# Real-time Estimation of DoS Duration and Frequency for Security Control

Yifan Sun, Jianquan Lu, , Daniel W. C. Ho, and Lulu Li This work was partially
supported by the National Natural Science Foundation of China under Grant No.
62373105, and the Research Grants Council of the Hong Kong Special
Administrative Region, China (CityU 11203521, 11213023). (Corresponding
Author: Jianquan Lu.)Yifan Sun and Jianquan Lu are with the School of
Mathematics, Southeast University, Nanjing 211189, Jiangsu, China (email:
<EMAIL_ADDRESS>jqluma@seu.edu.cn).Daniel W. C. Ho is with the Department
of Mathematics, City University of Hong Kong, Hong Kong, China (email:
madaniel@cityu.edu.hk).Lulu Li is with the School of Mathematics, Hefei
University of Technology, Hefei 230601, Anhui, China (email:
lululima@hfut.edu.cn).

###### Abstract

In this paper, we develop a new denial-of-service (DoS) estimator, enabling
defenders to identify duration and frequency parameters of any DoS attacker,
except for three edge cases, exclusively using real-time data. The key
advantage of the estimator lies in its capability to facilitate security
control in a wide range of practical scenarios, even when the attacker’s
information is previously unknown. We demonstrate the advantage and
application of our new estimator in the context of two classical control
scenarios, namely consensus of multi-agent systems and impulsive stabilization
of nonlinear systems, for illustration.

###### Index Terms:

DoS attack, real-time DoS estimation, security control, average consensus,
impulsive stabilization.

##  I Introduction

Denial-of-service (DoS) attack is a form of cyber-attack that makes a machine
or network resource unavailable to its intended users. In a networked control
system, it usually affects the timeliness of communication by blocking the
channel and inducing unexpected transmission failures, and thus invalidates
most classical control methods and causes the system to run into a hazardous
state [1, 2]. Therefore, it is necessary to take security into account when
designing controllers, or otherwise, the system will remain vulnerable to DoS
attacks [2].

In order to model the security control problem, a comprehensive
characterization of DoS attacks is essential. It is inadequate to assume a
specific probabilistic distribution [3, 4] or a certain action mode (e.g.,
periodicity [5]) of the attack because a malicious attacker does not have to
follow these assumptions and can exhibit quite irregular behavior. For better
universality, De Persis and Tesi [1] proposed a DoS model that does not assume
about the attacker’s underlying strategy. This model is capable of capturing
various types of DoS patterns, and therefore, has been widely used in follow-
up research on anti-DoS control [6, 7, 8, 9, 10].

Under the framework [1], DoS duration and DoS frequency are two core metrics
characterizing DoS attacks. The former metric evaluates the percentage of time
when a DoS attack is active, while the latter characterizes how frequently the
attack is launched (both in an average sense over the entire timeline). The
advantage of the framework [1] is to allow controller configurations based
only on the duration and frequency parameters regardless of any other detailed
attack information, thus leading to very concise control strategies[11, 8, 12,
13, 14, 15]. For example, if a duration- & frequency-based controller setting
is effective for the periodic DoS attack over $[0,1)$s, $[2,3)$s, $[4,5)$s,
$\cdots$, then it is also effective for an aperiodic attack over $[1,2)$s,
$[3,4-\ln{2})$s, $[5,6+\ln{\frac{3}{2}})$s, $\cdots$,
$[2n+1,2n+2+(-1)^{n}\ln{\frac{n+1}{n}})$s, $\cdots$. More precisely, both
these attacks have the same duration and frequency parameters, that is,
affecting around $50\%$ of the time and being launched every two seconds.

Unfortunately, the above advantage of the framework [1] can hardly be
implemented in most practical applications. It is because the framework relies
on directly assumed values of the duration and frequency parameters whose
correctness cannot be guaranteed in real cases where the defender usually has
little adversary information before the encounter. To validate the framework
[1] in these cases, the defender should estimate the parameters of interest
based only on real-time DoS data, which can be collected from past actions of
the attacker.

Motivated by the discussion above, the focus of this paper lies in achieving a
dependable estimation of DoS duration and frequency parameters utilizing
solely real-time attack data. As the theoretical support of our estimation, we
reveal a fundamental vulnerability inherent in DoS attackers: they unavoidably
disclose information about the duration and frequency through their actions.
This inherent weakness inspires the development of our novel DoS estimator,
which can identify the desired parameters by examining the past attack
durations and the historical count of DoS launches.

Despite some involved analyses ensuring theoretical validity, the core idea of
our estimation is remarkably intuitive. As defenders, we consider slightly
higher DoS duration and frequency than the worst previous scenario,
anticipating that such estimation will achieve reliability within a finite
time. We will see a satisfactory performance of the estimator against
virtually all types of DoS attackers except for three extreme edge cases.
These atypical scenarios correspond to three categories of extremely powerful
attackers, which are rarely encountered in real network environments. To
further elaborate on this point, we will provide a new and concise definition
in Section II, justifying it based on the physical meanings of the three
cases.

To the authors’ best knowledge, few existing works have considered the present
topic despite its undoubted importance. The reference closest to this paper
might be [14], in which available transmission bits are adaptively allocated
based on real-time DoS strength. Yet, how to estimate the attacker’s duration
& frequency parameters remains unresolved in that article. The work merely
provided a precise solution to the allocation problem of channel resources,
but the design of the sampling period there still relies on presupposed
attacker information. Another relevant reference is [16], in which the DoS
duration or frequency is not required to be known by the defender. However,
the controller design is based on the maximum duration of the DoS attack,
which still needs to be assumed before the design time and, therefore, does
not change the essence of the problem. In [17], a real-time scheme is proposed
to detect and estimate DoS attacks that lead to a differentiable transmission
delay. Most real-world cases cannot be covered because DoS-induced delays are
usually discontinuous. Li et al. proposed another viewpoint in [18],
formulating a game-theoretic framework to describe the interaction between
attackers and defenders. Although the reliance on duration & frequency
parameters can be avoided, such a framework requires that both sides know the
energy constraints of their adversaries before the game begins.

