Novel Results on Finite-Time Stability of Solutions for Stochastic Ψ -Hilfer Fractional System

Tis article studies a new kind of Ψ -Hilfer fractional system driven by m -dimensional Brownian motion. By utilizing the generalized Laplace transform and its inverse, the contraction mapping principle, and the properties of a semigroup, we establish the uniqueness of the solution. In addition, fnite-time stability is investigated by means of the properties of norm and inequalities scaling technique. As verifcation, an example is given to show the deduced conclusions.


Introduction
Actually, the fractional derivative can be expressed as a diferential-integral convolution operator, which is nonlocal.Te integral term defned by it refects the historical dependence of system development well.Te fractional time derivative operator has long-range correlation and memory.Terefore, fractional calculus has been widely used in the research of viscoelastic material, abnormal difusion, fuid mechanics, biomedicine, chaos and turbulence, control theory, and many other felds.One can refer to the monographs [1][2][3][4] for more information.
Te existence and uniqueness theorems for solutions are a primary subject of fractional system.Various methods were used to obtain the existence and uniqueness results in [5,6] and the following literatures.As for the non-Lipschitz condition, Abouagwa et al. [7,8] established the existence theorem for solutions by applying the Carathéodory approximation.Under global Lipshitz conditions, with the aid of Picard iteration method and contradiction method, Moghaddam et al. [9] and Umamaheswari et al. [10] deduced the existence and uniqueness results.In [11][12][13], the monotone iterative method was used to obtain the existence theorem of mild solutions.In addition, various fxed point theorems were proposed to show the existence and uniqueness results for solutions in [14][15][16].Jleli et al. [17] researched the uniqueness of solutions for a kind of coupled system by applying Perov's fxed point theorem together with a type of Lyapunov inequality.In [18], Baghani derived the uniqueness result for the Langevin equation with two orders by establishing a new type of norm in a Banach space and combining the contraction principle.In [19], the uniqueness theorem was deduced by establishing a new type of α-ψ-contractive mapping.In [20,21], Krasnoselskii-type fxed point theorem and the extended Krasnoselskii's fxed point theorem were used to deduce the existence and uniqueness theorem of the considered system.
Te fnite-time stability analysis of a system has the following two situations.Te frst case is to research the transient performance of the system over a fxed fnite time domain, that is, in a fnite time domain the state of the system remains within a given boundary, which is independent of Lyapunov stability.Te second case is to study steady-state performance over an infnite time domain; that is, the system converges to equilibrium over fnite time within the category of Lyapunov stability.Te second case is known as fnite time convergent stable.In the present article, we study the frst kind.Finite-time stability analysis plays an indispensable role in many practical problems, for example, the launch of a rocket, automatic active suspension system, trafc fow node control, and satellite sliding mode control Amato et al. [22,23].
Since the groundbreaking work of Dorato [24], subsequently, the basic defnition of fnite-time stability in stochastic system was frst proposed by Kushner [25].Te Grönwall approach was used to establish the fnite-time stability of stochastic fractional system in [26][27][28][29].Based on the properties of nabla diference for Riemann-Liouvilletype and the generalized Grönwall inequality, Lu et al. [26] researched fnite-time stability in the mean for the fractional diference equations with the nabla operator, in which contains uncertain term.With the aid of the Laplace transform and its inverse, Luo et al. [27] researched two kinds of stochastic fractional delay systems, and the fnitetime stability results were established by applying the generalized Henry-Grönwall delay inequality.Under some assumptions, Mathiyalagan and Balachandran [28] researched the fnite-time