Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

: In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative com-bines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufﬁcient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.


Introduction
Fractional calculus is a mathematical area that deals with the generalization of the classical notions of derivative and integral to a noninteger order. This fascinating theory has attracted the interest of the scientific community over the last few decades due to the fact that it is a powerful tool to deal with the dynamics of complex systems. Its importance is notable not only in Mathematics but also in Physics [1], Chemistry [2], Biology [3], Epidemiology [4], Control Theory [5], etc. (for completeness, we also point out that partial differential equations from classical calculus properly fit in the modeling of real problems; see, for instance, Refs. [6][7][8] for models from mathematical biology).
Since the beginning of the fractional calculus in 1695, numerous definitions of fractional integrals and derivatives were introduced by important mathematicians such as Leibniz, Euler, Fourier, Liouville, Riemann, Letnikov, etc. Many of these fractional derivatives can be related between them by an explicit formula [9,10]. Later on, in 1969, Caputo introduced the distributed-order fractional integrals and derivatives [11,12]. These operators can be seen as a new kind of generalization of the classical fractional operators, since these operators involve a weighted integral of different orders of differentiation. Another way that allows a generalization of the classical fractional operators is considering the notions of fractional integrals and derivatives with respect to another function [9,13,14].
The specificity of fractional calculus that can be considered the cause of its success in applications to real world problems is that the large number of fractional operators allows researchers to choose the most suitable one to model the problem under investigation.
In the recent paper [15], the authors introduced new notions of fractional derivatives combining the distributed-order derivatives and fractional derivatives with respect to an arbitrary smooth function, creating a new type of derivatives: distributed-order fractional derivatives with arbitrary kernels. In this paper, we are going to deal with these kinds of generalized fractional derivatives in order to study different types of problems of the calculus of variations.

Preliminaries and Notations
We assume that the reader is familiar with the definitions and properties of the Riemann-Liouville and Caputo fractional operators with respect to another function (cf. [9,13], resp.).
In this paper, we consider variational problems involving the new concepts of distributedorder fractional derivatives with respect to an arbitrary smooth kernel recently introduced in [15]. For the reader's convenience, we recall here the definitions introduced in [15].
Definition 1 ([15]). Let x : [a, b] → R be an integrable function and ψ ∈ C 1 ([a, b], R) be an increasing function such that ψ (t) = 0, for all t ∈ [a, b]. The left and right Riemann-Liouville distributed-order fractional derivatives of a function x with respect to ψ are defined by: 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 ([15]). Let x, ψ ∈ C 1 ([a, b], R) be two functions such that ψ is increasing and ψ (t) = 0, for all t ∈ [a, b]. The left and right Caputo distributed-order fractional derivatives of x with respect to 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. Now, we will extend the definitions introduced in [15] to the higher-order case. In the following, we assume that n ∈ N and φ : [n − 1, n] → [0, 1] is a continuous function such that To the best of our knowledge, this is the first work that deals with higher-order distributed-order fractional derivatives. Definition 3. Let x : [a, b] → R be an integrable function and ψ ∈ C n ([a, b], R) be an increasing function such that ψ (t) = 0, for all t ∈ [a, b]. The left and right Riemann-Liouville distributedorder fractional derivatives of a function x with respect to the kernel ψ are defined by: where D α,ψ a + and D α,ψ b − are the left and right ψ-Riemann-Liouville fractional derivatives of order α ∈ [n − 1, n], respectively. Definition 4. Let x, ψ ∈ C n ([a, b], R) be two functions such that ψ is increasing and ψ (t) = 0, for all t ∈ [a, b]. The left and right Caputo distributed-order fractional derivatives of x with respect to ψ are defined by In the following, we use the notations are, respectively, the left and right Riemann-Liouville fractional integrals of order n − α with respect to the kernel ψ. In addition, we fix two functions φ and ψ satisfying the assumptions above. In order to simplify notation, we will use the abbreviated symbol Next, we prove the integration by parts formulae, which are fundamental tools for the proofs of our main results. In our previous work, we proved a similar result when the fractional order is between 0 and 1 [15] [Theorem 3.1]. In this paper, we present a generalization of such result for the case when function φ is defined on the interval [n − 1, n].
Proof. Using the definition of the left ψ-Caputo distributed-order fractional derivative, Applying Dirichlet's formula, we get Integrating by parts, we have b a d ds y Using integration by parts in the last integral, we obtain b a −1 Repeating the process of integration by parts n − 3 more times, we prove the formula. Using similar techniques, we deduce the integration by parts formula involving the

