#  Predefined-time Stabilization for Nonlinear Stochastic Systems

Tianliang Zhang<EMAIL_ADDRESS>Shengyuan Xu<EMAIL_ADDRESS>School of
Automation, Nanjing University of Science and Technology, Nanjing 210094,
Jiangsu Province, China

###### Abstract

In this paper, a control scheme for stochastic predefined-time stabilization
is proposed, which improves the control effect compared with stochastic
finite-time or fixed-time stabilization. The stochastic predefined-time
stabilization allows the upper bound of the mathematical expectation of the
settling-time function below an any given positive value. Some Lyapunov-type
results for predefined-time stabilization of general stochastic Itô systems
are presented. Moreover, a state feedback control scheme is designed for a
class of stochastic nonlinear systems in strict-feedback form. Two simulation
examples are supplied to show the usefulness of the proposed stochastic
predefined-time stabilization.

###### keywords:

Stochastic predefined-time stabilization, nonlinear systems, stochastic
systems, settling-time function.

††thanks: Corresponding author: Shengyuan Xu.

,

##  1 Introduction

The study of convergence time of dynamic systems is of both practical and
theoretical importance. For some engineering requirements, we need to control
the convergence speed faster or slower. Different from Lyapunov stability,
which is to study the system asymptotic behavior in an infinite time horizon,
finite-time or fixed-time stability investigates the transient response of the
system state in a finite time horizon. Finite-time control has important
applications in robot manipulators [8]. Therefore, adaptive finite-time
tracking control [17], fixed-time stabilization [21] and global fast finite-
time stabilization of high-order nonlinear systems [25] have been extensively
studied in recent years. However, in many practical applications, it is
expected that the state trajectory of a controlled system can converge to the
equilibrium point at any admissible time through appropriately adjusting
control parameters [14, 27, 28]. A new concept called predefined-time
stability was proposed in [24], which can achieve some consequents that cannot
be provided by the traditional finite-time control schemes, such as the
arbitrarily adjustable upper bound of the convergence time independent of the
initial value. Recently, great progress has been made for predefined-time
control of deterministic nonlinear systems [12, 18, 20, 23].

Due to wide applications of stochastic systems, stochastic analysis and
synthesis have been popular research areas over the past few decades [3, 9,
16, 19, 29, 30, 31, 32, 33, 34, 35, 36]. Hence, it is very valuable to study
the finite-time convergence behavior of stochastic systems and generalize the
above mentioned works to stochastic systems. Recently, stochastic finite-time
stability/stabilization [2, 6, 7, 30, 32, 33, 34] and stochastic fixed-time
stability/stabilization [15, 26] have been studied extensively. Stochastic
finite-time stability was first strictly analyzed in [32], which means that
the stochastic settling-time function $T(x_{0})$ is finite almost surely and
the system is stable in probability. Additionally, if we also have
$\sup_{x_{0}\in{\mathcal{R}}^{n}}{\mathcal{E}}T(x_{0})<\infty$, then the
stochastic uncontrolled system is called fixed-time stable. Stochastic fixed-
time stability means that the system is not only finite-time stable but also
$\sup_{x_{0}\in{\mathcal{R}}^{n}}{\mathcal{E}}T(x_{0})<T_{\max}$ for a fixed
average time $T_{\max}$ with any initial state $x_{0}\in{\mathcal{R}}^{n}$. In
order to obtain a fast convergence speed, we expect that the upper bound of
the mathematical expectation of the settling-time function can be arbitrarily
adjusted by a control input. Therefore, it is necessary to generalize the
predefined-time concept of deterministic systems to stochastic systems.
However, to the best of our knowledge, up to now, there seems no work on
stochastic predetermined-time stabilization based on stochastic settling-time
function. It should be mentioned that a recent work [16] studied stochastic
nonlinear prescribed-time stabilization in mean square sense.

This paper investigates the problem of stochastic predefined-time
stabilization of nonlinear stochastic Itô systems. The main contributions of
this paper are highlighted as follows:

  * 1.

Stochastic predefined-time stability and stabilization are introduced and a
stochastic predefined-time stabilization theorem is obtained for general
nonlinear stochastic Itô systems; see Theorem 2.1. Because predefined-time
stability is stronger than finite-time stability and fixed-time stability,
Theorem 2.1 is applicable to stochastic finite-time/fixed-time stabilization
of [32, 26], which can also be viewed as extensions of [11, 12] to stochastic
systems. Theorem 2.1 is a Lyapunov-type theorem similarly to the results of
[32, 34]. Theorem 2.1 yields some useful corollaries that can be conveniently
used in practice.

  * 2.

The high-order nonlinear stochastic system is an important class of stochastic
systems that can be used to describe many phenomena arising from mechanical
systems. Its feedback stabilization has been investigated in [3, 26, 29, 30,
31]. Based on the results of Section 2, we also study the predefined-time
stabilization of high-order nonlinear stochastic systems and give a practical
controller design algorithm. Through adding the power integrator technique
together with a corollary given in Section 2, a state feedback control scheme
is given for a class of high-order stochastic nonlinear systems in strict-
feedback form, which guarantees that the closed-loop system is stochastically
predefined-time stabilizable. In addition, the proposed stochastic predefined-
time stabilization control scheme improves some existing results [26, 7, 6].

The rest of this paper is organized as follows: In Section 2, we make some
preliminaries by introducing some new definitions, theorems and corollaries,
Lyapunov-type theorems about the stochastic predetermined-time stabilization
are obtained. Section 3 presents the controller design procedure for high-
order nonlinear stochastic systems. In Section 4, two simulation examples are
given to show the effectiveness of our main results. Section 5 concludes this
paper with some remarks.

Notation: $\mathcal{R}^{n}$ denotes the $n$-dimensional real Euclidean vector
space. $\mathcal{R}_{+}:=[0,\infty)$. $D^{\prime}$ is the transpose of a
matrix or vector $D$. $\mathcal{C}^{2}$ stands for the set of real-valued
twice continuously differentiable functions. $sign(x):=1$ for $x>0$, $-1$ for
$x<0$, $0$ for $x=0$. For any $b\geq 0$, $a\in\mathcal{R}$, the function
$\lceil a\rfloor^{b}$ is defined as $\lceil a\rfloor^{b}=sign(a)|a|^{b}$.
$\mathcal{E}$ means the mathematical expectation operator. $\mathcal{P}(A)$
represents the probability of event $A$. $I_{A}(x)$ is the indicator function,
i.e., $I_{A}(x)=1$ for $x\in A$, otherwise, $I_{A}(x)=0$.
${\mathcal{K}}^{1}$-functions: A scalar continuous function $f$ defined from
${\mathcal{R}}_{+}$ to $[0,1)$ is said to be a ${\mathcal{K}}^{1}$-function if
it is strictly increasing, $f(0)=0$, and $f(x)\to 1$ as $x\to\infty$.

##  2 Stochastic predefined-time stabilization

In this section, we will consider the following continuous-time stochastic
system:

$\displaystyle dx(t)=f(x(t),u(t))\,dt+g(x(t),u(t))\,dw(t),$ (1) $\displaystyle
x(0)=x_{0}\in\mathcal{R}^{n}\setminus\\{0\\},$

where $x(t)\in\mathcal{R}^{n}$ represents the system state.
$u\in\mathcal{R}^{n_{u}}$ stands for the control input. $w(t)$ is a standard
one-dimensional Wiener process defined on the filtered probability space
$(\Omega,\mathcal{F},\\{{\mathcal{F}}_{t}\\}_{t\geq 0},\mathcal{P})$.
$u(t)\in\mathcal{R}^{n_{u}}$ stands for the control input. The admissible
control set $\mathcal{U}$ consists of all ${\mathcal{F}}_{t}$-adaptive control
processes $u(x(t))$, $u(0)=0$, which makes the closed-loop system

$dx(t)=f(x(t),u(x(t)))\,dt+g(x(t),u(x(t)))\,dw$

admits a unique weak solution in forward time for
$x_{0}\in\mathcal{R}^{n}\setminus\\{0\\}$. In the considered system, we assume
that $f:\mathcal{R}^{n}\times\mathcal{R}^{n_{u}}\rightarrow\mathcal{R}^{n}$
and $g:\mathcal{R}^{n}\times\mathcal{R}^{n_{u}}\rightarrow\mathcal{R}^{n}$ are
continuous in $x$ and $u$ satisfying $f(0,0)=0$ and $g(0,0)=0$. The purpose of
this section is to find an admissible control law $u(x)\in\mathcal{U}$ to
stabilize the stochastic system (1) before a predefined time, i.e., achieve
the stochastic predefined-time stabilization of system (1).

######  Remark 2.1.

When studying finite-time stable systems, we are interested in having a unique
solution in forward time [1], which means that, for any non-zero initial
condition $x_{0}$, $x(t,x_{0})$ is unique before reaching $0$. For stochastic
finite-time stable systems, the concept is extended to the solution in the
weak sense [34]. Based on the assumption of $f(0,0)=0$ and $g(0,0)=0$, the
origin is an equilibrium point of (1). The following lemma gives an existence
result of a weak solution to system (1). For the trajectory $x(t)$ after
reaching the zero equilibrium point, the finite-time attractiveness (in
Definition 2.1) is provided.

######  Lemma 2.1.

[33, 34] Suppose that there exits a nonnegative radially unbounded
$\mathcal{C}^{2}$-function $V(x)$ for system (1), if $\mathcal{L}V(x)\leq 0$,
then system (1) has a regular continuous solution for any initial value.

######  Remark 2.2.

The regular solution means that there is no finite explosion time with
probability $1$.

Before introducing stochastic predefined-time stabilization, we will first
review some well-known definitions on stochastic finite-time (fixed-time,
predefined-time, respectively) stabilization.

######  Definition 2.1.

[34, 26] System (1) is said to be stochastically finite-time stabilizable or
finite-time stabilizable in probability, if there exists a state feedback
control $u(t)=u^{*}(x(t))\in\mathcal{U}$, such that the closed-loop system

$\displaystyle dx(t)=f(x(t),u^{*}(x(t)))\,dt+g(x(t),u^{*}(x(t)))\,dw(t),$ (2)
$\displaystyle x(0)=x_{0}\in\mathcal{R}^{n}\setminus\\{0\\},$

is stochastically finite-time stable, i.e.,

  * •

Finite-time attractiveness: For any non-zero initial value $x_{0}$, there
exists the first settling-time function
$T(x_{0}):=\inf\\{t:x(t,u^{*},x_{0})=0\\}$, such that

$\mathcal{P}(T(x_{0})<\infty)=1,$

where $x(t,u^{*},x_{0})$ is the solution of (2). Moreover,
$x(t+T(x_{0}),u^{*},x_{0})\equiv 0$, a.s., for any $t\geq 0$.

  * •

Stability in probability: For any pair $(\alpha,\beta)$, $\alpha\in(0,1)$,
$\beta>0$, there exists a $\delta(\alpha,\beta)>0$, such that for all
$\|x_{0}\|<\delta(\alpha,\beta)$, we have

$\displaystyle\mathcal{P}(\|x(t,u^{*},x_{0})\|<\beta,\forall t>0)\geq
1-\alpha.$

Moreover, if there exists a positive constant $T_{max}$ such that

$\displaystyle\sup_{x_{0}\in\mathcal{R}^{n}\setminus\\{0\\}}\mathcal{E}T(x_{0})\leq
T_{max},$

then system (2) is said to be stochastically fixed-time stable, and system (1)
is said to be stochastically fixed-time stabilizable.

