# Improved finite-time stability and instability theorems for stochastic
nonlinear systems††thanks: This paper was not presented at any IFAC meeting.

Weihai Zhang , Liqiang Yao College of Electrical Engineering and Automation,
Shandong University of Science and Technology, Qingdao, Shandong Province,
266590, P. R. China. Corresponding author. Email<EMAIL_ADDRESS>of
Mathematics and Information Science, Yantai University, Yantai, Shandong
Province, 264005, P. R. China. Email<EMAIL_ADDRESS>
###### Abstract

This paper studies finite-time stability and instability theorems in
probability sense for stochastic nonlinear systems. Firstly, a new sufficient
condition is proposed to guarantee that the considered system has a global
solution. Secondly, we propose improved finite-time stability and instability
criteria that relax the constraints on $\mathcal{L}V$ (the infinitesimal
operator of Lyapunov function $V$) by the uniformly asymptotically stable
function(UASF). The improved finite-time stability theorems allow
$\mathcal{L}V$ to be indefinite (negative or positive) rather than just only
allow $\mathcal{L}V$ to be negative. Most existing finite-time stability and
instability results can be viewed as special cases of the obtained theorems.
Finally, some simulation examples verify the validity of the theoretical
results.

_Keywords._ Stochastic nonlinear systems; finite-time stability; finite-time
instability; uniformly asymptotically stable function(UASF).

##  I. Introduction

Stability plays a central role in systems theory and engineering applications,
and is always the most fundamental consideration in system analysis and
synthesis. The two most commonly used concepts in stability theory are
asymptotic stability [1-5] and finite-time stability [6-9]. The asymptotic
stability describes the asymptotic behavior of the system state when time
tends to infinity, however, the finite-time stability illustrates the
transient performance of the system state trajectory within a finite time
interval. For many engineering problems, one often pays more attention to
finite time stability instead of asymptotic stability; see, e.g., the tracking
control of robotic manipulators [10], the position and orientation control of
underwater vehicles [11]. Specially, the finite-time stable system usually
shows better robustness and faster convergence rate. So finite-time stability
and its related control problems have aroused great interest during the past
two decades; see [12-16].

Many practical systems are subject to stochastic perturbations such as
environmental noise. Actual systems in random environments are often modeled
as stochastic systems, which can be studied by stochastic analysis method. It
is natural that the finite-time stability and stabilization of stochastic
systems have attracted a great deal of attention. For example, [17] presented
the definition of finite-time stability and established finite-time stability
and instability criteria in probability sense under the hypothesis that the
considered system has a unique strong solution. Generally speaking, it is
difficult to ensure that there exists a unique strong solution to stochastic
nonlinear systems without Lipschitz condition. So researchers have to study
finite-time stability and stabilization problems for stochastic nonlinear
systems under the framework of strong solution or weak solution. [18]
generalized the finite-time stability criteria of [17] and relaxed the
constraint conditions on $\mathcal{L}V$ (the infinitesimal generator of
Lyapunov function) in strong solution or weak solution sense. Recently, [19]
has presented new finite-time stability results by multiple Lyapunov functions
for stochastic nonlinear time-varying systems. In addition, [20], [21], [22]
and [23] studied feedback stabilization problems for several classes of
stochastic nonlinear strict-feedback systems in finite time. [24] and [25]
discussed the finite-time stabilization for two classes of stochastic
nonlinear high-order systems with switching, respectively.

In this paper, we focus on the existence of the system solution and its
finite-time stability and instability. A new sufficient condition is given to
guarantee that the considered system has a global solutio, which is weaker
than the sufficient condition appeared in Lemma 1 of [19]. It can be found
that, in existing literature, most finite-time stability theorems are based on
$\mathcal{L}V$ to be negative (e.g., [17, 23]) or non-positive (e.g., [18]),
where $V$ is a Lyapunov function. Similarly, most finite-time instability
theorems of stochastic nonlinear systems also require $\mathcal{L}V$ to be
negative (e.g., [17, 18]). In order to relax the constraints on Lyapunov
function, this paper proposes improved finite-time stability and instability
theorems. By UASF proposed in [26, 27], we establish some improved finite-time
stability/instability theorems in which $\mathcal{L}V$ can be negative
definite or positive definite rather than just only negative definite. On the
other hand, the obtained results in this paper contain or cover many existing
sufficient conditions about finite-time stability and instability criteria as
special cases.

The rest of this paper is organized as follows. Mathematical preliminaries are
given in Section 2. Section 3 and Section 4 discuss finite-time stability and
instability of stochastic nonlinear systems, respectively. In Section 5, an
example is given to show the application of our improved finite-time stability
theorems. Section 6 concludes this paper with some remarks.

Notations: The set of all natural numbers is denoted by $\mathcal{N}$.
$\mathcal{R}_{+}$ denotes the family of all nonnegative real numbers and
$\mathcal{R}^{r}$ denotes the real $r$-dimensional space. $|x|$ is the
Euclidean norm of a vector $x$. The Frobenius norm of a real matrix $X$ is
$||X||=[\text{Tr}(X^{T}X)]^{1/2}$.
$C^{1,2}([t_{0},\infty)\times\mathcal{R}^{r};\mathcal{R}_{+})$ stands for the
set of all nonnegative functions $w(t,x)$ that are $C^{1}$ in $t$ and $C^{2}$
in $x$. $C^{1,2}_{0}([t_{0},\infty)\times\mathcal{R}^{r};\mathcal{R}_{+})$
denotes the set of all nonnegative functions $w(t,x)\in
C^{1,2}([t_{0},\infty)\times\mathcal{R}^{r};\mathcal{R}_{+})$ except possibly
at the point $x=0$. $\mathcal{K}$ represents the set of all strictly
increasing, continuous functions $\gamma(t)$ with $\gamma(0)=0$.
$\mathcal{K}_{\infty}$ denotes the set of all unbounded functions $\gamma(t)$
with $\gamma(t)\in\mathcal{K}$. $\mathcal{KL}$ denotes the set of all
functions $\beta(s,t)\in\mathcal{K}$ with $t$ being fixed and
$\lim_{t\rightarrow\infty}\beta(s,t)=0$ with $s$ being fixed.

##  II. Preliminaries

Consider the stochastic nonlinear Itô system

$dx(t)=f(t,x(t))dt+g(t,x(t))dW(t),$ (1)

in which system state $x(t)\in\mathcal{R}^{r}$ and $x(t_{0})=x_{0}$,
$W(t)\in\mathcal{R}^{d}$, defined on the filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq t_{0}},P)$, is a standard
Wiener process. The continuous functions
$f:[t_{0},\infty)\times\mathcal{R}^{r}\rightarrow\mathcal{R}^{r}$ and
$g:[t_{0},\infty)\times\mathcal{R}^{r}\rightarrow\mathcal{R}^{r\times d}$
satisfy $f(t,0)=0$ and $g(t,0)=0$ with $t\in[t_{0},\infty)$.

According to Lemma 3.2 in [Chapter 4, 3], the trivial solution of system (1)
is impossible to reach the origin in a finite time, if $f(t,x)$ and $g(t,x)$
satisfy local Lipschitz condition in $x$. Thus some conditions should be
imposed on system (1) such that system (1) has a solution, which may arrive at
the origin in a finite time. The following Lemma 1 ensures that system (1) has
a unique strong solution and it can be viewed as a special case of Theorem 170
in [28].

###### Lemma 1

For every $N=1,2,\cdots,$ and any $T\in[t_{0},\infty)$, if system (1)
satisfies that
H1: $|f(t,x)|\leq(1+|x|)h(t)$, $||g(t,x)||^{2}\leq(1+|x|^{2})h(t)$ and
H2:
$2(x_{1}-x_{2},f(t,x_{1})-f(t,x_{2}))+||g(t,x_{1})-g(t,x_{2})||^{2}\leq\psi_{T}^{N}(|x_{1}-x_{2}|^{2})h_{T}^{N}(t)$,
$|x_{1}|\vee|x_{2}|\leq N$
for any $t\in[t_{0},T]$, where the nonnegative functions $h_{T}^{N}(t)$ and
$h(t)$ satisfy $\int_{t_{0}}^{T}h_{T}^{N}(t)dt<\infty$ and
$\int_{t_{0}}^{T}h(t)dt<\infty$. Moreover, for any $v\geq 0$,
$\psi_{T}^{N}(v)\geq 0$ is a continuous, strictly increasing, non-random and
concave function satisfying $\int_{0+}dv/\psi_{T}^{N}(v)=\infty$. Then, system
(1) has a unique solution for any given initial value
$x_{0}\in\mathcal{R}^{r}$.

