# Novel Results on Finite-Time Stability of Solutions for StochasticΨ-Hilfer Fractional System

Authors: Mengquan Tian; Danfeng Luo; Zhiguo Luo
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

This article studies a new kind ofΨ-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, finite-time stability is investigated by means of the properties of norm and inequalities scaling technique. As verification, an example is given to show the deduced conclusions.


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## Body

## 1. Introduction

Actually, the fractional derivative can be expressed as a differential-integral convolution operator, which is nonlocal. The integral term defined by it reflects the historical dependence of system development well. The fractional time derivative operator has long-range correlation and memory. Therefore, fractional calculus has been widely used in the research of viscoelastic material, abnormal diffusion, fluid mechanics, biomedicine, chaos and turbulence, control theory, and many other fields. One can refer to the monographs [1–4] for more information.The 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 [11–13], the monotone iterative method was used to obtain the existence theorem of mild solutions. In addition, various fixed point theorems were proposed to show the existence and uniqueness results for solutions in [14–16]. Jleli et al. [17] researched the uniqueness of solutions for a kind of coupled system by applying Perov’s fixed 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 fixed point theorem and the extended Krasnoselskii’s fixed point theorem were used to deduce the existence and uniqueness theorem of the considered system.The finite-time stability analysis of a system has the following two situations. The first case is to research the transient performance of the system over a fixed finite time domain, that is, in a finite time domain the state of the system remains within a given boundary, which is independent of Lyapunov stability. The second case is to study steady-state performance over an infinite time domain; that is, the system converges to equilibrium over finite time within the category of Lyapunov stability. The second case is known as finite time convergent stable. In the present article, we study the first kind. Finite-time stability analysis plays an indispensable role in many practical problems, for example, the launch of a rocket, automatic active suspension system, traffic flow node control, and satellite sliding mode control Amato et al. [22, 23].Since the groundbreaking work of Dorato [24], subsequently, the basic definition of finite-time stability in stochastic system was first proposed by Kushner [25]. The Grönwall approach was used to establish the finite-time stability of stochastic fractional system in [26–29]. Based on the properties of nabla difference for Riemann–Liouville-type and the generalized Grönwall inequality, Lu et al. [26] researched finite-time stability in the mean for the fractional difference 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 finite-time stability results were established by applying the generalized Henry–Grönwall delay inequality. Under some assumptions, Mathiyalagan and Balachandran [28] researched the finite-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 finite-time stability of stochastic fractional linear delay system, in which the analysis method was the generalized Grönwall inequality. The analysis of finite-time stability for various systems had been investigated by applying different methods in [30] and the follows. By applying the Lyapunov functions approach, Luo et al. [31] derived finite-time stability results. In [32–34], based on a delayed Mittag–Leffler-type matrix, Li et al. deduced finite-time stability results of different systems. Moreover, the delayed exponential matrix method was also a useful tool to study finite-time stability in [35–37]. 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 finite-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 finite-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 ofΨ-Hilfer fractional system under boundary conditions in [39] and coupled differential equations in the sense of Ψ-Hilfer fractional derivative in [40]. With the aid of Grönwall approach, Luo and Luo [41] researched the finite-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 [41–44], 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 difficulties arising from considering the Ψ function in the analysis of stochastic Hilfer-type fractional system. These provide the main motivations for us to find a new method to investigate the stochastic Ψ-Hilfer fractional system.Based on the discussions above, in the present article, we will study the following stochasticΨ-Hilfer fractional system:(1)HD0+α,β;ΨXt=AXt+σt,Xt,HD0+p,q;ΨXt+gt,Xt,HD0+p,q;ΨXtdWtdt,t∈J=0,T,I0+1−γ;ΨX0=0,where HD0+α,β;Ψ⋅ and HD0+p,q;Ψ⋅ are the fractional Ψ-Hilfer derivative operators with order α−p>1/2, 0<α,p<1 and type 0≤β,q≤1. I0+1−γ;Ψ⋅ represents the fractional Ψ-Hilfer integral operator with order 1−γ, where γ=α+β−αβ. A∈Rn×Rn, σ:J×Rn×Rn⟶Rn, and g:J×Rn×Rn⟶Rn×m are continuously differentiable functions. Ω,F,P is the complete probability space, and Wt 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 differential 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. The 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. There are few literatures available for solving this type of considered system.(2)
By applying the generalized Laplace transform and its inverse, we make the first attempt to construct the form of solutions for stochasticΨ-Hilfer fractional system. This method is essentially new.(3)
In order to estimate equation∣HD0+p,q;ΨRXt−HD0+p,q;ΨRYt∣ in the process of proving the uniqueness of the solution, we construct a Ψ-Riemann–Liouville fractional integral for RXt at the first step, and then skillfully use its semigroup properties, which greatly simplify our proof.The vein of this article is developed as follows: In Section2, some basic definitions and their properties are introduced, which play an indispensable role in the subsequent derivation. Section 3 mainly proves the existence and finite-time stability results for our investigated system. As verification, an example is given to expound the derived conclusions in Section 4.


