Fractional Calculus of Variations for Composed Functionals with Generalized Derivatives

Abstract

This paper extends the fractional calculus of variations to include generalized fractional derivatives with dependence on a given kernel, encompassing a wide range of fractional operators. We focus on variational problems involving the composition of functionals, deriving the Euler–Lagrange equation for this generalized case and providing optimality conditions for extremal curves. We explore problems with integral and holonomic constraints and consider higher-order derivatives, where the fractional orders are free.

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 $1/2$. 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 $\alpha$ as a positive real number (the fractional order) and $\mathcal{K}:[a,b]\to \mathbb{R}$ as a smooth function with a positive derivative (the fractional kernel).

The (left) Caputo fractional derivative of a smooth function $y:[a,b]\to \mathbb{R}$ is defined as follows (see [25]):

$${}^C\mathbb{D}_{+}^{\alpha }y\left(t\right):={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_a^t{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(t\right)-\mathcal{K}\left(\tau \right))}^{n-\alpha -1}{\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(\tau \right)}}{\displaystyle \frac{d}{d\tau }}\right)}^{n}y\left(\tau \right)d\tau ,$$

where $n=\lfloor \alpha \rfloor +1$ if $\alpha \notin \mathbb{N}$, and $n=\alpha$ 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]

$${\mathbb{I}}_{-}^{\alpha }y\left(t\right):={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_t^b{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(\tau \right)-\mathcal{K}\left(t\right))}^{\alpha -1}y\left(\tau \right)d\tau ,$$

and the fractional derivative

$${\mathbb{D}}_{-}^{\alpha }y\left(t\right):={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\left(-{\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{\int }_t^b{\mathcal{K}}^{\prime }\left(\tau \right){(\mathcal{K}\left(\tau \right)-\mathcal{K}\left(t\right))}^{n-\alpha -1}y\left(\tau \right)d\tau ,$$

where $n=\lfloor \alpha \rfloor +1$.

We can recognize that, for specific choices of the kernel $\mathcal{K}$, we obtain some of the most important fractional operators. For instance, if $\mathcal{K}\left(t\right)=t$, we recover the usual Riemann–Liouville and Caputo fractional derivatives; if $\mathcal{K}\left(t\right)=ln\left(t\right)$, we obtain those of the Hadamard type; and if $\mathcal{K}\left(t\right)=t^{\sigma }$, 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 $\alpha =m\in \mathbb{N}$, then the fractional derivative is given by

$${}^C\mathbb{D}_{+}^{\alpha }y\left(t\right)={\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{m}y\left(t\right)$$

and, in particular, for $\mathcal{K}\left(t\right)=t$, we obtain ${}^C\mathbb{D}_{+}^{\alpha }y\left(t\right)=y^{\left(m\right)}\left(t\right)$.

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 $t-t_0$. 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 $\alpha$ lies between 0 and 1 and that the space of functions considered is $C^1[a,b]$. For each $k\in \{1,\dots ,n\}$, let $L_k:[a,b]\times {\mathbb{R}}^2\to \mathbb{R}$ be a function admitting continuous partial derivatives with respect to its second and third variables, denoted by ${\partial }_2{L}_k$ and ${\partial }_3{L}_k$, respectively. Furthermore, let $C:{\mathbb{R}}^n\to \mathbb{R}$ be a function with continuous partial derivatives, where we denote ${\partial }_{k}C$ for each $k\in \{1,\dots ,n\}$.

The functional under consideration is defined as

$$F\left(y\right):=C\left({\int }_a^b{L}_1(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt,\dots ,{\int }_a^b{L}_n(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt\right),$$

(1)

with domain $D_F=C^1[a,b]$.

Boundary conditions may be imposed on the problem:

$$y\left(a\right)=y_a,y\left(b\right)=y_b,y_a,y_b\in \mathbb{R}.$$

(2)

In this case, a curve z is said to be admissible if it satisfies the boundary conditions (2).

We define the norm

$$\parallel y\parallel :=\underset{t\in [a,b]}{max}\left|y\left(t\right)\right|+\underset{t\in [a,b]}{max}\left|{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right)\right|$$

and say that a curve $\overline{y}$ (locally) minimizes the functional F in (1) if there exists $r>0$ such that

$$F\left(\overline{y}\right)\le F\left(y\right),$$

for all admissible functions $y\in D_F$ satisfying $\parallel y-\overline{y}\parallel <r$.

To simplify the notation, given a curve y, we denote by $\left\{y\right\}$ the vector

$$\left\{y\right\}:=\left({\int }_a^b{L}_1(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt,\dots ,{\int }_a^b{L}_n(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt\right).$$

The main result is presented below and is known in the literature as the fractional Euler–Lagrange equation.

Theorem 1.

Let $\overline{y}$ be a curve that minimizes the functional F in (1). Then, this curve satisfies the following fractional differential equation:

$$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}\left({\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)=0,$$

(3)

for all $t\in [a,b]$.

Moreover, if $y\left(a\right)$ is unrestricted, the following transversality condition holds:

$$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)=0,$$

(4)

at $t=a$. Conversely, if $y\left(b\right)$ is arbitrary, then Equation (4) is satisfied at $t=b$.

Proof. 