######  Remark 2.3.

In recent literature, new and improved definitions called predefined-time
stable and stabilizable were proposed for deterministic systems based on
fixed-time stability [12, 24]. This paper aims to extend these concepts and
related results to stochastic system (1).

######  Definition 2.2.

System (1) is said to be stochastically predefined-time stabilizable if it is
stochastically fixed-time stabilizable and for any $\varsigma>0$, there exists
a control $u(t)=u^{*}(x(t))\in\mathcal{U}$, such that

$\displaystyle\sup_{x_{0}\in\mathcal{R}^{n}\setminus\\{0\\}}\mathcal{E}T(x_{0})\leq\varsigma.$

######  Remark 2.4.

Recently, another newly proposed definition called prescribed-time mean-square
stability was introduced [16], which is different from Definition 2.2.
Definition 2.2 is based on stochastic settling time function, when the system
degenerates into deterministic systems, Definition 2.2 is consistent with the
corresponding definition of deterministic systems.

######  Remark 2.5.

From Definitions 2.1 and 2.2, for the system (1) with non-zero initial value,
there must be $x(t+T(x_{0}),u^{*},x_{0})\equiv 0$ with $t\geq 0$ and
$T(x_{0})$ being the stochastic settling-time function. So, $0$ is an
absorbing state. Moreover, when discussing predefined-time stabilization or
finite-time stabilization control problems, a non-zero initial value is
usually assumed [12, 33].

The following theorem is a Lyapunov-type theorem about stochastic predefined-
time stabilization.

######  Theorem 2.1.

For any $\alpha>0$, if there exists a control input
$u^{*}(x(t))\in\mathcal{U}$, driving the state of system (1) to satisfy

$\mathcal{L}_{u=u^{*}}V(x)\leq-\frac{1}{\alpha}\beta(V(x)),\
x\in\mathcal{R}^{n}\setminus\\{0\\},$ (3)

where $\beta:\mathcal{R}_{+}\mapsto\mathcal{R}_{+}$ with
$\dot{\beta}(\cdot)\geq 0$, $\beta(s)>0$ for any $s>0$, and
$\int^{\infty}_{0}\frac{1}{\beta(s)}\,ds\leq 1$,
$V:\mathcal{R}^{n}\mapsto\mathcal{R}_{+}$ is a $\mathcal{C}^{2}$-positive
definite and radially unbounded function, and $\mathcal{L}_{u=u^{*}}$
represents the infinitesimal generator of system (2). Then system (1) is
stochastically predefined-time stabilizable, and
$\sup_{x_{0}\in\mathcal{R}^{n}\setminus\\{0\\}}\mathcal{E}T(x_{0})\leq\alpha.$.

Proof. By Theorem 1 of [34], system (1) is stochastically finite-time
stabilizable under the conditions of this theorem. Hence, in order to prove
stochastic predefined-time stabilization of system (1), we only need to show
that for any $\alpha>0$, the following holds:

$\sup_{x_{0}\in\mathcal{R}^{n}\setminus\\{0\\}}\mathcal{E}T(x_{0})\leq\alpha.$

Obviously, if $x_{0}=0$, it directly leads to $x(t,u^{*},x_{0})=0$ a.s. for
any $t\geq 0$. So we only need to consider the case of non-zero initial state.
From condition (3) and Lemma 2.1, it leads to that for each $x_{0}\neq 0$,
there exists a regular continuous solution $x(t,u^{*},x_{0})$ to (1).
Therefore, there exists a minimal positive integer $k_{0}\in\\{1,2,\cdots,\\}$
such that $\frac{1}{k_{0}}<\|x_{0}\|<k_{0}$. Define some sequences as follows:

$\displaystyle\tau_{k}=\inf\left\\{t\geq
0:\|x(t,u^{*},x_{0})\|\notin\left(\frac{1}{k},k\right)\right\\},$
$\displaystyle\tau_{1k}=\inf\left\\{t\geq
0:\|x(t,u^{*},x_{0})\|\in\left[0,\frac{1}{k}\right]\right\\},$
$\displaystyle\tau_{2k}=\inf\left\\{t\geq
0:\|x(t,u^{*},x_{0})\|\in\left[k,\infty\right)\right\\}.$

Considering Lemma 2.1 and the definition of the admissible control set
$\mathcal{U}$, according to the definition of stopping time, sequences
$\tau_{k}$, $\tau_{1k}$ and $\tau_{2k}$ are ${\mathcal{F}}_{\tau_{k}}$-,
${\mathcal{F}}_{\tau_{1k}}$\- and ${\mathcal{F}}_{\tau_{2k}}$-measurable,
respectively. So they are all stopping times. For convenience, in the sequel,
we denote the solution $x(t,u^{*},x_{0})$ by $x(t)$ for short. We introduce a
new Lyapunov function $P(x)=\int^{V(x)}_{0}\frac{\alpha}{\beta(s)}ds$. By Itô
formula, we get

$\displaystyle dP(x(t))$ $\displaystyle=\mathcal{L}_{u=u^{*}}P(x(t))\,dt$
$\displaystyle\ \ +\frac{\alpha}{\beta(V(x(t)))}\frac{\partial
V^{\prime}}{\partial x}g(x(t),u^{*}(x(t)))\,dw(t),$

where $\mathcal{L}_{u=u^{*}}P(x)$ can be computed as

$\displaystyle\mathcal{L}_{u=u^{*}}P(x)=\frac{\alpha\mathcal{L}_{u=u^{*}}V(x)}{\beta(V(x))}$
$\displaystyle-\frac{1}{2}\frac{\alpha\dot{\beta}(V)}{\beta^{2}(V)}\left(g^{\prime}(x,u^{*})\frac{\partial
V}{\partial x}\right)\left(\frac{\partial V^{\prime}}{\partial
x}g(x,u^{*})\right).$ (4)

Then, similar to Theorem 3.1 in [32], it follows that

$\displaystyle\mathcal{E}P(x(t\wedge\tau_{k}))-P(x_{0})=\mathcal{E}\int^{t\wedge\tau_{k}}_{0}\mathcal{L}_{u=u^{*}}P(x(v))dv.$
(5)

From (3) and (2), we have $\mathcal{L}_{u=u^{*}}P(x)\leq-1$ for any $x\neq 0$.
So
$\mathcal{E}P(x(t\wedge\tau_{k}))-P(x_{0})\leq-\mathcal{E}(t\wedge\tau_{k})$
and $\mathcal{E}(t\wedge\tau_{k})\leq P(x_{0})$ hold. Because
$\\{t\wedge\tau_{k}\\}_{k\in\\{k_{0},k_{0}+1,\cdots\\}}$ is an increasing
sequence of nonnegative random variables, by monotonic convergence theorem,

$\displaystyle\lim_{k\rightarrow\infty}\mathcal{E}(t\wedge\tau_{k})=\mathcal{E}(\lim_{k\rightarrow\infty}t\wedge\tau_{k})=\mathcal{E}(t\wedge\tau_{\infty}).$

Note that

$\tau_{\infty}=\tau_{1,\infty}\wedge\tau_{2,\infty}.$

Since $x(t,x_{0})$ is a regular solution, $\tau_{2,\infty}=\infty$ a.s.. So
$\mathcal{E}(t\wedge\tau_{\infty})=\mathcal{E}(t\wedge\tau_{1,\infty})\leq
P(x_{0})\leq\alpha$. Due to the arbitrariness of $x_{0}$ and
$\tau_{1,\infty}=T(x_{0})$, it yields that

$\sup_{x_{0}\in\mathcal{R}^{n}\setminus\\{0\\}}\mathcal{E}T(x_{0})\leq\alpha.$

From (3) and $V(x)\geq 0$, $V_{t}:=V(x(t,u^{*},x_{0}))$ is a nonnegative
continuous supermartingale with augmented filtration
$\\{{\mathcal{F}}_{t}\\}_{t\geq 0}$. Through Doob’s optional-sampling theorem
for continuous nonnegative supermartingales in [10],

$\displaystyle\mathcal{E}(V_{T(x_{0})+t}|\mathcal{F}_{T(x_{0})})\leq
V_{T(x_{0})}=0,\forall t\geq 0.$

Taking expectation on both sides of the above inequality, we have
$\mathcal{E}V_{T(x_{0})+t}\equiv 0$. Since $V$ is positive definite, it
follows that $x(T(x_{0})+t,u^{*},x_{0})\equiv 0$, a.s.. The proof is
completed.

Generally speaking, stochastic predefined-time stabilization is a special case
of stochastic fixed-time stabilization, which is stronger than stochastic
fixed-time stabilization. In order to illustrate the difference between these
two concepts, a numerical example is presented below.

######  Example 2.1.

Consider a scalar system

$dx=udt+xdw,\ x(0)=x_{0}.$ (6)

By selecting the control input

$u=u^{*}=-\frac{1}{2}x-x^{a}-x^{b}$ (7)

with $a\in(0,1)$ and $b>1$, then the feedback system of (6) becomes

$dx=(-\frac{1}{2}x-x^{a}-x^{b})\,dt+x\,dw,\ x(0)=x_{0}.$ (8)

For system (8) and the Lyapunov function $V(x)=x^{2}$, it is easy to compute

${\mathcal{L}}_{u=u^{*}}V(x)=-2x^{a+1}-2x^{b+1},$

where ${\mathcal{L}}_{u=u^{*}}$ is the infinitesimal generator of system (8).
According to Lemma 7 of [15], we have

$\displaystyle\sup_{x_{0}\in\mathcal{R}}\mathcal{E}T(x_{0})\leq
T_{max}:=\frac{1}{1-a}+\frac{1}{b-1}.$

Hence, system (6) is stochastically fixed-time stabilizable.

In addition, for any $\alpha>0$, if we choose the control input as

$u=u^{*}(x)=-\frac{\sqrt{\pi}}{2\alpha}sign(x)e^{x^{2}}-\frac{1}{2}x,$ (9)

and $\mathcal{C}^{2}$-positive definite and radially unbounded function
$V(x)=x^{2}$, then by Itô formula, we have

$\displaystyle\mathcal{L}_{u=u^{*}}V(x)$
$\displaystyle=2xu^{*}+x^{2}=x(-\frac{\sqrt{\pi}}{\alpha}sign(x)e^{x^{2}})$
$\displaystyle=-\frac{\sqrt{\pi}}{\alpha}|x|e^{x^{2}}=-\frac{\sqrt{\pi}}{\alpha}V(x)^{1/2}e^{V(x)}.$

Let $\beta(s)=\sqrt{\pi}s^{1/2}e^{s}$, then $\beta(s)$ satisfies the
conditions of Theorem 2.1. By Theorem 2.1, system (6) is also stochastically
predefined-time stabilizable, and (9) is a stochastic predefined-time
stabilizing controllor.

