On Atangana-Baleanu fuzzy-fractional optimal control problems

Abstract

Optimal control is a very important field of study, not only in theory but in applications, and fractional optimal control is also a significant branch of research in theory and applications. Based on the concept of fuzzy process, a fuzzy fractional optimal control problem is presented. In this article, we derived the necessary and sufficient optimality conditions for a class of fuzzy-fractional optimal control problems (FFOCPs) with gH-Atangana-Baleanu fuzzy-fractional derivative expressed in Caputo sense. The main aim is to find the best possible control that minimizes the fuzzy performance index and satisfies the related ABC fuzzy-fractional dynamical systems. We also presented some examples for more illustration of the subject.

Keywords

Fuzzy-fractional derivative Atangana-Baleanu fuzzy-fractional derivative generalized Hakahara differentiability 90C46 34K36 93C42

1 Introduction

1.1 Research background

Fractional calculus: Fractional derivatives, or more precisely derivatives of arbitrary orders, have played a significant role in engineering, science, and mathematics in recent years. Samko et al. in [29] provide an encyclopedic treatment for this subject. For further applications and surveys of this field in science and engineering, readers can see [20 , 34].

Riemann-Liouville and Caputo are the most famous fractional derivatives. But these fractional derivatives have some drawbacks. The Caputo derivative demands higher regularity conditions for differentiability, that are specified only for differentiable functions. However, the Riemann-Liouville derivative of a constant function is non-zero, and it also requires initial non-integer order conditions that are not possible to determine physically.

A new fractional derivative known as Caputo Fabrizio (CF) fractional derivative was introduced in [8] but in [6] and [22], authors concluded that this fractional derivative is not a fractional-order derivative but only a filter with a fractional parameter. After that, a new fractional-order operator based on the generalized Mittag-Leffler function was introduced in [5] by Atangana and Baleanu, to overcome the drawbacks of existing fractional derivatives.

Fuzzy calculus: To handle vagueness and uncertainty, which is very common in the input and output of most dynamical systems, Zadeh introduced fuzzy set theory as an extension of crisp set theory in [37]. Zadeh followed this up in [37], when he introduced the idea of a fuzzy derivative. In the past few decades, a lot of research has been done in various fields and applications of fuzzy theory. In particular, the interest in fuzzy optimal control problems has increased and fuzzy optimal control problems have attracted a great deal of attention.

Fuzzy fractional calculus: The concepts of Riemann–Liouville and Caputo fuzzy fractional differentiability, based on the Hukuhara difference, are introduced in [31] and [24], respectively. In [3, 30], the fuzzy fractional Caputo and Riemann–Liouville differentiability concepts, based on generalized-Hukuhara difference are investigated, which strongly generalizes fuzzy differentiability. Moreover, the fuzzy fractional gH-Atangana-Baleanu derivative was introduced in [39]. The new kernel of the gH-Atangana-Baleanu fractional derivative has no singularity and no locality, which was not precisely illustrated in the previous definitions.

Fuzzy fractional optimal control: An extension of the calculus of variations is considered as an optimal control theory, but it uses the control variables for the optimization of the function. In the early 1950s, the theory for optimal control was developed in response to various problems in economics, engineering, and science. The interested readers can see, [4] for some recent advances on fuzzy optimal control problems. In addition, precise modeling of many dynamic systems brings on a set of fractional differential equations. Fractional differential equations are essential for accurate modeling of several dynamical systems, see [23, 28]. When fractional differential equations are combined with initial conditions and a performance index, they lead to the fractional optimal control problems (FOCPs) discussed in [2]. As a result, the studies of fuzzy-fractional optimal problems, which are included fuzzy fractional differential equations, gain more attention. Under the CF differentiability, Zhang et al. [40] proposed the necessary and sufficient optimality conditions for problems with the fractional calculus of variations with a Lagrange function. However, Sheikh et al. [32, 33] pointed out that the kernel of the CF fractional derivative was nonsingular but was still nonlocal. Farhadinia [16, 17] was the first one who introduced the concept of fuzzy variational problems and studied the optimality conditions for this problem. Fard and Salehi [15] and Soolaki et al. [34] also established the necessary optimality conditions for fuzzy fractional variational problems using the concept of Caputo and combined Caputo differentiability based on Hukuhara difference of fuzzy functions. Recently, under the condition of gH-Atangana-Baleanu fractional differentiability, Zhang et al. [39] proposed the generalized necessary and sufficient optimality conditions for problems of the fuzzy fractional calculus of variations with a Lagrange function.

1.2 Research question

Let $\tilde{y} (x)$ be a fuzzy state variable, where x ∈ [0, 1] represents time, $\tilde{F}$ and $\tilde{G}$ be two orbitrary fuzzy functions. The fuzzy optimal control problem can be posed:

Mininmize the fuzzy performace index $\tilde{J} (\tilde{u}) = \int_{0}^{1} \tilde{F} (\tilde{y} (x), \tilde{u} (x), x) dx .$ (1) Subjected to a fuzzy-fractional dynamic constraint $_{0}^{ABC} D_{x}^{α, gH} \tilde{y} (x) = \tilde{G} (\tilde{y} (x), \tilde{u} (x), x),$ (2) and initial conditions $\tilde{y} (0) = x_{0}, \tilde{y} (1) is free .$ (3)

In this paper, we discuss the following questions:

In [4] Alinezhad and Allahviranloo provided the necessary and sufficient optimality conditions for fuzzy fractional optimal control problem with the Caputo derivative. What are the necessary and sufficient optimality conditions for fuzzy fractional optimal control problems with the gH-Atangana-Baleanu derivative?

