A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter

: In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufﬁcient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results. MSC: 26A33, 49K05, 34A08


Introduction
Non-integer calculus, known as fractional calculus, deals with integrals and derivatives with arbitrary real or complex orders [1,2]. It has developed in the past decades, becoming an important tool in applied sciences and engineering. Nowadays, fractional calculus is an important subject, e.g., in physics [3,4], robot trajectory controllers [5], heat diffusion [6], signal and image processing [7], or biology [8,9].
A question that arises when dealing with fractional calculus is which fractional integral or derivative should we choose? There are several definitions proposed, such as Riemann-Liouville, Caputo, Hadamard, Erdélyi-Kober, Grünwald-Letnikov, Weyl, or Marchaud fractional operators. However, there are ways to overcome this issue, considering a more general class of operators. In [2], we find the concept of fractional derivative with respect to another function. For particular choices of such function, we obtain some of the previous ones. We denote the fractional order by α ∈ R + , and let ψ ∈ C 1 ([a, b], R) be a function with ψ (t) > 0, for all t ∈ [a, b]. Given an integrable function x : [a, b] → R, the left-sided and the right-sided Riemann-Liouville fractional integrals of x with kernel ψ are defined as where n = [α] + 1. For simplicity, we call the last operators as ψ-Riemann-Liouville fractional derivatives of x of order α. It can be easily noticed that for certain choices of function ψ, we recover some important fractional derivatives. We also remark that when α = m ∈ N, we have It is worth mentioning that, opposite to the ordinary derivatives, fractional derivatives are non-local and, in the case of left-sided derivatives, take into account the past. This is particularly useful for problems in different areas, such as economics, epidemiology, and optimal control problems [10][11][12][13].
Recently, in [14], motivated by the concept of Caputo fractional derivative and by these generalized fractional operators, the following definition was presented. Let α > 0 and n ∈ N be defined 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], the left-and right-sided Caputo fractional derivatives of x with kernel ψ (or simply, ψ-Caputo fractional derivatives of x), are defined as  For α ∈ R + \ N, then We now present the following formulas (see Lemma 1 in [14]) that are useful in Section 3. If n < β ∈ R, then Next, we present the following fractional integration by parts formulas that are fundamental for the proofs of our results. For a more detailed study of the ψ-Caputo fractional derivatives, we refer to [14].
In particular, when 0 < α < 1, Fractional calculus of variations started with the pioneering works of Riewe [15,16]. Since then, numerous works have appeared for different types of fractional derivatives and integrals. To mention a few in such vast literature, we can refer the reader to the books [17][18][19]. The goal is to extremize (minimize or maximize) a given functional, depending on some fractional operator. Due to the large number of fractional operators to choose from, we found several works dealing with similar subjects (e.g., [20][21][22][23][24][25][26]). By considering a more general form of fractional derivative, such as the one given in [14], we can study different problems in a general form. In [27], some calculus of variation problems were addressed, with dependence on this fractional derivative. Necessary and sufficient conditions of optimality were proven, such as the Euler-Lagrange equation, and the isoperimetric problem was studied, among others.
The main goal of this paper is to generalize the fractional variational problem studied in [27], considering the case where the Lagrangian depends not only on the independent variable, an unknown function and its left-and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoints conditions and a free parameter. This type of generalized fractional variational problems cannot be solved using the classical theory. Our motivation for studying generalized variational problems where the Lagrangian explicitly depends on state values and a free parameter comes from interesting applications in economics [28] and in physics [29], respectively. It is worth mentioning that, since these types of fractional derivatives are generalizations of several fractional derivatives and our variational problem is a generalization of different types of fractional variational problems, many results available in the literature are corollaries of the results proven in this paper.
The organization of the paper is as follows. We start Section 2.1 considering the generalized fractional variational problem with fixed boundary conditions and proving a necessary optimality condition of Euler-Lagrange type and also a necessary condition which arises as a consequence of the Lagrangian dependence of the parameter. Then, we prove the natural boundary conditions for variational problems with free boundary conditions. In Section 2.2, we prove necessary optimality conditions for variational problems with integral constraints, with and without fixed boundary conditions. The variational problem with an holonomic constraint is studied in Section 2.3. In Section 2.4, we prove sufficient optimality conditions for the variational problems considered in the previous subsections. We conclude the paper with some illustrative examples and concluding remarks.