The main contributions of this paper are in three different aspects. Firstly,
we develop a new DoS estimator capable of addressing any DoS attacker apart
from three edge cases, identifying the duration and frequency parameters for
defensive purposes solely based on real-time data. This novel development has
the potential to significantly broaden the applicability of numerous existing
anti-DoS control strategies, specifically extending their reach from managing
rare, previously identified attackers to addressing all but a few extreme
ones.

The second goal of this paper is to provide clear guidance on exploiting the
DoS estimator for security controller design. To achieve this, we examine two
specific control scenarios, namely the consensus of multi-agent systems (MASs)
and impulsive stabilization of nonlinear systems, for illustration. Utilizing
the new estimator, we present concrete design schemes that exhibit
satisfactory control performances in both cases. Based on our estimator, the
design reflects the feasibility of expanding the application scope of previous
control strategies.

Finally, the design presented in this paper offers several parametric tuning
options to balance security against cost-efficiency. These options provide
flexibility for a conservative defender to allocate more resources for faster
estimation (resulting in enhanced security), or for an aggressive defender to
prioritize energy-saving over the speed of estimation. The choice between
these options depends on how skeptical the defender is about the attacker’s
behavior. The trade-off therefore reveals an interesting interplay between the
offensive and defensive sides during the estimation process, and it offers a
possibility for controller configuration to meet diverse requirements.

Standard notations will be used throughout this paper. We use $\mathbb{N}$
(resp., $\mathbb{N}_{+}$) to represent the set of all non-negative integers
(resp., positive integers), and use $\mathbb{R}^{n}$ (resp., $\mathbb{R}$) to
represent the $n$-dimensional (resp., $1$-dimensional) Euclidean vector space.
We use $\mathbb{R}_{\geqslant 0}$ to represent the set of non-negative real
numbers. For a set $\Omega$, we use $\operatorname{card}(\Omega)$ to represent
its cardinality. In addition, if $\Omega\subseteq\mathbb{R}$, then we use
$\lvert\Omega\rvert$ to represent its Lebesgue measure [19], use $\inf\Omega$
to represent its infimum, and use $\sup\Omega$ to represent its supremum [20].
We use
$\limsup_{n\rightarrow\infty}{a_{n}}\coloneqq\lim_{n\rightarrow\infty}\left(\sup_{m\geqslant
n}a_{m}\right)$ to represent the limit superior [20] of a sequence
$\\{a_{n}\\}_{n\in\mathbb{N}_{+}}$. For a function $f(t)$, we use
$\limsup_{t\rightarrow+\infty}{f(t)}\coloneqq\lim_{t\rightarrow+\infty}\left(\sup_{\tau\geqslant
t}f(\tau)\right)$ to represent its limit superior, use $\dot{f}(t)$ to
represent its derivative with respect to $t$, and use $D^{+}f(t)$ to represent
its upper-right Dini derivative [21]. We say that $f:\mathbb{R}_{\geqslant
0}\mapsto\mathbb{R}_{\geqslant 0}\in\mathcal{K}$, if $f$ is continuous,
$f(0)=0$ and $f(s)$ is strictly increasing in $s$. For a vector $x$, we use
$x^{T}$ to represent its transposition and use $\lvert x\rvert$ to represent
its Euclidean norm. We use $\mathbf{1}_{N}$ to represent an $N$-dimensional
vector $(1,1,\cdots,1)^{T}$.

##  II DoS estimation

###  II-A Problem formulation

Since the main purpose of this paper is to give reliable estimation of DoS
duration and frequency parameters, firstly, we need to review their
definitions proposed by [1].

Let $\\{h_{n}\\}_{n\in\mathbb{N}_{+}}$ represent the sequence of instants when
the DoS attack switches from off to on, where $h_{1}\geqslant 0$. Let

$\displaystyle H_{n}\coloneqq\\{h_{n}\\}\cup[h_{n},h_{n}+\tau_{n})$ (1)

represent the $n$-th DoS time interval of a length $\tau_{n}$, over which the
attack is on, and all transmission attempts are denied. Without loss of
generality, we assume that $h_{n+1}>h_{n}+\tau_{n}$ for any
$n\in\mathbb{N}_{+}$ so that $H_{i}\cap H_{j}=\emptyset$ for any $i\neq j$. We
use the notation

$\displaystyle\Xi(\tau,s)\coloneqq\bigcup_{n\in\mathbb{N}_{+}}H_{n}\cap[\tau,s]$
(2)

to denote the set of instants subject to DoS attack on $[\tau,s]$
($\tau\leqslant s$), and use

$\displaystyle\Theta(\tau,s)\coloneqq[\tau,s]\setminus\Xi(\tau,s)$ (3)

to denote the complement of $\Xi(\tau,s)$, over which the attack is off and
transmission is allowed. In this paper, we use
$\xi\coloneqq\\{H_{n}\\}_{n\in\mathbb{N}_{+}}$ to denote the sequence of all
the DoS time intervals, and we call $\xi$ a DoS sequence for brevity. For
notation consistency, we specify that $h_{m+1}=h_{m+1}+\tau_{m+1}=+\infty$ if
there are only $m$ elements in $\xi$, $m\in\mathbb{N}$. Finally, we denote by

$\displaystyle
n_{\xi}(\tau,s)\coloneqq\operatorname{card}\left(\\{h_{n}\\}_{n\in\mathbb{N}_{+}}\cap[\tau,s]\right)$
(4)

the number of DoS off-to-on transitions on $[\tau,s]$.

