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Optimality conditions for fractional variational problems with CaputoFabrizio fractional derivatives
Advances in Difference Equations volume 2017, Article number: 357 (2017)
Abstract
In this paper, we study the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrange function depending on a CaputoFabrizio fractional derivative. The new kernel of CaputoFabrizio fractional derivative has no singularity, which is critical to interpreting the memory aftermath of the system. This property was not precisely illustrated in the previous definitions. Two special cases of fractional variational problems are considered to demonstrate the application of the optimality conditions.
1 Introduction
Since the introduction of fractional calculus of variations by Riewe [1], fractional calculus has played an important role in dealing with many natural dynamical processes. Riewe showed that the traditional Lagrangian and Hamiltonian mechanics cannot be used with nonconservative forces such as friction. However, fractional calculus may be better to describe the behavior of the natural processes because of its memory property. So, 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. Agrawal [2–4] studied the fractional EulerLagrange equations for general fractional variational problems (FVP) involving RiemannLiouville, Caputo and Riesz fractional derivatives. Almeida investigated optimality conditions for fractional variational problems with a Lagrangian depending on the RieszCaputo derivative [5] and the CaputoKatugampola derivative [6]. In [7], Almeida exhibited the conditions of optimality for functionals depending on Caputo fractional integrals and derivatives, on indefinite integrals and on the presence of time delay. Xu and Agrawal [8] deduced the EulerLagrange equation of the fractional variational problem involving a modified Hilfer fractional derivative. Farhadinia [9], Fard [10] and Soolaki [11] established the necessary optimality conditions for fuzzy fractional variational problems by using the generalized Hukuhara differentiability concept.
The most popular fractional calculi are RiemannLiouville (RL) and Caputo type. But it is well known that the two derivatives have some drawbacks. For example, the RL derivative of a constant is not zero and it demands initial conditions of noninteger order which are not physically determined, and the Caputo derivative requires higher conditions of regularity for differentiability, which is specified only for differentiable functions.
In 2015, Caputo and Fabrizio [12] proposed a new fractional derivative with nonsingular kernel which conveniently portrays the performance of material heterogeneities and the structures with different scales. The main difference between the Caputo derivative and the CaputoFabrizio (CF) fractional derivative is that the new kernel has no singularity. Losada and Nieto studied some properties of the new fractional derivative [13], and several researchers tried to utilize it for solving fractional differential equations (see [14] and the references therein). Kumar et al. [15] studied a timefractional modified Kawahara equation with a CF fractional derivative. In [16], Singh et al. analyzed the El NinoSouthern Oscillation model in the global climate with the CF fractional derivative and obtained the solution by using the iterative method. By using the CF fractional derivative, Hristov [17–19] expressed the Cattaneo constitutive equation with Jeffrey’s fading memory naturally resulting in a heat conduction equation with a relaxation term, and this approach allowed to see the physical background of the CF time fractional derivative and demonstrate how other constitutive equations could be modified with nonsingular fading memories.
Fractional derivatives of variable order were also used to set up the mathematical models for engineering practice, especially in the fields of heat [20] and fluid flows [21, 22]. Zaky and Machado [23] derived the generalized necessary conditions for the fractional optimal control problems and proposed an efficient numerical scheme. Bhrawy and Zaky studied the accurate numerical schemes for the problems of variableorder fractional Schrodinger equations [24]. In [25], the Nabla EulerLagrange equations of the discrete fractional variational problems were given. Garra et al. [26] proposed the necessary conditions for the fractional Herglotz variational problems with generalized Caputo derivatives. Tavares et al. [27] studied the necessary conditions for the constrained FVP of variable order. However, it is difficult to solve the fractional differential equations. Some novel numerical techniques [28–33] were proposed to solve the class of problems. In [29–31], Kumar et al. introduced a new numerical algorithm, which was named qhomotopy analysis transform algorithm, to obtain the approximate solutions for the fractional model of regularized longwave equation, the nonlinear fractional dynamical model and the timefractional RosenauHyman equation.