###### Remark 1

Note that H2 in Lemma 1 is very restrictive for system (1), which limits the
application scope of finite-time stability theorems. In Lemma 1, H2 is used to
show the uniqueness of solution to system (1).

The following Lemma 2 and Lemma 3 give some sufficient conditions, which
ensure that system (1) has a continuous strong solution.

###### Lemma 2

If system (1) satisfies that H3: $|f(t,x)|^{2}+||g(t,x)||^{2}\leq
H(1+|x|^{2})$ where $H>0$ is a constant, then system (1) has a continuous
solution with probability 1.

###### Lemma 3

For system (1), assume that $f(t,x)$ and $g(t,x)$ are locally bounded in $x$
and are uniformly bounded in $t$. If there exist a function $U(t,x)\in
C^{1,2}([t_{0},\infty)\times\mathcal{R}^{r};\mathcal{R}_{+})$, a
$\mathcal{K}_{\infty}$ function ${\gamma}(\cdot)$, a continuous function
$l(t)$ and a constant $d_{U}\geq 0$ such that $\gamma(|x|)\leq U(t,x)$ and
$\mathcal{L}U(t,x)\leq l(t)U(t,x)+d_{U}$ hold for any $x\in\mathcal{R}^{r}$,
then system (1) has a continuous solution with probability 1.

_Proof:_ For each integer $k\geq 1$, let

$\displaystyle
f_{k}(t,x)=\begin{cases}f(t,\frac{kx}{|x|})&\textmd{if}~{}~{}|x|\geq k\\\
f(t,x)&\textmd{if}~{}~{}|x|<k\\\
\end{cases},~{}~{}~{}~{}g_{k}(t,x)=\begin{cases}g(t,\frac{kx}{|x|})&\textmd{if}~{}~{}|x|\geq
k\\\ g(t,x)&\textmd{if}~{}~{}|x|<k\\\ \end{cases},$

together with $f(t,x)$ and $g(t,x)$ being locally bounded in $x$ and uniformly
bounded in $t$. Then $|f_{k}(t,x)|\leq\sup_{|y|\leq k,t\geq
t_{0}}|f(t,y)|\triangleq c_{1k}<\infty,~{}~{}|g_{k}(t,x)|\leq\sup_{|y|\leq
k,t\geq t_{0}}|g(t,y)|\triangleq c_{2k}<\infty.$ According to Theorem 5.2 in
[29], we know that stochastic nonlinear system

$dx_{k}(t)=f_{k}(t,x_{k}(t))dt+g_{k}(t,x_{k}(t))dW(t)$ (2)

has a continuous solution $x_{k}(t)$ on $[t_{0},\infty)$. Further, define
$\tau_{k}=\inf\\{t\geq t_{0}:|x_{k}(t)|\geq k\\}$, then (2) can be rewritten
as

$dx_{k}(t)=f(t,x_{k}(t))dt+g(t,x_{k}(t))dW(t),$ (3)

where $t\in[t_{0},\tau_{k})$. For any $T\in[t_{0},\infty)$, let
$\tilde{U}(t,x)=U(t,x)\exp\\{-\int_{t_{0}}^{t}l(s)ds\\}$, then we have

$\displaystyle\mathcal{L}\tilde{U}(t,x_{k}(t))=(\mathcal{L}{U}(t,x_{k}(t))-l(t){U}(t,x_{k}(t)))e^{-\int_{t_{0}}^{t}l(v)dv}\leq
d_{U}e^{-\int_{t_{0}}^{t}l(v)dv}$

from which we can further obtain that

$\displaystyle E[U(\tau_{k}\wedge T,x(\tau_{k}\wedge T))]\leq$ $\displaystyle
E[{U}(t_{0},x_{0})e^{\int_{t_{0}}^{\tau_{k}\wedge
T}l(v)dv}]+d_{U}E[\int_{t_{0}}^{\tau_{k}\wedge
T}e^{{\int_{t_{0}}^{\tau_{k}\wedge T}l(v)dv}-\int_{t_{0}}^{s}l(v)dv}ds]$ (4)
$\displaystyle\leq$
$\displaystyle{U}(t_{0},x_{0})e^{\int_{t_{0}}^{T}|l(v)|dv}+d_{U}\int_{t_{0}}^{T}e^{\int_{s}^{T}|l(v)|dv}ds$
(5) $\displaystyle\leq$
$\displaystyle{U}(t_{0},x_{0})e^{M_{T}(T-t_{0})}+d_{U}Te^{M_{T}(T-t_{0})},$
(6)

where $M_{T}=\max_{t_{0}\leq t\leq T}\\{|l(t)|\\}$. Note that
$\inf_{|x|=k,t\geq t_{0}}U(t,x)P\\{\tau_{k}\leq T\\}\leq E[U(\tau_{k}\wedge
T,x(\tau_{k}\wedge T))],$ then (4) can be turned into

$P\\{\tau_{k}\leq T\\}\leq\frac{{U}(t_{0},x_{0})+d_{U}T}{\inf_{|x|=k,t\geq
t_{0}}U(t,x)}e^{M_{T}(T-t_{0})}.$ (7)

Let $k\rightarrow\infty$ in (7), then we get that
$\lim_{k\rightarrow\infty}P\\{\tau_{k}\leq T\\}=0$ for any
$T\in[t_{0},\infty)$, where

$\lim_{k\rightarrow\infty}\inf_{|x|=k,t\geq t_{0}}U(t,x)=\infty$

is used. This means that $\lim_{k\rightarrow\infty}\tau_{k}=\infty$ a.s..
Hence, for any $t\in[t_{0},\tau_{k})$, we define $x(t)=x_{k}(t)$ in (3), then
$x(t)$ is the global solution of system (1).

###### Remark 2

(i) Lemma 2 comes from Theorem 5.2 in [29]. Lemma 3 is an improved version of
Lemma 1 in [19]. In Lemma 1 of [19], $l(t)$ is a nonnegative function with
$\int_{t_{0}}^{\infty}|l(t)|dt<\infty$ and $d_{U}=0$. However, Lemma 3 removes
these strict constraints and allow $l(t)$ to be a continuous function and
$d_{U}\geq 0$.

###### Definition 1

([17]) If system (1) has a solution $x(t)$ and satisfies that
(i) For any $x_{0}\in\mathcal{R}^{r}\backslash\\{0\\}$, the stochastic
settling time $\rho_{x_{0}}=\inf\\{t\geq t_{0}:x(t)=0\\}$ is a finite time
a.s., i.e, $P\\{\rho_{x_{0}}<\infty\\}=1$.
(ii) There is a positive constant $\delta(\varepsilon,R)$ such that
$P\\{|x(t)|<R,~{}~{}t\geq t_{0}\\}\geq 1-\varepsilon$ with
$|x_{0}|<\delta(\varepsilon,R)$ for any $0<\varepsilon<1$ and $R>0$.
Then, the trivial solution of system (1) is called finite-time stable in
probability.

###### Definition 2

([19]) In Definition 1, if (i) or (ii) cannot be satisfied, then the trivial
solution of system (1) is finite-time instable in probability.

###### Definition 3

([26]) For system $\dot{z}(t)=\mu(t)z(t)$ with $t\in[t_{0},\infty)$, if there
exists a function $\beta(\cdot,\cdot)\in\mathcal{K}\mathcal{L}$ such that
$|z(t)|\leq\beta(|z(t_{0})|,t-t_{0})$, then this system is globally uniformly
asymptotically stable and $\mu(t)$ is called a UASF.

###### Remark 3

According to [27], $\mu(t)$ is a UASF if and only if there exist constants
$c_{\mu}>0$ and $d_{\mu}\geq 0$ such that $\int_{t_{0}}^{t}\mu(s)ds\leq
d_{\mu}-c_{\mu}(t-t_{0})$.

For any given $V(t,x)\in
C^{1,2}([t_{0},\infty)\times\mathcal{R}^{r};\mathcal{R}_{+})$ associated with
system (1), the infinitesimal operator $\mathcal{L}$ is defined as
$\mathcal{L}V(t,x)=V_{t}(t,x)+V_{x}(t,x)f(t,x)+\frac{1}{2}\text{Tr}\\{g^{T}(t,x)V_{xx}(t,x)g(t,x)\\}$.

