Approximate controllability of Hilfer fractional neutral stochastic systems of the Sobolev type by using almost sectorial operators

: The main aim of this work is to conduct an analysis of the approximate controllability of Hilfer fractional (HF) neutral stochastic di ﬀ erential systems under the condition of an almost sectorial operator with delay. The theoretical ideas linked to stochastic analysis, fractional calculus and semigroup theory, along with the ﬁxed-point technique, are utilized to establish the key results of this article. More precisely, the main theorem of this study is devoted to proving the fact that the relevant linear system is approximately controllable. Finally, to help this research be as comprehensive as possible, we provide a theoretical application and ﬁlter system.


Introduction
Fractional calculus was introduced as a significant area of advanced calculus in 1695.The idea of fractional calculus has been effectively applied to a number of fields.Researchers in the fields of physics and mathematics have demonstrated that this calculus may accurately reflect a variety of non-local dynamics.The most common domains in which fractional calculus is used include elasticity, kinetic oscillations in identical and homogeneous constructions, aqueous waterways, imaging, viscoelasticity and other areas.The success of fractional structures has caused several researchers to re-evaluate their mathematical estimation techniques, because diagnostic configurations may not be available in many domains.The readers can discover some interesting findings on fractional dynamical systems in many research works about the theory and applications of fractional differential systems [1][2][3][4][5][6].Particularly, partial neutral constructions with or without delays act as an overview of several partial neutral systems that appear in problems concerning heat transfer in components, viscoelasticity and a variety of natural events.Additionally, interested readers are able to review various books [7][8][9][10] and research articles [11][12][13][14][15][16][17] that focus on the most popular neutral structures.
Hilfer [18] pioneered fractional derivatives, including the Riemann-Liouville (RL) and Caputo derivatives.Additionally, some theoretical discussions on thermoelasticity in solid compounds, pharmaceutical manufacturing, rheological adaptive computing, mechanics and related areas have revealed the applicability of Hilfer fractional derivatives (HFDs).Gu and Trujillo [19], in 2015, used a measure of noncompactness method, along with the fixed-point criterion, to prove that the HFD evolution problem has an integral solution.They considered a new variable r ∈ [0, 1], together with a fractional variable s, to indicate the derivative's order.As a result, r = 0 gives the RL derivative, while r = 1 gives the Caputo derivative.Numerous papers have been written in the context of Hilfer fractional calculus [20][21][22][23].Jaiswal and Bahuguna [24] and Karthikeyan et al. [25] turned to the existence of a mild solution in relation to the Hilfer differential systems by using almost sectorial operators.
Due to the numerous applications of neutral differential equations in fields including electronics, chemical kinetics, biological modelling and fluid dynamics, this form of equation has attracted a lot of interest recently.We cite the publications [26][27][28] and the references therein for the theory and applications of neutral partial differential equations with non-local and classical circumstances.Due to the fact that neutral structures are prevalent in several areas of applied mathematics, recent years have seen an increase in interest in them.
According to what we already know, the condition of controllability is an essential qualitative and quantitative property of the control construction, and its characteristics are important in a range of control challenges for both restricted and limitless networks.Recently, this notion has sparked a lot of interest from researchers in the area of controllability of a wave equation of fractional order.See [29] for significant new findings on the exact and approximate controllability of nonlinear delay or nondelay dynamical systems.The approximate controllability of Atangana-Baleanu fractional neutral delay integro-differential stochastic systems with non-local conditions was established by Ma et al. [30] by using the fixed-point approach.A novel approach for such controllability of Sobolev-type Hilfer fractional (HF) differential equations was recently unveiled by Pandey et al. [31] in 2023.