What is the best possible fuzzy control which satisfies the related fuzzy fractional dynamic systems and minimizes the fuzzy performance index?

What is the application of this fuzzy-fractional optimal control problem? Is this problem providing better results for this application than other optimal control problems?

1.3 Objective of the work

Different performance indexes with fractional type dynamic constraint have many applications and many results are available for crisp range. But in many practical applications, it is seen that many situations cannot be modeled using the crisp range. In these cases, to handle such a situation, those performance indexes with fractional type dynamic constraints are needed to define in a fuzzy environment. FFOCP is an optimal control problem that is governed by a set of fuzzy fractional differential equations. To our best knowledge, there are few papers on the necessary optimality conditions for fuzzy fractional optimal control problems. Only limited work has been done in fuzzy fractional optimal control [4]. It is a new idea to define FFOCP regarding different definitions of fractional derivatives and derive the necessary optimality conditions for a defined class of FFOCP. In this paper, we modify the class of fuzzy-fractional optimal control problems with fuzzy gH-Atangana-Baleanu fractional derivative expressed in Caputo sense. Then we concentrate on providing a set of proper conditions that enable us to derive the necessary optimality conditions for FFOCPs. After that, we derive sufficient optimality conditions for this class.

1.4 Structure of the study

Section 2, provides basic definitions which are essential to develop our main results. In Section 3, first, we provided the fundamental lemma for gH-Atangana-Baleanu fuzzy-fractional integral. We derived the necessary and sufficient optimality conditions for a class of FFOCPs with gH-Atangana-Baleanu fuzzy-fractional derivative expressed in Caputo sense. We devote Section 4 to the result of an application to FFOCPs with gH-Atangana-Baleanu fuzzy-fractional derivative. We give some limitations of the results in Section 5. Finally, some conclusions are given.

2 Basic notions

This section presents some definitions and results from [1 , 37] which will be used in this paper.

Definition 2.1. A fuzzy set $\tilde{v} : R \to [0, 1]$ is known as the fuzzy number, if it satisfies the conditions mentioned below:

$\tilde{v}$ is normal.

$\tilde{v}$ is convex.

$\tilde{v}$ is an upper semi-continuous.

Support of $\tilde{v}$ is compact in R .

Suppose $\tilde{R}$ denotes the collection of fuzzy numbers on R. Obviously, $R \subset \tilde{R}$ . For each α ∈ [0, 1], α-level sets of $\tilde{v}$ are denoted by ${[\tilde{v}]}^{α}$ and define as, ${[\tilde{v}]}^{α} = {b \in R | \tilde{v} (b) \geq α}$ and ${[\tilde{v}]}^{0} = {b \in R | \tilde{v} (b) > 0}$ .

Definition 2.2. The H-difference (Hukuhara difference) of any two fuzzy numbers $\tilde{v}$ and $\tilde{u}$ is a fuzzy number (if exists) which is defined as: $\tilde{v} \underset{H}{⊖} \tilde{u} = \tilde{z} \Leftrightarrow \tilde{v} = \tilde{u} + \tilde{z} .$ If H-difference $\tilde{v} \underset{g}{⊖} \tilde{u}$ exists, its α-level sets are ${[\tilde{v} \underset{H}{⊖} \tilde{u}]}_{α} = [{\underline{v}}_{α} - {\underline{u}}_{α}, {\bar{v}}_{α} - {\bar{u}}_{α}] .$

Definition 2.3. The gH-difference (generalized Hukuhara difference) of any two fuzzy numbers $\tilde{v}$ and $\tilde{u}$ is a fuzzy number (if exists) which s defined as $\tilde{v} \underset{gH}{⊖} \tilde{u} = \tilde{z} \Leftrightarrow {\begin{matrix} (i) \tilde{v} = \tilde{u} + \tilde{z} \\ (ii) \tilde{u} = \tilde{v} + (- 1) \tilde{z} . \end{matrix}$ If $\tilde{v} \underset{gH}{⊖} \tilde{u}$ exists, then its α-level sets are $\begin{matrix} {[\tilde{v} \underset{gH}{⊖} \tilde{u}]}_{α} = [min {{\underline{v}}_{α} - {\underline{u}}_{α}, {\bar{v}}_{α} - {\bar{u}}_{α}}, \\ max {{\underline{v}}_{α} - {\underline{u}}_{α}, {\bar{v}}_{α} - {\bar{u}}_{α}}] . \end{matrix}$

Definition 2.4. A fuzzy function $\tilde{f} : [a, b] \subset R \to \tilde{R}$ is a mapping, if for all c ∈ [a, b], $\tilde{f} (c)$ belongs to $\tilde{R}$ .

Definition 2.5. A full-fuzzy function $\tilde{F} : [\tilde{a}, \tilde{b}] \subset \tilde{R} \to \tilde{R}$ is a mapping, if for all $\tilde{c} \in [\tilde{a}, \tilde{b}]$ , $\tilde{F} (\tilde{c})$ belongs to $\tilde{R}$ .

