1. 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
, and let
be a function with
, for all
. Given an integrable function
, the left-sided and the right-sided Riemann–Liouville fractional integrals of
x with kernel
are defined as
and
respectively, where
represents the Gamma function, and
is the order of the fractional integral. Moreover,
We can recognize that the Riemann–Liouville, the Hadamard and the Erdélyi–Kober fractional integrals are just particular cases of this more general definition. With respect to fractional differentiation, the left- and right-sided Riemann–Liouville fractional derivatives of
x, with kernel
, are given by the formulas
and
where
. For simplicity, we call the last operators as
-Riemann–Liouville fractional derivatives of
x of order
. It can be easily noticed that for certain choices of function
, we recover some important fractional derivatives. We also remark that when
, 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
and
be defined by
if
, and
if
. Given two functions
, with
, for all
, the left- and right-sided Caputo fractional derivatives of
x with kernel
(or simply,
-Caputo fractional derivatives of
x), are defined as
and
respectively. Thus, if
, then
For
, then
and
We now present the following formulas (see Lemma 1 in [
14]) that are useful in
Section 3. If
, then
and
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
-Caputo fractional derivatives, we refer to [
14].
Theorem 1. [14] Let and be two functions. Then,and In particular, when
, Theorem 1 becomes
and
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.
2. Main Results
In this work, we consider a functional depending on time, on the state function x, its fractional derivatives and of orders , the values and , and a free parameter . More specifically, we will study the following generalized fractional variational problem.
Problem: Determine
and
such that
where
and
We will consider problem
with fixed boundary conditions
for some
, and when
and
are free. We will also consider problem
subject to an isoperimetric constraint
for some
and to an holonomic constraint
for a given function
g.
Remark 1. We remark that:
If the function , and the Lagrange function does not depend on a free parameter ζ, then we get the fractional variational problem studied in [30]; Taking , and if ψ is the identity, the operators and can be replaced by and , respectively (see [1]). Hence, if α and β goes to 1, our functional tends to the generalized variational functional of the classical calculus of variations studied in [31]; If , α and β goes to 1, and the Lagrange function does not depend on the state values and on a free parameter, functional reduces to the functional from Lemma 2.2.2 in [32].
Next, we proceed with some basic definitions that are useful in what follows.
Definition 1. Let and . We say that is an admissible pair to subject to (2), if x satisfies (2). Function is an admissible variation to subject to (2) if . Definition 2. We say that is a local minimizer (resp. local maximizer) for the functional if there exists some , such that, for all with , we have (resp. ), where . The pair is called a local extremizer of .
If (resp. ) holds for all , then we say that is a global minimizer (resp. global maximizer). In these cases, we say that is a global extremizer of .
2.1. Generalized Fractional Variational Principle
The following result provides necessary conditions for an admissible pair
to be a local extremizer of functional
, where
x satisfies the boundary conditions (
2). The equation
is called the Euler–Lagrange equation. We will represent it by
. To simplify, consider the two following conditions:
and
where
H is a function and
i a positive integer.
Theorem 2 (Generalized fractional variational principle).
Suppose that L satisfies and . If is a local extremizer of functional , subject to the boundary conditions (2), then Equation (4) holds. Additionally,is verified. Proof. Let
be a local extremizer for functional
subject to (
2),
an admissible variation and
an arbitrary real number. Define the new function
by
. Since
is a local extremizer of
, then
. Therefore, the following condition holds
Using the fractional integration by parts formulas stated in Theorem 1, we get
Since
, then
Taking
and using the arbitrariness of
, by Lemma 2.2.2 in [
32], we obtain (
4).
Taking the admissible variation
to be null, we deduce from (
6) that
By the arbitrariness of
, we conclude that
proving (
5). □
Remark 2. In Theorem 2, since the state values and are fixed, the Lagrangian’s explicit dependence on and is irrelevant. However, in Theorem 3, since the state values can be free, this dependency is effective.
We remark that, although the functional
depends only on
-Caputo fractional derivatives, the Euler–Lagrange equation involves
-Riemann–Liouville fractional derivatives. Using the relations (see Theorem 3 in [
14])
and
it is possible to write Equation (
4) using only
-Caputo fractional derivatives.
Definition 3. We say that a pair is an extremal of functional if satisfies the Euler-Lagrange Equation (4). We now consider the case when the values and are not necessarily specified. For each boundary condition missing, there is a corresponding natural boundary condition, as given by Theorem 3.
Theorem 3(Generalized fractional natural boundary conditions).
Suppose that L satisfies and . If is a local extremizer of functional , then (4) and (5) hold. Moreover, Proof. Suppose that
is a local extremizer of functional
. Let
and
be an arbitrary real number. Let
. Since no boundary conditions are imposed,
do not need to be null at the endpoints. However, since Equation (
6) must be satisfied for all
, 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
satisfies the necessary conditions (
4) and (
5).
Suppose that
is free. Restricting
to be null at
, and substituting the necessary conditions (
4) and (
5) into (
6) it follows that
From the arbitrariness of
, we get
Suppose now that
is not fixed. Restricting
to be null at
, 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
, then condition (
5) is trivially satisfied and we get the following results.
Corollary 1. If x is a local extremizer ofthen x satisfies the generalized fractional equation , whereMoreover, If is free, then x satisfies the following condition If is free, then x satisfies
If the Lagrangian function does not depend on the state values and , and on a real parameter , then we get the following result.
Corollary 2. If x is a local extremizer of functionalthen x satisfies where Moreover, If is free, then x satisfies the following condition If is free, then x satisfies
Remark 3. Note that
If we consider , for all , in Corollary 1 and Corollary 2, we obtain Theorem 3.1 from [30] and Theorem 1 from [33], respectively; 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 and/or , are of a different nature when compared with the case where the Lagrangian does not depend on the endpoint conditions.
