Finite-Time Stability of Equilibrium Points of Nonlinear Fractional Stochastic Differential Equations

Abstract

This paper focuses on the problem, claimed in some works, of the non-existence of finite-time stable equilibria in nonlinear fractional differential equations. After dividing the equilibrium point into the initial equilibrium point and the finite-time equilibrium point, we provide sufficient conditions for the equilibrium point of a fractional stochastic differential equation. Then the finite-time stability of the equilibrium points of nonlinear fractional stochastic differential equations is presented. Finally, the correctness of the theoretical analysis is illustrated through an example.

1. Introduction

In mathematical modeling and system control, fractional calculus is used by mathematicians for the mathematical modeling and qualitative analysis of some complex dynamical systems with memory and genetic properties [1]. At present, fractional calculus has achieved mature development. Many basic theories about fractional differential equations can be found in monographs [2,3]. Additionally, some research results on iterative learning control for fractional systems can be found in [4,5,6,7]. On the other hand, stochastic phenomena are common in real-world dynamic processes and are referred to by practitioners of stochastic analysis as one of the inherent features of natural systems. No dynamic evolutionary process can avoid the influence of stochastic. Stochastic differential equations are important mathematical tools for describing stochastic models. After decades of development, Itô’s stochastic calculus has achieved significant progress in qualitative theory, stability theory, stochastic control, and other fields. Moreover, SDE and its applications in financial systems have also become increasingly mature [8,9].

In recent years, fractional stochastic differential equations (FSDEs) have attracted the attention and been extensively studied by many mathematicians, as they offer effective modeling capabilities for dynamic evolutionary processes that exhibit both memory and stochastic properties [10]. Doan et al. [11] studied a Caputo-type FSDE, and established existence results for their solutions. Subsequently, important theoretical frameworks such as the variation of constants formula [12], well-posedness [13,14], and regularity properties [15] have been developed for FSDEs. Moreover, some stability results for a Conformable FSDE have also been investigated [16,17].

In practical applications, stability is the most important theory in system control, and it is the basic mathematical tool for error analysis in system control. Finite-time control for nonlinear stochastic systems has always been a research hotspot. Lyapunov stability is the main method and tool for error analysis in control theory. Based on the existence of solutions [10,11], the stability [18] and averaging principle [19,20] of FSDEs have both been studied. Meanwhile, the asymptotic behavior and exponential stability of FSDEs have been studied in [21,22].

Currently, the stability theory for dynamical systems has reached a mature stage of development. From asymptotic stability to finite-time stability, numerous practical results have been achieved. As the stability theory for nonlinear systems continues to advance, the finite-time stability of stochastic nonlinear systems has also drawn considerable attention from researchers [23]. In recent years, a growing body of related research has been published. Notably, significant progress has been made in the area of predefined-time consensus control [24], and predefined-time control strategies for stochastic systems have been actively developed. Unfortunately, however, research on the finite-time stability of FSDEs has indeed experienced a period of stagnation.

Finite-time stability is the theoretical basis of control systems. It is very necessary to study the equilibrium points of FSDEs. Li et al. have undertaken a significant amount of important work on the equilibrium points of FSDEs, such as the definitions [25] and equations [26] of the equilibrium points of FSDEs. Using the Caputo fractional derivative, the equation of the equilibrium points of FSDEs is described as ${}^{C}D_{0^{+}}^{\alpha }{x}_e=f(t,x_e)\equiv 0$. Based on this equation of the equilibrium points, many previous works claim that the non-existence of finite-time equilibrium points of FSDEs [27,28,29]. This has led to the finite-time stability of FSDEs not being studied. Fortunately, Wei and Cao et al. [30] (Remark 1) pointed out that FSDEs might have a finite-time equilibrium point $x_e=0$. Meanwhile, Panda and Vijayakumar [31] (Corollary 4.3) pointed out that “the equilibrium point $x_e=0$ of the Caputo fractional differential equation is finite time stable”. Inspired by [30,31], Xiang et al. [32] proposed a new concept of finite-time equilibrium points and provided the finite-time stability criterion for FSDEs, which has brought great inspiration to the study of the finite-time stability of FSDEs.

Inspired by work [32], we consider the following Caputo FSDE:

$$\begin{array}{c}\hfill {}^{C}D_{0^{+}}^{ϵ}x\left(t\right)=f(t,x\left(t\right))+g(t,x\left(t\right)){\displaystyle \frac{dB\left(t\right)}{dt}},x\left(0\right)=x_0,t>0,\end{array}$$

(1)

where ${}^{C}D_{0^{+}}^{ϵ}$ denotes the Caputo fractional derivative and $\alpha \in ({\displaystyle \frac{1}{2}},1)$, $B(·)$ is a Brownian motion. $f,g:[0,+\infty )\times \mathbb{R}\to \mathbb{R}$ are measurable functions with $f(t,0)=0$ and $g(t,0)=0$, indicating that the system has a trivial zero solution $x\left(t\right)\equiv 0$. We aim to establish the concepts of the initial value equilibrium point and the finite-time equilibrium point in the stochastic sense, and present the Lyapnov criterion for finite-time stability.