Definition 2.6. The Housdorff distance on R_F is defined by $D (u, v) = sup_{α \in [0, 1]} {‖ {[u]}_{α} \underset{gH}{⊖} {[v]}_{α} ‖},$ where, for an interval [a, b], the norm is $‖ [a, b] ‖ = max {| a |, | b |} .$ (R_F, D) is a complete metric space.

Definition 2.7. A full fuzzy function $\tilde{F} : A \subset \tilde{R} \to \tilde{R}$ is continuous at $\tilde{b} \in A,$ if for any ɛ > 0, there exists δ > 0 such that $D (\tilde{c}, \tilde{b}) < δ$ for every $\tilde{c} \in A$ implies $D (\tilde{F} (\tilde{c}), \tilde{F} (\tilde{b})) < ɛ .$

A fuzzy function $\tilde{F} : A \subset \tilde{R} \to \tilde{R}$ is called continuous fuzzy function if it is continuous at all $\tilde{b} \in A$ .

Definition 2.8. Let $\tilde{c} \in (\tilde{a}, \tilde{b}) \subset \tilde{R}$ and δ ∈ R satisfying $\tilde{c} + δ \in (\tilde{a}, \tilde{b})$ , then gH-derivative (generalized Hukuhara derivative) of function $\tilde{F} : (\tilde{a}, \tilde{b}) \subset \tilde{R} \to \tilde{R}$ at $\tilde{c}$ is defined as ${\tilde{F}}_{gH}^{'} (\tilde{c}) = lim_{δ \to 0} \frac{\tilde{F} (\tilde{c} + δ) ⊖_{gH} \tilde{F} (\tilde{c})}{δ} .$ If ${\tilde{F}}_{gH}^{'} (\tilde{c}) \in \tilde{R}$ exists, then $\tilde{F}$ is called gH-differentiable at $\tilde{c} .$

Definition 2.9. Let $\tilde{w}, \tilde{v} \in \tilde{R}$ . We write $\tilde{w} ⪯ \tilde{v}$ , if ${\underline{w}}_{a} \leq {\underline{v}}_{a}$ and ${\bar{w}}_{a} \leq {\bar{v}}_{a}$ for all a ∈ [0, 1]. We also write $\tilde{w} ≽ v$ , if ${\underline{w}}_{a} \geq {\underline{v}}_{a}$ and ${\bar{w}}_{a} \geq {\bar{v}}_{a}$ for all a ∈ [0, 1]. Moreover, $\tilde{w} ≺ \tilde{v}$ , if $\tilde{w} ⪯ \tilde{v}$ and a ∈ [0, 1] such that ${\underline{w}}_{a} < {\underline{v}}_{a}$ or ${\bar{w}}_{a} < {\bar{v}}_{a}$ . Furthermore, $\tilde{w} \approx \tilde{v}$ if $\tilde{w} ⪯ \tilde{v}$ and $\tilde{w} ≽ \tilde{v}$ .

We say that $\tilde{w}, \tilde{v} \in \tilde{R}$ are comparable if either $\tilde{w} ⪯ \tilde{v}$ or $\tilde{w} ≽ \tilde{v}$ .

Now, we recall some basic definitions of the Atangana-Baleanu fractional operators [39].

Definition 2.10. Let y ∈ H¹ (a, b) , a < b and α ∈ (0, 1). The left ABC fractional derivative (left Atangana-Baleanu fractional derivative in Caputo sense) of order α is defined by $^{AB C} D_{a +}^{α} y (x) = \frac{β (α)}{1 - α} \int_{a}^{x} E_{α} (- \frac{α {(x - τ)}^{α}}{1 - α})^{y^{'}} (τ) d τ .$ Where the normalization function is β (α) with β (0) = β (1) = 1 and $E_{α} (ϖ) = \sum_{k = 0}^{\infty} \frac{ϖ^{k}}{Γ (α k + 1)}, Re (α) ≻ 0, ϖ \in C,$ is Mittag-Leffler function having one parameter.

Definition 2.11. The left ABR fractional derivative (left Atangana-Baleanu fractional derivative in the Riemann-Liouville sense) of order α is defined by $^{ABR} D_{a +}^{α} y (x) = \frac{β (α)}{1 - α} \frac{d}{dx} \int_{a}^{x} E_{α} (- \frac{α {(x - τ)}^{α}}{1 - α}) y (τ) d τ .$

Definition 2.12. The right ABC fractional derivative (right Atangana-Baleanu fractional derivative in the Caputo sense) of order α is defined by $^{ABC} D_{b -}^{α} y (x) = - \frac{β (α)}{1 - α} \int_{x}^{b} E_{α} (- \frac{α {(τ - x)}^{α}}{1 - α})^{y^{'}} (τ) d τ .$

Definition 2.13 The right ABR fractional derivative (right Atangana-Baleanu fractional derivative in the Riemann-Liouville sense) of order α is defined by $^{ABR} D_{b -}^{α} y (x) = - \frac{β (α)}{1 - α} \frac{d}{dx} \int_{x}^{b} E_{α} (- \frac{α {(τ - x)}^{α}}{1 - α}) y (τ) d τ .$ Proposition 2.14. The Right Atangana-Baleanu fractional derivative of order α with Mittag-Leffler kernel in Riemann-Liouville sense and in Caputo sense are related by the identity $^{ABC} D_{b -}^{α} y (x) =^{ABR} D_{b -}^{α} y (x) - \frac{β (α)}{1 - α} y (b) E_{α} (- \frac{α {(b - x)}^{α}}{1 - α}) .$

