Fractional Optimal Control Problem for Symmetric System Involving Distributed-Order Atangana–Baleanu Derivatives with Non-Singular Kernel

Abstract

The objective of this work is to discuss and thoroughly analyze the fractional variational principles of symmetric systems involving distributed-order Atangana–Baleanu derivatives. A component of distributed order, the fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems are studied concerning Atangana–Baleanu derivatives. We give a general formulation and a solution technique for a class of fractional optimal control problems (FOCPs) for such systems. The dynamic constraints are defined by a collection of FDEs, and the performance index of an FOCP is considered a function of the control variables and the state. The formula for fractional integration by parts, the Lagrange multiplier, and the calculus of variations are used to obtain the Euler–Lagrange equations for the FOCPs.

1. Introduction

Fractional calculus is still being developed, and its operators are used to model complicated systems, where the kernel of the fractional operators reflects the non-locality. Due to the singularity of fractional operators’ kernels, researchers have recently introduced new types of fractional operators with non-singular kernels and their discrete variations [1,2,3,4,5,6]. An extra technique for defining integrals and fractional derivatives was brought about by this recent tendency. By iterating weighted typical integrals or specific local-type derivatives, the fractionalizing process in classical fractional operators obtains the factorial function, which is then replaced with the gamma function. In [7], Sadek et al. introduce novel $\theta$-fractional operators. In [8], Sadek introduces the cotangent fractional derivative with its application.

Dirac delta functions are used as a limiting technique in the study of fractional calculus with operators having non-singular kernels. The function itself is obtained by first defining the fractional derivative with the non-singular kernel in the limiting case $\alpha \to 0$. When $\alpha \to 1$, we obtain the typical derivative of the function.

Laplace transforms are then used to assess the associated integral operators for functions whose convolution with the non-singular kernel disappears at the initial point. For a given dissipative system, Riewe first attempted to determine the fractional Lagrangian and Hamiltonian [9,10]. Agrawal et al. [11,12,13], Bahaa and Baleanu et al. [14,15,16,17] made significant contributions to the field of variational principles.

In [11,12,13,14,15,16,17,18,19,20,21,22,23,24], and the publications that have cited them, for example, several minimization problems of fractional optimal control problems (FOCPs) of discrete-order systems were investigated. In [25], Ounamane et al. employ the Caputo fractional derivative (CFD) approach and utilize the truncated exponential method to tackle linear fractional optimal control problems (FOCPs) with equality and inequality constraints in multi-dimensional settings. In [26], Nosrati et al. study a robust fractional-order singular Kalman filter. In [5], Sadek et al. used the Galerkin Bell method to solve fractional optimal control problems with inequality constraints. In [6], control theory for fractional differential Sylvester matrix equations with a Caputo fractional derivative is investigated.

The bang-bang property of fractional differential systems with Caputo fractional derivatives is examined by the authors in [21]. In [11,12,13], Agrawal presented a general formulation and solution method for fractional optimal control problems involving first- and second-order operators. The formulation was developed using the fractional variation concept and the Lagrange multiplier technique. In [19,20], Mophou applied the classical control theory to a fractional diffusion equation involving a second-order operator (Laplace operator) in a bounded domain with and without state limitations. In [22], Bahaa expanded the results in [19,20] to fractional optimal control of differential systems with discrete order. We also refer to [2,3,16,17,27,28,29,30,31,32,33,34,35,36,37,38,39] and the references therein for more literature on the optimum management of fractional evolution and diffusion equations.

Recently, Atangana and Baleanu, in [1], proposed a derivative with fractional order to tackle a number of open problems in the field of fractional calculus that have been brought up by many scholars. They state that the kernel of their derivative is non-singular and non-local. Their operators retain all the benefits of the Caputo–Fabrizio derivative while also having a kernel that is both non-local and non-singular. This non-locality provides a better description of memory effects within the structure across different scales and it is best suited for sub-diffusion and super-diffusion problems in complex media of heterogeneous self-similar aquifers, and has aroused interest from the scientific community.

In [40], Ma et al. study fractional optimal control problems with both integer-order and Atangana–Baleanu Caputo derivatives. First, the existence and uniqueness of the solution of a fractional Cauchy problem is given. Then, applying calculus of variations and the Lagrange multiplier method, they present the necessary optimality conditions of FOCPs and sufficient optimality conditions are also given under some assumptions. In [41], optimal control for a fractional-order nonlinear mathematical model of cancer treatment is presented. The suggested model is determined by a system of eighteen fractional differential equations. The fractional derivative is defined in the Atangana–Baleanu Caputo sense. In [42], Tajadodi et al. study optimal control problems with Atangana–Baleanu fractional derivatives. A computational method based on B-spline polynomials and their operational matrix of Atangana–Baleanu fractional integration is proposed for the numerical solution of this class of problems.

These methods have been applied to a variety of mathematical physics and engineering issues, with simulations using distributed-order fractional (D-OF) differential equations. The D-OF derivative was used by Lorenzo and Hartley [43] to investigate the rheological characteristics of composite materials. The D-OF differential equation ([44,45,46,47,48,49,50,51,52,53,54,55,56,57]) is a continuous extension of the multi-term and single-term fractional differential equations. Compared with single-term or multi-term fractional differential equations, the D-OF model provides a more flexible tool to describe some real physical phenomena in disordered, viscoelastic media and composite materials. Recently, the study of D-OF differential equations has attracted much attention from many researchers. Ford and Morgado, in [35], studied the existence and uniqueness of solutions of D-OF differential equations. In [58], they examined a Cauchy issue for the D-OF diffusion-wave equation and suggested a D-OF viscoelastic model for the uniaxial isothermal deformation of a viscoelastic body. In [59], they proved the uniqueness and continuous dependence on initial conditions of the D-OF diffusion equation on bounded domains. The analytical solutions of several distinct types of D-OF differential equations were studied in [35] and references therein.

Since the Atangana–Baleanu derivative incorporates both the left and right derivatives, applying it to fractional dynamics will open up new avenues for researching limited systems. Furthermore, a function’s fractional derivative, which depends on the function’s values throughout the entire interval, is determined by a defined integral. Therefore, fractional derivatives can be used to depict systems having long-range interactions in space and/or time (memory) and processes involving several scales of space and/or time.

Despite these attempts, the study of FOCPs with distributed order has received little attention, especially when it uses the fractional Atangana–Baleanu derivative. Based on the above motivation, in this paper, we propose to use the Atangana–Baleanu derivative with distributed order to extend the notion of comparable Lagrangians for the fractional scenario. Numerous techniques have been put forth to obtain the fractional Euler–Lagrange equations and the corresponding Hamiltonians for a given classical Lagrangian. However, the availability of multiple alternatives can be exploited to address a particular physical system because the fractional dynamics depends on the fractional derivatives used to create the Lagrangian in the first place.

