Approximate Controllability of Neutral Differential Systems with Fractional Deformable Derivatives

: This article deals with the existence and uniqueness of solutions, as well as the approximate controllability of fractional neutral differential equations (ACFNDEs) with deformable derivatives. The ﬁndings are achieved using Banach’s, Krasnoselskii’s, and Schauder’s ﬁxed-point theorems and semigroup theory. Three numerical examples are used to illustrate the application of the theories discussed in the conclusion.


Introduction
There are several branches of science and engineering in which the dynamic behaviour of natural processes is swiftly modelled using the concept of integrals, which have noninteger order and are covered in fractional calculus of functions and their derivatives, etc.Since FDEs include non-local relationships for both time and space, the study of fractional calculus has begun to advance more quickly all over the world.Additionally, fractional differential equations are an excellent tool for explaining memory and genetic traits in a variety of processes.The authors of [1,2] employed fractional differential operators for a different use.These operators were used to understand population growth models.Fractional calculus underwent significant growth after the 19th century, primarily as a result of its suitability for a wide range of disciplines and the emergence of numerous definitions for fractional derivatives, each of which had its own unique characteristics.Examples of these definitions include the Wieyl definition, the Riemann-Liouville definition, the Caputo definition, and others.Ironically, the integral form appears in the majority of the formulations of fractional derivatives.Readers are directed to [3][4][5] for a comprehensive understanding of fractional calculus and [6][7][8][9][10][11][12][13][14][15][16] for an overview of fractional differential equations (FDEs).
The majority of definitions of fractional derivatives utilise the integral form, as previously established.However, in 2014, R. Khalil et al. [17] proposed a limit-based approach, similar to the usual derivative, characterising it as conformable and analogous to a conventional derivative.Subsequently, F. Zulfeqarr et al. [18] introduced the new concept of deformable derivatives, which was notably more straightforward than Khalil's definition.This derivative was inspired by Khalil's work and addressed its limitations while also accommodating a broader range of applications.
It is common knowledge that engineering and mathematical control theory both strongly depend on the idea of controllability.As a result, several researchers have thoroughly examined the controllability of various nonlinear systems in recent years, for example [19][20][21][22][23][24][25], and the references therein.Finding an appropriate control function that will allow researchers to move the dynamical system's state towards the targeted final state is the controllability problem.While exact controllability steers towards an exact final state, approximate controllability allows researchers to steer the system towards an arbitrarily small neighbourhood of the final state.Approximate controllability, therefore, applies more to dynamical systems.
Since Hille and Yosida's discovery of generation theory in 1948, the study of semigroups of BLOs has undergone significant developments, making it a substantial area of mathematics that is widely employed in several analytical fields.To solve differential equations, it is necessary to understand the concept of semigroups of BLOs.It has been effective in resolving a considerable class of differential and integro-differential problems in recent years.Using semigroups, Pazy [26] investigated the EaU of classical solutions, strong solutions, and mild solutions to evolution systems.
Further investigation started with a discussion of previously published works [10,[27][28][29].Specifically, the existence and uniqueness of the mild solutions and approximate controllability of fractional evolution equations with deformable derivatives were investigated in [29,30] and the conclusions were made possible using Banach's and Schauder's fixed-point theorems in semigroup theory.
The authors of [10] examined the existence and uniqueness of solutions to the Cauchy problem for fractional differential equations with non-local conditions Also, the authors of [16] further investigated the properties of the deformable derivatives and used the results to study the existence of solutions to the integro-differential equation achieving their results using Weissinger's fixed-point theorem and Krasnoselskii's fixedpoint theorem.
Later, M. Etefa and Guerekata et al. [27] studied the results of sufficient conditions for the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the deformable fractional derivative Further, they obtained their results using the Banach contraction principle and the alternative Leray-Schauder fixed-point theorems.
Drawing inspiration from the above-mentioned works, we study the existence and uniqueness results for FNDE with DD of the model (1) and the corresponding controllability model where DD D ρ is the deformable fractional derivative of order ρ ∈ (0, 1) and where V is a Hilbert space, and B : V → X is a BLO.This study's primary findings are as follows: 1. We obtain the solutions to systems (1) and ( 2) and present them in Theorems 5 and 6.Also, we prove that systems (3) and ( 4) have approximate controllability.2. The results of this work improve and generalise other studies that have been reported in the literature [10,[27][28][29].
The rest of this article is organised as follows.In Section 2, we discuss the basic definitions, essential properties, and theorems.Our results were obtained using Krasnoselskii's fixed-point theorem and the Banach contraction principle.In Section 3, we discuss the main results, i.e., the existence and uniqueness of the solution to systems (1) and ( 2), using appropriate fixed-point theorems.Then, we show that systems (3) and ( 4) are approximately controllable.Moreover, in Section 4, we present three numerical examples to illustrate our results.