In contrast to deterministic models, stochastic ones should be investigated since both natural and artificial systems are prone to noise and uncontrolled perturbations.Differential equations with stochastic components contain unpredictability in their mathematical depiction of a specific event.Recently, much attention has been paid to the application of stochastic differential equations (SDEs) to describe a variety of occurrences in population motion, science, technological engineering, environment, neuroscience, biological science and several other domains of science and technology.Infinite and finite dimensions can both be employed with SDEs.An overview of SDEs and their applications may be found in [32,33].
Numerous physical phenomena, like fluid movement through fractured rocks and thermodynamics, have mathematical structures that often reveal the Sobolev differential system.The debate about the approximate controllability of Hilfer neutral fractional stochastic differential inclusions of the Sobolev type was presented and developed by Dineshkumar et al. [32] in 2022.Also, a study on the existence of mild solutions has been carried out for the Hilfer neutral fractional SDE by Sivasankar and Udhayakumar [34] with the help of almost sectorial operators with a delay.Nevertheless, as far as we are aware, the literature does not describe any research on the topic of the approximate controllability of Hilfer neutral fractional stochastic differential systems of the Sobolev type under the condition of an almost sectorial operator with delay.By taking inspiration from previous research, this study intends to address this gap.In other words, the aim of this manuscript is to establish the approximate controllability of Hilfer neutral fractional stochastic differential systems of the Sobolev type by using an almost sectorial operator with the delay in the following form: H(e, u e )dW(e), l ∈ I , where D r,s 0 + represents the HF derivative of order r ∈ (0, 1) and of type s Note that u l ∈ B, and it is defined axiomatically.The functions ℵ, G and multi-function H will be subject to satisfying some suitable criteria to be defined in the sequel.
The following describes the manuscript's structure: We give the theoretical principles in relation to fractional calculus that are relevant to our investigation in Section 2. We focus on the approximate controllability of the Hilfer neutral fractional stochastic differential system (1.1) in Section 3. To help our discussion be as applicable as possible, we offer the theoretical application in Section 4.1.

Preliminaries
The complete probability space (Λ, F, P) is introduced by a complete family of right-continuous, non-decreasing sub-σ-algebras {F l } l∈I fulfilling the condition that F l ∈ F. We denote a collection of all strongly measurable, mean square-integrable Z-valued random parameters by , where E denotes the expectation satisfying that Take a real-valued sequence {W n (l), l ≥ 0, n ∈ N} of one-dimensional standard Wiener processes, which are mutually independent in Λ.Let K be a real distinct Hilbert space, and define so that {β n ≥ 0, n ∈ N} and {δ n , n ∈ N} is a complete orthonormal basis of K.Moreover, take Q ∈ L(K, K) as an operator formulated by Qδ n = β n δ n , (n ∈ N), along with the finite trace T r If ψ L 0 2 < ∞, in this case, ψ will be called the Q-Hilbert-Schmidt operator.Here, L 0 2 (K, Z) is the space of all Hilbert-Schmidt operators endowed with the norm ψ 2 Some important properties of A γ are listed below.
[34] Let r ∈ (0, 1), γ ∈ (0, 1] and u ∈ D(A); then, some S γ > 0 exists such that 13.An F l -adopted and measurable stochastic process {u(l)} l∈I is named as a mild solution of the system (ω, u ω )dW(ω) de where M r,s (l) = I s(1−r) N r (l), and accordingly, We introduce the state value of (1.1) at the end time c related to the control κ and the actual value , which is the admissible set of system (1.1) at the end time c.Note that R(c, ξ) stands for the closure of R(c, ξ) in Z. Definition 2.14.[30] The Hilfer neutral fractional stochastic differential system of the Sobolev type (1.1) is approximately controllable on To conduct an analysis of the approximate controllability of the supposed nonlinear Sobolev-type Hilfer control system (1.1), in the first step, we should establish the property of the approximate controllability in the linear case, that is, To do this, we first need to introduce the pertinent operator and the set R(α, Γ c 0 ) = (αI + Γ c 0 ) −1 for α > 0. In the aforementioned notions, O * r (l) and Y * represent the adjoints of O r (l) and Y, respectively.It is notable that the linear operator Γ c 0 is easily proven to be bounded.Consider the following hypothesis: (H α ) αR(α, Γ c 0 ) → 0 as α → 0 + w.r.t. the strong operator topology.
