Variational Calculus With Conformable Fractional Derivatives

Ⅰ. INTRODUCTION

FRACTIONAL calculus is a generalization of (integer) differential calculus, allowing to define integrals and derivatives of real or complex order [1]-[3]. It had its origin in the 1600 s and for three centuries the theory of fractional derivatives developed as a pure theoretical field of mathematics, useful only for mathematicians. The theory took more or less finished form by the end of the 19th century. In the last few decades, fractional differentiation has been "rediscovered" by applied scientists, proving to be very useful in various fields: physics (classic and quantum mechanics, thermodynamics, etc.), chemistry, biology, economics, engineering, signal and image processing, and control theory [4]. One can find in the existent literature several definitions of fractional derivatives, including the Riemann-Liouville, Caputo, Riesz, Riesz-Caputo, Weyl, Grunwald-Letnikov, Hadamard, and Chen derivatives. Recently, a simple solution to the discrepancies between known definitions was presented with the introduction of a new fractional notion, called the conformable derivative [5]. The new definition is a natural extension of the usual derivative, and satisfies the main properties one expects in a derivative: the conformable derivative of a constant is zero; satisfies the standard formulas of the derivative of the product and of the derivative of the quotient of two functions; and satisfies the chain rule. Besides simple and similar to the standard derivative, one can say that the conformable derivative combines the best characteristics of known fractional derivatives [6]. For this reason, the subject is now under strong development: see [7]-[10] and references therein.

The fractional calculus of variations was introduced in the context of classical mechanics when Riewe [11] showed that a Lagrangian involving fractional time derivatives leads to an equation of motion with non-conservative forces such as friction. It is a remarkable result since frictional and non-conservative forces are beyond the usual macroscopic variational treatment [12]. Riewe generalized the usual calculus of variations for a Lagrangian depending on Riemann-Liouville fractional derivatives [11] in order to deal with linear non-conservative forces. Actually, several approaches have been developed to generalize the calculus of variations to include problems depending on Caputo fractional derivatives, Riemann-Liouville fractional derivatives, Riesz fractional derivatives and others [13]-[19] (see [20]-[22] for the state of the art). Among theses approaches, recently it was shown that the action principle for dissipative systems can be generalized, fixing the mathematical inconsistencies present in the original Riewe's formulation, by using Lagrangians depending on classical and Caputo derivatives [23].

In this paper we work with conformable fractional derivatives in the context of the calculus of variations and optimal control [20]. In order to illustrate the potential application of conformable fractional derivatives in physical problems we show that it is possible to formulate an action principle with conformable fractional calculus for the frictional force free from the mathematical inconsistencies found in the Riewe original approach and far simpler than the formulations proposed in [23]. Furthermore, we obtain a generalization of Noether's symmetry theorem for fractional variational problems and we also consider conformable fractional optimal control problems. Emmy Noether was the first who proved, in 1918, that the notions of invariance and constant of motion are connected: when a system is invariant under a family of transformations, then a conserved quantity along the Euler-Lagrange extremals can be obtained [24], [25]. All conservation laws of Mechanics, e.g., conservation of energy or conservation of momentum, are easily explained from Noether's theorem. In this paper we study necessary conditions for invariance under a family of continuous transformations, where the Lagrangian contains a conformable fractional derivative of order $\alpha\in (0, 1)$ . When $\alpha\to1$ , we obtain some well-known results, in particular the Noether theorem [25]. The advantages of our fractional results are clear. Indeed, the classical constants of motion appear naturally in closed systems while in practical terms closed systems do not exist: forces that do not store energy, so-called nonconservative or dissipative forces, are always present in real systems. Fractional dynamics provide a good way to model nonconservative systems [11]. Nonconservative forces remove energy from the systems and, as a consequence, the standard Noether constants of motion are broken [26]. Our results assert that it is still possible to obtain Noether-type theorems, which cover both conservative and nonconservative cases, and that this is done in a particularly simple and elegant way via the conformable fractional approach. This is in contrast with the approaches followed in [27]-[30].

The paper is organized as follows. In Section Ⅱ we collect some necessary definitions and results on the conformable fractional calculus needed in the sequel. In Section Ⅲ we obtain the conformable fractional Euler-Lagrange equation and in Section Ⅳ we formulate an action principle for dissipative systems, as an example of application and motivation to study the conformable calculus of variations. In Section V, we present an immediate consequence of the Euler-Lagrange equation that we use later in Sections Ⅵ and Ⅶ, where we prove, respectively, some necessary conditions for invariant fractional problems and a conformable fractional Noether theorem. We then review the obtained results using the Hamiltonian language in Section Ⅷ. In Section Ⅸ, we consider the conformable fractional optimal control problem, where the dynamic constraint is given by a conformable fractional derivative. Using the Hamiltonian language, we provide an invariant condition. In Section Ⅹ we consider the multi-dimensional case, for several independent and dependent variables.