The structure of this paper is organized as follows: Some mathematical definitions and important lemmas are presented in Section 2. The main results concerning equilibrium points and finite-time stability are discussed in Section 3. In Section 4, an example is provided to validate the theoretical stability results. Section 5 concludes this paper.

2. Preliminaries

Throughout the paper, let ${\mathcal{X}}_t:={\mathbb{L}}^2(\mathrm{\Omega },{\mathcal{F}}_t,P),t\in [0,+\infty )$ be the space of all ${\mathcal{F}}_t$-measurable process. Moreover, a process x is called $\mathcal{F}$-adapted if $x\left(t\right)\in {\mathcal{X}}_t,t\in [0,+\infty )$. $E\left[x\right]$ denotes the mathematical expectation of x. $a\vee b$ represents $max\{a,b\}$.

Definition 1.

(see [3]) Let $ϵ\in (0,1]$. The Caputo fractional derivative ${}^{C}D_{0^{+}}^{ϵ}$ of differentiable function $f(·)$ is defined as

$$\begin{array}{c}\hfill {}^{C}D_{0^{+}}^{ϵ}f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}{\int }_0^t{\displaystyle \frac{f^{\prime }\left(\tau \right)}{{(t-\tau )}^{ϵ}}}d\tau ,t\in [0,\infty ),\end{array}$$

where $\mathrm{\Gamma }\left(ϵ\right):={\int }_0^{\infty }{\tau }^{ϵ-1}{e}^{-\tau }d\tau$ is Gamma function. Hence, let $T_e>0$, we have

$$\begin{array}{c}\hfill {}^{C}D_{T_e}^{ϵ}f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}{\int }_{T_e}^t{\displaystyle \frac{f^{\prime }\left(\tau \right)}{{(t-\tau )}^{ϵ}}}d\tau ,t\in [T_e,\infty ).\end{array}$$

Moreover, the Caputo fractional integral ${\mathbb{I}}_{0^{+}}^{ϵ}$ of $f(·)$ is defined as

$$\begin{array}{c}\hfill {\mathbb{I}}_{0^{+}}^{ϵ}f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(ϵ\right)}}{\int }_0^t{(t-\tau )}^{ϵ-1}f\left(\tau \right)d\tau ,t\in [0,\infty ).\end{array}$$

Definition 2.

(see [3]) The Riemann–Liouville fractional derivative of $f(·)$ is defined as

$$\begin{array}{c}\hfill {}^{R}D_{0^{+}}^{ϵ}f\left(t\right)={\displaystyle \frac{d}{dt}}{\mathbb{I}}_{0^{+}}^{1-ϵ}f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}{\displaystyle \frac{d}{dt}}{\int }_0^t{\displaystyle \frac{f\left(\tau \right)}{{(t-\tau )}^{ϵ}}}d\tau ,ϵ\in (0,1].\end{array}$$

$\left(H1\right)$ For $\forall x,\overline{x}\in \mathbb{R}$, $t\in [0,+\infty ),L>0$.

$$\begin{array}{c}\hfill |f(t,x)-f(t,\overline{x})|\vee |g(t,x)-g(t,\overline{x})|\le L|x-\overline{x}|.\end{array}$$

$\left(H2\right)$ the function $g(·,0)$ satisfies ${sup}_{t\in [0,+\infty )}\left|g(t,0)\right|<+\infty$ and ${\int }_0^{+\infty }{\left|g(t,0)\right|}^{2}dt<+\infty .$

Lemma 1.

(See [11], Theorem 1). If (H1), (H2) hold, then for each $E\left(x_0\right)<\infty$, ${\displaystyle \frac{1}{2}}<ϵ<1$. Equation (1) has a unique solution:

$$\begin{array}{c}\hfill x\left(t\right):=x(t;x_0)=x_0+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ϵ\right)}}\left({\int }_0^t{\displaystyle \frac{f(\tau ,x(\tau \left)\right)}{{(t-\tau )}^{1-ϵ}}}d\tau +{\int }_0^t{\displaystyle \frac{g(\tau ,x(\tau \left)\right)}{{(t-\tau )}^{1-ϵ}}}dB\left(\tau \right)\right),t\in [0,T].\end{array}$$

(2)

Moreover, $x(·)$ is $\mathcal{F}$-adapted and $E\left({\int }_0^T\right|x\left(\tau \right){|}^{2}d\tau )<+\infty$.

Lemma 2.

