Hilfer Fractional Neutral Stochastic Differential Inclusions with Clarke’s Subdifferential Type and fBm: Approximate Boundary Controllability

: In this paper, the approximate boundary controllability of Hilfer fractional neutral stochastic differential inclusions with fractional Brownian motion (fBm) and Clarke’s subdifferential in Hilbert space is discussed. The existence of a mild solution of Hilfer fractional neutral stochastic differential inclusions with fractional Brownian motion and Clarke’s subdifferential is proved by using fractional calculus, compact semigroups, the fixed point theorem, stochastic analysis, and multivalued maps. The required conditions for the approximate boundary controllability of this system are defined according to a corresponding linear system that is approximately controllable. To demonstrate how our primary findings may be used, a final example is provided.


Introduction
Recent research has shown that fractional differential equations or inclusions with fractional order are effective modeling tools for a wide range of phenomena in physics, economics, engineering, and so on.In recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives.We consult the monographs for further information [1][2][3][4][5].The Riemann-Liouville fractional derivative and the Caputo fractional derivative are both parts of the Hilfer fractional derivative, which is a general fractional derivative that Hilfer introduced [6].The subject of fractional differential equations and inclusions has been extensively covered in publications; for example, Gu and Trujillo [7] established the existence of a mild solution for the evolution equation with Hilfer fractional derivative.The approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions was explored by Yang and Wang [8].Varun Bose and Udhayakumar [9] discussed the existence of Hilfer fractional differential inclusions with almost sectorial operators.
One of the fundamental concepts in the study of mathematical control theory is controllability.It is a characteristic of dynamical systems and has special significance in control theory.Controllability, stability, and stabilizability of deterministic and stochastic control systems have a number of significant connections.Any control system is considered to be controllable because it is probable to use the set of admissible controls to steer the system from an arbitrarily chosen starting point to an equally chosen ending point, where the starting point and the ending point may differ over the whole space.Ravikumar et al. [10] discussed the null controllability of nonlocal Sobolev-type Hilfer fractional stochastic differential system driven by fractional Brownian motion and Poisson jumps.Sivasankar et al. [11] investigated the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators.For further information on this topic, we refer to [12][13][14][15] and references therein.
Since they appear in a broad variety of issues in applied mathematics and biological models, including electronics, fluid dynamics, and chemical kinetics, neutral functional differential systems have attracted a lot of attention lately.Over the last decade, a significant number of academics have produced neutral fractional differential systems with or without delays, utilizing a variety of fixed-point procedures, mild solutions, noncompactness measures, and nonlocal conditions.For more details, we may refer to [16][17][18][19][20].Many researchers have extensively studied the existence of mild solutions for neutral stochastic differential systems in [21][22][23][24].The attention of several authors has been drawn to stochastic differential equations and inclusions driven by fractional Brownian motion (fBm).There have been significant developments about the existence, uniqueness, and controllability of the solution; for example, the existence of neutral stochastic functional differential equations driven by a fBm was examined by Boufoussi and Hajji [25].Introducing impulsive stochastic functional differential inclusions driven by a fBm with infinite delay was done by Boudaoui et al. [26].The best way to combine fractional Brownian motion and non-instantaneous impulsive fractional stochastic differential inclusion was explored by Balasubramaniam et al. in their study [27].Ahmed et al. [28] investigated the approximate controllability of nonlocal Sobolev-type neutral fractional stochastic differential equations with fractional Brownian motion and Clarke subdifferential.Mourad et al. [29] investigated stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic nonlocal conditions.Mourad [30] established the approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion.
Clarke's subdifferential arises from applied fields such as filtration in porous media and thermo-viscoelasticity and has fascinating applications in non-smooth analysis and optimization [31].Recent years have seen a rise in research activity in the study of control issues with Clarke's subdifferential; for example, Ahmed et al. [32] investigated fractional stochastic evolution inclusions with control on the boundary.Kavitha et al. [33] discussed the results on the approximate controllability of Sobolev-type fractional neutral differential inclusions of the Clarke subdifferential type.Sivasankar et al. [34] studied the optimal control problems for Hilfer fractional neutral stochastic evolution hemivariational inequalities.
Only a few authors have investigated the approximate boundary controllability; for example, Wang [35] investigated the approximate boundary controllability for semilinear delay differential equations.Olive [36] introduced the approximate boundary controllability of some linear parabolic systems.Ahmed et al. [37] established the approximate boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fBm and Poisson jumps.Ahmed [38] studied the approximate controllability of neutral fractional stochastic differential systems with control on the boundary.However, to the best of our knowledge, so far, no work has been reported in the literature about the approximate boundary controllability of Hilfer fractional neutral stochastic differential inclusions with fBm and Clarke's subdifferential.
Inspired by the above-mentioned works, this paper aims to fill this gap.The purpose of this paper is to show the existence of solutions and the approximate boundary controllability of Hilfer fractional neutral stochastic differential inclusions with fBm and Clarke's subdifferential of the form ( ) , ( ) , ( ) , (0, ], where D 0+ ℓ,y is the Hilfer fractional derivative (HFD) of order ℓ ∈ [0, 1] and type 1 , 1 2

