[ style=chinese ]

1]organization=Shanghai Center for Mathematical Sciences, Fudan University,
addressline=2005 Songhu Road, city=Shanghai, postcode=200438, country=China

2]organization=School of Mathematical Sciences, Fudan University,
addressline=220 Handan Road, city=Shanghai, postcode=200433, country=China

3]organization=Research Institute of Intelligent Complex Systems, Fudan
University, addressline=220 Handan Road, city=Shanghai, postcode=200433,
country=China

[style=chinese]

[1]

[1]

[cor1]Corresponding author

# Finite-time stochastic control for complex dynamical systems: The estimate
for control time and energy consumption

Xiaoxiao Peng<EMAIL_ADDRESS>[  [  [  Shijie Zhou<EMAIL_ADDRESS>
###### Abstract

Controlling complex dynamical systems has garnered significant attention
within academic circles in recent decades. While most existing works have
focused on closed-loop control schemes with infinite-time durations, this
paper introduces a novel finite-time, closed-loop stochastic controller that
considers control time, energy consumption, and their dependence on system
parameters. This stochastic control technique not only enables finite-time
control in chaotic dynamical systems but also facilitates finite-time
synchronization in unidirectionally coupled systems. Importantly, our proposed
scheme offers several advantages over existing deterministic finite-time
controllers, particularly in terms of time and energy consumption. Through
numerical experiments utilizing random ecosystems, neural networks, and Lorenz
systems, we provide evidence of the effectiveness of our analytical results.
We anticipate that this stochastic scheme will find wide-ranging applications
in the control of complex dynamical systems and in achieving network
synchronization.

###### keywords:

finite-time, stochastic control, control time and energy consumption

##  1 Introduction

In recent years, controlling complex dynamical systems has attracted much
interest. Among various techniques developed for controlling, the
methodologies of open-loop have received much attention (refer to Wang et al.
(2016); Liu et al. (2008); Sorrentino (2007); Yu et al. (2009); Lin (2008a);
Jia et al. (2013); Yuan et al. (2014)). In the open-loop control, the agent
selects several nodes in advance and applies predefined state-independent
control signals to drive the system from an initial state to a desired target
(refer to da Silva Jr and Tarbouriech (2006); Colonius et al. (2013);
Beauchard et al. (2007)). It is direct and can reduce the complexity in many
cases. However, such controls are usually energy-consuming, unsustainable, and
less robust. To overcome these difficulties, the closed-loop control methods
are proposed and applied in many complex dynamical systems. To be precise,
several feedback schemes of close-loop control, have been validated to be
effective numerically or theoretically in controlling various kinds of systems
with or without time delay/noise (refer to Lin and Ma (2010); Lin (2008b);
Pyragas (2006); Lin and Chen (2006); Lin et al. (2017); Appleby et al. (2008);
Liu and Shen (2012); Li et al. (2004); Sorrentino et al. (2007); Zhou et al.
(2022a, 2017); Zhou and Lin (2021)).

Most closed-loop control schemes are often assumed to be deterministic,
overlooking the presence of noise in real-world systems. Therefore, designing
closed-loop control schemes for systems that are affected by noise is
imperative. Numerous analytical and numerical investigations have shown that
closed-loop control schemes can facilitate stochastic stabilization or
synchronization of noisy systems (see Lin and Chen (2006); Lin et al. (2017);
Appleby et al. (2008); Liu and Shen (2012); Hu and Mao (2008); Mao (2007); Mao
et al. (2007); Pecora and Carroll (1990); Feng et al. (2006); Zhou et al.
(2019, 2022b, 2023)). However, current closed-loop control schemes have
limited control flexibility and require infinite control time. A few non-
closed stochastic control schemes have been developed that can control
specific systems within a finite time frame (see Yang and Cao (2010); Runzi
and Yinglan (2012); Sun et al. (2012)). Hence, it is natural to ask whether we
can design a finite-time, closed-loop stochastic controller for general
systems.

Numerous researchers have made significant strides in exploring finite-time
stabilization of stochastic dynamical systems. For instance, some studies have
established finite-time criteria for switched nonlinear stochastic systems, as
evidenced in Hu and Zhu (2023); Huang and Xiang (2016a, b). Others have
devised Lyapunov criteria for finite-time stability of stochastic systems, as
exemplified in Yin et al. (2011); Yu et al. (2021); Zhao et al. (2018).
Furthermore, certain studies have investigated stochastic controllers
utilizing backstepping design to realize finite-time stability, as elaborated
in Yin and Khoo (2015); Zhao et al. (2018). However, to our best knowledge,
there remains a lack of research examining the estimate for control time and
energy consumption for finite-time stochastic controllers applied to general
dynamical systems.

In this paper, the above questions are fully addressed. Drawing inspiration
from finite-time closed-loop deterministic control schemes, as documented in
prior literature such as Sun et al. (2017); Yu et al. (2019); Chen et al.
(2020); Zhu et al. (2020); Chen and Jiao (2010); Chen et al. (2019); Haddad
and Lee (2022); Zhu et al. (2023); Wang et al. (2022), we propose a stochastic
scheme that can be employed to stabilize complex dynamical systems with high
probability in a finite time frame. Notably, our work differs from past
literature (as highlighted in Chen et al. (2019); Sun et al. (2017); Zhu et
al. (2023)) in that the controller is added on the ${\rm d}B_{t}$ term rather
than the ${\rm d}t$ term, which makes the analytical estimation more difficult
(See Remark 6). The impact of the coupling gain and parameters on control time
and energy is studied using several real biophysical systems, including random
ecosystems, neural networks and Lorenz systems. We ascertain upper bounds for
control time and energy consumption via analytical means. Our scheme aims to
achieve not only finite-time control in chaotic dynamical systems, but also
finite-time synchronization in unidirectionally coupled systems. Intriguingly,
compared to deterministic finite-time controllers, our new scheme depicts
notable advantages from a physical perspective concerning time and energy
consumption.

The rest of this paper is structured as follows. In Section 2, we introduce
the stochastic model that we seek to modulate and the closed-loop control
schemes. In Section 3, we establish upper bounds for both control time and
energy consumption, and we utilize several illustrative examples to
demonstrate the applicability of our schemes in stabilization and
synchronization. In Section 4, we compare the proposed stochastic controller
with the corresponding deterministic controller in terms of time and energy
consumption. In Section 5, we provide mathematically rigorous proofs to
support the conclusions obtained in Section 3. Finally, in Section 6, we offer
concluding remarks and potential directions for future research.

Notations. For any $\bm{x},\bm{y}\in\mathbb{R}^{n}$, $\|\bm{x}\|$ represents
the standard Euclidean norm of $\bm{x}$, while $\langle\bm{x},\bm{y}\rangle$
refers to the standard Euclidean inner product of $\bm{x}$ and $\bm{y}$.
${B}_{t}(t\geq 0)$ is a one-dimensional Brownian motion. Moreover, we denote
by $(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$ the
complete probability space with a filtration $\mathscr{F}_{t}$ satisfying the
usual condition (${\rm i.e.}$, it is increasing and right continuous while
$\mathscr{F}_{0}$ contains all $\mathbb{P}-$null sets) where the one-
dimensional Brownian motion $B_{t}$ is defined (See Eq. (1)). For a random
variable $\xi$, $\mathbb{E}\xi$ denotes its expectation. For a measurable set
$\bm{A}$, $\bm{1}_{\bm{A}}$ denotes the indicator function of $\bm{A}$. ${\rm
a.s.}$ represents almost surely in abbreviation. For $x\in\mathbb{R}$, ${\rm
Sgn}(x)$ denotes the sign of $x$, which indicates ${\rm Sgn}(x)=1,0$ and $-1$
for $x>0,x=0$ and $x<0$, respectively.

##  2 Model Formulation

We consider a $n$-dimensional stochastic controlled dynamical systems as
follows.