Preliminaries
The objective of this section is to present a summary of the key concepts and results associated with deformable derivatives.These ideas and outcomes are instrumental in our efforts to derive our primary conclusions.Definition 1 (see [18]).The deformable derivative of order ρ ∈ [0, 1] for a function ω : (e 1 , e 2 ) → R is defined by where ρ + µ = 1.If this limit exists, we denote it by D ρ ω(β).
Definition 2 (see [18]).Let ω be a continuous function defined on the interval [e 1 , e 2 ].The ρ-fractional integral of ω is as follows: Theorem 1 (see [18]).Let ω be a differentiable function at a point β ∈ (e 1 , e 2 ) that is differentiable for all ρ at that point.So far, we have where Dω = d dβ ω.
Theorem 2 (see [18]).By assuming that ω is continuous over the interval [e 1 , e 2 ], it follows that I ρ e 1 ω is differentiable with respect to ρ in the open interval (e 1 , e 2 ), which can also be expressed as ω(e 1 ).
We refer the reader to [10,18,27] for further information on the properties and outcomes of deformable derivatives.
Assume that Q is a linear operator from D(Q) ⊂ X into X and ω 0 ∈ X.
From Theorem 3.5 [29], we have z As a result, By substituting the initial conditions for y and z into the equation = ω 0 , we can see that ω(β), as given by Equation ( 10), represents a mild solution to ( 8) and ( 9).

Main Results
The following section begins with a discussion of the EaU of a mild solution to systems (1) and (2), after which the researchers describe and demonstrate the requirements for the approximate controllability of systems (3) and (4).

Existence Results
Let X be a Banach space with the norm • , and C([0, χ], X) be a Banach space of all continuous functions from [0, χ] into X endowed with the supremum norm By using the information in the preceding Theorem 4, we are able to determine the solution to our addressed systems (1) and (2).Definition 3. A function ω ∈ C is considered to be a mild solution to systems (1) and ( 2) provided the integral exists.To study systems (1) and ( 2), the conditions that follow need to be listed: continuously differentiable and we can find the positive constants I A 2 , I A 1 in such a way that: differentiable, and the positive constants I A 2 , I A 1 , I A 2 , I A 1 can be found in such a way that: (ii) ∃ positive constants I H , I H in such a way that Theorem 5. Assume that A 1 and A 2 meet the requirements of (M0)-(M1) and that then, systems (1) and ( 2) have a unique solution on [0, χ].
Proof.We modify systems ( 1) and ( 2) into a fixed-point problem.Define Υ : C → C by We now show that where As a result, for β ∈ [0, χ] and ω ∈ B Q , we have The above results establish that the Υ causes the ball B Q to be transformed into itself.
Proof.Let us define two operators using system (11) as follows: and We now show that where Thus, for β ∈ [0, χ] and ω ∈ B Q , we have As a next step, we prove that Υ 1 is a contraction.As A 1 is continuous, so is Υ 1 .By letting ω, ω ∈ B Q , from ( 14) and (M0), (M1)(ii), we obtain Therefore, for β ∈ [0, χ] and ω ∈ B Q , the researchers arrived at Hence, Υ 1 is a contraction.The continuous operator Υ 2 has been deduced from the fact that the function A 2 is continuous.Moreover, Υ 2 is uniformly bounded on B Q as which suggests that Υ 2 ω C ≤ A. So, the value of Υ 2 is uniformly bounded, but it is still necessary to demonstrate that Υ 2 is equi-continuous to show that the operator is compact.Now, we find that for every κ 1 , κ 2 in [0, χ] with κ 1 , κ 2 , and By utilising (M0) and (M1)(i), we obtain For κ 1 = 0, it is simple to see that I 2 = 0.If κ 1 > 0 and > 0 are small enough, we have Because of the operator's compactness W(β), β > 0, the fact that Thus, Υ 2 is equi-continuous.Because Υ 2 (X) ⊂ X, Υ has at least one fixed point, according to Arzela-Ascoli's theorem, and Υ 2 is considered to be compact, then the problem obtained through the associated system has at least one solution.

Approximate Controllability
X is taken to be a Hilbert space for the sake of this subsection.In this subsection, we define and provide the requirements for the ACFNDE in ( 3) and ( 4).First, we define the mild solution to systems (3) and ( 4).Definition 4. A function ω ∈ C is termed a mild solution to ( 3) and ( 4) if for any v ∈ L 2 ([0, χ], X) and provided the integral exists.
Definition 5 (see [20]).Given any > 0, it is possible to steer from the point ω 0 to within a distance from all points in the state space X at time χ, and systems (3) and ( 4) are considered approximately controllable on [0, χ] if R(χ, ω 0 ) = X.
Proof.We must demonstrate that Υ θ has a fixed point to show that the FN control systems (( 3) and ( 4)) have a mild solution.The proof is divided into the following stages for simplicity: Step 1: For any θ > 0, we can find a constant then from Lemma 3, we obtain For this, we conclude that for large Λ > 0, Υ θ (B Λ ) ⊂ B Λ holds.
Step 3: A family of functions {Υ θ ω : Given Lemma 3 and Theorem 6, we have Given that the operator W(β) is compact and β > 0, we see that Thus, Υ 2 is equi-continuous.For the same reason that Υ 2 (X) ⊂ X, Υ 2 is considered to be compact.Hence, according to Schauder's fixed-point theorem (Theorem 2.8, [29]), we conclude that the operator Υ θ has a fixed point, which is a mild solution of the model in ( 3) and (4).Theorem 8. Suppose that (M0) * -(M4) are true.Furthermore, if the function A 2 is uniformly bounded by the positive constant C, then the FN model in (3) and ( 4) is AC on [0, χ].
Proof.Let ω θ be a fixed point of Υ θ in B Λ .Any fixed point of Υ θ is a mild solution of the system in (3) and ( 4) under the control where and fulfils the subsequent inequality Since A 2 is uniformly bounded, we have Then, from the above discussion, we have Consequently, this proves the approximate controllability of the model in ( 3) and (4).