However, some issues were pointed out against both derivatives, including the one in Caputo sense and the one in RiemannLiouville sense. As Sheikh [22] pointed out, the CF fractional derivative as the kernel in integral was nonsingular but was still nonlocal. Some researchers also concluded that the operator was not a derivative with fractional order but a filter with fractional parameter. The fractional parameter can then be viewed as a filter regulator. To overcome the above drawbacks, Yang et al. [34] proposed a new fractional derivative involving the normalized sinc function without singular kernel. Atangana and Baleanu introduced a new operator with fractional order based upon the generalized MittagLeffler function [35]. Their operators have all the benefits of that of the CF derivative in addition to the kernel being nonlocal and nonsingular. The nonlocality of the kernel gives better description of the memory within the structure with different scale.
The main aim of this paper is to present the optimality conditions for fractional variational problems involving the CF fractional derivative. This paper is structured as follows. In Section 2, the basic definitions and notations are introduced, including the CF fractional derivatives. In Sections 3 and 4, the optimality conditions for fractional variational problems are derived. In Section 5, the optimality condition for the fractional Herglotz problem is proposed. Finally, an example and conclusion are proposed in Sections 6 and 7, respectively.
2 Preliminaries
In this section, we recall some basic concepts with regard to the Caputo fractional derivative [36] and the CF fractional derivative [12]. Given a function \(x(t):[a,b]\rightarrow \mathbf{R}\), the Caputo fractional derivative of x of order \(\alpha \in (0,1)\) is defined as
If x is of class \(C^{1}\), then
The new CF fractional derivative [12, 13] can be obtained by changing the kernel \((t\tau)^{\alpha }\) into the function \(\exp (\alpha (t \tau)/(1\alpha))\) and \(1/\Gamma (1\alpha)\) into \(M(\alpha)/(1 \alpha)\). That is,
where \(M(\alpha)\) is a normalization function such that \(M(0) = M(1) = 1\). It is clear that if x is a constant function, then \({}^{CF}D ^{\alpha }x(t)=0\) as in the usual Caputo derivative, but contrary to the usual Caputo derivative, the kernel does not have singularity for \(t =\tau \).
Definition 1
([12])
Let \(x\in H^{1}(a,b)\), \(b>a\), \(\alpha \in (0,1)\), then the CF fractional derivative is described as (1), where \(M(\alpha)\) stands for a normalization function such that \(M(0)= M(1)=1\). If the function does not belong to \(x\in H^{1}(a,b)\), the derivative can be reconstructed as
Remark 1
The kernel function of the CF fractional derivative is an exponential function. As we introduced in Section 1, there are several nonsingular kernel functions such as the normalized sinc function [34], the generalized MittagLeffler function [35], Meijer Gfunction [37] and Fox Hfunction [38], which can be used to define the fractional derivative and integral. So there is a problem, and what kind of kernel function is better? Some researchers [20, 22, 39–41] compared the actual effects of CF derivatives with AtanganaBaleanu derivatives in the following practical problems.

(1)
For the generalized Casson fluid model with heat generation and chemical reaction, Sheikh [22, 40] pointed out that, for a unit time, the velocities obtained via AtanganaBaleanu and CF derivatives are identical. Velocities for the time less than 1 show little variation and for time bigger than 1 this variation increases. In [39], the AtanganaBaleanu and the CF derivatives were used to extend the model of reactiondiffusion known as AllenCahn model, and the modified models were both solved numerically and numerical simulations presented for different values of alpha.

(2)
Koca [41] pointed out that the CF derivative is a filter not a fractional derivative based upon the fact that the kernel used is local and may not be able to portray more accurately the complex system via which the flow of heat is taking place.
In summary, the existing studies for some practical problems have not consistently shown that the AtanganaBaleanu or the CF derivative is more effective. For fractional variational problems, which class of the fractional derivative is more effective? Further research is needed on the basis of practical fractional variational problems.
We denote an auxiliary fractional integral and a differential as
which is not the CF fractional integral as in [13], and
Proposition 1
Let x be a continuous function and y be of class \(C^{1}\). Then
Proof
From the definition of CF fractional derivative, we have
By Dirichlet’s formula and integrating by parts, we get
□
3 Optimality conditions for FVP
In this section, we consider the following problem with a CF fractional derivative. Given \(x\in C^{1}[a,b]\),
with \(x(a)=x_{a}\) and \(x(b)=x_{b}\), where \(x_{a},x_{b}\in \mathbf{R}\). The assumptions are as follows:

1.
