Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels

: In this work we study necessary and su ﬃ cient optimality conditions for variational problems dealing with a new fractional derivative. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with arbitrary kernels. After proving a fractional integration by parts formula, we obtain the Euler–Lagrange equation and natural boundary conditions for the fundamental variational problem. Also, fractional variational problems with integral and holonomic constraints are considered. We end with some examples to exemplify our results.


Introduction
The fractional calculus is an old subject and presents an extension of ordinary calculus [12,14]. It began at the same time with the works of Leibniz on differential calculus, where he questioned what could be a derivative of arbitrary real order α > 0. Since then, a large number of definitions of fractional order integral and derivative operators have appeared. Thus, we find in the literature numerous works dealing with similar topics, but for different fractional operators. One possible way to avoid such issue is to consider a more general class of fractional operators, like, for example, fractional integrals and derivatives with arbitrary kernels [3,12] or other types of general fractional derivatives [16][17][18][19][20].
Another possible approach to fractional calculus is, instead of fixing the fractional order α, the introduction of a new function that acts like a distribution of the orders of differentiation [9,10]. Our goal in this paper is to combine both ideas into a single operator, in order to obtain new results that will generalize some of the already known.
The main purpose of this paper is to prove optimality conditions for variational problems that depend on distributed-order fractional derivatives with arbitrary kernels. Namely, we will prove the Euler-Lagrange equation and the natural boundary conditions for variational problems with and without integral constraints and also with an holonomic constraint. Moreover, we provide sufficient optimality conditions for all the variational problems studied in this paper. With this work we generalize several existent works on fractional calculus of variations such as [1,2,7,8,13].
The structure of the paper is as follows. In Section 2, we recall the necessary definitions and results from fractional calculus that are needed to the present work. Our main contributions are presented in Section 3. We finalize the paper with some illustrative examples and concluding remarks.

Preliminaries
Throughout the paper, Γ represents the well-known Gamma function and [α] denotes the integer part of α ∈ R. We begin by recalling the definition of ψ-Riemann-Liouville fractional integrals of a function x of order α ∈ R + . Definition 2.2. [14] Let α ∈ R + , x : [a, b] → R an integrable function, and ψ ∈ C n ([a, b], R) with ψ (t) > 0, for all t ∈ [a, b]. The left and right Riemann-Liouville fractional derivatives of x with respect to the kernel ψ, of order α, are defined by The operators D α,ψ a+ x and D α,ψ b− x can be simply called ψ-Riemann-Liouville fractional derivatives of x of order α [5]. If we interchange the order of the ordinary derivative with the fractional integral, we obtain the definition of the ψ-Caputo fractional derivatives of x of order α. Definition 2.3. [3] Given α ∈ R + , let n ∈ N be given by n = [α] + 1 if α N, and n = α if α ∈ N. Given two functions x, ψ ∈ C n ([a, b], R) with ψ (t) > 0, for all t ∈ [a, b], we define the left and right Caputo fractional derivatives of x with respect to the kernel ψ, of order α, by respectively. Lemma 2.4. [3] Given α > 0, let n ∈ N be given by Definition 2.3. For β ∈ R with β > n, we have that Until the end of the work, the fractional order α belongs to the interval [0, 1] and the kernel ψ is a function on the set In order to introduce the new concepts of distributed-order fractional derivatives with respect to another function, in the Riemann-Liouville and in the Caputo sense, we consider a new continuous function φ : [0, 1] → [0, 1] that satisfies the condition Usually, function φ is called order-weighting or strength function. For some applications on distributed-order fractional derivatives, we suggest the paper [11].
where D α,ψ a+ and D α,ψ b− are the left and right ψ-Riemann-Liouville fractional derivatives of order α, respectively.
Definition 2.6. The left and right Caputo distributed-order fractional derivatives of a function x ∈ C 1 ([a, b], R) with respect to the kernel ψ are defined by where C D α,ψ a+ and C D α,ψ b− are the left and right ψ-Caputo fractional derivatives of order α, respectively.
For our work we will also need the concepts of distributed-order fractional integrals with respect to the kernel ψ: where I 1−α,ψ a+ and I 1−α,ψ b− are the left and right ψ-Riemann-Liouville fractional integrals of order 1 − α, respectively.
The goal of this work is to exhibit necessary and sufficient optimality conditions for the following fractional variational problem: Problem (P): Find a curve x ∈ C 1 ([a, b], R) that minimizes or maximizes the following functional where L : [a, b] × R 3 → R is assumed to be continuously differentiable with respect to the second, third and fourth variables. In our study, we will consider the variational problem with and without fixed boundary conditions, and also with an isoperimetric or holonomic constraints. Before proving our main results, we need to prove the following integration by parts formulae for the left and right Caputo distributed-order fractional derivatives with respect to another function.
Proof. By definition of the left ψ-Caputo distributed-order fractional derivative, we have the following: Reversing the order of integration, we get Using the standard integration by parts formula, we have Using similar techniques, we deduce the integration by parts formula involving the operator C D ϕ(α),ψ b− .