Main Results
In this work, we consider a functional depending on time, on the state function x, its fractional derivatives C D α,ψ a + x and C D β,ψ b − x of orders α, β ∈]0, 1[, the values x(a) and x(b), and a free parameter ζ. More specifically, we will study the following generalized fractional variational problem.
We will consider problem (P ) with fixed boundary conditions for some x a , x b ∈ R, and when x(a) and x(b) are free. We will also consider problem (P ) subject to an isoperimetric constraint for some γ ∈ R, and to an holonomic constraint for a given function g.
If the function ψ = Id, and the Lagrange function does not depend on a free parameter ζ, then we get the fractional variational problem studied in [30]; 2.
Taking α → 1 − , and if ψ is the identity, the operators C D α a + and C D α b − can be replaced by d dt and − d dt , respectively (see [1]). Hence, if α and β goes to 1, our functional J tends to the generalized variational functional Next, we proceed with some basic definitions that are useful in what follows.
Definition 2. We say that (x , ζ ) is a local minimizer (resp. local maximizer) for the functional J if there exists some δ > 0, such that, for all (x, ζ) is a global minimizer (resp. global maximizer). In these cases, we say that (x , ζ ) is a global extremizer of J .

Generalized Fractional Variational Principle
The following result provides necessary conditions for an admissible pair (x, ζ) to be a local extremizer of functional J , where x satisfies the boundary conditions (2). The equation is called the Euler-Lagrange equation. We will represent it by ELe{L[x, ζ]}. To simplify, consider the two following conditions: where H is a function and i a positive integer.
Proof. Let (x, ζ) be a local extremizer for functional J subject to (2), η an admissible variation and δ an arbitrary real number. Define the new function φ : Using the fractional integration by parts formulas stated in Theorem 1, we get Taking δ = 0 and using the arbitrariness of η, by Lemma 2.2.2 in [32], we obtain (4).

Remark 2.
In Theorem 2, since the state values x(a) and x(b) are fixed, the Lagrangian's explicit dependence on x(a) and x(b) is irrelevant. However, in Theorem 3, since the state values can be free, this dependency is effective.
We remark that, although the functional J depends only on ψ-Caputo fractional derivatives, the Euler-Lagrange equation involves ψ-Riemann-Liouville fractional derivatives. Using the relations (see Theorem 3 in [14]) it is possible to write Equation (4) using only ψ-Caputo fractional derivatives.
We now consider the case when the values x(a) and x(b) are not necessarily specified. For each boundary condition missing, there is a corresponding natural boundary condition, as given by Theorem 3. 3]. If (x, ζ) is a local extremizer of functional J , then (4) and (5) hold. Moreover,
, R) and δ be an arbitrary real number. Let φ( ) = J (x + η, ζ + δ). Since no boundary conditions are imposed, η do not need to be null at the endpoints. However, since Equation (6) must be satisfied for all η, it is also satisfied for those functions that vanish at the endpoints.
Using the same arguments used in the proof of Theorem 2, one can conclude that (x, ζ) satisfies the necessary conditions (4) and (5).

1.
Suppose that x(a) is free. Restricting η to be null at t = b, and substituting the necessary conditions (4) and (5) into (6) it follows that

2.
Suppose now that x(b) is not fixed. Restricting η to be null at t = a, and using similar arguments as previously, we get Equation (8).
From Theorem 3, we can obtain the following corollaries. Note that if L does not depend on the parameter ζ, then condition (5) is trivially satisfied and we get the following results. Moreover,

1.
If x(a) is free, then x satisfies the following condition If the Lagrangian function does not depend on the state values x(a) and x(b), and on a real parameter ζ, then we get the following result.
If x(a) is free, then x satisfies the following condition Remark 3. Note that The comparision of the natural boundary conditions (9) and (10) with (11) and (12) shows that the fractional problems of the calculus of variations, where the functional to extremize explicitly depends on x(a) and/or x(b), are of a different nature when compared with the case where the Lagrangian does not depend on the endpoint conditions.

Generalized Fractional Isoperimetric Problems
In this section, we deal with variational problems with integral constraints. Besides some possible boundary conditions, we impose on the set of admissible functions an integral restriction of type (3) (see, e.g., [34]). Such kinds of problems are known in the literature as isoperimetric problems. An example of this type of problem is Queen Dido's problem, probably the oldest problem in the calculus of variations, which consists of finding, among all the closed curves of the plane of a given perimeter, the curve that encloses the maximum area (see, e.g., [32]).
Before presenting necessary optimality conditions for such kind of variational problems, we first present the following definition.