The main goal of this paper is to present the improved finite-time stability
and instability theorems under the condition that system (1) has a solution,
which is in weak solution or strong solution sense. By stochastic analysis
technique and UASF, the established finite-time stability and instability
theorems can break through some strict constraints imposed on the existing
finite-time stability and instability results.

##  III. Finite-time stability theorems

In this section, our purpose is to build some improved finite-time stability
theorems of stochastic nonlinear systems. Compared with some existing relative
results, the improved finite-time stability theorems weaken the condition on
$\mathcal{L}V$ and permit $\mathcal{L}V$ to be indefinite. For this end, we
need the following Lemma 4, which plays an important role in the proof of
Theorem 1.

###### Lemma 4

For system (1), if there exist a function $V^{*}(t,x)\in
C^{1,2}_{0}([t_{0},\infty)\times U_{k};\mathcal{R}_{+})$ and a UASF
$\mu^{*}(t)$ such that for any $\varepsilon\in(0,k)$,

$\displaystyle\mathcal{L}V^{*}(t,x)\leq\mu^{*}(t),\forall x\in
U_{k,\varepsilon},$ (8)

where $U_{k}=\\{x\in\mathcal{R}^{r}:|x|<k\\}$ and
$U_{k,\varepsilon}=\\{x\in\mathcal{R}^{r}:\varepsilon<|x|<k\\}$. Then, the
trivial solution of system (1) with $x_{0}\in U_{k,\varepsilon}$ first reaches
the boundary of $U_{k,\varepsilon}$ in finite time a.s..

_Proof:_ Let $\varpi(t)=-\int_{t_{0}}^{t}\mu^{*}(s)ds$, then $\varpi(t)\geq
c_{\mu^{*}}(t-t_{0})-d_{\mu^{*}}$ and
$\lim_{t\rightarrow\infty}\varpi(t)=+\infty$ by Remark 3. Moreover, if we let
$T_{0}=\max\\{T\geq t_{0}:\int_{t_{0}}^{T}\mu^{*}(t)dt=0\\}$, then
$\varpi(T_{0})=0$ with $T_{0}<\infty$ and $\varpi(t)>0$ as $t>T_{0}$.

Let $\varrho^{*}_{k,\varepsilon}=\inf\\{t\geq t_{0}:|x(t)|\notin
U_{k,\varepsilon}\\}$, then we need to prove that
$P\\{\varrho^{*}_{k,\varepsilon}<\infty\\}=1$, i.e.,
$P\\{\varrho^{*}_{k,\varepsilon}=\bar{\varrho}_{k,\varepsilon}I_{\\{\varrho^{*}_{k,\varepsilon}\leq
T_{0}\\}}+\varrho_{k,\varepsilon}I_{\\{\varrho^{*}_{k,\varepsilon}>T_{0}\\}}<\infty\\}=1$
with $\bar{\varrho}_{k,\varepsilon}=\inf\\{t_{0}\leq t\leq T_{0}:|x(t)|\notin
U_{k,\varepsilon}\\}$ and $\varrho_{k,\varepsilon}=\inf\\{t>T_{0}:|x(t)|\notin
U_{k,\varepsilon}\\}$.

Firstly, we prove the case that
$P\\{\varrho^{*}_{k,\varepsilon}=\varrho_{k,\varepsilon}<\infty\\}=1$. From
Dynkin's formula and (8), we can get that for any $x\in U_{k,\varepsilon}$,

$\displaystyle EV^{*}(\varrho_{k,\varepsilon}\wedge
t,x(\varrho_{k,\varepsilon}\wedge t))=$ $\displaystyle
EV^{*}(T_{0},x(T_{0}))+E\int_{T_{0}}^{\varrho_{k,\varepsilon}\wedge
t}\mathcal{L}V^{*}(s,x(s))ds$ (9) $\displaystyle\leq$ $\displaystyle
EV^{*}(T_{0},x(T_{0}))+E\int_{T_{0}}^{\varrho_{k,\varepsilon}\wedge
t}\mu^{*}(s)ds,$ (10)

where $t>T_{0}$. Define $\varpi_{1}(t)=-\int_{T_{0}}^{t}\mu^{*}(s)ds$, then
$\varpi_{1}(t)=\varpi(t)>0$ as $t>T_{0}$ and
$\lim_{t\rightarrow\infty}\varpi_{1}(t)=\lim_{t\rightarrow\infty}\varpi(t)=+\infty$.
Further, (9) can be turned into $EV^{*}(\varrho_{k,\varepsilon}\wedge
t,x(\varrho_{k,\varepsilon}\wedge t))\leq
EV^{*}(T_{0},x(T_{0}))-E\varpi_{1}(\varrho_{k,\varepsilon}\wedge t),$ which
means that

$\displaystyle E\varpi_{1}(\varrho_{k,\varepsilon}\wedge t)\leq
EV^{*}(T_{0},x(T_{0})),~{}~{}t>T_{0}.$ (11)

Note that for any $t>T_{0}$,

$\displaystyle\varpi_{1}(t)P\\{\varrho_{k,\varepsilon}\geq
t\\}=\int_{\\{\varrho_{k,\varepsilon}\geq t\\}}\varpi_{1}(t)dP\leq
E\varpi_{1}(\varrho_{k,\varepsilon}\wedge t).$ (12)

Substituting (12) into (11) arrives at

$P\\{\varrho_{k,\varepsilon}\geq
t\\}\leq\frac{1}{\varpi_{1}(t)}EV^{*}(T_{0},x(T_{0})),$ (13)

where $t>T_{0}$. In (13), let $t\rightarrow\infty$, then
$P\\{\varrho_{k,\varepsilon}<\infty\\}=1$, that is to say,
$P\\{\varrho^{*}_{k,\varepsilon}=\varrho_{k,\varepsilon}<\infty\\}=1$ can be
obtained.

Secondly, we prove another case that
$P\\{\varrho^{*}_{k,\varepsilon}=\bar{\varrho}_{k,\varepsilon}<\infty\\}=1$.
In fact, it is natural because
$P\\{\varrho^{*}_{k,\varepsilon}=\bar{\varrho}_{k,\varepsilon}\leq
T_{0}\\}=1$.

Hence, we have
$P\\{\varrho^{*}_{k,\varepsilon}=\bar{\varrho}_{k,\varepsilon}I_{\\{\varrho^{*}_{k,\varepsilon}\leq
T_{0}\\}}+\varrho_{k,\varepsilon}I_{\\{\varrho^{*}_{k,\varepsilon}>T_{0}\\}}<\infty\\}=1$,
i.e., $P\\{\varrho^{*}_{k,\varepsilon}<\infty\\}=1$. This means that the
solution of system (1) with $x_{0}\in U_{k,\varepsilon}$ firstly arrives at
the boundary of $U_{k,\varepsilon}$ in finite time almost surely.

###### Theorem 1

Suppose that there exists a solution to system (1). If there are functions
$\underline{\gamma}(\cdot)\in\mathcal{K}_{\infty}$,
$\bar{\gamma}(\cdot)\in\mathcal{K}_{\infty}$, $V(t,x)\in
C^{1,2}([t_{0},\infty)\times\mathcal{R}^{r};\mathcal{R}_{+})$ and a UASF
$\mu(t)$, and a constant $\kappa\in(0,1)$ such that

$\displaystyle\underline{\gamma}(|x|)\leq V(t,x)\leq\bar{\gamma}(|x|),$ (14)
$\displaystyle\mathcal{L}V(t,x))\leq\mu(t)[V(t,x)]^{\kappa},$ (15)

then the solution of system (1) is finite-time stable in probability.

_Proof:_ If system initial value $x_{0}=0$, then $x(t)\equiv 0$ is the
solution of system (1), where $f(0,t)\equiv 0$ and $g(0,t)\equiv 0$ are
considered. In the following, we only discuss the case that
$x(t_{0})=x_{0}\in\mathcal{R}^{r}\backslash\\{0\\}$.