######  Remark 2.6.

Fixed-time stabilization is often difficult and sometimes impossible to adjust
the controller gain such that achieving stabilization within a required
predefined time. There is no guarantee that this upper bound can be adjusted
arbitrarily. On this point, we can refer to the discussion in [12, 23]. From
the above example, it is easier to tune the convergence time bound for
stochastic predefined-time stabilization than stochastic fixed-time
stabilization. Therefore, the stochastic fixed-time control discussed in [26]
cannot achieve the effect of the stochastic predefined-time control discussed
in this paper. Furthermore, this paper proves various Lyapunov-type theorems
for stochastic predefined-time stabilization, and presents a design scheme of
stochastic predefined-time controllers for higher-order nonlinear systems.

######  Remark 2.7.

Theorem 2.1 presents a Lyapunov-type result about stochastic predefined-time
stabilization. When $\int^{t}_{0}\frac{1}{\beta(s)}ds<\infty$ with $t\geq 0$
replaces $\int^{\infty}_{0}\frac{1}{\beta(s)}ds\leq 1$ in Theorem 2.1, then
system (1) is stochastically finite-time stabilizable [32]. In addition, if
the condition $\int^{\infty}_{0}\frac{1}{\beta(s)}ds\leq 1$ is changed as
$\int^{\infty}_{0}\frac{1}{\beta(s)}\,ds<\infty$, then system (1) is
stochastically fixed-time stabilizable.

######  Corollary 2.1.

For any $\alpha>0$, if there exists a control input
$u^{*}(x(t))\in\mathcal{U}$, driving the state of system (1) to satisfy

$\mathcal{L}_{u=u^{*}}V(x)\leq-\frac{\tilde{\beta}(V(x))^{p}}{(1-p)\alpha\dot{\tilde{\beta}}(V(x))},x\in\mathcal{R}^{n}\setminus\\{0\\},$
(10)

where $\tilde{\beta}(\cdot)$ is a function belonging to $\mathcal{K}^{1}$ with
$\ddot{\tilde{\beta}}(\cdot)\leq 0$, $0\leq p<1$.
$V:\mathcal{R}^{n}\mapsto\mathcal{R}_{+}$ is a $\mathcal{C}^{2}$-positive
definite and radially unbounded function. Then system (1) is stochastically
predefined-time stabilizable, and for any
$x_{0}\in\mathcal{R}^{n}\setminus\\{0\\}$, the first settling-time function
satisfies $\mathcal{E}T(x_{0})\leq\alpha$.

Proof. Corollary 2.1 can be directly obtained by Theorem 2.1 via setting

$\beta(s)=\frac{\tilde{\beta}(s)^{p}}{(1-p)\dot{\tilde{\beta}}(s)}.$

We can also give another Lyapunov-type theorem inspired by [11].

######  Corollary 2.2.

For any $\alpha>0$, if there exists a control input
$u^{*}(x(t))\in\mathcal{U}$, driving the state of system (1) to satisfy

$\mathcal{L}_{u=u^{*}}W(x)\leq-\frac{1}{\alpha},\
x\in\mathcal{R}^{n}\setminus\\{0\\},$ (11)

where $W:\mathcal{R}^{n}\mapsto\mathcal{R}_{+}$ is a
$\mathcal{C}^{2}$-positive definite function with $0\leq W(s)<1$,
$\lim_{s\rightarrow\infty}W(s)=1$. Then system (1) is stochastically
predefined-time stabilizable, and for any
$x_{0}\in\mathcal{R}^{n}\setminus\\{0\\}$, the first settling-time function
satisfies $\mathcal{E}T(x_{0})\leq\alpha$.

Proof. Set $V(x)=-\ln(1-W(x))$. By (11), it is easy to obtain that

$\mathcal{L}_{u=u^{*}}W(x)=e^{-V(x)}\mathcal{L}_{u=u^{*}}V(x)\leq-\frac{1}{\alpha}.$
(12)

This gives

$\mathcal{L}_{u=u^{*}}V(x)\leq-\frac{1}{\alpha}e^{V(x)}.$

By setting $\beta(s)=e^{s}$, and using Theorem 2.1, we have the desired
results immediately.

######  Corollary 2.3.

Assume there exist a $\mathcal{C}^{2}$-positive definite and radially
unbounded function $V$, positive constants $a>0$, $0<b_{1}<1$, and $b_{2}>1$
satisfying $a-ab_{1}-1>0$. If for any $\alpha>0$, there exists a control input
$u^{*}(x(t))\in\mathcal{U}$, such that

$\displaystyle\mathcal{L}_{u=u^{*}}V(x)\leq$
$\displaystyle-\frac{a-ab_{1}}{\alpha(a-ab_{1}-1)(b_{2}-1)}V(x)^{b_{2}}$
$\displaystyle-\frac{a}{\alpha}V(x)^{b_{1}}$ (13)

for $x\in\mathcal{R}^{n}\setminus\\{0\\}$, then system (1) is stochastically
predefined-time stabilizable, i.e., $\sup_{\forall
x_{0}}\mathcal{E}T(x_{0})\leq\alpha$.

Proof. In Theorem 2.1, set

$\beta(V(x))=aV(x)^{b_{1}}+\frac{a-ab_{1}}{(a-ab_{1}-1)(b_{2}-1)}V(x)^{b_{2}}.$

Obviously, $\beta(s)\geq 0$ for $s\in{\mathcal{R}}_{+}$, and
$\dot{\beta}(\cdot)\geq 0$. Moreover,

$\displaystyle\int^{\infty}_{0}\frac{1}{\beta(s)}ds=\int^{1}_{0}\frac{1}{\beta(s)}ds+\int^{\infty}_{1}\frac{1}{\beta(s)}\,ds$
$\displaystyle\leq$
$\displaystyle\int^{1}_{0}\frac{1}{as^{b_{1}}}\,ds+\int^{\infty}_{1}\frac{(a-ab_{1}-1)(b_{2}-1)}{(a-ab_{1})s^{b_{2}}}\,ds$
$\displaystyle=$
$\displaystyle\frac{1}{(1-b_{1})a}+\frac{a-ab_{1}-1}{a-ab_{1}}=1.$

Therefore, (2.3) implies that conditions of Theorem 2.1 are satisfied. This
corollary is proved.

##  3 Controller design in strict-feedback form

In this section, we address the stochastic predefined-time stabilization
problem for nonlinear stochastic systems based on backstepping method.
Consider the following high-order stochastic nonlinear system described by

$\displaystyle\begin{cases}dx_{i}=(h_{i}\lceil
x_{i+1}\rfloor^{q_{i}}+f_{i}(\bar{x}_{i}))dt+g_{i}(\bar{x}_{i})dw,\\\
dx_{n}=(h_{n}\lceil u\rfloor^{q_{n}}+f_{n}(x))dt+g_{n}(x)dw,\end{cases}$ (14)

where $i=1,2,\cdots,n-1$, $x=[x_{1},\cdots,x_{n}]^{\prime}\in\mathcal{R}^{n}$
stands for the system state, $u\in\mathcal{R}$ is the control input. $w$ is a
one-dimensional standard Wiener process defined on the filtered probability
space $(\Omega,{\mathcal{F}}$, $\\{{\mathcal{F}}_{t}\\}_{t\geq 0}$,
${\mathcal{P}})$. $\bar{x}_{i}$ is defined as
$\bar{x}_{i}=[x_{1},\cdots,x_{i}]^{\prime}$. $h_{1},\cdots,h_{n}$ are unknown
virtual control coefficients. The drift terms $f_{i}(x)$ and diffusion terms
$g_{i}(x)$, $i=1,2,\cdots,n$, are Borel measurable continuous functions with
$f_{i}(0)=0$ and $g_{i}(0)=0$. For any $i$, $1<q_{i}\in\mathcal{R}_{+}$ is
called the high-order of system (14).

The following assumption allows the functions $f_{i}(\bar{x}_{i})$ and
$g_{i}(\bar{x}_{i})$ to have high-order and low-order nonlinear growth rates.

######  Assumption 3.1.

For any $i=1,2,\cdots,n$, the drift term $f_{i}(\bar{x}_{i})$ and the
diffusion term $g_{i}(\bar{x}_{i})$ satisfy the following conditions:

$\displaystyle|f_{i}(\bar{x}_{i})|\leq\phi_{i}(\bar{x}_{i})\sum^{i}_{j=1}|x_{j}|^{\varpi_{ij}+\frac{r_{i}+\kappa}{r_{j}}},$
$\displaystyle|g_{i}(\bar{x}_{i})|\leq\varphi_{i}(\bar{x}_{i})\sum^{i}_{j=1}|x_{j}|^{\rho_{ij}+\frac{2r_{i}+\kappa}{2r_{j}}},$

where $\varpi_{ij}\geq 0$, $\rho_{ij}\geq 0$, $\phi_{i}(\bar{x}_{i})$ and
$\varphi_{i}(\bar{x}_{i})$ are non-negative smooth functions with
$\phi_{i}(0)=0$ and $\varphi_{i}(0)=0$,
$\kappa\in(-\frac{1}{1+\sum^{n-1}_{s=1}q_{1}\cdots q_{s}},0)$. $r_{i+1}$ is
recursively defined as $r_{i+1}=\frac{r_{i}+\kappa}{q_{i}}$ with $r_{1}=1$.

######  Remark 3.1.

Assumption 3.1 is a common nonlinear growth rate [6, 25]. the powers in growth
condition of $f_{i}(\bar{x}_{i})$ and $g_{i}(\bar{x}_{i})$ are defined as
$\varpi_{ij}+\frac{r_{i}+\kappa}{r_{j}}\in\mathcal{R}_{+}$ and
$\rho_{ij}+\frac{2r_{i}+\kappa}{2r_{j}}\in\mathcal{R}_{+}$, respectively.
$\varpi_{ij}+\frac{r_{i}+\kappa}{r_{j}}$ and
$\rho_{ij}+\frac{2r_{i}+\kappa}{2r_{j}}$ belonging to an interval $(0,\infty)$
allow $f_{i}(\bar{x}_{i})$ and $g_{i}(\bar{x}_{i})$ to have both both high-
order and low-order nonlinear growth rates. Assumption 3.1 includes the
counterparts in the closely related works as special cases. For example, when
$\varpi_{ij}=\rho_{ij}=0$, Assumption 3.1 degenerates to Assumption 1 in [5].
If $\phi_{i}(\bar{x}_{i})$ and $\varphi_{i}(\bar{x}_{i})$ are further
specialised as constants, Assumption 3.1 becomes the low-order growth rate
used in [30]. When $\varpi_{ij}=1-\frac{r_{i}+\kappa}{r_{j}}$ and
$\rho_{ij}=1-\frac{2r_{i}+\kappa}{2r_{j}}$, Assumption 3.1 reduces the linear-
like growth rate used in [13].

######  Assumption 3.2.

For any $i=1,2,\cdots,n$, there exist constants $\overline{h}_{i}>0$ and
$\underline{h}_{i}>0$ such that $\underline{h}_{i}\leq
h_{i}\leq\overline{h}_{i}$.

######  Lemma 3.1.

[29] For any $a\in\mathcal{R}$ and $b\geq 2$, we have that the function
$f(a):=\lceil a\rfloor^{b}\in\mathcal{C}^{2}$, $\frac{\partial f(a)}{\partial
a}=b|a|^{b-1}$ and $\frac{\partial^{2}f(a)}{\partial a^{2}}=b(b-1)\lceil
a\rfloor^{b-2}$.