Ⅱ. PRELIMINARIES

In this section we review the conformable fractional calculus [5]-[7]. The conformable fractional derivative is a new well-behaved definition of fractional derivative, based on a simple limit definition. We review in this section the generalization of [5] proposed in [6].

Definition 1: The left conformable fractional derivative of order $0 < \alpha \leq 1$ starting from $a \in {\mathbb R}$ of a function $f:[a, b]\rightarrow {\mathbb R}$ is defined by

If the limit (1) exist, then we say that f is left $\alpha$ -differentiable. Furthermore, if $f^{(\alpha)}_a (x)$ exist for $x \in (a, b)$ , then

and

The right conformable fractional derivative of order $\alpha \in (0, 1]$ terminating at $b \in {\mathbb R}$ of a function $f:[a, b]\rightarrow {\mathbb R}$ is defined by

If the limit (2) exist, then we say that f is right $\alpha$ -differentiable. Furthermore, if ${_bf^{(\alpha)}}(x)$ exist for $x\in (a, b)$ , then

and

It is important to note that for $\alpha=1$ the conformable fractional derivatives (1) and (2) reduce to first order ordinary derivatives. Furthermore, despite the definition of the conformable fractional derivatives (1) and (2) can be generalized for $\alpha>1$ [6], we consider only $0 < \alpha \leq 1$ in the present work. Is is also important to note that, differently from the majority of definitions of fractional derivative, including the popular Riemann-Liouville and Caputo fractional derivatives, the fractional derivatives (1) and (2) are local operators and are related to ordinary derivatives if the function is differentiable (see Remark 1). For more on local fractional derivatives, we refer the reader to [31], [32] and references therein.

Remark 1: If $f\in C^1[a, b]$ , then we have from (1) that

and from (2) that

where $f'(x)$ stands for the ordinary first order derivative of $f (x)$ .

From (3) and (4) it is easy to see that the conformable fractional derivative of a constant is zero, differently from the Riemann-Liouville derivative of a constant, and for the power functions $(x-a)^p$ and $(b-x)^p$ one has

and

for all $p\in {\mathbb R}$ .

The most remarkable consequence of definitions (1) and (2) is that the conformable fractional derivatives satisfy very simple fractional versions of chain and product rules.

Proposition 1 [5], [6]: Let $0 < \alpha < 1$ and f and $g$ be $\alpha$ -differentiable functions. Then,

1)

and

for all $c_1, c_2\in {\mathbb R}$ .

2)

and

3)

and

4) If $g (x) \ge a$ , then

5) If $g (x) \le b$ , then

6) If $g (x) < a$ , then

7) If $g (x) > b$ , then

The simple chain and product rules given in Proposition 1 justify the increasing interest in the study of the conformable fractional calculus, since it enable us to investigate its potential applications as a tool to practical modeling of complex problems in science and engineering.

The conformable fractional integrals are defined as follows [5], [6].

Definition 2: The left conformable fractional integral of order $0 < \alpha \leq 1$ starting from $a \in {\mathbb R}$ of a function $f\in L^1[a, b]$ is defined by

and the right conformable fractional integral of order $0 < \alpha \leq 1$ terminating at $b \in {\mathbb R}$ of a function $f\in L^1[a, b]$ is defined by

It is important to mention that the conformable fractional integrals (5) and (6) differ from the traditional fractional Riemann-Liouville integrals [1]-[3] only by a multiplicative constant. Moreover, for $\alpha=1$ , the conformable fractional integrals reduce to ordinary first order integrals.

In addition to these definitions, in the present work we make use of the following properties of conformable fractional derivatives and integrals.