## 2. Essential Definitions and Lemmas

For the convenience of reading and for the smooth derivation, some basic definitions and related lemmas are introduced.Definition 1 (see [45]).
IfΨt is a positive and monotonically increasing function on a,b, Ψ′t∈Ca,b. The fractional ψ-Riemann–Liouville integral can be written as(2)Ia+α;Ψft=1Γα∫atΨt−Ψsα−1fsΨ′sds.Remark 1.
Fractional integral operator has the following semigroup property, forα>0 and β>0(3)Ia+α;ΨIa+β;Ψft=Ia+α+β;Ψft.Definition 2 (see [45]).
Iff, Ψ∈Cna,b,R, Ψ is increasing and Ψ′t≠0, for ∀t∈a,b. The fractional Ψ-Hilfer derivative of order α and type 0≤β≤1 can be written as(4)HDa+α,β;Ψft=Ia+βn−α;Ψ1Ψ′tddtnIa+1−βn−α;Ψft,n−1<α<n,n∈N.Remark 2.
(1)
Iff∈C1a,b, 0<α<1 and 0≤β≤1, 1−α1−β=1−γ, we have(5)Ia+α;ΨHDa+α,β;Ψft=ft−Ψt−Ψaγ−1ΓγIa+1−β1−α;Ψfa.(2)
Iff∈C1a,b, α>0 and 0≤β≤1, then HDa+α,β;ΨIa+α;Ψft=ft.Definition 3.
(see [47]). We assume f:0,∞⟶R, and Ψ⋅ is a non-negative increasing function satisfying Ψ0=0. The generalized the Laplace transform of f can be written as(6)LΨft=Fs=∫0∞e−sΨtΨ′tftdt.Definition 4.
(see [47]). We assume f and g are piecewise continuous and Ψ-exponential order functions in 0,T. The Ψ-convolution with respect to f and g is defined as(7)f∗Ψgt=∫0tΨ−1Ψt−ΨτfgτΨ′τdτ.Lemma 1 (see [47]).
We assumeα>0,n=⌊α⌋+1,0≤β≤1, and a Ψ-exponential order function f⋅ satisfying ft, D0+i;ΨI0+1−βn−α;Ψft∈C0,∞, where i=0,1,2,…,n−1, while HD0+α,β;Ψft is piecewise continuous on 0,∞. Then,(8)LΨHD0+α,β;Ψft=sαLΨft−∑i=0n−1sn1−β+αβ−i−1I0+1−βn−α−i;Ψf0.Lemma 2 (see [47]).
Letα>0, a Ψ-exponential order function of f which is piecewise continuous on 0,T. Then,(9)LΨI0+α;Ψft=s−αLΨft.Lemma 3 (Jensen’s inequality [48]).
Letn∈N and a1,a2,…,an be real and nonnegative numbers. Then,(10)∑i=1naip≤np−1∑i=1naip,forp>1.