Lemma 2.15.[40,41] Let P cv,cl,bd (Z) be the collection of nonempty bounded closed and convex sets in Z and I be a compact real interval.Consider the L 2 -Caratheodory multi-valued function which is nonempty.Moreover, let Σ be a linear continuous function that maps L 2 (I, Z) to .Then, is a closed graph operator in × .

Approximate controllability
Here, the property of approximate controllability is studied in relation to the given nonlinear Sobolev type Hilfer stochastic control system (1.1).
The following hypotheses are required to prove the main theorems.
For almost every l ∈ I and any u ∈ B p , where q 1 ∈ L 1 (I, R + ) and we have the continuous increasing function is continuous, and for any u ∈ B p , the function H(•, u) : Note that S H : [0, ∞) → [0, ∞) is continuous and increasing.
(H I ) The following inequality holds: de, where Υ(l) = a 1 max{q 1 (e), q 2 (e)}, l ∈ I, Remark 3.1.[30] The following implications hold: Note that the Hilfer stochastic control system of the Sobolev type (1.1) is approximately controllable if a continuous function u exists such that ∀ α > 0: and , where We first state an auxiliary lemma (it will be used later).
Define the operator Ψ mapping from B p into 2 B p , denoted by Ψu, as the set y ∈ B p so that (ω, u ω )dW(ω) de where ∈ S H,u .We shall show that ∆ admits a fixed point that is the mild solution of the Hilfer stochastic control system of the Sobolev type (1.1).Obviously, u c = u(c) ∈ (∆u)(c), which means that κ u (u, l) gives (1.1) as u 0 → u c in the finite time c.Since ϕ ∈ B p , we introduce ϕ as follows: We consider v to satisfy (3.1) if and only if v satisfies v 0 = 0 and Hence, (B p , • ) is a Banach space.Set D r = {v ∈ B p : v 2 c ≤ r} for some r > 0. Accordingly, D r ⊆ B p has the uniform boundedness property.If v ∈ D r , from Lemma 2.7, we obtain Define ∆ : B p → 2 B p , denoted by ∆v, as the set y ∈ B p such that We begin the proofs by stating some theorems that will allow us to prove the main theorem on the approximate controllability.Proof.We know that a fixed point of ∆ exists if and only if a fixed point of Π exists.We break the proof into several steps for the sake of simplicity.
Step 1: ∆v is convex, ∀ v ∈ B p : Indeed, when ϕ 1 , ϕ 2 ∈ ∆v, then ∃ 1 , 2 ∈ S H,u such that Step 2: Boundedness of ∆v on the bounded sets of B p : It is enough to prove that some π > 0 exists such that ∀ ϕ ∈ ∆v, v ∈ D r , we have ϕ c ≤ π.Subject to ϕ ∈ ∆v, there exists ∈ S H,u such that, for any l ∈ I, and from (H ℵ ), (H G ), (H H ) and (H I ), we obtain where Thus, for all ϕ ∈ ∆(D r ), we have that ϕ c ≤ π.
Step 3: ∆ maps the bounded sets into equicontinuous sets of B P : Assume that 0 < l 1 < l 2 ≤ c.For every ϕ ∈ ∆v in which v belongs to D r = {v ∈ B p : v 2 c ≤ r}, there exists ∈ S H,u such that for any l ∈ I, we obtain  When l 2 → l 1 , the right-hand side of the above inequality tends to 0, because O r (l) is an operator with the strong continuity, and because the compactness of O r (l) requires uniform continuity.As a result, the set {∆v : v ∈ D r } is equicontinuous.The Arzela-Ascoli theorem and Steps 2 and 3 allow us to conclude that ∆ is compact.