With the preparations above, the duration and frequency parameters of a DoS
sequence $\xi$ can be defined as follows.

###### Definition 1

([1] DoS duration) For a DoS sequence $\xi$ and a constant $B_{d}\in[0,1]$, if
there exists a constant $0<\kappa(B_{d})<+\infty$ such that

$\displaystyle\lvert\Xi(0,t)\rvert\leqslant\kappa+B_{d}\,t,\quad\forall\,t\geqslant
0,$

then we say that $B_{d}$ is a duration-bound of $\xi$. $\square$

###### Definition 2

([1] DoS frequency) For a DoS sequence $\xi$ and a constant
$B_{f}\in[0,+\infty]$, if there exists an integer $0<\Lambda(B_{f})<+\infty$
such that

$\displaystyle n_{\xi}(0,t)\leqslant\Lambda+B_{f}\,t,\quad\forall\,t\geqslant
0,$

then we say that $B_{f}$ is a frequency-bound of $\xi$. $\square$

The duration-bound characterizes the ratio of time over which the DoS attack
is active, and the frequency-bound characterizes how frequently the attack is
launched. Generally, the values of these two metrics do not have to affect
each other, and therefore, we can deal with $B_{d}$ and $B_{f}$, respectively,
in order to simplify analysis.

According to Definitions 1 and 2, $B_{d}$ and $B_{f}$ are related to the
attacker’s action on $[0,+\infty)$. Their values, however, are directly
assumed by the defender when $t=0$ in most previous works [1, 6, 9, 11, 7, 10,
12, 8, 13, 14, 15]. Such presupposed values are reliable only if the defender
has accurate future information about the attacker, but not for most real
cases where nothing about $[t,+\infty)$ is known at time $t$. In order to
manage the latter cases, $B_{d}$ and $B_{f}$ should be estimated in real-time
based only on the attacker’s past actions over $[0,t]$.

Inspired by the discussion above, the first task of a defender is to give a
real-time estimation of $B_{d}$ and $B_{f}$ merely utilizing past knowledge
about the attacker. For the defense effectiveness, the estimation is required
to become reliable within finite time. This task can be written more formally
as follows, if we denote by $\hat{B}_{d}(t)$ and $\hat{B}_{f}(t)$,
respectively, the estimated value of $B_{d}$ and $B_{f}$ at time $t$, and by
$\mathscr{K}_{a}(t)$ the available knowledge about the attacker on $[0,t]$.

###### Problem 1

(Real-time DoS estimation) Design a DoS estimator as

$\displaystyle\hat{B}_{d}(t)=\hat{B}_{d}\left(\mathscr{K}_{a}(t)\right),\quad\hat{B}_{f}(t)=\hat{B}_{f}\left(\mathscr{K}_{a}(t)\right),$

such that there exists a finite $T$ satisfying

  * a)

$\hat{B}_{d}(t)$ is a duration-bound of $\xi$ for any $t\geqslant T$.

  * b)

$\hat{B}_{f}(t)$ is a frequency-bound of $\xi$ for any $t\geqslant T$.

###  II-B Solution to Problem 1: the DoS estimator

This subsection aims at solving Problem 1. To this end, we first need to find
connections between the attacker’s past knowledge $\mathscr{K}_{a}(t)$ and its
duration-bound $B_{d}$ & frequency-bound $B_{f}$. This is not easy since
$B_{d}$ and $B_{f}$ are not necessarily unique. In fact, they merely provide a
loose estimation of how strong the attacker can be. Defensive strategies based
on such a loose estimation could be inefficient because the defender must
prepare for the worst scenario, even if it never actually happens.

To make the defense more efficient, we have to investigate the sharp bounds of
$B_{d}$ and $B_{f}$. For this purpose, let us consider the set composed of all
duration-bounds of $\xi$, namely,

$\displaystyle\mathcal{D}(\xi)\coloneqq\left\\{B_{d}\in[0,1]\mid B_{d}\text{
is a {duration-bound} of $\xi$}\right\\}.$

By Definition 1, $\mathcal{D}(\xi)$ is nonempty because we always have
$1\in\mathcal{D}(\xi)$. Hence, the infimum of $\mathcal{D}(\xi)$ is well-
defined and satisfies $\inf\mathcal{D}(\xi)\leqslant 1$. Similarly, for the
frequency-bound, we can define

$\displaystyle\mathcal{F}(\xi)\coloneqq\left\\{B_{f}\in[0,+\infty]\mid
B_{f}\text{ is a {frequency-bound} of $\xi$}\right\\},$

and we have $\inf\mathcal{F}(\xi)\leqslant+\infty$. Although $B_{d}$ (resp.,
$B_{f}$) is not unique, there must be only one infimum of $\mathcal{D}(\xi)$
(resp., $\mathcal{F}(\xi)$). In addition, we have by a simple derivation that

$\displaystyle{\mathcal{D}^{\circ}(\xi)=(\inf\mathcal{D}(\xi),1)}$ (5)

and

$\displaystyle{\mathcal{F}^{\circ}(\xi)=(\inf\mathcal{F}(\xi),+\infty),}$ (6)

where $\Omega^{\circ}$ represents the interior of a set $\Omega$. From (5) and
(6), $\inf\mathcal{D}(\xi)$ and $\inf\mathcal{F}(\xi)$ are exactly the desired
sharp bounds of $B_{d}$ and $B_{f}$, respectively.

With the preparations above, finding the connections between
$\mathscr{K}_{a}(t)$ and $B_{d}$ & $B_{f}$ is equivalent to finding the ones
between $\mathscr{K}_{a}(t)$ and $\inf\mathcal{D}(\xi)$ &
$\inf\mathcal{F}(\xi)$. The following result can establish such connections.