For the Caputo fractional derivative ${}^{C}D_{0^{+}}^{ϵ}x\left(t\right)$ with $ϵ\in (0,1)$, one has

$$\begin{array}{c}\hfill {\mathbb{I}}_{0^{+}}^{ϵ}\left[{}^{C}D_{0^{+}}^{ϵ}x\left(t\right)\right]=x\left(t\right)-x\left(0\right).\end{array}$$

3. Main Results

Our main results are divided into two parts. Firstly, we will discuss the equation of the equilibrium point $x_e$, and then present the finite-time stability of Equation (1).

3.1. Finite-Time Equilibrium Point

In the control system, a state $x=x_e$ is defined as equilibrium point if, once the state of the system is equal to $x_e$, then it remains equal to $x_e$ for all future time. Unlike the integer-order system, the fractional derivative is a global derivative with a memory effect; the equation of the equilibrium points of fractional Equation (1) depends on the system initial state $x_0$. Therefore, it is necessary to classify the equilibrium points into two classes according to the initial state $x_0$.

Definition 3.

The state $x_e$ is the initial equilibrium point of Equation (1), if $x_0=x_e$ or equivalently $x\left(t\right)\equiv x_e$ hold for all $t\ge 0$.

Definition 4.

The state $x_e$ is the finite-time equilibrium point of Equation (1), if there is a $T_e>0$, such that $x\left(t\right)\ne x_e$ for all $t\in [0,T_e]$ and $x\left(t\right)\equiv x_e$ for all $t\ge T_e$.

The equation of the initial equilibrium point of Equation (1) can be depicted by $\dot{x}\left(t\right)=0$ as the case of integer-order systems. However, condition $\dot{x}\left(t\right)=0$ is not suitable to characterize the equation of the finite-time equilibrium point of Equation (1). Thus, the above two definitions of the equilibrium point of Equation (1) are classified necessarily. Denoting

$$\begin{array}{c}\hfill x\left(t\right)=\left\{\begin{array}{c}\overline{x}\left(t\right),t\in [0,T_e],\hfill \\ x_e,t\ge T_e.\hfill \end{array}\right.\end{array}$$

Hence, we have

$$\begin{array}{c}\hfill {}^{C}D_{0^{+}}^{ϵ}x\left(t\right)=\left\{\begin{array}{c}\varphi (t,x\left(t\right))=f(t,\overline{x}\left(t\right))+g(t,\overline{x}\left(t\right)){\displaystyle \frac{dB\left(t\right)}{dt}},t\in [0,T_e],\hfill \\ \overline{\varphi }(t,x\left(t\right))=f(t,x_e)+g(t,x_e)){\displaystyle \frac{dB\left(t\right)}{dt}},t\ge T_e.\hfill \end{array}\right.\end{array}$$

For the finite-time equilibrium point of Equation (1), for $\forall t\ge T_e$, we can obtain

$$\begin{array}{ccc}\hfill {}^{C}D_{0^{+}}^{ϵ}x\left(t\right)& =& {\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}\left[{\int }_0^t{\displaystyle \frac{\dot{x}\left(s\right)}{{(t-s)}^{ϵ}}}ds+{\int }_0^t{\displaystyle \frac{\dot{x}\left(s\right)}{{(t-s)}^{ϵ}}}dB\left(s\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}\left[{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}ds+{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}dB\left(s\right)\right]\hfill \\ & & +{\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}\left[{\int }_{T_e}^t{\displaystyle \frac{{\dot{x}}_e}{{(t-s)}^{ϵ}}}ds+{\int }_{T_e}^t{\displaystyle \frac{{\dot{x}}_e}{{(t-s)}^{ϵ}}}dB\left(s\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}\left[{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}ds+{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}dB\left(s\right)\right].\hfill \end{array}$$

(3)

It can be seen that (3) does not necessarily remain at 0 when $t\ge T_e$, but tends to 0 when $t\to \infty$. We will directly present this conclusion and prove it below.

Theorem 1.

If state $x_e$ is the finite-time equilibrium point of Equation (1), then

$$\begin{array}{c}\hfill {}^{C}D_{0^{+}}^{ϵ}x\left(t\right)=\varphi (t,x_e)=f(t,x_e)+g(t,x_e)){\displaystyle \frac{dB\left(t\right)}{dt}}\neg \not\equiv 0,\forall t\ge T_e.\end{array}$$

(4)

Proof. 