To prove the necessary conditions presented, we consider variations of the optimal curve in the form $\overline{y}+\eta y$, where $y\in C^1[a,b]$ and $\eta$ is a real number. Defining the real-valued function

$$f\left(\eta \right):=F(\overline{y}+\eta y),$$

and noting that $\overline{y}$ minimizes F, we conclude that $\eta =0$ is a minimizer of f. Consequently, it follows that $f^{\prime }\left(0\right)=0$, that is,

$$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))y\left(t\right)+{\partial }_3{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right)){}^C\mathbb{D}_{+}^{\alpha }y\left(t\right)dt=0.$$

Using the integration by parts formula (see [25], Theorem 12), we obtain

$$\begin{array}{c}\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b\left({\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)y\left(t\right)dt\hfill \\ \hfill +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)y\left(t\right)\right]}_{t=a}^{t=b}=0.\end{array}$$

(5)

Initially, assuming that the variations satisfy the conditions $y\left(a\right)=0$ and $y\left(b\right)=0$, and using the arbitrariness of the curve y in the interval $(a,b)$, we obtain condition (3). Substituting into (5), we obtain

$$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)y\left(t\right)\right]}_{t=a}^{t=b}=0.$$

Now, assuming first that $y\left(a\right)=0$ and $y\left(b\right)\ne 0$, 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 $[a,b]$.

Definition 1.

A curve $\overline{y}$ 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 $y=(y_1,\dots ,y_m)$, 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 $y_i$ for $i\in \{1,\dots ,m\}$.

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 $\overline{y}$ minimizes the functional

$$F\left(y\right):={\int }_a^b{L}_1(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))·{\int }_a^b{L}_2(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt.$$

Then, the Euler–Lagrange equation for this functional is given by

$$\begin{array}{c}{\int }_a^b{L}_2(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))·\left({\partial }_2{L}_1(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{L}_1(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)\hfill \\ \hfill +{\int }_a^b{L}_1(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))·\left({\partial }_2{L}_2(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{L}_2(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)=0,\end{array}$$

for all $t\in [a,b]$.

Additionally, if either $y\left(a\right)$ or $y\left(b\right)$ is free, then the transversality condition is given by

$$\begin{array}{c}{\int }_a^b{L}_2(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))·{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_3{L}_1(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)\hfill \\ \hfill +{\int }_a^b{L}_1(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))·{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_3{L}_2(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)=0,\end{array}$$

at $t=a$ or $t=b$, respectively.

Corollary 2.

Suppose that $\overline{y}$ minimizes the functional

$$F\left(y\right):={\displaystyle \frac{{\int }_a^b{L}_1(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))}{{\int }_a^b{L}_2(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt}}.$$

Then, the Euler–Lagrange equation for this functional is given by

$$\begin{array}{c}{\int }_a^b{L}_2(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))·\left({\partial }_2{L}_1(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{L}_1(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)\hfill \\ \hfill -{\int }_a^b{L}_1(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))·\left({\partial }_2{L}_2(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{L}_2(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)=0,\end{array}$$

for all $t\in [a,b]$.

Additionally, if either $y\left(a\right)$ or $y\left(b\right)$ is free, then the transversality condition is given by

$$\begin{array}{c}{\int }_a^b{L}_2(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))·{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_3{L}_1(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)\hfill \\ \hfill -{\int }_a^b{L}_1(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))·{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_3{L}_2(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)=0,\end{array}$$

at $t=a$ or $t=b$, 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 $\overline{y}$ is a minimizer of the functional. Indeed, let $\overline{y}$ be a curve satisfying conditions (3) and (4), and assume that for all $k\in \{1,\dots ,n\}$, one of the following two conditions holds:

• ${\partial }_{k}C\left\{\overline{y}\right\}>0$ and the Lagrangian function $L_k$ is convex with respect to its second and third variables.
• ${\partial }_{k}C\left\{\overline{y}\right\}<0$ and the Lagrangian function $L_k$ is concave with respect to its second and third variables.

If y is another function in a neighborhood of $\overline{y}$, then, by the Mean Value Theorem, we obtain

$$\begin{array}{c}F(\overline{y}+y)-F\left(\overline{y}\right)\hfill \\ \hfill =\sum _{k=1}^n{\partial }_{k}C\left\{v\right\}{\int }_a^b{L}_k(t,\overline{y}\left(t\right)+y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right)+{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))-L_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))dt,\end{array}$$

for some vector $v\in {\mathbb{R}}^n$ between $\{\overline{y}+y\}$ and $\left\{\overline{y}\right\}$. Consider a neighborhood of $\overline{y}$ such that, for all $k\in \{1,\dots ,n\}$, we have ${\partial }_{k}C\left\{v\right\}>0$ (or $<0$) whenever ${\partial }_{k}C\left\{\overline{y}\right\}>0$ (or $<0$). Then, using the given assumptions, we obtain

$$\begin{array}{cc}\hfill & F(\overline{y}+y)-F\left(\overline{y}\right)\hfill \\ \hfill & \ge \sum _{k=1}^n{\partial }_{k}C\left\{v\right\}{\int }_a^b{\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))y\left(t\right)+{\partial }_3{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right)){}^C\mathbb{D}_{+}^{\alpha }y\left(t\right)dt\hfill \\ \hfill & =\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b\left({\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)y\left(t\right)dt\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)y\left(t\right)\right]}_{t=a}^{t=b}=0,\hfill \end{array}$$

thus 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, $\alpha$, the problem extends beyond finding the minimizer function $\overline{y}$. Specifically, we seek to determine both the optimal function $\overline{y}$ and the corresponding fractional order $\overline{\alpha }$ that yields the minimal functional value.