The following is an outline of the paper: A new explanation of the integration by parts involving the Atangana–Baleanu fractional time derivative and the distributed-order Atangana–Baleanu fractional derivative in the Riemann–Liouville and Caputo senses is provided in Section 2. In Section 3, a few instances are discussed in detail and a quick review of the fractional Lagrangian and Hamiltonian techniques applied to discrete systems based on fractional distributed-order Atangana–Baleanu derivatives is given. In Section 4, we extend the fractional Lagrangian to n state equations and obtain the fractional canonical momenta. Section 5 further discusses and thoroughly examines a few examples of the limited system inside Atangana–Baleanu derivatives. Section 6 introduces the distributed-order Atangana–Baleanu derivatives in fractional optimal control problems (FOCPs). One FOCP for time-invariant FOCPs and another for time-variant FOCPs are stated. Our conclusions are given in Section 7.

2. Preliminaries

The Riemann–Liouville, Grunwald–Letnikov, Weyl, Caputo, Marchaud, and Riesz fractional derivatives are among the several definitions of fractional derivatives that have been offered. The problem will be formulated using the Atangana–Baleanu fractional time derivatives, which are provided as follows.

Let $H^m\left(\mathsf{\Omega }\right)$ and $H_0^m\left(\mathsf{\Omega }\right)$ be normal Sobolev spaces, and let $L^2\left(\mathsf{\Omega }\right)$ be the usual Hilbert space with the scalar product $(.,.)$.

We begin by reviewing the Mittag–Leffler function $E_{\alpha ,\beta }\left(z\right)$ for $\alpha \in (0,1)$, which is defined below and is utilized extensively in this study.

$$E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathsf{\Gamma }(k\alpha +\beta )}},E_{\alpha ,1}\left(z\right)=E_{\alpha }\left(z\right)z\in \mathcal{C},$$

where $\mathsf{\Gamma }(.)$ denotes the Gamma function, defined as

$$\mathsf{\Gamma }\left(z\right)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt,\mathcal{R}\left(z\right)>0.$$

A two-parameter family of complete functions of z of order ${\alpha }^{-1}$ is known as the Mittag–Leffler function. One specific instance of the Mittag–Leffler function is the exponential function, which is

$$E_{1,1}\left(z\right)=e^z,E_{2,1}\left(z\right)=cosh\sqrt{z},E_{1,2}\left(z\right)={\displaystyle \frac{e^z-1}{z}},E_{2,2}\left(z\right)={\displaystyle \frac{sinh\sqrt{z}}{\sqrt{z}}}.$$

Let us review some helpful Atangana–Baleanu definitions of fractional derivatives [1].

Definition 1

(see [1,4,23,24]). The Atangana–Baleanu fractional derivative (AB derivative) of u variable-order α in the Caputo sense ${}_a{}^{\mathit{ABC}}D_t^{\alpha }u\left(t\right)$ (where A denotes Atangana, B denotes Baleanu, and C denotes Caputo type) with a base point a is defined at a point $t\in (a,b)$ for a given function, $u\in H^1(a,b),b>a$, $\alpha \in (0,1)$.

$${}_a{}^{\mathit{ABC}}D_t^{\alpha }u\left(t\right)={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{\int }_a^t{u}^{\prime }\left(s\right)E_{\alpha }[-\gamma {(t-s)}^{\alpha }]ds,(\mathit{left}\mathit{ABCD})$$

(1)

where

$$\gamma ={\displaystyle \frac{\alpha }{(1-\alpha )}},$$

$E_{\alpha }(.)$ stands for the Mittag–Leffler function, and $B\left(\alpha \right)$ is a normalization function satisfying

$$B\left(\alpha \right)=(1-\alpha )+{\displaystyle \frac{\alpha }{\mathsf{\Gamma }\left(\alpha \right)}},\mathit{where}B\left(0\right)=B\left(1\right)=1,$$

and in the Riemann–Liouville sense with

$${}_a{}^{\mathit{ABR}}D_t^{\alpha }u\left(t\right)={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{\displaystyle \frac{d}{dt}}{\int }_a^{t}u\left(s\right)E_{\alpha }[-\gamma {(t-s)}^{\alpha }]ds,(\mathit{left}\mathit{ABRD}).$$

(2)

For $\alpha =1$ in (2), we consider the usual classical derivative ${\partial }_t$.

The associated left AB fractional integral ${}_a{}^{\mathit{AB}}I_t^{\alpha }u\left(t\right)$ is also defined as

$$\begin{array}{cc}\hfill {}_a{}^{\mathit{AB}}I_t^{\alpha }u\left(t\right)& ={\displaystyle \frac{1-\alpha }{B\left(\alpha \right)}}u\left(t\right)+{\displaystyle \frac{\alpha }{B\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}}{\int }_a^{t}u\left(s\right){(t-s)}^{\alpha -1}ds,(\mathit{left}\mathit{ABI})\hfill \\ & ={\displaystyle \frac{1-\alpha }{B\left(\alpha \right)}}u\left(t\right)+{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}{}_{a}I_t^{\alpha }u\left(t\right)\hfill \end{array}$$

(3)

Keep in mind that we recover the original function if $\alpha =0$ in (3); if $\alpha =1$ in (3), we take into consideration the standard ordinary integral. The work in [1,60] includes some new findings and characteristics pertaining to this operator. Consequently, we recall the following definition.

Definition 2

(see [1,4,23,24,61]). The definition states that for a given function, $u\in H^1(a,b)$, $b>t>a$, the right Atangana–Baleanu fractional derivative u of order α in the Caputo sense with base point b is defined at a point $t\in (a,b)$,

$${}_b{}^{\mathit{ABC}}D_t^{\alpha }g\left(t\right)=-{\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{\int }_t^b{g}^{\prime }\left(s\right)E_{\alpha }[-\gamma {(s-t)}^{\alpha }]ds,(\mathit{right}\mathit{ABCD}),$$

(4)

and in Riemann–Liouville sense with

$${}_b{}^{\mathit{ABR}}D_t^{\alpha }g\left(t\right)=-{\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{\displaystyle \frac{d}{dt}}{\int }_t^{b}g\left(s\right)E_{\alpha }[-\gamma {(s-t)}^{\alpha }]ds,(\mathit{right}\mathit{ABRD}).$$

(5)

The associated right AB fractional integral ${}_a{}^{\mathit{AB}}I_t^{\alpha }u\left(t\right)$ is also defined as

$$\begin{array}{cc}\hfill {}_t{}^{\mathit{AB}}I_b^{\alpha }u\left(t\right)& ={\displaystyle \frac{1-\alpha }{B\left(\alpha \right)}}u\left(t\right)+{\displaystyle \frac{\alpha }{B\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}}{\int }_t^{b}u\left(s\right){(s-t)}^{\alpha -1}ds,(\mathit{right}\mathit{ABI})\hfill \\ & ={\displaystyle \frac{1-\alpha }{B\left(\alpha \right)}}u\left(t\right)+{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}{}_{t}I_b^{\alpha }u\left(t\right)\hfill \end{array}.$$

(6)

The following formulas show that the respective AB fractional integrals and the left and right AB FDs in the Riemann–Liouville and Caputo senses have advantageous links.