We use proof by contradiction to prove this conclusion. Suppose the equation of the equilibrium point $x_e$ satisfies

$$\begin{array}{c}\hfill f(t,x_e)=g(t,x_e))\equiv 0,\forall t\ge T_e.\end{array}$$

Hence, for all $t\ge T_e$, we have

$$\begin{array}{ccc}\hfill E|x\left(t\right)-x_0{|}^2& =& E|{\mathbb{I}}_{0^{+}}^{ϵ}{}^{C}D_{0^{+}}^{ϵ}x\left(t\right){|}^2\hfill \\ & =& E|{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ϵ\right)}}{\int }_0^t{\displaystyle \frac{f(s,x(s\left)\right)}{{(t-s)}^{1-ϵ}}}ds+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ϵ\right)}}{\int }_0^t{\displaystyle \frac{g(s,x(s\left)\right)}{{(t-s)}^{1-ϵ}}}dB\left(s\right){|}^2\hfill \\ & \le & 4E|{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ϵ\right)}}{\int }_0^{T_e}{\displaystyle \frac{f(s,\overline{x}\left(s\right))}{{(t-s)}^{1-ϵ}}}ds{|}^2+4E|{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ϵ\right)}}{\int }_{T_e}^t{\displaystyle \frac{f(s,x_e)}{{(t-s)}^{1-ϵ}}}ds{|}^2\hfill \\ & & +4E|{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ϵ\right)}}{\int }_0^{T_e}{\displaystyle \frac{g(s,\overline{x}\left(s\right))}{{(t-s)}^{1-ϵ}}}dB\left(s\right){|}^2+4E|{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ϵ\right)}}{\int }_{T_e}^t{\displaystyle \frac{g(s,x_e)}{{(t-s)}^{1-ϵ}}}dB\left(s\right){|}^2\hfill \\ & =& {\displaystyle \frac{4}{{\mathrm{\Gamma }}^2\left(ϵ\right)}}E|{\int }_0^{T_e}{\displaystyle \frac{f(s,\overline{x}\left(s\right))}{{(t-s)}^{1-ϵ}}}ds{|}^2+{\displaystyle \frac{4}{{\mathrm{\Gamma }}^2\left(ϵ\right)}}E|{\int }_0^{T_e}{\left[{\displaystyle \frac{g(s,\overline{x}\left(s\right))}{{(t-s)}^{1-ϵ}}}\right]}^{2}ds|\hfill \\ & \le & {\displaystyle \frac{4+T_e}{{\mathrm{\Gamma }}^2\left(ϵ\right)}}E\left|{\int }_0^{T_e}{\left[{\displaystyle \frac{\psi (s,\overline{x}\left(s\right))}{{(t-s)}^{1-ϵ}}}\right]}^{2}ds\right|,\hfill \end{array}$$

where $\psi (s,\overline{x}\left(s\right))=|f(s,\overline{x}\left(s\right))|\vee |g(s,\overline{x}\left(s\right))|,t\in [0,T_e]$. Since $\varphi (s,\overline{x}\left(s\right))$ is continuous on $[0,T_e]$, it is bounded on $[0,T_e]$. Hence, we can claim that there is $M>0$, such that $E\left|\psi \right(s,\overline{x}\left(s\right)\left)\right|<M$ for $t\in [0,T_e]$. Thus, we have

$$\begin{array}{ccc}\hfill E|x\left(t\right)-x_0{|}^2& \le & {\displaystyle \frac{(4+T_e)M^2}{{\mathrm{\Gamma }}^2\left(ϵ\right)}}E\left|{\int }_0^{T_e}{\left[{(t-s)}^{ϵ-1}\right]}^{2}ds\right|\hfill \\ & =& {\displaystyle \frac{(4+T_e)M^2}{(2ϵ-1){\mathrm{\Gamma }}^2\left(ϵ\right)}}(t^{2ϵ-1}-{(t-T_e)}^{2ϵ-1})>0.\hfill \end{array}$$

Note that $ϵ\in ({\displaystyle \frac{1}{2}},1)$, so we can obtain $0<2ϵ-1<1$. Hence, we have

$$\begin{array}{c}\hfill t^{2ϵ-1}-{(t-T_e)}^{2ϵ-1}={\int }_{t-T_e}^t(2ϵ-1){\tau }^{2ϵ-2}d\tau .\end{array}$$

Using mean value theorem, we can find a number $\xi \in [t-T_e,t]$ that satisfies

$$\begin{array}{c}\hfill t^{2ϵ-1}-{(t-T_e)}^{2ϵ-1}=(2ϵ-1){\int }_{t-T_e}^t{\tau }^{2ϵ-2}d\tau =(2ϵ-1)T_e{\xi }^{2ϵ-2}.\end{array}$$

Now, letting $t\to \infty$, we can obtain $\xi \to \infty$. Hence, we can derive

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}E|x\left(t\right)-x_0{|}^2\le \underset{t\to \infty }{lim}{\displaystyle \frac{(4+T_e)M^2{T}_e}{{\mathrm{\Gamma }}^2\left(ϵ\right){\xi }^{2-2ϵ}}}=\underset{\xi \to \infty }{lim}{\displaystyle \frac{(4+T_e)M^2{T}_e}{{\mathrm{\Gamma }}^2\left(ϵ\right){\xi }^{2-2ϵ}}}\to 0.\end{array}$$

which implies that $x_e=x_0$—that is, $x_e$ is an initial equilibrium point of Equation (1). This contradicts the assumption that $x_e$ is the finite-time equilibrium point of Equation (1). Therefore, for $\forall t\ge T_e$, we have

$$\begin{array}{c}\hfill {}^{C}D_{0^{+}}^{ϵ}x\left(t\right)=f(t,x_e)+g(t,x_e){\displaystyle \frac{dB\left(t\right)}{dt}}\neg \not\equiv 0.\end{array}$$

□

It must be pointed out that the Equation (4) is not only a necessary condition, but also a sufficient condition of the finite-time equilibrium point of Equation (1).

