Calculus of variations with fractional derivatives and fractional integrals

Transcription

1 Anis do CNMAC v.2 ISSN X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro Aveiro, Portugl E-mils: Astrct: We prove Euler-Lgrnge frctionl equtions nd sufficient optimlity conditions for prolems of the clculus of vritions with functionls contining oth frctionl derivtives nd frctionl integrls in the sense of Riemnn-Liouville. Mthemtics Suject Clssifiction: 49K05, 26A33. Key words: Euler-Lgrnge eqution, Riemnn-Liouville frctionl derivtive, Riemnn-Liouville frctionl integrl. 1 Introduction In recent yers numerous works hve een dedicted to the frctionl clculus of vritions. Most of them del with Riemnn-Liouville frctionl derivtives 3, 4, 5, 6, few with Cputo or Riesz derivtives 1, 2. Depending on the type of functionl eing considered, different frctionl Euler-Lgrnge type equtions re otined. Here we propose new kind of functionl with Lgrngin contining not only Riemnn-Liouville frctionl derivtive (RLFD) ut lso Riemnn-Liouville frctionl integrl (RLFI). We prove necessry conditions of Euler-Lgrnge type for the fundmentl frctionl prolem of the clculus of vritions nd for the frctionl isoperimetric prolem. Sufficient optimlity conditions re lso otined under pproprite conveity ssumptions. 2 Frctionl Clculus In this section we review the necessry definitions nd fcts from frctionl clculus. For more on the suject we refer the reder to 7. Let f e function defined on the intervl,. Let α e positive rel nd n := α + 1. Definition 2.1 The left RLFI is defined y nd the right RLFI y The left RLFD is defined y I α f() = 1 Γ(α) I α f() = 1 Γ(α) D α f() = dn d n I n α f() = ( t) α 1 f(t)dt, (t ) α 1 f(t)dt. d n 1 Γ(n α) d n ( t) n α 1 f(t)dt, Work prtilly supported y the R&D unit CEOC cofinnced y FCT nd the EC fund FEDER/POCI

2 while the right RLFD is given y D α dn f() = ( 1)n d n I n α f() = ( 1)n Γ(n α) d n d n (t ) n α 1 f(t)dt. The opertors of Definition 2.1 re oviously liner. We now present the rules of frctionl integrtion y prts for RLFI nd RLFD. Let p 1, q 1, nd 1/p+1/q 1+α. If g L p (, ) nd f L q (, ), then g() I α f()d = f() I α g()d ; if f, g, nd the frctionl derivtives D α g nd D α f re continuous on,, then 0 < α < 1. g() D α f()d = f() D α g()d, Remrk 1 The left RLFD of f is infinite t = if f() 0. Similrly, the right RLFD is infinite if f() 0. Thus, ssuming tht f possesses continuous left nd right RLFD on,, then f() = f() = 0 must e stisfied. This fct restrin the oundry conditions of the frctionl prolems of the clculus of vritions henceforth. 3 The Euler-Lgrnge eqution Let us consider the following prolem: J (y) = suject to the oundry conditions L(, I y(), D β y()) d min (1) y() = 0 nd y() = 0 (2) (recll Remrk 1). We ssume tht L(,, ) C 1 (, R 2 ; R), 2 L(, I y(), Dy()) β hs continuous right RLFI of order 1 α nd 3 L(, I y(), Dy()) β hs continuous right RLFD of order β, where α nd β re rel numers in the intervl (0, 1). Remrk 2 We re ssuming tht the dmissile functions y re such tht I y() nd Dy() β eist on the closed intervl,. We lso note tht s α nd β goes to 1 our frctionl functionl J tends to the clssicl functionl L(, y(), y ()) d of the clculus of vritions. Remrk 3 We consider functionls J contining the left RLFI nd the left RLFD only. This comprise the importnt cses in pplictions. The results of the pper re esily generlized for functionls contining lso the right RLFI nd/or right RLFD. Theorem 3.1 (The frctionl Euler-Lgrnge eqution) Let y( ) e locl minimizer of prolem (1) (2). Then, y( ) stisfies the frctionl Euler-Lgrnge eqution I for ll,. 2 L(, I y(), Dy()) β + D β 3L(, I y(), Dy()) β = 0 (3) Remrk 4 Condition (3) is only necessry for n etremum. The question of sufficient conditions for n etremum is considered in Section