$${}_a{}^{\mathit{AB}}I_t^{\alpha }\left\{{}_a{}^{ABR}D_t^{\alpha }u\left(t\right)\right\}={}_t{}^{AB}I_b^{\alpha }\left\{{}_t{}^{ABR}D_b^{\alpha }u\left(t\right)\right\}=u\left(t\right)$$

$${}_a{}^{\mathit{AB}}I_t^{\alpha }\left\{{}_a{}^{ABC}D_t^{\alpha }u\left(t\right)\right\}=u\left(t\right)-u\left(a\right)$$

$${}_t{}^{\mathit{AB}}I_b^{\alpha }\left\{{}_t{}^{ABC}D_b^{\alpha }u\left(t\right)\right\}=u\left(t\right)-u\left(b\right)$$

$${}_b{}^{ABC}D_t^{\alpha }u\left(t\right)=-{}_0{}^{ABC}D_t^{\alpha }u\left(t\right).$$

Thus, the forward-in-time operator with the non-singular Mittag–Leffler kernel and fractional time derivative is $-{}_0{}^{\mathit{ABC}}D_t^{\alpha }$. At the basis point T, this is equal to the backward-in-time operator with the fractional time derivative using the non-singular Mittag–Leffler kernel.

We define the following variable-order fractional-integral-type operators in accordance with [61].

Definition 3

(see [4,61]).

$$E_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},a^{+}}\phi \left(t\right)={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{\int }_a^t{E}_{\alpha }\left[{\displaystyle \frac{-\alpha }{1-\alpha }}{(t-s)}^{\alpha }\right]\phi \left(s\right)ds,t>a.$$

(7)

Similarly, one may define the appropriate generalized fractional integral operator by

$$E_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b^{-}}\phi \left(t\right)={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{\int }_t^b{E}_{\alpha }\left[{\displaystyle \frac{-\alpha }{1-\alpha }}{(s-t)}^{\alpha }\right]\phi \left(s\right)ds,t<b.$$

(8)

The Atangana–Baleanu fractional derivatives in the Riemann–Liouville sense (ABR derivative) of order $\alpha$ with base point a are determined at a point $t\in (a,b)$ for a given function $\phi \left(t\right)\in H^1(a,b),b>t>a$ (where R stands for Riemann).

$${}_a{}^{\mathit{ABR}}D_t^{\alpha }\phi \left(t\right)={\displaystyle \frac{d}{dt}}{E}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},a^{+}}\phi \left(t\right)(leftABRD)$$

(9)

$${}_t{}^{\mathit{ABR}}D_b^{\alpha }\phi \left(t\right)=-{\displaystyle \frac{d}{dt}}{E}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b^{-}}\phi \left(t\right)(rightABRD),$$

(10)

and the Atangana–Baleanu fractional derivatives in the Caputo sense (ABC derivative) of order $\alpha$ with base point a are defined for a point $t\in (a,b)$ for a given function $\phi \left(t\right)\in H^1(a,b)$, $b>t>a$ (where C stands for Caputo).

$${}_a{}^{\mathit{ABC}}D_t^{\alpha }\phi \left(t\right)=E_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},a^{+}}{\phi }^{\prime }\left(t\right)(leftABRD)$$

(11)

$$\begin{array}{cc}\hfill {}_t{}^{\mathit{ABC}}D_b^{\alpha }\phi \left(t\right)& =-E_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b^{-}}{\phi }^{\prime }\left(t\right)(rightABRD),\hfill \end{array}$$

(12)

The following proposition, which provides the integration by parts, is then stated.

Proposition 1

(Integration by parts; see [60]). Let, $\alpha >0,r\ge 1,s\ge 1$, and ${\displaystyle \frac{1}{r}}+{\displaystyle \frac{1}{s}}\le 1+\alpha (r\ne 1$ and $s\ne 1\mathit{in}\mathit{the}\mathit{case}{\displaystyle \frac{1}{r}}+{\displaystyle \frac{1}{s}}=1+\alpha )$. Then, for any $\varphi \left(x\right)\in L^r(a,b),\psi \left(x\right)\in L^s(a,b)$, we have

$${\int }_a^b\varphi {\left(x\right)}^{\mathit{AB}}{}_{a}I_t^{\alpha }\psi \left(x\right)dx={\int }_a^b\psi {\left(x\right)}^{\mathit{AB}}{}_{t}I_b^{\alpha }\varphi \left(x\right)dx$$

(13)

$${\int }_a^b\varphi {\left(t\right)}^{\mathit{AB}}{}_{t}I_b^{\alpha }\psi \left(t\right)dt={\int }_a^b\psi {\left(t\right)}^{\mathit{AB}}{}_{a}I_t^{\alpha }\varphi \left(t\right)dt$$

(14)

$\mathit{if}\varphi \left(t\right){\in }^{\mathit{AB}}{}_{t}I_b^{\alpha }\left(L^r\right)and\psi \left(x\right)\in {}_a{}^{\mathit{AB}}I_t^{\alpha }\left(L^s\right),$ then

$${\int }_a^b\varphi \left(t\right){}_a{}^{\mathit{ABR}}D_t^{\alpha }\psi \left(t\right)dt={\int }_a^b\psi \left(t\right){}_t{}^{\mathit{ABR}}D_b^{\alpha }\varphi \left(t\right)dt$$

(15)

$${\int }_a^b\varphi \left(t\right){}_a{}^{\mathit{ABC}}D_t^{\alpha }\psi \left(t\right)dt={\int }_a^b\psi \left(t\right){}_t{}^{\mathit{ABR}}D_b^{\alpha }\varphi \left(t\right)dt+{\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}\psi \left(t\right)E_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b}^1{\varphi \left(t\right)|}_a^b$$

(16)

$${\int }_a^b\varphi \left(t\right){}_t{}^{\mathit{ABC}}D_b^{\alpha }\psi \left(t\right)dt={\int }_a^b\psi \left(t\right){}_a{}^{\mathit{ABR}}D_t^{\alpha }\varphi \left(t\right)dt-{\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}\psi \left(t\right)E_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},a}^1{\varphi \left(t\right)|}_a^b$$

(17)

where the left generalized fractional integral operator is

$$E_{\gamma ,\mu ,\omega ,a}^{\alpha }x\left(t\right)={\int }_a^t{(t-\tau )}^{\mu -1}{E}_{\gamma ,\mu }^{\alpha }\left[\omega {(t-\tau )}^{\gamma }\right]x\left(\tau \right)d\tau ,t>a,$$

and the right generalized fractional integral operator is

$$E_{\gamma ,\mu ,\omega ,b}^{\alpha }x\left(t\right)={\int }_t^b{(\tau -t)}^{\mu -1}{E}_{\gamma ,\mu }^{\alpha }\left[\omega {(\tau -t)}^{\gamma }\right]x\left(\tau \right)d\tau ,t<b.$$

