Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives∗ Ricardo Almeida Delfim F. M. Torres [email protected] [email protected] Department of Mathematics University of Aveiro 3810-193 Aveiro, Portugal Abstract We prove optimality conditions for different variational functionals containing left and right Caputo fractional derivatives. A sufficient condi- tion of minimization under an appropriate convexity assumption is given. An Euler–Lagrange equation for functionals where the lower and upper bounds of the integral are distinct of the bounds of the Caputo derivative is also proved. Then, the fractional isoperimetric problem is formulated with an integral constraint also containing Caputo derivatives. Normal and abnormal extremals are considered. Mathematics Subject Classification 2010: 49K05, 26A33. Keywords: Euler–Lagrange equations, isoperimetric problems, Caputo fractional derivatives. 1 Introduction arXiv:1007.2937v1 [math.OC] 17 Jul 2010 Fractional Calculus is the branch of Mathematics that generalizes the deriva- tive and the integral of a function to a non-integer order. The subject is not recent and it is as old as the calculus itself. In a letter dated 30th September 1695, Leibniz proposed the following problem to L’Hopital: ”Can the meaning of derivatives with integer order be generalized to derivatives with non-integer orders?” Since then, several mathematicians studied this question, among them Liouville, Riemann, Weyl and Letnikov. There are many fields of applications where we can use the fractional calculus, like viscoelasticity, electrochemistry, diffusion processes, control theory, heat conduction, electricity, mechanics, chaos ∗Submitted 6/March/2010 to Communications in Nonlinear Science and Numerical Sim- ulation; revised 12/July/2010; accepted for publication 16/July/2010. 1 and fractals (see some references at the end, e.g., [10,17,21,22,24,27,30,31]). To solve fractional differential equations, there exist several methods: Laplace and Fourier transforms, truncated Taylor series, numerical methods, etc. (see [6] and references therein). Recently, a lot of attention has been put on the fractional calculus of variations (see, e.g., [1, 3, 4, 8, 9, 12–14, 19, 20, 25, 26, 28, 29]. We also mention [7], were necessary and sufficient conditions of optimality for functionals containing fractional integrals and fractional derivatives are presented. For re- sults on fractional optimal control see [2,15]. In the present paper we work with the Caputo fractional derivative. For problems of calculus of variations with boundary conditions, Caputo’s derivative seems to be more natural, since for a given function y to have continuous Riemann–Liouville fractional derivative on a closed interval [a,b], the function must satisfy the conditions y(a)= y(b) = 0 [8]. We also mention that, if y(a) = 0, then the left Riemann–Liouville derivative is equal to the left Caputo derivative. The paper is organized as follows. In Section 2 we present the necessary definitions and some necessary facts about fractional calculus. Section 3 is dedicated to our original results. We study fractional Euler–Lagrange equations and the fractional isoperimetric problem within Caputo’s fractional derivative context for different kinds of functionals. We also give sufficient conditions of optimality for fractional variational problems. 