Variational Problems with Time Delay
We begin this section by studying variational problems involving distributed-order fractional derivatives with time delay. For clarity of presentation, we restrict ourselves to the case where α ∈ [0, 1], that is, considering the definitions introduced in [15].
Consider two continuous functions φ, ϕ : [0, 1] → [0, 1] satisfying the following conditions In what follows, a, b ∈ R are such that a < b and τ is a fixed real number satisfying We are now in position to present the first problem under study.
, R) is a given initial function, that minimizes or maximizes the following functional: where L : [a, b] × R 4 → R is assumed to be continuously differentiable with respect to the second, third, fourth, and fifth variables. We will consider the variational problem (P τ ) with and without fixed terminal boundary condition, and also with isoperimetric or holonomic constraints.
Let us fix the following notations: by ∂ i L, we denote the partial derivative of L with respect to its ith-coordinate and To simplify the presentation of our results, we consider the following conditions: where H is a function and i ∈ N.
Theorem 2 (Fractional Euler-Lagrange equations and natural boundary condition for problem (P τ )). Suppose that L satisfies the conditions is an extremizer of functional J , then x satisfies the following Euler-Lagrange equations and is free, then the following natural boundary condition holds: Replacing (6) into (5) (8) and, for all Using Theorem 1 and (8) Once again, by Theorem 1 and (9), we obtain Replacing (10) and (11) into (7), we get that From the arbitrariness of h, we get the desired Equations (2)-(4).
Next, we consider the case where we add to problem (P τ ) an isoperimetric restriction.
Problem 2 ((P I τ )). The isoperimetric problem with a time delay τ can be formulated in the following way: minimize or maximize the functional J in (1) subject to an integral constraint of type where k ∈ R is fixed and G : [a, b] × R 4 → R is a continuously differentiable function with respect to the second, third, fourth, and fifth variables.
The following theorem presents necessary conditions for x to be a solution of the fractional isoperimetric problem (P I τ ) under the assumption that x is not an extremal for G. Theorem 3 (Necessary optimality conditions for problem (P I τ )-Case I). Let x ∈ C 1 ([a − τ, b], R) be a curve such that J attains an extremum at x, when subject to the integral constraint (13). Assume that x does not satisfy the Euler-Lagrange Equation (2) or (3) with respect to G. Moreover, suppose that L satisfies the conditions C Then, there exists λ ∈ R such that x is a solution of the equations and (15) where H := L + λG.
If x(b) is free, then Proof. The proof follows from the ideas presented in Theorem 2 and Theorem 3.3 of [15]. Now, we present necessary optimality conditions for the case when the solution of the isoperimetric problem is an extremal for the fractional isoperimetric functional (13).
Theorem 4 (Necessary optimality conditions for fractional problem (P I τ )-Case II). Let x be a curve such that J attains an extremum at x, when subject to the integral constraint (13). Moreover, suppose that L satisfies the conditions C Then, there exists a vector (λ 0 , λ) ∈ R 2 \ {(0, 0)} such that x is a solution of Equations (14) and (15), with the Hamiltonian H defined as H := λ 0 L + λG. If x(b) is free, then x must satisfy Equation (16).
Proof. The result is an immediate consequence of Theorem 3.
In the following, we study variational problems with a holonomic constraint. For this purpose, we now assume that x is a two-dimensional vector function and L : [a, b] × R 8 → R is assumed to be continuously differentiable with respect to the ith variable, with i = 2, . . . , 9.
Theorem 5 (Necessary optimality conditions for problem (P C τ )). Consider the functional , R) and subject to the constraints (17) and (18).

Suppose that L satisfies the conditions C
If x is an extremizer of functional J and if then there exists a continuous function λ : and If x(b) is free, then, for i = 1, 2, Proof. The proof follows combining the ideas from Theorem 2 above with Theorem 3.5 from [15]. Now, we focus our attention on sufficient optimality conditions for all the variational problems studied previously.