###### Lemma 1

Given any DoS sequence $\xi$, we have

$\displaystyle\limsup_{t\rightarrow+\infty}{\frac{\lvert\Xi(0,t)\rvert}{t}}=\inf\mathcal{D}(\xi)$
(7)

and

$\displaystyle\limsup_{t\rightarrow+\infty}{\frac{n_{\xi}(0,t)}{t}}=\inf\mathcal{F}(\xi).$
(8)

Proof. See Appendix -A. $\square$

###### Remark 1

As revealed by Lemma 1, the values of $\inf\mathcal{D}(\xi)$ &
$\inf\mathcal{F}(\xi)$ have explicit connections with certain behaviors of the
attacker. More specifically, in the sense of limit superior, the ratio of time
under attack converges to $\inf\mathcal{D}(\xi)$, and the frequency of
launches converges to $\inf\mathcal{F}(\xi)$. The results in (7) and (8)
provide the possibility of estimating the duration & frequency bounds,
respectively, based on real-time DoS data $\lvert\Xi(0,t)\rvert$ and
$n_{\xi}(0,t)$, and thus are crucial for addressing Problem 1.

Based on Lemma 1, we point out three edge cases of $\xi$ which are beyond our
ability before providing a solution to Problem 1. These edge cases correspond
to three types of extremely powerful DoS attackers that rarely exist in real
networks. More formally, we have the following definition of these cases.

###### Definition 3

In this paper, a DoS sequence $\xi$ is called an edge case, if it satisfies
any of the following.

  * i)

$\inf\mathcal{D}(\xi)=1$.

  * ii)

$\inf\mathcal{F}(\xi)=+\infty$.

  * iii)

For any $\Gamma\in(0,+\infty)$, there exists an $n\in\mathbb{N}_{+}$ such that
$\tau_{n}>\Gamma.$ Here, $\tau_{n}$ is the length of the $n$-th DoS interval
as defined in (1). $\square$

In this paper, the defender becomes powerless under the edge cases in
Definition 3. Nevertheless, this is acceptable because, from an attacker’s
perspective, being an edge case means being extremely powerful and quite
demanding. We can use Lemma 1 to elaborate on this point. For i) in the
definition, it means that $\xi$ should attack and corrupt almost the entire
interval $[0,t]$ even when $t\rightarrow+\infty$ according to (7). For ii),
the attack should be launched at an infinitely high rate, according to (8).
For iii), it means that $\tau_{n}$ cannot be upper-bounded by any finite
constant, indicating the unlimited storage capacity of the attacker. Since it
is rare to encounter such strong opponents in practice, the inability to deal
with them has little impact on the universality of our solution. Hence, the
following assumption is imposed on $\xi$ hereafter.

###### Assumption 1

The DoS sequence $\xi$ is not an edge case.

Now, we proceed to address Problem 1. Let us design the DoS estimator as

$\displaystyle\hat{B}_{d}(t)\coloneqq\left\\{\begin{array}[]{lcl}\epsilon_{0},\qquad
t\in[0,h_{\ell}+\tau_{\ell}),\\\ \mbox{max}_{\ell\leqslant i\leqslant
n}\left\\{\epsilon_{0},\theta B_{d}(i)+(1-\theta)\right\\},\\\ \qquad\quad
t\in[h_{n}+\tau_{n},h_{n+1}+\tau_{n+1})\end{array}\right.$ (12)

and

$\displaystyle\hat{B}_{f}(t)\coloneqq\left\\{\begin{array}[]{ll}\epsilon_{0},&t\in[0,h_{\ell}),\\\
\mbox{max}_{\ell\leqslant i\leqslant
n}\left\\{\epsilon_{0},\frac{B_{f}(i)}{\theta}\right\\},&t\in[h_{n},h_{n+1}).\end{array}\right.$
(15)

Here, $2\leqslant\ell\in\mathbb{N}_{+}$, $0<\epsilon_{0}<1$ and $0<\theta<1$
are tunable parameters. They can be flexibly chosen and are independent of
$\xi$. In addition,
$B_{d}(i)\coloneqq\frac{\lvert\Xi(0,h_{i}+\tau_{i})\rvert}{h_{i}+\tau_{i}}$,
and $B_{f}(i)\coloneqq\frac{n_{\xi}(0,h_{i})}{h_{i}}=\frac{i}{h_{i}}$, where
the notations $h_{n}$, $\Xi(\tau,s)$ and $n_{\xi}(\tau,s)$ are defined,
respectively, in (1), (2) and (4). For any $t\geqslant 0$, it is clear that
$\hat{B}_{d}(t)$ and $\hat{B}_{f}(t)$ generated by (12)–(15) only rely on past
information over $[0,t]$.

The following result shows that DoS estimator (12)–(15) is a satisfactory
solution to Problem 1.

###### Theorem 1

Let the estimation $\hat{B}_{d}(t)$ and $\hat{B}_{f}(t)$ be generated by DoS
estimator (12)–(15). Then under Assumption 1, there must exist a finite
$T\geqslant 0$, such that $\hat{B}_{d}(t)\in\mathcal{D}(\xi)$ and
$\hat{B}_{f}(t)\in\mathcal{F}(\xi)$ hold for any $t\geqslant T$.

Proof. See Appendix -B. $\square$

###### Remark 2

Unlike many conventional approaches [1, 6, 7, 8, 9, 10] that rely on pre-
assumed values of $B_{d}$ and $B_{f}$, DoS estimator (12)–(15) allows for
controller design based on their real-time estimations, namely
$\hat{B}_{d}(t)$ and $\hat{B}_{f}(t)$. According to Theorem 1, such an
estimator-based design will become reliable after a finite period of time as
long as edge cases do not occur. We will further elaborate on this point in
Sections III and IV by examining two specific control scenarios and providing
concrete design schemes for them based on the DoS estimator.