Definition 4.
We say that (x, ζ) ∈ C 1 ([a, b], R) × R is a normal extremizer of the isoperimetric problem (1) and (3) if (x, ζ) is a local extremizer of functional J and not an extremal of functional I; if (x, ζ) ∈ C 1 ([a, b], R) × R is a local extremizer of functional J and an extremal of functional I, we say that (x, ζ) is an abnormal extremizer.
In the next two results, we prove necessary optimality conditions for generalized fractional isoperimetric problems, with and without fixed boundary conditions, respectively, for the particular case of normal extremizers.

If x(b) is not fixed then
Proof. The idea of the proof is to combine the methods presented in the proofs of Theorem 3 and Theorem 4.

Generalized Fractional Holonomic Constrained Problems
We now turn our attention to what is called in the literature as holonomic constrained problems. Suppose that the state variable x is a two-dimensional vector x = (x 1 , x 2 ). Thus, functional J is defined as

Boundary conditions are
x(a) = x a and x(b) = x b (20) for some x a , x b ∈ R 2 and the holonomic constraint is where g is a given C 1 function.

Sufficient Optimality Conditions
Now, we focus on sufficient conditions that guarantee the existence of extremizers of functional J . We consider fractional variational problems with or without an isoperimetric and holonomic restrictions. Our results are presented in the general case where the state values are not fixed.
Proof. We shall give the proof only for the case where L is jointly convex; the other case is similar. Let η ∈ C 1 ([a, b], R) and δ ∈ R be arbitrary. Since L is jointly convex in [a, b] × R 6 , we get Introducing (4) and (5) and (7) and (8) into the last inequality, we conclude that proving the desired result.
A similar result can be proved for isoperimetric problems.
Proof. We shall give the proof only for the case where L and λF are jointly convex; the proof of the other case is analogous. Since M is jointly convex then, by Theorem 9, (x, ζ) is a global minimizer of functional M defined by Hence, for any η ∈ C 1 ([a, b], R) and δ ∈ R, one has and, therefore, If we restrict to the integral constraint (3), we can conclude that proving the desired result.
Finally, a sufficient condition of optimality is proven when in the presence of an holonomic constraint. Theorem 11. Suppose L is jointly convex (resp. jointly concave) in [a, b] × R 11 . Let function λ be defined by (28), where g is a C 1 function, such that ∂ 3 g{x, ζ}(t) = 0, for all t ∈ [a, b]. If (x, ζ) satisfies the necessary conditions (23) and (24) and the natural boundary conditions (31) and (32), then (x, ζ) is a global minimizer (resp. global maximizer) of functional J as in (19), subject to the holonomic constraint (21).
Proof. Again, we only consider the case when L is jointly convex. In such situation, From Equations (26) and (28), we get, for all t ∈ [a, b], Replacing the last formula into (34) and using conditions (23) and (24) and (31) and (32), we prove the desired result.

Examples
In this section we present three examples in order to illustrate some results developed in the previous section. In all the examples, we suppose that functions L and ψ satisfy the needed assumptions. in the class of functions C 1 ([0, 1], R), subject to the restriction x(0) = 0 (x(1) is free). From Theorem 3, every local extremizer of functional J satisfies the following necessary conditions: Since the Lagrangian function is jointly convex, by Theorem 9, the solution of this system is actually a minimizer of J . Observe that, when α → 1 and ψ(t) = t, t ∈ [0, 1], our problem tends toJ with x(0) = 0, and the necessary conditions are 1.
we can conclude that c = 9 4 . Hence, is the only candidate to be a local extremizer of functionalJ , and, sinceJ is jointly convex, (x,ζ) is the global minimizer. Solving the fractional problem analytically is very difficult, and thus a numerical technique is applied. In Figure 1, we show the results. Four different fractional orders are considered, and, as can be observed, the solution converges to (x, 1) as α goes to one.

Conclusions and Future Work
In this work, we proved necessary and sufficient conditions of optimality, where the Lagrangian function depends on a general form of fractional derivative, a free parameter, and the state values. The Euler-Lagrange equation was deduced, for the fundamental problem, as well when in presence of constraints. With some examples, we show the applicability of the procedure.
For future, direct methods can be studied to deal with such generalized fractional variational problems. One possible direction is to study discretizations of the fractional derivative and then convert the problem as a finite dimensional case. In addition, other optimization conditions could be obtained, e.g., with arbitrary fractional orders α, β ∈ R + , or optimal control problems where the state equation involves the ψ-Caputo fractional derivative.