######  Lemma 3.2.

[4] Suppose $0<p<1$ and $q>1$, then for any $x,y\in\mathcal{R}$, we have

$\displaystyle|\lceil x\rfloor^{pq}-\lceil y\rfloor^{pq}|\leq 2^{1-p}|\lceil
x\rfloor^{q}-\lceil y\rfloor^{q}|^{p}.$

######  Lemma 3.3.

[22] For any positive real numbers $p$, $q$ and any real-valued function
$f(x,y)>0$, the following relationship holds:

$\displaystyle|x|^{p}|y|^{q}\leq\frac{p}{p+q}f(x,y)|x|^{p+q}+\frac{q}{p+q}f^{-\frac{p}{q}}(x,y)|y|^{p+q}.$

######  Lemma 3.4.

[22] For any $x,y\in\mathcal{R}$, $a\geq 1$, we have
$(|x|+|y|)^{\frac{1}{a}}\leq|x|^{\frac{1}{a}}+|y|^{\frac{1}{a}}\leq
2^{\frac{a-1}{a}}(|x|+|y|)^{\frac{1}{a}}$.

######  Lemma 3.5.

[22]
$(\sum^{j}_{i=1}a_{i})^{b}\leq\max\\{j^{b-1},1\\}(\sum^{j}_{i=1}a_{i}^{b})$
always holds for any positive numbers $a_{1},a_{2},\cdots,a_{j}$ and $b$.

Now, we are ready to present our main theorem of this section.

######  Theorem 3.1.

If the high-order stochastic nonlinear system (14) satisfies Assumptions 3.1
and 3.2, then there exists a state-feedback controller such that the closed-
loop system is stochastically predefined-time stabilizable.

Proof. The proof is based on inductive arguments. Step 1: Firstly, by the
power integrator technique, we choose the Lyapunov function for the first
subsystem as $V_{1}(x_{1})=\int^{x_{1}}_{x_{1}^{*}}\lceil\lceil
s\rfloor^{\frac{r}{r_{1}}}-\lceil
x_{1}^{*}\rfloor^{\frac{r}{r_{1}}}\rfloor^{\frac{4r-\kappa-r_{1}}{r}}\,ds$,
where $r\geq\max_{1\leq i\leq n}\\{2r_{i}\\}$ and $x^{*}_{1}=0$. By Itô’s
formula, Assumption 3.1, and Lemma 3.1, we can get that

$\displaystyle\mathcal{L}_{1}V_{1}(x_{1})$ $\displaystyle=$
$\displaystyle\frac{\partial V_{1}}{\partial x_{1}}(h_{1}\lceil
x_{2}\rfloor^{q_{1}}+f_{1}(\bar{x}_{1}))+\frac{1}{2}\frac{\partial^{2}V_{1}}{\partial
x_{1}^{2}}|g_{1}(\bar{x}_{1})|^{2}$ $\displaystyle=$
$\displaystyle\lceil\xi_{1}\rfloor^{\frac{4r-\kappa-r_{1}}{r}}(h_{1}\lceil
x_{2}\rfloor^{q_{1}}+f_{1}(\bar{x}_{1}))+\frac{4r-\kappa-
r_{1}}{2r}|\xi_{1}|^{\frac{3r-\kappa-r_{1}}{r}}$
$\displaystyle\frac{r}{r_{1}}|x_{1}|^{\frac{r}{r_{1}}-1}|g_{1}(\bar{x}_{1})|^{2}$
$\displaystyle\leq$ $\displaystyle\lceil\xi_{1}\rfloor^{\frac{4r-\kappa-
r_{1}}{r}}h_{1}(\lceil x_{2}\rfloor^{q_{1}}-\lceil
x_{2}^{*}\rfloor^{q_{1}})+\lceil\xi_{1}\rfloor^{\frac{4r-\kappa-
r_{1}}{r}}h_{1}\lceil x_{2}^{*}\rfloor^{q_{1}}$
$\displaystyle+\frac{4r-\kappa-r_{1}}{2r}|\xi_{1}|^{\frac{3r-\kappa-
r_{1}}{r}}\frac{r}{r_{1}}|x_{1}|^{\frac{r}{r_{1}}-1}(\bar{\varphi}_{1}(\bar{x}_{1})|x_{1}|^{\frac{2r_{1}+\kappa}{2r_{1}}})^{2}$
$\displaystyle+\lceil\xi_{1}\rfloor^{\frac{4r-\kappa-
r_{1}}{r}}\bar{\phi}_{1}(x_{1})|x_{1}|^{\frac{r_{1}+\kappa}{r_{1}}}$
$\displaystyle\leq$ $\displaystyle\lceil\xi_{1}\rfloor^{\frac{4r-\kappa-
r_{1}}{r}}h_{1}(\lceil x_{2}\rfloor^{q_{1}}-\lceil
x_{2}^{*}\rfloor^{q_{1}})+\lceil\xi_{1}\rfloor^{\frac{4r-\kappa-
r_{1}}{r}}h_{1}\lceil x_{2}^{*}\rfloor^{q_{1}}$
$\displaystyle+|\xi_{1}|^{4}\bar{\phi}_{1}(x_{1})+\frac{4r-\kappa-
r_{1}}{2r_{1}}|\xi_{1}|^{4}\bar{\varphi}_{1}^{2}(\bar{x}_{1}),$

where $\xi_{1}=\lceil x_{1}\rfloor^{\frac{r}{r_{1}}}$,
$\bar{\varphi}_{1}(x_{1})\geq\varphi_{1}(x_{1})|x_{1}|^{\rho_{11}}$ and
$\bar{\phi}_{1}(x_{1})\geq\phi_{1}(x_{1})|x_{1}|^{\varpi_{11}}$ are smooth
functions. $\mathcal{L}_{1}$ is the infinitesimal generator of the first
subsystem. The definitions of $\bar{\varphi}_{i}(\bar{x}_{i})$ and
$\bar{\phi}_{i}(\bar{x}_{i})$ used in the later proof are similar and will not
be repeated. Designing
$x_{2}^{*}=-\beta_{1}\lceil\xi_{1}\rfloor^{\frac{r_{2}}{r}}$ and
$\beta_{1}>0$, it leads to

$\displaystyle\lceil x_{2}^{*}\rfloor^{q_{1}}=$ $\displaystyle
sign(-\beta_{1}\lceil\xi_{1}\rfloor^{\frac{r_{2}}{r}})|\beta_{1}\lceil\xi_{1}\rfloor^{\frac{r_{2}}{r}}|^{q_{1}}$
$\displaystyle=$ $\displaystyle-
sign(\lceil\xi_{1}\rfloor^{\frac{r_{2}}{r}})|\beta_{1}\lceil\xi_{1}\rfloor^{\frac{r_{2}}{r}}|^{q_{1}}$
$\displaystyle=$
$\displaystyle-\beta_{1}^{q_{1}}\lceil\xi_{1}\rfloor^{\frac{r_{2}q_{1}}{r}}.$

Note that $\lceil\xi_{1}\rfloor^{\frac{4r-\kappa-
r_{1}}{r}}\lceil\xi_{1}\rfloor^{\frac{r_{2}q_{1}}{r}}=|\xi_{1}|^{4}$. Set
$\beta_{1}(x_{1})=\left(\frac{4r-\kappa-
r_{1}}{2r_{1}\overline{h}_{1}}\bar{\varphi}_{1}^{2}(x_{1})+\frac{nk_{1}}{\overline{h}_{1}k_{4}}+\frac{k_{2}}{\overline{h}_{1}k_{4}}|\xi_{1}|^{k_{3}}+\frac{\bar{\phi}_{1}(x_{1})}{\overline{h}_{1}}\right)^{\frac{1}{q_{1}}}$
with $k_{1}$, $k_{2}$, $k_{3}$, $k_{4}$ to be designed in the future. Then

$\displaystyle\mathcal{L}_{1}V_{1}(x_{1})\leq$
$\displaystyle\lceil\xi_{1}\rfloor^{\frac{4r-\kappa-r_{1}}{r}}h_{1}(\lceil
x_{2}\rfloor^{q_{1}}-\lceil x_{2}^{*}\rfloor^{q_{1}})$
$\displaystyle-\frac{nk_{1}}{k_{4}}|\xi_{1}|^{4}-\frac{k_{2}}{k_{4}}|\xi_{1}|^{4+k_{3}}.$

Step 2-Inductive assumption: Suppose for $j\in\\{3,4,\cdots,n\\}$, there is a
$\mathcal{C}^{2}$\- positive definite radially unbounded function $V_{j-1}$,
and a set of virtual controllers $x_{1}^{*},x_{2}^{*},\cdots,x_{j}^{*}$
defined by

$\displaystyle\begin{array}[]{cccc}x_{1}^{*}=0,&\xi_{1}=\lceil
x_{1}\rfloor^{\frac{r}{r_{1}}}-\lceil x_{1}^{*}\rfloor^{\frac{r}{r_{1}}};\\\
x_{2}^{*}=-\beta_{1}(\bar{x}_{1})\lceil\xi_{1}\rfloor^{\frac{r_{2}}{r}},&\xi_{2}=\lceil
x_{2}\rfloor^{\frac{r}{r_{2}}}-\lceil x_{2}^{*}\rfloor^{\frac{r}{r_{2}}};\\\
\vdots&\vdots\\\
x_{j}^{*}=-\beta_{j-1}(\bar{x}_{j-1})\lceil\xi_{j-1}\rfloor^{\frac{r_{j}}{r}},&\xi_{j}=\lceil
x_{j}\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}\end{array}$

with functions $\beta_{i}(\bar{x}_{i})>0$, $i\in\\{1,2,\cdots,j-1\\}$, such
that

$\displaystyle\mathcal{L}_{j-1}V_{j-1}(\bar{x}_{j-1})\leq$
$\displaystyle\lceil\xi_{j-1}\rfloor^{\frac{4r-\kappa-
r_{j-1}}{r}}h_{j-1}(\lceil x_{j}\rfloor^{q_{j-1}}-\lceil
x_{j}^{*}\rfloor^{q_{j-1}})$
$\displaystyle-(n-j+2)\sum^{j-1}_{i=1}\frac{k_{1}}{k_{4}}|\xi_{i}|^{4}-\sum^{j-1}_{i=1}\frac{k_{2}}{k_{4}}|\xi_{i}|^{4+k_{3}},$
(15)

where $\mathcal{L}_{j-1}$ is the infinitesimal generator of the first $j-1$
subsystems.