Proof.
We prove the case when L is convex. The other case is similar. Consider h ∈ C 1 ([a − τ, b], R) an arbitrary function. Since L is convex, we can conclude that Using the same techniques used in the proof of Theorem 2, we get If x(b) is fixed then h(a) = h(b) = 0, and so from (23) we obtain Since x is a solution of the fractional Euler-Lagrange Equations (2) and (3), then we conclude that J (x + h) − J (x) ≥ 0. The case when x(b) is free follows by considering h(t) = 0, t ∈ [a − τ, a] and h(b) non-zero in (23).
Using similar techniques as the ones used in the proof of the last theorem, we can prove the following two results.
Theorem 7 (Sufficient optimality conditions for problem (P I τ )). Let us assume that, for some constant λ, the functions L and λG are convex (resp. concave) in [a, b] × R 4 and define the function H as H = L + λG. Then, each solution x of the fractional Equations (14) and (15) minimizes (resp. maximizes) the functional J given in (1), subject to the restrictions x(t) = µ(t), t ∈ [a − τ, a] and x(b) = x(b), and the integral constraint (13). If x(b) is free, then each solution x of the fractional Equations (14)-(16) minimizes (resp. maximizes) J subject to (13).

Higher-Order Variational Problems
In this section, we consider the general case with respect to fractional orders. Thus, The problem is formulated as follows: Problem 4 ((P n )). Find a curve x ∈ C n ([a, b], R) for which the functional attains a minimum or a maximum value, where L : [a, b] × R 2n+1 → R is a continuously differentiable function. In addition, the following boundary conditions may be assumed.
We will consider the variational problem (P n ) with and without fixed boundary conditions (25), and also with isoperimetric or holonomic constraints.
As done previously, we use the abbreviations and where H is a function and i, j ∈ N.
When in the presence of an isoperimetric or holonomic contraint, similar results are proven for this new variational problem. To simplify, we will assume that the boundary conditions (25) hold. In addition, the proofs will be omitted since they follow the same pattern as the ones presented before.
Problem 5 ( (P I n )). The isoperimetric problem can be formulated as follows: minimize or maximize the functional J in (24) assuming the boundary conditions (25) and also an integral restriction where G : [a, b] × R 2n+1 → R is a C 1 function.
Theorem 10 (Necessary optimality conditions for problem (P I n )-Case I). Let x ∈ C n ([a, b], R) be a solution of problem (P I n ). Suppose that there exists some t ∈ [a, b] such that hold, for all i ∈ {1, ..., n}, 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.
Theorem 11 (Necessary optimality conditions for problem (P I n )-Case II). Let x ∈ C n ([a, b], R) be a solution of problem (P I n ). To finish this section, we will study problem (P n ) with a holonomic constraint. Problem 6 ((P C n )). The objective is to find x ∈ C n ([a, b], R) × C n ([a, b], R) that minimizes or maximizes the functional defined on C n ([a, b], R) × C n ([a, b], R) and subject a constraint where g : [a, b] × R 2 → R is a C n function. In addition, boundary conditions are imposed on the variational problem.
Theorem 12 (Necessary optimality conditions for problem (P C n )). Let x be an extremizer of functional J defined by (32) and subject to the constraints (33)-(34).

Remark 1.
In a similar way, we can prove that, in case function L is convex (resp. concave), then the conditions given in Theorems 9-12 are also sufficient conditions to ensure that the candidates of extremizers are indeed minimizers (resp. maximizers) of the functional.

Illustrative Examples
Some illustrative examples are provided to demonstrate the applicability of our results.
Note that x satisfies the assumptions of Theorem 2 and also the Euler-Lagrange Equations (2) and (3), as well as the transversality condition (4), proving that x is a candidate to be a local minimizer of J . Since the Lagrangian function is convex, we conclude by Theorem 6 that x is a minimizer of J .  0)) .

Example 3. Determine x that minimizes the functional
In addition, observe that and therefore C D We can easily verify that x satisfies assumptions of Theorem 9, the Euler-Lagrange Equation (26), and the natural boundary condition (28), proving that x is a candidate to be a local minimizer of J . Since the Lagrangian function is convex, we conclude that x is a minimizer of J .

Conclusions and Future Work
In this article, we continue the study started in [15], considering now new problems in the calculus of variations. Namely, two distinct types are considered: when the Lagrangian function involves a time delay and derivatives of order greater than 1. Necessary and sufficient optimization conditions are proved, for the basic problem and when in the presence of additional constraints to the problem. The study is formulated in the context of fractional calculus, where the derivative of the state curve is of the fractional type involving distributed-orders and the kernel involves an arbitrary smooth function.
In the future, we intend to study variational problems of Herglotz type and some generalizations involving distributed-order fractional derivatives with arbitrary smooth kernels.