Defining ${W}(V)=\int_{0}^{V}s^{-\kappa}ds$ with $V\in(0,+\infty)$ along with
system (1) and using Lemma 3.1 in [30], then

$\displaystyle\mathcal{L}{W}(V(t,x))=\frac{\mathcal{L}V}{V^{\kappa}}-\frac{\kappa}{2}\frac{\textmd{Tr}\\{(V_{x}g)^{T}V_{x}g\\}}{V^{\kappa+1}}\leq
V^{-\kappa}\mathcal{L}V\leq\mu(t),~{}~{}x\in\mathcal{R}^{r}\backslash\\{0\\}.$
(16)

Let $\rho_{k}=\varrho_{k,\frac{1}{k}}=\inf\\{t\geq t_{0}:|x(t)|\notin
U_{k,\frac{1}{k}}\\}$ with
$U_{k,\frac{1}{k}}=\\{x\in\mathcal{R}^{r}:\frac{1}{k}<|x|<k\\}$ and
$k\in\\{2,3,\cdots\\}$, then by (16), we must have

$\displaystyle\mathcal{L}{W}(V(t,x))\leq\mu(t),~{}~{}x\in U_{k,\frac{1}{k}}.$
(17)

By Dynkin's formula, it follows from (17) that

$\displaystyle-{W}(V(t_{0},x_{0}))\leq
E{W}(V(t\wedge\rho_{k},x(t\wedge\rho_{k})))-{W}(V(t_{0},x_{0}))\leq
E\int_{t_{0}}^{t\wedge\rho_{k}}\mu(s)ds.$ (18)

Because $\mu(t)$ is a UASF, there exist constants $c_{\mu}>0$ and $d_{\mu}\geq
0$ such that $\int_{t_{0}}^{t}\mu(s)ds\leq d_{\mu}-c_{\mu}(t-t_{0})$. Hence,
(18) can be turned into

$\displaystyle-\frac{1}{1-\kappa}V^{1-\kappa}(t_{0},x_{0})=$
$\displaystyle-\int_{0}^{V(t_{0},x_{0})}s^{-\kappa}ds\leq-{c_{\mu}}E(t\wedge\rho_{k})+{c_{\mu}}t_{0}+{d_{\mu}},$

that is to say,

$\displaystyle E(t\wedge\rho_{k})\leq
t_{0}+\frac{d_{\mu}}{c_{\mu}}+\frac{\bar{\gamma}^{1-\kappa}(|x_{0}|)}{c_{\mu}(1-\kappa)},$
(19)

where (14) is considered.

Let ${W}=V^{*}$ and $\mu(t)=\mu^{*}(t)$ in (16), then we have
$\rho_{k}\rightarrow\rho_{\infty}=\rho_{x_{0}}$ a.s. as $k\rightarrow\infty$
and $P\\{\rho_{x_{0}}<\infty\\}=1$ by Lemma 4 and Definition 1. Further, let
$t=k$ and $k\rightarrow\infty$, then $k\wedge\rho_{k}\rightarrow\rho_{x_{0}}$
a.s.. Accordingly, (19) can be changed into

$E(\rho_{x_{0}})\leq
t_{0}+\frac{d_{\mu}}{c_{\mu}}+\frac{\bar{\gamma}^{1-\kappa}(|x_{0}|)}{c_{\mu}(1-\kappa)}.$
(20)

This implies that the trivial solution of system (1) is finite-time attractive
in probability.

Applying Dynkin's formula for (16), together with (14), leads to

$\displaystyle
E[\underline{\gamma}^{1-\kappa}(|x(t)|)I_{\\{t\in[t_{0},\rho_{x_{0}})\\}}]\leq
E[V^{1-\kappa}(t,x(t))I_{\\{t\in[t_{0},\rho_{x_{0}})\\}}]\leq\bar{\gamma}^{1-\kappa}(|x_{0}|)+d_{\mu}(1-\kappa).$
(21)

Meanwhile,

$E[\underline{\gamma}^{1-\kappa}(|x(t)|)I_{\\{t\in[\rho_{x_{0}},\infty)\\}}]\equiv
0.$ (22)

Hence, we have

$\displaystyle E[\underline{\gamma}^{1-\kappa}(|x(t)|)]=$ $\displaystyle
E[\underline{\gamma}^{1-\kappa}(|x(t)|)I_{\\{t\in[t_{0},\rho_{x_{0}})\\}}+\underline{\gamma}^{1-\kappa}(|x(t)|)I_{\\{t\in[\rho_{x_{0}},\infty)\\}}]$
(23) $\displaystyle=$ $\displaystyle
E[\underline{\gamma}^{1-\kappa}(|x(t)|)I_{\\{t\in[t_{0},\rho_{x_{0}})\\}}]+E[\underline{\gamma}^{1-\kappa}(|x(t)|)I_{\\{t\in[\rho_{x_{0}},\infty)\\}}]$
(24) $\displaystyle\leq$
$\displaystyle\bar{\gamma}^{1-\kappa}(|x_{0}|)+d_{\mu}(1-\kappa),~{}~{}t\geq
t_{0}$ (25)

from (21) and (22).

For any $\varepsilon\in(0,1)$, let
$\bar{R}>\varepsilon^{-1}\bar{\gamma}^{1-\kappa}(|x_{0}|)+\varepsilon^{-1}d_{\mu}(1-\kappa)$,
then from (23) and Chebyshev's inequality, we can get that

$\displaystyle
P\\{\underline{\gamma}^{1-\kappa}(|x(t)|)\geq\bar{R},~{}~{}t\geq t_{0}\\}\leq
E[\underline{\gamma}^{1-\kappa}(|x(t)|)]\bar{R}^{-1}<\varepsilon,$

which means that

$P\\{|x(t)|<R,~{}~{}t\geq t_{0}\\}\geq 1-\varepsilon,$ (26)

where $R=\underline{\gamma}^{-1}(\bar{R}^{\frac{1}{1-\kappa}})$,
$|x_{0}|<\delta(\varepsilon,R)$ and
$\delta(\varepsilon,R)=\bar{\gamma}^{-1}([\varepsilon\underline{\gamma}^{1-\kappa}(R)-d_{\mu}(1-\kappa)]^{\frac{1}{1-\kappa}})$.
This signifies that the trivial solution of system (1) is stable in
probability.

###### Theorem 2

Suppose that there is a unique solution to system (1). If there are functions
$\underline{\gamma}\in\mathcal{K}_{\infty}$,
$\bar{\gamma}\in\mathcal{K}_{\infty}$, a UASF $\mu(t)$ and $V(t,x)\in
C^{1,2}([t_{0},\infty)\times\mathcal{R}^{r};\mathcal{R}_{+})$ such that

$\displaystyle\underline{\gamma}(|x|)\leq V(t,x)\leq\bar{\gamma}(|x|),$ (27)
$\displaystyle\mathcal{L}V(t,x)\leq\mu(t),$ (28)

then the trivial solution of system (1) is finite-time stable in probability.

_Proof:_ Similar to Theorem 1, we also only consider the case that
$x(t_{0})=x_{0}\in\mathcal{R}^{r}\backslash\\{0\\}$ in the following.

Applying Dynkin's formula for (28) and the property of the UASF $\mu(t)$, we
have

$\displaystyle
EV(t\wedge\sigma_{n}\wedge\rho_{x_{0}},x(t\wedge\sigma_{n}\wedge\rho_{x_{0}}))\leq
V(t_{0},x_{0})-c_{\mu}E(t\wedge\sigma_{n}\wedge\rho_{x_{0}}-t_{0})+d_{\mu},$
(29)

where $\sigma_{n}=\inf\\{t\geq t_{0}:|x(t)|>n\\}$ and $\rho_{x_{0}}$
represents the stochastic settling time. Note that
$\sigma_{n}\rightarrow\infty$ as $n\rightarrow\infty$. So by Fatou's lemma,
(27) and (29), we have

$\displaystyle c_{\mu}E(t\wedge\rho_{x_{0}})\leq
EV(t\wedge\rho_{x_{0}},x(t\wedge\rho_{x_{0}}))+c_{\mu}E(t\wedge\rho_{x_{0}})\leq\bar{\gamma}(|x_{0}|)+c_{\mu}t_{0}+d_{\mu},$

which indicates that

$\displaystyle
E(t\wedge\rho_{x_{0}})\leq\frac{\bar{\gamma}(|x_{0}|)}{c_{\mu}}+\frac{d_{\mu}}{c_{\mu}}+t_{0}.$
(30)

Applying Fatou's lemma for (30), we have

$\displaystyle
E(\rho_{x_{0}})\leq\frac{\bar{\gamma}(|x_{0}|)}{c_{\mu}}+\frac{d_{\mu}}{c_{\mu}}+t_{0},$
(31)

which signifies that $P\\{\rho_{x_{0}}<\infty\\}=1$.

The proof of stability in probability is similar to that of Theorem 1 and is
thus omitted here.

