A Result Regarding Finite-Time Stability for Hilfer Fractional Stochastic Differential Equations with Delay

: Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the existence and uniqueness of understanding, respectively. Further, ﬁnite-time stability is obtained using the generalized Grönwall–Bellman inequality. As veriﬁcation, an example is provided to support the theoretical results


Introduction
Fractional calculus is favored by many researchers because of its genetic characteristics. In simulation processes related to control theory, physics, chaos and turbulence, fluid mechanics and visco-elastic materials, fractional differential equations obtain better results than integer-order differential equations. For related content, readers can refer to the monographs [1][2][3][4]. Therefore, in recent decades, fractional calculus has gradually become a powerful tool for discussing and resolving the problems of modern production technology in relation to the rapid development of such technology and natural science. It is worth noting that, in the study of control theory, some researchers use fractional calculus to obtain better simulation results [5]. There are also some related findings that can be referred to in the articles [6][7][8]. Significantly, as an improvement of Riemann-Liouville fractional calculus and Caputo fractional calculus, Hilfer fractional calculus is more widely used in practical life. Gou [9] studied the monotonic iterative technique for a kind of Hilfer fractional system. Random disturbances are well-known and unavoidable factors in the study of real systems. They are also one of the important factors in research on system stability and play an indispensable role. Fractional Brownian motion is widely used to describe uncertainty because of its excellent properties, such as self-similarity and long-distance correlation. In many fields of stochastic analysis, fractional Brownian motion has attracted much attention because of these good properties. On this basis, investigators have carried out many interesting studies and obtained many useful conclusions [10][11][12][13][14][15][16][17][18][19][20][21]. In general, the study of stochastic systems is very meaningful and challenging.
In order to get closer to real life, investigators tend to study systems with delays in their research processes. A system is affected by its current state and its past state as well. In general, delay often causes a system to oscillate and become unstable. Therefore, it is necessary to study systems with delay. Some researchers have studied systems with delays and achieved some results. For example, Luo et al. [16] analyzed a class of stochastic fractional differential equations with time delays and proved the results obtained using numerical simulation. Xu et al. [22] studied a class of stochastic delay fractional differential equations powered by Brownian motion. Ahmed et al. investigated differential equations where D γ,δ 0 + expresses the Hilfer fractional derivative equipped with 0 ≤ γ ≤ 1, are all continuous measurable functions. B¯h is n-dimensional Brownian motion defined on a complete probability space (Ω, F , P). Ψ: [−τ, 0] → R d is a continuous function. Let the norm of R d be · and satisfy E Ψ(h) 2 < ∞.
The major contributions of this paper can be described as the following: (i) The system we encountered is almost affected by the current states and the past states. Compared with [12,14], System (1) contains delay, which brings the states of the system closer to the states in real life and is more convenient for practical applications; (ii) The model mentioned in this paper is more general than the model in [41]. The delay term is considered in a fractional stochastic system of the Hilfer type. There is relatively little research on this type of system; (iii) The difference between this manuscript and [16][17][18][19] is that we adopt the method of the Laplace transformation and its inverse when we derive the solutions to the fractional delay stochastic differential equations, and the Mittag-Leffler function is included in the derived processes, which is helpful for subsequent derivations.
The rest of the paper is arranged as follows. We provide some basic preparatory work in Section 2. In Section 3, we first give the solution to the system under consideration using the Laplace transformation and its inverse and then prove the existence and uniqueness of the solution using the Picard iteration technique and the contradiction method. The results for finite-time stability are given in Section 4. Finally, an example is provided to demonstrate the theoretical results in Section 5.

Essential Definitions and Lemmas
Definition 1 ([15]). For a function σ, the fractional integral operator of order δ can be described as where Γ(·) denotes the Gamma function.

Definition 3 ([15]
). The form of the Mittag-Leffler function can be defined as In particular, we have

Lemma 3 ([44]
). Given a random variable ω, ∀l > 0 and 1 ≤ m < ∞, and we can deduce that the formula mentioned below is true

Definition 5 ([36]
). The positive constants , ξ, andV satisfy < ξ. Thus, the system is finite- We postulate the following hypotheses to facilitate the smooth development of the following work.
, and we can find a corresponding constant µ 1 > 0 such that , and we can find a constant µ 2 > 0,

Existence and Uniqueness
In this section, the equivalent form of system (1) under consideration is derived by means of the Laplace transformation and its inverse and expressed with the Mittag-Leffler function.