Proposition 2.15. (ABC fractional deravative integration by parts). Suppose φ is continuous funtion and ψ is from class C¹. Then, $\begin{matrix} \int_{0}^{b} φ {(x)}^{ABC} D_{a +}^{α} ψ (x) dx & = \int_{0}^{b} ψ {(x)}^{ABR} D_{b -}^{α} φ (x) dx \\ + \frac{β (α)}{1 - α} {[ψ (x) E_{α, 1, \frac{- α}{1 - α}, b -}^{1} φ (x)] |}_{x = 0}^{x = b} . \end{matrix}$ $\begin{matrix} \int_{0}^{b} φ {(x)}^{ABC} D_{b -}^{α} ψ (x) dx & = \int_{0}^{b} ψ {(x)}^{ABR} D_{0 +}^{α} φ (x) dx \\ - \frac{β (α)}{1 - α} {[ψ (x) E_{α, 1, \frac{- α}{1 - α}, 0 +}^{1} φ (x)] |}_{x = 0}^{x = b}, \end{matrix}$ where, left generalized fractional integral operator is defined by $E_{α, μ, ω, a +}^{γ} φ (x) = \int_{a}^{x} {(x - τ)}^{μ - 1} E_{α, μ}^{γ} [ω {(x - τ)}^{α}] φ (τ) d τ, x ≻ a,$ and right generalized fractional integral operator is defined by $E_{α, μ, ω, b -}^{γ} φ (x) = \int_{x}^{b} {(τ - x)}^{μ - 1} E_{α, μ}^{γ} [ω {(τ - x)}^{α}] φ (τ) d τ, x ≺ b,$ where $E_{α, μ}^{γ} (ξ) = \sum_{k = 0}^{\infty} ({(γ)}_{k} ξ^{k} / Γ (α k + μ) k!)$ is generalized Mittag-Leffler function that is defined for complex number α, μ, γ, (Re(α) > 0).

$(\tilde{C} [a, b], \tilde{R})$ is a collection of continuous fuzzy-valued functions on interval [a, b] ⊂ R, $({\tilde{C}}^{1} [a, b], \tilde{R})$ is a space of fuzzy-valued functions having continuous first derivatives on [a, b] ⊂ R, and the class of Lebesgue integrable fuzzy-valued functions on interval [a, b] ⊂ R is $(\tilde{L} [a, b], \tilde{R})$ , as in [3].

Definition 2.16. Suppose $D_{gH} \tilde{y} \in (\tilde{C} [a, b], \tilde{R}) \cap (\tilde{L} [a, b], \tilde{R})$ . The ABC fuzzy-fractional derivative (gH-Atangana-Baleanu fuzzy-fractional derivative in the Caputo sense) of $\tilde{y} (x)$ ( ${[gH]}_{α}^{ABC}$ -differentiable) is defined as follows: $^{ABC} D_{a +}^{α, gH} \tilde{y} (x) = \frac{β (α)}{1 - α} \int_{a}^{x} E_{α} (- \frac{α {(x - τ)}^{α}}{1 - α}) (D_{gH} \tilde{y} (τ)) d τ .$ Where 0 < α ≤ 1, x > a .

Theorem 2.17. Suppose $D_{gH} \tilde{y} (x) \in (\tilde{C} [a, b], \tilde{R}) \cap (\tilde{L} [a, b], \tilde{R})$ and ${[\tilde{y} (x)]}^{r} = [\underline{y} (x, r), \bar{y} (x, r)]$ for all x ∈ [a, b] and 0 ≤ r ≤ 1. If the real-valued functions $\underline{y} (x, r)$ and $\bar{y} (x, r)$ are ABC fractional diffrentiable at x ∈ [a, b], then the fuzzy function $\tilde{y}$ is ${[gH]}_{α}^{ABC}$ -differentiable at x ∈ [a, b] and $\begin{matrix} {[^{ABC} D_{a +}^{α, gH} \tilde{y} (x)]}_{r} & = [min {^{ABC} D_{a +}^{α} \underline{y} (x, r),^{ABC} D_{a +}^{α} \bar{y} (x, r)}, \\ max {^{ABC} D_{a +}^{α} \underline{y} (x, r),^{ABC} D_{a +}^{α} \bar{y} (x, r)}] . \end{matrix}$

Definition 2.18. Suppose $D_{gH} \tilde{y} \in (\tilde{C} [a, b], \tilde{R}) \cap (\tilde{L} [a, b], \tilde{R})$ . The ABR fuzzy-fractional derivative (gH-Atangana-Baleanu fuzzy fractional derivative in the Riemann-Liouville sense) of $\tilde{y}$ ( ${[gH]}_{α}^{ABR}$ -differentiable) is defined as follows: $^{ABR} D_{b -}^{α} \tilde{y} (x) = - \frac{β (α)}{1 - α} {(\int_{x}^{b} E_{α} (- \frac{α {(τ - x)}^{α}}{1 - α}) \tilde{y} (τ) d τ)}_{gH}^{'} .$