Necessary optimality conditions
In what follows, we will denote by ∂ i L the partial derivative of L with respect to its ith-coordinate and use the notation: We are now in a position to prove our first main result.
are continuous on [a, b], then x satisfies the following Euler-Lagrange equation for all t ∈ [a, b]. Also, if x(a) is free, then

3)
and if x(b) is free, then , R) be an arbitrary function and define the function j by j( ) : Since h is arbitrary, from the Fundamental Lemma of Calculus of Variations (see [15]), we get Since h(a) is arbitrary, we get that, at t = a, proving the natural boundary condition (3.3). Similarly, if x(b) is free, considering h(a) = 0 and h(b) 0 in (3.5) and using (3.2), we deduce the natural boundary condition (3.4).

Remark 1.
It is clear that the variational problem (P) can be easily extended to functionals depending on a vector function x := (x 1 , ..., x n ). More precisely, let L : [a, b] × R 3n → R be a continuously differentiable function with respect to the j-th variable, for j = 2, . . . , 3n+1, and consider the functional It follows from the proof of Theorem 3.2 that, if functional J attains an extremum at x = (x 1 , ..., x n ), then, for all t ∈ [a, b] and i = 1, ..., n, If the state values x(a) and x(b) are free, then we get the following 2n natural boundary conditions: for all i = 1, ..., n.
Next, we consider problem (P) subject to an integral constraint of type where k ∈ R is fixed and G : [a, b]×R 3 → R is a continuously differentiable function with respect to the second, third and fourth variables. This type of problems are known in the literature as isoperimetric problems.
Theorem 3.3. (Necessary optimality conditions for isoperimetric problems I) Let x be a curve such that J attains an extremum at x, when subject to the integral constraint (3.6). Assume that x does not satisfies the equation If the maps are continuous on [a, b], then there exists a real number λ such that x is a solution of the equation for all t ∈ [a, b], where H := L + λG. Also, if the state variable x(a) is free, then