Theorem 2.

Let $(\overline{y},\overline{\alpha })$ be a solution to the following variational problem: minimize the functional

$$F(y,\alpha ):=C\left({\int }_a^b{L}_1(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt,\dots ,{\int }_a^b{L}_n(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt\right),$$

over the set $C^1[a,b]\times (0,1)$. Define the vector

$$\left\{\overline{y}\right\}:=\left({\int }_a^b{L}_1(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\left(t\right))dt,\dots ,{\int }_a^b{L}_n(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\left(t\right))dt\right)$$

and introduce the function ${\Lambda }_t:(0,1)\to \mathbb{R}$, where for each $t\in [a,b]$, ${\Lambda }_t\left(\alpha \right):={}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right)$. Then, the following four necessary optimality conditions are derived:

• Generalized Euler–Lagrange equation: for all $t\in [a,b]$,

  $$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}\left({\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\overline{\alpha }}\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)=0.$$
• Transversality condition: if $y\left(a\right)$ or $y\left(b\right)$ is free, then

  $$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\mathbb{I}}_{-}^{1-\overline{\alpha }}\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)=0,$$

  at $t=a$ or $t=b$, respectively.
• Condition for the variation of the fractional order $\overline{\alpha }$: the fractional parameter $\overline{\alpha }$ satisfies

  $$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_3{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right))·{\Lambda }_t^{\prime }\left(\overline{\alpha }\right)dt=0.$$

Proof. 

The variation of the optimal solution now depends on two coordinates $(\overline{y}+\eta y,\overline{\alpha }+\eta \alpha )$, where $\alpha$ is a real number. Considering the perturbed functional

$$f\left(\eta \right):=F(\overline{y}+\eta y,\overline{\alpha }+\eta \alpha ),$$

and using the necessary optimality condition $f^{\prime }\left(0\right)=0$, we obtain

$$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right))y\left(t\right)+{\partial }_3{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right))\left({}^C\mathbb{D}_{+}^{\overline{\alpha }}y\left(t\right)+\alpha {\Lambda }_t^{\prime }\left(\overline{\alpha }\right)\right)dt=0.$$

Applying the integration by parts formula to the fractional derivative term, we obtain

$$\begin{array}{c}\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b\left({\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\overline{\alpha }}\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)y\left(t\right)dt\hfill \\ +\alpha ·\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_3{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right))·{\Lambda }_t^{\prime }\left(\overline{\alpha }\right)dt\\ \hfill +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[{\mathbb{I}}_{-}^{1-\overline{\alpha }}\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)y\left(t\right)\right]}_{t=a}^{t=b}=0.\end{array}$$

Due to the arbitrariness of the parameter $\alpha$ and the function y on $[a,b]$, 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 $k\in \{1,\dots ,n\}$, let $M_k:[a,b]\times {\mathbb{R}}^2\to \mathbb{R}$ be a function with continuous partial derivatives with respect to its second and third variables. Additionally, let $D:{\mathbb{R}}^n\to \mathbb{R}$ be a continuously differentiable function.

The functional of interest is given by

$$G\left(y\right):=D\left({\int }_a^b{M}_1(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt,\dots ,{\int }_a^b{M}_n(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt\right).$$

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 $G\left(y\right)=\Lambda$, where $\Lambda$ is a fixed real number.

Theorem 3.

Let $\overline{y}$ be a solution to the isoperimetric problem (IP). Then, there exists a nonzero vector $({\lambda }_C,{\lambda }_D)$ such that

$$\begin{array}{c}{\lambda }_C\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}\left({\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)\hfill \\ \hfill +{\lambda }_D\sum _{k=1}^n{\partial }_{k}D\left\{\overline{y}\right\}\left({\partial }_2{M}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{M}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)=0,\end{array}$$

(6)

for all $t\in [a,b]$.

Proof. 

The proof will be divided into two distinct cases.

In the first case, we assume that $\overline{y}$ is an extremal curve for the functional G. In this scenario, Equation (6) is immediately satisfied by choosing $({\lambda }_C,{\lambda }_D)=(0,1)$.

Now, suppose that y is not an extremal curve for G. Consider a variation of the optimal curve depending on two parameters, given by $\overline{y}+{\eta }_1{y}_1+{\eta }_2{y}_2,$ where $y_1,y_2\in C^1[a,b]$ satisfy $y_1\left(a\right)=y_1\left(b\right)=0$ and $y_2\left(a\right)=y_2\left(b\right)=0$, and ${\eta }_1,{\eta }_2$ are real numbers. Define the function

$$g({\eta }_1,{\eta }_2):=G(\overline{y}+{\eta }_1{y}_1+{\eta }_2{y}_2)-\Lambda .$$

Since $y_2\left(a\right)=0$ and $y_2\left(b\right)=0$, it follows that

$$\begin{array}{c}{\displaystyle \frac{\partial g}{\partial {\eta }_2}}(0,0)=\hfill \\ \hfill \sum _{k=1}^n{\partial }_{k}D\left\{\overline{y}\right\}{\int }_a^b\left({\partial }_2{M}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{M}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)y_2\left(t\right)dt.\end{array}$$

Since $\overline{y}$ is not an extremal curve with respect to the functional G, we conclude that there must exist a function $y_2$ such that $\partial g/\partial {\eta }_2(0,0)\ne 0$. Furthermore, since $g(0,0)=0$, we can ensure the existence of a function $\varphi$, defined in a neighborhood of zero, such that $g({\eta }_1,\varphi \left({\eta }_1\right))=0$. In other words, there exists an infinite subfamily of variations that satisfy the integral constraint of the isoperimetric problem (IP).