stability of fractional stochastic singular delay system driven by white noise by using the Laplace transform and its inverse and based on the Grönwall method.Mchiri et al. [29] studied the fnite-time stability of stochastic fractional linear delay system, in which the analysis method was the generalized Grönwall inequality.Te analysis of fnite-time stability for various systems had been investigated by applying diferent methods in [30] and the follows.By applying the Lyapunov functions approach, Luo et al. [31] derived fnite-time stability results.In [32][33][34], based on a delayed Mittag-Lefer-type matrix, Li et al. deduced fnite-time stability results of diferent systems.Moreover, the delayed exponential matrix method was also a useful tool to study fnite-time stability in [35][36][37].In [38], with the aid of variation of constants method and fractional order cosine and sine delayed matrices, Liang et al. obtained the representation of the solution, and fnite-time stability results were subsequently deduced by norm estimates and Caputo derivative properties.Zhang and Wang [16] studied a kind of Hadamard-type fractional nonlinear system, in which fnite-time stability result was derived by means of Hadamard-type impulsive Grönwall inequality.
At the same time, the Hilfer-type fractional system was also favored by many scholars.Harikrishnan et al. researched a class of Ψ-Hilfer fractional system under boundary conditions in [39] and coupled diferential equations in the sense of Ψ-Hilfer fractional derivative in [40].With the aid of Grönwall approach, Luo and Luo [41] researched the fnite-time stability of Ψ-Hilfer fractional impulsive delay system.In addition, Zhou et al. [42] and Luo et al. [43] established the existence of solutions and stability results for Ψ-Hilfer fractional system.Under the non-Lipschitz assumption, using the Laplace transform and its inverse, Luo et al. [44] considered a kind of stochastic Hilfer-type fractional system.Under non-local conditions, Gou [12] studied a kind of Hilfer fractional system.For more knowledge about Hilfer fractional calculus, one can refer to [1,45].Compared with the references [41][42][43][44], in this article, we will investigate the stochastic Ψ-Hilfer fractional system.According to all the studies we known, there are few results on Ψ-Hilfer fractional system driven by random process.In addition, we are particularly interested in the difculties arising from considering the Ψ function in the analysis of stochastic Hilfer-type fractional system.Tese provide the main motivations for us to fnd a new method to investigate the stochastic Ψ-Hilfer fractional system.
{ } is the complete probability space, and W(t) denotes m-dimensional Brownian motion on it.
Up to now, there exist a lot of literatures using the Laplace transform and its inverse to solve Caputo fractional diferential equations [27] and the Hilfer fractional system [44,46], but few literatures have used this kind of technique to solve Ψ-Hilfer fractional system.In this article, we apply a new type generalized Laplace transform and its inverse to solve this kind of stochastic Ψ-Hilfer fractional system.Te main contributions and innovations of this article are at least as follows: (1) Compared with [42], the proposed model in present manuscript is more generalized, in which the random term is considered in Ψ-Hilfer fractional system.Tere are few literatures available for solving this type of considered system.(2) By applying the generalized Laplace transform and its inverse, we make the frst attempt to construct the form of solutions for stochastic Ψ-Hilfer fractional system.Tis method is essentially new.
| in the process of proving the uniqueness of the solution, we construct a Ψ-Riemann-Liouville fractional integral for RX(t) at the frst step, and then skillfully use its semigroup properties, which greatly simplify our proof.
Te vein of this article is developed as follows: In Section 2, some basic defnitions and their properties are introduced, which play an indispensable role in the subsequent derivation.Section 3 mainly proves the existence and fnite-time stability results for our investigated system.As verifcation, an example is given to expound the derived conclusions in Section 4.