Definition 4

(see [29,35,62]). For $p\left(\alpha \right)\ge 0$, $p\left(\alpha \right)\ne 0,{\int }_1^{0}p\left(\alpha \right)d\alpha <\infty$, and $\alpha \in (0,1)$, the left- and right-sided D-OF derivatives in the Riemann–Liouville sense are defined by

$${}_a{}^{\mathit{ABR}}D_t^{p\left(\alpha \right)}g\left(t\right)={\int }_0^{1}p\left(\alpha \right){}_a{}^{\mathit{ABR}}D_t^{\alpha }g\left(t\right)d\alpha$$

(18)

$${}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}g\left(t\right)={\int }_0^{1}p\left(\alpha \right){}_t{}^{\mathit{ABR}}D_b^{\alpha }g\left(t\right)d\alpha$$

(19)

Definition 5

(see [29,35,62]). For $p\left(\alpha \right)\ge 0$, $p\left(\alpha \right)\ne 0,{\int }_1^{0}p\left(\alpha \right)d\alpha <\infty$, and $\alpha \in (0,1)$, the left- and right-sided D-OF derivatives in the Caputo sense are defined by

$${}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}g\left(t\right)={\int }_0^{1}p\left(\alpha \right){}_a{}^{\mathit{ABC}}D_t^{\alpha }g\left(t\right)d\alpha$$

(20)

$${}_t{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}g\left(t\right)={\int }_0^{1}p\left(\alpha \right){}_t{}^{\mathit{ABC}}D_b^{\alpha }g\left(t\right)d\alpha$$

(21)

Lemma 1

(see [29,35,62]). For functions u and v defined on $[a,b]$ and $0<\alpha \le 1$, we have

$${\int }_a^{b}u\left(t\right){}_{a}I_t^{p\left(\alpha \right)}v\left(t\right)dt={\int }_a^{b}v\left(t\right){}_{t}I_b^{p\left(\alpha \right)}u\left(t\right)dt,$$

(22)

$${\int }_a^{b}u\left(t\right){}_{t}I_b^{p\left(\alpha \right)}v\left(t\right)dt={\int }_a^{b}v\left(t\right){}_{a}I_t^{p\left(\alpha \right)}u\left(t\right)dt.$$

(23)

Proof. 

From Definitions 3 and 4, and by changing the order of integrals, we have

$$\begin{array}{ccc}\hfill {\int }_a^{b}u\left(t\right){}_{a}I_t^{p\left(\alpha \right)}v\left(t\right)dt& =& {\int }_a^{b}u\left(t\right)\left[{\int }_0^{1}p\left(\alpha \right){}_{a}I_t^{\alpha }v\left(t\right)d\alpha \right]dt\hfill \\ & =& {\int }_0^{1}p\left(\alpha \right)\left[{\int }_a^{b}u\left(t\right){}_{a}I_t^{\alpha }v\left(t\right)dt\right]d\alpha \hfill \\ \hfill & =& {\int }_0^{1}p\left(\alpha \right)\left[{\int }_a^{b}v\left(t\right){}_{t}I_b^{\alpha }u\left(t\right)dt\right]d\alpha \hfill \\ & =& {\int }_a^{b}v\left(t\right)\left[{\int }_0^{1}p\left(\alpha \right){}_{t}I_b^{\alpha }u\left(t\right)d\alpha \right]dt\hfill \\ & =& {\int }_a^{b}v\left(t\right){}_{t}I_b^{p\left(\alpha \right)}u\left(t\right)dt.\hfill \end{array}$$

From Definitions 3 and 4, and by changing the order of integrals, we have

$$\begin{array}{ccc}\hfill {\int }_a^{b}u\left(t\right){}_{t}I_b^{p\left(\alpha \right)}v\left(t\right)dt& =& {\int }_a^{b}u\left(t\right)\left[{\int }_0^{1}p\left(\alpha \right){}_{t}I_b^{\alpha }v\left(t\right)d\alpha \right]dt\hfill \\ & =& {\int }_0^{1}p\left(\alpha \right)\left[{\int }_a^{b}u\left(t\right){}_{t}I_b^{\alpha }v\left(t\right)dt\right]d\alpha \hfill \\ \hfill & =& {\int }_0^{1}p\left(\alpha \right)\left[{\int }_a^{b}v\left(t\right){}_{a}I_t^{\alpha }u\left(t\right)dt\right]d\alpha \hfill \\ & =& {\int }_a^{b}v\left(t\right)\left[{\int }_0^{1}p\left(\alpha \right){}_{a}I_t^{\alpha }u\left(t\right)d\alpha \right]dt\hfill \\ & =& {\int }_a^{b}v\left(t\right){}_{a}I_t^{p\left(\alpha \right)}u\left(t\right)dt.\hfill \end{array}$$

□

Now, we can use Lemma 1 to show the following integration by parts for $AB$ fractional integrals with distributed order.

Proposition 2

(Distributed-order integration by parts; see [60]). Suppose that $\alpha >0,r\ge 1,s\ge 1$, and that ${\displaystyle \frac{1}{r}}+{\displaystyle \frac{1}{s}}\le 1+\alpha (r\ne 1$ and $s\ne 1\mathit{in}\mathit{the}\mathit{case}{\displaystyle \frac{1}{r}}+{\displaystyle \frac{1}{s}}=1+\alpha )$ holds true. Consequently, for each $\varphi \left(x\right)\in L^r(a,b),\psi \left(x\right)\in L^s(a,b)$, we have

$${\int }_a^b\varphi {\left(x\right)}^{\mathit{AB}}{}_{a}I_t^{p\left(\alpha \right)}\psi \left(x\right)dx={\int }_a^b\psi {\left(x\right)}^{\mathit{AB}}{}_{t}I_b^{p\left(\alpha \right)}\varphi \left(x\right)dx$$

(24)

$${\int }_a^b\varphi {\left(x\right)}^{\mathit{AB}}{}_{t}I_b^{p\left(\alpha \right)}\psi \left(x\right)dx={\int }_a^b\psi {\left(x\right)}^{\mathit{AB}}{}_{a}I_t^{p\left(\alpha \right)}\varphi \left(x\right)dx$$

(25)

$\mathit{if}\varphi \left(x\right){\in }^{\mathit{AB}}{}_{t}I_b^{p\left(\alpha \right)}\left(L^r\right)\mathit{and}\psi \left(x\right){\in }^{\mathit{AB}}{}_{a}I_t^{\alpha }\left(L^s\right),$ then

$${\int }_a^b\varphi \left(x\right){}_a{}^{\mathit{ABR}}D_t^{p\left(\alpha \right)}\psi \left(x\right)dx={\int }_a^b\psi \left(x\right){}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}\varphi \left(x\right)dx$$

(26)

$${\int }_a^{b}u\left(t\right){}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}v\left(t\right)dt={\int }_a^{b}v\left(t\right){}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}u\left(t\right)dt+v\left(t\right){\mathcal{E}}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b}^1{u\left(t\right)|}_a^b$$

(27)

$${\int }_a^{b}u\left(t\right){}_t{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}v\left(t\right)dt={\int }_a^{b}v\left(t\right){}_a{}^{\mathit{ABR}}D_t^{p\left(\alpha \right)}u\left(t\right)dt-v\left(t\right){\mathcal{E}}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},a}^1{u\left(t\right)|}_a^b$$

(28)

where the left generalized fractional integral operator is

$${\mathcal{E}}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},a}^{1}u\left(t\right)={\int }_0^{1}p\left(\alpha \right){\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{E}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},a}^{1}u\left(t\right)d\alpha$$

and the right generalized fractional integral operator is

$${\mathcal{E}}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b}^{1}u\left(t\right)={\int }_0^{1}p\left(\alpha \right){\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{E}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b}^{1}u\left(t\right)d\alpha$$

Proof. 

