which is descried y the followig iiizatio prole: Miiize J ( x u) = f ( t x( t) u( t)) dt suject to xt ( )= yt ( ) () ktsus ( ( )) xsds ( ) aeo. [ a ] a where f C([ a ]) R R ad y(.) C ([ a ]) are give fuctios xt () ut () C ([ a ]]) are the trajectory () ad cotrol fuctios respectively which to e deteried ad the give kerel fuctio ktsus ( ( )) is sooth i C ([ a ]) C([ a ]) R. Here we assue that the prole ()-() has a uique solutio. Correspodig Author: Mahood Sachooli Youg Researchers Clu Islaic Azad Uiversity Aliaad Katul Brach Aliaad Katul Ira. Eail: sachooli@gail.co 5675

2 Aust. J. Basic & Appl. Sci. 4(): Due to the lack of existig a appropriate uerical ethod for solvig this kid of optial cotrol proles the ai purpose of this study is to preset a direct uerical schee for otaiig approxiate solutios of the prole ()-() y usig paraeterizatio ad Berstei polyoials. A Approxiate-Aalytical Solutio to FIE: I this sectio followig the work of Madal (Madal et ai. 7) it is assued that for a give u(s) the Fredhol itegral equatio () with k a cotiuous ad square itegrale fuctio has a uique solutio. To fid a appropriate solutio of () x(t) is approxiated i the Berstei polyoial asis i [a ] as x()= t a B () t (3) i i= i where Bi ( t) ( i =... ) are Berstei polyoials of degree defied o [a ] as i i ( ta) ( t) Bi ( t) = i =... i ( a) (4) ad a ( i =... ) i are ukow costats to e deteried. Sustitutig (3) i () we otai ab ()= t yt () a k( t su ( s)) B ( s) a< t< i i i a i i= i= (5) Multiplyig oth side y Bi ( t) ( i =... ) ad itegratig oth sides with respect to t etwee t = ad t = we otai the liear syste ac i ij= j j=... i= (6) where ij i a a i j c = [ B ( t ) k ( t s u ( s )) B ( s ) ds ] B ( t ) dt i j =... ad = y( t) B ( t) dt. j a j The liear syste (6) ca e solved y ay stadard ethod to produce ai ( i =... ). These a 's whe sustituted i (3) produce x(t) approxiately. Here we take [a ] = []. 3 The Solutio to the Optial Cotrol Prole: Let Q e the suset of the product space C ([]) C ([]) cotais all pairs ( x(.) u(.)) which satisfy the equatio (). Also let Q e the suset of Q cosistig of all pairs ( x (.) u (.)) where u (.) is a paraeterized cotrol fuctio as the followig polyoial (7) (8) i u ()= t at i i= (9) 5676

3 ad x (.) Aust. J. Basic & Appl. Sci. 4(): is the extracted solutio of the itegral equatio () which is cosidered as a polyoial of degree at ost j x ( t) = e ( a a... a ) t. j j= () Here e : R R j =... are cotiuous fuctios. Now we cosider the iiizig of J o j Q with { a } k k = as ukows. This is oviously a optiizatio prole i + diesioal space k k = {( a a... a ) R : a = u () = u a = u () = u } ad J( x u ) ay e cosidered as a fuctio Ja ( a... a ). Suppose ( x (.) u (.)) e the solutio of iiizig J o Q =...; =... the the polyoial for of u (.) =... i (9) ad the Eq. () allow us to apply the preseted ethod (Sectio ) for extractig polyoial solutio of () which results i otaiig a sequece of trajectory fuctios { x (...)} = as Berstei polyoials ad fially to achieve a iiizig sequece {( x (.) u (.)}. Lea 3.: If = if Q for =... the { } is a coverget sequece. J Proof. See (S. Fard et al. ). Theore 3.: = = = if QJ If li the. ˆ Proof. By Lea 3. let { } coverges to aely. By cotradictio if ˆ > the ˆ = > ˆ J( x(.) u(.)) < =.. Hece there exists ( x(.) u(.)) such that () Fro the cotiuity of J there is a > Jv ( (.) w(.)) Jx ( (.) u(.)) < wheever ( v(.) w(.)) ( x(.) u(.)) < Where () (3) Here. is a or o the vector space C ([]) C ([]) which ca e defied as follows: 5677