Theorem 1: Let $f\in C [a, b]$ and $0 < \alpha \leq 1$ . Then,

and

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

Theorem 2 (Fundamental theorem of conformable fractional calculus): Let $f\in C^1[a, b]$ and $0 < \alpha \leq 1$ . Then,

and

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

Theorem 3 (Integration by parts): Let $f, g:[a, b]\rightarrow {\mathbb R}$ be two functions such that $fg$ is differentiable. Then,

and

If $f, g:[a, b]\rightarrow {\mathbb R}$ are differentiable functions, then

The proof of Theorem 1 follows directly from (3), (4), (5) and (6) since $I^{\alpha}_a f (x)$ and ${_bI^{\alpha}} f (x)$ are differentiable. On the other hand, the fundamental theorem of the conformable fractional calculus (Theorem 2) is a direct consequence of (3), (4) and definitions (5) and (6) since $f, g:[a, b]\rightarrow {\mathbb R}$ are differentiable functions. Finally, the integration by parts (7) and (8) follow from Proposition 1 and Theorem 1. We also need the following result.

Theorem 4 (Chain rule for functions of several variables): Let $f:{{\mathbb{R}}^{N}}\to \mathbb{R}$ ( $N\in \mathbb{N}$ )} be a differentiable function in all its arguments and $y_1, ..., y_N:{\mathbb R} \to {\mathbb R}$ be $\alpha$ -differentiable functions. Then,

and

Proof : For simplicity, we prove (9) only for $N=2$ . The proofs for a general $N$ and of (10) are similar. From (1) we have for $N=2$ that (by writing $\overline{x}=x+\epsilon (x-a)^{1-\alpha}$ for simplicity)

since f is differentiable.

Ⅲ. THE CONFORMABLE EULER-LAGRANGE EQUATION

Let us first consider the fractional variational integral

defined on the set of continuous functions $y:[a, b]\to {\mathbb R}$ such that $y^{(\alpha)}_a$ exists on $[a, b]$ , where the Lagrangian $L$ , $L (x, y, y^{(\alpha)}_a):[a, b]\times{\mathbb R}^2 \to {\mathbb R}$ , is of class $C^1$ in each of its arguments. The fundamental problem of the calculus of variations consists in finding which functions extremize functional (11). In order to obtain a necessary condition for the extremum of (11) we need the following Lemma.

Lemma 1 (Fundamental lemma for the conformable calculus of variations): Let $M$ and $\eta$ be continuous functions on $[a, b]$ .

If

for any $\eta \in C [a, b]$ with $\eta (a)=\eta (b)=0$ , then

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

Proof : We do the proof by contradiction. From (12) we have that

Suppose that there exist an $x_0 \in (a, b)$ such that $M (x_0)\neq 0$ . Without loss of generality, let us assume that $M (x_0)>0$ . Since $M$ is continuous on $[a, b]$ , there exists a neighborhood $N^{\delta}(x_0) \subset (a, b)$ such that

Let us choose

Clearly, $\eta (x)$ given by (15) is continuous and satisfy $\eta (a)=0$ and $\eta (b)=0$ . Inserting (15) into (14), we obtain that

which contradicts our hypothesis. Thus,

Since $(x-a)^{1-\alpha}>0$ for $x\in (a, b)$ , and since $M\in C [a, b]$ , we get

Theorem 5 (The conformable fractional Euler-Lagrange equation): Let $\mathcal{J}$ be a functional of form (11) with $L\in C^1$ $\left ([a, b]\times {\mathbb R}^2\right)$ , and $0 < \alpha \leq 1$ . Let $y:[a, b]\rightarrow {\mathbb R}$ be a $\alpha$ -differentiable function with $y (a)=y_a$ and $y (b)=y_b$ , $y_a,$ $y_b\in {\mathbb R}$ . Furthermore, let $y \frac{\partial L}{\partial y^{(\alpha)}_a}$ be a differentiable function and $\frac{\partial L}{\partial y^{(\alpha)}_a}$ be $\alpha$ -differentiable. If $y$ is an extremizer of $\mathcal{J}$ , then $y$ satisfies the following fractional Euler-Lagrange equation:

Proof : Let $y^{\ast}$ give an extremum to (11). We define a family of functions

where $\epsilon$ is a constant and $\eta$ is an arbitrary $\alpha$ -differentiable function satisfying $\eta\frac{\partial L}{\partial {y^{\ast}}^{(\alpha)}_a} \in C^1$ and the boundary conditions $\eta (a)=\eta (b)=0$ (weak variations). From (17), the boundary conditions $\eta (a)=\eta (b)=0$ , and the fact that $y^{\ast}(a)=y_a$ and $y^{\ast}(b)=y_b$ , it follows that function $y$ is admissible: $y$ is $\alpha$ -differentiable with $y (a)=y_a$ , $y (b)=y_b$ , and $y\frac{\partial L}{\partial {y^{\ast}}^{(\alpha)}_a}$ is differentiable. Let the Lagrangian $L$ be $C^1([a, b] \times {\mathbb R}^2)$ . Because $y^{\ast}$ is an extremizer of functional $\mathcal{J}$ , the Gateaux derivative $\delta \mathcal{J}(y^{\ast})$ needs to be identically null. For the functional (11),