###### Remark 3

The choice of $\theta$ ($0<\theta<1$) is crucial and clearly reflects our
basic idea. The left-hand side inequality $\theta>0$ means that we surrender
under edge cases, while the right-hand side $\theta<1$ means that we keep
being skeptical about the observed data $B_{d}(i)$ & $B_{f}(i)$ and consider
scenarios slightly worse than that. It is important that $\theta$ should be
strictly less than $1$, or otherwise, the wrong estimation might be obtained.
For example, for $\xi=\\{[2n+1,2n+2)\\}_{n\in\mathbb{N}_{+}}$, we have
$\inf\mathcal{D}(\xi)=0.5$, but
$B_{d}(i)=\frac{\lvert\Xi(0,h_{i}+\tau_{i})\rvert}{h_{i}+\tau_{i}}=\frac{i}{2i+2}<0.5$
for any $i\in\mathbb{N}_{+}$. In such a case, if we let $\theta=1$, then the
estimator $\hat{B}_{d}(t)$ fails to deduce the duration-bound of $\xi$ even
when $t\rightarrow+\infty$. We will further illustrate this counterexample in
Example 3.

###### Remark 4

In the DoS estimator (12) and (15), selecting a larger value of $\ell$ means
letting the estimation process in the second line begin later, and selecting a
larger $\theta$ means being less skeptical about the attacker’s past actions
(because that makes the generated bounds closer to the observed $B_{d}(i)$ and
$B_{f}(i)$, respectively). Intuitively, setting them larger will eventually
lead to tighter duration & frequency bounds (and consequently, fewer resources
are required for defense), but we might have to sacrifice some transient
security as the trade-off. Hence, this will be further illustrated in Example
2.

##  III Anti-DoS consensus of MASs

In this section, we show how DoS estimator (12)–(15) can help regulate a
multi-agent system (MAS) to reach an average consensus [22, 23] without pre-
assuming the duration and frequency parameters of the attacker.

###  III-A Problem formulation

We consider the following MAS

$\displaystyle\left\\{\begin{array}[]{lcl}\dot{x}_{i}=u_{i},&i=1,2,\cdots,N,\\\
x_{i}(0)=x_{i0}.&\end{array}\right.$ (18)

where $x_{i}\in\mathbb{R}$ represents the state of the $i$-th agent and
$u_{i}\in\mathbb{R}$ represents the control input to be designed. A common
setting of $u_{i}$ is as follows [24, 25]:

$\displaystyle
u_{i}(t)=\sum_{j=1}^{N}a_{ij}\left(x_{j}(t_{k})-x_{i}(t_{k})\right),\,t\in[t_{k},t_{k+1}),$

where $a_{ij}=1$ if there is a connection between agents $i$ and $j$, or
$a_{ij}=0$ otherwise. In addition, $\\{t_{k}\\}_{k\in\mathbb{N}_{+}}$ is the
sequence of sampling instants. Due to the impact of the DoS attack, some
transmissions are denied, and the actual input of the agent $i$ therefore
becomes

$\displaystyle
u_{i}(t)=\left\\{\begin{array}[]{ll}0,&t_{k}\in\Xi(0,+\infty),\\\
\sum_{j=1}^{N}a_{ij}\left(x_{j}(t_{k})-x_{i}(t_{k})\right),&t_{k}\in\Theta(0,+\infty)\end{array}\right.$
(21)

for any $t\in[t_{k},t_{k+1})$. Here, the notations $\Xi(\tau,s)$ and
$\Theta(\tau,s)$ are, respectively, defined in (2) and (3).

For the convenience of the statement, we denote by
$\mathscr{L}=(l_{ij})_{N\times N}$ the Laplacian of the multi-agent network,
where

$\displaystyle l_{ij}=\left\\{\begin{array}[]{lcl}-a_{ij},&i\neq j,\\\
\sum_{i\neq j}a_{ij},&i=j.\end{array}\right.$

We assume $\mathscr{L}$ to satisfy the following assumption.

###### Assumption 2

[22] $\mathscr{L}$ is symmetric and irreducible. As a result, the eigenvalues
of $\mathscr{L}$ can be arranged as

$0=\lambda_{1}(\mathscr{L})<\lambda_{2}(\mathscr{L})\leqslant\lambda_{3}(\mathscr{L})\leqslant\cdots\leqslant\lambda_{N}(\mathscr{L}).$

Our control objective is to make MAS (18) reach an average consensus, which is
defined as follows.

###### Definition 4

([22] Average consensus) MAS (18) is said to reach average consensus, if

$\lim_{t\rightarrow+\infty}{\left(x_{i}(t)-\frac{1}{N}\sum_{j=1}^{N}x_{j0}\right)}=0,\;\forall\,i=1,\cdots,N.$

$\square$

In a classical sampled data setting, the sampling is usually carried out
periodically, that is, $t_{k+1}-t_{k}=\Delta$ with some constant $\Delta>0$.
Under such a setting, average consensus can be reached if $\lvert
1-\Delta\lambda_{N}(\mathscr{L})\rvert<1$ [26] and

$\displaystyle B_{d}+\Delta B_{f}<1\text{
\cite[cite]{[\@@bibref{}{WXu_distributedDoS_tcyb2020}{}{}]}}.$ (23)

However, the design of $\Delta$ in (23) is seldom reliable because it relies
on presupposed values of the attacker’s duration-bound $B_{d}$ & frequency-
bound $B_{f}$.