4 ( v(.) w(.)) = v(.) w(.) Aust. J. Basic & Appl. Sci. 4(): where the or properties ca e checked easily. O the other had the set of all polyoial pairs are dese i C ([]) C ([]) so there is a pair of polyoials p () t of degree at ost ad q () t of degree at ost such that ( p(.) q(.)) ( x(.) u(.)) <. 3 (4) Whereas the pair ( p (.) q (.)) does ot satisfy ( p () q ()) = ( x u )( p () q ()) = ( x u ) so we have to defie aother polyoials v ( t) = p ( t) ( x p ())( t) ( x p ()) t w ( t) = q ( t) ( u q ())( t) ( u q ()) t that satisfy ( v() w()) = ( x u) ad ( v() w()) = ( x u ) so ( v w) Q. Fro (5) for t = we have ( p() q()) ( x u) < ( p() q()) ( x u) < 3 3 Now for t [] y defiitio v (.) ad (.) we have w ( v (.) w (.)) ( x(.) u(.)) ( p ( t) q ( t)) ( x( t) u( t)) ( p () q ()) ( x u ) ( t) ( p () q ()) ( x u ) t < = Therefore ( v (.) w (.)) ( x(.) u(.)) < ad (3)-(4) iply that Jv ( (.) w(.)) Jx ( (.) u(.)) < ad so fro () Jv ( (.) (.)) < ( (.) (.)) < ˆ w Jx u a cotradictio appears with ( v (.) w (.)) Q so ˆ =. Now we ca suarize the aove results i a uerical algorith for otaiig approxiate optial cotrol of () suject to Eq. (). 5678

5 Aust. J. Basic & Appl. Sci. 4(): Nuerical Experiets: I order to validate the optial cotrol forulatio ad to test the proposed uerical solutio procedures we preset results of uerical experiets with two test proles ad the uerical calculatios are all udertake y MATLAB software. Exaple 4.: I the first exaple we cosider the followig optial cotrol prole = ( ( ) ) ( ( ) ) Miiize J x t t u t t dt suject to Fredhol itegral equatio [3] ()= () ( ( )) ( ). x t y t t u s x s ds (5) (6) where yt ()= t. The Exact optial solutios of (5)-(6) are * * x ( t) = t ad u ( t) = t. with the optial criterio * * J = J( x ( t) u ( t))=. Usig the aove ethod we have the uerical results otaied i Tale. As ca e see fro Tale ad Figures -3 the rapid covergece of the schee is issued whe the itegers ad are icreased util. Tale : The Approxiate-Aalytical results for Exaple 4. Iteratio x(t) u(t) J(x(t) u(t)) t t t t-.653t t t Fig. : Trajectory ad cotrol fuctios for Exaple 4. = =. 5679

6 Aust. J. Basic & Appl. Sci. 4(): Fig. : Trajectory ad cotrol fuctios for Exaple 4. = =. Fig. 3: Trajectory ad cotrol fuctios for Exaple 4. = =. Exaple 4.: Here i secod exaple we cosider agai the optial cotrol prole govered y Fredhol itegral equatio selected fro [3] = ( ().) ( () ) Miiize J x t t u t t dt suject to ()= () ( ( )) ( ). x t y t ts t u s x s ds (7) (8) where yt ()=. The Exact optial solutios of (7)-(8) are * * x ( t) =. t ad u ( t) = t. with the optial criterio * * J = J( x ( t) u ( t))=. 568