Step 3: In the following, we shall show that (3) still holds when $j-1$ is
replaced by $j$ with

$\displaystyle V_{j}(\bar{x}_{j})=V_{j-1}(\bar{x}_{j-1})+W_{j}(\bar{x}_{j})$
(16)

and

$\displaystyle W_{j}(\bar{x}_{j})=\int^{x_{j}}_{x_{j}^{*}}\lceil\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}\rfloor^{\frac{4r-\kappa-r_{j}}{r}}\,ds.$
(17)

With the help of Proposition 6.1 in Appendix, it can be deduced from
$V_{j}(\bar{x}_{j})$ that

$\displaystyle\mathcal{L}_{j}V_{j}(\bar{x}_{j})$ $\displaystyle\leq$
$\displaystyle\lceil\xi_{j-1}\rfloor^{\frac{4r-\kappa-
r_{j-1}}{r}}h_{j-1}(\lceil x_{j}\rfloor^{q_{j-1}}-\lceil
x_{j}^{*}\rfloor^{q_{j-1}})$
$\displaystyle-(n-j+2)\sum^{j-1}_{i=1}\frac{k_{1}}{k_{4}}|\xi_{i}|^{4}-\sum^{j-1}_{i=1}\frac{k_{2}}{k_{4}}|\xi_{i}|^{4+k_{3}}$
$\displaystyle+\lceil\xi_{j}\rfloor^{\frac{4r-\kappa-r_{j}}{r}}h_{j}\lceil
x_{j+1}\rfloor^{q_{j}}+\lceil\xi_{j}\rfloor^{\frac{4r-\kappa-
r_{j}}{r}}f_{j}(\bar{x}_{j})$ $\displaystyle+\sum^{j-1}_{i=1}\frac{\partial
W_{j}}{\partial x_{i}}(h_{i}\lceil x_{i+1}\rfloor^{q_{i}}+f_{i}(\bar{x}_{i}))$
$\displaystyle+\frac{1}{2}\sum^{j}_{i,m=1}\left|\frac{\partial^{2}W_{j}}{\partial
x_{i}\partial
x_{m}}\right|\left|g_{i}(\bar{x}_{i})^{\prime}g_{m}(\bar{x}_{m})\right|.$ (18)

Based on Propositions 6.2-6.5 in Appendix, we estimate some terms in (3) and
obtain that

$\displaystyle\mathcal{L}_{j}V_{j}(\bar{x}_{j})$ $\displaystyle\leq$
$\displaystyle(\alpha_{1,j}+\alpha_{2,j}+\alpha_{3,j}+\alpha_{4,j})|\xi_{j}|^{4}-(n-j+1)\sum^{j-1}_{i=1}\frac{k_{1}}{k_{4}}|\xi_{i}|^{4}$
$\displaystyle-\sum^{j-1}_{i=1}\frac{k_{2}}{k_{4}}|\xi_{i}|^{4+k_{3}}+\lceil\xi_{j}\rfloor^{\frac{4r-\kappa-
r_{j}}{r}}h_{j}\lceil x_{j+1}^{*}\rfloor^{q_{j}}$
$\displaystyle+\lceil\xi_{j}\rfloor^{\frac{4r-\kappa-r_{j}}{r}}h_{j}(\lceil
x_{j+1}\rfloor^{q_{j}}-\lceil x_{j+1}^{*}\rfloor^{q_{j}}).$

It is easy to see that the virtual controller

$x_{j+1}^{*}=-\beta_{j}\lceil\xi_{j}\rfloor^{\frac{r_{j+1}}{r}}$

and

$\displaystyle\beta_{j}=$
$\displaystyle\left(\frac{(n-j+1)k_{1}}{\overline{h}_{j}k_{4}}+\frac{\alpha_{1,j}+\alpha_{2,j}+\alpha_{3,j}+\alpha_{4,j}}{\overline{h}_{j}}\right.$
$\displaystyle\left.+\frac{k_{2}}{k_{4}\overline{h}_{j}}|\xi_{j}|^{k_{3}}\right)^{\frac{1}{q_{j}}}.$

Then

$\displaystyle\mathcal{L}_{j}V_{j}(\bar{x}_{j})$ $\displaystyle\leq$
$\displaystyle-(n-j+1)\sum^{j}_{i=1}\frac{k_{1}}{k_{4}}|\xi_{i}|^{4}-\sum^{j}_{i=1}\frac{k_{2}}{k_{4}}|\xi_{i}|^{4+k_{3}}$
$\displaystyle+\lceil\xi_{j}\rfloor^{\frac{4r-\kappa-r_{j}}{r}}h_{j}(\lceil
x_{j+1}\rfloor^{q_{j}}-\lceil x_{j+1}^{*}\rfloor^{q_{j}}).$ (19)

From Steps 1-3, we have proved that (19) hold for any
$j\in\\{1,2,\cdots,n\\}$. Hence, at the $n$th step, one concludes that

$\displaystyle\mathcal{L}_{n}V_{n}(\bar{x}_{n})$ $\displaystyle\leq$
$\displaystyle-\sum^{n-1}_{i=1}\frac{k_{1}}{k_{4}}|\xi_{i}|^{4}-\sum^{n-1}_{i=1}\frac{k_{2}}{k_{4}}|\xi_{i}|^{4+k_{3}}+\lceil\xi_{n}\rfloor^{\frac{4r-\kappa-
r_{n}}{r}}h_{n}\lceil u\rfloor^{q_{n}}$
$\displaystyle+(\alpha_{1,n}+\alpha_{2,n}+\alpha_{3,n}+\alpha_{4,n})|\xi_{n}|^{4}$
$\displaystyle\leq$
$\displaystyle-\sum^{n}_{i=1}\frac{k_{1}}{k_{4}}|\xi_{i}|^{4}-\sum^{n}_{i=1}\frac{k_{2}}{k_{4}}|\xi_{i}|^{4+k_{3}}$

with the Lyapunov function as

$V_{n}(\bar{x}_{n})=V_{n-1}(\bar{x}_{n-1})+W_{n}(\bar{x}_{n})$

and

$W_{n}(\bar{x}_{n})=\int^{x_{n}}_{x_{n}^{*}}\lceil\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{n}^{*}\rfloor^{\frac{r}{r_{n}}}\rfloor^{\frac{4r-\kappa-r_{n}}{r}}\,ds.$

Consequently, the control $u$ can be designed as

$\displaystyle u$ $\displaystyle=$
$\displaystyle-\beta_{n}\lceil\xi_{n}\rfloor^{\frac{r_{n+1}}{r}}$
$\displaystyle=$
$\displaystyle-\left(\frac{k_{1}}{\bar{h}_{n}k_{4}}+\frac{k_{2}}{k_{4}\bar{h}_{n}}|\xi_{n}|^{k_{3}}+\frac{\alpha_{1,n}+\alpha_{2,n}+\alpha_{3,n}+\alpha_{4,n}}{\bar{h}_{n}}\right)^{\frac{1}{q_{n}}}$
$\displaystyle\cdot\lceil\xi_{n}\rfloor^{\frac{r_{n+1}}{r}}.$

Based on the proof of Proposition 6.4, we can get that

$\displaystyle\int^{x_{j}}_{x_{j}^{*}}\lceil\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}\rfloor^{\frac{4r-\kappa-r_{j}}{r}}ds\leq
2^{1-\frac{r_{j}}{r}}|\xi_{j}|^{\frac{4r-\kappa}{r}}$

and

$\displaystyle V_{n}\leq 2\sum^{n}_{i=1}|\xi_{i}|^{\frac{4r-\kappa}{r}}.$

Setting $k_{3}\geq-\frac{\kappa}{r}$ and using Lemma 3.5 arrive at

$\displaystyle-\sum^{n}_{i=1}|\xi_{i}|^{4+k_{3}}\leq-n^{\frac{-rk_{3}-\kappa}{4r-\kappa}}(\sum^{n}_{i=1}|\xi_{i}|^{\frac{4r-\kappa}{r}})^{\frac{r(4+k_{3})}{4r-\kappa}}$
$\displaystyle\leq$
$\displaystyle-n^{\frac{-rk_{3}-\kappa}{4r-\kappa}}\frac{1}{2}^{\frac{r(4+k_{3})}{4r-\kappa}}V_{n}^{\frac{r(4+k_{3})}{4r-\kappa}}$

and

$-\sum^{n}_{i=1}|\xi_{i}|^{4}\leq-\frac{1}{2}^{\frac{4r}{4r-\kappa}}V_{n}^{\frac{4r}{4r-\kappa}}.$

Accordingly, one has

$\displaystyle\mathcal{L}_{n}V_{n}(\bar{x}_{n})\leq$
$\displaystyle-\frac{k_{1}}{k_{4}}\left(\frac{V_{n}}{2}\right)^{\frac{4r}{4r-\kappa}}-\frac{k_{2}}{k_{4}}n^{\frac{-rk_{3}-\kappa}{4r-\kappa}}\left(\frac{V_{n}}{2}\right)^{\frac{4r+rk_{3}}{4r-\kappa}}.$

Set $k_{3}>-\frac{\kappa}{r}$, $b_{1}=\frac{4r}{4r-\kappa}$,
$b_{2}=\frac{4r+rk_{3}}{4r-\kappa}$, $a=2^{-b1}k_{1}$,
$k_{1}>\frac{2^{b1}}{1-b_{1}}$,
$k_{2}=\frac{a-ab_{1}}{(a-ab_{1}-1)(b_{2}-1)}n^{\frac{rk_{3}+\kappa}{4r-\kappa}}2^{b_{2}}$
. The control parameter $k_{4}$ can be chosen as an arbitrary positive number.
Then, considering Corollary 2.3, the origin of the closed-loop system is
stochastically predefined-time stabilizable and
$\sup_{x_{0}\in\mathcal{R}^{n}}\mathcal{E}T(x_{0})\leq k_{4}$.

##  4 Simulation Examples

In this section, we present two examples to illustrate the validity of our
main results.

######  Example 4.1.

In this example, we consider the following one-dimensional system

$\displaystyle dx=(x^{\frac{5}{3}}+u)dt+x^{2}dw.$ (20)

For system (20), the existing design schemes [12, 26] can not solve its
stochastic predefined-time stabilization. Through the design method proposed
in Corollary 2.3, the controller can be constructed as

$\displaystyle
u=-x^{\frac{5}{3}}-\frac{1}{2}x^{3}-\frac{k_{1}}{2k_{4}}x^{\frac{1}{3}}-\frac{k_{2}}{2k_{4}}x^{k_{3}},$

where $k_{1}>4$, $k_{2}=\frac{k_{1}}{3(\frac{k_{1}}{4}-1)\frac{k_{3}-1}{2}}$,
$k_{3}>1$, and $k_{4}$ is a positive control parameter that can be adjusted
arbitrarily. Then, the mathematical expectation of the first time to reach the
equilibrium point must be less than $k_{4}$. Here we choose $x(0)=1$,
$k_{1}=4.1$, $k_{3}=3$. In order to compare the convergence speed, $k_{4}$ is
selected as a variety of different values of $4$, $2$ and $0.5$. The
corresponding trajectories of the states for these groups of control
experiments are all described in Figure 1.

Figure 1:  The state $x(t)$ of the closed-loop system (20) in the cases of
$k_{4}=4,2,0.5$.

We have completed $25$ random experiments for $k_{4}=4,2,0.5$, respectively.
From Figure 1, we can find that smaller $k_{4}$ leads to faster convergence
speed. It is obvious that the rapidity of the convergence time meets the
requirement of the stochastic fixed-time stabilization proposed in this paper.

######  Example 4.2.

Figure 2: $x(t)$ of the closed-loop system (21) in the cases of $k_{4}=1$.