Theorem 2.19. Suppose $D_{gH} \tilde{y} \in (\tilde{C} [a, b], \tilde{R}) \cap (\tilde{L} [a, b], \tilde{R})$ and ${[\tilde{y} (x)]}^{r} = [\underline{y} (x, r), \bar{y} (x, r)]$ for all x ∈ [a, b] and 0 ≤ r ≤ 1. If the real-valued functions $\underline{y} (x, r)$ and $\bar{y} (x, r)$ are ABR fractional differentiable at x ∈ [a, b], then the fuzzy function $\tilde{y} (x)$ is ${[gH]}_{α}^{ABC}$ differentiable at x ∈ [a, b] and $\begin{matrix} {[^{ABR} D_{b -}^{α, gH} \tilde{y} (x)]}_{α} & = [min {^{ABR} D_{b -}^{α} \underline{y} (x, r),^{ABR} D_{b -}^{α} \bar{y} (x, r)}, \\ max {^{ABR} D_{b -}^{α} \underline{y} (x, r),^{ABR} D_{b -}^{α} \bar{y} (x, r)}] . \end{matrix}$

3 Main results

In this section, we are interested to derive necessary and sufficient conditions for fuzzy fractional optimal control problems.

Lemma 3.1. Let $\tilde{λ}, \tilde{y} : [0, 1] \to \tilde{R}$ and $\tilde{λ}, \tilde{y}$ continuous on [0, 1] and α ∈ (0, 1) . If $\tilde{λ} (1) = 0$ and $\tilde{y} (0) = {\tilde{y}}_{0}$ , then $\int_{0}^{1} \tilde{λ} ⊙_{0}^{ABC} D_{x}^{α, gH} δ \tilde{y} dx = \int_{0}^{1} δ \tilde{y} ⊙_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} dx .$

Proof. Left hand side integral in parametric form is defined as follow, ${[\int_{0}^{1} \tilde{λ} ⊙_{0}^{ABC} D_{x}^{α, gH} δ \tilde{y} dx]}^{r} =$ (4) $\begin{matrix} [\int_{0}^{1} min {\underline{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \underline{y} (x, r), \underline{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \bar{y} (x, r), \\ \bar{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \underline{y} (x, r), \bar{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \bar{y} (x, r)}, \\ \int_{0}^{1} max {\underline{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \underline{y} (x, r), \underline{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \bar{y} (x, r), \\ \bar{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \underline{y} (x, r), \bar{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \bar{y} (x, r)}] \end{matrix}$

By using Proposition 2.15 (integration by parts for ABC fractional derivatives) and conditions $\tilde{λ} (1) = \tilde{0}$ , $\tilde{y} (0) = x_{0} .$ We get $\begin{matrix} \int_{0}^{1} \underline{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \underline{y} (x, r) dx \\ = \int_{0}^{1} δ \underline{y} (x, r) ._{x}^{ABR} D_{1}^{α, gH} \underline{λ} (x, r) dx \end{matrix}$ (5) $\begin{matrix} + \frac{B (α)}{1 - α} {[δ \underline{y} (x, r) E_{α, 1, \frac{- α}{1 - α}, 1 -}^{1} \underline{λ} (x, r)]}_{0}^{1} \\ = \int_{0}^{1} δ \underline{y} (x, r) ._{x}^{ABR} D_{1}^{α, gH} \underline{λ} (x, r) dx . \end{matrix}$ Similarly,

$\begin{matrix} \int_{0}^{1} \underline{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \bar{y} (x, r) dx \\ = \int_{0}^{1} δ \bar{y} (x, r) ._{x}^{ABR} D_{1}^{α, gH} \underline{λ} (x, r) dx, \\ \int_{0}^{1} \bar{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \underline{y} (x, r) dx \\ = \int_{0}^{1} δ \underline{y} (x, r) ._{x}^{ABR} D_{1}^{α, gH} \bar{λ} (x, r) dx, \\ \int_{0}^{1} \bar{λ} (x, r) ._{0}^{ABC} D_{x}^{α, gH} δ \bar{y} (x, r) dx \\ = \int_{0}^{1} δ \bar{y} (x, r) ._{x}^{ABR} D_{1}^{α, gH} \bar{λ} (x, r) dx . \end{matrix}$ (6) Using equations (4), (5) and (6), we obtain $\begin{matrix} {[\int_{0}^{1} \tilde{λ} ⊙_{0}^{ABC} D_{x}^{α, gH} δ \tilde{y} dx]}^{r} \\ = [\int_{0}^{1} min {δ \underline{y} (x, r) ._{x}^{ABR} D_{1}^{α, gH} \underline{λ} (x, r), δ \bar{y} (x, r) ._{x}^{ABR} D_{1}^{α, gH} \underline{λ} (x, r), \\ δ \underline{y} {(x, r)}_{x}^{ABR} D_{1}^{α, gH} \bar{λ} (x, r), δ \bar{y} (x, r) ._{x}^{ABR} D_{1}^{α, gH} \bar{λ} (x, r)}, \\ \int_{0}^{1} max {δ \underline{y} (x, r) ._{x}^{ABR} D_{1}^{α, gH} \underline{λ} (x, r), δ \bar{y} (x, r) ._{x}^{ABR} D_{1}^{α, gH} \underline{λ} (x, r), \\ δ \underline{y} {(x, r)}_{x}^{ABR} D_{1}^{α, gH} \bar{λ} (x, r), δ \bar{y} (x, r) ._{x}^{ABR} D_{1}^{α, gH} \bar{λ} (x, r)}] . \end{matrix}$ Hence, $\int_{0}^{1} \tilde{λ} ⊙_{0}^{ABC} D_{x}^{α, gH} δ \tilde{y} dx = \int_{0}^{1} δ \tilde{y} ⊙_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} dx .$ ■

