Euler–Lagrange equations for variational problems involving the Riesz–Hilfer fractional derivative

In this paper, we obtain the Euler-Lagrange equations for different kind of variational problems with the Lagrangian function containing the Riesz-Hilfer fractional derivative. Since the Riesz-Hilfer fractional derivative is a generalization for the Riesz-Riemann-Liouville and the Riesz-Caputo derivative, then our results generalize many recent works in which the Lagrangian function involving the Riesz-Riemann-Liouville or the Riesz-Caputo derivative. We also study the problem in the presence of delay derivatives and establish a version for Noether theorem in the Riesz-Hilfer sense. In order to achieve our aims we derive some formulas to integration by parts for the Riesz-Hilfer fractional derivative. In the last section, examples are given to clarify the possibility of applicability of our results. In order to clarify the significant conclusions of the paper, we refer to our techniques enable to study many different variational problems containing the Riesz-Hilfer derivative.

In this paper, and in order to generalize many results cited above, we give the notations for the Riesz-Hilfer fractional derivative (RHFD) which includes Riesz-Riemann-Liouville fractional derivative (RRLFD) and the Riesz-Caputo fractional derivatives (RCFD), and hence we obtain the Euler-Lagrange equations for different kind of fractional variational problems, with a Lagrangian containing (RHFD). In Section 2, we derive some formulas to integration by parts for (RHFD). In Section 3, we consider a simple fractional variational problem, then we take the case when the interval of integration of the functional is contained in the interval of fractional derivative. In Section 4, we obtain the Euler-Lagrange equations for isoperimetric problems, in which the function eligible for the extremization of a given definite integral is required to conform with certain restrictions. In Section 5, we study the problem in the presence of delay derivatives. In Section 6, we obtain the conditions that assure a pair of function-time to be an optimal solution. In Section 7, we establish a version for Noether theorem in the Riesz-Hilfer sense is given. Finally, we give examples to clarify the possibility of applicability of our results.
To the best of our knowledge, up to now, no work has reported on fractional variational problems, with a Lagrangian function involving (RHFD). Moreover, since (RHFD) is generalization for (RRLFD) and (RCFD), then our work generalizes many works mentioned above and allow to consider other variational problems, where the Lagrangian function involving (RHFD).

Preliminaries and notations
For any natural number m let We recall some concepts on fractional calculus [1,4]. (1) where is the Euler gamma function.

Definition 2.2:
The (RRLFI)of order μ > 0 for a Lebesgue integrable function h : [a, b] → R is defined as:

Definition 2.3:
The (RFI) of order μ > 0 for a function h ∈ L 1 [a, b], is defined as: Notice that In what follows μ denotes to a positive real number and m is the smallest natural number such that μ ≤ m.
Definition 2.7: The left-sided Caputo fractional deriva- Definition 2.8: The right-sided Caputo fractional deriva- and Definition 2.12: Remark 2.2: From the above definitions it follows that: In the following lemmas we derive formulas to integration by parts for (RHFD) Proof: Let q = (1 − β)(1 − μ). Using Lemma 2.1, then ordinary integration by parts and then again by From the definition of RH a D μ,β b η(t), (17) and (18) it yields (16).
We need to the following lemma in the third section.

Remark 2.3: If we add Equations
We need to the following basic Lemma [18].
defined on the set of functions q 1 , . . . , q n which are continuously differentiable and such that RH Then a necessary condition for the functional (29) attains an extremum at q i , i = 1, 2, . . . , n is that q i satisfy the following Euler-Lagrange equations: If β = 1, then the functions q i should be satisfy the transversality condition: Proof: It is known that the necessary condition for q i , i = 1, 2, . . . , n to be extremum, is given by where η i are arbitrary continuously differentiable functions for which That is where Notice that if β = 0, then by (34) we get If β ∈ (0, 1), then q = 0 = β(1 − μ), and consequently, the continuity of Then, by (37) Moreover, Equations (34) It yields from (35), (40) and (41) that and this should be true for all admissible functions η i . Since the functions η i are independent, then we can choose for any fixed i, and the proof is completed.   (30) to be an extremum of the functional given by (31) are that q i , i = 1, 2, . . . , n, satisfy the following Euler-Lagrange equations: If β = 1, then q i , i = 1, 2, . . . , n, should be satisfy the transversality condition Proof: As above, the necessary condition for q i , i = 1, 2, . . . , n to be extremum, is given by where η i , i = 1, 2, . . . , n, are arbitrary continuously differentiable functions for which Utilizing the rule of integration by parts (21) and (22) and taking into account the assumptions η i (A) = η i (B) = 0, i = 1, 2, . . . , n, we get Notice that if β = 0, then equation (40) leads to , and consequently, the continuity of ∂L/(∂ RH a D μ,β b q i ) and η(t) implies to If β = 1, then equation (48) implies that Using equations (53)-(55), equation (52) becomes and hence Since the functions η i are independent, then for appropriate choices of η i , we derive, by applying Lemma 2.4, the necessary conditions (45)- (47).

Remark 3.2:
If we put β = 0 in the previous theorem, then we obtain Theorem 3 in [24].