Our task in this section is to design the sampling sequence
$\\{t_{k}\\}_{k\in\mathbb{N}}$, such that control law (21) drives MAS (18) to
reach an average consensus. The values of $B_{d}$ & $B_{f}$ are not allowed to
be directly used in the design, and only the past estimation $\hat{B}_{d}(s)$
& $\hat{B}_{f}(s)\,(s\leqslant t)$ generated by (12)–(15) is available at any
time $t$. Our basic idea is letting $t_{k+1}-t_{k}$ be adaptive to
$\hat{B}_{d}$ & $\hat{B}_{f}$ so that it converges to some $\Delta$ satisfying
(23). Such adaptive sampling intervals should have a strictly positive lower
bound to guarantee a finite sampling rate. If we denote $\Delta_{k}\coloneqq
t_{k+1}-t_{k}$, this task can be written more formally as follows.

###### Problem 2

(Consensus design of the sampling sequence) Design the sampling interval
$\Delta_{k}$ as

$\Delta_{k}=\Delta_{k}(\hat{B}_{d}(s),\hat{B}_{f}(s))$

where $s\leqslant t_{k}$, such that

  * a)

Control law (21) regulates MAS (18) to reach an average consensus.

  * b)

There exists a constant $\underline{\Delta}>0$ satisfying

$\Delta_{k}\geqslant\underline{\Delta},\;\forall\,k\in\mathbb{N}.$

###  III-B Solution to Problem 2: the adaptive sampling sequence

This subsection gives a detailed design of $\Delta_{k}$, which solves Problem
2. We let $t_{1}=0$ and initialize $\Delta_{k}$ as

$\displaystyle\Delta_{k}$
$\displaystyle=\Delta_{0},\,t_{k}\in[0,h_{1}+\tau_{1}),$ (24)

where $\Delta_{0}$ satisfies $\lvert
1-\Delta_{0}\lambda_{N}(\mathscr{L})\rvert<1$. When $t=h_{n}+\tau_{n}$,
$n\in\mathbb{N}_{+}$, we update $\Delta_{k}$ as

$\displaystyle\Delta_{k}$
$\displaystyle=\min\left\\{\Delta_{0},\frac{1}{\gamma_{1}\hat{B}_{f}(h_{n})}\left(1-\hat{B}_{d}(h_{n}+\tau_{n})\right)\right\\},$
$\displaystyle\qquad\qquad\qquad\qquad
t_{k}\in[h_{n}+\tau_{n},h_{n+1}+\tau_{n+1}).$ (25)

Here, $\hat{B}_{d}(t)$ and $\hat{B}_{f}(t)$ are generated by the DoS estimator
(12)–(15), and $\gamma_{1}>1$ ensures that $\Delta_{k}$ satisfies an
“estimated” version of (23), that is,

$\hat{B}_{d}(t_{k})+\Delta_{k}\hat{B}_{f}(t_{k})<1.$

The following result shows that the sampling logic determined by (24) and
(III-B) can well address Problem 2.

###### Theorem 2

Under Assumptions 1 and 2, MAS (18)–(21) subject to a DoS sequence $\xi$ can
reach average consensus if the sampling intervals
$\\{\Delta_{k}\\}_{k\in\mathbb{N}_{+}}$ are determined by (24)–(III-B).

Proof. See Appendix -C. $\square$

###### Remark 5

For consensus of MASs, Theorem 2 significantly extends the reach of the
security design (23) [10]. In particular, even if $B_{d}$ or $B_{f}$ is
previously unknown to the defender (and therefore (23) cannot be directly
implemented), satisfactory resilience is still preserved if we design the
sampling sequence as in (24)–(III-B). The theorem is, therefore, an
illustration of how we broaden the applicability of existing anti-DoS control
strategies based on our new DoS estimator.

##  IV Anti-DoS impulsive stabilization of nonlinear systems

In this section, we show that the DoS estimator is also effective in the
impulsive stabilization of nonlinear systems [21, 27, 28] in the presence of
previously unknown DoS attacks.

To avoid overcomplicating notations, the symbols $t_{k}$ and $\Delta_{k}$ in
this section will represent different physical meanings compared to the
previous section. Specifically, this section uses $t_{k}$ to represent the
$k$-th impulsive control instant and uses $\Delta_{k}\coloneqq t_{k+1}-t_{k}$
to represent the $k$-th control interval.

###  IV-A Problem formulation

We consider the following nonlinear system [21]

$\displaystyle\left\\{\begin{array}[]{lcl}\dot{X}(t)=f(t,X(t)),&t\neq
t_{k},\\\ X(0^{-})=X_{0}.&\end{array}\right.$ (28)

with an impulsive control law

$\displaystyle
X(t_{k}^{+})=X(t_{k}^{-})+U(X(t_{k}^{-})),\,t_{k}\in\Theta(0,+\infty).$ (29)

Here, $X\in\mathbb{R}^{n}$ is a right-continuous state variable,
$f:\mathbb{R}_{\geqslant 0}\times\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ is
continuous, $f(t,0)=0,\,\forall\,t\geqslant 0$, and $U(\cdot)$ is the
impulsive control input. Due to the DoS attack, the impulsive input cannot be
operated over $\Xi(0,+\infty)$, and therefore

$\displaystyle X(t_{k}^{+})=X(t_{k}^{-}),\,t_{k}\in\Xi(0,+\infty).$ (30)

Impulsive control system (28)–(30) is assumed to satisfy the following
assumption.

###### Assumption 3

[29, 21] There exists a function $V:\mathbb{R}_{\geqslant
0}\times\mathbb{R}^{n}\mapsto\mathbb{R}_{\geqslant 0}$, such that

  1. 1.

$V$ is continuous in $[t_{k},t_{k+1})\times\mathbb{R}^{n}$, and for each
$X\in\mathbb{R}^{n}$ and $k=1,2,\cdots$, the limit

$\displaystyle\lim_{(t,y)\rightarrow(t_{k}^{-},X)}V(t,Y)=V(t_{k},X)$

exists.

  2. 2.

$V$ is locally Lipschitzian in its second argument.

  3. 3.