${\rm d}\bm{x}_{t}=\bm{f}(\bm{x}_{t}){\rm d}t+\bm{u}(\bm{x}_{t}){\rm d}B_{t}$
(1)

with initial state $\bm{x}(0)=\bm{x}_{0}$ is a random variable, where
$\bm{x}_{t}\in\mathbb{R}^{n}$ is the state vector, $\bm{f}(\bm{x})$ denotes
the vector field and $\bm{u}(\bm{x})$ denotes the stochastic controller to be
designed. ${B}_{t}$ is a one-dimensional Brownian motion and the filtration
$\mathscr{F}_{t}(t\geq 0)$ is generated by $\bm{x}_{0}$ and the Brownian
motion $B_{s}(s\leq t)$. According to the classical theory of stochastic
differential equations, we need locally Lipschitizian conditions for vector
$\bm{f}$ as follows.

###### Condition 1

(Locally Lipschitzian condition) For any given positive number $m$, there
exists a positive number $K_{m}$ such that

$\|\bm{f}(\bm{x})-\bm{f}(\bm{y})\|\leq K_{m}\|\bm{x}-\bm{y}\|$

for any $\|\bm{x}\|\leq m$ and $\|\bm{y}\|\leq m$. $\|\cdot\|$ represents for
Euclidean norm in $\mathbb{R}^{n}$.

We suppose that $\bm{x}_{t}\equiv\bm{0}$ is a solution for origin system. We
also need conditions as follows, which has been proposed and investigated in
past literatures (see Hu (2006); Abbaszadeh and Marquez (2010); Zhao et al.
(2010); Beikzadeh and Marquez (2016)).

###### Condition 2

(Globally one-sided Lipschitzian condition) There exists a positive number $L$
such that

$\langle\bm{x},\bm{f}(\bm{x})\rangle\leq L\|\bm{x}\|^{2}$

for any $\bm{x}\in\mathbb{R}^{n}$.

Denote by the stopping time when the trajectory first hits zero.

$\tau\triangleq\inf\bigg{\\{}t>0\bigg{|}\bm{x}_{t}=\bm{0}\bigg{\\}}.$

###### Definition 1

Eq. (1) is said to be finite-time stable if $\mathbb{P}(\tau<+\infty)=1$.
Equivalently speaking, $\tau$ exists finitely in the sense of probability one.

###### Definition 2

If Eq. (1) is finite-time stable, for $q>0$, denote by the $L^{q}$ energy
consumption

$\mathcal{E}_{q}\triangleq\int_{0}^{\tau}\|\bm{u}(\bm{x}_{s})\|^{q}{\rm d}s.$

If $\mathbb{E}\mathcal{E}_{q}<+\infty$, we say that the energy of the system
(1) is $L_{q}$ integrable.

###### Remark 1

In framework of deterministic control, the $L_{q}$ energy is usually defined
for $q>1$. However, in our framework of stochastic control, we usually
consider the situation $q<1$ (See Theorem 2).

In most existing closed-loop stochastic control systems, the feedback
controller has been designed as $\bm{u}(\bm{x})=k\bm{x}$, where the coupling
gain $k$ satisfies the condition $k>\sqrt{2L}$ (refer to Mao (1994); Lin and
Chen (2006)). This type of stochastic controller can asymptotically
exponentially stabilize the solution of (1) to the zero state under Condition
2. However, the convergence time is infinite. Therefore, we aim to find a
method to achieve a finite control time. As has been done in deterministic
dynamical systems (refer to Sun et al. (2017)), we propose the following
controller:

$\bm{u}(\bm{x})=k\bm{x}\cdot\bm{1}_{\|\bm{x}\|\geq
1}+k\|\bm{x}\|^{\alpha-1}\bm{x}\cdot\bm{1}_{\|\bm{x}\|<1},$ (2)

where $\alpha\in(0,1)$ denotes the steepness exponent, and $k>\sqrt{2L}$ is
the coupling gain.

By replacing the diffusion term $\bm{u}(\bm{x}_{t}){\rm d}\bm{B}_{t}$ with
$-\bm{u}(\bm{x}_{t}){\rm d}t$, the equation (1) can be transformed into a
closed-loop control for a deterministic system, which has been proven to
effectively stabilize the original ordinary differential equations ${\rm
d}\bm{x}_{t}=\bm{f}(\bm{x}_{t}){\rm d}t$ within a finite time duration. It is
important to note that this deterministic controller is similar to, but
distinct from, the one established in Sun et al. (2017) (refer to Remark 3).
The physical basis of $\bm{u}(\bm{x})$ is illustrated in Figure 1. Moreover,
it can be applied to achieve synchronization. But to our best knowledge, there
have not been systematic investigations on the time and energy consumption for
the stochastic closed-loop controller (2).

###### Remark 2

Since Brownian motion $\bm{B}_{t}$ is symmetric about zero, the control
$\bm{u}(\bm{x}_{t}){\rm d}\bm{B}_{t}$ is equivalent to
$-\bm{u}(\bm{x}_{t}){\rm d}\bm{B}_{t}$. This represents a significant
distinction between the stochastic controller and the deterministic controller
$-\bm{u}(\bm{x}_{t}){\rm d}t$.

###### Remark 3

In Sun et al. (2017), the authors established the following finite-time
closed-loop control

$\dot{\bm{x}}=\bm{f}(\bm{x})-\bm{v}(\bm{x}),$

where the controller
$\bm{v}(\bm{x})=[{v}_{1}(\bm{x}),{v}_{2}(\bm{x}),\cdots,{v}_{n}(\bm{x})]^{\rm
T}$ is denoted by

${v}_{i}(\bm{x})\triangleq\left\\{\begin{aligned} &kx_{i},\|\bm{x}\|>1,\\\
&k({\rm sgn}~{}x_{i})|x_{i}|^{\alpha},\|\bm{x}\|\leq 1.\end{aligned}\right.$

It is worthwhile to note that $\bm{u}(\bm{x})$ is similar to, but distinct
from $\bm{v}(\bm{x})$. Additionally, it should be emphasized that
$\bm{u}(\bm{x})$ is continuous in $\mathbb{R}^{n}$, whereas $\bm{v}(\bm{x})$
is not continuous on the surface of the unit ball $\|\bm{x}\|=1$. The reason
why we choose $\bm{u}(\bm{x})$ instead of $\bm{v}(\bm{x})$ lies in that the
major mathematical tool used in the proof of Theorem 1 and 2 is the
generalized It$\hat{o}$’s formula (Lemma 1), which requires the vector fields
to be continuous.

Figure 1: Physical underpinning of $\bm{u}(\bm{x})$ for the three- and one-
dimension case, respectively. Here,
$\bm{u}^{a}(\bm{x})=k\bm{x}\|\bm{x}\|^{\alpha-1}$,
$\bm{u}^{b}(\bm{x})=k\bm{x}$. Since controller (2) is designed as a stochastic
controller, it is possible for the trajectory to enter and exit the unit ball
multiple times before eventually converging to zero within a finite time
(refer to Remark 6). This behavior is significantly different from the
deterministic controller discussed in Sun et al. (2017).

Here, we first show that it can achieve stabilization as well as
synchronization via some examples and then give the detailed proofs.

##  3 Realization of Stabilization and Synchronization

###  3.1 Stabilization

To begin with, we will introduce two theorems that facilitate our
characterization of convergence time and energy consumption estimates. The
proof of the theorems will be provided in Section 5.

###### Theorem 1

Assume that Conditions 1,2 are valid. Then Eq. (1) with stochastic controller
(2) is finite-time stable for $k>\sqrt{2L}$. Furthermore, the time estimation
upper bound $T_{f}^{\rm Sup}$ satisfies that

$\mathbb{E}\tau\leq T_{f}^{\rm Sup}\triangleq\left\\{\begin{aligned}
&\dfrac{2\mathbb{E}(\log\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|>1})}{{k^{2}}-2L}\\\
&+\dfrac{8k^{2}}{(k^{2}-2L)^{2}}[\mathbb{P}(\|\bm{x}_{0}\|>1)\\\
&+\mathbb{E}(\|\bm{x}_{0}\|^{p^{*}};\bm{1}_{\|\bm{x}_{0}\|\leq
1})],\alpha<\dfrac{3}{4}+\dfrac{L}{2k^{2}},\\\
&\dfrac{2\mathbb{E}(\log\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|>1})}{{k^{2}}-2L}\\\
&+\dfrac{[\mathbb{P}(\|\bm{x}_{0}\|>1)+\mathbb{E}(\|\bm{x}_{0}\|^{2-2\alpha};\bm{1}_{\|\bm{x}_{0}\|\leq
1})]}{(1-\alpha)[(2\alpha-1)k^{2}-2L]},\\\
&\alpha\geq\dfrac{3}{4}+\dfrac{L}{2k^{2}},\end{aligned}\right.$

where $p^{*}\triangleq\dfrac{1}{2}-\dfrac{L}{k^{2}}$.

