Controllability of Mild Solution of Nonlocal Conformable Fractional Differential Equations

Hannabou, Mohamed; Bouaouid, Mohamed; Hilal, Khalid

doi:https://doi.org/10.1155/2022/3671909

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 3671909 | https://doi.org/10.1155/2022/3671909

Controllability of Mild Solution of Nonlocal Conformable Fractional Differential Equations

Mohamed Hannabou,¹Mohamed Bouaouid,¹and Khalid Hilal¹

Academic Editor: John D. Clayton

Received31 Oct 2021

Revised04 Aug 2022

Accepted06 Aug 2022

Published05 Sept 2022

Abstract

In many research works Bouaouid et al. have proved the existence of mild solutions of an abstract class of nonlocal conformable fractional Cauchy problem of the form: The present paper is a continuation of these works in order to study the controllability of mild solution of the above Cauchy problem. Precisely, we shall be concerned with the controllability of mild solution of the following Cauchy problem where is the vectorial conformable fractional derivative of order in a Banach space and is the infinitesimal generator of a semigroup on . The element is a fixed vector in and , are given functions. The control function is an element of with is a Banach space and is a bounded linear operator from into .

1. Introduction

Mathematical models based on factional derivatives with respect to time have been the focus of many studies due to their recent applications in various areas of science [1–5]. Many concrete applications prove that the fractional derivative is a very good approaches to deal better with modeling of dynamical systems with memories [6–17]. Regarding to the literature of fractional calculus, it is well known that there are many approaches to define fractional derivatives including the Riemann-Liouville and Caputo definitions. Unfortunately, these definitions have some shortcomings. For example, they do not satisfy derivative formulas for the product and quotient of two functions. In consequence, many researchers have paid attention to propose a best and simple definition of fractional derivative [18, 19]. For example in the work [18], the authors have proposed a new definition of fractional derivative named conformable fractional derivative. This novel fractional derivative is very simple and verifies all the properties of the classical derivative. Actually, the conformable fractional derivative becomes the subject of many research contributions [20–39].

For example in [20–22], the authors have proved the existence of mild solution for the following nonlocal conformable fractional Cauchy problem: where represents the conformable fractional derivative of order , and is the infinitesimal generator of a semigroup on a Banach space ([40]). The element is a fixed vector in and , are given functions, with is the Banach space of continuous functions defined from into equipped with the norm . The expression means the so-called nonlocal condition, which can be applied in physics with better effects than the classical initial condition [41–43].

Recently, the study of control problems has attracted the attention of many mathematicians and physicists in various fields of science [44–50]. For example, in theory of differential equations, the controllability consists to control evolution systems from the initial position to the desired position. Motivated by the fact that the controllability is a most important qualitative behavior of a dynamical system, we will be concerned with the controllability of the Cauchy problem (1). Precisely, we will prove a controllability result for the following Cauchy problem where the control function is an element of with is a Banach space and is a bounded linear operator from into .

The rest of this paper is organized as follows. In Section 2, we briefly recall some tools related to the conformable fractional calculus. In Section 3, we present the main result. Section 4 is devoted to a concert application.

2. Preliminaries

Recalling some preliminary facts on the conformable fractional calculus.

Definition 1 (see [18]). For , the conformable fractional derivative of order of a function is defined as provided that the limits exist.
The conformable fractional integral of a function is defined by

Theorem 2 (see [21]). If is a continuous function in the domain of , then, we have

Theorem 3 (see [23]). If is a differentiable function, then, we have

Definition 4 (see [23]). The conformable fractional Laplace transform of order of a function is defined as follows The following proposition gives us the actions of the conformable fractional integral and the conformable fractional Laplace transform on the conformable fractional derivative, respectively.

Proposition 5 (see [23]). If is a differentiable function, then, we have the following results

According to [28], we have the following remark.

Remark 6. For two functions and , we have provided that the both terms of each equality exist.

3. Main Result

Lemma 7. If is a solution of Cauchy problem (2), then, the function satisfies the following integral equation

The proof of this result is essentially based on the conformable fractional Laplace transform. For the complete proof, one can see the works [20–22].

Definition 8 (see [20–22]). A function is called a mild solution of Cauchy problem (2) if Now, we deal with the controllability of Cauchy problem (2).

Definition 9. The Cauchy problem (2) is said to be controllable on , if for every , there exists a control such that the mild solution of (2) satisfies .
In the sequel of this paper, we will need the following assumptions:
(H₁) The function is continuous and there exist positive constants , such that and for all , .
(H₂) The function is continuous for all .
(H₃) The function is continuous.
(H₄) There exist positive constants and such that (H₅) The bounded linear operator defined by has an induced inverse operator , which takes values in , and there exist positive constants , such that and .

Theorem 10. Assume that hold, then Cauchy problem (2) is controllable on , provided that

Proof. By using hypothesis for an arbitrary function , we can define a control as follows For this control, we define the operator by

We also introduce for a radius the ball , and we denote by the norm in the space of bounded operators defined from into itself.

We will show that the operator has a fixed point, which is a mild solution of the control problem (2). To do so, we will give the proof in two steps.

Step 1. Prove that there exists a radius such that .
For and , we have Then, one has By using hypothesis , , and , we obtain On the other hand, we have known that In view of assumptions , , and , we obtain By replacing this estimate in , we get Separating the terms containing the expression , one has By using a simple factorization, we obtain Hence, it suffices to consider as a solution in of the following inequality Precisely, we can choose such that

Step 2. We show that is a contraction operator on .
For , , we have According to , , and , we obtain In the other hand, we know that Then, one has By replacing this estimate in , we obtain Taking the supremum, we get Since , then, is a contraction operator on . Hence, there exists a unique element such that for all .
It remains to show that the mild solution is controllable. To this end, we have Thus, Cauchy problem (2) is controllable on .

4. Application

Let be equipped with the inner product and norm defined by and . Define the operator by

As well known, the operator has a discrete spectrum, and the eigenvalues are with the corresponding normalized eigenvectors , The operator generates a contraction semigroup given explicitly by

Then, we have

Next, define the control operator as follows

We have

Then, and thus the operator is bounded. Now return back to the operator , we obtain

Hence, the right inverse of the operator may defined as follows

For the operator , we get

Define the functions and by

For the function , we have

Here in this application example, we have , , , , , , and . Then, the contraction condition assumed in Theorem 10 becomes

For in the last contraction condition, we get . Thus, by using Theorem 10, we conclude that the following Cauchy problem

Has a unique controllable mild solution.

5. Conclusion and Comments

The existence of mild solutions of a Cauchy problem of nonlocal differential equations with conformable fractional derivative is largely studied in several works [20–22]. Our contribution in this present work is the study of the controllability of mild solutions for such Cauchy problems by means of the Banach fixed point theorem combined with theory of semigroups of linear operators. We notice that the constants of increases of the norms of the bounded operators and in the previous application are given directly in a simple way in terms of the exponential function, however, for the Caputo fractional derivative in the application of the nice work [51] are given in terms of the so-called Mittag-Leffler function.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

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Copyright © 2022 Mohamed Hannabou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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