\(L:[a,b]\times \mathbf{R}^{2}\rightarrow \mathbf{R}\) is continuously differentiable with respect to the second and third arguments;

2.
Given any x, the map \(t\mapsto{}^{CF}D^{\alpha }_{b}(\partial_{3}L(t,x(t),{}^{CF}D ^{\alpha }_{a+}x(t)))\) is continuous.
Hereafter, we denote \(\partial_{i}f(x_{1},\ldots,x_{n}):=\frac{ \partial f}{\partial x_{i}}(x_{1},\ldots,x_{n})\) for a function \(f:S\subseteq \mathbf{R}^{n}\rightarrow \mathbf{R}\). We propose the EulerLagrange equation of (2). At the solutions of (2), the first variation of the functional must be vanished.
Theorem 1
Let x be a solution of (2). Then x is a solution of the fractional EulerLagrange equation for all \(t\in [a,b]\)
Proof
Consider \(x+\epsilon h\) to be a variation of x, and \(h:[a,b]\rightarrow \mathbf{R}\) is a function of class \(C^{1}[a,b]\) such that the boundary conditions of \(h(a)=h(b)=0\) hold. Let \(j(\epsilon)=J(x+ \epsilon h)\), since x is a solution of (2), then \(j^{\prime}(0)=0\).
Computing \(j^{\prime}(\epsilon)\vert _{\epsilon =0}\) and using Proposition 1, we have
From the boundary conditions of \(h(a)=h(b)=0\) and h is arbitrary elsewhere, we get
□
Definition 2
A function x that is a solution of (3) is called an extremal for J.
Remark 2
The EulerLagrange equation (3) is easily extended to the case of several variations.
Definition 3
We say that \(L(t,x,y)\) is convex in \(K\subseteq \mathbf{R}^{3}\) if \(\partial_{2}L\) and \(\partial_{3}L\) exist and are continuous, and the condition
holds for every \((t,x,y),(t,x+x_{1},y+y_{1})\in K\).
Theorem 2
If the function L as in (2) is convex in \([a,b]\times \mathbf{R}^{2}\), then each solution of the fractional EulerLagrange equation (3) minimizes J, when restricted to the boundary conditions of \(x(a)=x_{a}\) and \(x(b)=x_{b}\).
Proof
Let x be a solution of (3) and \(x+\epsilon h\) be a variation of x with \(\vert \epsilon \vert \ll 1\), and \(h\in C^{1}[a,b]\) with \(h(a)=h(b)=0\). Using Proposition 1, we get
Since x is a solution of (3), then
Thus, \(J(x+\epsilon h)\geq J(x)\), x is a local minimizer of J. □
Next, we consider a more general class of fractional variational problems for \(A\in (a,b)\) and the functional
with \(x(t)\in C^{1}[a,b]\) and \(x(a)=x_{a}\), \(x(b)=x_{b}\), where \(x_{a},x_{b}\in \mathbf{R}\). The assumptions are as previous ones for \(L(t,x(t),{}^{CF}D^{\alpha }_{a+}x(t))\).
Theorem 3
If x is a solution of (4), then x satisfies
on \([a,A]\),
on \([A,b]\) and
at \(t=a\).
Proof
Let x be a solution of (4) and \(x+\epsilon h\) be a variation of x with \(\vert \epsilon \vert \ll 1\), and \(h\in C^{1}[a,b]\) with \(h(A)=h(b)=0\). Let \(j(\epsilon)=J(x+\epsilon h)\), since x is a solution of (4), then \(j^{\prime}(0)=0\).
Computing \(j^{\prime}(\epsilon)\vert _{\epsilon =0}\) and using Proposition 1, we have
From \(h(A)=h(b)=0\), the above equation deduces the following:
Since h is arbitrary elsewhere, we get the three necessary conditions. □
4 The fractional variational problem with holonomic constraint
Let \((x_{1}(t),x_{2}(t))\in C^{1}[a,b]\times C^{1}[a,b]\) and \(x_{a}^{1}, x_{a}^{2}, x_{b}^{1}, x_{b}^{2}\in \mathbf{R}\) be fixed such that \((x_{1}(a),x_{2}(a))=(x_{a}^{1}, x_{a}^{2})\) and \((x_{1}(b),x _{2}(b))=(x_{b}^{1}, x_{b}^{2})\). Consider the following problem:
Assume that the following conditions hold:

1.