Let

$$f({\eta }_1,{\eta }_2):=F(\overline{y}+{\eta }_1{y}_1+{\eta }_2{y}_2).$$

We can convert the variational problem into a finite-dimensional optimization problem. Indeed, $(0,0)$ is a solution of the following problem: minimizing f subject to the constraint $g\equiv 0$.

As previously proven, $\nabla g(0,0)\ne (0,0)$. By the method of Lagrange multipliers, there exists a scalar $\lambda$ such that

$$\nabla (f+\lambda g)(0,0)=(0,0).$$

Computing

$${\displaystyle \frac{\partial (f+\lambda g)}{\partial {\eta }_1}}(0,0)=0,$$

we obtain

$$\begin{array}{c}\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b\left({\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)y_1\left(t\right)dt\hfill \\ \hfill +\lambda \sum _{k=1}^n{\partial }_{k}D\left\{\overline{y}\right\}{\int }_a^b\left({\partial }_2{M}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_3{M}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)y_1\left(t\right)dt=0.\end{array}$$

(7)

The result follows from the arbitrariness of the function $y_1$ and by choosing $({\lambda }_C,{\lambda }_D)=(1,\lambda )$. □

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, $L_k:[a,b]\times {\mathbb{R}}^4\to \mathbb{R}$ is a function admitting continuous partial derivatives ${\partial }_i{L}_k$ for $k\in \{1,\dots ,n\}$ and $i\in \{2,3,4,5\}$. The functional depends on two functions $y_1,y_2\in C^1[a,b]$ and is defined by

$$\begin{array}{c}F(y_1,y_2):=C({\int }_a^b{L}_1(t,y_1\left(t\right),y_2\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{y}_1\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{y}_2\left(t\right))dt,\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\dots ,{\int }_a^b{L}_n(t,y_1\left(t\right),y_2\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{y}_1\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{y}_2\left(t\right))dt).\end{array}$$

The holonomic problem (HP) is defined as follows: minimize F subject to the boundary constraints $y_i\left(a\right)=y_{ia}$ and $y_i\left(b\right)=y_{ib}$, with $y_{ia},y_{ib}\in \mathbb{R}$ for $i=1,2$, and the condition

$$\Lambda (t,y_1\left(t\right),y_2\left(t\right))=0,t\in [a,b],$$

where $\Lambda :[a,b]\times {\mathbb{R}}^2\to \mathbb{R}$ is a function admitting continuous partial derivatives with respect to its second and third variables.

Theorem 4.

Let $\overline{y}:=({\overline{y}}_1,{\overline{y}}_2)$ be a solution to problem (HP) that satisfies the conditions

$${\partial }_3\Lambda (t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right))\ne 0,\forall t\in [a,b],and\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}\ne 0.$$

(8)

Then, there exists a function $\lambda :[a,b]\to \mathbb{R}$ such that, for $i=1,2$ and for all $t\in [a,b]$, the following condition holds:

$$\begin{array}{c}\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}({\partial }_{i+1}{L}_k(t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_1\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_2\left(t\right))\hfill \\ \hfill +{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_{i+3}{L}_k(·,{\overline{y}}_1,{\overline{y}}_2,{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_1,{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_2)}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)+\lambda \left(t\right){\partial }_{i+1}\Lambda (t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right)))=0.\end{array}$$

(9)

Proof. 

Let $\lambda$ be the solution of Equation (9) for $i=2$. It remains to prove that the equation also holds for $i=1$.

Consider a variation of the solution curve given by $({\overline{y}}_1+\eta y_1,{\overline{y}}_2+\eta y_2)$, where $y_1,y_2\in C^1[a,b]$ and $\eta \in \mathbb{R}$. Due to the boundary conditions, we assume that $y_1\left(t\right)=0$ and $y_2\left(t\right)=0$ for $t=a,b$.

These variations must satisfy the holonomic constraint, i.e.,

$$\Lambda \left(t,{\overline{y}}_1\left(t\right)+\eta y_1\left(t\right),{\overline{y}}_2\left(t\right)+\eta y_2\left(t\right)\right)=0,\forall t\in [a,b].$$

Differentiating with respect to $\eta$ at $\eta =0$, we obtain

$${\partial }_2\Lambda (t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right))y_1\left(t\right)+{\partial }_3\Lambda (t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right))y_2\left(t\right)=0,\forall t\in [a,b].$$

Next, we define the function

$$f\left(\eta \right)=F({\overline{y}}_1+\eta y_1,{\overline{y}}_2+\eta y_2).$$

Since $f^{\prime }\left(0\right)=0$, we obtain

$$\begin{array}{c}\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b({\partial }_2{L}_k(t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_1\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_2\left(t\right))\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,{\overline{y}}_1,{\overline{y}}_2,{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_1,{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_2)}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right))y_1\left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+({\partial }_3{L}_k(t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_1\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_2\left(t\right))\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_5{L}_k(·,{\overline{y}}_1,{\overline{y}}_2,{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_1,{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_2)}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right))y_2\left(t\right)dt=0.\end{array}$$

