1. Introduction and Basic Concepts
Fractional calculus is a generalization of classical calculus that focuses on the integration and differentiation of non-integer orders. The concept of fractional operators was introduced almost concurrently with classical operators, highlighted by an exchange of letters between Leibniz and l’Hôpital, in which they questioned the meaning of a derivative of order
. This concept captured the attention of several prominent mathematicians, including Liouville and Riemann. Since then, numerous studies have emerged, not only in the domain of pure mathematics but also across various applied sciences. Indeed, fractional calculus has found applications in diverse fields, such as the rheological behavior of viscoelastic materials [
1,
2], acoustic wave equations [
3,
4], control theory [
5,
6], physics [
7,
8], signal processing [
9,
10], and robotics [
11,
12]. One key reason for its success is its ability to more accurately model real-world phenomena by incorporating memory effects and external features inherent to the dynamics of the systems. The flexibility afforded by fractional derivatives, which can take any real value, allows for a more efficient representation of complex phenomena.
Another field where fractional calculus has proven highly applicable is in the calculus of variations (see, for example, the seminal works in [
13,
14,
15]). The goal is to find the curves that minimize a given functional, such that in a neighborhood of these curves, the functional’s value is lower than that evaluated on other nearby curves. In this context, functionals are modeled not by the first-order derivative of a path, but by fractional derivatives. This approach was first explored in the work of Riewe [
16,
17], who observed that “traditional Lagrangian and Hamiltonian mechanics cannot be used with nonconservative forces, such as friction”. Following this crucial insight, extensive research on the fractional calculus of variations has been conducted. For example, we can refer the works [
18,
19,
20,
21].
Due to the vast variety of fractional operators, the need for a more general formulation of fractional derivatives led to the pioneering work of Osler in the 1970s. These fractional operators are defined through a chosen kernel, allowing the recovery of various well-known fractional operators as special cases. This approach provides a more unified theoretical framework and expands the possibilities for defining and analyzing fractional operators. Significant contributions in this area include results on the Leibniz and chain rules, as well as extensions to Taylor and Laurent series [
22,
23,
24]. More recently, the corresponding Caputo-type operator was introduced in [
25]. This new fractional derivative incorporates a kernel function, allowing for the recovery of several well-known fractional derivatives as special cases when specific kernel functions are chosen. This generalized operator has proven to be highly effective in modeling real-world problems, offering improvements over classical models [
26,
27]. Subsequently, in [
28], the corresponding variational problems associated with this fractional derivative were investigated. The study established several necessary optimality conditions for different classes of problems, further expanding the applicability and theoretical foundation of the operator. Our goal is to extend previous work to the case where the functional to be optimized is a composition of functionals. This topic was first introduced in [
29], with applications in economics. Later, it was generalized within the framework of time scales [
30,
31]. We also highlight the contribution of [
32], which explored this concept, including applications to Noether’s theorem.
Throughout this work, we consider
as a positive real number (the fractional order) and
as a smooth function with a positive derivative (the fractional kernel).
The (left) Caputo fractional derivative of a smooth function
is defined as follows (see [
25]):
where
if
, and
otherwise.
From this definition, we observe that, in contrast to classical derivatives, which are local in nature, the evaluation of a fractional derivative at a specific time t is influenced by the entire history of the process, incorporating all previous memory.
Additionally, we require the concepts of right fractional operators in the Riemann–Liouville sense, including the fractional integral [
33]
and the fractional derivative
where
.
We can recognize that, for specific choices of the kernel
, we obtain some of the most important fractional operators. For instance, if
, we recover the usual Riemann–Liouville and Caputo fractional derivatives; if
, we obtain those of the Hadamard type; and if
, we arrive at the Erdélyi–Kober operator. Thus, these fractional operators not only generalize well-known cases but also have the potential to define new ones by appropriately selecting the kernel.
We would also like to point out that when the order of the fractional derivative is an integer number, it reduces to a standard derivative. In fact, if
, then the fractional derivative is given by
and, in particular, for
, we obtain
.
Remark 1. We emphasize that the choice of the kernel function must adhere to certain mathematical properties and physical interpretations to ensure that the resulting operator can be classified as a fractional derivative. Specifically, the corresponding fractional differential equation should not be reducible to one with a finite number of integer-order derivatives, it should preserve the principle of nonlocality [34], and it must satisfy the criteria outlined in [35]. This paper is structured as follows.
Section 2 introduces the main problem and derives the first necessary optimality condition, the Euler–Lagrange equation. Additionally, we investigate the determination of the optimal fractional order that minimizes the functional.