Theorem 2.

If the state $x_e$ is the finite-time stability equilibrium point of Equation (1), then there exists $T_e>0$, such that, for $\forall t\ge T_e$,

$$\begin{array}{c}\hfill {}^{C}D_{0^{+}}^{ϵ}x\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}ds+{\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}dB\left(s\right).\end{array}$$

(5)

Proof. 

We only prove the sufficiency; if there is a $T_e>0$ satisfying (5), then $x_e=x\left(T_e\right)$ is a finite-time equilibrium point of Equation (1). For all $t\ge T_e$, we have

$$\begin{array}{ccc}\hfill {}^{C}D_{0^{+}}^{ϵ}x\left(t\right)& =& {\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}\left[{\int }_0^t{\displaystyle \frac{\dot{x}\left(s\right)}{{(t-s)}^{ϵ}}}ds+{\int }_0^t{\displaystyle \frac{\dot{x}\left(s\right)}{{(t-s)}^{ϵ}}}dB\left(s\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}\left[{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}ds+{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}dB\left(s\right)\right]\hfill \\ & & +{\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}\left[{\int }_{T_e}^t{\displaystyle \frac{\dot{x}\left(s\right)}{{(t-s)}^{ϵ}}}ds+{\int }_{T_e}^t{\displaystyle \frac{\dot{x}\left(s\right)}{{(t-s)}^{ϵ}}}dB\left(s\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}\left[{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}ds+{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}dB\left(s\right)\right]+{}^{C}D_{T_e}^{ϵ}x\left(t\right)\hfill \\ & =& {\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}\left[{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}ds+{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}dB\left(s\right)\right].\hfill \end{array}$$

(6)

Letting ${}^{C}D_{T_e}^{ϵ}x\left(t\right)=\overline{\varphi }(t,x\left(t\right))$, we have

$$\begin{array}{c}\hfill {}^{C}D_{T_e}^{ϵ}x\left(t\right)=\overline{\varphi }(t,x\left(t\right))\equiv 0,t>T_e.\end{array}$$

Now, from Lemma 2, we have

$$\begin{array}{ccc}\hfill E|x\left(t\right)-x_e|& =& E\left[{\mathbb{I}}_{T_e}^{ϵ}{}^{C}D_{T_e}^{ϵ}x\left(t\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(ϵ\right)}}E\left[{\int }_{T_e}^t{\displaystyle \frac{\overline{\varphi }(t,x\left(t\right))}{{(t-s)}^{1-ϵ}}}ds\right]+{\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}E\left[{\int }_0^{T_e}{\displaystyle \frac{\dot{\overline{x}}\left(s\right)}{{(t-s)}^{ϵ}}}dB\left(s\right)\right]\hfill \\ & =& 0.\hfill \end{array}$$

Hence, we have $P\{x\left(t\right)=x_e\}=1$ hold for $\forall t\ge T_e$; this implies that $x_e$ is the finite-time equilibrium point of Equation (1). □

From the above proof, we can see that $x_e$ is an equilibrium point of Equation (1), including the initial equilibrium point and finite-time equilibrium point, if and only if there is a $T_e>0$ satisfying

$$\begin{array}{c}\hfill {}^{C}D_{0^{+}}^{ϵ}x\left(t\right)\equiv 0,x_e\left(t\right)=x\left(T_e\right),\forall t>T_e.\end{array}$$

Thus, Definition 5 depicting the equilibrium point in a more general and convenient way, which can be provided as follows:

Definition 5.

The state $x_e$ is an equilibrium point of Equation (1), if there exists a number $T_e>0$, such that, for all $t\ge T_e$,

$$\begin{array}{c}\hfill {}^{C}D_{0^{+}}^{ϵ}x\left(t\right)={}^{C}D_{T_e}^{ϵ}{x}_e=\overline{\varphi }(t,x\left(t\right))\equiv 0.\end{array}$$

Remark 1.

When $T_e=0$, $x_e$ in Definition 5 is the initial equilibrium point, and when $T_e>0$, $x_e$ is the finite-time equilibrium point. Thus, Definition 5 is suitable to depict the equilibrium point of the Equation (1), including the initial equilibrium point and finite-time equilibrium point.

3.2. Finite-Time Stability

We firstly give a natural definition for the finite-time stability of the equilibrium point of Equation (1).

Definition 6.