###### Remark 4

(i) Theorem 2 signifies that Theorem 1 also holds for $\kappa=0$, that is to
say, Theorem 1 holds for any $\kappa\in[0,1)$. In the proof of Theorem 1, (20)
also holds for $\kappa\in[0,1)$ since (31) is consistent with (20) with
$\kappa=0$. (ii) If $\mu(t)=-c(c>0)$ that is a UASF, then Theorem 1 and
Theorem 2 are in line with Theorem 3.1 and Theorem 3.2 in [17], respectively.
Note that Theorem 3.1 in [17] does not hold for $\kappa=1$ according to
Example 3.3 in [17]. Therefore, Theorem 1 also does not hold for $\kappa=1$.

###### Remark 5

In some existing results, $\mathcal{L}V$ must be negative definite such as
[17, 23] or negative semi-definite such as [18]. Theorem 1 and Theorem 2
generalize the existing results and relax some constraints on $\mathcal{L}V$
such that $\mathcal{L}V$ can be negative or positive.

Example 1. For stochastic nonlinear system

$\displaystyle
dx(t)=0.5\mu_{1}(t)x^{\frac{1}{3}}(t)dt-0.5x(t)dt+x(t)\cos(x(t))dW(t),$ (32)

where $\mu_{1}(t)=2/(1+t)-|\sin 2t|$ which is a UASF by [31] and
$W(t)\in\mathcal{R}$ is a standard Wiener process. Note that

$\displaystyle|f(t,x)|^{2}+|g(t,x)|^{2}=$
$\displaystyle|0.5\mu_{1}(t)x^{\frac{1}{3}}(t)-0.5x(t)|^{2}+|x(t)\cos(x(t))|^{2}$
(33) $\displaystyle\leq$ $\displaystyle
0.25\mu_{1}^{2}(t)x^{\frac{2}{3}}(t)+1.25|x(t)|^{2}+0.5|\mu_{1}(t)|x^{\frac{4}{3}}(t)$
(34) $\displaystyle\leq$ $\displaystyle
x^{\frac{2}{3}}(t)+\frac{5}{4}|x(t)|^{2}+x^{\frac{4}{3}}(t)$ (35)
$\displaystyle\leq$
$\displaystyle\frac{1}{3}|x(t)|^{2}+\frac{2}{3}+\frac{5}{4}|x(t)|^{2}+\frac{2}{3}|x(t)|^{2}+\frac{1}{3}$
(36) $\displaystyle\leq$ $\displaystyle H(|x(t)|^{2}+1),$ (37)

where $H=9/4$, Lemma 3 in [24] and $\mu_{1}(t)=2/(1+t)-|\sin 2t|\leq 2$ are
used. Hence, system (32) has a continuous solution by Lemma 2. Let
$V_{1}(x)=x^{2}$, then we have

$\displaystyle\mathcal{L}V_{1}(x(t))=$ $\displaystyle
2x(t)(0.5\mu_{1}(t)x^{\frac{1}{3}}(t)-0.5x(t))+x^{2}(t)\cos^{2}(x(t))$ (38)
$\displaystyle\leq$ $\displaystyle\mu_{1}(t)x^{\frac{4}{3}}(t)$ (39)
$\displaystyle=$ $\displaystyle\mu_{1}(t)V_{1}^{\frac{2}{3}}(x(t)).$ (40)

This implies that the trivial solution of system (32) is finite-time stable in
probability by Theorem 1.

Figure 1. The state curve of system (32).

In simulation, let $x_{0}=0.6$, then the state curve of system (32) is shown
in Figure 1, which shows that system state trajectory beginning from non-zero
initial value converges to the origin in finite time.

Example 2. For stochastic nonlinear system

$\begin{cases}dx_{1}(t)=f_{1}(t,x)dt+g_{1}(t,x)dW_{1}(t),\\\
dx_{2}(t)=f_{2}(t,x)dt+g_{2}(t,x)dW_{2}(t),\end{cases}$ (41)

where $f_{1}(t,x)=-x_{1}(t)+(\psi(t)-0.5)x_{1}^{\frac{4}{5}}(t)$,
$g_{1}(t,x)=\sqrt{2}x_{2}(t)\cos(x_{1}(t))$,
$f_{2}(t,x)=-x_{2}(t)+(\psi(t)-0.5)x_{2}^{\frac{4}{5}}(t)$,
$g_{2}(t,x)=\sqrt{2}x_{1}(t)\sin(x_{2}(t))$, $\psi(t)=t\sin t/(1+t)$,
$W_{i}(t)\in\mathcal{R}(i=1,2)$ are mutually independent standard Wiener
processes. System (41) satisfies the condition in Lemma 2, so there is a
continuous solution to system (41).

Figure 2. The trajectory of state $x(t)$.

Let $V_{2}(x)=x_{1}^{2}+x_{2}^{2}$, then

$\displaystyle\mathcal{L}V_{2}(x)\leq$ $\displaystyle
2(\psi(t)-0.5)(x_{1}^{\frac{9}{5}}+x_{2}^{\frac{9}{5}})=2\bar{\mu}_{2}(t)[(x_{1}^{2})^{\frac{9}{10}}+(x_{2}^{2})^{\frac{9}{10}}]\leq\mu_{2}(t)V_{2}^{\frac{9}{10}}(x),$

where Lemma 4 in [24] is used, $\bar{\mu}_{2}(t)=\psi(t)-0.5$ and

${\mu}_{2}(t)=\begin{cases}2^{\frac{11}{10}}\bar{\mu}_{2}(t)~{}~{}&\textmd{if}~{}~{}\psi(t)\geq
0.5\\\ 2\bar{\mu}_{2}(t)~{}~{}&\textmd{if}~{}~{}\psi(t)<0.5\\\ \end{cases}.$

Since $\bar{\mu}_{2}(t)=\psi(t)-0.5$ is a UASF and
$\int_{t_{0}}^{t}\bar{\mu}_{2}(s)ds\leq 5-0.5(t-t_{0})$, it is easy to verify
that ${\mu}_{2}(t)$ is also a UASF. Therefore, the solution of system (41)
with $x_{0}\in\mathcal{R}^{2}\backslash\\{0\\}$ is finite-time stable in
probability by Theorem 1 with $\kappa=9/10$. For simulation, let
$x_{0}=(0.4,-0.6)^{T}$, then the trajectories of the state $x_{1}(t)$ and
$x_{2}(t)$ are described in Figure 2. This signifies that the trivial solution
of system (41) is finite-time stable in probability.

###### Remark 6

(i) Since system (32) and (41) satisfy the condition of Lemma 2, system (32)
and (41) have a continuous solution, respectively. We can also prove that
Example 1 and Example 2 satisfy Lemma 3. For example, we know that $f(x,t)$
and $g(x,t)$ in Example 1 are locally bounded in $x$ and are uniformly bounded
in $t$ by (33). Furthermore, let $U_{1}(x)=V_{1}(x)=x^{2}=|x|^{2}$ in Example
1, then $\mathcal{L}U_{1}(x(t))=\mu_{1}(t)U_{1}^{\frac{2}{3}}(x(t))\leq
2U_{1}^{\frac{2}{3}}(x(t))\leq{4}/{3}U_{1}+2/3\triangleq l(t)U_{1}+2/3$. (ii)
Example 1 and Example 2 show that the finite-time stability of stochastic
nonlinear systems can be analyzed by Theorem 1. (iii) In order to further show
the validity of Theorem 1, we focus on finite-time stabilization problem for a
stochastic nonlinear system, which is given in Section V.

##  IV. Finite-time instability theorem

In this section, we discuss some sufficient conditions which ensure that
stochastic nonlinear system (1) is finite-time instable in probability.
Compared with some existing stochastic finite-time instability results (such
as [17] and [19]), the given finite-time instability criterion relaxes the
condition imposed on $\mathcal{L}V$ and allows $\mathcal{L}V$ to be
indefinite.

###### Theorem 3

Assuming that there exists a solution to system (1) with any given non-zero
initial value. If there exist a function $V(t,x)\in
C^{1,2}([t_{0}-\tau,\infty)\times\mathcal{R}^{r};\mathcal{R}_{+})$,
$\mathcal{K}_{\infty}$ functions $\underline{\gamma}$ and $\bar{\gamma}$, and
a UASF $\mu(t)$ such that

$\displaystyle\underline{\gamma}(|x|)\leq V(t,x)\leq\bar{\gamma}(|x|),$ (42)
$\displaystyle\mathcal{L}V(t,x)=\mu(t)V(t,x),$ (43)
$\displaystyle|V_{x}(t,x)g(t,x)|^{2}\leq a(t)V^{2}(t,x),$ (44)

where $a(t)\geq 0$ and $\int_{t_{0}}^{\infty}a(t)dt<\infty$, then the trivial
solution of system (1) is finite-time instable in probability.