A simulation example is given to illustrate how to apply Theorem 3.1. Consider
the following nonlinear stochastic system:

$\displaystyle\begin{cases}dx_{1}=(h_{1}\lceil
x_{2}\rfloor^{\frac{5}{3}}-\lceil
x_{1}\rfloor^{\frac{3}{4}})dt+\sin(x_{1})|x_{1}|dw,\\\ dx_{2}=h_{2}\lceil
u\rfloor^{\frac{4}{3}}\,dt.\end{cases}$ (21)

Clearly, Assumptions 3.1 and 3.2 are satisfied with $\underline{h}_{2}=1$,
$\overline{h}_{2}=2$, $\phi_{1}=\varphi_{1}=1$, $\kappa=-\frac{1}{4}$,
$\varpi_{11}=0$, $\rho_{11}=\frac{1}{8}$. According to Theorem 3.1, we can
select the control parameters as $k_{1}=65.6$, $k_{2}=494.6$, $k_{3}=3.1$,
$k_{4}=3$, and the control input can be designed as
$u=-(10.9+3.5\beta_{1}(x_{1})+82.4|\xi_{2}|^{0.75})\lfloor\xi_{2}\rceil^{\frac{1}{8}}$
with $\beta_{1}(x_{1})=(47.1+165|\xi_{1}|^{3.1})^{0.6}$. Figure 2 shows that
the designed controller renders the origin of system (21) stochastic
predefined-time stabilization with $\mathcal{E}T_{f}\leq k_{4}$. This
demonstrates the usability of the given design method.

##  5 Conclusion

In this paper, we have obtained several Lyapunov-type results about stochastic
predefined-time stabilization of stochastic nonlinear systems, which extend
the results of finite-time and fixed-time stabilization. Furthermore, we have
presented feasible conditions and provided a constructive solution to
stochastic predefined-time stabilization of a class of stochastic nonlinear
systems in strict-feedback form. Two examples have been given to demonstrate
the validness of the obtained results.

## References

  * [1] Bhat, S. P., & Bernstein, D. S. (2000). Finite-time stability of continuous autonomous systems. _SIAM Journal on Control and optimization_ , 38(3): 751-766.
  * [2] Chen, W., & Jiao, L. C. (2010). Finite-time stability theorem of stochastic nonlinear systems. _Automatica_ , 46(12): 2105-2108.
  * [3] Cui, R. H., & Xie, X. J. (2021). Adaptive state-feedback stabilization of state-constrained stochastic high-order nonlinear systems. _Science China Information Sciences_ , 64(10):200203:1-200203:11.
  * [4] Ding, S., Li, S., & Zheng, W. (2012). Nonsmooth stabilization of a class of nonlinear cascaded systems. Automatica , 48: 2597-2606.
  * [5] Fang, L., Ma, L., Ding, S., & Zhao, D. (2019). Robust finite-time stabilization of a class of high-order stochastic nonlinear systems subject to output constraint and disturbances. International Journal of Robust and Nonlinear Control, 29(16): 5550-5573.
  * [6] Gao, F., Wu, Y., & Liu, Y. (2019). Finite-time stabilisation of a class of stochastic nonlinear systems with dead-zone input. International Journal of Control, 92(7): 1541-1550.
  * [7] Huang, S., & Xiang, Z. (2016). Finite-time stabilization of a class of switched stochastic nonlinear systems under arbitrary switching. International Journal of Robust and Nonlinear Control, 26(10): 2136-2152.
  * [8] Hong, Y., Xu, Y., Huang, J. (2002). Finite-time control for robot manipulators. Systems and Control letters, 46(4): 243-253.
  * [9] Has’minskii, R. Z. (1980). Stochastic Stability of Differential Equations. Alphen: Sijtjoff and Noordhoff.
  * [10] Rogers, L. C. G., & Williams, D. (2000). _Diffusions, Markov processes and martingales: volume 1, foundations_. Cambridge university press.
  * [11] Jiménez-Rodríguez, E., Muñoz-Vázquez, A. J., Sánchez-Torres, J. D., & Loukianov, A. G. (2018). A note on predefined-time stability. _IFAC-PapersOnLine_ , 51(13): 520-525.
  * [12] Jiménez-Rodríguez, E., Muñoz-Vázquez, A. J., Sánchez-Torres, J. D., Defoort, M., & Loukianov, A. G. (2020). A Lyapunov-like characterization of predefined-time stability. IEEE Transactions on Automatic Control, 65(11): 4922-4927.
  * [13] Khoo, S., Yin, J., Man, Z., & Yu, X. (2013). Finite-time stabilization of stochastic nonlinear systems in strict-feedback form. Automatica, 2013, 49(5): 1403-1410.
  * [14] Krishnamurthy, P., Khorrami, F., & Krstic, M. (2020). Robust adaptive prescribed-time stabilization via output feedback for uncertain nonlinear strict-feedback-like systems. European Journal of Control, 55: 14-23.
  * [15] Liang, Y., Li, Y. X., & Hou, Z. (2021). Adaptive fixed-time tracking control for stochastic pure-feedback nonlinear systems. International Journal of Adaptive Control and Signal Processing, 35(9): 1712-1731.
  * [16] Li, W., & Krstic, M. (2021). Stochastic nonlinear prescribed-time stabilization and inverse optimality, IEEE Transactions on Automatic Control, doi: 10.1109/TAC.2021.3061646.
  * [17] Li. H., Zhao. S., He. W., & Lu, R. (2019). Adaptive finite-time tracking control of full state constrained nonlinear systems with dead-zone. Automatica, 100: 99-107.
  * [18] Muñoz-Vázquez, A. J., Fernández-Anaya G, Sánchez-Torres, J. D., & Meléndez-Vázquez, F. (2021). Predefined-time control of distributed-order systems. Nonlinear Dynamics, 103(3): 2689-2700.
  * [19] Mao. X. (2007). Stochastic differential equations and applications. 2nd Edition, Horwood.
  * [20] Ni, J., Liu, L., Tang, Y., & Liu, C. (2021). Predefined-time consensus tracking of second-order multiagent systems. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 51(4): 2550-2560.
  * [21] Polyakov, A. (2011). Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Transactions on Automatic Control, 57(8): 2106-2110.
  * [22] Qian, C., & Lin, W. (2001). A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Transactions on Automatic Control, 46: 1061-1079.
  * [23] Sánchez-Torres, J. D., Muñoz-Vázquez, A. J., Defoort, M., Jiménez-Rodríguez, E., & Loukianov, A. G. (2020). A class of predefined-time controllers for uncertain second-order systems. European Journal of Control, 53: 52-58.
  * [24] Sánchez-Torres, J. D., Sanchez, E. N., & Loukianov, A. G. (2015). Predefined-time stability of dynamical systems with sliding modes. 2015 American Control Conference, July 1-3, 2015, 5842-5846.
  * [25] Sun, Z. Y., Yun, M. M., & Li, T. (2017). A new approach to fast global finite-time stabilization of high-order nonlinear system. Automatica, 81: 455-463.
  * [26] Song, Z., Li. P., Zhai, J., Wang. Z., Huang. X. (2020). Global fixed-time stabilization for switched stochastic nonlinear systems under rational switching powers. Applied Mathematics and Computation, 387: 124856.
  * [27] Song. Y., Wang. Y., Holloway. J., & Krstic, M. (2017). Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time. Automatica, 83: 243-251.
  * [28] Wang, Y., Song, Y., Hill, D. J., & Krstic, M., (2019). Prescribed-time consensus and containment control of networked multiagent systems. IEEE Transactions on Cybernetics, 49(4): 1138-1147.
  * [29] Xie, X. J., Duan. N., & Zhao, C. R. (2014). A combined homogeneous domination and sign function approach to output-feedback stabilization of stochastic high-order nonlinear systems, IEEE Transactions on Automatic Control 59(5): 1303-1309.
  * [30] Xie, X. J., $\&$ Li, G. J. (2019). Finite-time output-feedback stabilization of high-order nonholonomic systems. International Journal of Robust and Nonlinear Control, 29(9): 2695-2711.
  * [31] Xie, X. J., & Liu, L. (2013). A Homogeneous domination approach to state feedback of stochastic high-order nonlinear systems with time-varying delay. IEEE Transactions on Automatic Control, 58(2): 494-499.
  * [32] Yin, J., Khoo, S., Man, Z., & Yu, X. (2011). Finite-time stability and instability of stochastic nonlinear systems. _Automatica_ , 47: 1288-1292.
  * [33] Yin. J., & Khoo, S. (2015). Continuous finite-time state feedback stabilizers for some nonlinear stochastic systems, International Journal of Robust and Nonlinear Control, 25(11): 1581-1600.
  * [34] Yu. X., Yin, J., $\&$ Khoo, S. (2019). Generalized Lyapunov criteria on finite-time stability of stochastic nonlinear systems. _Automatica_ , 107: 183-189.
  * [35] Zhang, W., Xie, L., $\&$ Chen, B. S. (2017). Stochastic $H_{2}/H_{\infty}$ Control: A Nash Game Approach. CRC Press.
  * [36] Zhang, T., Deng, F., $\&$ Zhang, W. (2021). Robust $H_{\infty}$ filtering for nonlinear discrete-time stochastic systems. Automatica, 123: 109343.

##  6 Appendix

The following proposition can be obtained from [26].

######  Proposition 6.1.

The following statements accordingly hold when the positive definite and
proper Lyapunov function $V_{j}$ and $W_{j}$ are defined by (16) and (17)
respectively.

$\displaystyle\frac{\partial W_{j}}{\partial
x_{j}}=\lceil\xi_{j}\rfloor^{\frac{4r-\kappa-r_{j}}{r}},$
$\displaystyle\frac{\partial W_{j}}{\partial x_{i}}=-\frac{4r-\kappa-
r_{j}}{r}\int^{x_{j}}_{x_{j}^{*}}|\lceil s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}|^{\frac{3r-\kappa-r_{j}}{r}}ds$
$\displaystyle\ \ \cdot\frac{\partial\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial x_{i}},$
$\displaystyle\frac{\partial^{2}W_{j}}{\partial^{2}x_{j}}=\frac{4r-\kappa-
r_{j}}{r_{j}}|x_{j}|^{\frac{r-r_{j}}{r_{j}}}|\xi_{j}|^{\frac{3r-\kappa-
r_{j}}{r}},$ $\displaystyle\frac{\partial^{2}W_{j}}{\partial x_{i}\partial
x_{j}}=\frac{\partial^{2}W_{j}}{\partial x_{j}\partial
x_{i}}=-\frac{4r-\kappa-r_{j}}{r}|\xi_{j}|^{\frac{3r-\kappa-
r_{j}}{r}}\frac{\partial\lceil x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial
x_{i}},$ $\displaystyle\frac{\partial^{2}W_{j}}{\partial x_{i}\partial
x_{m}}=\frac{(4r-\kappa-r_{j})(3r-\kappa-r_{j})}{r^{2}}\frac{\partial\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial x_{i}}\frac{\partial\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial x_{m}}$
$\displaystyle\cdot\int^{x_{j}}_{x_{j}^{*}}\lceil\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}\rfloor^{\frac{2r-\kappa-
r_{j}}{r}}\,ds-\frac{4r-\kappa-r_{j}}{r}\frac{\partial^{2}\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial x_{i}\partial x_{m}}$
$\displaystyle\cdot\int^{x_{j}}_{x_{j}^{*}}|\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}|^{\frac{3r-\kappa-r_{j}}{r}}\,ds,$

where $i,m<j$.

Next, we need to prove some necessary propositions listed below in order to
estimate some terms in (3).