###### Theorem 2

Assume that Conditions 1,2 are valid. The system (1) with the controller (2)
is finite-time stable. When
$0<q<\min\left\\{2-2\alpha,1-\dfrac{2L}{k^{2}}\right\\}$, the energy is
$L_{q}$-integrable. Furthermore, we have the estimation of the $L_{q}$ energy
consumption as follows.

$\displaystyle\mathbb{E}\mathcal{E}_{q}\leq\mathcal{E}_{q}^{\rm Sup}$
$\displaystyle\triangleq-\dfrac{k^{q}}{H_{2}(q)}[\mathbb{E}(\|\bm{x}_{0}\|^{q})+2\mathbb{P}(\|\bm{x}_{0}\|>1)$
$\displaystyle+\mathbb{E}(\|\bm{x}_{0}\|^{q};\|\bm{x}_{0}\|\leq 1)],$

where $H_{2}(q)\triangleq qL+\dfrac{q(q-1)k^{2}}{2}<0.$

It can be easily checked $T_{f}^{\rm Sup}$ and $\mathcal{E}_{q}^{\rm Sup}$ are
decreasing with respect to the parameter $k$ when $\alpha$ and $\bm{x}_{0}$
are fixed. Let $k\to\sqrt{2L}+$, the time estimation $T_{f}^{\rm Sup}$ will
diverge, which aligns with our intuition. For the scalar system ${\rm
d}x_{t}=Lx_{t}+kx_{t}{\rm d}B_{t}$, it follows from It$\hat{o}$’s formula that
${\rm d}\log x_{t}=(L-\dfrac{k^{2}}{2}){\rm d}t+k{\rm d}B_{t}$, indicating
that the system converges or diverges exponentially at a rate of
$L-\dfrac{k^{2}}{2}$. As $k\to\sqrt{2L}+$, the system converges at a slower
rate. Since our controller $\eqref{2}$ aligns with $kx$ when $|x|>1$, it is
natural for the time estimation $T_{f}^{\rm Sup}$ to diverge.

We now discuss how $\alpha$ influences the convergence time. We firstly note
that $T_{f}^{\rm Sup}$ is continuous over $\alpha$. When
$\dfrac{3}{4}+\dfrac{L}{2k^{2}}\leq\alpha<1$, we first analyze the
monotonicity of the function
$J(\alpha)\triangleq(1-\alpha)[(2\alpha-1)k^{2}-2L]$. It can be calculated
that

$\frac{{\rm d}J}{{\rm d}\alpha}=-4k^{2}\alpha+3k^{2}+2L.$

So $\alpha=\dfrac{3}{4}+\dfrac{L}{2k^{2}}$ is the maximum point of function
$J(\alpha)$. And
$\mathbb{E}(\|\bm{x}_{0}\|^{2-2\alpha};\bm{1}_{\|\bm{x}_{0}\|\leq 1})]$ is
monotonically increasing over $\alpha\in(0,1)$ obviously, which implies that
$T_{f}^{\rm Sup}$ achieves minimum when
$\alpha\in\left(0,\dfrac{3}{4}+\dfrac{L}{2k^{2}}\right]$. Addtionally, when
$\alpha=1$, the controller specified in (2) becomes a infinite-time controller
and this aligns with the fact that the time estimation $T_{f}^{\rm Sup}$
diverges when $\alpha\to 1-$ (See Figure. 2). However, for
$\alpha\in[0,\dfrac{3}{4}+\dfrac{L}{2k^{2}})$, $T_{f}^{\rm Sup}$ is a
constant, which implies that the time estimation is conservative, leaving room
for further improvement.

Figure 2: The solid (blue) cureves indicate the dependence of the $T_{f}^{\rm
Sup}$ on $k$ and $\alpha$. And the dashed (red) curves indicate
$\alpha=3/4+L/(2k^{2})$. The parameters are set as $L=8$, $\|\bm{x}_{0}\|=3$.

We give some illustrative examples in the following.

###### Example 1

Consider a one-dimensional linear system described as

${\rm d}x_{t}=Lx_{t}{\rm d}t+u(x_{t}){\rm d}B_{t},$ (3)

where $L=2$. According to Theorem 1, when $k>2$, the system (3) is finite-time
stable. As shown in Figure. 3, the convergence time and energy consumption
show a decreasing tendency as $k$ increases when fix $\alpha$. As seen in
Figure. 3, the convergence time shows increasing tendency as $\alpha$
increases from $0$ to $1$ and diverges when $\alpha\to 1-$, while the energy
consumption does not have an obvious monotonic trend when $\alpha$ varies
between $(0,1)$. All these align with our estimations obtained in Theorem 1
and 2.

Figure 3: The convergence time and energy consumption for system (3) for
different values of $\alpha$ and $k$. The solid (blue) curves represent the
numerical evaluation of convergence time and energy consumption, which is
counted as the average of 1000 random realization of system (3). The dashed
(red) curves indicate the upper bound obtained from Theorem 1 and 2. (a) When
fix $\alpha$, convergence time and energy consumption shows a decreasing
tendency when $k$ increases. The parameters are $\alpha=0.5$, $q=1/2$. (b)When
fix $k$, convergence time shows an increasing tendency when $\alpha$ increases
while energy consumption does not have an obvious monotonic trend when
$\alpha$ varies. The parameters are $k=5,q=1/2$. For both (a) and (b), the
initial value is $x_{0}=10$. Here, Euler-Maruyama scheme (see Higham (2001);
Maruyama (1954)) with step size $\triangle t=10^{-6}$ is used for integrating
the system of stochastic differentian equation (3).

###### Example 2

Consider a two-dimensional nonlinear neural networks described as

${\rm d}\bm{x}_{t}=-\bm{C}\bm{x}_{t}{\rm d}t+\bm{A}\bm{g}(\bm{x}_{t}){\rm
d}t+\bm{u}(\bm{x}_{t}){\rm d}B_{t},$ (4)

where

$\bm{C}=\left[\begin{array}[]{ll}1&2\\\
3&4\end{array}\right],\bm{A}=\left[\begin{array}[]{ll}3&3\\\
1&3\end{array}\right],$

and the activation function
$\bm{g}(\bm{x})=[{g}_{1}(\bm{x}_{1}),{g}_{2}(\bm{x}_{2})]^{\rm T}$ satisfies
that $g_{1}(x)=\tanh(x)$ and $g_{2}(x)=\tanh(2x)$.

Using $|\tanh(x)|\leq|x|$ and $|\tanh(2x)|\leq 2|x|$, we obtain that

$\displaystyle\langle\bm{x},-\bm{C}\bm{x}+\bm{A}\bm{g}(\bm{x})\rangle$
$\displaystyle=-x_{1}^{2}-4x_{2}^{2}-5x_{1}x_{2}+3x_{1}\tanh(x_{1})$
$\displaystyle+3x_{2}\tanh(2x_{2})+x_{2}\tanh(x_{1})+3x_{1}\tanh(2x_{2})$
$\displaystyle\leq 2x_{1}^{2}+2x_{2}^{2}+12|x_{1}x_{2}|\leq
8(x_{1}^{2}+x_{2}^{2}).$

Thus, the Lipschitzian constant value $L$, which is defined in Condition 2,
can be taken as $8$. According to Theorem 1, we obtain that for each pair of
$k>4$ and $\alpha\in(0,1),$ the controlled system (4) is finite-time stable.
As shown in Figure 4, the numerical evaluation of the expectation of
convergence time, $\tau_{k}$, shows a decreasing tendency with the increase of
$k$. The numerical average value, $\tau_{k}$, is always much smaller than the
analytical upper bounds derived in Theorem 1, indicating that estimation
presented in our theorem is only in an approximating manner, leaving room for
further improvement.