Given that

$$\begin{array}{c}\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b({\partial }_3{L}_k(t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_1\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_2\left(t\right))\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_5{L}_k(·,{\overline{y}}_1,{\overline{y}}_2,{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_1,{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_2)}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right))y_2\left(t\right)dt\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=-\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b\lambda \left(t\right){\partial }_3\Lambda (t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right))y_2\left(t\right)dt\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b\lambda \left(t\right){\partial }_2\Lambda (t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right))y_1\left(t\right)dt\end{array}$$

it follows that

$$\begin{array}{c}\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b({\partial }_2{L}_k(t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_1\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_2\left(t\right))\hfill \\ \hfill +{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,{\overline{y}}_1,{\overline{y}}_2,{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_1,{}^C\mathbb{D}_{+}^{\alpha }{\overline{y}}_2)}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)+\lambda \left(t\right){\partial }_2\Lambda (t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right)))y_1\left(t\right)dt=0.\end{array}$$

From the arbitrariness of the function $y_1$, we obtain the desired result for $i=1$. □

Remark 5.

We note that condition (8) may be trivially satisfied. For example, when $n=2$, $\Lambda (t,{\overline{y}}_1\left(t\right),{\overline{y}}_2\left(t\right))={\overline{y}}_1\left(t\right)+{\overline{y}}_2\left(t\right)$, and $C(x,y)=x+y$, this condition holds for any curve.

4. Necessary Conditions with Time Delays

The calculus of variations with time delays investigates problems in which the functional to be optimized depends not only on the function and its derivatives but also on its past values. These problems commonly arise in dynamic systems with delayed responses, where the presence of time delays modifies the optimality conditions, leading to adjusted Euler–Lagrange equations that incorporate the influence of past states on the system’s evolution. Due to the complexity of solving such fractional differential equations, various methods can be found to numerically solve such problems. For example, we can cite [36,37].

Let $L_k:[a,b]\times {\mathbb{R}}^3\to \mathbb{R}$, with $k\in \{1,\dots ,n\}$, be a function admitting continuous partial derivatives ${\partial }_2{L}_k$, ${\partial }_3{L}_k$, and ${\partial }_4{L}_k$. The time delay is denoted by $t_0\in (0,b-a)$. The domain of the functional is the set $D_F=C^1[a-t_0,b]$ and is defined by the expression

$$F\left(y\right):=C\left({\int }_a^b{L}_1(t,y\left(t\right),y(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt,\dots ,{\int }_a^b{L}_n(t,y\left(t\right),y(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt\right).$$

In this section,

$$\left\{y\right\}:=\left({\int }_a^b{L}_1(t,y\left(t\right),y(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt,\dots ,{\int }_a^b{L}_n(t,y\left(t\right),y(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right))dt\right).$$

The time delay problem (DP) is formulated as follows: minimize F over $D_F$, subject to the constraint $y\left(t\right)=\psi \left(t\right)$ for $t\in [a-t_0,a]$, where $\psi :[a-t_0,a]\to \mathbb{R}$ is a $C^1$-class function. The necessary optimality conditions are presented in the following result.

Theorem 5.

Let $\overline{y}$ be a solution of (DP). Then,

$$\begin{array}{c}\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}({\partial }_2{L}_k(t,\overline{y}\left(t\right),\overline{y}(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+{\partial }_3{L}_k(t+t_0,\overline{y}(t+t_0),\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}(t+t_0))\hfill \\ +{}^{t_0}\mathbb{D}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},\overline{y}(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\\ \hfill -{\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dt}}{\int }_{b-t_0}^b{(\mathcal{K}\left(\tau \right)-\mathcal{K}\left(t\right))}^{-\alpha }{\partial }_4{L}_k(\tau ,\overline{y}\left(\tau \right),\overline{y}(\tau -t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(\tau \right))d\tau )=0,\end{array}$$

for all $t\in [a,b-t_0]$, and

$$\begin{array}{c}\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}({\partial }_2{L}_k(t,\overline{y}\left(t\right),\overline{y}(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))\hfill \\ \hfill +{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},\overline{y}(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right))=0,\end{array}$$

for all $t\in [b-t_0,b]$, where ${}^{t_0}\mathbb{D}_{-}^{\alpha }$ represents the right-sided fractional derivative for functions defined on the interval $[a,b-t_0]$. Also,

$$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},\overline{y}(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(b\right)=0.$$

Proof. 

The variation of the optimal curve is given by $\overline{y}+\eta y$, where $y\in C^1[a-t_0,b]$ and $y\left(t\right)=0$ for all $t\in [a-t_0,a]$. The first variation is given by

$$\begin{array}{c}\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_2{L}_k(t,\overline{y}\left(t\right),\overline{y}(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))y\left(t\right)\hfill \\ \hfill +{\partial }_3{L}_k(t,\overline{y}\left(t\right),\overline{y}(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))y(t-t_0)+{\partial }_4{L}_k(t,\overline{y}\left(t\right),y(t-t_0){}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right)){}^C\mathbb{D}_{+}^{\alpha }y\left(t\right)dt.\end{array}$$

Due to the boundary conditions imposed in the problem, we have

$$\begin{array}{c}\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_3{L}_k(t,\overline{y}\left(t\right),\overline{y}(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))y(t-t_0)dt\hfill \\ \hfill =\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^{b-t_0}{\partial }_3{L}_k(t+t_0,\overline{y}(t+t_0),\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}(t+t_0))y\left(t\right)dt.\end{array}$$