\(L:[a,b]\times \mathbf{R}^{4}\rightarrow \mathbf{R}\) is continuously differentiable with respect to its ith arguments for \(i=2,3,4,5\);

2.
Given any \(\mathbf{x}=(x_{1},x_{2})\), the map \(t\mapsto{}^{CF}D^{ \alpha }_{b}(\partial_{i}L(t,x_{1}(t),x_{2}(t),{}^{CF}D^{\alpha }_{a+}x _{1}(t), {}^{CF}D^{\alpha }_{a+}x_{2}(t)))\) is continuous for \(i=4,5\);

3.
The admissible function \(g:[a,b]\times \mathbf{R}^{2}\) is continuously differentiable with respect to its ith arguments for \(i=2,3\).
Next, we denote that
Theorem 4
Let the function x be a solution of (5)(6). If \(\partial_{3}g(t,\mathbf{x})\neq 0\) for all \(t\in [a,b]\), then there is a continuous function \(\lambda (t):[a,b]\rightarrow \mathbf{R}\) such that x is a solution of the fractional differential equation
and
on \([a,b]\).
Proof
Let x be a solution of (5)(6) and \(\mathbf{x}+ \epsilon \mathbf{h}\) be a variation of x with \(\vert \epsilon \vert \ll 1\), and \(\mathbf{h}=(h_{1}(t),h_{2}(t))\in C^{1}[a,b]\times C^{1}[a,b]\) with \(\mathbf{h}(a)=\mathbf{h}(b)=(0,0)\). From the assumption of \(\partial_{3}g(t,\mathbf{x})\neq 0\) for all \(t\in [a,b]\) and the implicit function theorem, there exists a unique function \(h_{2}( \epsilon, h_{1})\) such that \((x_{1}+\epsilon h_{1}, x_{2}+\epsilon h _{2})\) satisfies (6). So, we have the following equation satisfied for all \(t\in [a,b]\):
Then
that is,
Since \(\partial_{3}g(t,\mathbf{x})\neq 0\) for all \(t\in [a,b]\), we denote
Let \(j(\epsilon)=J(\mathbf{x}+\epsilon \mathbf{h})\), since x is a solution of (5), the first variation of J must vanish, then \(j^{\prime}(0)=0\). Computing \(j^{\prime}(\epsilon)\vert _{\epsilon =0}\), we have
Using Proposition 1 and \(\mathbf{h}(a)=\mathbf{h}(b)=(0,0)\), we obtain
Inserting (12)(13) into the above equation, we obtain
Since \(h_{1}\) is arbitrary elsewhere, we get the necessary condition as follows:
From (11), we have another necessary condition
□
Theorem 5
Let the function \(L(t,\mathbf{x},{}^{CF}D_{a+}^{\alpha }\mathbf{x})\) as in (5) be convex in \([a,b]\times \mathbf{R}^{4}\), \(g:[a,b]\times \mathbf{R}^{2}\rightarrow \mathbf{R}\) is continuously differentiable with respect to its ith arguments for \(i=2,3\). For the continuous function \(\lambda (t):[a,b]\rightarrow \mathbf{R}\) be given in (11) and \(\partial_{3}g(t,\mathbf{x})\neq 0\) for all \(t\in [a,b]\), if x is a solution of (7) subject to (6), then x is also a solution of (5)(6).