2 Preliminaries 2.1 Review on fractional calculus There are several definitions of fractional derivatives and fractional integrals, like Riemann–Liouville, Caputo, Riesz, RieszCaputo, Weyl, Grunwald–Letnikov, Hadamard, Chen, etc. We will present the definitions of the first two of them. Except otherwise stated, proofs of results may be found in [18]. Let f :[a,b] → R be a function, α a positive real number, n the integer satisfying n − 1 ≤ α<n, and Γ the Euler gamma function. Then, 1. the left and right Riemann–Liouville fractional integrals of order α are defined by x 1 − Iαf(x)= (x − t)α 1f(t)dt, a x Γ(α) a and b 1 − Iαf(x)= (t − x)α 1f(t)dt, x b Γ(α) x respectively; 2. the left and right Riemann–Liouville fractional derivatives of order α are defined by n n x d − 1 d − − Dαf(x)= In αf(x)= (x − t)n α 1f(t)dt, a x dxn a x Γ(n − α) dxn a 2 and n n n b d − (−1) d − − Dαf(x) = (−1)n In αf(x)= (t − x)n α 1f(t)dt, x b dxn x b Γ(n − α) dxn x respectively; 3. the left and right Caputo fractional derivatives of order α are defined by n x − d 1 − − C Dαf(x)= In α f(x)= (x − t)n α 1f (n)(t)dt, a x a x dxn Γ(n − α) a and n b − d 1 − − C Dαf(x) = (−1)n In α f(x)= (−1)n(t−x)n α 1f (n)(t)dt, x b x b dxn Γ(n − α) x respectively. There exists a relation between the Riemann–Liouville and the Caputo frac- tional derivatives: − n 1 (k) f (a) − C Dαf(x)= Dαf(x) − (x − a)k α a x a x Γ(k − α + 1) k =0 and − n 1 (k) f (b) − C Dαf(x)= Dαf(x) − (b − x)k α. x b x b Γ(k − α + 1) k =0 Therefore, ′ (n−1) C α α if f(a)= f (a)= ... = f (a)=0, then a Dx f(x)= aDx f(x) and ′ (n−1) C α α if f(b)= f (b)= ... = f (b)=0, then x Db f(x)= xDb f(x). These fractional operators are linear, i.e., P(µf(x)+ νg(x)) = µ Pf(x)+ ν Pg(x), α α C α C α α α where P is aDx , xDb , a Dx , x Db , aIx or xIb , and µ and ν are real numbers. If f ∈ Cn[a,b], then the left and right Caputo derivatives are continuous on [a,b]. The main advantage of Caputo’s approach is that the initial conditions for fractional differential equations with Caputo derivatives take on the same form as for integer-order differential equations. Some properties valid for integer differentiation and integer integration re- main valid for fractional differentiation and fractional integration; namely the Caputo fractional derivatives and the Riemann–Liouville fractional integrals are inverse operations: 3 1. If f ∈ L∞(a,b) or f ∈ C[a,b], and if α> 0, then C α α C α α a Dx aIx f(x)= f(x) and x Db xIb f(x)= f(x). 2. If f ∈ Cn[a,b] and if α> 0, then n−1 f (k)(a) IαC Dαf(x)= f(x) − (x − a)k a x a x k! k =0 and n− 1 (−1)kf (k)(b) IαC Dαf(x)= f(x) − (b − x)k. x b x b k! k =0 We also need for our purposes integration by parts formulas. For α> 0, we have (cf. [4]) b b C α α g(x) · a Dx f(x)dx = f(x) · xDb g(x)dx a a − n 1 b α+j−n n−1−j + xDb g(x) · xDb f(x) a j=0 and b b C α α g(x) · x Db f(x)dx = f(x) · aDx g(x)dx a a n−1 n+j α+j−n n−1−j b + (−1) aDx g(x) · aDx f(x) a , j=0 k −k k −k where aDxg(x)= aIx g(x) and xDb g(x)= xIb g(x) if k< 0. Therefore, if 0 <α< 1, we obtain b b C α α 1−α b g(x) · a Dx f(x)dx = f(x) · xDb g(x)dx + xIb g(x) · f(x) a (1) a a and b b C α α 1−α b g(x) · x Db f(x)dx = f(x) · aDx g(x)dx − aIx g(x) · f(x) a . (2) a a Moreover, if f is a function such that f(a)= f(b) = 0, we have simpler formulas: b b C α α g(x) · a Dx f(x)dx = f(x) · xDb g(x)dx (3) a a and b b C α α g(x) · x Db f(x)dx = f(x) · aDx g(x)dx. (4) a a 4 Remark 1. As α goes to 1, expressions (1) and (2) reduce to the classical integration by parts formulas: b b ′ ′ b g(x) · f (x)dx = − f(x) · g (x)dx + [g(x) · f(x)]a a a and b b ′ ′ b g(x) · (−f (x))dx = f(x) · g (x)dx − [g(x) · f(x)]a , a a respectively. Remark 2. Observe that the left member of equations (3) and (4) contains a Caputo fractional derivative, while the other one contains a Riemann–Liouville fractional derivative. However, since there exists a relation between the two derivatives, we could present this formula with only one fractional derivative, although in this case the resulting equation will contain some extra terms. 2.2 Fractional Euler–Lagrange equations From now on we fix 0 <α,β < 1. Also, to simplify, we denote C α C β [y](x) := (x, y(x), a Dx y(x), x Db y(x)). Let α β R C α C β a Eb = y :[a,b] → | a Dx y and x Db y exist and are continuous on [a,b] . Definition 2.1. The space of variations C V ar(a,b) for the Caputo derivatives is defined by C α β V ar(a,b)= h ∈ a Eb | h(a)= h(b)=0 . We now present first order necessary conditions of optimality for functionals, α β defined on a Eb , of the type b J(y)= L[y](x)dx.