## 3. 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, finite-time stability results are obtained by means of the properties of norm and inequalities scaling technique. We define the following space:(11)ℵ=XXt∈C1J,Rn,HD0+p,q;ΨXt∈C1J,Rn,with norm(12)Xℵ=maxEsupt∈JXt2,Esupt∈J∣HD0+p,q;ΨXt∣2,where E is the mathematical expectation. We can readily verify that ℵ,⋅ℵ is a Banach space, and see [49, 50] for more details.Definition 5.
(see [22]). Assuming that there exist positive constants T, δ, ε with δ<ε, then system (1) is finite-time stable if X0ℵ≤δ implies Xℵ≤ε for ∀t∈0,T, where X0=X0.Before starting our proof of the main conclusions, we make the following assumptions on the coefficients of system (1):H1 As for any Xi,Yi∈Rn, there exists a positive bounded function L1⋅ satisfying(13)σt,X1,Y1−σt,X2,Y22∨gt,X1,Y1−gt,X2,Y22≤L1tX1−Y12+X2−Y22.H2 We assume that there is a positive constant M such that supt∈JΨ′t2≤M.H3 For ∀t∈J, let Ξ=sups∈0,tEα,αAΨt−Ψsα, where Eα,α⋅ denotes the two-parameters Mittag–Leffler function, and see [44] for details.H4 As for any X,Y∈Rn, there exists a bounded positive function L2⋅ satisfying(14)∣∣2∨gt,X,Y2≤L2t1+∣X∣2+∣Y∣2.⋅ is the norm of Rn, a∨b=maxa,b, supt∈JLit=Li, i=1,2, t∈J.Theorem 1.
We assume that hypothesesH1–H3 hold, then system (1) has a unique solution in ℵ,⋅ℵ, if there is a constant C, 0<C<1, and(15)maxT+44MΞ2L1ΨT−Ψ02α−12α−1,T+44MΞ2L1ΨT−Ψ02α−p−12α−p−1≤C.Proof.
Taking the generalized Laplace transform on (1), we get the following with the aid of Lemma 1:(16)LΨXt=LΨσt,Xt,HD0+p,q;ΨXtsα−A+LΨgt,Xt,HD0+p,q;ΨXtdWt/dtsα−A.
Subsequently, using the generalized inverse Laplace transform, we obtain the solution of system (1) is(17)Xt=Eα,αAΨtαΨtα−1∗Ψσt,Xt,HD0+p,q;ΨXt+Eα,αAΨtαΨtα−1∗Ψgt,Xt,HD0+p,q;ΨXtdWtdt=∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′sσs,Xs,HD0+p,q;ΨXsds+∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′sgs,Xs,HD0+p,q;ΨXsdWs.
We define operatorR:ℵ,⋅ℵ⟶ℵ,⋅ℵ as(18)RXt=∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′sσs,Xs,HD0+p,q;ΨXsds+∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′sgs,Xs,HD0+p,q;ΨXsdWs.
According to the properties of fractionalΨ-Hilfer derivative and fractional Ψ-Riemann–Liouville integral, it is readily to verify that above operator R is well-defined. 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 H3(19)∣RXt−RYt∣≤∣∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′s⋅σs,Xs,HD0+p,q;ΨXs−σs,Ys,HD0+p,q;ΨYsds∣+∣∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′s⋅gs,Xs,HD0+p,q;ΨXs−gs,Ys,HD0+p,q;ΨYsdWs∣≤Ξ∣∫0tΨt−Ψsα−1Ψ′s⋅σs,Xs,HD0+p,q;ΨXs−σs,Ys,HD0+p,q;ΨYsds∣+Ξ∣∫0tΨt−Ψsα−1Ψ′s⋅gs,Xs,HD0+p,q;ΨXs−gs,Ys,HD0+p,q;ΨYsdWs∣.