Using the integration by parts formula (7) ( $\eta\frac{\partial L}{\partial {y^\ast}^{(\alpha)}_a}$ is differentiable), we get

since $\eta (a)=\eta (b)=0$ . The fractional Euler-Lagrange equation (16) follows from (18) by using the fundamental Lemma 1.

Definition 3: A continuous function $y$ solution of (16) is said to be an extremal of (11).

Remark 2: For $\alpha=1$ , the functional $\mathcal{J}$ given by (11) reduces to the classical variational functional

and the associated Euler-Lagrange equation (16) is

Let us now consider the more general case where the Lagrangian depends on both integer order and fractional order derivatives. In this case the following theorem holds.

Theorem 6 (The generalized conformable fractional Euler-Lagrange equation): Let $\mathcal{J}$ be a functional of form

with $L\in C^1\left ([a, b]\times {\mathbb R}^3\right)$ , and $0 < \alpha \leq 1$ . Let $y:[a, b]\rightarrow {\mathbb R}$ be a differentiable function with $y (a)=y_a$ and $y (b)=y_b$ , $y_a, y_b\in {\mathbb R}$ . If $y$ is an extremizer of $\mathcal{J}$ , then $y$ satisfies the following fractional Euler-Lagrange equation:

where $\tilde{L}\left (x, y, y', y^{(\alpha)}_a\right) =(x-a)^{1-\alpha}L\left (x, y, y', y^{(\alpha)}_a\right)$ .

Proof : Let $y^{\ast}$ give an extremum to (20). We define a family of functions as in (17) but with $y\in C^1[a, b]$ . From (17) and the boundary conditions $\eta (a)=\eta (b)=0$ , and the fact that $y^{\ast}(a)=y_a$ and $y^{\ast}(b)=y_b$ , it follows that function $y$ is admissible. Because $y^{\ast}$ is an extremizer of $\mathcal{J}$ , the Gateaux derivative $\delta \mathcal{J}(y^{\ast})$ needs to be identically null. For the functional (20) we have

where we performed an integration by parts in the second term in the first integral (since $\eta (a)=\eta (b)=0$ ), and we rewrote the second integral as a conformable integral by using definition (5). Using the integration by parts formula (7) ( $\eta\frac{\partial L}{\partial {y^\ast}^{(\alpha)}_a}$ is differentiable), we get

since $\eta (a)=\eta (b)=0$ . The fractional Euler-Lagrange equation (21) follows from (22) by using the fundamental Lemma 1.

Ⅳ. LAGRANGIAN FORMULATION FOR FRICTIONAL FORCES

As an example of potential application of the variational calculus with conformable fractional derivatives, we formulate an action principle for dissipative systems free from the mathematical inconsistencies found in the Riewe approach [23] and far simpler than the formulation proposed in [23]. The action principle we propose states that the equation of motion for dissipative systems is obtained by taking the limit $a\rightarrow b$ in the extremal of the action

that satisfy the fractional Euler-Lagrange equation (see (21))

where $\tilde{L}\left (x, x', x^{(\alpha)}_a\right) =(t-a)^{1-\alpha}L\left (x, x', x^{(\alpha)}_a\right)$ , $x (t)$ is the path of the particle and $t$ is the time. It is important to emphasize that the condition $a\rightarrow b$ (also considered in the original Riewe's approach) applied to the action principle does not imply any restrictions for conservative systems, since in this case $x (t)$ is the action's extremal for any time interval $[a, b]$ , even when $a\rightarrow b$ . Furthermore, our action principle is simpler than the formulation in [23] and free from the mathematical inconsistencies present in Riewe's approach (see [23] for a detailed discussion). In order to show that our method provides us with physical Lagrangians, let us consider the simple problem of a particle under a frictional force proportional to velocity. A quadratic Lagrangian for a particle under a frictional force proportional to the velocity is given by

where the three terms in (25) represent the kinetic energy, potential energy, and the fractional linear friction energy, respectively. Note that, differently from Riewe's Lagrangian [11], our Lagrangian (25) is a real function with a linear friction energy, which is physically meaningful. Since the equation of motion is obtained in the limit $a\rightarrow b$ , if we consider the last term in (25) up to first order in $\Delta t=t-a$ , we get

that coincides, apart from the multiplicative constant 1/2, with the work from the frictional force $\gamma x'$ in the displacement $\Delta x \approx x'\Delta t$ . The appearance of an additional multiplicative constant is a consequence of the use of fractional derivatives in the Lagrangian and does not appear in the equation of motion after we apply the action principle [23].