_Proof:_ Firstly, let $\tilde{V}(t,x)=e^{-\int_{t_{0}}^{t}\mu(s)ds}V(t,x)$
along system (1) and apply Itô's formula for $\tilde{V}$, then

$\displaystyle
d\tilde{V}=\mathcal{L}\tilde{V}dt+\tilde{V}_{x}^{T}g(t,x(t))dW(t),$ (45)

where $\tilde{V}_{x}=e^{-\int_{t_{0}}^{t}\mu(s)ds}V_{x}(t,x)$ and
$\mathcal{L}\tilde{V}=e^{-\int_{t_{0}}^{t}\mu(s)\,ds}(\mathcal{L}V(t,x)-\mu(t)V(t,x))=0.$
It follows from (45) that

$\displaystyle
e^{-\int_{t_{0}}^{t}\mu(s)ds}V(t,x(t))-V(t_{0},x_{0})=\int_{t_{0}}^{t}e^{-\int_{t_{0}}^{s}\mu(v)dv}V_{x}^{T}gdW(s).$
(46)

From (44), (46) and Burkholder-Davis-Gundy inequality (Theorem 7.3 in [Chapter
1, 3]), we can obtain that

$\displaystyle E\Big{[}\sup_{t_{0}\leq t\leq
T}\Big{(}e^{-\int_{t_{0}}^{t}\mu(s)ds}V(t,x(t))\Big{)}^{2}\Big{]}$ (47)
$\displaystyle\leq$ $\displaystyle
8E\int_{t_{0}}^{T}a(t)\Big{(}e^{-\int_{t_{0}}^{t}\mu(v)dv}V(t,x(t))\Big{)}^{2}dt+2V^{2}(t_{0},x_{0})$
(48) $\displaystyle\leq$ $\displaystyle
8\int_{t_{0}}^{T}a(t)E\Big{[}\sup_{t_{0}\leq s\leq
T}\Big{(}e^{-\int_{t_{0}}^{s}\mu(v)dv}V(s,x(s))\Big{)}^{2}\Big{]}dt+2V^{2}(t_{0},x_{0}).$
(49)

Further, by Gronwall's inequality (Theorem 8.1 in [Chapter 1, 3]), it can be
derived from (47) that

$\displaystyle E\Big{[}\sup_{t_{0}\leq t\leq
T}\Big{(}e^{-\int_{t_{0}}^{t}\mu(s)ds}V(t,x(t))\Big{)}^{2}\Big{]}\leq
2e^{\int_{t_{0}}^{\infty}8a(t)dt}V^{2}(t_{0},x_{0})\leq
2e^{\int_{t_{0}}^{\infty}8a(t)dt}\bar{\gamma}^{2}(|x_{0}|)\triangleq H_{0}.$
(50)

Let $T\rightarrow\infty$, then by Fatou's lemma, we have

$E\Big{[}\sup_{t\in[t_{0},\infty)}\Big{(}e^{-\int_{t_{0}}^{t}\mu(s)ds}V(t,x(t))\Big{)}^{2}\Big{]}\leq
H_{0}<\infty,$

which means that

$\displaystyle
E\Big{[}\sup_{t\in[t_{0},\infty)}\Big{(}e^{-\int_{t_{0}}^{t}\mu(s)ds}V(t,x(t))\Big{)}\Big{]}<\infty.$
(51)

For any given $x_{0}\in\mathcal{R}^{r}\backslash\\{0\\}$, let
$\sigma_{n}=\inf\\{t\geq t_{0}:|x(t)|>n\\}$. It follows from (46) that

$\displaystyle
E\Big{\\{}\Big{[}e^{-\int_{t_{0}}^{t}\mu(s)ds}V(t,x(t))\Big{]}\Big{|}_{t=\sigma_{n}\wedge\sigma_{b}}\Big{\\}}=V(t_{0},x(t_{0})),$
(52)

where $\sigma_{b}$ represents any bounded stopping time. Moreover,

$\displaystyle 0\leq
e^{-\int_{t_{0}}^{\sigma_{n}\wedge\sigma_{b}}\mu(s)ds}V(\sigma_{n}\wedge\sigma_{b},x(\sigma_{n}\wedge\sigma_{b}))\leq\sup_{t\in[t_{0},\infty)}\Big{(}e^{-\int_{t_{0}}^{t}\mu(s)ds}V(t,x(t))\Big{)}.$
(53)

From (51) and (53), apply Lebesgue's dominated convergence theorem for (52),
then we have

$\displaystyle
E\Big{[}e^{-\int_{t_{0}}^{\sigma_{b}}\mu(s)ds}V(\sigma_{b},x(\sigma_{b}))\Big{]}=V(t_{0},x(t_{0})),$
(54)

where $\sigma_{n}\rightarrow\infty$ as $n\rightarrow\infty$ is used. This
signifies that $e^{-\int_{t_{0}}^{t}\mu(s)ds}V(t,x(t))$ is a uniformly
integrable martingale. Hence, by martingale convergence theorem (Theorem 7.11
in [32]), we have

$\displaystyle
0\leq\lim_{t\rightarrow\infty}e^{-\int_{t_{0}}^{t}\mu(s)ds}V(t,x(t))=\vartheta(\omega)<\infty,~{}~{}~{}a.s.,$
(55)

where $\vartheta(\omega)$ is a random variable.

Secondly, the conditions (42) and (43) ensure that there is a stable solution
to system (1) in probability. In fact, for every given
$x_{0}\in\mathcal{R}^{r}\backslash\\{0\\}$, let $T^{*}=\max\\{T\geq
t_{0}:\int_{t_{0}}^{T}\mu(t)dt=0\\}$. If $t\in[t_{0},T^{*}]$, from (42), (46)
and Remark 3, we obtain that

$\displaystyle E[\underline{\gamma}(|x(t)|)]\leq
EV(t,x(t))\leq\bar{\gamma}(|x_{0}|)e^{d_{\mu}}.$ (56)

If $t\in(T^{*},\infty)$, by (46), we have

$\displaystyle E[\underline{\gamma}(|x(t)|)]\leq$ $\displaystyle
EV(t,x(t))=V(t_{0},x_{0})e^{\int_{t_{0}}^{t}\mu(s)ds}\leq\bar{\gamma}(|x_{0}|),$
(57)

where
$\int_{t_{0}}^{t}\mu(s)ds=\int_{t_{0}}^{T^{*}}\mu(s)ds+\int_{T^{*}}^{t}\mu(s)ds=\int_{T^{*}}^{t}\mu(s)ds<0$
is used.

Hence, for any $t\geq t_{0}$, we have

$\displaystyle
E[\underline{\gamma}(|x(t)|)]\leq\bar{\gamma}(|x_{0}|)e^{d_{\mu}}.$

From Chebyshev's inequality, for any $\varepsilon\in(0,1)$ and $R>0$, we have

$\displaystyle P\\{|x(t)|\geq R,t\geq
t_{0}\\}=P\\{\underline{\gamma}(|x(t)|)\geq\underline{\gamma}(R),t\geq
t_{0}\\}\leq\underline{\gamma}^{-1}(R)E[\underline{\gamma}(|x(t)|)]\leq\underline{\gamma}^{-1}(R)\bar{\gamma}(|x_{0}|)e^{d_{\mu}}\leq\varepsilon$
(58)

as
$|x_{0}|<\tilde{\delta}(\varepsilon,R)=\bar{\gamma}^{-1}(\varepsilon\underline{\gamma}(R)e^{-d_{\mu}})$.
This implies that $P\\{|x(t)|<R,t\geq t_{0}\\}\geq 1-\varepsilon$ as
$|x_{0}|<\tilde{\delta}(\varepsilon,R)$.

Finally, consider (55), we can prove that it is not finite-time attractive in
probability for the solution of system (1) by the same method as Theorem 4.1
in [17].

###### Remark 7

In Theorem 4.1 of [17], $\mathcal{L}V(t,x)=-c_{3}V(t,x)$ with $c_{3}>0$ is
used. It is clear that $\mu(t)=-c_{3}(c_{3}>0)$ is a UASF. In Theorem 3, the
constraint condition of $\mathcal{L}V(t,x)=-c_{3}V(t,x)$ is replaced by
$\mathcal{L}V(t,x)=\mu(t)V(t,x)$. Hence, Theorem 3 improves Theorem 4.1 of
[17].

