Existence and controllability of Hilfer fractional neutral di ﬀ erential equations with time delay via sequence method

: This paper deals with the existence and approximate controllability outcomes for Hilfer fractional neutral evolution equations. To begin, we explore existence outcomes using fractional computations and Banach contraction ﬁxed point theorem. In addition, we illustrate that a neutral system with a time delay exists. Further, we prove the considered fractional time-delay system is approximately controllable using the sequence approach. Finally, an illustration of our main ﬁndings is o ﬀ ered. form: Considering A : H H , 2 , which

Thermal science, chemical engineering, and mechanics all use the time-fractional advection-reaction-diffusion equation. An analytic solution to this equation is nearly impossible to find. Recently, numeral modalities are provided, including a finite differentiation optimization approach and a homotope perturbation method. The Taylor's formula, also known as the Delta function, was employed for three decades to build the replicating kernel space, which has proven to be an excellent technique for three decades, the Taylor's formula, also known as the Delta function, was used to construct the replicating kernel space and it has proven to be a useful method for resolving different forms. In [1], the authors proposed various new reproductive kernel spaces for numerical approaches to time-fractional advection-reaction-diffusion equations based on Legendre polynomials.
References [2,9] explored the approximate controllability of semilinear inclusions with respect to HFD. Furati, et al. [7] discussed the existence and uniqueness of a problem involving HFD.
Neutral systems have gotten increasing attention in the present generation because among their widespread applicability in various domains of pragmatic mathematics. Several neutral systems, including heat flow in materials, visco-elasticity, wave propagation, and several natural developments, benefit from neutral systems with or without delay. To know more details on neutral system and its application reader can refer [4,20,21,53].
The advancement of current mathematical control theory has been aided by approximate controllability. The difficulties of approximation controllability of differential systems are extensively used in theory connected to system analysis with control. The system with fractional order generated by the fractional evolution system has attracted attention in recent years, list of these distributions may be found in [21,52]. Li et al. [26] and He et al. [12] developed a fractal differential model as well as a fractal Duffing-Van der Pol oscillator (DVdP) with two-scale fractal derivatives.
An analytic approximate solution can be obtained using two-scale transforms and the He-Laplace method. He and Ji [1] focused on two-scale mathematics and fractional calculus for thermodynamics, and found it is required to show the information lost owing to the reduced dimensional method. In general, one scale is set by utlization, in which case regular calculus is used, and the other scale is determined by the need to reveal lost information, in which case the continuity assumption is allowed and fractional or fractal calculus must be used. For numerical results of space fractional variable coefficient kdv-modified kdv equation via Fourier spectral approach, see [49,50]. Many academics are now using the Sequence method to represent the approximate controllability outcomes using Riemann-Liouville fractional derivative, fractional evolution with damping, and an impulsive system. See articles [4-6, 18, 19, 26, 28, 29, 31, 34-42, 44-48, 54, 55, 57] for further information. Consider (1.1) The Hilfer fractional derivative is symbolized by D α,β 0 + , whose order and type are 0 < β < 1, 0 ≤ α ≤ 1 on Hilbert space H, A refers to a C 0 semigroup {S (ϑ)} ϑ≥0 's infinitesimal generator.
On a Hilbert space H , A 1 denotes a bounded linear operator. We choose, K is a function space associate to W, and Y is the space of values ϑ(·), then the control function w(·) ∈ W, B : W → Y; We split this work into the sections below: The fundamentals of fractional differential systems, semigroup and control systems are addressed in Section 2. Existence outcomes for the system (1.1) is given in Section 3. The filter diagram is included in Section 4. Further we evaluated the results in Section 5 with respect to approximate controllability, 6 we establish the outcomes with time delay by utilizing the sequence method and nonlocal conditions. In 7, we provide an application to demonstrate our main arguments and some inference are established in the end.