Novelties of the work
The contributions of this paper exist in the following aspects: • Hilfer fractional neutral stochastic differential inclusions with fBm and Clarke's subdifferential are introduced.
• The primary outcomes for the systems (1) are derived by applying fractional calculus, compact semigroups, the fixed point theorem, stochastic analysis and multivalued maps.
• Approximate boundary controllability for Hilfer fractional neutral stochastic differential inclusions with fBm and Clarke's subdifferential is an unexplored topic in the literature, and this is an additional motivation for writing this manuscript.
• The main findings are demonstrated using an example.

Structure of the paper
The following describes how the paper is organized: We present some definitions, lemmas, and theorems in Section 2 that are helpful in proving the major results.The sufficient condition to demonstrate approximate boundary controllability for the system (1) is examined in Section 3. In Section 4, we present an example to verify the results of the theoretical work.

Preliminaries
The lemmas, theorems, and definitions required to support the primary results are presented in this part of the paper.
Definition 2.1 [2,6] The HFD of order 0 ≤ ℓ ≤ 1 and 0 < y < 1 is characterized as Assuming that (Ω, S, {S t } t ≥ 0 , P) indicates a complete probability space equipped with a normal filtering {S t }, t ∈ (0, ξ ], because S t corresponds to the σ-algebra produced by {w(s), B H (s), s ∈ (0, ξ ]} and all P-null sets.Let Θ ∈ (Y, Y ) be an operator denoting Θe k = λ k e k along with finite trace The fBm on Y in infinite dimensions with covariance Θ is defined as where β k H (t ) are real, independent of fBm.In order to define the Wiener integrals with respect to the Θ-fBm, we introduce the space of all Θ-Hilbert Schmidt operators φ : Y → Λ as L 2 0 := L 2 0 (Y, Λ).We remember that φ ∈ L(Y, Λ) is known as Θ-Hilbert Schmidt operators, if , and that the space L 2 0 equipped with the inner product Let φ(s); s ∈ (0, ξ ] be a function with values in L 0 2 (Y, Λ), the Wiener integral of φ with respect to B H is defined by where β k denotes the standard Brownian motion.
where ∂B(ι) is the Clarke's generalized gradient of B at ι ∈ O. Here, The Banach space of all continuous functions m from T into L 2 (Ω, S, P, Λ) is denoted by λ := t(T, L 2 (Ω, S, P, Λ)) We require the following hypotheses to be verified in the findings: (A1) Dom(F) ⊂ Dom(δ) and the restriction of δ to Dom(F) are continuous with regard to the graph norm of Dom(F).
(A4) For all t ∈ (0, ξ] and for t ∈ T, m 1 , m 2 ∈ Λ, and the inequality ( ) The following conditions are fulfilled by the function Let m(t) be the solution of (1).Then, we define x(t) = m(t) − ϱV(t).From our hypotheses, it is obvious that x(t) ∈ Dom(A).As a result, (1) may be represented in terms of A and ϱ as follows: Hence, the integral inclusion of ( 1) is provided by Lemma 2.4 [32] An S t -adapted stochastic process m(•) ∈ L S 2 (T, Λ) is said to be a mild solution of the control system (1), and provided that , 0 1, (0, ), ( 1)! ( 1) the Wright-type function that meets the following inequality Lemma 2.5 [7] The operators ℵ ℓ, y and P y possess the following characteristics: (i) {P y (t) : t > 0} is continuous in the uniform operator topology.(ii) ℵ ℓ, y (t) and P y (t) are linear bounded operators, ( ) (1 ) (iii) {P y (t), ℵ ℓ, y (t)} t >0 are strongly continuous.
Lemma 2.6 [37] If the assumption (A4) is satisfied, then Lemma 2.7 [39] If (A9) is satisfied, then the set S(m) has nonempty, convex, and weakly compact values for every m ∈ L S 2 (T, Λ).Lemma 2.8 [39] The operator S verifies: if Theorem 2.9 [40] Let D be a locally convex Banach space and P ε : D → 2 D be a compact convex valued (CCV), upper semicontinuous multivalued maps such that there exists a closed neighborhood L of 0 for which