3 Proof. Since y is n etremizer of J, y well known result of the clculus of vritions the first vrition of J ( ) is zero t y, i.e., if η is such tht η() = η() = 0, then Integrting y prts, nd 0 = δj (η, y) = I η 2 L d = D β η 3 L d = ( I η 2 L + D β η 3 L) d. (4) η I 2 L d (5) η D β 3L d. (6) Sustituting (5) nd (6) into eqution (4), we find tht ( I 2 L + D β 3L)η d = 0 for ech η. Since η is n ritrry function, y the fundmentl lemm of the clculus of vritions we deduce tht I 2 L + D β 3L = 0. Remrk 5 As α nd β goes to 1, the frctionl Euler-Lgrnge eqution (3) ecomes the clssicl Euler-Lgrnge eqution 2 L d/d 3 L = 0. A curve tht is solution of the frctionl differentil eqution (3) will e clled n etreml of J. Etremls ply lso n importnt role in the solution of the frctionl isoperimetric prolem (see Section 5). We note tht eqution (3) contins right RLFI nd right RLFD, which re not present in the formultion of prolem (1) (2). 4 Some generliztions We now give some generliztions of Theorem Etension to vritionl prolems of non-commensurte order We now consider prolems of the clculus of vritions with Riemnn-Liouville derivtives nd integrls of non-commensurte order, i.e., we consider functionls contining RLFI nd RLFD of different frctionl orders. Let J (y) = L(, I 1 y(),..., I n y(), D β 1 y(),..., D βm y()) d, (7) where n nd m re two positive integers nd α i, β j (0, 1), i = 1,..., n nd j = 1,..., m. Following the proof of Theorem 3.1, we deduce the following result. Theorem 4.1 If y( ) is locl minimizer of (7) suject to the oundry conditions (2), then y( ) stisfies the Euler-Lgrnge eqution for ll,. n i=1 I i i+1 L + m D β j j+n+1l = 0 j=1 1224

4 4.2 Etension to severl dependent vriles We now study the cse of multiple unknown functions y 1,..., y n. Theorem 4.2 Let J e the functionl given y the epression J (y 1,..., y n ) = L(, I y 1 (),..., I y n (), Dy β 1 (),..., Dy β n ()) d, defined on the set of curves such tht y i () = y i () = 0 for ll i = 1,..., n. If y 1 ( ),..., y n ( ) is locl minimizer of J, then it stisfies for ll, the following system of n frctionl differentil equtions: I 2 L + D β n+2l = 0 I 3 L + D β n+3l = 0. I n+1 L + D β 2n+1L = 0. Proof. Denote y y nd η the vectors (y 1,..., y n ) nd (η 1,..., η n ), respectively. In this cse, η() = η() = 0. For prmeter ɛ, we consider new function J(ɛ) = J (y + ɛη) (8) Since y 1 ( ),..., y n ( ) is n etremizer of J, J (0) = 0. Differentiting eqution (8) with respect to ɛ, t ɛ = 0, we otin I Integrting y prts leds to η 1 2 L + + I η n n+1 L + Dη β 1 n+2 L + + Dη β n 2n+1 L d = 0. I 2 L + D β n+2l η I n+1 L + D β 2n+1L η n d = 0. Considerer vrition η = (η 1, 0,..., 0), η 1 ritrry; then y the fundmentl lemm of the clculus of vritions we otin I 2 L + D β n+2l = 0. Selecting pproprite vritions η, one deduce the remining formuls. 5 The frctionl isoperimetric prolem We consider now the prolem of minimizing the functionl J given y (1) suject to the oundry conditions (2) nd to n integrl constrint I(y) = g(, I y(), Dy()) β d = l, where l is prescried vlue. This prolem ws solved in 2 for functionls contining Cputo frctionl derivtives nd RLFI. Using similr techniques s the ones discussed in 2, one proves the following: Theorem 5.1 Consider the prolem of minimizing the functionl J s in (1) on the set of functions y stisfying conditions (2) nd I(y) = l. Let y e locl minimum for the prolem. Then, there eist two constnts λ 0 nd λ, not oth zero, such tht y stisfies the Euler-Lgrnge eqution I 2 K + D β 3K = 0 for ll,, where K = λ 0 L + λg. Remrk 6 If y is not n etreml for I, then one cn choose λ 0 = 1 in Theorem 5.1: there eists constnt λ such tht y stisfies I 2 F + D β 3F = 0 for ll,, where F = L + λg. 1225