######  Proposition 6.2.

There exists a non-negative smooth function $\alpha_{1j}(\bar{x}_{j})$ such
that

$\displaystyle\lceil\xi_{j-1}\rfloor^{\frac{4r-\kappa-
r_{j-1}}{r}}h_{j-1}(\lceil x_{j}\rfloor^{q_{j-1}}-\lceil
x_{j}^{*}\rfloor^{q_{j-1}})$ $\displaystyle\leq$
$\displaystyle\frac{k_{1}}{4k_{4}}|\xi_{j-1}|^{4}+\alpha_{1j}(\bar{x}_{j})|\xi_{j}|^{4}.$

######  Proposition 6.3.

There exists a non-negative smooth function $\alpha_{2j}(\bar{x}_{j})$ such
that

$\displaystyle\lceil\xi_{j}\rfloor^{\frac{4r-\kappa-
r_{j}}{r}}f_{j}(\bar{x}_{j})\leq\frac{k_{1}}{4k_{4}}\sum^{j-1}_{i=1}|\xi_{i}|^{4}+\alpha_{2j}(\bar{x}_{j})|\xi_{j}|^{4}.$

######  Proposition 6.4.

There exists a non-negative smooth function $\alpha_{3j}(\bar{x}_{j})$ such
that

$\sum^{j-1}_{i=1}\frac{\partial W_{j}}{\partial x_{i}}(h_{i}\lceil
x_{i+1}\rfloor^{q_{i}}+f_{i}(\bar{x}_{i}))\leq\frac{k_{1}}{4k_{4}}\sum^{j-1}_{i=1}|\xi_{i}|^{4}+\alpha_{3j}(\bar{x}_{j})|\xi_{j}|^{4}.$

######  Proposition 6.5.

There exists a non-negative smooth function $\alpha_{4j}(\bar{x}_{j})$ such
that

$\frac{1}{2}\sum^{j}_{i,m=1}\left|\frac{\partial^{2}W_{j}}{\partial
x_{i}\partial
x_{m}}\right|\left|g_{i}(\bar{x}_{i})^{\prime}g_{m}(\bar{x}_{m})\right|\leq\frac{k_{1}}{4k_{4}}\sum^{j-1}_{i=1}|\xi_{i}|^{4}+\alpha_{4,j}|\xi_{j}|^{4}.$

Proof of Proposition 6.2

$\displaystyle\lceil\xi_{j-1}\rfloor^{\frac{4r-\kappa-
r_{j-1}}{r}}h_{j-1}(\lceil x_{j}\rfloor^{q_{j-1}}-\lceil
x_{j}^{*}\rfloor^{q_{j-1}})$ $\displaystyle\leq$
$\displaystyle|\xi_{j-1}|^{\frac{4r-\kappa-r_{j-1}}{r}}\bar{h}_{j-1}|(\lceil
x_{j}\rfloor^{q_{j-1}\frac{r}{r_{j}}})^{\frac{r_{j}}{r}}-(\lceil
x_{j}^{*}\rfloor^{q_{j-1}\frac{r}{r_{j}}})^{\frac{r_{j}}{r}}|.$

Based on Lemmas 3.2 and 3.3, and $q_{j-1}r_{j}=r_{j-1}+\kappa$, the above
inequality leads to

$\displaystyle\lceil\xi_{j-1}\rfloor^{\frac{4r-\kappa-
r_{j-1}}{r}}\bar{h}_{j-1}(\lceil x_{j}\rfloor^{q_{j-1}}-\lceil
x_{j}^{*}\rfloor^{q_{j-1}})$ $\displaystyle\leq$
$\displaystyle|\xi_{j-1}|^{\frac{4r-\kappa-
r_{j-1}}{r}}\bar{h}_{j-1}2^{1-\frac{r_{j-1}+\kappa}{r}}|\lceil
x_{j}\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}|^{\frac{r_{j-1}+\kappa}{r}}$
$\displaystyle\leq$
$\displaystyle\frac{k_{1}}{4k_{4}}|\xi_{j-1}|^{4}+\bar{h}_{j-1}2^{1-\frac{r_{j-1}+\kappa}{r}}\frac{r_{j-1}+\kappa}{4r}\Phi_{j}^{-\frac{4r-\kappa-
r_{j-1}}{r_{j-1}+\kappa}}|\xi_{j}|^{4},$

where the real-valued function $\Phi_{j}$ can be selected as
$\Phi_{j}=\frac{rk_{1}}{k_{4}(4r-\kappa-
r_{j-1})\bar{h}_{j-1}2^{1-\frac{r_{j-1}+\kappa}{r}}}$. The proof is completed.

Proof of Proposition 6.3 In view of Assumption 3.1, we have

$\displaystyle\lceil\xi_{j}\rfloor^{\frac{4r-\kappa-
r_{j}}{r}}f_{j}(\bar{x}_{j})$ $\displaystyle\leq$
$\displaystyle|\xi_{j}|^{\frac{4r-\kappa-
r_{j}}{r}}\bar{\phi}_{j}(\bar{x}_{j})\sum^{j}_{i=1}||\xi_{i}|+|\beta_{i-1}|^{\frac{r}{r_{i}}}|\xi_{i-1}||^{\frac{r_{j}+\kappa}{r}}$

with $\xi_{i}=0$. Note that $r_{j}+\kappa<r$. It is obtained from Lemma 3.4
that

$\displaystyle\lceil\xi_{j}\rfloor^{\frac{4r-\kappa-
r_{j}}{r}}f_{j}(\bar{x}_{j})$ $\displaystyle\leq$
$\displaystyle|\xi_{j}|^{\frac{4r-\kappa-
r_{j}}{r}}\bar{\phi}_{j}(\bar{x}_{j})\left(\sum^{j-1}_{i=1}(1+|\beta_{i}|^{\frac{r_{j}+\kappa}{r_{i+1}}})|\xi_{i}|^{\frac{r_{j}+\kappa}{r}}+|\xi_{j}|^{\frac{r_{j}+\kappa}{r}}\right).$

From Lemma 3.3,

$\displaystyle\lceil\xi_{j}\rfloor^{\frac{4r-\kappa-
r_{j}}{r}}f_{j}(\bar{x}_{j})\leq\frac{k_{1}}{4k_{4}}\sum^{j-1}_{i=1}|\xi_{i}|^{4}+\alpha_{2j}(\bar{x}_{j})|\xi_{j}|^{4}$

where $\alpha_{2j}$ is a non-negative smooth function. The proof is completed.

Proof of Proposition 6.4

According to Lemma 3.4 and the proof of Proposition 6.3, we have
$f_{i}(\bar{x}_{i})\leq\bar{\phi}_{i}(\bar{x}_{i})\left(\sum^{i-1}_{m=1}(1+|\beta_{m}|^{\frac{r_{i}+\kappa}{r_{m+1}}})|\xi_{m}|^{\frac{r_{i}+\kappa}{r}}+|\xi_{i}|^{\frac{r_{i}+\kappa}{r}}\right)$,
and $\lceil
x_{i+1}\rfloor^{q_{i}}\leq|\xi_{i+1}|^{\frac{r_{i}+\kappa}{r}}+|\beta_{i}|^{\frac{r_{i}+\kappa}{r_{i+1}}}|\xi_{i}|^{\frac{r_{i}+\kappa}{r}}.$
Considering Proposition 6.1, we can get that

$\displaystyle\sum^{j-1}_{i=1}\frac{\partial W_{j}}{\partial
x_{i}}(h_{i}\lceil x_{i+1}\rfloor^{q_{i}}+f_{i}(\bar{x}_{i}))$
$\displaystyle\leq$ $\displaystyle\sum^{j-1}_{i=1}\left|\frac{4r-\kappa-
r_{j}}{r}\frac{\partial\lceil x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial
x_{i}}\int^{x_{j}}_{x_{j}^{*}}|\lceil s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}|^{\frac{3r-\kappa-r_{j}}{r}}ds\right|$
$\displaystyle\Bigg{[}\bar{\phi}_{i}(\bar{x}_{i})\left(\sum^{i-1}_{m=1}(1+|\beta_{m}|^{\frac{r_{i}+\kappa}{r_{m+1}}})|\xi_{m}|^{\frac{r_{i}+\kappa}{r}}+|\xi_{i}|^{\frac{r_{i}+\kappa}{r}}\right)$
$\displaystyle+|\xi_{i+1}|^{\frac{r_{i}+\kappa}{r}}+|\beta_{i}|^{\frac{r_{i}+\kappa}{r_{i+1}}}|\xi_{i}|^{\frac{r_{i}+\kappa}{r}}\Bigg{]}.$
(22)

Using Lemma 3.2, one has

$\displaystyle\int^{x_{j}}_{x_{j}^{*}}\lceil\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}\rfloor^{\frac{3r-\kappa-r_{j}}{r}}ds\leq
2^{1-\frac{r_{j}}{r}}|\xi_{j}|^{\frac{3r-\kappa}{r}}.$ (23)

Next, by induction, we can estimate

$\displaystyle\left|\frac{\partial\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial
x_{i}}\right|\leq\Theta_{1ji}\sum^{j-1}_{l=1}|\xi_{l}|^{\frac{r-r_{i}}{r}},\
j>i\geq 1,$ (24)

where $\Theta_{1ji}$ is a non-negative smooth function. According to (6), (23)
and (24), we have

$\displaystyle\sum^{j-1}_{i=1}\frac{\partial W_{j}}{\partial
x_{i}}(h_{i}\lceil x_{i+1}\rfloor^{q_{i}}+f_{i}(\bar{x}_{i}))$
$\displaystyle\leq$ $\displaystyle\sum^{j-1}_{i=1}\frac{4r-\kappa-
r_{j}}{r}2^{1-\frac{r_{j}}{r}}|\xi_{j}|^{\frac{3r-\kappa}{r}}\Theta_{1ji}\sum^{j-1}_{l=1}|\xi_{l}|^{\frac{r-r_{i}}{r}}\Theta_{2i}$
$\displaystyle\sum^{j}_{m=1}|\xi_{m}|^{\frac{r_{i}+\kappa}{r}}$
$\displaystyle\leq$
$\displaystyle\frac{k_{1}}{4k_{4}}\sum^{j-1}_{i=1}|\xi_{i}|^{4}+\alpha_{3j}(\bar{x}_{j})|\xi_{j}|^{4}.$

where $\alpha_{3j}$ is a non-negative smooth function. Proposition 6.4 is
proved.