Isoperimetric problem
In this section we consider a problem in which the eligible function for the extremization of a given definite integral is required to satisfy certain restrictions that are added to the usual conditions.  1, 2, . . . , n) is continuous on [a, b]. Then, necessary conditions for J[q 1 , q 2 , . . . , q n ] to have an extremum at q * i , i = 1, 2, . . . , n, which satisfies the boundary conditions (30), such that are that q * i satisfy the following Euler-Lagrange equations If β = 1, then q * i should satisfy the transversality conditions where λ is the Lagrange's multiplier whose value can be determine by the conditions on L and z.
Proof: To derive the necessary conditions let where η i , and ζ i i = 1, 2, . . . , n, are arbitrary continuously differentiable functions for which (65) Inserting (64) in (59) and (60), respectively, we get and 2 ζ 1 (t), . . . , q * n (t) Clearly, the parameters 1 and 2 are not independent because I[ 1 , 2 ] = l.Since q * i , i = 1, 2, . . . , n are assumed to be the actual extermizing functions, we have J[ 1 , 2 ] is extremum with respect to 1 and 2 which satisfy (65), when 1 = 2 = 0. According to the method of Lagrange multipliers we introduce where λ is the Lagrange's multiplier. Then, according to the method of Lagrange multipliers we must have It follows, by applying the rule of integration by parts (16), As in the proof of Theorem 3.1, one can show that if β ∈ [0, 1), then by (66) one obtains Notice that equations (62) and (63) imply the validity of (72) when β = 1. Since setting 1 = 2 = 0 is equivalent to replacing q i and RH a D Consequently from (57)-(59), we get Since the functions η i and ς i are independent, the proof is finished.

Fractional variational problem with delay
We study the case when there is a delay on the system. Let τ ∈ (0, b − a) and consider the functional where achieves an extremum at q i , i = 1, 2, . . . , n, is that q i sat-isfy following Euler-Lagrange equations If β = 1, then q i should be verify the transversality condition Proof: We follow the approach discussed in the proof of Theorem 3.1, the necessary condition for q i , i = 1, 2, . . . , n to be extremum, is given by where η i are arbitrary continuously differentiable functions for which In the fourth and fifth term making the change of variables for t−r and taking into account that It follows from (82) and (83) that Using the usual rule integrating by parts and Equations (21), (22), equation (84) becomes This equation reduces to If β ∈ [0, 1), then by (81) and if β = 1, then by condition (79), equation (87) still true. Therefore, (86) becomes

Optimal time problem
In this section, we find the necessary conditions for a variational problem to have a extremum on an optimal time.
t ∈ [a, S], If β = 0, then the following transversality condition should be hold If β = 1, then the following transversality condition should be hold Proof: Let > 0 and define a family of curves q(t) = q * (t) + υ(t), where ν is an arbitrary continuously differentiable functions for which ν(a) = 0. Let T be a positive real number. Then the function depends on only. Since J admits an extremum at (q * , S) then the necessary condition for which J[ ] achieves a minimum, is Applying Lieibniz integral rule we get From (97) and (96) Since setting equal to zero is equivalent to replacing q i and R a D μ b q i by q * i and R a D μ b q i , the last equation becomes Now, to simplify the notations we put It follows from (99) and (100) that Remark that If β ∈ (0, 1), then by the continuity of ν and If β = 0, then from the assumption ν(a) = 0 and (93) and If β = 1, it follows from (94) that and Using equations (102)

Results and discussion
As mentioned earlier, variational problems and fractional calculus have many applications in different branches in engineering and mathematics, moreover, the Riesz-Hilfer fractional derivative (RHFD) is a generalization for the Riesz-Riemann-Liouville and the Riesz-Caputo derivative. The results that we obtained are to find Euler-Lagrange equations for various of fractional variational problems with the Lagrangian function containing (RHFD), and hence our results generalize many recent papers in the literature, for example, [22][23][24][25]36]. On other hand, our technique allows to generalize some works, such as the obtained results in [26] to the case when the functional involving (RHFD). As we mentioned in Corollary 2.1, relations (20) and (21) in [22] are particular cases of Lemma 2.1, and if β = 0 and i = 1 in Theorem 3.1, then we obtain Theorem 3.1, in [22]. Moreover, If we put β = 0 in both Theorems 3.2 and 6.1, we obtain Theorems 4.1 and 7.1, respectively, in [24].

Conclusion
Euler-Lagrange equations for different kind of fractional variational problems with the Lagrangian function containing the Riesz-Hilfer fractional derivative are obtained. Since the Riesz-Hilfer fractional derivative is a generalization for the Riesz-Riemann-Liouville and the Riesz-Caputo derivative, then our results generalize many recent works in which the Lagrangian function involving the Riesz-Riemann-Liouville or the Riesz-Caputo derivative. Fractional variational problem in the presence of delay derivatives is considered. Moreover, a version for Noether theorem in the Riesz-Hilfer sense is established. Necessary conditions for a pair function-time to be an optimal solution to the problem are investigated. Examples are given to illustrate the applicability of the obtained results. Furthermore, our obtained results generalize some existing results such as Theorem 1 in [22] and Theorems 3 and 6 in [24]. Also, the technique used in the present paper enable to extend the results in [26,28,31] when the treated problems in these works involving Riesz-Hilfer fractional derivative. Moreover, this work, may be, encourages to study partial differential equations containing Riesz-Hilfer fractional derivative.