5 6 Sufficient conditions In this section we prove sufficient conditions tht ensure the eistence of minimums. Similrly to wht hppens in the clssicl clculus of vritions, some conditions of conveity re in order. Definition 6.1 Given function L, we sy tht L(, u, v) is conve in S R 3 if 2 L nd 3 L eist nd re continuous nd verify the following condition: L(, u + u 1, v + v 1 ) L(, u, v) 2 L(, u, v)u L(, u, v)v 1 for ll (, u, v), (, u + u 1, v + v 1 ) S. Similrly, we define conveity for L(, u, v). Theorem 6.2 Let L(, u, v) e conve function in, R 2 nd let y 0 e curve stisfying the frctionl Euler-Lgrnge eqution (3), i.e., I 2 L(, I y 0 (), Dy β 0 ()) + D β 3L(, I y 0 (), Dy β 0 ()) = 0 for ll,. Then, y 0 minimizes (1) suject to y() = y 0 () = 0 nd y() = y 0 () = 0. Proof. Let η e such tht η() = η() = 0. Then, the following holds: J (y 0 + η) J (y 0 ) = = L(, I y 0 () + I η(), Dy β 0 () + Dη()) β L(, I y 0 (), Dy β 0 ()) d 2 L(, I y 0 (), Dy β 0 ()) I η + 3 L(, I y 0 (), Dy β 0 ()) Dη β d 2 L + D β 3L η d = 0. I (, I y 0 (), D β y 0 ()) Thus, J (y 0 + η) J (y 0 ). We now present sufficient condition for conve Lgrngins on the third vrile only. First we recll the notion of ect field. Definition 6.3 Let D R 2 nd let Φ : D R e function of clss C 1. We sy tht Φ is n ect field for L covering D if there eists function S C 1 (D, R) such tht 1 S(, y) = L(, y, Φ(, y)) 3 L(, y, Φ(, y))φ(, y), 2 S(, y) = 3 L(, y, Φ(, y)). Remrk 7 This definition is motivted y the clssicl Euler-Lgrnge eqution. Indeed, every solution y 0 C 2, of the differentil eqution y = Φ(, y()) stisfies the (clssicl) Euler- Lgrnge eqution 2 L d d 3L = 0. Theorem 6.4 Let L(, u, v) e conve function in, R 2, Φ n ect field for L covering, R D, nd y 0 solution of the frctionl eqution D α y() = Φ(, I y()). (9) Then, y 0 is minimizer for J (y) = L(, I y(), D α y()) d suject to the constrint { y :, R I y() = I y 0 (), I y() = I y 0 () }. (10) 1226

6 Proof. Let E(, y, z, w) = L(, y, w) L(, y, z) 3 L(, y, z)(w z). First oserve tht d d S(, I y()) = 1 S(, I y()) + 2 S(, I y()) d d I y() = 1 S(, I y()) + 2 S(, I y()) D α y(). Since E 0, it follows tht J (y) = = = E(, I + 3 L(, I L(, I + 3 L(, I 1 S(, I y, Φ(, I d d S(, I y) d = S(, I Since y 0 is solution of (9), E(, I s efore, one hs J (y 0 ) = S(, I y, Φ(, I y, Φ(, I y)) y, Φ(, I y), D α y) + L(, I y, Φ(, I y)) y))( D α y Φ(, I y)) d y))( D α y Φ(, I y)) d y) + 2 S(, I y) D α y d y()) S(, I y()). J (y) when suject to the constrint (10). References y 0, Φ(, I y 0 ()) S(, I y 0 ), D α y 0 ) = 0. With similr clcultions y 0 ()). We just proved tht J (y 0 ) 1 O. P. Agrwl, Frctionl vritionl clculus in terms of Riesz frctionl derivtives, J. Phys. A 40(24) (2007), R. Almeid nd D. F. M. Torres, Necessry nd sufficient conditions for the frctionl clculus of vritions with Cputo derivtives, sumitted. 3 T. M. Atncković, S. Konjik nd S. Pilipović, Vritionl prolems with frctionl derivtives: Euler Lgrnge equtions, J. Phys. A: Mth. Theor., 41 (9) (2008), R. A. El-Nulsi nd D. F. M. Torres, Necessry optimlity conditions for frctionl ctionlike integrls of vritionl clculus with Riemnn-Liouville derivtives of order (α, β), Mth. Methods Appl. Sci., 30 (15) (2007), G. S. F. Frederico nd D. F. M. Torres, A formultion of Noether s theorem for frctionl prolems of the clculus of vritions, J. Mth. Anl. Appl., 334 (2) (2007), G. S. F. Frederico nd D. F. M. Torres, Frctionl conservtion lws in optiml control theory, Nonliner Dynm., 53 (3) (2008), A. A. Kils, H. M. Srivstv nd J. J. Trujillo, Theory nd pplictions of frctionl differentil equtions, Elsevier, Amsterdm,