The state $x_e$ is the equilibrium point of Equation (1), if there exists $T_e>0$, such that

$$\begin{array}{c}\hfill {}^{C}D_{T_e}^{ϵ}x\left(t\right)=\overline{\varphi }(t,x\left(t\right))\equiv 0,t>T_e.\end{array}$$

Definition 7.

(see [33]) The trivial solution $x\left(t\right)$ of (1) is said to be stochastically finite-time stable, if the process $x(·)$ satisfies the following:

• (i) Finite-time attractiveness in probability: For all $x_0\in {\mathbb{R}}^n\setminus \left\{\mathbf{0}\right\}$, the first hitting time $T\left(x_0\right)=inf\{t>0;x\left(t\right)=\mathbf{0})\}$ satisfies $\mathbf{P}\left(T\left(x_0\right)<\infty \right)=1$.
• (ii) Stability in probability: For every pair of $r>0$ and $\epsilon \in (0,1)$, there exists a constant $\delta =\delta (\epsilon ,r)>0$ such that

$$\begin{array}{c}\hfill \mathbf{P}\left(\left|\right|x\left(t\right)\left|\right|<r,\forall t\ge 0\right)\ge 1-\epsilon ,\end{array}$$

whenever $\left|\right|x_0\left|\right|<\delta$.

Theorem 3.

If there is a differentiable function $V(t,x\left(t\right)):[0,+\infty )\times \mathrm{\Omega }\to {\mathbb{R}}^{+}$, three class $\mathcal{K}$ functions are ${\gamma }_1(·),{\gamma }_2(·),{\gamma }_3(·)$, and a $\delta >0$, such that

$$\begin{array}{ccc}& \left(i\right)& {\gamma }_1\left(\right|\left|x\left(t\right)\right|\left|\right)\le V(t,x\left(t\right))\le {\gamma }_2\left(\right|\left|x\left(t\right)\right|\left|\right),\hfill \\ & \left(ii\right)& \dot{V}(t,x\left(t\right))\le -{\gamma }_3(V(t,x\left(t\right)),\hfill \\ & \left(iii\right)& {\int }_0^{\epsilon }{\displaystyle \frac{1}{{\gamma }_3\left(s\right)}}ds<+\infty ,\forall 0\le \epsilon \le \delta ,\hfill \end{array}$$

then $x_e=0$ is a finite-time stability equilibrium point of Equation (1).

Proof. 

Letting ${}^{C}D_{0^{+}}^{ϵ}V(t,X\left(t\right))=f_v(t,x\left(t\right))$. Note that ${}^{R}D_{0^{+}}^{1-ϵ}{f}_v(t,x\left(t\right))={\displaystyle \frac{dV(t,x(t\left)\right)}{dt}}$; by condition $\left(ii\right)$, one has

$$\begin{array}{c}\hfill {}^{R}D_{0^{+}}^{1-ϵ}{f}_v(t,x\left(t\right))=\dot{V}(t,x\left(t\right))\le -{\gamma }_3(V(t,x\left(t\right)).\end{array}$$

(7)

Obviously, $V(t,x(t\left)\right)$ is monotone, decreasing on $t\ge 0$.

According to Definition 1, utilizing fractional integral operator ${\mathbb{I}}_{T_e}^{1-ϵ}$ on both sides of (7) yields

$$\begin{array}{ccc}\hfill {}^{C}D_{0^{+}}^{ϵ}V(t,x\left(t\right))={\mathbb{I}}_{T_e}^{1-ϵ}\dot{V}(t,x\left(t\right))& =& {\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}{\int }_0^t\dot{V}(t,x\left(t\right)){(t-s)}^{-ϵ}ds\hfill \\ & \le & -{\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}{\int }_0^t{\gamma }_3{(V(t,x\left(t\right))(t-s)}^{-ϵ}ds\hfill \\ & \le & -{\displaystyle \frac{1}{\mathrm{\Gamma }(1-ϵ)}}{\int }_0^t{\gamma }_3({\gamma }_1\left(\right|\left|x\left(t\right)\right|\left|\right){(t-s)}^{-ϵ}ds\hfill \\ & \le & -{\displaystyle \frac{{\gamma }_3({\gamma }_1\left(\right|\left|x\left(t\right)\right|\left|\right))}{\mathrm{\Gamma }(1-ϵ)}}{\displaystyle \frac{t^{1-ϵ}}{1-ϵ}}.\hfill \end{array}$$

When $t\ge 1$, we can obtain

$$\begin{array}{c}\hfill {}^{C}D_{0^{+}}^{ϵ}V(t,x\left(t\right))\le -{\displaystyle \frac{{\gamma }^{*}\left(\right|\left|x\left(t\right)\right|\left|\right)}{(1-ϵ)\mathrm{\Gamma }(1-ϵ)}}<0,\end{array}$$

where ${\gamma }^{*}(·)={\gamma }_3\left({\gamma }_1(·)\right)$. Note that ${\gamma }_1(·),{\gamma }_3(·)$ are class $\mathcal{K}$ functions; one has the compound function ${\gamma }_3\left({\gamma }_1(·)\right)$ is also class $\mathcal{K}$ function. Hence, we have the equilibrium point $x_e=0$ of Equation (1), which is uniformly asymptotically stable.