Figure 4: Dependence of the mean value of the convergence time $\tau_{k}$ on
the coupling gain $k$ and $\alpha$. The solid (blue) curves indicate the
numerical evaluation of $\tau_{k}$, which is counted as the average of 100
random realizations of system (4). The dashed (red) curves indicate upper
bound obtained from Theorem 1. The initial states are $\bm{x}_{0}=[10,10]^{\rm
T}$.

###### Example 3

Consider May’s classic ecosystem May (1972, 2019) described by

${\rm d}\bm{x}_{t}=\bm{C}\bm{x}_{t}{\rm d}t+\bm{u}(\bm{x}_{t}){\rm d}B_{t},$
(5)

where each species $x_{i}$ is one-dimensional, $\bm{C}=(c_{ij})_{N\times N}$
describes the random mutual interactions with $c_{ii}=-r$, and $N$ is the
population size. The off-diagonal elements $c_{ij}$ are set as mutually
independent Gaussian random variables $\mathscr{N}(0,\sigma^{2})$ with
probability $p$ and the probability for the elements to be zero is $1-p$.

Denote each element of matrix $\bm{E}\triangleq\dfrac{1}{2}\left(\bm{C}^{\rm
T}+\bm{C}\right)$ by $e_{ij}=\dfrac{1}{2}(c_{ij}+c_{ji})$. The expectation is
$\mathbb{E}e_{ij}=0$ and the variance is given by

$\mathbb{D}e_{ij}=\dfrac{1}{4}\mathbb{D}(e_{ij}+e_{ji})=\dfrac{1}{2}\mathbb{D}e_{ij}=\dfrac{1}{2}\mathbb{E}e^{2}_{ij}=\dfrac{1}{2}p\sigma^{2}.$

It follows from the semicircle law for random matrices that the eigenvalues of
$\bm{E}$ are located in $[-r-\sqrt{2Np}\sigma,-r+\sqrt{2Np}\sigma]$ in a
probabilistic sense as $N\to+\infty$.

Since
$\left\langle\bm{x},\bm{C}\bm{x}\right\rangle=\left\langle\bm{x},\dfrac{1}{2}\left(\bm{C}^{\rm
T}+\bm{C}\right)\bm{x}\right\rangle$, the Lipschitzian constant $L$ can be
taken as $\eta_{\max}\triangleq-r+\sqrt{2Np}\sigma$ approximately. According
to Theorem 1, we obtain that when $k>\sqrt{2\eta_{\max}}$, the controlled
system (5) is finite-time stable.

Figure 5: The convergence time and the required energy consumption for (5),
vary respectively, with the increase of $N$. The parameters are
$p=\dfrac{1}{3}$, $\sigma=1$, $r=1$, $q=0.1$, $k=1.1\sqrt{2\eta_{\max}}$, and
the initial states are $\bm{x}_{0}=[1,1\cdots,1]^{\rm T}$. The solid (blue)
curves indicate the numerical evaluation of $\tau$ and $\mathcal{E}_{q}$,
which is counted as the average of 100 random realizations of system (5). The
dashed (red) curves indicate upper bound obtained from Theorem 1 and 2.

###### Remark 4

As shown in Figure. 3, our analytical upper bound slightly exceeds numerical
results in Example 1. However, Figures 4 and 5 demonstrate that in Examples 2
and 3, respectively, our analytical upper bound significantly surpasses the
numerical results. The reason for this phenomenon lies in our use of the
inequality $\langle\bm{x},\bm{f}(\bm{x})\rangle\leq L\|\bm{x}\|^{2}$ in
obtaining the estimations in Theorems 1 and 2. However, this estimation is
excessively conservative for Examples 2 and 3. Specifically, as
$k\to\sqrt{2L}-$, the upper bound estimation for convergence time diverges,
aligning with the numerical results in Example 1. Nevertheless, in Examples 2
and 3, $\langle\bm{x},\bm{f}(\bm{x})\rangle$ is considerably smaller than
$L\|\bm{x}\|^{2}$ in most of the time. Thus, our estimation is rough and
leaves room for further improvement.

###  3.2 Synchronization

Given that the synchronization problem can be viewed as a specific type of
stabilization problem, it is possible to utilize the stochastic controller (2)
for the purpose of achieving synchronization. More precisely, for the ordinary
differential equations

${\rm d}{\bm{x}}_{t}=\bm{f}(\bm{x}_{t}){\rm d}t,$ (6)

we can rewrite (1) as

${\rm d}\bm{y}_{t}=\bm{f}(\bm{y}_{t}){\rm
d}t+\bm{u}(\bm{y}_{t}-\bm{x}_{t}){\rm d}B_{t}$ (7)

to achieve $\lim_{t\rightarrow+\infty}\bm{z}_{t}=0$ with
$\bm{z}_{t}=\bm{y}_{t}-\bm{x}_{t}$. And we apply

$\mathcal{E}_{q}\triangleq\int_{0}^{\tau}\|\bm{u}(\bm{y}_{s}-\bm{x}_{s})\|^{q}{\rm
d}s.$

as the energy consumption here.

###### Example 4

We consider the following Hindmarsh-Rose model Hindmarsh and Rose (1984);
Storace et al. (2008):

$\displaystyle{\rm d}{\bm{x}}_{t}$ $\displaystyle=$
$\displaystyle{\bm{f}}({\bm{x}}_{t}){\rm d}t,$ $\displaystyle{\rm
d}{\bm{y}}_{t}$ $\displaystyle=$ $\displaystyle{\bm{f}}({\bm{y}}_{t}){\rm
d}t+{\bm{u}}({\bm{y}}_{t}-{\bm{x}}_{t}){\rm d}{B}_{t},$

where

$\bm{f}(\bm{x})=[x_{2}-x_{1}^{3}+3x_{1}^{2}-x_{3}+3,1-5x_{1}^{2}-x_{2},\epsilon(4x_{1}+6.4-x_{3})]^{\top}.$

By defining $\bm{z}=\bm{y}-\bm{x}$, it can be observed that
$\bm{F}(\bm{z}(t))=\bm{f}(\bm{y})-\bm{f}(\bm{x})=\bm{f}(\bm{z}+\bm{x}(t))-\bm{f}(\bm{x}(t))$
is not one-sided Lipschitzian with respect to $\bm{z}$. Nevertheless, the
achievement of synchronization can be demonstrated through Figures 6 and 7. In
Figure 7, it can be observed that in the absence of coupling term, two systems
exhibiting different initial values exhibit asynchronous spiking behavior.
However, with the introduction of the coupling diffusion term
$\bm{u}(\bm{y}_{t}-\bm{x}_{t})$, rapid synchronization between the two systems
is achieved. Furthermore, the convergence time and energy consumption display
a fluctuating decreasing trend as $k$ increases. It is likely that this
behavior is influenced by errors stemming from the Euler-Maruyama scheme.

Figure 6: The convergence time and the required energy consumption for (4),
respectively, vary with the increase of $k$. The parameters are
$\epsilon=0.005$, $\alpha=0.5$, $q=0.1$, and the initial states are
$\bm{x}_{0}=[0,0,1]^{\rm T}$, $\bm{x}_{0}=[0,0,2]^{\rm T}$. The solid (blue)
curves indicate the numerical evaluation of $\tau$ and $\mathcal{E}_{q}$,
which is counted as the average of 100 random realizations of the system (4).
Figure 7: The dynamics of membrane potential $x_{1}$ and $y_{1}$ with (upper)
and without (lower) the coupling diffusion term, respectively. The parameters
are $\epsilon=0.005$, $\alpha=0.5$, $q=0.1$, and the initial states are
$\bm{x}_{0}=[0,0,1]^{\rm T}$, $\bm{x}_{0}=[0,0,2]^{\rm T}$. Here, we only show
one numerical realization of the the system (4).

##  4 Comparison with Deterministic Scheme

In this section, we compare our stochastic control scheme with the
corresponding deterministic scheme. That is the substitution of (1) as
follows.

${\rm d}\bm{x}_{t}=\bm{f}(\bm{x}_{t}){\rm d}t-\bm{u}(\bm{x}_{t}){\rm d}t,$ (8)

where $\bm{u}$ is also set in the form of (2). This scheme has been rigorously
proved to be feasible for $k>L$, where $L$ represents the one-sided Lipschitz
constant. When $L>2$, we observe $\sqrt{2L}<L$, indicating that the demand for
$k$ is more stringent analytically for the deterministic scheme. Furthermore,
convergence time and energy consumption, which are discussed and compared in
the following example, serve as additional criteria for determining the
superiority of either scheme.

