Abstract

In this paper, the necessary and sufficient conditions of optimality for variational problems with Caputo partial fractional derivative are established. Fractional Euler-Lagrange equations are obtained. The Legendre condition and Noether’s theorem are also presented.

1. Introduction

Since the introduction of fractional calculus of variations by Riewe [1], fractional calculus has been a subject of interest not only among mathematicians, but also among fluid mechanics, electricity and finance specialists, chemical physicists, biomedical engineering specialists, and control theory specialists.

Considerable progress has been made to determine necessary and sufficient conditions that any extremal for the variational functional with fractional calculus must satisfy in recent years. R. Almeida [2] provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Almeida established the fractional Euler-Lagrange equations for the fundamental problem and when in presence of an integral constraint and Almeida obtained a Legendre condition. In [3] Almeida studied certain problems of calculus of variations that are dependent upon a Lagrange function on a Caputo-type fractional derivative; sufficient and necessary conditions of the first- and second-order are presented. In [4] Zhang Jianke, Ma Xiaojue, and Li Lifeng studied the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrange function depending on a Caputo-Fabrizio fractional derivative. In [5] Almeida et al. obtained necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative and indefinite integral. There has been a significant development in ordinary and partial fractional differential equations in recent years [6–9].

D. Tavares et al. in [10] presented two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of variable fractional order and presented a new variational problem subject to holonomic constraint.

Noether’s symmetry, namely, the invariance of Hamilton action under the infinitesimal transformations, is put forward for the first time by Noether [11]. In [12] Frederico et al. obtained a generalization of the Noether theorem for Lagrangians depending on mixed classical and Caputo derivatives that can be used to obtain constants of motion for dissipative systems. In [13] the Noether theorem and its inverse theorem for the nonlinear dynamical systems with nonstandard Lagrangians are studied. In [14] a variational principle for Lagrangian densities containing derivatives of real order was formulated and the invariance of this principle is studied in two characteristic cases. In [15] Yan B. et al. studied Noether’s symmetries and conserved quantities of the Birkhoffian systems in terms of fractional derivatives of variable order.

In this paper, we extend the study to variational problems involving partial fractional derivatives. Our aim is to obtain the necessary and sufficient conditions for the minimizer. Moreover, we will establish Noether’s theorem for these problems.

The paper is organized as follows. In Section 2, the basic definitions and notations are given. In Sections 3.1 and 3.2, the first-order necessary condition and the second-order necessary condition for the minimizer are established. Then, the variational problem subject to an integral constraint is investigated in Section 3.3 and the isoperimetric problem is discussed in Section 3.4. In Section 3.5, Noether’s theorem for this system is proved. In Section 4, as applications of our results, some examples are presented.

2. Preliminaries

In this section, we will recall some basic concepts and preliminary results on fractional calculus, needed in the sequel. To fix notation, in the following and is the boundary of .

Definition 1 (Riemann-Liouville fractional integrals, [12]). The left and right Riemann-Liouville fractional integrals of order for the function are defined, respectively, as where is the gamma function.

Definition 2 (Riemann-Liouville fractional derivatives, [12]). The left and right Riemann-Liouville fractional derivatives of order for the function are defined, respectively, as

Remark 3. The Riemann-Liouville fractional derivative of a constant need not to be zero.

Definition 4 (Caputo fractional derivatives, [12]). The left Caputo fractional derivative of of order is defined by If is of class , then we have the equivalent form while the right Caputo fractional derivative of of order is given by

Remark 5. The Caputo derivative exhibits an important feature: the derivative of a constant is zero.

Definition 6 (partial Riemann-Liouville integrals, [9]). Let and . The left and right partial Riemann-Liouville integrals of order of with respect to are defined, respectively, by the expression for almost all Analogously, we define the integrals for almost all .

Definition 7 (partial Riemann-Liouville derivatives, [9]). Let and . The left and right partial Riemann-Liouville derivatives of order of with respect to are defined, respectively, by the expression for almost all . Analogously, we define the derivatives for almost all .

Definition 8 (partial Caputo fractional derivative, [9]). Let and . The left and right partial Caputo fractional derivatives of order of with respect to are defined, respectively, by the expression for almost all . Analogously, we define the derivatives for almost all .

Theorem 9. Let be a continuous function and be of class ; then and

Proof. Analogously, we obtain

Theorem 10 (see [16]). If satisfies for all with , then on .

3. The Variational Problem

The aim of this section is to study problems of fractional calculus of variations, where the integral functional depends on the partial Caputo fractional derivative. Given , we define the functional with the following assumptions:

(i) is a continuous function, such that , and exist and are continuous.

(ii) Given any , and are continuous.

For simplification, we consider the operator

3.1. The Fundamental Problem

Theorem 11. Suppose that is a local minimizer for as in (17), defined on where and . Then is a solution to the equation for almost all .

Proof. Let be a variation of , with , , and . We define the function in a neighborhood of zero by the expression Since is a minimizer of , then is a minimizer of , and so . Differentiating at and using Theorem 9, we obtainSince and is arbitrary elsewhere, we conclude that for almost all .

Remark 12. Equation (20) is called the Euler-Lagrange equation associated with the functional . Solutions of this equation are called extremals.

Remark 13. If we define functional then we obtain the Euler-Lagrange equation for almost all .