On the other hand, for all $t\in [a,b-t_0]$, we have

$$\begin{array}{c}{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},\overline{y}(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)={}^{t_0}\mathbb{D}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},\overline{y}(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)\hfill \\ \hfill -{\displaystyle \frac{1}{\Gamma (1-\alpha ){\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{\int }_{b-t_0}^b{(\mathcal{K}\left(\tau \right)-\mathcal{K}\left(t\right))}^{-\alpha }{\partial }_4{L}_k(\tau ,\overline{y}\left(\tau \right),\overline{y}(\tau -t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(\tau \right))d\tau .\end{array}$$

Thus, by applying the fractional integration by parts formula, we obtain

$$\begin{array}{cc}\hfill \sum _{k=1}^n& {\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_4{L}_k(t,\overline{y}\left(t\right),y(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right)){}^C\mathbb{D}_{+}^{\alpha }y\left(t\right)dt\hfill \\ \hfill & =\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},y(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)y\left(t\right)dt\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},y(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(b\right)y\left(b\right)\hfill \\ \hfill & =\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^{b-t_0}({}^{t_0}\mathbb{D}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},\overline{y}(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dt}}{\int }_{b-t_0}^b{(\mathcal{K}\left(\tau \right)-\mathcal{K}\left(t\right))}^{-\alpha }{\partial }_4{L}_k(\tau ,\overline{y}\left(\tau \right),\overline{y}(\tau -t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(\tau \right))d\tau )y\left(t\right)dt\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_{b-t_0}^b{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},y(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)y\left(t\right)dt\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},y(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(b\right)y\left(b\right)\hfill \end{array}$$

By combining all the terms, we arrive at the equation

$$\begin{array}{cc}\hfill \sum _{k=1}^n& {\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^{b-t_0}({\partial }_2{L}_k(t,\overline{y}\left(t\right),\overline{y}(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))\hfill \\ \hfill & +{\partial }_3{L}_k(t+t_0,\overline{y}(t+t_0),\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}(t+t_0))\hfill \\ \hfill & +{}^{t_0}\mathbb{D}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},\overline{y}(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dt}}{\int }_{b-t_0}^b{(\mathcal{K}\left(\tau \right)-\mathcal{K}\left(t\right))}^{-\alpha }{\partial }_4{L}_k(\tau ,\overline{y}\left(\tau \right),\overline{y}(\tau -t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(\tau \right))d\tau )y\left(t\right)dt\hfill \\ \hfill +\sum _{k=1}^n& {\partial }_{k}C\left\{\overline{y}\right\}{\int }_{b-t_0}^b({\partial }_2{L}_k(t,\overline{y}\left(t\right),\overline{y}(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))\hfill \\ \hfill & +{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},y(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right))y\left(t\right)dt\hfill \\ \hfill +\sum _{k=1}^n& {\partial }_{k}C\left\{\overline{y}\right\}{\mathbb{I}}_{-}^{1-\alpha }\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},y(·-t_0),{}^C\mathbb{D}_{+}^{\alpha }\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(b\right)y\left(b\right)=0\hfill \end{array}$$

Again, the result follows from the arbitrariness of the function y on the interval $(a,b]$. □

Remark 6.

We note that, in our definition of the functional, the time delay does not affect the kernel. However, this could be incorporated by defining a new functional that includes the time delay dependence in the fractional derivative, as follows:

$$\begin{array}{c}F\left(y\right):=C({\int }_a^b{L}_1(t,y\left(t\right),y(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y(t-t_0))dt,\dots ,\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\int }_a^b{L}_n(t,y\left(t\right),y(t-t_0),{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }y(t-t_0))dt),\end{array}$$

where, in the last term, the initial point of the fractional derivative is shifted to $a-t_0$.

Example 1.

Consider $n=2$, $C(x,y)=xy$, a fractional order $\alpha \in (0,1)$, a time delay $t_0=1$, and the functional

$$F\left(y\right):={\int }_0^{10}{y}^2\left(t\right)+{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right)dt\times {\int }_0^{10}{y}^3(t-1)+{}^C\mathbb{D}_{+}^{\alpha }y\left(t\right)dt.$$

The Euler–Lagrange equations associated with this time delay problem are given by

$$\begin{array}{c}{\int }_0^{10}{y}^3(\tau -1)+{}^C\mathbb{D}_{+}^{\alpha }y\left(\tau \right)d\tau ·(2y\left(t\right)+{}^1\mathbb{D}_{-}^{\alpha }\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\hfill \\ -{\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dt}}{\int }_9^{10}{(\mathcal{K}\left(\tau \right)-\mathcal{K}\left(t\right))}^{-\alpha }d\tau )\\ +{\int }_0^{10}{y}^2\left(\tau \right)+{}^C\mathbb{D}_{+}^{\alpha }y\left(\tau \right)d\tau ·(3y^2\left(t\right)+{}^1\mathbb{D}_{-}^{\alpha }\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\\ \hfill -{\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dt}}{\int }_9^{10}{(\mathcal{K}\left(\tau \right)-\mathcal{K}\left(t\right))}^{-\alpha }d\tau )=0,t\in [0,9],\end{array}$$

and

$$\begin{array}{c}{\int }_0^{10}{y}^3(\tau -1)+{}^C\mathbb{D}_{+}^{\alpha }y\left(\tau \right)d\tau ·\left(2y\left(t\right)+{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)\hfill \\ \hfill +{\int }_0^{10}{y}^2\left(\tau \right)+{}^C\mathbb{D}_{+}^{\alpha }y\left(\tau \right)d\tau ·{\mathbb{D}}_{-}^{\alpha }\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)=0,t\in [9,10].\end{array}$$

5. Necessary Conditions with Higher-Order Derivatives

Until now, we have considered only problems where the fractional order $\alpha$ 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 ${\left({\alpha }_i\right)}_{i\in \{1,\dots ,m\}}$, where $m\in \mathbb{N}$ and ${\alpha }_i\in (i-1,i)$ for all $i\in \{1,\dots ,m\}$.