From Definition 3, and by applying the first part of Lemma 1, we have

$$\begin{array}{ccc}\hfill {\int }_a^{b}u\left(t\right){}_a{}^{\mathit{AB}}I_t^{p\left(\alpha \right)}v\left(t\right)dt& =& {\int }_a^{b}u\left(t\right)\left[{\int }_0^{1}p\left(\alpha \right){}_a{}^{\mathit{AB}}I_t^{\alpha }v\left(t\right)d\alpha \right]dt\hfill \\ & =& {\int }_0^{1}p\left(\alpha \right)\left[{\int }_a^{b}u\left(t\right){}_a{}^{\mathit{AB}}I_t^{\alpha }v\left(t\right)dt\right]d\alpha \hfill \\ \hfill & =& {\int }_0^{1}p\left(\alpha \right)\left[{\int }_a^{b}v\left(t\right){}_t{}^{\mathit{AB}}I_b^{\alpha }u\left(t\right)dt\right]d\alpha \hfill \\ & =& {\int }_a^{b}v\left(t\right)\left[{\int }_0^{1}p\left(\alpha \right){}_t{}^{\mathit{AB}}I_b^{\alpha }u\left(t\right)d\alpha \right]dt\hfill \\ & =& {\int }_a^{b}v\left(t\right){}_t{}^{\mathit{AB}}I_b^{p\left(\alpha \right)}u\left(t\right)dt.\hfill \end{array}$$

The second statement, which is analogous to the first component, is proved using the second part of Lemma 1.

$$\begin{array}{ccc}\hfill {\int }_a^{b}u\left(t\right){}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}v\left(t\right)dt& =& {\int }_a^{b}u\left(t\right)\left[{\int }_0^{1}p\left(\alpha \right){}_a{}^{\mathit{ABC}}D_t^{\alpha }v\left(t\right)d\alpha \right]dt\hfill \\ & =& {\int }_0^{1}p\left(\alpha \right)\left[{\int }_a^{b}u\left(t\right){}_a{}^{\mathit{ABC}}D_t^{\alpha }v\left(t\right)dt\right]d\alpha \hfill \\ \hfill & =& {\int }_0^{1}p\left(\alpha \right)\left[{\int }_a^{b}v\left(t\right){}_t{}^{\mathit{ABR}}D_b^{\alpha }u\left(t\right)dt\right.\hfill \\ & +& \left.{[{\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}v\left(t\right)E_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b}^{1}u\left(t\right)]}_a^b\right]d\alpha \hfill \\ & =& {\int }_a^{b}v\left(t\right)\left[{\int }_0^{1}p\left(\alpha \right){}_t{}^{\mathit{ABR}}D_b^{\alpha }u\left(t\right)d\alpha \right]dt\hfill \\ & +& {\left[v\left(t\right){\int }_0^{1}p\left(\alpha \right){\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{E}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b}^{1}u\left(t\right)d\alpha \right]}_a^b\hfill \\ & =& {\int }_a^{b}v\left(t\right){}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}u\left(t\right)dt+{\left[v\left(t\right){\mathcal{E}}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b}^{1}u\left(t\right)\right]}_a^b.\hfill \end{array}$$

The second statement, which is analogous to the first component, is proved using the second part of Lemma 1. □

3. Distributed-Order Fractional Atangana–Baleanu Derivatives Using Fractional Variational Principles

Below is a concise overview of distributed-order Atangana–Baleanu derivatives, including the fractional Hamilton and fractional Euler–Lagrange equations.

Theorem 1.

Let $J\left[x\right]$ be a functional of the form

$$J\left[x\right]={\int }_a^{b}L(t,x,{}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right))dt$$

(29)

characterized as the set of functions with a continuous variable-order Atangana–Baleanu fractional derivative in the Caputo sense on the set of order $p\left(\alpha \right)$ in $[a,b]$ that satisfy the boundary constraints $x\left(a\right)=x_a$ and $x\left(b\right)=x_b$. Then, for $J\left[x\right]$ to have a maximum for the given function $x\left(t\right)$, $x\left(t\right)$ must satisfy the following Euler–Lagrange equation:

$${\displaystyle \frac{\partial L}{\partial x}}+{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}\left({\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)}}\right)=0.$$

(30)

Proof. 

To determine the requirements needed for the extremum, assume that the desired function is $x^{\ast }\left(t\right)$. Assume $\epsilon \in R$ and create a family of curves.

$$x\left(t\right)=x^{\ast }\left(t\right)+\epsilon \eta \left(t\right)$$

(31)

where $\eta \left(t\right)$ is an arbitrary curve except that it satisfies the boundary conditions; i.e., we require that

$$\eta \left(a\right)=\eta \left(b\right)=0.$$

(32)

The Euler–Lagrange equation can be obtained by inserting Equation (31) into Equation (29), differentiating the result with respect to $\epsilon$, and setting the result to 0. Consequently, the following condition is experienced by the extremum:

$${\int }_a^b\left[{\displaystyle \frac{\partial L}{\partial x}}\eta \left(t\right)+{\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)}}{}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}\eta \left(t\right)\right]dt=0,$$

(33)

Using Equation (27), Equation (33) can be written as

$${\int }_a^b\left[{\displaystyle \frac{\partial L}{\partial x}}{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)}}\right]\eta dt+\eta \left(t\right){\mathcal{E}}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b}^1\left({\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{\alpha }x\left(t\right)\right)}}{|}_a^b\right)=0,$$

(34)

we call ${\mathcal{E}}_{\alpha ,1,{\displaystyle \frac{-\alpha }{1-\alpha }},b}^1\left({\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{\alpha }x\left(t\right)\right)}}{|}_a^b\right)=0$ the natural boundary conditions. Since $\eta \left(t\right)$ is arbitrary, it follows from a well-established result in calculus of variations that

$${\displaystyle \frac{\partial L}{\partial x}}+{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)}}=0$$

(35)

Equation (35) is the generalized Euler–Lagrange equation (GELE) for the fractional calculus variation (FCV) problem, which is expressed in terms of the distributed-order Atangana–Baleanu fractional derivatives (ABFDs). Note that the Atangana–Baleanu derivatives in the Caputo and Riemann–Liouville senses are present in the resulting differential equations. □

Example 1.

