Calculus of Variations with Fractional and Classical Derivatives

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1 Clculus of Vritions with Frctionl nd Clssicl Derivtives Ttin Odzijewicz Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro Aveiro, Portugl e-mil: Deprtment of Mthemtics, University of Aveiro Aveiro, Portugl e-mil: Abstrct: We give proper frctionl extension of the clssicl clculus of vritions. Necessry optimlity conditions of Euler-Lgrnge type for vritionl problems contining both frctionl nd clssicl derivtives re proved. The fundmentl problem of the clculus of vritions with mixed integer nd frctionl order derivtives s well s isoperimetric problems re considered. Keywords: vritionl nlysis; optimlity; Riemnn-Liouville frctionl opertors; frctionl differentition; isoperimetric problems. 1. INTRODUCTION One of the clssicl problems of mthemtics consists in finding closed plne curve of given length tht encloses the gretest re: the isoperimetric problem. The legend sys tht the first person who solved the isoperimetric problem ws Dido, the queen of Crthge, who ws offered s much lnd s she could surround with the skin of bull. Dido s problem is nowdys prt of the clculus of vritions [Gelfnd nd Fomin, 1963, vn Brunt, 24]. Frctionl clculus is generliztion of integer differentil clculus, llowing to define derivtives nd integrls of rel or complex order [Kilbs et l., 26, Miller nd Ross, 1993, Podlubny, 1999]. The first ppliction of frctionl clculus belongs to Niels Henrik Abel nd goes bck to 1823 [Abel, 1965]. Abel pplied the frctionl clculus to the solution of n integrl eqution which rises in the formultion of the tutochrone problem. This problem, sometimes lso clled the isochrone problem, is tht of finding the shpe of frictionless wire lying in verticl plne such tht the time of bed plced on the wire slides to the lowest point of the wire in the sme time regrdless of where the bed is plced. The cycloid is the isochrone s well s the brchistochrone curve: it gives the shortest time of slide nd mrks the born of the clculus of vritions. The study of frctionl problems of the clculus of vritions nd respective Euler-Lgrnge type equtions is subject of current strong reserch due to its mny pplictions in science nd engineering, including mechnics, chemistry, biology, economics, nd control theory. In Riewe obtined version of the Euler Lgrnge equtions for frctionl vritionl problems combining the conservtive nd nonconservtive cses [Riewe, 1996, 1997]. Since then, numerous works on the frctionl clculus of vritions, frctionl optiml control This work is prt of the first uthor s PhD project, which is crried out t the University of Aveiro under the Doctorl Progrmme Mthemtics nd Applictions of Universities of Aveiro nd Minho. nd its pplictions hve been written see, e.g., [Agrwl, 22, Agrwl nd Blenu, 27, Almeid nd Torres, 29, 21, Atncković et l., 28, Blenu, 28, El- Nbulsi nd Torres, 27, Frederico nd Torres, 27, 28, Klimek, 22, Mlinowsk nd Torres, 21] nd references therein. For the study of frctionl isoperimetric problems see [Almeid et l., 29]. In the pioneering pper [Agrwl, 22], nd others tht followed, the frctionl necessry optimlity conditions re proved under the hypothesis tht dmissible functions y hve continuous left nd right frctionl derivtives on the closed intervl [, b]. By considering tht the dmissible functions y hve continuous left frctionl derivtives on the whole intervl, then necessrily y = ; by considering tht the dmissible functions y hve continuous right frctionl derivtives, then necessrily yb =. This fct hs been independently remrked, in different contexts, t lest in [Almeid et l., 29, Almeid nd Torres, 21, Atncković et l., 28, Jelicic nd Petrovcki, 29]. In our work we wnt to be ble to consider rbitrrily given boundry conditions y = y nd yb = y b nd isoperimetric constrints. For tht we consider vritionl functionls with integrnds involving not only frctionl derivtive of order α, 1 of the unknown function y, but lso the clssicl derivtive y. More precisely, we consider dependence of the integrnds on the independent vrible t, unknown function y, nd y + k D α t y with k rel prmeter. As consequence, one gets proper extension of the clssicl clculus of vritions, in the sense tht the clssicl theory is recovered with the prticulr sitution k =. We remrk tht this is not the cse with ll the previous literture on the frctionl vritionl clculus, where the clssicl theory is not included s prticulr cse nd only s limit, when α 1. The text is orgnized s follows. In Section 2 we briefly recll the necessry definitions nd properties of the frctionl clculus in the sense of Riemnn-Liouville. Our results re stted, proved, nd illustrted through n exmple, in Section 3. We end with Section 4 of conclusion.