Proof
Since x is a solution of (7) subject to (6) and \(g:[a,b]\times \mathbf{R}^{2}\rightarrow \mathbf{R}\) is continuously differentiable with respect to its ith arguments for \(i=2,3\), then x satisfies (10). From \(\partial_{3}g(t,\mathbf{x})\neq 0\) for all \(t\in [a,b]\), we get
On the other hand, if \(\mathbf{x}+\epsilon \mathbf{h}\) is a variation of x, we have
Integrating by parts for the righthand side of the above inequation, using Proposition 1 and \(\mathbf{h}(a)=\mathbf{h}(b)=(0,0)\), we obtain
Using (14) and since x is a solution of (7), we get
Thus, \(J(\mathbf{x}+\epsilon \mathbf{h})\geq J(\mathbf{x})\), x is a solution of J. □
5 The fractional Herglotz problem
The fractional Herglotz problem is to determine a curve \(x\in C^{1}[a,b]\) subject to \(x(a)=x_{a}\) and \(x(b)=x_{b}\) such that z is the solution of the following system:
and \(z(t)\vert _{t=b}\) is a minimum. This problem was studied in [6] based on the CaputoKatugampola fractional derivative. However, contrary to the CaputoFabrizio derivative in this paper, the CaputoKatugampola fractional derivative has singularity for \(t =\tau \). For any function \(x(t)\), the map \(t\mapsto{}^{CF}D_{a+}^{\alpha }x(t)\) is continuously differentiable and the map \(t\mapsto{}^{CF}D_{b}^{\alpha }(\lambda (t) \partial_{3}L(t,x(t),{}^{CF}D_{a+}^{\alpha }x(t), z(t)))\) is continuous with
It can be known that the solution z depends on t and x. If we consider the function of \(h(t)\in C^{1}[a,b]\) with \(h(a)=h(b)=0\) and any sufficiently small real number ϵ, then \(x+\epsilon h\in C ^{1}[a,b]\) lies in the neighborhood \(N_{\epsilon }(x)\). We substitute x by \(x+\epsilon h\), the solution z also depends on ϵ, and it is also differentiable with respect to ϵ.
Theorem 6
Let the function x be such that \(z(b)\) as in (15)(16) attains a minimum. Then x is a solution of the fractional differential equation
Proof
Let the function of \(h(t)\in C^{1}[a,b]\) with \(h(a)=h(b)=0\) and any sufficiently small real number ϵ. Using \(x+\epsilon h\) is a variation of x and the solution z is given by
and
Thus, we have
We denote that
and
Then the above differential equation can be deduced as
which is
then
Replacing t by b in the above equation, since \(z(a)\) is fixed and \(z(b)\) is the minimum, we have \(\phi (a)=\phi (b)=0\). By \(h(a)=h(b)=0\) and the arbitrariness of h in \((a,b)\), we get
for all \(t\in [a,b]\). □
6 Example
Example
Let us consider the following unconstrained fractional variational problem for \(0<\alpha <1\):
where \(f(t)=\frac{M(\alpha)}{\alpha }[e^{\frac{\alpha t}{1\alpha }}1]\) and \(M(\alpha)=1+\mathrm{sin}(\alpha \pi)\).
For this problem, according to the EulerLagrange equation as in (3), we get
By direct substitution, it can be shown that \(x(t)=t\) is the unique solution to this problem.
In fact, for the case of \(x(t)=t\), we have
It can be observed that as \(\alpha \rightarrow 1\), the fractional variational problem becomes
where \(f(t)=1\) since \(M(1)=1\). \(x(t)=t\) is obviously the unique solution to this problem.
7 Conclusions
In this paper, we have discussed the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrange function depending on a CaputoFabrizio fractional derivative. The advantage of the new fractional derivative has no singularity, which was not precisely illustrated in the previous definitions. Two classes of FVP are considered to demonstrate the application of the optimality conditions. However, the EulerLagrange equations for FVP are in general difficult to solve. As the future works, we should develop numerical methods to solve this problem.
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Acknowledgements
The author would like to express his gratitude to the anonymous reviewers for their very valuable remarks and comments. This work is supported by the National Natural Science Foundation of China (Grant No. 11701446, 11601420, 11401469, 60974082) and the Science Plan Foundation of the Education Bureau of Shaanxi Province (No. 2013JK 1130).
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Zhang, J., Ma, X. & Li, L. Optimality conditions for fractional variational problems with CaputoFabrizio fractional derivatives. Adv Differ Equ 2017, 357 (2017). https://doi.org/10.1186/s1366201713887
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DOI: https://doi.org/10.1186/s1366201713887