Optimization Conditions for Some Fractional Problems

The purpose of this study is to present necessary conditions for calculus of variations problems, where the Lagrange function involves a Caputo fractional derivative with nonconstant order. The fractional operator depends on another function, and for particular choices of that function, we obtain some well-known fractional derivatives. Several necessary conditions of optimality are proven, namely the Euler-Lagrange equation.


I. INTRODUCTION
Non-integer order derivatives are an old branch of mathematics, and go back to the birth of differential calculus [29], [35]. In 1695, Leibniz asked L'Hôpital: "Can the meaning of derivatives with integer order be generalized to derivatives with non-integer orders?" Later, L'Hôpital answered him: "What if the order is 1/2?". Since then, the field of fractional calculus has been studied extensively, though at the beginning mainly in pure mathematics. More recently, it has been applied in several areas such as science, economy, engineering, etc.
3) Riemann-Liouville derivative (⌫ > 0): where n is the ceiling of ⌫. 4) Caputo derivative [13] (⌫ > 0): However, these are not the only existent definitions, and many others can be found in the literature. It is worth to mention that, when ⌫ is an integer number, let us say ⌫ = m 2 N, the fractional derivative operator is simply the usual derivative: . One way to reduce such number of concepts is to define fractional derivatives/integrals with respect to another function [35]. For more on the subject, we advice the reader to see [5]. Let ⌫ > 0 be a real, ⌥, k : [a, b] ! R be two functions with k differentiable and k 0 (t) > 0, for all t. The operators we are going to study are defined using another function. They are the following: 1) the k-fractional integral: 2) the k-Riemann-Liouville fractional derivative: where n = [⌫] + 1. 3) the k-Caputo fractional derivative: where n = d⌫e if ⌫ / 2 N, and n = ⌫ otherwise. We mention that, if ⌫ = m is an integer number, then If ⌫ is not an integer, then For special choices of k, we obtain the Caputo (when we consider k(t) = t), the Caputo-Hadamard (when the kernel is the function k(t) = ln(t)), the Caputo-Katugampola (k(t) = t ⇢ ) and the Caputo-Erdelyi-Kober (similar to the previous case) fractional derivatives. Recently, the purposed general form of differentiation has found applications (see [4], [7], [17], [39]). One extension of these works is to consider the order of the derivative not constant along the process [36]. Since most problems are dynamic processes, it makes sense that the order of the operator varies with the process. Recently, it has found applications in control, physics, etc [14], [19], [32], [38].
The objective is to formulate conditions for variational problems, where the derivative operator is defined with respect to another function and the order of the derivative is nonconstant. In this way, we present some generalizations of known theorems in the calculus of variations, for the fractional context. For special cases of k and ⌫, we obtain classic results of this theory. We present and prove several necessary conditions that allow us to find the possible extremizers of the functionals. This study started with Riewe [33], [34]. As Riewe refered, "Traditional Lagrangian and Hamiltonian mechanics cannot be used with nonconservative forces such as friction", and after it numerous works have come up dealing with different problems. Applications are being found nowadays, in particular in fractional optimal control problems [12], [21], [31], [40] The organization is the following. In Section II we present the needed definitions and results. Next, in Section III, we present some results on necessary conditions for minimization of a functional, and we end with a Conclusion section.

II. SOME DEFINITIONS ON FRACTIONAL CALCULUS
Throughout this paper, ⌫ is a function with domain [a, b] taking values between zero and one (⌫ is the fractional order). Also, x and k are two differentiable functions, defined on [a, b], and k is increasing with k 0 > 0. The k-Caputo fractional derivative (of order ⌫(·)), is given by For the power function ⌥(t) when considering k(t) = t 3/2 and k(t) = ln(t + 1), respectively.
Next, we present and prove an integration by parts formula, needed for the sequel. First, let us present two more definitions: Consequently, we obtain proving the desired formula.