2 2. PRELIMINARIES hs n extremum. We ssume tht k is fixed rel number, F C 2 [, b] R 2 ; R, nd 3 F the prtil derivtive In this section some bsic definitions nd properties of of F,, with respect to its third rgument hs frctionl clculus re presented. For more on the subject continuous right Riemnn-Liouville frctionl derivtive of we refer the reder to the books [Kilbs et l., 26, Miller order α. nd Ross, 1993, Podlubny, 1999]. Definition 5. A function y C 1 [, b] tht stisfies the Definition 1. Left nd right Riemnn-Liouville derivtives. given boundry conditions 3 is sid to be dmissible for Let f be function defined on [, b]. The opertor D α t, problem 2-3. t D α 1 t ft = Γn α Dn t τ n α 1 fτdτ, is clled the left Riemnn-Liouville frctionl derivtive of order α, nd the opertor t D α b, td α 1 b ft = Γn α Dn τ t n α 1 fτdτ, t is clled the right Riemnn-Liouville frctionl derivtive of order α, where α R + is the order of the derivtives nd the integer number n is such tht n 1 α < n. Definition 2. Mittg-Leffler function. Let α, β >. The Mittg-Leffler function is defined by z k E α,β z = Γαk + β. k= Theorem 3. Integrtion by prts. If f, g nd the frctionl derivtives D α t g nd t D α b f re continuous t every point t [, b], then ft D α t gtdt = gt t D α b ftdt 1 for ny < α < 1. Remrk 4. If f, then D α t ft t= =. Similrly, if fb, then t D α b ft t=b =. Thus, if f possesses continuous left nd right Riemnn-Liouville frctionl derivtives t every point t [, b], then f = fb =. This explins why the usul term ftgt b does not pper on the right-hnd side of MAIN RESULTS Following [Jelicic nd Petrovcki, 29], we prove optimlity conditions of Euler-Lgrnge type for vritionl problems contining clssicl nd frctionl derivtives simultneously. In Section 3.1 the fundmentl vritionl problem is considered, while in Section 3.2 we study the isoperimetric problem. Our results cover frctionl vritionl problems subject to rbitrrily given boundry conditions. This is in contrst with [Agrwl, 22, 28, Agrwl nd Blenu, 27, Blenu et l., 29], where the necessry optimlity conditions re vlid for pproprite zero vlued boundry conditions cf. Remrk 4. For discussion on this mtter see [Almeid nd Torres, 21, Atncković et l., 28, Jelicic nd Petrovcki, 29]. 3.1 The Euler-Lgrnge eqution Let < α < 1. Consider the following problem: find function y C 1 [, b] for which the functionl F t, yt, y t + k D α t yt dt 2 subject to given boundry conditions y = y, yb = y b, 3 For simplicity of nottion we introduce the opertor [y] α k defined by [y] α k t = t, yt, y t + k D α t yt. With this nottion we cn write 2 simply s F [y] α k tdt. Theorem 6. The frctionl Euler-Lgrnge eqution. If y is n extremizer minimizer or mximizer of problem 2-3, then y stisfies the Euler-Lgrnge eqution 2 F [y] α k t d dt 3F [y] α k t + k t D α b 3 F [y] α k t = 4 for ll t [, b]. Proof. Suppose tht y is solution of 2-3. Note tht dmissible functions ŷ cn be written in the form ŷt = yt+ɛηt, where η C 1 [, b], η = ηb =, nd ɛ R. Let Jɛ = F t, yt + ɛηt, d dt yt + ɛηt + k D α t yt + ɛηtdt. Since D α t is liner opertor, we know tht D α t yt + ɛηt = D α t yt + ɛ D α t ηt. On the other hnd, dj b d dɛ = ɛ= dɛ F [ŷ]α k tdt ɛ= = 2 F [y] α k t ηt + 3 F [y] α k t dηt 5 dt + k 3 F [y] α k t D α t ηt dt. Using integrtion by prts we get nd 3 F dη dt dt = 3F η b k 3 F D α t ηdt = η d dt 3F dt 6 η t D b 3 F dt. 7 Substituting 6 nd 7 into 5, nd hving in mind tht η = ηb =, it follows tht dj b dɛ = ηt 2 F [y] α k t d ɛ= dt 3F [y] α k t + k t D α b 3 F [y] α k t dt. A necessry optimlity condition is given by dj dɛ ɛ= =. Hence, ηt 2 F [y] α k t d dt 3F [y] α k t + k t D α b 3 F [y] α k t dt =. 8 We obtin 4 pplying the fundmentl lemm of the clculus of vritions to 8.