Lemma 7. If a function χ(·)
is the solution to the following integral equations, it is also said to be the solution to System (1) (2) Taking the Laplace transformation of both sides of System (1) where χ L (t), Ξ(t), and Λ(t) represent the Laplace transformation of If we take the inverse Laplace transformation of both sides of the above formula, then we can obtain the following form Next, we aim to demonstrate the existence and uniqueness of the solution. In order to get the desired result, we use the Picard iteration technique and the contradiction method.
Assume V > 0 is sufficiently small and satisfies M = 4(V+4)P 2 µ 1 V 2γ−1 2γ−1 < 1 2 . Then, we use the Picard iteration technique and write the stochastic process {χ n (h), n ≥ 0} as follows (3) Step 1. Prove that the sequence {χ n (h)} is bounded. By using the Jensen inequality, we can obtain From the Hölder inequality and assumption (H 2 ), it is simple to find that By means of hypothesis (H 2 ) and the Burkholder-Davis-Gundy (B-D-G) inequality, we can derive In general, with Equations (4)-(6), we have Subsequently, we can draw a conclusion that there is a constant C that satisfies Step 2. Prove the sequence {χ n (h)} is a Cauchy sequence. From Equation (3), we get the following formula In particular, for n = 0, from (H 2 ) and the B-D-G inequality, Jensen inequality, and Hölder inequality, we have

It can be found from (H 1 ) and the B-D-G inequality, Jensen inequality, and Hölder inequality that
).
Suppose E sup 0≤ h≤h which are constants and only depend on γ, V, µ 1 . It can be obtained from mathematical induction combined with Equation (7) that The sum of both sides of the inequality above is We know that converges to 0. This also implies that χ n is a Cauchy sequence. Thus, χ n converges almost surely and uniform on [−τ, V] to a limit χ(h) defined by Combining the bounds of E χ n+1 (h) 2 and by using Fatou's lemma, we can obtain Therefore, from Equation (3), we can obtain for allh ∈ [−τ, V].

Finite-Time Stability
Theorem 3. Assume that conditions (H 1 ) and (H 2 ) are established and there are positive constants and ξ satisfying < ξ, Ψ(0) 2 ≤ . We can then deduce that System (1) is finite-time-stable if it satisfies Proof. From Section 3, we have understood that System (1) has a unique solution, as shown below Forh ∈ [0, V], by applying the Jensen inequality, we can obtain This is given by combining the Hölder inequality and (H 2 ) From the B-D-G inequality and by using hypothesis (H 2 ) again, we can derive Thus, from Equations (10) and (11), we can deduce that and

=Θ( ).
Thus, Then, we have Therefore, By applying Lemma 6, and according to the condition given by Theorem 3, we can obtain Therefore, We can thus know that System (1) is finite-time-stable on [−τ, V] from Definition 5.

Example
Consider the equations given below By simple calculation, we can get µ 1 = 0.0002 and µ 2 = 0.0002. Let = 0.1, ξ = 0.3, and A = 0.1I (I is the one-dimensional identity matrix). It can be obtained by using mathematical software that P = max 0≤h≤V E γ,γ (Ah γ ) = 2.0224.
We can find that the assumptions of Theorems 1 and 3 are satisfied through simple verification. Thus, System (12) has a unique solution, and we can conclude that the norm of the solution does not reach beyond ξ = 0.3 above [0, 10]. From Definition 5, the conclusion that System (12) is finite-time-stable on [0, 10] can be drawn.

Conclusions
In this manuscript, we focused on the existence, uniqueness, and finite-time stability of a kind of Hilfer-type fractional stochastic system with delay. Firstly, the existence of solutions was derived using the Picard iteration technique. Secondly, the uniqueness result was obtained using the the contradiction method. Subsequently, the finite-time stability was obtained by combining the Hölder inequality, B-D-G inequality, Jensen inequality, and generalized Grönwall-Bellman inequality. Finally, an example was provided to account for the validity of the theoretical results.
In the future, the focus of our research will be fuzzy differential equations. How to deal with random terms in fuzzy differential equations has been an inconvenient point in our following work and this will make our research more interesting.
Author Contributions: All authors of this paper participated in the exploration and in obtaining the main results, and the contributions of all authors were equal. All authors have read and agreed to the published version of the manuscript.