Remark 3: It is important to stress that the order of the fractional derivative should be fixed to $\alpha=1/2$ in order to obtain, by a fractional Lagrangian, a correct equation of motion of a dissipative system. For $\alpha$ different from $1/2$ , the Lagrangian does not describe a frictional system under a frictional force proportional to the velocity. Consequently, the fractional linear friction energy makes sense only for $\alpha=1/2$ .

The Lagrangian (25) is physical in the sense it provides physically meaningful relations for the momentum and the Hamiltonian. If we define the canonical variables

and

we obtain the Hamiltonian

From (26) we can see that the Lagrangian (25) is physical in the sense it provides us a correct relation for the momentum $p_1=m\dot{x}$ and a physically meaningful Hamiltonian (it is the sum of all energies). Furthermore, the additional fractional momentum $p_{\frac{1}{2}}=\gamma x^{(\frac{1}{2})}_a$ goes to zero when we take the limit $a\rightarrow b$ , since $x\in C^2[a, b]$ .

Finally, the equation of motion for the particle is obtained by inserting our Lagrangian (25) into the Euler-Lagrange equation (24),

where we have used (3) since $x\in C^2[a, b]$ and

is the external force. By taking the limit $a\rightarrow b$ with $t\in [a, b]$ , we finally obtain the correct equation of motion for a particle under a frictional force:

Ⅴ. THE CONFORMABLE DUBOIS-REYMOND CONDITION

In the remainder of the present work, we are going to consider only the simplest case where we have no mixed integer and fractional derivatives. We now present the DuBois-Reymond condition in the conformable fractional context. It is an immediate consequence of the chain rule (9) and the Euler-Lagrange equation (16).

Theorem 7 (The conformable fractional DuBois-Reymond condition): If $y$ is an extremal of $\mathcal{J}$ as in (11), then

Proof : By the chain rule (9) and the Leibniz rule in Proposition 1:

Corollary 1: If (11) is autonomous, that is, if $L=L (y, y^{(\alpha)}_a)$ does not depend on $x$ , then

along any extremal $y$ .

Remark 4: When $\alpha = 1$ and $y\in C^1$ , Theorem 7 is the classical DuBois-Reymond condition: if $y\in C^1$ is an extremal of $\mathcal{J}(y)=\int_0^1L (x, y, y') dx$ (i.e., $y$ satisfies (19)), then

Ⅵ. FRACTIONAL INVARIANT CONDITIONS

We consider invariance transformations in the $(x, y)$ -space, depending on a real parameter $\epsilon$ . To be more precise, we consider transformations of type

where the generators $\tau$ and $\xi$ are such that $\overline x \ge a$ and there exist $\tau^{(\alpha)}_a$ and $\xi^{(\alpha)}_a$ .

Definition 4: We say that the fractional variational integral (11) is invariant under the family of transformations (29) up to the Gauge term $\Lambda$ , if a function $\Lambda=\Lambda (x, y)$ exists such that for any function $y$ and for any real $x\in[a, b]$ , we have

for all $\epsilon$ in some neighborhood of zero, where $\frac{d^\alpha_a\overline x}{d^\alpha_a x}$ stands for

We note that for $\alpha=1$ our Definition 4 coincides with the standard approach (see, e.g., [33]). When $\Lambda\equiv 0$ , one obtains the concept of absolute invariance. The presence of a new function $\Lambda$ is due to the presence of external forces in the dynamical system, like friction. The function $\Lambda$ is called a Gauge term. In fact, many phenomena are nonconservative and this has to be taken into account in the conservation laws [26], [34]. We give an example.

Example 1: Consider the transformation

and the functional

Since

it is easy to verify that (33) is invariant under (32) up to the Gauge function $\Lambda=y$ .

Definition 5: Given a function $C=C (x, y, y^{(\alpha)}_a)$ , we say that $C$ is a conserved quantity for (11) if

along any solution $y$ of (16) (i.e., along any extremal of (11)).