Preliminary results
C(K, H) : K → H symbolizes the continuous function throughout this paper along with x * C = sup ϑ∈K e −rϑ x(ϑ) , where r is a fixed positive constant. Now characterize Following are the properties of A κ : A κ is a fractional power, 0 < κ ≤ 1, as a closed linear operator on D(A κ ) along inverse A −κ .
Definition 2.7. [14] Two-scale fractal derivative: The standard differential derivatives and the two-scale fractal derivative are conformable. The twoscale transform is used to convert the nonlinear Zhiber-Shabat oscillator with the fractal derivatives to the traditional model.
where x 0 is the smallest scale beyond which there is no physical understanding and it is the porous size. Refer [10,11,14], for the variational iteration method refer [13].

Existence results
In order to obtain the existence of mild solution for the system (1.1), the following assumptions are made.
For convenience Proof. Γ has a fixed point in H: Step Combining J 1 to J 7 , we get A positive constant q appearing from the norm · * C , 2) and the radius of the sphere From (3.2) and (3.3) we are getting a contradiction to F 3 . Therefore Γx * ≤ q.
Step 2: Contraction: For every ϑ ∈ (0, d] using (F 2 ) and there exists constants From the definition of r from (3.2), we obtain Therefore Γ is contraction on C([−σ, d]; H). Hence x has a fixed point of Γ, i.e., it is a mild solution of (1.1).

Filter system
By referring the articles [44,58], we have given a filter design for our system (1.1) shown in Figure 1 and it shows a rough diagram format, it contributes to the structure's practicality by reducing the number of input sources.
(a) Product modulators 1 and 2 accept the A and g(r, x r ), u(r) and B gives the outputs as Ag(r, x r ) and Bu(r). (c) A 1 and x(r − σ), produced A 1 x(r − σ).
(d) Q β (ϑ − r), G(r, x(r − σ)) are the inputs. Over ϑ, the inputs are joined and multiplied with an integrator output. (e) Q β (ϑ−r), A 1 x(r−σ) are the inputs. Over ϑ, the inputs are joined and multiplied with an integrator output. (g) Qβ(ϑ − r), Ag(r, x r ) are the inputs. Over ϑ, the inputs are joined and multiplied with an integrator output. (h) Q β (ϑ − r), Bu(r) are the inputs. Over ϑ, the inputs are joined and multiplied with an integrator output. (f) The following integrators sum up with the above mentioned modulators over the period ϑ, Finally, we move all integrator outputs to the network. As a result, we have our output result x(ϑ).

Approximate controllability results
Nonlinear control systems with approximate controllability are operated by fractional-order with time delay.
Definition 5.1. Let E(G) = {x(d; w) : u(·) ∈ U} be the reachable set of (1.1) at time d. Suppose G is identically zero then (1.1) is said to be corresponding linear system and E(0) is defined as the reachable set of (1.1). Following hypotheses are used to prove the main outcome.
As a result, for every µ > 0, there exists a positive integer number N, such that Hence, we get Therefore (1.1) is approximate controllability. Thus this ends the proof .

Nonlocal condition
Byszewski [15,16] investigated the idea of "nonlocal conditions", proving the existence and uniqueness of mild, strong, and classical nonlocal Cauchy problem solutions for semilinear evolution equations. In [51] the authors considered the controllability with nonlocal conditions by utilizing fixed point methods and fractional calculus. A valuable conversation about the nonlocal conditions are given in [25,27,51].
Apparently, the controllability of neutral differential problems in particular of time delay with nonlocal conditions with respect to Hilfer fractional differential equations has not been explored at this point. Motivated by the articles [25,53,56], consider Where K is a positive real, 0 < t 1 < t 2 < t 3 < · · · < t n ≤ d, p : C([0, K], H) → H and satisfying the following assumption: For every λ k , γ k ∈ K and consider N h = sup{ p(λ 1 , λ 2 , · · · , λ n ) : λ k ∈ K}.
As a result, any µ > 0, there is a positive integer number N, then Hence, we get As a consequence, the system (6.1) is approximately controllable. This ends the proof.

Conclusions
Our study investigates the existence and approximate controllability of HF neutral evolution equations with time delay. The Sequence method was used to derive the approximate controllability outcomes for HFD equations with time delay. An illustration is offered to support the analytical findings at the end and also given a filter diagram to represent the mild solution of the system with neutral term. Next, new research may use the sequence method with infinite delay to extend the Hilfer fractional stochastic differential evolution equations to approximate control results.