Let us look at the fractional Lagrangian that is provided by

$$L={\displaystyle \frac{1}{2}}{(x+{}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x)}^2,$$

(36)

Consequently, the independent fractional Euler–Lagrange Equation (35) is provided by

$$x+{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}\left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\right)=0$$

(37)

Example 2.

We consider now a fractional Lagrangian of the oscillatory system

$$L={\displaystyle \frac{1}{2}}m{\left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\right)}^2-{\displaystyle \frac{1}{2}}k{x}^2,$$

(38)

where m, the mass, and k are constant. Then, the fractional Euler–Lagrange equation is

$$m{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}\left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\right)-kx=0$$

(39)

When $p\left(\alpha \right)\to 1$, this equation simplifies to the harmonic oscillator’s equation of motion.

4. The Canonical Momenta of Fractions

The fractional Lagrangian $L=L(x,{}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right))$ is provided. The definition of the fractional canonical momentum is

$$P={\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)\right)}}$$

(40)

Thus, we generate the corresponding fractional Hamiltonian as follows:

$$H(x,P)=P{}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)-L.$$

(41)

Then, we have

$$\begin{array}{ccc}\hfill dH& =& Pd{}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)+{}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)dP-{\displaystyle \frac{\partial L}{\partial x}}dx-{\displaystyle \frac{\partial L}{\partial t}}dt\hfill \\ & -& {\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)\right)}}d{}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)\hfill \end{array}$$

(42)

This suggests that H is a function of $t,x$, and P only. Therefore, we can write

$$dH={\displaystyle \frac{\partial H}{\partial t}}dt+{\displaystyle \frac{\partial H}{\partial x}}dx+{\displaystyle \frac{\partial H}{\partial P}}dP$$

(43)

By using Equations (35), (40) and (41), we obtain the fractional Hamiltonian’s equations as follows:

$${}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)={\displaystyle \frac{\partial H}{\partial P}},{}_a{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}P=-{\displaystyle \frac{\partial H}{\partial x}},{\displaystyle \frac{\partial H}{\partial t}}=-{\displaystyle \frac{\partial L}{\partial t}}.$$

(44)

Indeed,

$${}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)dP={\displaystyle \frac{\partial H}{\partial P}}dP\Rightarrow {}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)={\displaystyle \frac{\partial H}{\partial P}},$$

from Equation (35), we have

$${\displaystyle \frac{\partial L}{\partial x}}=-{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)}}={\displaystyle \frac{\partial H}{\partial x}}$$

from Equation (40), we have

$$P={\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)\right)}}\Rightarrow {}_a{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}P=-{\displaystyle \frac{\partial H}{\partial x}},$$

and

$${\displaystyle \frac{\partial H}{\partial t}}=-{\displaystyle \frac{\partial L}{\partial t}}.$$

Equation (44) characterizes two fractional differential equations of order $p\left(\alpha \right)$ for the system that is equivalent to the system in Equation (35). Because of its similarity to the canonical Euler equations for integer-order systems, the system in Equation (44) is referred to as the fractional canonical system of Euler equations, or simply the fractional canonical Euler equations.

Example 3.

The function L in Equation (29) can be thought of as a function that incorporates both the left and the right. In the context of Atangana–Baleanu Caputo fractional derivatives (ABCFDs),

$$L=L(x,{}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right),{}_t{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right))$$

for which the GELE is given as ([17]),

$${\displaystyle \frac{\partial L}{\partial x}}+{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)}}+{}_a{}^{\mathit{ABR}}D_t^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_t{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)\right)}}=0.$$

(45)

Furthermore, the function L in Equation (29) can be thought of as a function that contains the left and right Caputo fractional derivatives (CFDs).

$$L=L(x,{}_a{}^{C}D_t^{p\left(\alpha \right)}x\left(t\right),{}_t{}^{C}D_b^{p\left(\alpha \right)}x\left(t\right))$$

for which the GELE is given as ([17])

$${\displaystyle \frac{\partial L}{\partial x}}+{}_{t}D_b^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_a{}^{C}D_t^{p\left(\alpha \right)}x\left(t\right)\right)}}+{}_{a}D_t^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_t{}^{C}D_b^{p\left(\alpha \right)}x\left(t\right)\right)}}=0.$$

(46)

For $p\left(\alpha \right)=1$, the Euler–Lagrange equation is given as

$${\displaystyle \frac{\partial L}{\partial x}}-{\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial L}{\partial x}}=0.$$

(47)

Equations (35) and (46) (and its ABCFD counterpart), and Equation (47) contain both forward and backward derivatives. Recall that $-d/dt$ is just a derivative in reverse orientation. As a result, backward derivatives are visible in Equations (35) and (46), but they are present in Equation (47) in a disguised form.

We can easily generalize to issues involving several unknown functions. ${\mathcal{F}}_n$ is a collection of all functions with continuous variables $x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)$. The ABC fractional derivatives of order $p\left(\alpha \right)$ on the left and $\beta$ on the right satisfy the conditions for $t\in [a,b]$.

$$x_i\left(a\right)=x_{ia},x_i\left(b\right)=x_{ib},i=1,2,\dots ,n.$$

The problem can be defined as follows: Identify the functions from ${\mathcal{F}}_n$ that are $x_1,x_2,\dots ,x_n$, for which the functional

$$J[x_1,x_2,\dots ,x_n]=$$

$${\int }_a^{b}L\left[t,x_1\left(t\right),\dots ,x_n\left(t\right),{}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}{x}_1\left(t\right),\dots ,{}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}{x}_n\left(t\right),{}_t{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}{x}_1\left(t\right),\dots {}_t{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}{x}_n\left(t\right)\right]dt$$

has an extremum where the first and second partial derivatives of the function

$$L(t,x_1,\dots ,x_n,u_1,\dots ,u_n,v_1,\dots ,v_n)$$

are continuous with respect to all of its arguments. To allow an extremum, $J[x_1,x_2,\dots ,x_n]$ must satisfy Euler–Lagrange equations for $x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)$.

$${\displaystyle \frac{\partial L}{\partial x_i}}+{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}{x}_i\left(t\right)\right)}}+{}_a{}^{\mathit{ABR}}D_t^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_t{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}{x}_i\left(t\right)\right)}}=0,i=1,2,\dots ,n.$$

(48)

Example 4.