The uniform asymptotical stability of $x_e=0$ and the continuity of function ${\gamma }_2\left(\right|\left|x\left(t\right)\right|\left|\right)$ guarantee that there exists $r>0$, such that for $\forall x_0\in {\mathbb{B}}_r=\{x\left(t\right)\in R^n\left|\right|\left|x\left(t\right)\right||<r\}\subset \mathrm{\Omega }$, and $x\left(t\right)=x(t,t_0,x_0)$ satisfies

$$\begin{array}{c}\hfill 0\le \underset{t\to \infty }{lim}V(t,x\left(t\right))\le \underset{t\to \infty }{lim}{\gamma }_2\left(\right|\left|x\left(t\right)\right|\left|\right)={\gamma }_2\left(0\right)=0,\end{array}$$

As a result, $0\le {lim}_{t\to \infty }V(t,x\left(t\right))=0$. Taking the monotonicity of $V(t,x(t\left)\right)$ into account, there is a time instant $T>0$, such that for $\forall t\ge T,V(t,x(t\left)\right)\le r$. Suppose $V(t,x(t\left)\right)$ reaches 0 at moment $t=T+\tilde{T}$. We next prove that $\tilde{T}<+\infty$.

For the condition $\left(iii\right)$, note the monotonicity of $V(t,x(t\left)\right)$; we can change the integral variable x into $V(t,x(t\left)\right)$ and the integral interval $[0,V(T,x\left(T\right)\left)\right]$ becomes $[T,T+\tilde{T}]$. Taking condition $\left(iii\right)$ and (7) into account, condition $\left(iii\right)$ is changed into

$$\begin{array}{ccc}\hfill +\infty >{\int }_0^{V(T,x(T\left)\right)}{\displaystyle \frac{1}{{\gamma }_3\left(x\right)}}dx& =& {\int }_T^{T+\tilde{T}}{\displaystyle \frac{1}{{\gamma }_3\left(V(t,x\left(t\right))\right)}}dV(t,x\left(t\right))\hfill \\ & =& {\int }_T^{T+\tilde{T}}{\displaystyle \frac{\dot{V}(t,x\left(t\right))}{{\gamma }_3\left(V(t,x\left(t\right))\right)}}dt={\int }_{T+\tilde{T}}^T{\displaystyle \frac{{}^{R}D_0^{1-ϵ}{f}_v(t,x\left(t\right))}{{\gamma }_3\left(V(t,x\left(t\right))\right)}}dt\hfill \\ & \ge & {\int }_T^{T+\tilde{T}}dt=\tilde{T}.\hfill \end{array}$$

Thus, for $\forall x_0\in {\mathbb{B}}_r$, $x\left(t\right)\to x_e=0$ within $[0,T+\tilde{T}]$. □

4. An Example

Example 1.

Consider the following FSDE extend from [32]:

$$\begin{array}{ccc}\hfill {}^{C}D_{0^{+}}^{ϵ}x\left(t\right)& =& {\displaystyle \frac{1}{\mathrm{\Gamma }(3-ϵ)}}{(\sqrt{\eta }-\sqrt{\left|x\right(t\left)\right|})}^{2-ϵ}-{\displaystyle \frac{\sqrt{\eta }}{\mathrm{\Gamma }(2-ϵ)}}{(\sqrt{\eta }-\sqrt{\left|x\right(t\left)\right|})}^{1-ϵ}+{\displaystyle \frac{1}{\mathrm{\Gamma }(3-ϵ)}}{t}^{2-ϵ}\hfill \\ & & -{\displaystyle \frac{1}{\mathrm{\Gamma }(3-ϵ)}}{(t-\sqrt{\eta })}^{2-ϵ}-{\displaystyle \frac{\sqrt{\eta }}{\mathrm{\Gamma }(2-ϵ)}}{t}^{1-ϵ}\hfill \\ & & +sgn(t-\sqrt{\eta })[-{\displaystyle \frac{1}{\mathrm{\Gamma }(3-ϵ)}}{(\sqrt{\eta }-\sqrt{\left|x\right(t\left)\right|})}^{2-ϵ}+{\displaystyle \frac{\sqrt{\eta }}{\mathrm{\Gamma }(2-ϵ)}}{(\sqrt{\eta }-\sqrt{\left|x\right(t\left)\right|})}^{1-ϵ}\hfill \\ & & +{\displaystyle \frac{1}{\mathrm{\Gamma }(3-ϵ)}}{t}^{2-ϵ}-{\displaystyle \frac{1}{\mathrm{\Gamma }(3-ϵ)}}{(t-\sqrt{a})}^{2-ϵ}-{\displaystyle \frac{\sqrt{\eta }}{\mathrm{\Gamma }(2-ϵ)}}{t}^{1-ϵ}]+2t{\displaystyle \frac{dB\left(t\right)}{dt}},t>0,\hfill \end{array}$$

(8)

where $x\left(0\right)=\eta >0$, $ϵ\in ({\displaystyle \frac{1}{2}},1)$.