Then, by means of Jensen’s inequality, Hölder inequality, and Doob’s martingale inequality, we get for allt∈J(20)Esup0≤u≤t∣RXu−RYu∣2≤2Ξ2Esup0≤u≤t∣∫0uΨu−Ψsα−1Ψ′s⋅σs,Xs,HD0+p,q;ΨXs−σs,Ys,HD0+p,q;ΨYsds∣2+2Ξ2Esup0≤u≤t∣∫0uΨu−Ψsα−1Ψ′s⋅gs,Xs,HD0+p,q;ΨXs−gs,Ys,HD0+p,q;ΨYsdWs∣2≤2Ξ2Esup0≤u≤t∫0uΨ′s2Ψu−Ψs2α−2ds⋅∫0u∣σs,Xs,HD0+p,q;ΨXs−σs,Ys,HD0+p,q;ΨYs∣2ds+8Ξ2E∫0tΨt−Ψs2α−2Ψ′s2⋅∣gs,Xs,HD0+p,q;ΨXs−gs,Ys,HD0+p,q;ΨYs∣2ds≔I1+I2.
By assumptionsH1-H2, we obtain(21)I1≤2MΞ2Ψt−Ψ02α−12α−1⋅∫0tL1sE∣Xs−Ys∣2+∣HD0+p,q;ΨXs−HD0+p,q;ΨYs∣2ds≤2MΞ2L1ΨT−Ψ02α−12α−1∫0tEsup0≤s1≤s∣Xs1−Ys1∣2+Esup0≤s1≤s∣HD0+p,q;ΨXs1−HD0+p,q;ΨYs1∣2ds≤2MΞ2L1ΨT−Ψ02α−12α−12TX−Yℵ.
Similarly, one can obtain(22)I2≤8Ξ2E∫0tΨt−Ψs2α−2Ψ′s2L1s⋅E∣Xs−Ys∣2+∣HD0+p,q;ΨXs−HD0+p,q;ΨYs∣2ds≤8Ξ2L1∫0tΨt−Ψs2α−2Ψ′s2Esup0≤s1≤s∣Xs1−Ys1∣2+Esup0≤s1≤s∣HD0+p,q;ΨXs1−HD0+p,q;ΨYs1∣2≤8Ξ2L1∫0tΨt−Ψs2α−2Ψ′s22X−Yℵds≤8MΞ2L1ΨT−Ψ02α−12α−12X−Yℵ.
Then, it is easy to obtain(23)Esup0≤u≤t∣RXu−RYu∣2≤4TMΞ2L1ΨT−Ψ02α−12α−1X−Yℵ+16MΞ2L1ΨT−Ψ02α−12α−1X−Yℵ=T+44MΞ2L1ΨT−Ψ02α−12α−1X−Yℵ≤CX−Yℵ.
On the contrary, we have(24)RXt≤ΞΓα1Γα∫0tΨt−Ψsα−1Ψ′s∣σs,Xs,HD0+p,q;ΨXs∣ds+ΞΓα1Γα∫0tΨt−Ψsα−1Ψ′s∣gs,Xs,HD0+p,q;ΨXs∣dWs=ΞΓαI0+α;Ψ∣σt,Xt,HD0+p,q;ΨXt∣+ΞΓαI0+α;Ψ∣gt,Xt,HD0+p,q;ΨXt∣dWtdt.
Then, with the aid of semigroup property introduced in Remark1, we can readily derive the following:(25)∣HD0+p,q;ΨRXt−HD0+p,q;ΨRYt∣=∣HD0+p,q;Ψ∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′s⋅σs,Xs,HD0+p,q;ΨXs−σs,Ys,HD0+p,q;ΨYsds+∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′s⋅gs,Xs,HD0+p,q;ΨXs−gs,Ys,HD0+p,q;ΨYsdWs∣≤∣HD0+p,q;ΨΞΓαI0+α;Ψ∣σt,Xt,HD0+p,q;ΨXt−σt,Yt,HD0+p,q;ΨYt∣+ΞΓαI0+α;Ψ∣gt,Xt,HD0+p,q;ΨXt−gt,Yt,HD0+p,q;ΨYt∣dWtdt∣≤∣ΞΓαI0+α−p;Ψ∣σt,Xt,HD0+p,q;ΨXt−σt,Yt,HD0+p,q;ΨYt∣∣+∣ΞΓαI0+α−p;Ψ∣gt,Xt,HD0+p,q;ΨXt−gt,Yt,HD0+p,q;ΨYt∣dWtdt∣=Ξ∣∫0tΨtΨsα−p−1Ψ′s⋅∣σs,Xs,HD0+p,q;ΨXs−σs,Ys,HD0+p,q;ΨYs∣ds∣+Ξ∣∫0tΨt−Ψsα−p−1Ψ′s⋅∣gs,Xs,HD0+p,q;ΨXs−gs,Ys,HD0+p,q;ΨYs∣dWs∣.
Similarly, by means of Jensen’s inequality, Hölder inequality, and Doob’s martingale inequality, we get(26)Esup0≤u≤t∣HD0+p,q;ΨRXu−HD0+p,q;ΨRYu∣2≤T+44MΞ2L1ΨT−Ψ02α−p−12α−p−1X−Yℵ≤CX−Yℵ.
Therefore, by the definition of⋅ℵ, for ∀X,Y∈ℵ(27)RXt−RYtℵ≤CX−Y‖ℵ<X−Yℵ,t∈J,then R is a contraction mapping within ℵ,⋅ℵ, which implies that R has a fixed point. Therefore, system (1) has a unique solution.Theorem 2.
We assume that the assumptionsH2–H4 hold, and there exist positive constants δ, ε satisfying δ<ε and X0ℵ≤δ. Then, system (1) is finite-time stable on 0,T, provided that(28)K≤ε1+2ε,where(29)K=maxT+42MΞ2L2ΨT−Ψ02α−12α−1,T+42MΞ2L2ΨT−Ψ02α−p−12α−p−1.Proof.