Remark 5: Applying the conformable integral (5) to both sides of equation (34), Definition 5 is equivalent to $C (x, y (x), y^{(\alpha)}_a (x)) \equiv const$ .

We now provide a necessary condition of invariance.

Theorem 8: If $\mathcal{J}$ given by (11) is invariant under a family of transformations (29), then

Proof : By the fractional chain rule (see Proposition 1),

Substituting this formula into (30), differentiating with respect to $\epsilon$ and then putting $\epsilon=0$ , we obtain relation (35).

Remark 6: Allowing $\alpha$ to be equal to 1, for $\Lambda\equiv0$ our equation (35) becomes the standard necessary condition of invariance (cf., e.g., [24]):

For $\alpha=1$ and an arbitrary $\Lambda$ , see [33].

In particular, if we consider "time invariance" (i.e., $\tau\equiv0$ ), we obtain the following result.

Corollary 2: Let $\overline y=y+\epsilon \xi (x, y (x))$ be a transformation that leaves invariant $\mathcal{J}$ in the sense that

Then,

Ⅶ. THE CONFORMABLE NOETHER THEOREM

Noether's theorem is a beautiful result with important implications and applications in optimal control [35]-[37]. We provide here a conformable fractional Noether theorem in the context of the calculus of variations. Later, in Section Ⅸ, we provide a conformable fractional optimal control version (see Theorem 11).

Theorem 9 (The conformable fractional Noether theorem): If $\mathcal{J}$ given by (11) is invariant under (29) and if $y$ is an extremal of $\mathcal{J}$ , then

Proof : From Theorem 8, and using the conformable fractional Euler-Lagrange equation (16) and the DuBois-Reymond condition (28), we deduce successively that

Thus, we obtain equation (36).

Remark 7: When $\alpha=1$ , equation (36) is simply Noether's conservation law in the presence of external forces: for any extremal of $\mathcal{J}$ and for any family of transformations $(\overline x, \overline y)$ for which $\mathcal{J}$ is invariant, the conservation law

holds [33, Theorem 2.1]. In addition, if system is conservative ( $\Lambda\equiv 0$ ), then one has the classical Noether theorem

Corollary 3 (The conformable fractional Noether theorem under the presence of an external force f): If $\mathcal{J}$ given by (11) is invariant under (29), $y$ is an extremal of $\mathcal{J}$ , and the function $f=f (x, y, y^{(\alpha)}_a)$ satisfies the equation

then

is a conserved quantity.

Corollary 4: If $\mathcal{J}$ given by (11) is invariant under the transformation $\overline x=x$ , $\overline y=y+\epsilon \xi (x, y (x))$ , and if $y$ is an extremal of $\mathcal{J}$ , then

is a conserved quantity.

Proof : Given that

the result follows immediately from Theorem 9.

Ⅷ. THE HAMILTONIAN FORMALISM

The Hamiltonian formalism is related to the Lagrangian one by the so called Legendre transformation, from coordinates and velocities to coordinates and momenta. Let the momenta be given by

and the Hamiltonian function by

To simplify notation, $[y](x)$ and $\{y\}(x)$ will denote $(x, y (x), y^{(\alpha)}_a (x))$ and $(x, y (x), y^{(\alpha)}_a (x), p (x))$ , respectively. Differentiating (38), and using definition (37), it follows that

On the other hand, by the definition of Hamiltonian (38), one has immediately that

and so we can write (39) in the form

If $y$ is an extremal of $\mathcal{J}$ , then by the conformable fractional Euler-Lagrange equation (16) one has

and we can write

The system (41) is nothing else than the conformable fractional Euler-Lagrange equation in Hamiltonian form. Substituting the expressions of (41) into (40), we get the analog to the DuBois-Reymond condition (28) in Hamiltonian form:

If the Lagrangian $L$ is autonomous, i.e., $L$ does not depend on $x$ , then

and, consequently, by (42) $H$ is a conserved quantity. If the Lagrangian $L$ does not depend on $y$ , then

and so $p^{(\alpha)}_a=0$ , i.e., $p$ is a conserved quantity.

We now exhibit Corollary 3 within the Hamiltonian framework.

Theorem 10 (Conformable fractional Noether's theorem in Hamiltonian form under the presence of an external force f): If $\mathcal{J}$ given by (11) is invariant under (29), $y$ is an extremal of $\mathcal{J}$ , and function $f=f (x, y (x), y^{(\alpha)}_a (x))$ satisfies the equation

then

is a conserved quantity.