Simplifying the Equation (8), we can obtain

$$\begin{array}{c}\hfill {}^{C}D_{0^{+}}^{ϵ}x\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{2}{\mathrm{\Gamma }(3-ϵ)}}{t}^{2-ϵ}-{\displaystyle \frac{2\sqrt{\eta }}{\mathrm{\Gamma }(2-ϵ)}}{t}^{1-ϵ}+2t{\displaystyle \frac{dB\left(t\right)}{dt}},t\in (0,\sqrt{\eta }),\hfill \\ {\displaystyle \frac{2}{\mathrm{\Gamma }(3-ϵ)}}{t}^{2-ϵ}-{\displaystyle \frac{2\sqrt{\eta }}{\mathrm{\Gamma }(2-ϵ)}}{t}^{1-ϵ}-{\displaystyle \frac{2}{\mathrm{\Gamma }(3-ϵ)}}{(t-\sqrt{\eta })}^{2-ϵ}+2t{\displaystyle \frac{dB\left(t\right)}{dt}},t\ge \sqrt{\eta }.\hfill \end{array}\right.\end{array}$$

(9)

Note that

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{\mathrm{\Gamma }\left(ϵ\right)}}{\int }_0^t{(t-s)}^{ϵ-1}[{\displaystyle \frac{2}{\mathrm{\Gamma }(3-ϵ)}}{t}^{2-ϵ}-{\displaystyle \frac{2\sqrt{\eta }}{\mathrm{\Gamma }(2-ϵ)}}{t}^{1-ϵ}]ds={(t-\sqrt{\eta })}^2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{\mathrm{\Gamma }\left(ϵ\right)}}{\int }_{\sqrt{\eta }}^t{(t-s)}^{ϵ-1}[{\displaystyle \frac{2}{\mathrm{\Gamma }(3-ϵ)}}{t}^{2-ϵ}-{\displaystyle \frac{2\sqrt{\eta }}{\mathrm{\Gamma }(2-ϵ)}}{t}^{1-ϵ}-{\displaystyle \frac{2}{\mathrm{\Gamma }(3-ϵ)}}{(t-\sqrt{\eta })}^{2-ϵ}]ds=0.\hfill \end{array}$$

(11)

Hence, we can get the solution of Equation (8) as follows:

$$\begin{array}{c}\hfill x\left(t\right)=\left\{\begin{array}{c}{(t-\sqrt{\eta })}^2+{\displaystyle \frac{2}{\mathrm{\Gamma }\left(ϵ\right)}}{\int }_0^t{(t-s)}^{ϵ-1}sdB\left(s\right),t\in (0,\sqrt{\eta }),\hfill \\ {\displaystyle \frac{2}{\mathrm{\Gamma }\left(ϵ\right)}}{\int }_0^t{(t-s)}^{ϵ-1}sdB\left(s\right),t\ge \sqrt{\eta }.\hfill \end{array}\right.\end{array}$$

(12)

Although $x_e=0$ is an equilibrium points of Equation (8), $E\left[{}^{C}D_{0^{+}}^{ϵ}x\left(t\right)\right]=E\left[\varphi (t,x)\right]\ne 0,t\in (0,\sqrt{\eta })$, $E\left[{}^{C}D_{0^{+}}^{ϵ}x\left(t\right)\right]=E\left[\varphi (t,x)\right]=0,t\ge \sqrt{\eta }$. When $ϵ=0.7$, we used MATLAB (https://matlab.mathworks.com/, accessed on 4 July 2025) to visualize the trajectory of the solution of Equation (8), and obtained its trajectory and characteristics as shown in Figure 1 and Figure 2, respectively.

Figure 1. The trajectory of the solution to FSDE (8) with $ϵ=0.7$.

Figure 2. The characteristics of solutions to FSDE (8) with $ϵ=0.7$.

5. Conclusions

Our results extend the work presented in [32] by rigorously proving the finite-time stability of equilibrium points in FSDE. These findings provide theoretical foundations for subsequent studies on synchronous control and controller error analysis. Notably, our research establishes a criterion for assessing the finite-time stability of equilibrium points in fractional systems, which can also be applied to address fixed-time stability and predefined-time stability. Moreover, the proposed framework can be extended to analyze finite-time stability in nonlinear fractional systems that incorporate time delays and impulsive effects. Future research will focus on control applications. Interested readers may refer to the works of [4,5,33] for further insights into fixed-time stability, predefined-time stabilization, and formation control in fractional stochastic systems.