3.1 Necessary optimality conditions

To find fuzzy optimal control $\tilde{u}$ , we adopt traditional aproach of Lagrange multiplier. We formulate a modified fuzzy performance index as $\tilde{J} (\tilde{u}) = \int_{0}^{1} [\tilde{F} (\tilde{y}, \tilde{u}, x) + \tilde{λ} ⊙ (_{0}^{ABC} D_{x}^{α, gH} \tilde{y} \underset{gH}{⊖} \tilde{G} (\tilde{y}, \tilde{u}, x))] dx .$ Now we derive variation of $\tilde{J}$ from increment $\begin{matrix} ▵ \tilde{J} & = \tilde{J} (\tilde{u} + δ \tilde{u}) \underset{gH}{⊖} \tilde{J} (\tilde{u}) \\ = \int_{0}^{1} [\tilde{F} (\tilde{y} + δ \tilde{y}, \tilde{u} + δ \tilde{u}, x) + (\tilde{λ} + δ \tilde{λ}) ⊙ (_{0}^{ABC} D_{x}^{α, gH} (\tilde{y} + δ \tilde{y}) \\ \underset{gH}{⊖} \tilde{G} (\tilde{y} + δ \tilde{y}, \tilde{u} x + δ \tilde{u}, x))] dx \underset{gH}{⊖} \int_{0}^{1} [\tilde{F} (\tilde{y}, \tilde{u}, x) \\ + \tilde{λ} ⊙ (_{0}^{ABC} D_{x}^{α, gH} \tilde{y} \underset{gH}{⊖} \tilde{G} (\tilde{y}, \tilde{u}, x))] dx \\ = \int_{0}^{1} [[\tilde{F} (\tilde{y} + δ \tilde{y}, \tilde{u} + δ \tilde{u}, x) \underset{gH}{⊖} \tilde{F} (\tilde{y}, \tilde{u}, x)] \\ + [(\tilde{λ} + δ \tilde{λ}) ⊙ (_{0}^{ABC} D_{x}^{α, gH} (\tilde{y} + δ \tilde{y}) \underset{gH}{⊖} \tilde{G} (\tilde{y} + δ \tilde{y}, \tilde{u} + δ \tilde{u}, x)) \\ \underset{gH}{⊖} \tilde{λ} ⊙ (_{0}^{ABC} D_{x}^{α, gH} \tilde{y} \underset{gH}{⊖} \tilde{G} (\tilde{y}, \tilde{u}, x))]] dx . \end{matrix}$ By using Proposition 3.1 given in [4], we get $\begin{matrix} ▵ \tilde{J} & = \int_{0}^{1} [[\tilde{F} (\tilde{y} + δ \tilde{y}, \tilde{u} + δ \tilde{u}, x) \underset{gH}{⊖} \tilde{F} (\tilde{y}, \tilde{u}, x)] \\ + [(\tilde{λ} + δ \tilde{λ}) ⊙ (_{0}^{ABC} D_{x}^{α, gH} (\tilde{y} + δ \tilde{y})) \underset{gH}{⊖} \tilde{λ} ⊙_{a} D_{x}^{α, gH} \tilde{y}] \\ \underset{gH}{⊖} [(\tilde{λ} + δ \tilde{λ}) ⊙ \tilde{G} (\tilde{y} + δ \tilde{y}, \tilde{u} + δ \tilde{u}, x) \underset{gH}{⊖} \tilde{λ} ⊙ \tilde{G} (\tilde{y}, \tilde{u}, x)]] dx \end{matrix}$ To find linear part $δ \tilde{J}$ of $▵ \tilde{J}$ with respect to $δ \tilde{y}$ , $δ \tilde{u}$ and $δ \tilde{λ}$ we can use fuzzy Taylor theorem (Theorem 3.14 in [4]) $\begin{matrix} δ \tilde{J} = \int_{0}^{1} [{\tilde{F}}_{\tilde{y}}^{gH} ⊙ δ \tilde{y} + {\tilde{F}}_{\tilde{u}}^{gH} ⊙ δ \tilde{u}] \\ + [\tilde{λ} ⊙ δ (_{0}^{ABC} D_{x}^{α, gH} \tilde{y}) + δ \tilde{λ} ⊙_{0}^{ABC} D_{x}^{α, gH} \tilde{y}] \end{matrix}$ (7) $\underset{gH}{⊖} [\tilde{λ} ⊙ {\tilde{G}}_{\tilde{y}}^{gH} ⊙ δ \tilde{y} + \tilde{λ} ⊙ {\tilde{G}}_{\tilde{u}}^{gH} ⊙ δ \tilde{u} + δ \tilde{λ} ⊙ \tilde{G}] dx .$ Now consider, $\int_{0}^{1} \tilde{λ} ⊙ δ (_{0}^{ABC} D_{x}^{α, gH} \tilde{y}) dx = \int_{0}^{1} \tilde{λ} ⊙_{0}^{ABC} D_{x}^{α, gH} δ \tilde{y} dx$

By using Lemma 3.1, we have $= \int_{0}^{1} δ \tilde{y} ⊙_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} dx$ (8)

Since $\tilde{y} (0)$ is specified, therefore $δ \tilde{y} (0) = \tilde{0}$ and $\tilde{y} (1)$ is free, we need $\tilde{λ} (1) = \tilde{0} .$