2
Mathematical Problems in Engineering

Essential Definitions and Lemmas
For the convenience of reading and for the smooth derivation, some basic defnitions and related lemmas are introduced.

Main Results
In the present section, we shall deduce the existence and uniqueness of solutions for system (1) by applying the contraction mapping principle.Furthermore, fnite-time stability results are obtained by means of the properties of norm and inequalities scaling technique.We defne the following space: Mathematical Problems in Engineering with norm where E is the mathematical expectation.We can readily verify that (ℵ, ‖ • ‖ ℵ ) is a Banach space, and see [49,50] for more details.
Defnition 5 (see [22]).Assuming that there exist positive constants T, δ, ε with δ < ε, then system (1 where Before starting our proof of the main conclusions, we make the following assumptions on the coefcients of system (1): [(H 2 )] We assume that there is a positive constant , where E α,α (•) denotes the two-parameters Mittag-Lefer function, and see [44] for details.
[(H 4 )] As for any X, Y ∈ R n , there exists a bounded positive function L 2 (•) satisfying Theorem 1.We assume that hypotheses (H 1 )-(H 3 ) hold, then system (1) has a unique solution in (ℵ, ‖ • ‖ ℵ ), if there is a constant C, 0 < C < 1, and Proof.Taking the generalized Laplace transform on (1), we get the following with the aid of Lemma 1: Subsequently, using the generalized inverse Laplace transform, we obtain the solution of system (1) is

Mathematical Problems in Engineering
We defne operator R: According to the properties of fractional Ψ-Hilfer derivative and fractional Ψ-Riemann-Liouville integral, it is readily to verify that above operator R is well-defned.
Moreover, we need to deduce that the operator R is a contraction mapping on ℵ for all X, Y ∈ ℵ.For ∀t ∈ J, we get the following by Ten, by means of Jensen's inequality, Hölder inequality, and Doob's martingale inequality, we get for all t ∈ J

Mathematical Problems in Engineering
By assumptions (H 1 )-(H 2 ), we obtain Similarly, one can obtain Ten, it is easy to obtain

Mathematical Problems in Engineering
On the contrary, we have Ten, with the aid of semigroup property introduced in Remark 1, we can readily derive the following: Mathematical Problems in Engineering Similarly, by means of Jensen's inequality, Hölder inequality, and Doob's martingale inequality, we get Terefore, by the defnition of then R is a contraction mapping within (ℵ, ‖ • ‖ ℵ ), which implies that R has a fxed point.Terefore, system (1) has a unique solution.
□ Theorem 2. We assume that the assumptions (H 2 )-(H 4 ) hold, and there exist positive constants δ, ε satisfying δ < ε and ‖X 0 ‖ ℵ ≤ δ.Ten, system ( 1) is fnite-time stable on [0, T], provided that where Proof.From Teorem 1, system (1) has a unique solution that has the following form: By applying Jensen's inequality, Hölder inequality, and Doob's martingale inequality, we have for ∀t ∈ J 8 Mathematical Problems in Engineering By assumptions (H 2 )-(H 4 ), we obtain Similarly, we obtain Terefore, we obtain On the other hand, we can readily derive the following: According to the defnition of ‖ • ‖ ℵ , we have then by the conditions of Teorem 2, yields which implies that system (1) is fnite-time stable on [0, T] .
(1) Taking the initial value to I 1−c;Ψ 0 + X(0) � X 0 ≠ 0 and applying the generalized Laplace transform and its inverse, we derive that system has the following solution: Mathematical Problems in Engineering We will fnd that the frst term is a singular function, then the estimation of E(sup 0≤u≤t | X(u) | 2 ), for all t ∈ J, will be unbounded.
(2) Taking the initial value to X(0) � X 0 , similarly, the system has the following solution: In order to estimate E(sup 0≤u≤t | H D p,q;Ψ 0 + X(u) | 2 ), we need to take the Ψ-Hilfer fractional derivative of the frst term, which will produce a singular function Λ[Ψ(t) − Ψ(0)] − α , where Λ represents a constant.Tis prevents us from considering the stability analysis.

Example
We consider the following stochastic Ψ-Hilfer fractional system.

Conclusion
In this article, we study a new kind of stochastic Ψ-Hilfer fractional system and apply the generalized Laplace transform and its inverse to solve this kind of system.We have established existence and uniqueness theories as well as fnite-time stability results for the solutions of the considered problem.Te nonlinear analysis method we used is essentially new, and yet there are few literatures available for solving this type of considered system.Te obtained results have been expounded via an example.