There exists a $\beta>0$, such that for any $X\in\mathbb{R}^{n}$ and $t\neq
t_{k}$, there holds that

$\displaystyle D^{+}V(t,X)\leqslant\beta V(t,X).$

  4. 4.

There exists a $\mu\in(0,1)$ such that

$\displaystyle V(t,X+U(t,X))\leqslant\mu
V(t^{-},X),\,\forall\,X\in\mathbb{R}^{n},\,t=t_{k}.$

  5. 5.

There exist functions $a(\cdot),b(\cdot)\in\mathcal{K}$, such that $a(\lvert
X\rvert)\leqslant V(t,X)\leqslant b(\lvert X\rvert)$ holds on
$\mathbb{R}_{\geqslant 0}\times\mathbb{R}^{n}$.

Our control objective is to stabilize the system (28) exponentially, which can
be defined as follows.

###### Definition 5

([30] Exponential stabilization) It is said that impulsive controller
(29)–(30) exponentially stabilizes system (28), if there exist constants
$C_{0}>0$ and $\zeta>0$ satisfying

$\lvert X(t)\rvert\leqslant C_{0}\lvert X_{0}\rvert e^{-\zeta
t},\;\forall\,t\geqslant 0.$

Our task in this subsection is to design $\Delta_{k}$ to achieve the objective
as defined in Definition 5 above. Similar to Section III, we can only use the
past estimation $\hat{B}_{d}$ & $\hat{B}_{f}$. Furthermore, $\Delta_{k}$
should have a strictly positive lower bound. This task can be written more
formally as follows.

###### Problem 3

(Stabilization design of the impulsive control sequence) Design the control
interval $\Delta_{k}$ as

$\Delta_{k}=\Delta_{k}(\hat{B}_{d}(s),\hat{B}_{f}(s))$

where $s\leqslant t_{k}$, such that

  * a)

Impulsive controller (29)–(30) exponentially stabilizes system (28).

  * b)

There exists a constant $\underline{\Delta}>0$ satisfying

$\Delta_{k}\geqslant\underline{\Delta},\;\forall\,k\in\mathbb{N}.$

###  IV-B Solution to Problem 3: the adaptive control sequence

Now, we provide a solution to Problem 3. We let $t_{1}=0$ and initialize
$\Delta_{k}$ as

$\displaystyle\Delta_{k}$
$\displaystyle=\Delta_{0}\coloneqq-\frac{\chi}{\gamma_{3}\beta},\,t_{k}\in[0,h_{1}+\tau_{1}),$
(31)

where $\chi\coloneqq\ln\mu<0$, $\gamma_{3}>1$ is a constant, $\mu<1$ and
$\beta>0$ are given in Assumption 3. When $t=h_{n}+\tau_{n}$,
$n\in\mathbb{N}_{+}$, $\Delta_{k}$ is updated as

$\displaystyle\Delta_{k}$
$\displaystyle=\frac{\chi\left(1-\hat{B}_{d}(h_{n}+\tau_{n})\right)}{\gamma_{3}\left(\hat{B}_{f}(h_{n})\chi-\beta\right)},$
$\displaystyle\qquad\qquad\qquad t_{k}\in[h_{n}+\tau_{n},h_{n+1}+\tau_{n+1}).$
(32)

where $\hat{B}_{d}(t)$ and $\hat{B}_{f}(t)$ are generated by (12)–(15).

The following result shows that (31)–(IV-B) can well address Problem 3.

###### Theorem 3

Under Assumptions 1 and 3, nonlinear system (28)–(30) subject to a DoS
sequence $\xi$ can be exponentially stabilized if the control intervals
$\\{\Delta_{k}\\}_{k\in\mathbb{N}_{+}}$ are determined by (31)–(IV-B).

Proof. See Appendix -D. $\square$

##  V Numerical simulations

In this section, we consider a DoS sequence $\xi$ with
$\inf\mathcal{D}(\xi)=2/3$ and $\inf\mathcal{F}(\xi)=0.5$. To test the
performance of the DoS estimator (12)–(15), we let the attacker disguise
itself by appearing weaker when $t\in[0,12]$s. After $t=12$s, the attack
gradually becomes consistent with the true duration & frequency bounds.

The two control scenarios in Sections III and IV are studied in this part. Our
new estimator-based controllers have satisfactory control performances in both
cases. Particularly in the impulsive stabilization scenario, we provide an
extra simulation result to justify the discussion in Remark 4. The trade-off
between security and cost-efficiency is clearly manifested by the extra
simulation.

The end of this section considers a special DoS sequence $\xi$ as a
counterexample, illustrating the necessity of the design $\theta<1$, as
discussed in Remark 3.

###### Example 1

In this example, both scenarios in Sections III and IV share the same DoS
estimator. In particular, we let the estimated duration & frequency bounds be
generated by (12)–(15) with $\epsilon_{0}=0.01$, $\theta=2/3$ and $\ell=2$.
Fig. 1 shows the estimation’s performance. As can be seen, $\hat{B}_{d}(t)$
converges to $\mathcal{D}(\xi)=[2/3,1]$ after $n=10$, and $\hat{B}_{f}(t)$
converges to $\mathcal{F}(\xi)=[0.5,+\infty]$ after only $2$ steps.

Figure 1: Top: The estimated duration-bound $\hat{B}_{d}(t)$ is generated by
(12). Bottom: The estimated frequency-bound $\hat{B}_{f}(t)$ is generated by
(15). (With $\theta=2/3$ and $\ell=2$.)