Using equation (7) and equation (8), we obtain $\begin{matrix} δ \tilde{J} = \int_{a}^{b} ([{\tilde{F}}_{\tilde{y}}^{gH} +_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} \underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{y}}^{gH}] \\ ⊙ δ \tilde{y} + [{\tilde{F}}_{\tilde{u}}^{gH} \underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{u}}^{gH}] ⊙ δ \tilde{u} \\ + δ \tilde{λ} ⊙ [_{0}^{ABC} D_{x}^{α, gH} \tilde{y} \underset{gH}{⊖} \tilde{G}]) dx . \end{matrix}$

According to fundamental theorem of calculus of variation $δ \tilde{J} = 0$

$\begin{matrix} 0 = \int_{a}^{b} [[{\tilde{F}}_{\tilde{y}}^{gH} +_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} \underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{y}}^{gH}] \\ ⊙ δ \tilde{y} + [{\tilde{F}}_{\tilde{u}}^{gH} \underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{u}}^{gH}] ⊙ δ \tilde{u} \\ + δ \tilde{λ} ⊙ [_{0}^{ABC} D_{x}^{α, gH} \tilde{y} \underset{gH}{⊖} \tilde{G}]] dx . \end{matrix}$

This leads to ${\tilde{F}}_{\tilde{y}}^{gH} +_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} \underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{y}}^{gH} = 0;$

${\tilde{F}}_{\tilde{u}}^{gH} \underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{u}}^{gH} = 0;$

$_{0}^{ABC} D_{x}^{α, gH} \tilde{y} \underset{gH}{⊖} \tilde{G} = 0 .$

Now we can state the necessary optimality conditions for the FFOCPs.

Theorem 3.2. The fuzzy optimal solution of FFOCP is $(\tilde{y}, \tilde{u}),$ if following conditions are satisfied

${\tilde{F}}_{\tilde{y}}^{gH} +_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} \underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{y}}^{gH} = 0,$

${\tilde{F}}_{\tilde{u}}^{gH} \underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{u}}^{gH} = 0,$

$_{0}^{ABC} D_{x}^{α, gH} \tilde{y} \underset{gH}{⊖} \tilde{G} = 0,$

$\tilde{λ} (b) = 0 .$ Where admissible fuzzy state is $\tilde{y}$ and $\tilde{u}$ is an admissible fuzzy control.

3.2 Sufficient optimality conditions

Theorem 3.3. If $({\tilde{y}}^{*}, {\tilde{u}}^{*}, {\tilde{λ}}^{*})$ satisfying the following conditions

${\tilde{F}}_{\tilde{y}}^{gH} +$ $_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} \underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{y}}^{gH} = 0$

${\tilde{F}}_{\tilde{u}}^{gH} \underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{u}}^{gH} = 0$

$_{0}^{ABC} D_{x}^{α, gH} \tilde{y} \underset{gH}{⊖} \tilde{G} = 0$

$\tilde{λ} (1) = \tilde{0}$

$\tilde{F} (\tilde{y}, \tilde{u}, x) \underset{gH}{⊖} \tilde{F} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) \geq {\tilde{F}}_{\tilde{y}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{y} \underset{gH}{⊖} {\tilde{F}}_{\tilde{u}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{u}$

$\tilde{G} (\tilde{y}, \tilde{u}, x) \underset{gH}{⊖} \tilde{G} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) \geq {\tilde{G}}_{\tilde{y}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{y} \underset{gH}{⊖} {\tilde{G}}_{\tilde{u}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{u}$

$\tilde{λ} \geq 0$ for all $\tilde{λ} \in [0, 1]$

Proof. Let $(\tilde{y}, \tilde{u})$ be an admisible fuzzy solution. Then we have $\tilde{J} (\tilde{u}) \underset{gH}{⊖} \tilde{J} ({\tilde{u}}^{*}) = \int_{a}^{b} [\tilde{F} (\tilde{y}, \tilde{u}, x) \underset{gH}{⊖} \tilde{F} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x)] dx .$ By using condition (5), we get $\begin{matrix} \tilde{J} (\tilde{u}) \underset{gH}{⊖} \tilde{J} ({\tilde{u}}^{*}) \\ \geq \int_{a}^{b} [{\tilde{F}}_{\tilde{y}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{y} \underset{gH}{⊖} {\tilde{F}}_{\tilde{u}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{u}] dx . \end{matrix}$ Conditions (1) and (2) implies that $\begin{matrix} \tilde{J} (\tilde{u}) \underset{gH}{⊖} \tilde{J} ({\tilde{u}}^{*}) \\ \geq \int_{a}^{b} [(\tilde{λ} ⊙ {\tilde{G}}_{\tilde{y}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) \underset{gH}{⊖}_{x}^{ABR} D_{1}^{α, gH} \tilde{λ}) ⊙ δ \tilde{y} \end{matrix}$ (9) $\underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{u}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{u}] dx .$ By using Lemma 3.1 and noting $\tilde{y} (a) = x_{a},$ we obtain $\begin{matrix} \int_{a}^{b}_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} ⊙ δ \tilde{y} dx & = \int_{a}^{b} \tilde{λ} ⊙_{0}^{ABC} D_{x}^{α, gH} δ \tilde{y} dx \\ = \int_{a}^{b} \tilde{λ} ⊙_{0}^{ABC} D_{x}^{α, gH} (\tilde{y} \underset{gH}{⊖} {\tilde{y}}^{*}) dx . \end{matrix}$ By using condition (3) we get $\begin{matrix} \int_{a}^{b}_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} ⊙ δ \tilde{y} dx \\ = \int_{a}^{b} \tilde{λ} ⊙ (\tilde{G} (\tilde{y}, \tilde{u}, x) \underset{gH}{⊖} \tilde{G} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x)) dx . \end{matrix}$ By using condition (5) $\begin{matrix} \int_{a}^{b}_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} ⊙ δ \tilde{y} dx = \int_{a}^{b} \tilde{λ} \\ ⊙ ({\tilde{G}}_{\tilde{y}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{y} \underset{gH}{⊖} {\tilde{G}}_{\tilde{u}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{u}) dx . \end{matrix}$ (10) Equation (9) and equation (10) gives $\begin{matrix} \tilde{J} (\tilde{u}) \underset{gH}{⊖} \tilde{J} ({\tilde{u}}^{*}) \\ \geq \int_{a}^{b} [\tilde{λ} ⊙ {\tilde{G}}_{\tilde{y}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{y} \underset{gH}{⊖} \tilde{λ} ⊙ \\ ({\tilde{G}}_{\tilde{y}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{y} \underset{gH}{⊖} {\tilde{G}}_{\tilde{u}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{u}) \\ \underset{gH}{⊖} \tilde{λ} ⊙ {\tilde{G}}_{\tilde{u}}^{gH} ({\tilde{y}}^{*}, {\tilde{u}}^{*}, x) ⊙ δ \tilde{u}] dx \end{matrix}$ $\tilde{J} (\tilde{u}) \underset{gH}{⊖} \tilde{J} ({\tilde{u}}^{*}) \geq 0 .$ $\tilde{J}$ has minimum at $({\tilde{y}}^{*}, {\tilde{u}}^{*}) .$ □