Main results
To investigate the approximate boundary controllability for (1), we consider the linear stochastic system with HFD and control on the boundary , 0 .
We introduce the operators associated with (5) as Assume that m(ξ; m 0 , V ) represents the state value of (1) at the terminal state ξ, which corresponds to the control V and the initial value m 0 .Indicate by N(ξ, m 0 ) = {m(ξ; m 0 , V ) : V ∈ L 2 (T, Y )} is the reachable set of the system (1) at terminal state ξ and its closure in Λ is marked by [41] The fractional linear control system (5) Now, for any m ξ ∈ L 2 (Ω, Λ), as described above, the control function provides ( ) Proof.Consider the map P ε : λ → 2 λ as follows For t ∈ T, and from (A1)-(A9), we have

1) ( )
In order to verify that P ε has a fixed point, the proof is divided into six steps.
Step 1: For all m ∈ λ, P ε (m) be nonempty, convex, and weakly compact values.We use Lemma 2.7 to show that P ε (m) is nonempty and has weakly compact values.Moreover, as S(m) has convex values, if k 1 , k 2 ∈ S(m), then lk 1 + (1 − l)k 2 ∈ S(m) for every l ∈ (0, 1), which implies clearly that P ε (m) is convex.
Step 2: P ε is bounded on a subset of λ.

E P t s F AP t s V s ds P t s F AP t s V s ds
) ( ) From the compactness of ℵ(t) (t > 0), Hence, P ε (m)(t) is continuous in T. Also, for t 1 = 0 and t 2 ∈ T, we can show that E||Φ(t 2 ) − Φ(t 1 )|| λ 2 → 0 as t 1 → 0. As a result, {P ε (m) : m ∈ A r } is equicontinuous.
Step 4: P ε is completely continuous.
We show that χ(t) = {Φ(t) : Φ ∈ P ε (A r )} is relatively compact in Λ for all t ∈ T, r > 0. Undoubtedly, χ(0) is relatively compact in A r .Let 0 < t ≤ ξ be fixed, 0 < η < t, for m ∈ A r , we define 49 sup 49 sup As a result, the set χ(t) is relatively compact in Λ.We may infer that P ε is completely continuous from the Arzela-Ascoli theorem and Step 3.
Step 5: The closed graph of

P t s s m s dw s P t s g s m s dB s
From the compactness of ℵ(t), (7) and ( 8), and we get Note that Φ k → Φ * in λ and  k ∈ S(m k ).From (9) and Lemma (2.8), we obtain  * ∈ S(m * ).Therefore, Φ * ∈ P ε (m * ), which implies that P ε has a closed graph and that P ε is a completely continuous multi-valued map with compact value.Thus, from [32], P ε is upper semicontinuous.