Step 4: ∆ has a closed graph: Suppose that {v n } ⊂ B p is a sequence such that v n → v * , and assume that {ϕ n } is a sequence belonging to ∆v n for any n ∈ N such that ϕ n → ϕ * .We shall demonstrate that ϕ * ∈ ∆v * .Since ϕ n ∈ ∆v n , then there exists n ∈ S H,u n such that We must show that ∃ * ∈ S H,u * such that Consider the linear continuous operator 0 : L 2 (I; Z) → C(I; Z) by Accordingly, by referring to Lemma 2.15, 0 • S is a closed graph.Moreover, Since v n → v * , because of Lemma 2.15, we may write Thus, ∆ has a closed graph.
In view of the four previous steps, ∆ is a completely continuous multi-valued map with upper semicontinuity and closed values that are convex.Now, in order to use the Martelli fixed-point theorem, we choose a parameter η > 1 and establish the following auxiliary problem: H(e, u e )dW(e), l ∈ I , Thus, by Definition 2.13, the mild solution of (3.2) can be defined in the following form: where Now, in this lemma, we can be sure of the above structure in relation to the mild solution of (3.2).
Lemma 3.4.Assume that (H O ), (H ℵ ), (H G ), (H H ) and (H I ) are satisfied.Then, u is a mild solution of (3.2).Moreover, the priori bound > 0 exists such that u l B p ≤ , ∀ l ∈ I, where is only dependent on c, q 1 (•), q 2 (•), S G and S H .
Proof.From the structure (3.3), we may write Therefore, by Lemma 2.7, we get + ξ B p .
We can conclude from the right-hand side of the above inequality that where Υ(l) = a 1 max{q 1 (e), q 2 (e)}.
These inequalities imply, for each l ∈ I, that Proof.Let Φ = {v ∈ B p : ηv ∈ Λv, for some η > 1}.Then, for all v ∈ Φ, we have Then, the function u = v + ϕ will be a mild solution of the system (3.3);thus, by Lemmas 3.4 and 2.7, we estimate the following: which gives the boundedness of Φ.Therefore, it gives, by Lemma 2.7 and the Martelli fixed-point theorem, that Λ admits a fixed point Then, u is a fixed point of Ψ, which is a mild solution of the Hilfer stochastic control system of the Sobolev type (1.1).
By considering the previous theorems, we can now prove the approximate controllability for the main given stochastic system.Proof.Let u α (•) ∈ D r be a fixed point of the operator Π.But, based on Theorem 3.3, we know that every fixed point of Π is a mild solution of the Hilfer stochastic control system of the Sobolev type (1.1).This shows that there is a u α such that u α ∈ Π(u α ); that is, by the stochastic Fubini theorem, Moreover, using the Dunford-Pettis theorem and the existing conditions on ℵ, G and , we find that ℵ(c, u c ), G(e, u c ) and (ω, u ω ) are weakly compact, respectively, in L 2 (I, Z), L 2 (I, Z), and L 2 (L Q (K, Z)).So, there are subsequences, denoted by ℵ(c, u c ), G(e, u c ) and (ω, u ω ), weakly converging to ℵ, G and , respectively, in L 2 (I, Z), L 2 (I, Z) and L 2 (L Q (K, Z)).Now, we write From (H α ), for each 0 ≤ e ≤ c, we get that α(αI + Γ c 0 ) −1 → 0 strongly as α → 0 + .Accordingly, α(αI + Γ c 0 ) −1 ≤ 1.Consequently, we have that E u α (c) − ūc 2 → 0 as α → 0 + from Lebesgue's dominated convergence theorem and the compactness of O r (l).Hence, the Hilfer stochastic control system of the Sobolev type (1.1) is approximately controllable which completes the proof.Moreover, A and F respectively take the following forms: where κ n (y) = 2 π sin(ny), n = 1, 2, 3, • • • denotes the orthonormal vectors of A. Additionally, for u ∈ Z, we have Note that Y admits the eigenvalues β n = −n 2 , n ∈ N, and that the corresponding eigenfunction is given by κ n .Therefore, the spectral representation of Y is formulated by Further, define Specify that where U is a space with the infinite dimension under the norm .