##  V. A simulation example

Example 3. Consider the stochastic nonlinear system with stochastic inverse
dynamics $(\Sigma)$

$\begin{cases}d\chi(t)=\varphi(t)\chi^{\beta_{1}}(t)dt+\cos(x_{1}(t))\chi^{\beta_{2}}(t)dB_{0}(t),\\\
dx_{1}(t)=x_{2}(t)dt\\\
dx_{2}(t)=u(t)dt+x_{2}^{\beta_{3}}(t)\sin(\chi(t))dB(t),\end{cases}$ (59)

where $\beta_{1}=(2l-1)/(2l+1)\in(0.5,1)$, $\beta_{2}=2l/(2l+1)\in(0.5,1)$ and
$\beta_{3}=(2l-2)/(2l-1)\in(0.5,1)$ with $l\in\mathcal{N}$ and $l\geq 2$.
$B_{0}(t)\in\mathcal{R}$ and $B(t)\in\mathcal{R}$ are mutually independent
standard Brownian motions(Wiener processes). $\varphi(t)=0.5({t\cos
t}/{(1+t)}-1.5)$.

For $\chi$-subsystem, we select $V_{0}=\chi^{2}$, then
$\mathcal{L}V_{0}=2\varphi(t)\chi^{\beta_{1}+1}+\chi^{2\beta_{2}}\cos^{2}(x_{1})\leq(2\varphi(t)+1)\chi^{2\beta_{2}}=\mu_{0}(t)V_{0}^{\beta_{2}},$
where $\beta_{1}+1=2\beta_{2}$ and $\mu_{0}(t)=2\varphi(t)+1$ is a UASF since
$\int_{t_{0}}^{t}\mu_{0}(s)ds\leq-0.5(t-t_{0})+5$.

For system $\Sigma$, we introduce $V_{1}=V_{0}+W_{0}$ with
$W_{0}=0.5z_{1}^{2}$ and $z_{1}=x_{1}$, then

$\displaystyle\mathcal{L}V_{1}=$
$\displaystyle\mathcal{L}V_{0}+\mathcal{L}W_{0}=\mathcal{L}V_{0}+z_{1}x_{2}\leq\mu_{0}(t)V_{0}^{\beta_{2}}+z_{1}(x_{2}-\alpha)+z_{1}\alpha.$
(60)

Let the stabilizing function $\alpha=-c_{1}z_{1}^{\lambda}$ with
$\lambda=\beta_{1}$ and $c_{1}>0$ being a design constant, then (60) can be
changed into

$\displaystyle\mathcal{L}V_{1}\leq\mu_{0}(t)V_{0}^{\beta_{2}}+z_{1}(x_{2}-\alpha)-c_{1}z_{1}^{1+\lambda}.$
(61)

Further, we introduce
$V_{2}=V_{1}+W_{1},W_{1}=\int_{\alpha}^{x_{2}}(v^{\frac{1}{\lambda}}-\alpha^{\frac{1}{\lambda}})^{2-\lambda}dv,$
and $z_{2}=x_{2}^{\frac{1}{\lambda}}-\alpha^{\frac{1}{\lambda}}$, then it
follows from Proposition B.1-B.2 in [33] that $V_{2}$ is a positive definite
function and

$\displaystyle W_{0}+W_{1}\leq 2(z_{1}^{2}+z_{2}^{2}).$ (62)

Meanwhile, we can deduce that

$\displaystyle\mathcal{L}V_{2}=\mathcal{L}V_{1}+\mathcal{L}W_{1}=\mathcal{L}V_{1}+\frac{\partial
W_{1}}{\partial x_{1}}x_{2}+\frac{\partial W_{1}}{\partial
x_{2}}u+\frac{1}{2}\frac{\partial^{2}W_{1}}{\partial
x_{2}^{2}}x_{2}^{2\beta_{3}}\sin^{2}(\chi).$ (63)

Note that

$\displaystyle\frac{\partial W_{1}}{\partial x_{1}}x_{2}=$
$\displaystyle-(2-\lambda)\frac{\partial\alpha^{\frac{1}{\lambda}}}{\partial
x_{1}}x_{2}\int_{\alpha}^{x_{2}}(v^{\frac{1}{\lambda}}-\alpha^{\frac{1}{\lambda}})^{1-\lambda}dv$
(64) $\displaystyle=$ $\displaystyle
c_{1}^{\frac{1}{\lambda}}(2-\lambda)x_{2}\int_{\alpha}^{x_{2}}(v^{\frac{1}{\lambda}}-\alpha^{\frac{1}{\lambda}})^{1-\lambda}dv$
(65) $\displaystyle\leq$ $\displaystyle
c_{1}^{\frac{1}{\lambda}}(2-\lambda)|z_{2}|^{1-\lambda}|x_{2}-\alpha||x_{2}|$
(66) $\displaystyle\leq$ $\displaystyle
c_{1}^{\frac{1}{\lambda}}(2-\lambda)2^{1-\lambda}|z_{2}|^{1-\lambda}|z_{2}|^{\lambda}|x_{2}|$
(67) $\displaystyle\leq$ $\displaystyle
c_{1}^{\frac{1}{\lambda}}(2-\lambda)2^{1-\lambda}|z_{2}|(|x_{2}-\alpha|+|\alpha|)$
(68) $\displaystyle\leq$ $\displaystyle
c_{1}^{\frac{1}{\lambda}}(2-\lambda)2^{1-\lambda}(2^{1-\lambda}z_{2}^{1+\lambda}+c_{1}|z_{2}||z_{1}|^{\lambda})$
(69) $\displaystyle\leq$ $\displaystyle
c_{1}^{\frac{1}{\lambda}}(2-\lambda)2^{1-\lambda}\Big{(}2^{1-\lambda}z_{2}^{1+\lambda}+\frac{c_{1}h_{1}^{-\lambda}}{1+\lambda}z_{2}^{1+\lambda}+\frac{c_{1}h_{1}\lambda}{1+\lambda}z_{1}^{1+\lambda}\Big{)}$
(70) $\displaystyle=$ $\displaystyle
c_{1}^{\frac{1}{\lambda}}(2-\lambda)[2^{2(1-\lambda)}z_{2}^{1+\lambda}+\frac{2^{1-\lambda}c_{1}h_{1}\lambda}{1+\lambda}z_{1}^{1+\lambda}]+\frac{2^{1-\lambda}c_{1}^{1+\frac{1}{\lambda}}h_{1}^{-\lambda}}{1+\lambda}(2-\lambda)z_{2}^{1+\lambda},$
(71) $\displaystyle\frac{1}{2}\frac{\partial^{2}W_{1}}{\partial
x_{2}^{2}}x_{2}^{2\beta_{3}}\sin^{2}(\chi)=$
$\displaystyle\frac{2-\lambda}{2}z_{2}^{1-\lambda}\frac{1}{\lambda}x_{2}^{\frac{1}{\lambda}-1}x_{2}^{2\beta_{3}}\sin^{2}(\chi)$
(72) $\displaystyle\leq$
$\displaystyle\frac{2-\lambda}{2\lambda}z_{2}^{1-\lambda}x_{2}^{2}$ (73)
$\displaystyle=$
$\displaystyle\frac{2-\lambda}{2\lambda}z_{2}^{1-\lambda}(x_{2}^{\frac{1}{\lambda}})^{2\lambda}$
(74) $\displaystyle\leq$
$\displaystyle\frac{2-\lambda}{2\lambda}z_{2}^{1-\lambda}2^{2\lambda-1}(z_{2}^{2\lambda}+\alpha^{2})$
(75) $\displaystyle\leq$
$\displaystyle\frac{2-\lambda}{\lambda}z_{2}^{1-\lambda}(z_{2}^{2\lambda}+\alpha^{2})$
(76) $\displaystyle=$
$\displaystyle\frac{2-\lambda}{\lambda}z_{2}^{1+\lambda}+\frac{2-\lambda}{\lambda}c_{1}^{2}z_{2}^{1-\lambda}z_{1}^{2\lambda}$
(77) $\displaystyle\leq$
$\displaystyle\frac{2-\lambda}{\lambda}z_{2}^{1+\lambda}+\frac{2-\lambda}{\lambda}c_{1}^{2}\Big{(}\frac{2\lambda
h_{2}}{1+\lambda}z_{1}^{1+\lambda}+\frac{1-\lambda}{1+\lambda}h_{2}^{-\frac{2\lambda}{1-\lambda}}z_{2}^{1+\lambda}\Big{)}$
(78) $\displaystyle=$
$\displaystyle\frac{2-\lambda}{\lambda}z_{2}^{1+\lambda}+\frac{2-\lambda}{\lambda}\frac{1-\lambda}{1+\lambda}c_{1}^{2}h_{2}^{-\frac{2\lambda}{1-\lambda}}z_{2}^{1+\lambda}+\frac{2(2-\lambda)}{1+\lambda}c_{1}^{2}h_{2}z_{1}^{1+\lambda},$
(79) $\displaystyle\frac{\partial W_{1}}{\partial x_{2}}u=z_{2}^{2-\lambda}u,$
(80)

where $1-\lambda=2/(2l+1)$ and $1+\lambda=4l/(2l+1)$, $h_{1}>0$ and $h_{2}>0$
are to be designed constants and Lemma 4 in [24] is used.