The Lagrangian function $L_k:[a,b]\times {\mathbb{R}}^{m+1}\to \mathbb{R}$, for $k\in \{1,\dots ,n\}$, is such that the partial derivatives ${\partial }_i{L}_k$ exist and are continuous for all $i\in \{2,\dots ,m+2\}$. To simplify notation, we introduce the following:

$$\overline{\alpha }:=({\alpha }_1,\dots ,{\alpha }_m)\mathrm{and}{}^C\mathbb{D}_{+}^{\overline{\alpha }}y\left(t\right):=({}^C\mathbb{D}_{+}^{\alpha }y\left(t\right),\dots ,{}^C\mathbb{D}_{+}^{{\alpha }_m}y\left(t\right)).$$

The functional to be minimized is defined as

$$F\left(y\right):=C\left({\int }_a^b{L}_1(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}y\left(t\right))dt,\dots ,{\int }_a^b{L}_n(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}y\left(t\right))dt\right),$$

on the set $D_F=C^m[a,b]$. Here, $\left\{y\right\}$ denotes the vector

$$\left({\int }_a^b{L}_1(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}y\left(t\right))dt,\dots ,{\int }_a^b{L}_n(t,y\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}y\left(t\right))dt\right).$$

The necessary optimality condition is presented below.

Theorem 6.

Let $\overline{y}$ be a minimizing curve of the functional F. Then, for all $t\in [a,b]$,

$$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}\left({\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+\sum _{i=1}^m{\mathbb{D}}_{-}^{{\alpha }_i}\left({\displaystyle \frac{{\partial }_{i+2}{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)=0.$$

Furthermore, for $j\in \{0,\dots ,m-1\}$, if ${\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{j}y\left(t\right)$ is arbitrary at $t=a$ or $t=b$, then

$$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}\left(\sum _{i=j+1}^m{\left(-{\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{i-j-1}{\mathbb{I}}_{-}^{i-{\alpha }_i}\left({\displaystyle \frac{{\partial }_{i+2}{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)\right)=0,$$

at $t=a$ or $t=b$, respectively.

Proof. 

Let us consider the variation $\overline{y}+\eta y$, where $y\in C^m[a,b]$. Since the first variation of the functional must vanish when evaluated at $\overline{y}$, we obtain

$$\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right))y\left(t\right)+\sum _{i=1}^m{\partial }_{i+2}{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right)){}^C\mathbb{D}_{+}^{{\alpha }_i}y\left(t\right)dt=0.$$

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

$$\begin{array}{cc}\hfill & \sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_3{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right)){}^C\mathbb{D}_{+}^{{\alpha }_1}y\left(t\right)dt\hfill \\ \hfill & =\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\mathbb{D}}_{-}^{{\alpha }_1}\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)y\left(t\right)dt\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[{\mathbb{I}}_{-}^{1-{\alpha }_1}\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)y\left(t\right)\right]}_{t=a}^{t=b}.\hfill \end{array}$$

For the second term, we obtain

$$\begin{array}{cc}\hfill & \sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_4{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right)){}^C\mathbb{D}_{+}^{{\alpha }_2}y\left(t\right)dt\hfill \\ \hfill & =\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\mathbb{D}}_{-}^{{\alpha }_2}\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)y\left(t\right)dt\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[\sum _{k=0}^1{\left(-{\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^k{\mathbb{I}}_{-}^{2-{\alpha }_2}\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{1-k}y\left(t\right)\right]}_{t=a}^{t=b}.\hfill \end{array}$$

Repeating this procedure up to the last term of the integral, we obtain

$$\begin{array}{cc}\hfill & \sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_{m+2}{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right)){}^C\mathbb{D}_{+}^{{\alpha }_m}y\left(t\right)dt\hfill \\ \hfill & =\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\mathbb{D}}_{-}^{{\alpha }_m}\left({\displaystyle \frac{{\partial }_{m+2}{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)y\left(t\right)dt\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[\sum _{k=0}^{m-1}{\left(-{\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^k{\mathbb{I}}_{-}^{m-{\alpha }_m}\left({\displaystyle \frac{{\partial }_{m+2}{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{m-k-1}y\left(t\right)\right]}_{t=a}^{t=b}.\hfill \end{array}$$

