New Results Concerning Approximate Controllability of Conformable Fractional Noninstantaneous Impulsive Stochastic Evolution Equations via Poisson Jumps

: We introduce the conformable fractional (CF) noninstantaneous impulsive stochastic evolution equations with fractional Brownian motion (fBm) and Poisson jumps. The approximate controllability for the considered problem was investigated. Principles and concepts from fractional calculus, stochastic analysis, and the ﬁxed-point theorem were used to support the main results. An example is applied to show the established results.


Introduction
The field of fractional calculus is constantly expanding, and applications range from engineering and natural phenomena to financial perspectives (see [1][2][3][4][5][6][7]). Recently, there seems to be much enthusiasm for the use of stochastic differential equations to describe a variety of phenomena in population dynamics, physics, electrical engineering, geography, psychology, biochemistry, and other areas of physics and technology (see [8][9][10][11][12][13][14]).Stochastic impulsive differential equations arise in a very natural way as mathematical models (see [15][16][17][18][19][20]). The introduction of drugs into the bloodstream and the consequent absorption into the body are gradual and continuous processes that can be described by noninstantaneous impulsive differential equations (see [21,22]).Now, several authors have discussed different types of controllability for fractional stochastic systems (see [23][24][25][26][27][28][29]).To the best of our knowledge, the approximate controllability of CF noninstantaneous impulsive stochastic evolution equations via fBm and Poisson jump mentioned in this study is an area of research that appears to give extra incentive for completing this research.
Assume that the CF noninstantaneous impulsive stochastic evolution equation via fBm, Poisson jump, and the control function has the following form:  (1) where D £ is the conformable fractional derivative (CFD) of order 1  2 < £ < 1 and −∆ generates semigroup Θ(℘), ℘ ≥ 0, on ℵ.Here, ℵ and G are separable Hilbert spaces with • and u(•) ∈ L 2 (Υ, U) is the control function, where L 2 (Υ, U) is the Hilbert space of control functions with U a Hilbert space.B is a bounded linear operator from U into ℵ, and hi is a noninstantaneous impulsive function for all i = 1, 2, . . ., m. Suppose that the time interval is Υ = (0, b], where ℘ i , k i are fixed numbers verifying 0 = k Assume {ω(℘)} ℘≥0 is a K-Wiener process on (F, S, {S ℘ } ℘≥0 , P) with values in G and {B H (℘)} ℘≥0 is fBm with Hurst parameter H ∈ ( 1 2 , 1) defined on (F, Sp, {S ℘ } ℘≥0 , P) with values in Q; Q is a Hilbert space with . .In this paper, L(G, ℵ) and L(Q, ℵ) are the space of all bounded linear operators from G into ℵ and from Q into ℵ, respectively, with . .M, V, Ω, , and hi are defined in Section 2.
The contributions of the present work are as follows: • The conformable fractional noninstantaneous impulsive stochastic evolution equation with fractional Brownian motion and Poisson jump is presented.

•
To the best of the author's knowledge, there has not been any research that has studied the approximate controllability of (1).

•
An example is applied to show the established results.

Preliminaries
Here, we collect the basic concepts, definitions, theorems, and lemmas that are used in the paper.
Definition 1 (See [7]).The CFD of order 0 < £ < 1 of z(℘) for ℘ > 0 is defined as Furthermore, the conformable integral is defined as Suppose (F, S, P) is a full probability area connected with a normal filtration Sp ℘ , ℘ ∈ [0, b], where Sn ℘ is the σ-algebra generated by random variables {ω(k), B H (k), s ∈ [0, b]} and all P-null sets.Let (F, ξ, ς(d f )) be a σ-finite measurable space.The stationary Poisson point process (p ℘ ) ℘≥0 is defined on (F, S, P) with values in F and characteristic measure ς.The counting measure of p ℘ is denoted by We introduce the space Lemma 1 (see [30]).
Theorem 1 (see [31]).Assume ( ; A) is a compact metric space.For a family of functions Z ∈ C( ), then the following statements are equivalent: (i) Z is relatively compact; (ii) Z is equicontinuous and uniformly bounded.

Approximate Controllability
Here, we investigate the approximate controllability of (1).Consider the linear conformable fractional evolution equation in the following form: We present the operators associated with (3) as and The state value of (1) at terminal state b, corresponding to the control u and the initial value N 0 , is denoted by N(b; N 0 , u).Furthermore, the reachable set of (1) Lemma 2 ([33]).The linear system (3) is approximately controllable on Υ if and only if We define the control function, for any δ > 0 and Nb ∈ L 2 (F, ℵ), in the following form: In this paper, we set = sup •∈Υ Θ(•) , B = B and B * = B * .Theorem 2. Suppose (A1)-(A6) holds, then (1) has a mild solution on Υ, such that and Proof.Consider the map Λ on C defined by to be verified: From (A1) and Hölder's inequality, we obtain ) is integrable on Υ, and by Bochner's theorem, Λ is defined on B ε .
Proof.Assume N is a fixed point of Λ.By the stochastic Fubini theorem, we obtain From the condition on M, V, and Ω, there exists D > 0 such that By Lemma 2, x(xI + Ξ b 0 ) −1 → 0 strongly as x → 0 + for all k m < k ≤ b, and furthermore, x(xI + Ξ b 0 ) −1 ≤ 1.Thus, E N(b) − Nb 2 → 0 as x → 0 + by the Lebesguedominated convergence theorem and the compactness of Θ(k).Hence, the system (1) is approximate controllable.

Conclusions
By using fractional calculus, a compact semigroup, Sadovskii's fixed-point theorem, and stochastic analysis, we investigated the approximate controllability of the given system (1).The obtained theoretical conclusions were illustrated in the later portion with an example.The results can be extended to a fractional stochastic inclusion system.

Author
Contributions: Conceptualization, Y.A. and H.M.A.; formal analysis, Y.A. and H.M.A.; investigation, Y.A.; resources, H.M.A.; writing-original draft preparation, Y.A.; writing-review and editing, H.M.A.All authors have read and agreed to the published version of the manuscript.

Funding:
The APC was funded by the Deanship of Scientific Research, Qassim University.Data Availability Statement: Not applicable.