Proof of Proposition 6.5

$\displaystyle\frac{1}{2}\sum^{j}_{i,m=1}\left|\frac{\partial^{2}W_{j}}{\partial
x_{i}\partial
x_{m}}\right|\left|g_{i}(\bar{x}_{i})^{\prime}g_{m}(\bar{x}_{m})\right|$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left|\frac{\partial^{2}W_{j}}{\partial^{2}x_{j}}\right|\left|g_{j}(\bar{x}_{j})^{\prime}g_{j}(\bar{x}_{j})\right|+\sum^{j-1}_{i=1}\left|\frac{\partial^{2}W_{j}}{\partial
x_{j}\partial
x_{i}}\right|\left|g_{j}(\bar{x}_{j})^{\prime}g_{i}(\bar{x}_{i})\right|$
$\displaystyle+\frac{1}{2}\sum^{j-1}_{i,m=1}\left|\frac{\partial^{2}W_{j}}{\partial
x_{i}\partial
x_{m}}\right|\left|g_{i}(\bar{x}_{i})^{\prime}g_{m}(\bar{x}_{m})\right|.$

Firstly, based on Propositions 6.1, 6.3 and Lemma 3.5, we have

$\displaystyle\frac{1}{2}\left|\frac{\partial^{2}W_{j}}{\partial^{2}x_{j}}\right|\left|g_{j}(\bar{x}_{j})^{\prime}g_{j}(\bar{x}_{j})\right|$
$\displaystyle\leq$ $\displaystyle\frac{(4r-\kappa-
r_{j})j}{2r_{j}}\Bigg{(}|\xi_{j}|^{\frac{r-r_{j}}{r}}+|\beta_{j-1}|^{\frac{r-r_{j}}{r_{j}}}|\xi_{j-1}|^{\frac{r-r_{j}}{r}}\Bigg{)}\bar{\varphi}_{j}(\bar{x}_{j})^{2}$
$\displaystyle|\xi_{j}|^{\frac{3r-\kappa-
r_{j}}{r}}\Bigg{(}\sum^{j-1}_{s=1}(1+|\beta_{s}|^{\frac{2r_{j}+\kappa}{r_{s+1}}})|\xi_{s}|^{\frac{2r_{j}+\kappa}{r}}+|\xi_{j}|^{\frac{2r_{j}+\kappa}{r}}\Bigg{)}$
$\displaystyle\leq$
$\displaystyle\Xi_{1j}(\zeta_{1j},\bar{x}_{j})|\xi_{j}|^{4}+\zeta_{1j}\sum^{j-1}_{s=1}|\xi_{s}|^{4},$
(25)

where $\Xi_{1j}(\zeta_{1j},\bar{x}_{j})$ is a positive real function and
$\zeta_{1j}$ can be an any positive real number. Next, we will consider

$\displaystyle\sum^{j-1}_{i=1}\left|\frac{\partial^{2}W_{j}}{\partial
x_{j}\partial
x_{i}}\right|\left|g_{j}(\bar{x}_{j})^{\prime}g_{i}(\bar{x}_{i})\right|$
$\displaystyle\leq$ $\displaystyle\sum^{j}_{i=1}\frac{4r-\kappa-
r_{j}}{r}|\xi_{j}|^{\frac{3r-\kappa-
r_{j}}{r}}\Theta_{1ji}\sum^{j-1}_{l=1}|\xi_{l}|^{\frac{r-r_{i}}{r}}\left|g_{j}(\bar{x}_{j})^{\prime}g_{i}(\bar{x}_{i})\right|.$

Note that

$\left|g_{j}(\bar{x}_{j})^{\prime}g_{i}(\bar{x}_{i})\right|\leq\bar{\varphi}_{i}(\bar{x}_{i})\bar{\varphi}_{j}(\bar{x}_{j})\Theta_{2ji}\sum^{j}_{s=1}|\xi_{s}|^{\frac{r_{j}+r_{i}+\kappa}{r}}$

with $\Theta_{2ji}$ being a positive continuous function. Summarizing the
above analysis leads to that

$\displaystyle\sum^{j-1}_{i=1}\left|\frac{\partial^{2}W_{j}}{\partial
x_{j}\partial
x_{i}}\right|\left|g_{j}(\bar{x}_{j})^{\prime}g_{i}(\bar{x}_{i})\right|$
$\displaystyle\leq$ $\displaystyle\sum^{j}_{i=1}\frac{(4r-\kappa-
r_{j})j}{r}|\xi_{j}|^{\frac{3r-\kappa-
r_{j}}{r}}\Theta_{1ji}\Theta_{2ji}\bar{\varphi}_{i}(\bar{x}_{i})\bar{\varphi}_{j}(\bar{x}_{j})$
$\displaystyle\sum^{j-1}_{l=1}|\xi_{l}|^{\frac{r-r_{i}}{r}}\sum^{j}_{s=1}|\xi_{s}|^{\frac{r_{j}+r_{i}+\kappa}{r}}$
$\displaystyle\leq$
$\displaystyle\Xi_{2j}(\zeta_{2j},\bar{x}_{j})|\xi_{j}|^{4}+\zeta_{2j}\sum^{j-1}_{l=1}|\xi_{l}|^{4},$
(26)

where $\Xi_{2j}(\zeta_{2j},\bar{x}_{j})$ is a positive real function and
$\zeta_{2j}$ can be an any positive real number.

Finally, we have the following estimation:

$\displaystyle\frac{1}{2}\sum^{j-1}_{i,m=1}\left|\frac{\partial^{2}W_{j}}{\partial
x_{i}\partial
x_{m}}\right|\left|g_{i}(\bar{x}_{i})^{\prime}g_{m}(\bar{x}_{m})\right|$
$\displaystyle\leq$ $\displaystyle\sum^{j-1}_{i,m=1}\Bigg{|}\frac{(4r-\kappa-
r_{j})(3r-\kappa-r_{j})}{r^{2}}\frac{\partial\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial x_{i}}\frac{\partial\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial x_{m}}$
$\displaystyle\int^{x_{j}}_{x_{j}^{*}}\lceil\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}\rfloor^{\frac{2r-\kappa-
r_{j}}{r}}ds-\frac{4r-\kappa-r_{j}}{r}\frac{\partial^{2}\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial x_{i}\partial x_{m}}$
$\displaystyle\int^{x_{j}}_{x_{j}^{*}}|\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}|^{\frac{3r-\kappa-
r_{j}}{r}}ds\Bigg{|}\bar{\varphi}_{i}(\bar{x}_{i})\bar{\varphi}_{m}(\bar{x}_{m})$
$\displaystyle\Theta_{2im}\sum^{j-1}_{s=1}|\xi_{s}|^{\frac{r_{m}+r_{i}+\kappa}{r}}.$

According to (23) and (24), we have $\int^{x_{j}}_{x_{j}^{*}}\lceil\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}\rfloor^{\frac{2r-\kappa-r_{j}}{r}}ds\leq
2^{1-\frac{r_{j}}{r}}|\xi_{j}|^{\frac{2r-\kappa}{r}}$ and
$\left|\frac{\partial\lceil x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial
x_{i}}\right|\left|\frac{\partial\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial
x_{m}}\right|\leq\Theta_{3jim}\sum^{j-1}_{l=1}|\xi_{l}|^{\frac{2r-r_{i}-r_{m}}{r}}$
with $\Theta_{3jim}$ being a positive real function. Therefore, we can get
that

$\displaystyle\sum^{j-1}_{i,m=1}\frac{(4r-\kappa-r_{j})(3r-\kappa-
r_{j})}{r^{2}}\frac{\partial\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial x_{i}}\frac{\partial\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial x_{m}}$
$\displaystyle\int^{x_{j}}_{x_{j}^{*}}\lceil\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}\rfloor^{\frac{2r-\kappa-
r_{j}}{r}}ds\bar{\varphi}_{i}(\bar{x}_{i})\bar{\varphi}_{m}(\bar{x}_{m})$
$\displaystyle\Theta_{2im}\sum^{j-1}_{s=1}|\xi_{s}|^{\frac{r_{m}+r_{i}+\kappa}{r}}$
$\displaystyle\leq$
$\displaystyle\Xi_{3j}(\zeta_{3j},\bar{x}_{j})|\xi_{j}|^{4}+\zeta_{3j}\sum^{j-1}_{l=1}|\xi_{l}|^{4},$
(27)

where $\Xi_{3j}(\zeta_{3j},\bar{x}_{j})$ is a positive real function and
$\zeta_{3j}$ can be an any positive real number. Now, we are in a position to
consider $\sum^{j-1}_{i,m=1}\Bigg{|}\frac{4r-\kappa-
r_{j}}{r}\frac{\partial^{2}\lceil x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial
x_{i}\partial x_{m}}\int^{x_{j}}_{x_{j}^{*}}|\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}|^{\frac{3r-\kappa-
r_{j}}{r}}ds\Bigg{|}\bar{\varphi}_{i}(\bar{x}_{i})\bar{\varphi}_{m}(\bar{x}_{m})\Theta_{3im}\sum^{j-1}_{s=1}|\xi_{s}|^{\frac{r_{m}+r_{i}+\kappa}{r}}$.
Through calculation, we can obtain $\frac{\partial^{2}\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial x_{i}\partial
x_{m}}\leq\sum^{j-1}_{s=1}\Theta_{4jims}|\xi_{s}|^{\frac{r-r_{i}-r_{m}}{r}}$
with
$\Theta_{4jims}=\prod^{j-1}_{l=s+1}|\beta_{l}^{\frac{r}{r_{l+1}}}|\Bigg{|}\frac{\partial^{2}\beta_{s}^{\frac{r}{r_{s+1}}}}{\partial
x_{i}\partial
x_{m}}\Bigg{|}|\xi_{s}|^{\frac{r_{i}+r_{m}}{r}}+I_{i=m}\prod^{j-1}_{l=s}|\beta_{l}^{\frac{r}{r_{l+1}}}|.$
As a consequence,

$\displaystyle\sum^{j-1}_{i,m=1}\Bigg{|}\frac{4r-\kappa-
r_{j}}{r}\frac{\partial^{2}\lceil x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}}{\partial
x_{i}\partial x_{m}}\int^{x_{j}}_{x_{j}^{*}}\lceil\lceil
s\rfloor^{\frac{r}{r_{j}}}-\lceil
x_{j}^{*}\rfloor^{\frac{r}{r_{j}}}\rfloor^{\frac{3r-\kappa-
r_{j}}{r}}ds\Bigg{|}$
$\displaystyle\bar{\varphi}_{i}(\bar{x}_{i})\bar{\varphi}_{m}(\bar{x}_{m})\sum^{m}_{s=1}|x_{s}|^{\frac{2r_{m}+\kappa}{2r_{s}}}\sum^{i}_{s=1}|x_{s}|^{\frac{2r_{i}+\kappa}{2r_{s}}}$
$\displaystyle\leq\sum^{j-1}_{i,m=1}\frac{4r-\kappa-
r_{j}}{r}2^{1-\frac{r_{j}}{r}}|\xi_{j}|^{\frac{3r-\kappa}{r}}\bar{\varphi}_{i}(\bar{x}_{i})\bar{\varphi}_{m}(\bar{x}_{m})\sum^{j-1}_{s=1}\Theta_{4jims}$
$\displaystyle|\xi_{s}|^{\frac{r-r_{i}-r_{m}}{r}}\Theta_{3im}\sum^{j-1}_{s=1}|\xi_{s}|^{\frac{r_{m}+r_{i}+\kappa}{r}}$
$\displaystyle\leq\Xi_{4j}(\zeta_{4j},\bar{x}_{j})|\xi_{j}|^{4}+\zeta_{4j}\sum^{j-1}_{s=1}|\xi_{s}|^{4}.$
(28)

where $\Xi_{4j}(\zeta_{4j},\bar{x}_{j})$ is a positive real function and
$\zeta_{4j}$ can be an any positive real number. Based on (6), (6), (6) and
(6), we can find a suitable $\zeta_{ij}$ such that
$\sum^{4}_{i=1}\zeta_{ij}=\frac{k_{1}}{4k_{4}}$, which completes the proof of
Proposition 6.5.