P t s s m s ds P t s s ds
Step 6: A priori estimate.

E P t s F AP t s V s ds E P t s s s ds
(2 1) ( ) where ( ) ) ( ) , we get , consequently, Ψ is bounded.By Theorem (2.9), P ε has a fixed point.As a result, the inclusion system ( 1) is approximately controllable on T. Theorem 3.5 Assuming that (A1)-(A9) are fulfilled.Additionally, if the functions f, σ, ℘, g and  are all uniformly bounded, then (1) is approximately controllable on T.
Proof.Using the stochastic Fubini theorem and m ζ as a fixed point on P ε , it is evident that Then ∆ can be written as is a sequence of one-dimensional fBm that are independent of one another.
It is obvious that ∆ makes the compact semigroup {ℵ(t)} t ≥ 0 on Λ.
Next, we verify that the hypothesis (A1)-(A9) for the above system (13) one by one.Verification of A1 Dom(F ) ⊂ Dom(δ) and the restriction of δ to Dom(F) are continuous with regard to the graph norm of Dom(F).Therefore, A1 is verified.

Verification of A4
For all t ∈ (0, ξ ] and V ∈ Y, ℵ( t)ϱV ∈ Dom(A).Additionally, there exists a constant Π It is now possible to write (13) in the form of (1).Clearly, all the assumptions of the Theorem 3.4 and 3.5 are satisfied.

Remark
In this paper, we discuss the approximate boundary controllability of Hilfer fractional neutral stochastic differential inclusions with fBm and Clarke's subdifferential.The primary results were obtained by using fractional calculus, stochastic analysis theory, and the fixed point theorem.Then, the proposed systems can be extended with impulsive effects and nonlocal conditions.

Conclusion
In this paper, we investigated the approximate boundary controllability of Hilfer fractional neutral stochastic differential inclusions with fBm and Clarke's subdifferential.Initially, we worked with stochastic analysis, nonsmooth analysis, semigroup theory, and the fixed point theorem of multivalued mappings to show that there is a mild solution to (1).Then, we offered a sufficient condition for the approximate boundary controllability of Hilfer fractional neutral stochastic differential inclusions with fBm and Clarke's subdifferential.The primary findings were finally shown using an example.In the future, we will use a fixed point technique to examine the approximate boundary controllability of

2
), the Hilbert space of admissible control functions on Y. σ : T × Λ → 2 Λ is a nonempty, bounded, closed, and convex (BCC) multivalued map.Let L Θ (D, Λ) be the space of all Θ-Hilbert Schmidt operators from D to Λ.The nonlinear operators f : be bounded linear operator and δ : Λ → D be a linear operator, where D be separable Hilbert space.ϱ 1 : Y → Λ denotes a bounded linear operator, where Y and Λ are Hilbert spaces.The state m(•) takes the value in Λ.Let A : Λ → Λ be a linear operator defined by Dom(A) = {m ∈ Dom(F) : δm = 0}, Am = Fm, for m ∈ Dom(A).Let {w(t)} t ≥ 0 be a Wiener process that has a finite trace nuclear covariance operator Θ ≥ 0 specified on a (Ω, S,{S t } t ≥0 , P) with values in Hilbert space D. {B H (t)} t ≥ 0 is a fBm t } t ≥ 0 , P) with values in Hilbert space Λ.Also, || • || for L(D, Λ), where L(D, Λ) is the space of all bounded linear operators.∂B(t, m(t)) denotes the Clarke's subdifferential of B(t, m( t)).V(•) is the control function in L 2 (T, Y is approximately controllable on T if and only if ζ(ζI+ Γ 0 ξ ) −1 → 0 as ζ → 0 + .