III. CALCULUS OF VARIATIONS OF FRACTIONAL ORDER
In this section, we intend to present some conditions to determine the optimal solutions. The Lagrangian depends of the Caputo derivative, and the order of the derivative is not constant. The problem is the following: The function Z : is continuously differentiable for x and v, @ 2 Z and @ 3 Z. The boundary conditions may be imposed to the problem. The next theorem presents a necessary condition to determine the candidates for minimizers of the functional. Such type of equations are usually called Euler-Lagrange equations. Along our work we will assume that the map t 7 ! D , where x is a minimizer of the functional. Also, we will use the following notations. By {x}(t), we mean the 3-dimensional vector (t, x(t), C D ⌫(t),k a+ x(t)). By @ i f (z 1 , . . . , z m ), we mean @ i f (z 1 , . . . , z m ) := @f @z i (z 1 , . . . , z m ).

Theorem 2.
Suppose that x is a curve that minimizes F (1), defined on a set of functions that fulfill the condition (2). Then, x satisfies the following equation Proof. To prove this result, we start by considering a new function $ 2 C 1 [a, b]. Since x is fixed at the endpoints a and b, we must impose the conditions $(a) = 0 and also applying Theorem 1, we can deduce that Since $ is arbitrary in the open interval ]a, b[, we prove formula (3).
Theorem 3. If F attains a minimum value at x, then x satisfies Eq. (3) and the two transversality conditions Proof. Observe that $ is arbitrary in the closed interval [a, b], obtaining the desired formulas.
The case of functionals depending on a vector function x := (x 1 , . . . , x m ) is similar to the previous ones. Now, the fractional derivative of the vector x is defined as C D Then, for every t, where the index i = 1, . . . , m. Moreover, if x i (a) is free, then So far, both the fractional derivative and the cost integral (1) start at the same point t = a. The next theorem generalizes Theorem 2, by considering the case when the integral starts at a point t = A > a.
where A > a is a fixed real. If x minimizes F, then, for all t 2 [a, A], and for all t 2 [A, b], we also have the following: Moreover, we have following transversality conditions hold: Proof. Using the following relations and for appropriate choices of $, we obtain the desired result.
For our next result, besides the boundary conditions (2), we impose an integral constraint of type for a fixed real K. Again, we assume that P is differentiable with respect to x and v, and t 7 ! D Theorem 6. Suppose that F (1), subject to the restrictions (2) and (4), attains a minimum value at x. If x is not a solution of then there is 2 R for which x is a solution of where F := Z + P .
Proof. Define the two following auxiliary functions and and x does not satisfies Eq. (5), there is a function On the other hand, (0, 0) is a minimizer of j, with the restriction p(·, ·) = 0, Thus, we ensure the existence of 2 R with r(j + p)(0, 0) = (0, 0).
Next, we consider an holonomic constraint of type The function x is a two dimensional vector, that is, we are dealing with a function x of form x(t) = (x 1 (t), x 2 (t)). The function g : [a, b] ⇥ R 2 ! R is differentiable with respect to x 1 and x 2 . We also consider the conditions Theorem 7. Let subject to the constraints (6)- (7). Suppose that x is a minimizer for F and that, for every t 2 [a, b], The existence of a continuous function ⌅ : [a, b] ! R is ensured for which x verifies the equation for all t and i = 1, 2.

IV. EXAMPLES Let
Since J (x) 0, for an arbitrary x, and J (◆) = 0 for ◆ := (k(t) k(a)) 8 , we conclude easily that the solution is the power function ◆. Let us see that ◆ satisfies the Euler-Lagrange equation (3). In fact, Eq.
it is an easy exercise to verify that ◆ is a candidate for solution.

V. CONCLUSION
We studied some calculus of variations problems. Instead of considering Lagrangians depending on the first order derivative, we replace it by a fractional derivative with respect to k. Also, the order of the derivative is variable in time and takes any value in the interval (0, 1). An Euler-Lagrange equation was obtained, as well some other cases, namely the ones when in presence of an integral and holonomic constraints. The author wants to thank the reviewers for the positive remarks that helped to improve this work.