###### Example 5

Consider the Lorenz system Lorenz (1963); Robinson (1998), which reads

$\displaystyle{\rm d}{\bm{x}}_{t}$ $\displaystyle=$
$\displaystyle{\bm{f}}({\bm{x}}_{t}){\rm d}t,$

where

$\bm{f}(\bm{x})=[\sigma x_{2}-\sigma x_{1},\rho
x_{1}-x_{3}x_{1}-x_{2},x_{1}x_{2}-\beta x_{3}]^{\top}.$

Now, we have

$\displaystyle\langle{\bm{x}},\bm{f}(\bm{x})\rangle$ $\displaystyle=$
$\displaystyle(\sigma+\rho)x_{1}x_{2}-\sigma x_{1}^{2}-x_{2}^{2}-\beta
x_{3}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{\sigma+\rho}{2}(x_{1}^{2}+x_{2}^{2})$ $\displaystyle\leq$
$\displaystyle\frac{\sigma+\rho}{2}|\bm{x}|^{2}.$

So the Lipschitzian constant $L$ can be taken as $(\sigma+\beta)/2$ here. And
if we take $k>\sqrt{\sigma+\rho}$ and $k>(\sigma+\beta)/2$, the Lorenz system
can be stabilized in a finite time duration for the stochastic and
deterministic scheme, respectively. Figure 8 indicates the numerical
simulation of convergence time and energy consumption for two different
schemes.

It is clear that the convergence time of stochastic scheme is strictly smaller
than the deterministic one. Besides, except for some specific values of $k$,
the stochastic scheme consumes less energy than the deterministic one. This
implies that the stochastic closed-loop controllers are generally better than
the deterministic ones. But we also note that our scheme shows a large
fluctuation for each trial, and it is greatly influenced by the time interval
$\Delta t$ in Euler-Maruyama scheme (see Higham (2001); Maruyama (1954)). So
there is much room for the improvement of robustness in our control scheme.

Figure 8: The convergence time and the required energy consumption for (5),
vary respectively, with the increase of $k$. The parameters are $\sigma=6$,
$\rho=10$, $\beta=3$, $\alpha=0.5$, $q=0.5$, and the initial states are
$\bm{x}_{0}=[0,0,1]^{\rm T}$. The blue and red curves indicate the numerical
evaluation of $\tau$ and $\mathcal{E}_{q}$ for the stochastic and
deterministic control schemes, respectively, which is counted as the average
of 200 random realizations of system (5).

###### Example 6

We now consider the following two-dimension systems:

$\left\\{\begin{matrix}\frac{\mathrm{d}x_{1}}{\mathrm{d}t}=\cos x_{2},\\\
\frac{\mathrm{d}x_{2}}{\mathrm{d}t}=\sin x_{1},\end{matrix}\right.$

which starts from the state $[0,1]^{\top}$. It obviously satisfies the global
Lipschitzian condition. For another system with the same drift term but starts
from different initial value $\bm{y}_{t}$, we apply the scheme ${\rm
d}{\bm{y}}_{t}={\bm{f}}({\bm{y}}_{t}){\rm
d}t+{\bm{u}}({\bm{y}}_{t}-{\bm{x}}_{t}){\rm d}{B}_{t}$ and ${\rm
d}{\bm{y}}_{t}={\bm{f}}({\bm{y}}_{t}){\rm
d}t-{\bm{u}}({\bm{y}}_{t}-{\bm{x}}_{t}){\rm d}{t}$, respectively, to achieve
synchronization. We compare their synchronization time and the energy
consumption in Figure 9.

Figure 9: The synchronization time and the required energy consumption for
(6), vary respectively, with the increase of $k$. The parameters are
$\alpha=0.5$, $q=0.1$, and the initial states are $\bm{x}_{0}=[0,1]^{\rm T}$
and $\bm{y}_{0}=[1,0]^{\rm T}$. The blue and red curves indicate the numerical
evaluation of $\tau$ and $\mathcal{E}_{q}$ for the stochastic and
deterministic control schemes, respectively, which is counted as the average
of 100 random realizations of system (6).

The stochastic scheme here still shows great progress compared to the
deterministic scheme. It always shows a shorter transition to synchronization
as well as less energy consumption. This probably comes from some large value
of $\Delta B_{t}$, which can synchronize two systems drastically in a short
period. And a shorter synchronization time often means a less energy
dissipation. Thus, the volatility here contributes greatly to the close-loop
control.

##  5 Analytical Validation

The following generalized It$\hat{o}$’s formula, which can be found in
(Karatzas and Shreve, 1991, Problem 7.3), will be used in this section.

###### Lemma 1

Let

$X_{t}=X_{0}+M_{t}+V_{t},0\leq t<+\infty$

be a continuous semimartingale, where $M=\\{M_{t},\mathscr{F}_{t};0\leq
t<+\infty\\}$ is a local martingale, and $V=\\{V_{t},\mathscr{F}_{t};0\leq
t<+\infty\\}$ is the difference of continuous, nondecreasing adapted process
with $V_{0}=0$, and $\\{\mathscr{F}_{t}\\}$ satisfies the usual conditions. If
$f:\mathbb{R}\to\mathbb{R}$ be a function whose derivative is absolutely
continuous. Then $f^{\prime\prime}$ exists Lebesgue-almost everywhere, and we
have the generalized It$\hat{o}$’s formula:

$\displaystyle f(X_{t})=$ $\displaystyle
f(X_{0})+\int_{0}^{t}f^{\prime}(X_{s}){\rm
d}M_{s}+\int_{0}^{t}f^{\prime}(X_{s}){\rm d}V_{s}$
$\displaystyle+\dfrac{1}{2}\int_{0}^{t}f^{\prime\prime}(X_{s}){\rm d}\langle
M\rangle_{s}~{}~{}{\rm a.s.},$

where $\langle M\rangle$ denotes by the quadratic process of $M$.

###  5.1 Estimation of Control Time

Proof of Theorem 1: We introduce the following function,

$V_{p}(x)=\left\\{\begin{aligned} &p\log x+1,x>1,\\\ &x^{p},0\leq
x<1.\end{aligned}\right.$

We note that $V$ is a continuous function defined on $[0,+\infty)$ with
absolutely continuous first derivative on $(0,+\infty)$. Here, $p\in(0,1)$ is
a parameter to be identified later. Therefore, an application of the
generalized It$\hat{o}$’s formula (Lemma 1) yields that

$V_{p}(\|\bm{x}_{t}\|)-V_{p}(\|\bm{x}_{0}\|)=\int_{0}^{t}\mathscr{L}V_{p}(\bm{x}_{s}){\rm
d}s+\int_{0}^{t}\mathscr{D}V_{p}(\bm{x}_{s}){\rm d}B_{s},$

where

$\mathscr{L}V_{p}(\bm{x})\triangleq\left\\{\begin{aligned}
&p\Bigg{[}\dfrac{\mathscr{G}(\bm{x})}{\|\bm{x}\|^{2}}-\dfrac{k^{2}}{2}\Bigg{]},\|\bm{x}\|\geq
1,\\\
&p\|\bm{x}\|^{p-2}\mathscr{G}(\bm{x})+\dfrac{p(p-1)k^{2}}{2}\|\bm{x}\|^{p-2+2\alpha},\\\
&\|\bm{x}\|<1.\end{aligned}\right.$

$\mathscr{D}V_{p}(\bm{x})\triangleq\left\\{\begin{aligned} &kp,\|\bm{x}\|\geq
1,\\\ &kp\|\bm{x}\|^{\alpha+p-1},\|\bm{x}\|<1.\end{aligned}\right.$

Here, $\mathscr{G}(\bm{x})\triangleq\langle\bm{x},\bm{f}(\bm{x})\rangle\leq
L\|\bm{x}\|^{2}$. Choosing $0<p\leq\min\left\\{1,2-2\alpha\right\\}$ and using
$k>\sqrt{2L}$ yield following estimation:

$\mathscr{L}V_{p}(\bm{x})\leq\left\\{\begin{aligned}
&-p\left(\dfrac{k^{2}}{2}-L\right)\triangleq H_{1}(p),\|\bm{x}\|\geq 1,\\\
&H_{2}(p),\|\bm{x}\|<1,\end{aligned}\right.$ (9)

where $H_{2}(p)\triangleq pL+\dfrac{p(p-1)k^{2}}{2}$. Denote by a stopping
time $\sigma\triangleq\inf\Bigg{\\{}t>0\Bigg{|}\|\bm{x}_{t}\|=1\Bigg{\\}}$.
Apparently, we obtain that $\sigma\leq\tau$. Then we give the estimation for
$\tau$ on two disjoint measurable sets $\bm{1}_{\|\bm{x}_{0}\|>1}$ and
$\bm{1}_{\|\bm{x}_{0}\|\leq 1}$ seperately.