4 Example

This section contain illustrative examples of our results.

Example 4.1. The problem is defined as follows: $minimize : \tilde{J} (\tilde{u}) = \int_{0}^{1} [{\tilde{y}}^{2} (x) + {\tilde{u}}^{2} (x)] dx .$ Subjected to $_{0}^{ABC} D_{x}^{α, gH} \tilde{y} = \tilde{u}, \tilde{y} (0) = (- 1, 0, 1) .$ The necessary conditions are $\begin{matrix} 2 ⊙ \tilde{y} (x) +_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} & = 0, \\ 2 ⊙ \tilde{u} (x) & = \tilde{λ}, \\ shitasen 0^{ABC} D_{x}^{α, gH} \tilde{y} & = \tilde{u} . \end{matrix}$ Following fuzzy-fractional boundary-value problem needs to be solved to find optimal solution. $2 ⊙ \tilde{y} (x) +_{x}^{ABR} D_{1}^{α, gH} (2 ⊙_{0}^{ABC} D_{x}^{α, gH} \tilde{y}) = 0,$ $\tilde{y} (0) = (- 1, 0, 1), \tilde{λ} (1) = (0, 0, 0) .$

Example 4.2. The problem is defined as follows: $minimize : \tilde{J} (\tilde{u}) = \int_{0}^{1} [\tilde{y} (x) \tilde{u} (x) + 4] dx .$ Subjected to $_{0}^{ABC} D_{x}^{α, gH} \tilde{y} = \tilde{y} (x) + \tilde{u} (x), \tilde{y} (0) = (0, 1, 3) .$ The necessary conditions are $\begin{matrix} \tilde{u} (x) +_{x}^{ABR} D_{1}^{α, gH} \tilde{λ} & = \tilde{λ}, \\ \tilde{y} (x) & = \tilde{λ}, \\ shitasen 0^{ABC} D_{x}^{α, gH} \tilde{y} & = \tilde{y} (x) + \tilde{u} (x) . \end{matrix}$ Following fuzzy-fractional boundary-value problem needs to be solved to find optimal solution. $_{0}^{ABC} D_{x}^{α, gH} \tilde{y} \underset{gH}{⊖} \tilde{y} (x) +_{x}^{ABR} D_{1}^{α, gH} (\tilde{y} (x)) = 0,$ $\tilde{y} (0) = (0, 1, 3), \tilde{λ} (1) = (0, 0, 0) .$

5 Limitations of the paper

The limitations of the paper are:

(i) We present two examples for more illustration and effectiveness of our subjects. According to our method, we need to solve a system of fuzzy-fractional differential equations to find an optimal solution for the fuzzy-fractional optimal control problem. But solving such a system, analytically, is usually impossible.

(ii) The existence of optimal solutions to fuzzy fractional optimal control problems based on new fractional derivatives is a challenging task.

6 Conclusion

In this article, we have presented a novel technique to solve a class of fuzzy fractional optimal control problems with gH-Atangana-Baleanu fuzzy-fractional derivative expressed in Caputo sense. We derived a set of necessary optimality conditions for this class of FFOCPs. We show that these necessary optimality conditions are sufficient under some more assumptions. From the example section, we summarized and highlighted the key feature of the article.

We have discussed necessary and sufficient optimality conditions. Much remains to be done and we end by mentioning potential lines of research. The got fuzzy fractional optimality conditions are, in general, difficult to solve and it would be good to develop specific numerical methods to address the issue. In the future, we will discuss the existence of the optimal solutions to fuzzy fractional optimal control problems based on gH-Atangana-Baleanu fuzzy-fractional derivative.

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