In this step, we can define Y : U → U as so that Y is a linear continuous map.Now, by the above definitions, consider the following Hilfer stochastic control system of the Sobolev type as follows: H(e, u(e, z))dW(e), 0 ≤ l ≤ e, u(l, 0) = u(l, π) = 0, l > 0, where W(l) is the standard one-dimensional Brownian motion in Z belonging to the filtered probability space (Λ, F, P).Obviously, all assumptions (H O ), (H ℵ ), (H G ), (H H ) and (H I ) hold; thus, the above Hilfer stochastic control system of the Sobolev type (4.1) is approximately controllable based on Theorem 3.6.

Example II
In this part, we examine the approximate controllability of Hilfer neutral fractional stochastic differential systems of the Sobolev type by using an almost sectorial operator with delay.Consider the mild solution of the system (2.Motivated by the filter system presented in [22,42,43], we present the digital filter system corresponding to the mild solution in Figure 1.Digital filters are the backbone for any signal processing applications.Many biomedical signals related to the human body are currently being acquired for various informative feature extractions.Most of the aforementioned signals generally possess a low frequency by nature.These signals describe the information pertaining to various disorders or diseases for which the accuracy is of high concern.The efficiency of any digital signal-processing filtering system relies on the ability to reject the noise.Figure 1 describes the following: (1) The product modulator 1 accepts the input [F ξ(0) − ℵ(0, ξ)], and M r,s at time l = 0 produces the output M r,s (l)[F ξ(0) − ℵ(0, ξ)].
(3) The product modulator 3 accepts the input κ(e) and Y and produces the output Yκ(e).
(4) The product modulator 4 accepts the input u(e) and ℵ and gives output ℵ(e, u e ).
(5) The product modulator 5 accepts the input u(e) and G and gives output G(e, u e ). ( 6) The product modulator 6 accepts the input u(ω) and and gives output (ω, u ω ).(2) Inputs F −1 (l − e) r−1 O r (l − e) and G(e, u e ) are combined and multiplied with an output of the integrator over (0, e).
(4) Inputs F −1 (l − e) r−1 O r (l − e) and Yκ(e) are combined and multiplied with an output of the integrator over (0, e).Finally, we move all of the outputs from the integrators to the summer network.Therefore, the output of u(l) is attained; it is bounded and controllable.

Conclusions
This paper focuses on the approximate controllability of a Hilfer stochastic neutral control system of the Sobolev type by using an almost sectorial operator with delay.The concepts of stochastic analysis, fractional calculus, semigroup theory and fixed-point technique are used to find the mild solutions of the mentioned system.More precisely, by defining some operators, and under some control conditions, we could prove the existence result for the mild solutions.Finally, we provided a theoretical example and filter system to effectively analyse our results.In future works, one can extend the control Hilfer stochastic neutral systems under some well-known boundary value conditions.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The linear operators A, F : D(A) ⊂ Z → Z are identified now based on the following criteria[36]: (A1) F is bijective and D(F ) ⊂ D(A).(A2) A and F are closed.(A3) F −1 : Z → D(F ) is continuous.

Theorem 3 . 3 .
If the hypotheses (H O ), (H ℵ ), (H G ), (H H ) and (H I ) are to be held, then the multi-valued map ∆ : B p → 2 B p has the complete continuity and upper semi-continuity properties with the closed and convex values.

Theorem 3 . 6 .
If the hypotheses (H O ), (H ℵ ), (H G ), (H H ) and (H I ) are satisfied and G and H have the uniform boundedness property, then the Hilfer stochastic control system of the Sobolev type (1.1) is approximately controllable on I.