3 Remrk 7. Note tht for k = our necessry optimlity condition 4 reduces to the clssicl Euler-Lgrnge eqution [Gelfnd nd Fomin, 1963, vn Brunt, 24]. 3.2 The frctionl isoperimetric problem As before, let < α < 1. We now consider the problem of extremizing functionl F t, yt, y t + k D α t yt dt 9 in the clss y C 1 [, b] when subject to given boundry conditions y = y, yb = y b, 1 nd n isoperimetric constrint Gt, yt, y t + k D α t ytdt = ξ. 11 We ssume tht k nd ξ re fixed rel numbers, F, G C 2 [, b] R 2 ; R, nd 3 F nd 3 G hve continuous right Riemnn-Liouville frctionl derivtives of order α. Definition 8. A function y C 1 [, b] tht stisfies the given boundry conditions 1 nd isoperimetric constrint 11 is sid to be dmissible for problem Definition 9. An dmissible function y is n extreml for I if it stisfies the frctionl Euler-Lgrnge eqution 2 G[y] α k t d dt 3G[y] α k t + k t D α b 3 G[y] α k t = for ll t [, b]. The next theorem gives necessry optimlity condition for the frctionl isoperimetric problem Theorem 1. Let y be n extremizer to the functionl 9 subject to the boundry conditions 1 nd the isoperimetric constrint 11. If y is not n extreml for I, then there exists constnt λ such tht 2 H[y] α k t d dt 3H[y] α k t + k t D α b 3 H[y] α k t = 12 for ll t [, b], where Ht, y, v = F t, y, v λgt, y, v. Proof. We introduce the two prmeter fmily ŷ = y + ɛ 1 η 1 + ɛ 2 η 2, 13 in which η 1 nd η 2 re such tht η 1, η 2 C 1 [, b] nd they hve continuous left nd right frctionl derivtives. We lso require tht η 1 = η 1 b = = η 2 = η 2 b. First we need to show tht in the fmily 13 there re curves such tht ŷ stisfies 11. Substituting y by ŷ in 11, Iŷ becomes function of two prmeters ɛ 1, ɛ 2. Let Îɛ 1, ɛ 2 = Gt, ŷ, ŷ + k D α t ŷdt ξ. Then, Î, = nd = η ɛ G d dt 3G + k t D α b 3 G dt., Since y is not n extreml for I, by the fundmentl lemm of the clculus of vritions there is function η 2 such tht. ɛ 2, By the implicit function theorem, there exists function ɛ 2 defined in neighborhood of, such tht Îɛ 1, ɛ 2 ɛ 1 =. Let Ĵɛ 1, ɛ 2 = J ŷ. Then, by the Lgrnge multiplier rule, there exists rel λ such tht Becuse Ĵ = ɛ 1, nd = ɛ 1, Ĵ, λî, =. η 1 2 F d dt 3F + k t D α b 3 F dt η 1 2 G d dt 3G + k t D α b 3 G dt, one hs [ b η 1 2 F d dt 3F + k t D α b 3 F λ 2 G d dt 3G + k t D α b 3 G ] dt =. Since η 1 is n rbitrry function, 12 follows from the fundmentl lemm of the clculus of vritions. 3.3 An exmple Let α, 1 nd k, ξ R. Consider the following frctionl isoperimetric problem: y =, y1 = y + k D α t y 2 dt min y + k D α t y dt = ξ E 1 α,1 k 1 τ 1 α ξdτ. 14 In this cse the ugmented Lgrngin H of Theorem 1 is given by Ht, y, v = v 2 λv. One cn esily check tht yt = t E 1 α,1 k t τ 1 α ξdτ 15 is not n extreml for I; stisfies y +k D α t y = ξ see, e.g., [Kilbs et l., 26, p. 297, Theorem 5.5]. Moreover, 15 stisfies 12 for λ = 2ξ, i.e., d dt 2 y + k D α t y 2ξ + k t D α 1 2 y + k D α t y 2ξ =. We conclude tht 15 is the extreml for problem 14. Exmple 11. Choose k =. In this cse the isoperimetric constrint is trivilly stisfied, 14 is reduced to the clssicl problem of the clculus of vritions y t 2 dt min y =, y1 = ξ, 16 nd our generl extreml 15 simplifies to the well-known minimizer yt = ξt of 16.