From Theorem1, system (1) has a unique solution that has the following form:(30)Xt=∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′sσs,Xs,HD0+p,q;ΨXsds+∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′sgs,Xs,HD0+p,q;ΨXsdWs.
By applying Jensen’s inequality, Hölder inequality, and Doob’s martingale inequality, we have for∀t∈J(31)Esup0≤u≤tXu2≤2Esup0≤u≤t∣∫0uEα,αAΨu−ΨsαΨu−Ψsα−1⋅Ψ′sσs,Xs,HD0+p,q;ΨXsds∣2+2Esup0≤u≤t∣∫0uEα,αAΨu−ΨsαΨu−Ψsα−1⋅Ψ′sgs,Xs,HD0+p,q;ΨXsdWs∣2≤2Esup0≤u≤t∫0uEα,αAΨu−Ψsα]2Ψu−Ψs2α−2Ψ′s2ds⋅∫0u∣σs,Xs,HD0+p,q;ΨXs∣2ds+8E∫0tEα,αAΨt−Ψsα2Ψt−Ψs2α−2Ψ′s2⋅∣gs,Xs,HD0+p,q;ΨXs∣2ds≔=I3+I4.
By assumptionsH2–H4, we obtain(32)I3≤2MΞ2ΨT−Ψ02α−12α−1E∫0t∣σs,Xs,HD0+p,q;ΨXs∣2ds≤2MΞ2ΨT−Ψ02α−12α−1∫0tL2sE1+∣Xs∣2+∣HD0+p,q;ΨXs∣2ds≤2MΞ2L2ΨT−Ψ02α−12α−1∫0t1+2Xℵds≤2TMΞ2L2ΨT−Ψ02α−12α−11+2Xℵ.
Similarly, we obtain(33)I4≤8Ξ2∫0tΨt−Ψs2α−2Ψ′s2L2sE1+∣Xs∣2+∣HD0+p,q;ΨXs∣2ds≤8Ξ2L2∫0tΨt−Ψs2α−2Ψ′s21+2Xℵds≤8MΞ2L2ΨT−Ψ02α−12α−11+2Xℵ.
Therefore, we obtain(34)Esup0≤u≤tXu2≤T+42MΞ2L2ΨT−Ψ02α−12α−11+2Xℵ.
On the other hand, we can readily derive the following:(35)Esup0≤u≤t∣HD0+p,q;ΨXu∣2≤T+42MΞ2L2ΨT−Ψ02α−p−12α−p−11+2Xℵ.
According to the definition of⋅ℵ, we have(36)Xℵ≤K1+2Xℵ,then by the conditions of Theorem 2, yields(37)Xℵ≤ε,which implies that system (1) is finite-time stable on 0,T.Remark 3.
Taking the initial valueI0+1−γ;ΨX0=0 is somewhat strict. One can let the initial value to I0+1−γ;ΨX0=X0 or X0=X0. However, when performing stability estimation, we will meet the following difficulties.(1)
Taking the initial value toI0+1−γ;ΨX0=X0≠0 and applying the generalized Laplace transform and its inverse, we derive that system has the following solution:(38)Xt=Eα,αAΨt−Ψ0αΨt−Ψ0γ−1X0+∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′sσs,Xs,HD0+p,q;ΨXsds+∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′sgs,Xs,HD0+p,q;ΨXsdWs.We will find that the first term is a singular function, then the estimation ofEsup0≤u≤t∣Xu∣2, for all t∈J, will be unbounded.(2)
Taking the initial value toX0=X0, similarly, the system has the following solution:(39)Xt=X0ΓγΓ2−γEαAΨt−Ψ0α+∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′sσs,Xs,HD0+p,q;ΨXsds+∫0tEα,αAΨt−ΨsαΨt−Ψsα−1Ψ′sgs,Xs,HD0+p,q;ΨXsdWs.
In order to estimateEsup0≤u≤t∣HD0+p,q;ΨXu∣2, we need to take the Ψ-Hilfer fractional derivative of the first term, which will produce a singular function ΛΨt−Ψ0−α, where Λ represents a constant. This prevents us from considering the stability analysis.