I: The estimation for the set $\bm{1}_{\|\bm{x}_{0}\|>1}$. An application of
the generalized It$\hat{o}$’s formula ( Lemma 1) for $V_{p}(\|\bm{x}_{t}\|)$
on the time interval $[0,t\wedge\sigma]$ yields that

$\displaystyle
V_{p}(\|\bm{x}_{t\wedge\sigma}\|)-V_{p}(\|\bm{x}_{0}\|)=\int_{0}^{t\wedge\sigma}\mathscr{L}V_{p}(\bm{x}_{s}){\rm
d}s$ $\displaystyle+\int_{0}^{t\wedge\sigma}\mathscr{D}V_{p}(\bm{x}_{s}){\rm
d}B_{s}.$

Taking the expectation of both sides on the measurable set
$\bm{1}_{\|\bm{x}_{0}\|>1}$ yields that

$\displaystyle\mathbb{P}(\|\bm{x}_{0}\|>1)\leq$
$\displaystyle\mathbb{E}V_{p}(\|\bm{x}_{t\wedge\sigma}\|;\bm{1}_{\|\bm{x}_{0}\|>1})$
$\displaystyle\leq$
$\displaystyle\mathbb{E}V_{p}(\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|>1})$
$\displaystyle+H_{1}(p)\mathbb{E}(t\wedge\sigma;\bm{1}_{\|\bm{x}_{0}\|>1}),$

which further indicates that

$\mathbb{E}(t\wedge\sigma;\bm{1}_{\|\bm{x}_{0}\|>1})\leq\dfrac{2\mathbb{E}(\log\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|>1})}{{k^{2}}-2L}.$

Letting $t\to+\infty$ leads to that

$\mathbb{E}(\sigma;\bm{1}_{\|\bm{x}_{0}\|>1})\leq\dfrac{2\mathbb{E}(\log\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|>1})}{{k^{2}}-2L}$

and $\sigma<+\infty$ almost surely.

Moreover, an application of the generalized It$\hat{o}$’s formula (Lemma 1)
for $V_{p}(\|\bm{x}_{t}\|)$ on the time interval
$[\sigma,(\sigma+t)\wedge\tau]$ yields that

$\displaystyle
V_{p}(\|\bm{x}_{(\sigma+t)\wedge\tau}\|)-V_{p}(\|\bm{x}_{\sigma}\|)=$
$\displaystyle\int_{\sigma}^{(\sigma+t)\wedge\tau}\mathscr{L}V_{p}(\bm{x}_{s}){\rm
d}s$ (10)
$\displaystyle+\int_{\sigma}^{(\sigma+t)\wedge\tau}\mathscr{D}V_{p}(\bm{x}_{s}){\rm
d}B_{s}.$

Note that even for $t>\sigma$, the trajectory may be outside the unit ball.
However, we have the estimation $H_{1}(p)<H_{2}(p)$, which implies that
$\mathscr{L}V_{p}(\bm{x}_{s})\leq H_{2}(p)~{}~{}{\rm a.s.}.$ Taking the
expectation of both sides yields that

$\displaystyle 0$
$\displaystyle\leq\mathbb{E}V_{p}(\|\bm{x}_{(\sigma+t)\wedge\tau}\|;\bm{1}_{\|\bm{x}_{0}\|>1})$
$\displaystyle\leq\mathbb{P}(\|\bm{x}_{0}\|>1)+H_{2}(p)\mathbb{E}\left[(\sigma+t)\wedge\tau-\sigma;\bm{1}_{\|\bm{x}_{0}\|>1}\right].$

When $H_{2}(p)<0$, we obtain that

$\mathbb{E}\left[(\sigma+t)\wedge\tau-\sigma;\bm{1}_{\|\bm{x}_{0}\|>1}\right]\leq-\dfrac{\mathbb{P}(\|\bm{x}_{0}\|>1)}{H_{2}(p)}.$

Letting $t\to+\infty$ leads to that

$\mathbb{E}\left[\tau-\sigma;\bm{1}_{\|\bm{x}_{0}\|>1}\right]\leq-\dfrac{\mathbb{P}(\|\bm{x}_{0}\|>1)}{H_{2}(p)}$

and $\tau<+\infty$ almost surely. We obtain that

$\mathbb{E}(\tau;\bm{1}_{\|\bm{x}_{0}\|>1})\leq\dfrac{2\mathbb{E}(\log\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|>1})}{{k^{2}}-2L}-\dfrac{\mathbb{P}(\|\bm{x}_{0}\|>1)}{H_{2}(p)}.$

II: The estimation for the set $\bm{1}_{\|\bm{x}_{0}\|\leq 1}.$ On this set we
have $\sigma=0$. Eq. (10), constrained within the set
$\bm{1}_{\|\bm{x}_{0}\|\leq 1}$, yields that

$\displaystyle V_{p}(\|\bm{x}_{t\wedge\tau}\|;\bm{1}_{\|\bm{x}_{0}\|\leq
1})-V_{p}(\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|\leq 1})$ (11)
$\displaystyle=\int_{0}^{t\wedge\tau}\mathscr{L}V_{p}(\bm{x}_{s})\bm{1}_{\|\bm{x}_{0}\|\leq
1}{\rm d}s$
$\displaystyle+\int_{0}^{t\wedge\tau}\mathscr{D}V_{p}(\bm{x}_{s})\bm{1}_{\|\bm{x}_{0}\|\leq
1}{\rm d}B_{s}.$

Note that the estimation (9) and $H_{1}(p)\leq H_{2}(p)$ still apply, we have
$\mathscr{L}V_{p}(\bm{x}_{s})\leq H_{2}(p)~{}~{}{\rm a.s.}$ even $\bm{x}_{t}$
may be outside the unit ball. By taking expectation on Eq. (11), we have

$\displaystyle\hskip
14.22636pt-\mathbb{E}V_{p}(\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|\leq 1})$
$\displaystyle\leq\mathbb{E}[V_{p}(\|\bm{x}_{t\wedge\tau}\|;\bm{1}_{\|\bm{x}_{0}\|\leq
1})-V_{p}(\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|\leq 1})]$ $\displaystyle\leq
H_{2}(p)\mathbb{E}(t\wedge\tau;\bm{1}_{\|\bm{x}_{0}\|\leq 1}).$

When $H_{2}(p)<0$, we obtain that

$\displaystyle\mathbb{E}(t\wedge\tau;\bm{1}_{\|\bm{x}_{0}\|\leq 1})$
$\displaystyle\leq-\dfrac{\mathbb{E}V_{p}(\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|\leq
1})}{H_{2}(p)}$
$\displaystyle=-\dfrac{\mathbb{E}(\|\bm{x}_{0}\|^{p};\bm{1}_{\|\bm{x}_{0}\|\leq
1})}{H_{2}(p)}.$

Letting $t\to+\infty$ yields that

$\mathbb{E}(\tau;\bm{1}_{\|\bm{x}_{0}\|\leq
1})\leq-\dfrac{\mathbb{E}(\|\bm{x}_{0}\|^{p};\bm{1}_{\|\bm{x}_{0}\|\leq
1})}{H_{2}(p)}.$

Consequently, we have

$\displaystyle\mathbb{E}\tau\leq$
$\displaystyle\dfrac{2\mathbb{E}(\log\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|>1})}{{k^{2}}-2L}-\dfrac{\mathbb{P}(\|\bm{x}_{0}\|>1)}{H_{2}(p)}$
$\displaystyle-\dfrac{\mathbb{E}(\|\bm{x}_{0}\|^{p};\bm{1}_{\|\bm{x}_{0}\|\leq
1})}{H_{2}(p)}.$

Now we come to investigate $H_{2}(p)$. Since
$H_{2}^{\prime}(p)=pk^{2}-\left(\dfrac{k^{2}}{2}-L\right)$, $H_{2}(p)$
decreases on the interval $(0,p^{*})$ and increases on the interval
$(p^{*},+\infty)$, where $p^{*}\triangleq\dfrac{1}{2}-\dfrac{L}{k^{2}}$.
Therefore, we obtain that

$\min_{0<p\leq\min\\{1,2-2\alpha\\}}H_{2}(p)=\left\\{\begin{aligned}
-\dfrac{{(kp^{*})}^{2}}{2},p^{*}<2-2\alpha,\\\ H_{2}(2-2\alpha),p^{*}\geq
2-2\alpha.\end{aligned}\right.$

If we choose $p=p^{*}$ when $\alpha<\dfrac{3}{4}+\dfrac{L}{2k^{2}}$ and
$p=2-2\alpha$ when $\alpha>\dfrac{3}{4}+\dfrac{L}{2k^{2}}$, we get the
following estimations.