Substituting these equalities into the first variation of the functional, we obtain

$$\begin{array}{cc}\hfill & 0=\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y}\left(t\right))y\left(t\right)\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\mathbb{D}}_{-}^{{\alpha }_1}\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)y\left(t\right)dt\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[{\mathbb{I}}_{-}^{1-{\alpha }_1}\left({\displaystyle \frac{{\partial }_3{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)y\left(t\right)\right]}_{t=a}^{t=b}\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\mathbb{D}}_{-}^{{\alpha }_2}\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)y\left(t\right)dt\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[\sum _{k=0}^1{\left(-{\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^k{\mathbb{I}}_{-}^{2-{\alpha }_2}\left({\displaystyle \frac{{\partial }_4{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{1-k}y\left(t\right)\right]}_{t=a}^{t=b}\hfill \\ \hfill & +\dots +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b{\mathbb{D}}_{-}^{{\alpha }_m}\left({\displaystyle \frac{{\partial }_{m+2}{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)y\left(t\right)dt\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[\sum _{k=0}^{m-1}{\left(-{\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^k{\mathbb{I}}_{-}^{m-{\alpha }_m}\left({\displaystyle \frac{{\partial }_{m+2}{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{m-k-1}y\left(t\right)\right]}_{t=a}^{t=b}.\hfill \end{array}$$

By factoring out the terms $y\left(t\right),\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right),\cdots ,{\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{m-1}$, we obtain the following expression:

$$\begin{array}{cc}\hfill & \sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\int }_a^b\left({\partial }_2{L}_k(t,\overline{y}\left(t\right),{}^C\mathbb{D}_{+}^{\alpha }\overline{y}\left(t\right))+\sum _{i=1}^m{\mathbb{D}}_{-}^{{\alpha }_i}\left({\displaystyle \frac{{\partial }_{i+2}{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)y\left(t\right)dt\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[\sum _{i=1}^m{\left(-{\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{i-1}{\mathbb{I}}_{-}^{i-{\alpha }_i}\left({\displaystyle \frac{{\partial }_{i+2}{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)y\left(t\right)\right]}_{t=a}^{t=b}\hfill \\ \hfill & +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[\sum _{i=2}^m{\left(-{\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{i-2}{\mathbb{I}}_{-}^{i-{\alpha }_i}\left({\displaystyle \frac{{\partial }_{i+2}{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right)\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)y\left(t\right)\right]}_{t=a}^{t=b}\hfill \\ \hfill & +\dots +\sum _{k=1}^n{\partial }_{k}C\left\{\overline{y}\right\}{\left[{\mathbb{I}}_{-}^{m-{\alpha }_m}\left({\displaystyle \frac{{\partial }_{m+2}{L}_k(·,\overline{y},{}^C\mathbb{D}_{+}^{\overline{\alpha }}\overline{y})}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{m-1}y\left(t\right)\right]}_{t=a}^{t=b}=0.\hfill \end{array}$$

The result follows again from the arbitrariness of the function y. Indeed, assuming that $y\left(a\right)=0$, $y\left(b\right)=0$, and, for all $j\in \{0,\dots ,m-1\}$, ${\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{j}y\left(t\right)=0$ at $t=a$ and $t=b$, we obtain the Euler–Lagrange equation. To derive the transversality condition for $j\in \{0,\dots ,m-1\}$, it is sufficient to assume that ${\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{j}y\left(t\right)\ne 0$ at $t=a$ or $t=b$, 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 $y^{\left(j\right)}\left(t\right)$, being fixed at $t=a$ and $t=b$ for $j\in \{0,\dots ,m-1\}$, then in Theorem 6, we would obtain only the Euler–Lagrange equation.

Example 2.

Consider $n=2$, $C(x,y)=xy$, fractional orders ${\alpha }_1\in (0,1)$ and ${\alpha }_2\in (1,2)$, and the cost functional

$$F\left(y\right):={\int }_0^1{y}^2\left(t\right)+{}^C\mathbb{D}_{+}^{{\alpha }_1}y\left(t\right)·{}^C\mathbb{D}_{+}^{{\alpha }_2}y\left(t\right)dt\times {\int }_0^1{}^C\mathbb{D}_{+}^{{\alpha }_1}y\left(t\right)dt.$$

The Euler–Lagrange equation associated with this problem is given by

$$\begin{array}{c}{\int }_0^1{}^C\mathbb{D}_{+}^{{\alpha }_1}y\left(\tau \right)d\tau ·\left(2y\left(t\right)+{\mathbb{D}}_{-}^{{\alpha }_1}\left({\displaystyle \frac{{}^C\mathbb{D}_{+}^{{\alpha }_2}y}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)+{\mathbb{D}}_{-}^{{\alpha }_2}\left({\displaystyle \frac{{}^C\mathbb{D}_{+}^{{\alpha }_1}y}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)\right)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_0^1{y}^2\left(\tau \right)+{}^C\mathbb{D}_{+}^{{\alpha }_1}y\left(\tau \right)·{}^C\mathbb{D}_{+}^{{\alpha }_2}y\left(\tau \right)d\tau ·{\mathbb{D}}_{-}^{{\alpha }_1}\left({\displaystyle \frac{1}{{\mathcal{K}}^{\prime }}}\right)\left(t\right){\mathcal{K}}^{\prime }\left(t\right)=0,t\in [0,1].\end{array}$$

6. Conclusions and Future Work

In this work, we have established necessary optimality conditions for various fractional variational problems, extending classical results to the fractional setting. We derived a general fractional Euler–Lagrange equation for functionals dependent on a generalized Caputo derivative. Our analysis also considered optimization problems with constraints, such as isoperimetric and holonomic constraints, as well as problems incorporating time delays and higher-order derivatives.

For future research, one potential direction is the development of efficient numerical methods for solving the derived fractional differential equations, as analytical solutions are often challenging to obtain. Another possible direction is to establish conditions that guarantee the existence and uniqueness of solutions for these variational problems.