9)
and if x(b) is free, then x must satisfy Proof. Suppose that x is an extremizer of functional J subject to the integral constraint (3.6). Let , and define the two functions i and j in the following way Using similar techniques as the ones used in the proof of Theorem 3.2, we get Since h 2 (a) = h 2 (b) = 0, we conclude that Since x does not satisfies equation (3.7), one concludes that there exists t 0 ∈ [a, b] such that, Then, there exists some function h 2 for which ∂ 2 i(0, 0) 0. Also, i(0, 0) = 0 and so, applying the Implicit Function Theorem, we conclude that there exists a continuously differentiable function g defined on an open set U ⊆ R containing 0, such that g(0) = 0 and i( 1 , g( 1 )) = 0, for all 1 ∈ U. Hence, there exists an infinity subfamily of functions x + 1 h 1 + g( 1 )h 2 that satisfies the integral restriction (3.6). From now on we will consider such subfamily of variations. Observe that the vector (0, 0) is an extremizer of j, subject to the constraint i(·, ·) = 0. Since ∇i(0, 0) (0, 0), by the Lagrange Multiplier Rule, there exists a real number λ such that ∇( j + λi)(0, 0) = (0, 0). Hence, ∂ 1 ( j + λi)(0, 0) = 0, and, therefore, Since h 1 is an arbitrary function and considering h 1 (a) = h 1 (b) = 0, it follows from the Fundamental Lemma of Calculus of Variations that are continuous on [a, b], then there exists a vector (λ 0 , λ) ∈ R 2 \ {(0, 0)} such that x is a solution of the equation Proof. The proof is similar to the one of Theorem 3.3. Since the vector (0, 0) is an extremizer of j, subject to the constraint i(·, ·) = 0, the Lagrange Multiplier Rule guarantees the existence of two constants λ 0 , λ ∈ R, not both zero, such that ∇(λ 0 j+λi)(0, 0) = (0, 0). Computing ∂ 1 (λ 0 j+λi)(0, 0) = 0, we obtain the desired result. Now, we consider problem (P) but in presence of an holonomic restriction. Suppose that the state variable x is a two-dimensional vector function x = (x 1 , x 2 ), where x 1 , x 2 ∈ C 1 ([a, b], R). We impose the following restriction to our variational problem: where g : [a, b] × R 2 → R is a continuously differentiable function. Also, boundary conditions may be imposed to the variational problem.
Also, if x(a) is free, then, for i = 1, 2, and if x(b) is free, then, for i = 1, 2, ([a, b], R). To prove Eqs (3.15), first assume that h(a) = (0, 0) = h(b) and let ∈ R. Since the variations must fulfill the holonomic restriction (3.12), then Differentiating (3.18) with respect to and taking = 0, we conclude that Define the function λ : [a, b] → R by the rule From the definition of λ, we prove equation ( Let us define the new function j by the rule j( ) : Since h 1 is arbitrary, from the Fundamental Lemma of Calculus of Variations, we get for all t ∈ [a, b], proving Eq (3.15) for i = 1. We now prove the transversality conditions (3.16) and (3.17). If x(a) is free, then by considering h(a) (0, 0) and h(b) = (0, 0) in (3.22) and using (3.15), (3.16) is proved. If x(b) is free, then consider h(a) = (0, 0) and h(b) (0, 0) to deduce (3.17).

Sufficient optimality conditions
Now we will prove sufficient optimality conditions for all the variational problems studied in the last subsection.
Definition 3.6. We say that f (t, x 2 , x 3 , ..., x n ) is a convex (resp. concave) function in U ⊆ R n if ∂ i f (t, x 2 , x 3 , ..., x n ), i = 2, . . . , n, exist and are continuous, and if Proof. Let η ∈ C 1 ([a, b], R) be an arbitrary function. Since L is convex, we have Applying Theorem 3.1, we obtain x is a solution of the fractional Euler-Lagrange equation (3.2). If x(a) is free, then by considering η(a) 0 and η(b) = 0 in (3.23), we have Proof. First, assume that functions L and λG are convex. It is easy to verify that function H is convex.

Illustrative examples
In this section we provide three examples in order to illustrate our results.
Thus, C D ϕ(α),ψ 1− and therefore x satisfies condition (4.4), the Euler-Lagrange equation (4.5), and the natural boundary condition (4.6). Since the Lagrangian function is convex, by Theorem 3.7, x is indeed a minimizer of J.  (ψ(1) − ψ(t)) 3 − ψ(1) + ψ(t) ln(ψ(1) − ψ(t)) . Let Therefore, x satisfies the Euler-Lagrange equation with respect to the Hamiltonian H: for all t ∈ [0, 1] and the transversality condition Thus, x satisfies the necessary conditions of Theorem 3.3 with λ = −2. Since the Hamiltonian function H is convex, by Theorem 3.8, a solution of equations (4.7) and (4.8) is actually a minimizer of J subject to the previous integral constraint. Hence, is a solution of the proposed problem.

Conclusions and future work
In this work we generalized some of the results presented in [4] and [6], by considering in the Lagrangian functional a new fractional derivative that combines the two ones given in those papers. Namely, we deduced necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions.
For future, we intend to generalize the results presented in this paper, by considering variational problems with higher-order derivatives and delayed arguments. Also, we intend to study variational problems of Herglotz type involving the new distributed-order fractional derivatives with arbitrary kernels introduced in this paper.