$\mathbb{E}\tau\leq T_{f}^{\rm Sup}\triangleq\left\\{\begin{aligned}
&\dfrac{2\mathbb{E}(\log\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|>1})}{{k^{2}}-2L}\\\
&+\dfrac{8k^{2}}{(k^{2}-2L)^{2}}[\mathbb{P}(\|\bm{x}_{0}\|>1)\\\
&+\mathbb{E}(\|\bm{x}_{0}\|^{p^{*}};\bm{1}_{\|\bm{x}_{0}\|\leq 1})],\
\alpha<\dfrac{3}{4}+\dfrac{L}{2k^{2}},\\\
&\dfrac{2\mathbb{E}(\log\|\bm{x}_{0}\|;\bm{1}_{\|\bm{x}_{0}\|>1})}{{k^{2}}-2L}\\\
&+\dfrac{[\mathbb{P}(\|\bm{x}_{0}\|>1)+\mathbb{E}(\|\bm{x}_{0}\|^{2-2\alpha};\bm{1}_{\|\bm{x}_{0}\|\leq
1})]}{(1-\alpha)[(2\alpha-1)k^{2}-2L]},\\\
&\alpha\geq\dfrac{3}{4}+\dfrac{L}{2k^{2}}.\end{aligned}\right.$

$\blacksquare$

###  5.2 Estimation of Control Energy

Proof of Theorem 2: The energy consumption of the control is given by

$\displaystyle\mathcal{E}_{q}$ $\displaystyle=$
$\displaystyle\int_{0}^{\tau}\|\bm{u}_{s}\|^{q}{\rm d}s\leq
k^{q}\int_{0}^{\sigma}\|\bm{x}_{s}\|^{q}{\rm d}s$
$\displaystyle+k^{q}\int_{\sigma}^{\tau}(1\vee\|\bm{x}_{s}\|^{q}){\rm d}s$
$\displaystyle\leq$ $\displaystyle
k^{q}\int_{0}^{\sigma}\|\bm{x}_{s}\|^{q}{\rm
d}s+k^{q}\int_{\sigma}^{\tau}(1+\|\bm{x}_{s}\|^{q}){\rm d}s$

Denote by $W_{q}(x)\triangleq x^{q}$. An application of the generalized
It$\hat{o}$’s formula (Lemma 1) yields that

$\displaystyle W_{q}(\|\bm{x}_{t}\|)-W_{q}(\|\bm{x}_{0}\|)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\mathscr{L}W_{q}(\bm{x}_{s}){\rm d}s$
$\displaystyle+\int_{0}^{t}\mathscr{D}W_{q}(\bm{x}_{s}){\rm d}B_{s},$

where

$\mathscr{L}W_{q}(\bm{x})=\left\\{\begin{aligned}
&q\|\bm{x}\|^{q-2}\mathscr{G}(\bm{x})+\dfrac{q(q-1)k^{2}}{2}\|\bm{x}\|^{q},\
\|\bm{x}\|\geq 1,\\\
&\|\bm{x}\|^{q-2}\mathscr{G}(\bm{x})+\dfrac{q(q-1)k^{2}}{2}\|\bm{x}\|^{q-2+2\alpha},\\\
&\|\bm{x}\|<1.\end{aligned}\right.$ (12)

and

$\mathscr{D}W_{q}(\bm{x})\triangleq\left\\{\begin{aligned}
&kq\|\bm{x}\|^{q},\|\bm{x}\|\geq 1,\\\
&kq\|\bm{x}\|^{\alpha+q-1},\|\bm{x}\|<1.\end{aligned}\right.$

Choosing $0<q<\min\Bigg{\\{}2-2\alpha,1-\dfrac{2L}{k^{2}}\Bigg{\\}}$ yields
following estimations

$\mathscr{L}W_{q}(\bm{x})\leq\left\\{\begin{aligned}
&H_{2}(q)\|\bm{x}\|^{q},\|\bm{x}\|\geq 1,\\\
&H_{2}(q),\|\bm{x}\|<1.\end{aligned}\right.$ (13)

I: We first do the estimation on the set $\bm{1}_{\|\bm{x}_{0}\|>1}$. Thus, on
the time interval $[0,t\wedge\sigma]$, we have

$\displaystyle W_{q}(\|\bm{x}_{t\wedge\sigma}\|)$ $\displaystyle=$
$\displaystyle
W_{q}(\|\bm{x}_{0}\|)+\int_{0}^{t\wedge\sigma}\mathscr{L}W_{q}(\bm{x}_{s}){\rm
d}s$ $\displaystyle+\int_{0}^{t\wedge\sigma}\mathscr{D}W_{q}(\bm{x}_{s}){\rm
d}B_{s}.$

Taking the expectation of both sides on $\bm{1}_{\|\bm{x}_{0}\|>1}$ leads to
that

$\displaystyle\mathbb{E}\bigg{[}\|\bm{x}_{t}\|^{q}\bm{1}_{t<\sigma,\|\bm{x}_{0}\|>1}\bigg{]}$
$\displaystyle\leq\mathbb{E}\left[W_{q}(\|\bm{x}_{t\wedge\sigma}\|);\bm{1}_{\|\bm{x}_{0}\|>1}\right]$
$\displaystyle\leq\mathbb{E}\left[W_{q}(\|\bm{x}_{0}\|);\bm{1}_{\|\bm{x}_{0}\|>1}\right]$
$\displaystyle+H_{2}(q)\int_{0}^{t}\mathbb{E}\bigg{[}\|\bm{x}_{s}\|^{q}\bm{1}_{s<\sigma,\|\bm{x}_{0}\|>1}\bigg{]}{\rm
d}s,$

this, together with Gronwell inequality, leads to that

$\displaystyle\mathbb{E}\bigg{[}\|\bm{x}_{t}\|^{q}\bm{1}_{t<\sigma,\|\bm{x}_{0}\|>1}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left[W_{q}(\|\bm{x}_{0}\|);\bm{1}_{\|\bm{x}_{0}\|>1}\right]\exp\bigg{[}H_{2}(q)t\bigg{]}.$

Note that $H_{2}(q)<0$, we obtain

$\displaystyle\mathbb{E}\Bigg{[}\int_{0}^{\sigma}\|\bm{x}_{s}\|^{q}{\rm
d}s;\|\bm{x}_{0}\|>1\Bigg{]}$ $\displaystyle=$
$\displaystyle\int_{0}^{+\infty}\mathbb{E}\bigg{[}\|\bm{x}_{t}\|^{q}\bm{1}_{t<\sigma,\|\bm{x}_{0}\|>1}\bigg{]}{\rm
d}t$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\left[W_{q}(\|\bm{x}_{0}\|);\bm{1}_{\|\bm{x}_{0}\|>1}\right]\int_{0}^{+\infty}\exp\bigg{[}H_{2}(q)t\bigg{]}{\rm
d}t$ $\displaystyle\leq$
$\displaystyle-\dfrac{\mathbb{E}\left[W_{q}(\|\bm{x}_{0}\|);\bm{1}_{\|\bm{x}_{0}\|>1}\right]}{H_{2}(q)}.$