3.2. The Legendre Condition

In [17], a second-order necessary condition had been proved for functionals involving Riemann-Liouville fractional derivatives.

In this section, we give a second-order necessary condition, usually called Legendre condition, for functionals involving Riemann-Liouville partial fractional derivatives.

We introduce the functional with the same assumptions on as in Section 3.1.

Theorem 14. Suppose that is a local minimizer for in . If exists and is continuous for , then satisfies for almost all .

Proof. Let be a variation of , with , , and .
We define the function in a neighborhood of zero by the expression . Then, we have ; that is, Assume that the Legendre condition is violated at some ; i.e., Then, there exists a rectangle and six real constants with , such that for all .
Define the function as follows: Then, is of class , and Moreover, for every , and Let By the properties of function , we have , and Bringing this variation into (28), we get If , there arises a contradiction.
Hence, (27) holds. In the same manner, we can get (26).

3.3. The Fractional Variational Problem with Holonomic Constraint

Consider the functional defined by on the space subject to where and . For simplicity, we denote

Assume the Lagrangian in (38) satisfies the following conditions:

(iii) is continuously differentiable with respect to its th argument, for ;

(iv) Given any functions , the maps are continuous.

We consider the variational problem when in presence of a holonomic constraint. Assume that the admissible functions lie on the surface where is continuously differentiable with respect to its th argument, for .

Theorem 15. Let be a minimizer of as in (38), under the constraint (43). If then there is a continuous function such that and

Proof. Consider a variation of of type , with , and , satisfying the boundary conditions . By and the implicit function theorem, there exists a subfamily of variations satisfying restriction equation (43). That is, there exists a unique function such that satisfies (43). Hence, for all , we have Differentiating (48) with respect to and putting , we get Define the function On the other hand, since is a minimizer of , the first variation of must vanish; Integrating by parts and , we obtain Since is arbitrary, we have that is a solution of the equation At the same time, by (50), we obtain the second condition

3.4. Isoperimetric Problem

Let be constant and be a continuously differentiable function with respect to its -th argument, for . For any function , the maps and are continuous.

We now consider a new isoperimetric type problem with the isoperimetric constraint as follows:

Theorem 16. Suppose that is a local minimizer for as in (25) on the space , subject to the integral constraint (57). If is not an extremal of , then there exists a real such that is a solution of the equation where the function is defined by .

Proof. Consider a variation of two parameters , with and satisfying . Define the functions and with two parameters in a neighborhood of zero as and When , the derivative is Integrating by parts and , we obtainSince is not an extremal of , there exists a function such that . An application of the implicit function theorem implies that there exists a subfamily of variations satisfying the integral constraint.
On the other hand, is a minimize of , under the restriction , and we just proved that . By the Lagrange multiplier rule, there exists a real number such that . In particular, . Since , we have Hence, This ends the proof.

Theorem 17. Suppose that the functions and as in (25) and (57) are convex in . Let be a constant and . Then, each solution of the fractional Euler-Lagrange equation minimizes in , subject to the integral constraint (57).

Proof. Consider a variation of of type , with and , satisfying the boundary conditions . Noting that is convex, We have According to the integral constraint, we obtain As a consequence, This ends the proof.

3.5. Noether Symmetry

Suppose that the configuration of a dynamical system is

The action is defined by (25); i.e.,

The isochronal variation principle with fixed end-point conditions is called the Hamilton principle on the action. From the principle equations (72)-(73) and Theorem 9, it is easy to obtain Euler-Lagrange equations . Let us introduce the infinitesimal transformations of the one-parameter finite transformation group ; their expansion formulas are where is infinitesimal parameter; , and are infinitesimal generators or generating functions of the infinitesimal transformations. We introduce the following notation [14], , and .

So we have and

Theorem 18. Let and be a local one-parameter group of transformations by (74). Then and

Proof. By the definitions, we have Analogously, we obtain

Under the infinitesimal transformations, the action equation (25) will be transformed into

The difference of the fractional action before and after the infinitesimal transformation is where the space is adjacent to the space .

Moreover, where

By Theorem 9 and relation (76), we get

Definition 19. If the fractional action is invariant under the infinitesimal transformations (74) of group, i.e., for each of the infinitesimal transformations, the formula holds, then the infinitesimal transformations are called the symmetric transformations in sense of Noether for the fractional system.
By Definition 19 and (73) and (85), we can get the following theorem.

Theorem 20. For the infinitesimal transformations (74) of group, if the condition is satisfied, then the transformations (74) are the Noether symmetric transformations for the dynamical system based on the action.

4. Examples

Example 21. Consider the functional subject to the restriction and . From Theorem 11, the necessary condition that every minimizer of the functional must fulfill is the following: We see that satisfies these conditions. Also, using the Legendre condition, a local minimizer for the functional must verify (26) and (27), which for our example is verified. In fact, and .

Example 22. Let subject to the restrictions and where and
The function satisfies the necessary conditions, as stated in Theorem 15.

Example 23. Let us consider the following fractional problem of the calculus of variations: subject to the restriction and .
If the transformation is , where is arbitrary constant, then the fractional integral functional is invariant under Definition 19. In fact, the transformation satisfies (87).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by the Doctoral Fund of Education Ministry of China (20134219120003), the Natural Science Foundation (61473338), and Hubei Province Key Laboratory of Systems Science in Metallurgical Process (Y201705).