Ⅸ. CONFORMABLE FRACTIONAL OPTIMAL CONTROL

The conformable fractional optimal control problem is stated as follows: find a pair of functions $(y (\cdot), v (\cdot))$ that minimizes

when subject to the (nonautonomous) fractional control system

A pair $(y (\cdot), v (\cdot))$ that minimizes functional (43) subject to (44) is called an optimal process. The reader interested on the fractional optimal control theory is referred to [28], [29], [38]. Here we note that if $\alpha=1$ , then (43) and (44) is the standard optimal control problem: to minimize

subject to the control system

We assume that the Lagrangian $L$ and the velocity vector $\varphi$ are functions at least of class $C^1$ in their domain $[a, b]\times {\mathbb R}^2$ . Also, the admissible state trajectories $y$ are such that $y^{(\alpha)}_a$ exist.

Remark 8: In case $\varphi \equiv v$ , the previous problem (43) and (44) reduces to the fundamental problem of the conformable fractional variational calculus (11), as stated in Section Ⅲ.

Following the standard approach [37], [39], we consider the augmented conformable fractional functional

where $p$ is such that $p^{(\alpha)}_a$ exists. Consider a variation vector of type $(y+\epsilon y_1, v+\epsilon v_1, p+\epsilon p_1)$ with $|\epsilon| \ll 1$ . For convenience, we restrict ourselves to the case $y_1(a)=y_1(b)=0$ . If $(y (\cdot), v (\cdot))$ is an optimal process, then the first variation is zero when $\epsilon=0$ . Thus, using the conformable fractional integration by parts formula (Theorem 3), we obtain that

By the arbitrariness of the the variation functions, we obtain the following system, called the Euler-Lagrange equations for the conformable fractional optimal control problem:

These equations give necessary conditions for finding the optimal solutions of problem (43) and (44). We remark that they are similar to the standard ones, in case of integer order derivatives, but in this case they contain conformable fractional derivatives, as expected. The solution can be stated using the Hamiltonian formalism. Consider the Hamiltonian function

Then (46) gives:

1) The fractional Hamiltonian system

2) The stationary condition

Definition 6: Any triplet $(y, v, p)$ satisfying system (48) and (49) is called a conformable fractional Pontryagin extremal.

Remark 9: In the particular case $\varphi \equiv v$ , i.e., when the conformable fractional optimal control problem is reduced to the fundamental conformable fractional problem of the calculus of variations, we obtain

and the equations

Therefore, we obtain the conformable fractional Euler-Lagrange equation (16):

Let us now considerer the augmented fractional variational functional (45) written in the Hamiltonian form:

where $H$ is given by expression (47). For a parameter $\epsilon$ , with $|\epsilon| \ll 1$ , consider the family of transformations

We now define the notion of invariance of (43)-(44) in terms of the Hamiltonian $H$ and the augmented conformable fractional variational functional (50).

Definition 7: The conformable fractional optimal control problem (43) and (44) is invariant under the transformations (51) up to the Gauge term $\Lambda$ , if a function $\Lambda=\Lambda (x, y)$ exists such that for any functions $y, v$ and $p$ , and for any real $x\in[0,1]$ , the following equality holds:

for all $\epsilon$ in some neighborhood of zero, where as in Definition 4 $\frac{d^\alpha_a\overline x}{d^\alpha_a x}$ stands for (31).

Theorem 11 (Fractional Noether's theorem for the fractional optimal control problem (43)-(44)): If (43) and (44) is invariant under (51) in the sense of Definition 7, and if $(y, v, p)$ is a conformable fractional Pontryagin extremal, then

Proof : Differentiating (52) with respect to $\epsilon$ , then choosing $\epsilon=0$ , we get

Equation (53) follows because $(y, v, p)$ is a conformable fractional Pontryagin extremal.

Remark 10: When $\alpha=1$ and $\Lambda=0$ , equation (53) becomes

Using relations (48) and (49) with $\alpha=1$ , we deduce that

which is the optimal control version of Noether's theorem [35]-[37]. For $\alpha\in (0, 1)$ , Theorem 11 extends the main result of [28].

Ⅹ. THE MULTI-DIMENSIONAL CASE

In this section, we show a necessary condition of invariance, when the Lagrangian depends on two independent variables $x_1$ and $x_2$ and on $m$ functions $y_1, ..., y_m$ . First, we define conformable fractional partial derivatives and conformable multiple fractional integrals in a natural way, similarly as done in the integer case. In addition, we are going to use the following properties.