Furthermore, we have

$\displaystyle W_{q}(\|\bm{x}_{(t+\sigma)\wedge\tau}\|)=$ $\displaystyle
W_{q}(\|\bm{x}_{\sigma}\|)+\int_{\sigma}^{(t+\sigma)\wedge\tau}\mathscr{L}W_{q}(\bm{x}_{s}){\rm
d}s$
$\displaystyle+\int_{\sigma}^{{(t+\sigma)\wedge\tau}}\mathscr{D}W_{q}(\bm{x}_{s}){\rm
d}B_{s},$

which implies that

$\displaystyle\mathbb{E}\bigg{[}\|\bm{x}_{t+\sigma}\|^{q}\bm{1}_{t<\tau-\sigma,\|\bm{x}_{0}\|>1}\bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\bigg{[}W_{q}(\|\bm{x}_{(t+\sigma)\wedge\tau}\|)\bm{1}_{t<\tau-\sigma,\|\bm{x}_{0}\|>1}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}[\bm{1}_{t<\tau-\sigma,\|\bm{x}_{0}\|>1}].$

Therefore, we obtain that

$\displaystyle\mathbb{E}\left[\int_{\sigma}^{\tau}\|\bm{x}_{s}\|^{q}{\rm
d}s;\|\bm{x}_{0}\|>1\right]$
$\displaystyle=\int_{0}^{\tau-\sigma}\mathbb{E}\bigg{[}\|\bm{x}_{t+\sigma}\|^{q}\bm{1}_{0<t<\tau-\sigma}\bigg{]}{\rm
d}t$
$\displaystyle\leq\int_{0}^{+\infty}\mathbb{E}\bigg{[}\bm{1}_{0<t<\tau-\sigma,\|\bm{x}_{0}\|>1}\bigg{]}{\rm
d}t$
$\displaystyle=\mathbb{E}\bigg{[}\int_{0}^{+\infty}\bm{1}_{0<t<\tau-\sigma}{\rm
d}t;\|\bm{x}_{0}\|>1\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\tau-\sigma;\|\bm{x}_{0}\|>1\bigg{]}\leq-\dfrac{\mathbb{P}(\|\bm{x}_{0}\|>1)}{H_{2}(q)}.$

This results in that

$\displaystyle\mathbb{E}[\mathcal{E}_{q};\|\bm{x}_{0}\|>1]$
$\displaystyle\leq$ $\displaystyle
k^{q}\mathbb{E}\Bigg{[}\int_{0}^{\sigma}\|\bm{x}_{s}\|^{q}{\rm
d}s;\|\bm{x}_{0}\|>1\Bigg{]}$
$\displaystyle+k^{q}\mathbb{E}\Bigg{[}\int_{\sigma}^{\tau}(1+\|\bm{x}_{s}\|^{q}){\rm
d}s;\|\bm{x}_{0}\|>1\Bigg{]}$ $\displaystyle\leq$
$\displaystyle-k^{q}\dfrac{\mathbb{E}[\|\bm{x}_{0}\|^{q}+2;\|\bm{x}_{0}\|>1]}{H_{2}(q)}.$

II: Secondly, we do estimations on the set $\bm{1}_{\|\bm{x}_{0}\|\leq 1}$. A
similar argument leads to that

$\mathbb{E}[\mathcal{E}_{q};\|\bm{x}_{0}\|\leq
1]\leq-\dfrac{2k^{q}\mathbb{E}[\|\bm{x}_{0}\|^{q};\|\bm{x}_{0}\|\leq
1)]}{H_{2}(q)}.$

Above all, we have the estimation

$\displaystyle\mathbb{E}\mathcal{E}_{q}\leq\mathcal{E}_{q}^{\rm Sup}$
$\displaystyle\triangleq-\dfrac{k^{q}}{H_{2}(q)}[\mathbb{E}(\|\bm{x}_{0}\|^{q})+2\mathbb{P}(\|\bm{x}_{0}\|>1)$
$\displaystyle+\mathbb{E}(\|\bm{x}_{0}\|^{q};\|\bm{x}_{0}\|\leq 1)]$

This completes the proof of the whole theorem. $\blacksquare$

###### Remark 5

The conclusion that Eq. (1) is finite-time stable
($\mathbb{P}(\tau<+\infty)=1$) is obtained by using the result of Theorem 1,
which states that $\mathbb{E}\tau<+\infty$. In most literatures (for example,
see Yin et al. (2011); Yu et al. (2021)), the definition for finite-time
stable is $\mathbb{P}(\tau<+\infty)=1$ (refer to Definition 1) rather than
$\mathbb{E}\tau<+\infty$. This definition is commonly used because it is less
restrictive and easier to verify in certain cases. For example, we consider a
time-varying one-dimensional SDE (refer to (Yu et al., 2021, Example 3.1))

${\rm d}x=(-\dfrac{a}{2(1+t)}x^{\frac{1}{3}}+\dfrac{b}{1+t^{2}}x){\rm
d}t+\dfrac{d}{\sqrt{1+t}}x^{\frac{2}{3}}{\rm d}B_{t}$ (14)

where the parameters $a\geq d^{2},b\geq 0$. Authors in Yu et al. (2021) show
that Eq. (14) is finite-time stable by using (Yu et al., 2021, Corollary 3.2)
and obtaining that $\mathbb{E}\beta(\tau)<+\infty,$ where
$\beta(t)=\int_{0}^{t}c(s)({\rm e}^{-\int_{0}^{s}l(u){\rm d}u})^{1-\gamma}{\rm
d}s,c(t)=\dfrac{a}{2(1+t)},\gamma=\frac{1}{3},l(t)=\dfrac{b}{1+t^{2}}$.
However, it is challenging to demonstrate that $\mathbb{E}\tau<+\infty$ in
this particular case.

###### Remark 6

We need to emphasize that even for $t>\sigma$, the trajectory may be outside
the unit circle because it is a stochastic system. Therefore, we cannot
seperate the time interval into two parts and do the estimation using
different Lyapunov functions seperately. This is totally different from the
past work on the deterministic systems (See Sun et al. (2017)). This is reason
why we construct a piecewise smooth Lyapnounv function and use the generalized
It$\hat{o}$’s formula (Lemma 1), which is the major difficulty for the proof.

##  6 Concluding Remarks

In this paper, we propose a finite-time, closed-loop stochastic feedback
controller. The usefulness of the scheme in chaos control and network
synchronization is illustrated not only by analytical validations but also by
numerically controlling several representative stochastic dynamical models.
Additionally, we investigate convergence time and energy consumption as well
as its dependence on parameters. And we also numerically validate the our
estimation of the upper bound.

As for the future research directions, the estimation of convergence time and
energy consumption is much larger than the exact value by the simulation.
Moreover, our findings suggest that under certain conditions of $\alpha$, the
values of $T_{f}^{\rm Sup},\mathcal{E}_{q}^{\rm Sup}$ remains a constant when
$\alpha$ satisfies particular conditions.Thus, there is a need for more
accurate estimation techniques to address this issue.

In addition, there are practical implementation challenges associated with our
proposed controller. One particular challenge is the reliance on instant-time
information, which may not always be available in practical applications. To
address this issue, event-triggered schemes have been developed for
stabilizing deterministic dynamical systems (See Nair et al. (2017); Liu et
al. (2019, 2018, 2020)). Additionally, schemes incorporating time delay have
also been explored (See Yang and Wang (2012)). Furthermore, in practical
applications, uncertainties such as unknown parameters, unmodeled dynamics,
uncertain functions, and disturbances are inevitable. For stabilizing
deterministic systems, several finite-time adaptive control designs have been
established to address uncertainty (See Kong et al. (2020); Jin (2018)). It
would be worthwhile to investigate how to extend these results to stochastic
controllers in future research.

## Data availability statement

The data that supports the findings of this study are computationally
generated and available upon request.

## Acknowledgments

The authors thank Luan Yang for plotting Figure 1. S.Z. is supported by the
National Natural Science Foundation of China (No. 11925103) and by the STCSM
(Nos. 22JC1402500, 22JC1401402, and 2021SHZDZX0103). He is also supported by
the Research on the Assessment and Prevention Techniques for Telecom Network
Fraud Victims Based on Big Data (No. 21DZ1201402).

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