7 Aust. J. Basic & Appl. Sci. 4(): Usig the aove ethod we have the uerical results otaied i Tale. As ca e see fro Tale ad Figures 4-6. Tale : The Approxiate-Aalytical results for Exaple 4. Iteratio x(t) u(t) J(x(t) u(t)) t t-3.39t t /9 t. Fig. 4: Trajectory ad cotrol fuctios for Exaple 4.= =. Fig. 5: Trajectory ad cotrol fuctios for Exaple 4. = =. Fig. 6: Trajectory ad cotrol fuctios for Exaple 4. = =. 568

8 Aust. J. Basic & Appl. Sci. 4(): Coclusios: A iterative schee for uerical solutio of optial cotrol proles govered y Fredhol itegral equatios usig paraetrizatio ad Berstei polyoials has ee proposed. The covergece ad uiqueess of the ethod has ee proved. The efficiecy of it discussed i soe exaples. REFERENCES Adou M.A. 3. O the solutio of liear ad oliear itegral equatio. Appl. Math. Coput. 46: Baolia E. J. Biazar ad A.R. Vahidi 4. The decopositio ethod applied to systes of Fredhol itegral equatios of the secod kid. Appl. Math. Coput. 48: Chakraarti A. ad S.C. Martha 9. Approxiate solutios of Fredhol itegral equatios of the secod kid. Appl. Math. Coput. : Huag S.C. ad R.P. Shaw 995. The Trefftz ethod as a itegral equatio. Adv. Eg. Software 4: Hoescu C. ad I.M. Navo 3. Optial cotrol of flow with discotiuous Joural of coputatioal physics 87: Jiag S. ad V. Rokhli 4. Secod kid itegral equatios for the classical potetial theory o ope surface II. J. Coput. Phys. 95: -6. Kythe P.K. ad P. Puri. Coputatioal Methods of Liear Itegral Equatios. Birkhauser Boste c/o Spriger-Verlag New York USA. Liag D. ad B. Zhag 4. Nuerical aalysis of graded esh ethods for a class of secod kid itegral equatios o real lie. J. Math. Aal. Appl. 94: Malekejad K. ad Y. Mahoudi 4. Nuerical solutio of liear Fredhol itegral equatios y usig hyrid Taylor ad Block-Pulse fuctios. Appl. Math. Coput. 49: Malekejad K. ad M. Karai 5. Usig the WPG ethod for solvig itegral equatios of the secod kid. Appl. Math. Coput. 66: 3-3. Malekejad K. ad M. Karai 5. Nuerical solutio of o-liear Fredhol itegral equatios y usig ultiwavelets i the Petrov Galerki ethod. Appl. Math. Coput. 68: -. Madal B.N. ad Suhra Bhattacharya 7. Nuerical solutios of soe classes of itergral equatios usig Berstei polyoials. Appl. Math. Coput. 9: Fard S. Oid Mahood Sachooli ad Akar H. Borzaadi. Taylor solver for Fredhol optial cotrol proles. Joural of Advaced Research i Differetial Equatios : -. Stes Ilse Y.M. K.J.E. Versyck ad J.F.M. Va Ipe. Optial cotrol theory: a geeric tool for idetificatio ad cotrol of (Bio-)cheical reactors. Aul reviews of cotrol. 6: Wag W. 6. A ew echaical algorith for solvig the secod kid of Fredhol itegral equatio. Appl. Math. Coput. 7: Yag S.C.. A ivestigatio ito itegral equatio ethods ivolvig early sigular kerels for acoustic scatterig. J. Soud Vi. 34: Zdeek T. T. Vosegaard C. Kehlet N. Khaeja S.J. Glaser ad N.C. Nielse 9. Optial cotrol i NMR spectroscopy: Nuerical ipleetatio i SIMPSON. Joural of Magetic Resoace 97: -34. Zhag S Itegral Equatios. Chogqig press Chogqig Chia. 568