Imagine a system of two planar pendula, each with length l and mass m, that are strung on a horizontal line at the same distance from one another and move in the same plane. The fractional version of the Lagrangian is given by

$$L(t,x_1,x_2,{}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}{x}_1,{}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}{x}_2)={\displaystyle \frac{1}{2}}m\left[{\left({}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}{x}_1\right)}^2+{\left({}_a{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}{x}_2\right)}^2\right]-{\displaystyle \frac{1}{2}}{\displaystyle \frac{mg}{l}}(x_1^2+x_1^2).$$

(49)

To obtain the fractional Euler–Lagrange equation, we use

$${\displaystyle \frac{\partial L}{\partial x}}+{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)}}+{}_a{}^{\mathit{ABR}}D_t^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_t{}^{\mathit{ABC}}D_b^{p\left(\alpha \right)}x\left(t\right)\right)}}=0,i=1,2.$$

(50)

It follows that

$${}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}\left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}{x}_1\right)+{\displaystyle \frac{g}{l}}{x}_1=0,{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}\left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}{x}_2\right)+{\displaystyle \frac{g}{l}}{x}_2=0$$

(51)

When $p\left(\alpha \right)\to 1$, these equations reduce to the harmonic oscillator’s equation of motion.

$$x_1+{\displaystyle \frac{g}{l}}{x}_1=0,x_2+{\displaystyle \frac{g}{l}}{x}_2=0$$

(52)

5. Constrained Systems and Fractional Variational Principles in Distributed-Order Atangana–Baleanu Derivatives

The above-mentioned fractional canonical equations hold when there are no primary constraints, i.e., when all canonical momenta are linearly independent. Many dynamical systems of physical importance have constraints. The problem can be defined as follows. Identify the functional extremum

$$J\left[x\right]={\int }_a^{b}L(t,x,{}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right))dt,$$

subject to the dynamic constraint

$${}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)=\varphi \left(x\right),$$

with the boundary conditions

$$x\left(a\right)=x_a,x\left(b\right)=x_b.$$

In this case, we define the functional

$$S\left[x\right]={\int }_a^b[L+\lambda \mathsf{\Phi }]dt,$$

where

$$\mathsf{\Phi }(t,x,{}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right))=\varphi \left(x\right)-{}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)=0$$

and $\lambda$ is the Lagrange multiplier. Then, Equation (35) in this case takes the form

$${\displaystyle \frac{\partial S}{\partial x}}+{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}{\displaystyle \frac{\partial S}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)}}=0$$

(53)

which can be written as

$${\displaystyle \frac{\partial L}{\partial x}}+{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}{\displaystyle \frac{\partial L}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)}}+\lambda [{\displaystyle \frac{\partial \mathsf{\Phi }}{\partial x}}+{}_t{}^{\mathit{ABR}}D_b^{p\left(\alpha \right)}{\displaystyle \frac{\partial \mathsf{\Phi }}{\partial \left({}_a{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)}}]=0$$

(54)

Example 5.

Let us take

$$J\left[x\right]={\int }_0^1({{}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right))}^{2}dt,$$

with the boundary conditions

$$x\left(0\right)=0,x\left(1\right)=0,$$

$$K_1\left[x\right]={\int }_0^{1}xdt=0,K_2\left[x\right]={\int }_0^{1}txdt=1.$$

Then, we have

$$S\left[x\right]={\int }_0^1\left[({{}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right))}^2+{\lambda }_{1}x+{\lambda }_{2}tx\right]dt,$$

where ${\lambda }_1$ and ${\lambda }_2$ are the Lagrange multipliers. Then, Equation (53) takes the form

$$2{}_t{}^{\mathit{ABR}}D_1^{p\left(\alpha \right)}\left({}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)-{\lambda }_1-{\lambda }_{2}t=0$$

(55)

which can be written as

$${}_t{}^{\mathit{ABR}}D_1^{p\left(\alpha \right)}\left({}_0{}^{\mathit{ABC}}D_1^{p\left(\alpha \right)}x\left(t\right)\right)={\displaystyle \frac{1}{2}}{\lambda }_1+{\displaystyle \frac{1}{2}}{\lambda }_{2}t$$

(56)

After some calculations, we obtain

$$x=-60t+180t^2-120t^3$$

In Figure 1, we show the solution for x.

6. Problem of Fractional Optimal Control with Distributed-Order Derivatives of Atangana–Baleanu Derivatives

Using the concepts provided above, the fractional optimal control problem (FOCP) under examination can be defined as follows. Choose the most effective control, $u\left(t\right)$, for an FDS that minimizes the performance index

$$J\left[u\right]={\int }_0^{1}F(x,u,t)dt,$$

(57)

subject to the dynamic constraint

$${}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)=G(x,u,t),$$

(58)

with the boundary conditions

$$x\left(0\right)=x_0.$$

(59)

$x\left(t\right)$ is the state variable, t is the time, and F and G are two arbitrary functions. Note that there may be more terms with state variables at the end of Equation (57). This phrase is not utilized here for simplicity’s sake. When $p\left(\alpha \right)=1$, the previously indicated problem turns into a standard optimum control problem. In this instance, 0 and 1 have been selected as the integration limits.

We also consider the case $0<p\left(\alpha \right)<1$. These are not problems with the method. Any order is acceptable for the derivative, and any limitations can be considered. These considerations are considered for the sake of simplicity. The optimal control is found by applying the traditional approach and creating a modified performance index.

Here, we specify the functional

$$\mathcal{J}\left[x\right]={\int }_0^1[F(x,u,t)+\lambda (G(x,u,t)-{}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right))]dt,$$

(60)

where $\lambda$ is the Lagrange multiplier, sometimes called an adjoint variable or costate. Using a variant of Equation (60), we arrive at

$$\delta \mathcal{J}\left[u\right]={\int }_0^1[{\displaystyle \frac{\partial F}{\partial x}}\delta x+{\displaystyle \frac{\partial F}{\partial u}}\delta u+\delta \lambda (G(x,u,t)-{}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right))$$

$$+\lambda \left({\displaystyle \frac{\partial G}{\partial x}}\delta x+{\displaystyle \frac{\partial F}{\partial u}}\delta u-\delta \left({}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)\right)]dt.$$

(61)

Using Equation (7), the last integral in Equation (61) can be written as

$${\int }_0^1\lambda \delta \left({}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)\right)dt={\int }_0^1\delta x\left(t\right)\left({}_t{}^{\mathit{ABC}}D_1^{p\left(\alpha \right)}\lambda \right)dt$$

(62)

provided $\delta x\left(0\right)=0$ or $\lambda \left(0\right)=0$, and $\lambda x\left(1\right)=0$ or $\lambda \left(1\right)=0$. We have $\delta x\left(0\right)=0$ since $x\left(0\right)$ is specified, and since $x\left(1\right)$ is not, we need $\lambda \left(1\right)$ to be zero. Equation (62)’s identity is met under these assumptions. It should be noted that we have assumed that the fractional derivative and the order of variation are interchangeable. Equations (61) and (62) are used to obtain the following:

$$\delta \mathcal{J}\left[u\right]={\int }_0^1[\delta \lambda (G(x,u,t)-{}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right))+\delta x[{\displaystyle \frac{\partial F}{\partial x}}+\lambda {\displaystyle \frac{\partial G}{\partial x}}$$

$$-{}_t{}^{\mathit{ABC}}D_1^{p\left(\alpha \right)}\lambda ]+\delta u[{\displaystyle \frac{\partial F}{\partial u}}+\lambda {\displaystyle \frac{\partial G}{\partial u}}]]dt,$$

(63)