2.2. 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 is a normal extremizer of the isoperimetric problem (1) and (3) if is a local extremizer of functional and not an extremal of functional ; if is a local extremizer of functional and an extremal of functional , we say that 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.
Theorem 4 (Necessary optimality conditions for normal extremizers to fractional isoperimetric problems I).
Suppose that , , , and are verified. Let be a normal extremizer of problem (1) and (3), subject to the boundary conditions (2). Then there exists , such that defining , satisfies the Euler–Lagrange equationand the necessary condition Proof. Consider variations
, where
are admissible variations, with
, and
are arbitrary real numbers, for
. Let
be the functions defined by
and
Since
, we conclude that
Observing that
is not an extremal of functional
, one concludes that
for some
. Then we can choose
, such that
. Since
, there exists an open set
, such that
and there exists
such that
and
, for all
. This means that there exists an infinite family of pairs
where
and
, which satisfies the isoperimetric constraint (
3). Now, we proceed proving the necessary conditions. Observe that
is a local extremizer of function
, subject to the constraint
, and we just proved that
. Then, by the Lagrange Multiplier Rule, there exists a real number
such that
. Observe that
and
Denoting
, we get
Since the last equation must hold for any
, then, in particular, we can conclude that
From Lemma 2.2.2 in [
32], we obtain
proving Equation (
13). Introducing the Euler–Lagrange Equation (
13) into (
16), one gets
From the arbitrariness of
, we get
proving (
14). □
Theorem 5 (Necessary optimality conditions for normal extremizers to fractional isoperimetric problems II).
Suppose that , , , and hold. Let be a normal extremizer of functional , subject to (3). Then, there exists , such that defining , satisfies Equations (13) and (14). Moreover,- 1.
If is not fixed then - 2.
If 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. □
Theorem 6 (Necessary optimality conditions for normal and abnormal extremizers to fractional isoperimetric problems).
Suppose that , , , and hold. Let be a local extremizer of functional , subject to the integral constraint (3). Then, there exists a vector , such that, defining the Lagrangian function , then Equations (13) and (14) hold, as well the natural boundary conditions (17) and (18). Proof. First, suppose that is a normal extremizer. Then, the results follow from Theorem 5 fixing . Otherwise, we consider . □
2.3. 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
. Thus, functional
is defined as
where
, , and ;
and ;
.
Boundary conditions are
for some
and the holonomic constraint is
where
g is a given
function.
Theorem 7 (Generalized fractional variational principle with holonomic constraint).
Assume , , , and hold. Let be an extremizer of functional , as in (19), subject to (20) and to the holonomic constraint (21). Ifthen there exists satisfyingfor , and Proof. Consider variations of
of type
, where
is a differentiable function with
. Since variations must fulfill the holonomic constraint, we have that
for all
. Differentiating Equation (
25) with respect to
and taking
, we get that
Using the boundary conditions of
, Equation (
26) becomes
Define function
as
Hence, Equation (
23) is proven for the case
. Now, we prove the remaining conditions. Since
is an extremizer of functional
, its first variation must vanish when evaluated at
, that is,
By relations (
27) and (
28), we obtain
Taking into account the boundary conditions of
, and since
is arbitrary on
, and parameter
is free, we prove Equation (
22) for the case i = 2 and Equation (
24), ending the proof. □
Now, we deduce the natural boundary conditions, for the case where and are free.
Theorem 8 (Natural boundary conditions with an holonomic constraint).
Let be an extremizer of functional , as in (19), subject to the holonomic constraint (21) such that Equation (22) holds, and to the remaining assumptions of Theorem 7. Then, there exists satisfying (23) and (24). Moreover,If is not fixed, then, for , If is not fixed, then, for ,
Proof. Consider an arbitrary variation
of the optimal solution. Assuming that
, following the proof of Theorem 7, we deduce Equations (
23) and (
24). Introducing Equations (
23) and (
24) and (
26)–(
28) into (
29), we obtain
First, suppose that
is free. Assuming that
, we get that
Since
and
are arbitrary real numbers, we obtain conditions (
31). Conditions (
32) are proven in a similar way. □
2.4. Sufficient Optimality Conditions
Now, we focus on sufficient conditions that guarantee the existence of extremizers of functional . 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.
Definition 5. We say that is jointly convex (resp. jointly concave) in if, for all , exist and are continuous and verifyfor all and where Theorem 9. Suppose that L is jointly convex (resp. jointly concave) in . If satisfies the necessary conditions (4) and (5) and (7) and (8), then is a global minimizer (resp. global maximizer) of functional . Proof. We shall give the proof only for the case where
L is jointly convex; the other case is similar. Let
and
be arbitrary. Since
L is jointly convex in
, 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.
Theorem 10. Suppose L is jointly convex (resp. jointly concave) in , and also that there exists a real number λ such that is jointly convex (resp. jointly concave) in . Let . If satisfies the necessary conditions (13) and (14) and (17) and (18), then is a global minimizer (resp. global maximizer) of functional subject to the isoperimetric constraint (3). Proof. We shall give the proof only for the case where
L and
are jointly convex; the proof of the other case is analogous. Since
M is jointly convex then, by Theorem 9,
is a global minimizer of functional
defined by
Hence, for any
and
, 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 . Let function λ be defined by (28), where g is a function, such that , for all . If satisfies the necessary conditions (23) and (24) and the natural boundary conditions (31) and (32), then is a global minimizer (resp. global maximizer) of functional 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
,
and
Replacing the last formula into (
34) and using conditions (
23) and (
24) and (
31) and (
32), we prove the desired result. □
4. 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 , or optimal control problems where the state equation involves the -Caputo fractional derivative.