Section 3 examines variational problems with additional constraints, including isoperimetric (that is, when there is an integral constraint in the space of admissible functions) and holonomic restrictions (that is, when the constraint is expressed as an algebraic equation involving the variables). In
Section 4, we explore cases where the state function incorporates a time delay, so that the functional depends on the state function at time
t, as well as at a previous time
. Finally,
Section 5 extends the analysis to scenarios where the fractional order can assume any positive real value, extending the previous studies that were limited to fractional orders between 0 and 1. We note that connecting two distinct topics is feasible using the techniques presented in each section of this work, though it increases the overall complexity.
2. The Euler–Lagrange Equation
We assume that the order of the derivative
lies between 0 and 1 and that the space of functions considered is
. For each
, let
be a function admitting continuous partial derivatives with respect to its second and third variables, denoted by
and
, respectively. Furthermore, let
be a function with continuous partial derivatives, where we denote
for each
.
The functional under consideration is defined as
with domain
.
Boundary conditions may be imposed on the problem:
In this case, a curve
z is said to be admissible if it satisfies the boundary conditions (
2).
We define the norm
and say that a curve
(locally) minimizes the functional
F in (
1) if there exists
such that
for all admissible functions
satisfying
.
To simplify the notation, given a curve
y, we denote by
the vector
The main result is presented below and is known in the literature as the fractional Euler–Lagrange equation.
Theorem 1. Let
be a curve that minimizes the functional F in (
1)
. Then, this curve satisfies the following fractional differential equation: for all
. Moreover, if
is unrestricted, the following transversality condition holds:at
. Conversely, if
is arbitrary, then Equation (
4)
is satisfied at
. Proof. To prove the necessary conditions presented, we consider variations of the optimal curve in the form
, where
and
is a real number. Defining the real-valued function
and noting that
minimizes
F, we conclude that
is a minimizer of
f. Consequently, it follows that
, that is,
Using the integration by parts formula (see [
25], Theorem 12), we obtain
Initially, assuming that the variations satisfy the conditions
and
, and using the arbitrariness of the curve
y in the interval
, we obtain condition (
3). Substituting into (
5), we obtain
Now, assuming first that
and
, and then the reverse case, we obtain the two transversality conditions as presented in Equation (
4). □
Remark 2. We recall that the kernel function is assumed to be smooth with a strictly positive derivative, ensuring that Equations (
3)
and (
4)
are well defined throughout the entire interval
. Definition 1. A curve
that satisfies Equation (
3)
is said to be an extremal with respect to the functional F in (
1)
. Remark 3. Theorem 1 can be extended to higher-dimensional functions. If
, then we obtain m necessary conditions, each analogous to those in the previous theorem. These conditions are derived by considering the partial derivatives with respect to each component function
for
.
Next, two particular cases of Theorem 1 are presented, where the function to be minimized is either the product or the quotient of two functionals.
Corollary 1. Suppose that
minimizes the functionalThen, the Euler–Lagrange equation for this functional is given byfor all
. Additionally, if either
or
is free, then the transversality condition is given byat
or
, respectively. Corollary 2. Suppose that
minimizes the functionalThen, the Euler–Lagrange equation for this functional is given byfor all
. Additionally, if either
or
is free, then the transversality condition is given byat
or
, respectively. Remark 4. Under certain conditions, we can demonstrate that the necessary conditions presented in Theorem 1 are, in fact, sufficient to ensure that the curve
is a minimizer of the functional. Indeed, let
be a curve satisfying conditions (
3)
and (
4)
, and assume that for all
, one of the following two conditions holds: and the Lagrangian function
is convex with respect to its second and third variables.
and the Lagrangian function
is concave with respect to its second and third variables.
If y is another function in a neighborhood of
, then, by the Mean Value Theorem, we obtainfor some vector
between
and
. Consider a neighborhood of
such that, for all
, we have
(or
) whenever
(or
). Then, using the given assumptions, we obtainthus proving the desired result. A fundamental challenge in fractional variational calculus is the identification of the optimal order of differentiation that minimizes a given functional. Since fractional derivatives incorporate an additional parameter,
, the problem extends beyond finding the minimizer function
. Specifically, we seek to determine both the optimal function
and the corresponding fractional order
that yields the minimal functional value.
Theorem 2. Let
be a solution to the following variational problem: minimize the functionalover the set
. Define the vectorand introduce the function
, where for each
,
. Then, the following four necessary optimality conditions are derived: Generalized Euler–Lagrange equation: for all
, Transversality condition: if
or
is free, thenat
or
, respectively. Condition for the variation of the fractional order
: the fractional parameter
satisfies
Proof. The variation of the optimal solution now depends on two coordinates
, where
is a real number. Considering the perturbed functional
and using the necessary optimality condition
, we obtain
Applying the integration by parts formula to the fractional derivative term, we obtain
Due to the arbitrariness of the parameter
and the function
y on
, we derive the four necessary optimality conditions. □
3. Necessary Conditions Under Constraints
In this section, we will study necessary optimization conditions, but the admissible functions for the problem must satisfy some additional conditions. Two types of problems will be considered: the first involves an integral constraint (isoperimetric problem), and the second involves a predefined relationship between time and spatial coordinates (holonomic constraint). Similarly to what was carried out in the previous section, we will consider a new functional, and for this purpose, some new terms are now introduced.