4 Exmple 12. When α 1 the isoperimetric constrint is redundnt with the boundry conditions, nd the frctionl problem 14 simplifies to the clssicl vritionl problem k y t 2 dt min y =, y1 = ξ k Our frctionl extreml 15 gives yt = ξ k+1t, which is exctly the minimizer of 17. Exmple 13. Choose k = ξ = 1. When α one gets from 14 the clssicl isoperimetric problem y t + yt 2 dt min ytdt = 1 e y =, y1 = 1 1 e. 18 Our extreml 15 is then reduced to the clssicl extreml yt = 1 e t of 18. Exmple 14. Choose k = 1 nd α = 1 2. Then 14 gives the following frctionl isoperimetric problem: y + D t y dt min y =, y1 = ξ y + D 1 2 t y dt = ξ 1 erfc1 + 2, π 19 where erfc is the complementry error function. The extreml 15 for the prticulr frctionl problem 19 is yt = ξ 1 e t erfc t + 2 t. π 4. CONCLUSION Frctionl vritionl clculus provides very useful frmework to del with nonlocl dynmics in Mechnics nd Physics [Blenu nd Trujillo, 21]. Motivted by the results nd insights of [Almeid et l., 29, Almeid nd Torres, 29, Jelicic nd Petrovcki, 29], in this pper we generlize previous frctionl Euler-Lgrnge equtions by proving optimlity conditions for frctionl problems of the clculus of vritions where the highest derivtive in the Lgrngin is of integer order. This pproch voids difficulties with the given boundry conditions when in presence of Riemnn Liouville derivtives [Jelicic nd Petrovcki, 29]. We focus our ttention to problems subject to integrl constrints frctionl isoperimetric problems, which hve recently found brod clss of importnt pplictions [Almeid nd Torres, 29b, Blåsjö, 25, Curtis, 24]. For k = our results re reduced to the clssicl ones [vn Brunt, 24]. This is in contrst with the stndrd pproch to frctionl vritionl clculus, where the clssicl integer-order cse is obtined only in the limit. ACKNOWLEDGEMENTS The first uthor is supported by the Portuguese Foundtion for Science nd Technology FCT through the PhD fellowship SFRH/BD/33865/29; the second uthor by FCT through the Center for Reserch nd Development in Mthemtics nd Applictions CIDMA. REFERENCES N. H. Abel. Euvres completes de Niels Henrik Abel. Christin: Imprimerie de Grondhl nd Son; New York nd London: Johnson Reprint Corportion. VIII, 621 pp., O. P. Agrwl. Formultion of Euler-Lgrnge equtions for frctionl vritionl problems. J. Mth. Anl. Appl., 272 1: , 22. O. P. Agrwl. A generl finite element formultion for frctionl vritionl problems. J. Mth. Anl. Appl., 337 1: 1 12, 28. O. P. Agrwl nd D. Blenu. A Hmiltonin formultion nd direct numericl scheme for frctionl optiml control problems. J. Vib. Control, : , 27. R. Almeid nd D. F. M. Torres. Clculus of vritions with frctionl derivtives nd frctionl integrls. Appl. Mth. Lett., 22 12: , 29. R. Almeid nd D. F. M. Torres. Hölderin vritionl problems subject to integrl constrints. J. Mth. Anl. Appl., 359 2: , 29b. R. Almeid nd D. F. M. Torres. Leitmnn s direct method for frctionl optimiztion problems. Appl. Mth. Comput., in press, 21. DOI: 1.116/j.mc R. Almeid, R. A. C. Ferreir, nd D. F. M. Torres. Isoperimetric problems of the clculus of vritions with frctionl derivtives. Submitted, 29. T. M. Atncković, S. Konjik, nd S. Pilipović. Vritionl problems with frctionl derivtives: Euler-Lgrnge equtions. J. Phys. A, 41 9: 9521, 12, 28. D. Blenu. New pplictions of frctionl vritionl principles. Rep. Mth. Phys., 61 2: , 28. D. Blenu nd J. I. Trujillo. A new method of finding the frctionl Euler-Lgrnge nd Hmilton equtions within Cputo frctionl derivtives. Communictions in Nonliner Science nd Numericl Simultion, 15 5: , 21. D. Blenu, O. Defterli, nd O. P. Agrwl. A centrl difference numericl scheme for frctionl optiml control problems. J. Vib. Control, 15 4: , 29. V. Blåsjö. The isoperimetric problem. Amer. Mth. Monthly, 112 6: , 25. J. P. Curtis. Complementry extremum principles for isoperimetric optimiztion problems. Optim. Eng., 5 4: , 24. R. A. El-Nbulsi nd D. F. M. Torres. Necessry optimlity conditions for frctionl ction-like integrls of vritionl clculus with Riemnn-Liouville derivtives of order α, β. Mth. Methods Appl. Sci., 3 15: , 27. G. S. F. Frederico nd D. F. M. Torres. A formultion of Noether s theorem for frctionl problems of the clculus of vritions. J. Mth. Anl. Appl., 334 2: , 27.

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