Substituting (61), (64)-(80) into (63), we can obtain that

$\displaystyle\mathcal{L}V_{2}\leq\mu_{0}(t)V_{0}^{\beta_{2}}+z_{1}(x_{2}-\alpha)-c_{1}z_{1}^{1+\lambda}+\tilde{d}_{2}z_{2}^{1+\lambda}+\tilde{d}_{1}z_{1}^{1+\lambda}+z_{2}^{2-\lambda}u,$
(81)

where
$\tilde{d}_{2}=(2-\lambda){2^{1-\lambda}c_{1}^{1+\frac{1}{\lambda}}h_{1}^{-\lambda}}/{(1+\lambda)}+(2-\lambda)c_{1}^{\frac{1}{\lambda}}2^{2(1-\lambda)}+{(1-\lambda)(2-\lambda)c_{1}^{2}h_{2}^{-\frac{2\lambda}{1-\lambda}}}/{\lambda(1+\lambda)}+({2-\lambda})/{\lambda}$,
$\tilde{d}_{1}={2^{1-\lambda}(2-\lambda)c_{1}^{1+\frac{1}{\lambda}}h_{1}\lambda}/{(1+\lambda)}+{2(2-\lambda)c_{1}^{2}h_{2}}/{(1+\lambda)}$.
In addition,

$\displaystyle
z_{1}(x_{2}-\alpha)\leq|z_{1}||(x_{2}^{\frac{1}{\lambda}})^{\lambda}-(\alpha^{\frac{1}{\lambda}})^{\lambda}|\leq
2^{1-\lambda}|z_{1}||z_{2}|^{\lambda}\leq\frac{2^{1-\lambda}}{1+\lambda}[h_{3}z_{1}^{1+\lambda}+\lambda
h_{3}^{-\frac{1}{\lambda}}z_{2}^{1+\lambda}],$ (82)

where $h_{3}>0$ is a constant that will be designed, and Lemma 2.3 in [34] is
applied. Hence, (81) and (82) mean that

$\displaystyle\mathcal{L}V_{2}\leq\mu_{0}(t)V_{0}^{\beta_{2}}-c_{1}z_{1}^{1+\lambda}+{d}_{2}z_{2}^{1+\lambda}+{d}_{1}z_{1}^{1+\lambda}+z_{2}^{2-\lambda}u,$
(83)

where $d_{1}=\tilde{d}_{1}+{2^{1-\lambda}h_{3}}/{(1+\lambda)}$ and
$d_{2}=\tilde{d}_{2}+{2^{1-\lambda}\lambda
h_{3}^{-\frac{1}{\lambda}}}/{(1+\lambda)}$.

Let
$h_{1}=(1+\lambda)2^{\lambda-1}c_{1}^{-\frac{1}{\lambda}}/6\lambda(2-\lambda)$,
$h_{2}=(1+\lambda)c_{1}^{-1}/12(2-\lambda)$ and
$h_{3}=2^{\lambda}c_{1}(1+\lambda)/12$, then $d_{1}=c_{1}/2$ and (83) can be
rewritten as

$\displaystyle\mathcal{L}V_{2}\leq\mu_{0}(t)V_{0}^{\beta_{2}}-0.5{c_{1}}z_{1}^{1+\lambda}+{d}_{2}z_{2}^{1+\lambda}+z_{2}^{2-\lambda}u.$
(84)

Further, we choose the controller

$\displaystyle u=-{d}_{2}z_{2}^{2\lambda-1}-0.5{c_{2}}z_{2}^{2\lambda-1},$
(85)

where $c_{2}>0$ is a design constant. Then, together with
$1+\lambda=1+\beta_{1}=2\beta_{2}$, (84) and (85) lead to

$\displaystyle\mathcal{L}V_{2}\leq$
$\displaystyle\mu_{0}(t)V_{0}^{\beta_{2}}-0.5{c_{1}}(z_{1}^{2})^{\beta_{2}}-0.5{c_{2}}(z_{2}^{2})^{\beta_{2}}$
(86) $\displaystyle\leq$
$\displaystyle\mu_{0}(t)V_{0}^{\beta_{2}}-0.5{c_{0}}((z_{1}^{2})^{\beta_{2}}+(z_{2}^{2})^{\beta_{2}})$
(87) $\displaystyle\leq$
$\displaystyle\mu_{0}(t)V_{0}^{\beta_{2}}-0.5{c_{0}}(z_{1}^{2}+z_{2}^{2})^{\beta_{2}}$
(88) $\displaystyle\leq$
$\displaystyle\mu_{0}(t)V_{0}^{\beta_{2}}-0.25{c_{0}}(W_{0}+W_{1})^{\beta_{2}}$
(89) $\displaystyle\leq$
$\displaystyle\tilde{\mu}(t)(V_{0}^{\beta_{2}}+(W_{0}+W_{1})^{\beta_{2}})$
(90) $\displaystyle\leq$ $\displaystyle{\mu}(t)V_{2}^{\beta_{2}},$ (91)

where

$\mu(t)=\begin{cases}2^{1-\beta_{2}}\tilde{\mu}(t)~{}~{}&\textmd{if}~{}~{}\tilde{\mu}(t)\geq
0\\\ \tilde{\mu}(t)~{}~{}&\textmd{if}~{}~{}\tilde{\mu}(t)<0\\\
\end{cases},~{}~{}~{}~{}\tilde{\mu}(t)=\begin{cases}\mu_{0}(t)~{}~{}&\textmd{if}~{}~{}\mu_{0}(t)\geq-0.25c_{0}\\\
-0.25c_{0}~{}~{}&\textmd{if}~{}~{}\mu_{0}(t)<-0.25c_{0}\\\ \end{cases},$

$c_{0}=\min\\{c_{1},c_{2}\\}$. Moreover, Lemma 4 in [24], (62) and
$0.5^{\beta_{2}}>0.5$ are used in (86). Note that $\mu_{0}(t)$ is a UASF, so
$\tilde{\mu}(t)$ and $\mu(t)$ are UASFs by Remark 3.

Therefore, let $\xi=(\chi,z_{1},z_{2})^{T}$ and Lyapunov function
$V=V_{0}+W_{0}+W_{1}$ for system $\Sigma$, then the positive function
$V=V_{0}+W_{0}+W_{1}\leq\chi^{2}+2(z_{1}^{2}+z_{2}^{2})\leq 2|\xi|^{2}$ and
$\mathcal{L}V\leq{\mu}(t)V^{\beta_{2}}$, where (62) is considered. By Theorem
1, stochastic nonlinear system (59) is finite-time stabilizable in probability
with the controller (85).

Figure 3. The state curves of system (59) with (85).

In simulation, we choose design parameter $l=4$, $c_{1}=0.3$ and $c_{2}=0.3$,
system initial value $\chi(0)=0.2$, $x_{1}(0)=0.1$ and $x_{2}(0)=-0.2$, then
system control input trajectory and system state curves are given in Figure 3.

##  VI. Conclusions

This paper has further studied the finite-time stability and instability in
probability for stochastic nonlinear systems. A weaker sufficient condition,
which ensures that the considered system has a global solution, has been
presented. Some improved finite-time stability and instability theorems have
been given by the UASF. The obtained stability and instability results relax
the strict constraint conditions on $\mathcal{L}V$ existing in previous
references. Some examples are given to illustrate that the obtained stability
theorems can be used to analysis and synthesis of stochastic nonlinear systems
including autonomous systems and forced systems.

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