For each
, let
be a function with continuous partial derivatives with respect to its second and third variables. Additionally, let
be a continuously differentiable function.
The functional of interest is given by
For simplicity, we assume that some boundary conditions are satisfied. If they are not, transversality conditions can be derived in a manner analogous to Theorem 1.
The isoperimetric problem (IP) is formulated as follows: find a curve
y that minimizes the functional
F given by (
1), subject to the boundary conditions (
2) and the integral constraint
, where
is a fixed real number.
Theorem 3. Let
be a solution to the isoperimetric problem (IP). Then, there exists a nonzero vector
such thatfor all
. Proof. The proof will be divided into two distinct cases.
In the first case, we assume that
is an extremal curve for the functional
G. In this scenario, Equation (
6) is immediately satisfied by choosing
.
Now, suppose that
y is not an extremal curve for
G. Consider a variation of the optimal curve depending on two parameters, given by
where
satisfy
and
, and
are real numbers. Define the function
Since
and
, it follows that
Since
is not an extremal curve with respect to the functional G, we conclude that there must exist a function
such that
. Furthermore, since
, we can ensure the existence of a function
, defined in a neighborhood of zero, such that
. In other words, there exists an infinite subfamily of variations that satisfy the integral constraint of the isoperimetric problem (IP).
We can convert the variational problem into a finite-dimensional optimization problem. Indeed,
is a solution of the following problem: minimizing f subject to the constraint
.
As previously proven,
. By the method of Lagrange multipliers, there exists a scalar
such that
The result follows from the arbitrariness of the function
and by choosing
. □
Next, we present what is known as a holonomic problem (HP). These problems in the calculus of variations refer to optimization problems in which the admissible functions are constrained by an algebraic relation between the independent and dependent variables.
In what follows,
is a function admitting continuous partial derivatives
for
and
. The functional depends on two functions
and is defined by
The holonomic problem (HP) is defined as follows: minimize
F subject to the boundary constraints
and
, with
for
, and the condition
where
is a function admitting continuous partial derivatives with respect to its second and third variables.
Theorem 4. Let
be a solution to problem (HP) that satisfies the conditionsThen, there exists a function
such that, for
and for all
, the following condition holds: Proof. Let
be the solution of Equation (
9) for
. It remains to prove that the equation also holds for
.
Consider a variation of the solution curve given by
, where
and
. Due to the boundary conditions, we assume that
and
for
.
These variations must satisfy the holonomic constraint, i.e.,
Differentiating with respect to
at
, we obtain
Next, we define the function
Since
, we obtain
Given that
it follows that
From the arbitrariness of the function
, we obtain the desired result for
. □
Remark 5. We note that condition (
8)
may be trivially satisfied. For example, when
,
, and
, this condition holds for any curve. 5. Necessary Conditions with Higher-Order Derivatives
Until now, we have considered only problems where the fractional order
is between 0 and 1. In this final part, we will consider functionals depending on derivatives of any real order. To this end, we consider a sequence
, where
and
for all
.
The Lagrangian function
, for
, is such that the partial derivatives
exist and are continuous for all
. To simplify notation, we introduce the following:
The functional to be minimized is defined as
on the set
. Here,
denotes the vector
The necessary optimality condition is presented below.
Theorem 6. Let
be a minimizing curve of the functional F. Then, for all
, Furthermore, for
, if
is arbitrary at
or
, thenat
or
, respectively. Proof. Let us consider the variation
, where
. Since the first variation of the functional must vanish when evaluated at
, we obtain
We now apply fractional integration by parts to each of the last
m terms in this integral. For the first of these terms, we have
For the second term, we obtain
Repeating this procedure up to the last term of the integral, we obtain
Substituting these equalities into the first variation of the functional, we obtain
By factoring out the terms
, we obtain the following expression:
The result follows again from the arbitrariness of the function
y. Indeed, assuming that
,
, and, for all
,
at
and
, we obtain the Euler–Lagrange equation. To derive the transversality condition for
, it is sufficient to assume that
at
or
, while the others are all zero. □
Remark 7. If, in the formulation of the problem with higher-order derivatives, we assume that in the space of admissible functions, the boundary conditions satisfy
, being fixed at
and
for
, then in Theorem 6, we would obtain only the Euler–Lagrange equation.
Example 2. Consider
,
, fractional orders
and
, and the cost functionalThe Euler–Lagrange equation associated with this problem is given by