Theorem 12 (Conformable Green's theorem for a rectangle): Let f and $g$ be two continuous and $\alpha$ -differentiable functions whose domains contain $R=[a, b]\times[c, d]\subset {\mathbb R}^2$ . Then,

Proof : By Theorem 2, we have

and

Therefore,

Remark 11: From Definition 2 and Remark 1, it is easy to verify that for $C^1$ functions our fractional Green's theorem over a rectangular domain (Theorem 12) reduces to the conventional Green's identity for

and

Lemma 2: Let f, $G$ and $h$ be $\alpha$ -differentiable continuous functions whose domains contain $R=[a, b]\times[c, d]$ . If $h=0$ on the boundary $\partial R$ of $R$ , then

Proof : By choosing $f=Fh$ and $g=Gh$ in Green's formula (54), we obtain that

Since $h=0$ on the boundary $\partial R$ of $R$ , we have

Remark 12: In the very recent and general paper [40], a vector calculus with deformed derivatives (as the conformable derivative) is formally introduced. We refer the reader to [40] for a detailed discussion of a vector calculus with deformed derivatives and more properties on the multi-dimensional conformable calculus.

Let us now consider the fractional variational integral

where for simplicity we choose $R=[a, b] \times [a, b]$ , and where $x=(x_1, x_2)$ , $y=(y_1, ..., y_m)$ , $d^\alpha_a x=d^\alpha_a x_1d^\alpha_a x_2$ , and

We are assuming that the Lagrangian

is at least of class $C^1$ , that the domains of $y_k$ , $k\in\{1, ..., m\}$ , contain $R$ , and that all these partial conformable fractional derivatives exist.

Theorem 13 (The multi-dimensional fractional Euler-Lagrange equation): Let $y$ be an extremizer of (56) with

for some given function $\psi=(\psi_1, ..., \psi_m)$ . Then, the following equation holds:

for all $k\in\{1, ..., m\}$ .

Proof : Let $y^{\ast}=(y_1^{\ast}, ..., y_m^{\ast})$ give an extremum to (56). We define $m$ families of functions

where $k\in \{1, ..., m\}$ , $\epsilon$ is a constant, and $\eta_k$ is an arbitrary $\alpha$ -differentiable function satisfying the boundary conditions $\eta_k|_{\partial R}=0$ (weak variations). From (58), the boundary conditions $\eta_k|_{\partial R}=0$ and $y_k|_{\partial R}=\psi_k (x_1, x_2)$ , it follows that function $y_k$ is admissible. Let the Lagrangian $L$ be $C^1$ . Because $y^{\ast}$ is an extremizer of functional $\mathcal{J}$ , the Gateaux derivative $\delta \mathcal{J}(y^{\ast})$ needs to be identically null. For the functional (56),

Using (55), we get that

since $\eta_k|_{\partial R}=0$ . The fractional Euler-Lagrange equation (57) follows from (59) by using the fundamental lemma of the conformable fractional calculus of variations (Lemma 1).

Let $\epsilon$ be a real, and consider the following family of transformations:

where $\tau_i$ and $\xi_k$ are such that there exist $\frac{\partial^\alpha_a\tau_i}{\partial {x_j}^\alpha_a}$ and $\frac{\partial ^\alpha_a\xi_k}{\partial {x_j}^\alpha_a}$ for all $i, j \in \{1, 2\}$ and all $k \in \{1, ..., m\}$ . Denote by $\left[\frac{\partial^\alpha_a \overline x}{\partial^\alpha_a x} \right]$ the matrix

Definition 8: Functional $\mathcal{J}$ as in (56) is invariant under the family of transformation (60) if for all $y_k$ and for all $x_i\in[0,1]$ we have

for all $\epsilon$ in some neighborhood of zero.

Using the same techniques as in the proof of Theorem 8, we obtain a necessary condition of invariance for the fractional variational problem (56).

Theorem 14: If $\mathcal{J}$ given by (56) is invariant under transformations (60), then

Proof : Using relations

and

we conclude that (61) holds.

Remark 13: When $\alpha=1$ and $\Lambda\equiv0$ , Theorem 14 reduces to the standard one [24]: equality (61) simplifies to

Corollary 5: If $\mathcal{J}$ given by (56) is invariant under (60), $\tau_1\equiv0\equiv\tau_2$ , and no Gauge term is involved (i.e., $\Lambda\equiv0$ ), then

It remains an open question how to obtain a Noether constant of motion for the conformable fractional multi-dimensional case.