## 4. Example

We consider the following stochasticΨ-Hilfer fractional system.Example 1.
(40)HD0+0.8,0.5;0.1tXt=AXt+0.1e−tXt+0.1HD0+0.2,0.5;0.1tXt+0.1costXt+0.1sintHD0+0.2,0.5;0.1tXtdWtdt,t∈0,10,I0+0.1;0.1tX0=0.
We know that(41)σt,Xt,HD0+p,q;ΨXt=0.1e−tXt+0.1HD0+0.2,0.5;0.1tXt,gt,Xt,HD0+p,q;ΨXt=0.1costXt+0.1sintHD0+0.2,0.5;0.1tXt.
We can readily calculate thatL1=0.02, L2=0.02. Taking M=0.1, C=0.8, δ=0.1, ε=1.5, and A=0.2I, where I denotes the identity matrix, using mathematical software, we calculate the following results:(42)Ξ=sups∈0,tEα,αAΨt−Ψsα=1.119,t∈0,10,maxT+44MΞ2L1ΨT−Ψ02α−12α−1,T+44MΞ2L1ΨT−Ψ02α−p−12α−p−1=0.7012<C,K=0.3506,K<ε1+2ε=0.375.
Then, it can be easily verified that all the conditions in Theorem1 and 2 are satisfied. Therefore, system (40) has a unique solution, and the norm of solution will not exceed the given bound ε=1.5 over the finite time interval [0, 10]. Then, we can conclude that system (40) is finite-time stable on [0, 10].Remark 4.
When the system evolves beyond this given finite time domain, for example, let us considerT=12. By simple calculation, we can obtain Ξ=1.167 and C=0.95 and the other values are the same as in Example 1, and the following results are obtained:(43)maxT+44MΞ2L1ΨT−Ψ02α−12α−1,T+44MΞ2L1ΨT−Ψ02α−p−12α−p−1=0.904<C.
It is verified that system (40) has a unique solution, according to Theorem 1. However, K=0.452>ε/1+2ε=0.375; then, with the aid of Theorem 2, we conclude that system (40) is not finite time stable on [0, 12].


## 5. Conclusion

In this article, we study a new kind of stochasticΨ-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 finite-time stability results for the solutions of the considered problem. The nonlinear analysis method we used is essentially new, and yet there are few literatures available for solving this type of considered system. The obtained results have been expounded via an example.



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*Source: 3642394-2023-01-31.xml*