For $\mathcal{J}\left[u\right]$ to be minimized (and consequently for $J\left(u\right)$ to be minimized), $\delta x,\delta u$, and $\delta \lambda$ in Equation (63) must all be zero. This results in

$${}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)=G(x,u,t)$$

(64)

$${}_t{}^{\mathit{ABR}}D_1^{p\left(\alpha \right)}\lambda ={\displaystyle \frac{\partial F}{\partial x}}+\lambda {\displaystyle \frac{\partial G}{\partial x}}$$

(65)

$${\displaystyle \frac{\partial F}{\partial u}}+\lambda {\displaystyle \frac{\partial G}{\partial u}}=0.$$

(66)

and

$$x\left(0\right)=x_0\mathrm{and}\lambda \left(1\right)=0.$$

(67)

Equations (64)–(66) display the Euler–Lagrange equations for the FOCP. The requirements for the FOCP under discussion to be optimal are given by these formulas. They are quite similar to the Euler–Lagrange equations for traditional optimal control problems, except for the differential equations that arise, which contain the left and right fractional derivatives. Furthermore, the derivation of these equations is quite similar to that of an integral-order derivative optimum control issue. Solving Equations (64)–(67) is necessary to obtain the optimal control for the fractional system.

Equation (64) has left Atangana–Baleanu in Caputo FD, while Equation (65) has right Atangana–Baleanu. This clearly shows that to solve optimum control problems and consider end conditions, one must comprehend both forward and backward derivatives. Conventional theories of optimum control either ignore or fail to mention this issue. The order-1 backward derivative is the negative of the order-1 forward derivative, which is the main cause of this.

Example 6.

As an example, suppose that the state and control’s quadratic forms are integrals of the performance index

$$J\left[u\right]={\displaystyle \frac{1}{2}}{\int }_0^1[q\left(t\right)x^2\left(t\right)+r\left(t\right)u^2]dt,$$

(68)

where $q\left(t\right)\ge 0$ and $r\left(t\right)>0$, and the dynamics of the system is described by the following linear fractional differential equation:

$${}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)=a\left(t\right)x+b\left(t\right)u,$$

(69)

Formulations and solutions for this linear system within the conventional Riemann–Liouville and Caputo derivatives have been well documented in numerous textbooks and journal articles (see, for example, [11,63]). A lot of research has been performed on this linear system with $p\left(\alpha \right)=1$ and $0<p\left(\alpha \right)<1$. We study it in relation to the distributed-order Atangana–Baleanu derivatives that have been created recently. The Euler–Lagrange Equations (64)–(66) result in Equation (69) for $0<p\left(\alpha \right)<1$.

$${}_0{}^{\mathit{ABR}}D_t^{p\left(\alpha \right)}\lambda =q\left(t\right)x+a\left(t\right)\lambda ,$$

(70)

and

$$r\left(t\right)u+b\left(t\right)\lambda =0.$$

(71)

From Equations (64) and (70), we obtain

$${}_0{}^{\mathit{ABR}}D_t^{p\left(\alpha \right)}x=a\left(t\right)x-r^{-1}\left(t\right)b^2\left(t\right)\lambda .$$

(72)

Under the terminal circumstances given by Equation (66), the state $x\left(t\right)$ and the costate $\lambda \left(t\right)$ are obtained by solving the fractional differential Equations (70) and (72). Once $\lambda \left(t\right)$ is known, the control variable $u\left(t\right)$ can be found using Equation (71).

Example 7.

Time-invariant FOCP with distributed order

Take the following time-invariant FOCP as a first example: To minimize the quadratic performance index, find the control u(t):

$$J\left[u\right]={\displaystyle \frac{1}{2}}{\int }_0^1[x^2\left(t\right)+u^2]dt,$$

(73)

subject to the system dynamics

$${}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)=-x+u,$$

(74)

and the initial condition

$$x\left(0\right)=1.$$

(75)

Note that in this example,

$$q\left(t\right)=r\left(t\right)=-a\left(t\right)=b\left(t\right)=x_0=1,$$

(76)

and Equations (70) and (71) are given as

$${}_t{}^{\mathit{ABR}}D_1^{p\left(\alpha \right)}\lambda =x-\lambda .$$

(77)

and

$$u+\lambda =0,$$

(78)

In Figure 2, the control u and the states x and $\alpha =1$ are shown.

As a result, the optimal control function $u\left(t\right)$ is negative and the costate variable $\lambda \left(t\right)$ is negative. Refs. [11,63] and the references therein include numerical and analytical results using the conventional Riemann-Liouville and Caputo fractional derivatives for this situation for $p\left(\alpha \right)=1$ and for $p\left(\alpha \right)\in (0,1)$.

Example 8.

Distributed-order time-varying FOCP

Consider a linear time-varying system as a second example. It has the same beginning condition and performance index as those in Example 7, but it is also subject to the following dynamic restriction:

$${}_0{}^{\mathit{ABC}}D_t^{p\left(\alpha \right)}x\left(t\right)=tx+u,$$

(79)

For this case,

$$q\left(t\right)=r\left(t\right)=b\left(t\right)=x_0=1,a\left(t\right)=t,$$

(80)

$${}_t{}^{\mathit{ABR}}D_1^{p\left(\alpha \right)}\lambda =x+t\lambda .$$

(81)

and

$$u+\lambda =0,$$

(82)

In Figure 3, we show the control u and the states x and $\alpha =1$.

Several researchers have previously examined this problem for $p\left(\alpha \right)=1$ and for $p\left(\alpha \right)\in (0,1)$ within the standard classical Riemann–Liouville and Caputo fractional derivatives (see, for example, [11,63] and references therein).

7. Conclusions

Novel distributed-order Atangana–Baleanu fractional derivatives have been used to provide a general formulation for a class of fractional optimal control issues. The Euler–Lagrange equations for the fractional optimal control problems were derived using the formula for fractional integration by parts, the calculus of variations, and the Lagrange multiplier technique. The resulting equations and the provided framework are very similar to those for conventional optimum control problems.

The formulation is intended for a system with a fractional system dynamic constraint and a quadratic performance index. The aforementioned and additional fractional calculus literature shows that many concepts from ordinary calculus can be applied to fractional calculus with just a few modifications. One benefit of the new fractional derivative is that, unlike the earlier formulations, it does not exhibit a singularity.

Finally, we observe that the new distributed-order Atangana–Baleanu fractional derivatives have not seen much development in the field of FOCPs. The main cause of this is a dearth of research on the mathematics behind fractional derivatives. This obstacle has been removed by recent advancements in the realm of fractional derivatives. Numerous ideas from classical control theory are readily applicable to FOCPs, as demonstrated by the prior formulation. The formulation can be extended to many more FOCPs using the unique distributed-order Atangana–Baleanu fractional derivatives, even though just one class of FOCPs using these derivatives was examined here. This comment is meant to pique curiosity about fractional optimal control and fractional variational calculus.

In future work, some fractional optimal control problems for systems governed by some new fractional derivatives, such as $\theta$-fractional derivatives and a cotangent fractional derivative, will be studied; see [5,6,7,8].