Now, we configure security controllers based on the results in Sections III
and IV. First, we study the sampled data control performance of MAS (18) with
$7$ agents and a ring topology. By simple calculation,
$\lambda_{N}(\mathscr{L})=3.8019$, and thus $\Delta_{0}$ should be strictly
less than $0.5260$ according to Theorem 2. In addition, the initial values are
randomly chosen from $[-10,10]$, and the average consensus value is $-3/7$. By
taking $\Delta_{0}=0.4208$, $\gamma_{1}=1.3$ and letting $\Delta_{k}$ be
generated by (24)–(III-B), average consensus can be achieved as shown at the
top of Fig. 2. At the bottom of Fig. 2, we plot the adaptive sampling
intervals $\Delta_{k}$ generated by (24)–(III-B), as well as its theoretical
supremum determined by Theorem 2. As can be seen, the generated sampling
sequence is over two times more frequent than necessary. Hence, this suggests
that our strategy is somewhat conservative, which is consistent with our
intuition because we always consider a stronger opponent than observed. We
will discuss how to reduce such conservatism later in Example 2.

Figure 2: Top: Consensus of MAS (18) regulated by the sample-data controller
(21). Bottom: The values of $\Delta_{k}$ generated by (24)–(III-B) at each
sampling instance, as well as the theoretical supremum of the intervals
determined by Theorem 2.

Next, we study the impulsive control performance of the nonlinear system (28)
with a $2$-dimensional state and $f(t,X)=AX$. Here, we have an unstable matrix
$A=\begin{bmatrix}1&0.3\\\ 0&1\end{bmatrix}$ and the impulsive input
$U(X)=0.7X$. By choosing $V(t,X)=\lvert X\rvert$, the parameters in Assumption
3 are obtained as $\mu=0.7$ and
$\beta=\sqrt{\lambda_{\mbox{max}}(A^{T}A)}=1.1612$. Taking $\gamma_{3}=1.2$
and letting $\Delta_{k}$ be generated by (31)–(IV-B), impulsive stabilization
can be achieved as shown at the top of Fig. 3.

Figure 3: Top: Stabilization of system (28) regulated by the impulsive
controller (29)–(30). Bottom: The values of $\Delta_{k}$ generated by
(31)–(IV-B) at each control instants, as well as the theoretical supremum of
the intervals determined by Theorem 3.

###### Example 2

In this example, the DoS sequence is the same as in Example 1, and we tune the
parameters in the DoS estimator (12)–(15) to be $\theta=0.9$ and $\ell=3$. As
can be seen in Fig. 4, the estimator generates tighter duration & frequency
bounds of the DoS sequence $\xi$ compared with the previous example. This is
because we are now less skeptical about the historical data $B_{d}(i)$ and
$B_{f}(i)$, and thus the generated bounds are closer to the actual ones.

Figure 4: The estimated duration-bound (top) & frequency-bound (bottom) of
$\xi$ generated by the DoS estimator (12)–(15) with $\ell=3$ and $\theta=0.9$,
and a comparison with Example 1 in which $\ell=2$ and $\theta=0.67$.

In Remark 4, we have claimed that transient security might be sacrificed as
the trade-off for tighter DoS bounds. In order to demonstrate this, we use the
adjusted estimator with $\theta=0.9$ and $\ell=3$ to generate the impulsive
control sequence and observe the transient response of the system (28). For
comparison fairness, the system is set to be the same as in Example 1 except
for the control instants $t_{k}$. As shown in Fig. 5, the intervals
$\Delta_{k}$ are larger than those in Example 1, while the system takes much
longer to stabilize. Moreover, when $t=50$s, the state peak is near $500$,
which is an abnormal value for many practical systems.

Figure 5: Top: The transient response of the system (28) under impulsive
control (29)–(30). The control sequence is generated based on the adjusted DoS
estimator with $\theta=0.9$ and $\ell=3$. Bottom: The values of $\Delta_{k}$
generated at each control instant and a comparison to Example 1.

The reason why we do not compare the case of MAS consensus is that it is not
sensitive to the DoS estimator. More specifically, we can see from Fig. 2 that
the range of $\Delta_{k}$ determined by $\lvert
1-\Delta_{0}\lambda_{N}(\mathscr{L})\rvert<1$ (the region below the red dotted
line) is a subset of that determined by (23) (the region below the black
dotted line). For this reason, the initial value $\Delta_{0}$ is already small
enough to defend against the DoS attack and achieve consensus. As
$\Delta_{k}\leqslant\Delta_{0}$, the follow-up values of $\Delta_{k}$ become
inconsequential.

###### Example 3

Here, we provide a counterexample to illustrate that $\theta=1$ can lead to
the wrong estimation, as discussed in Remark 3. Specifically, we consider
$\xi=\\{[2n+1,2n+2)\\}_{n\in\mathbb{N}_{+}}$, and it can be observed that
$\inf\mathcal{D}(\xi)=0.5$ and $\inf\mathcal{F}(\xi)=0.5$. If we set
$\theta=1$ in DoS estimator (12)–(15), then the estimation $\hat{B}_{d}(t)$ &
$\hat{B}_{f}(t)$ is generated as in Fig. 6. The figure clearly shows the
incorrect estimation of $\xi$’s duration & frequency bounds.

Figure 6: The incorrect estimation of the duration-bound and frequency-bound
by setting $\theta=1$ and $\ell=2$.

The reason why we fail to estimate the bounds is intuitive: An attacker can
hide its strength by behaving as if it is weak, but setting $\theta=1$ means
not having any doubt about the attacker’s behavior. The necessity of
$\theta<1$ suggests that the defender must always be stronger than the
attacker in order to defend successfully. Simply matching the attacker’s
strength could result in failure.

##  VI Conclusion

This paper has addressed the real-time estimation problem of DoS duration and
frequency, enabling security control when the attacker is previously unknown
to the defender. A new DoS estimator has been developed to identify the
duration and frequency parameters of any DoS attacker except for three edge
cases, and it has been successfully applied to two specific control scenarios
for illustration.

###  -A Proof of Lemma 1

We first prove (7). From (5), we know that, for $\inf\mathcal{D}(\xi)<1$ and
any $\varepsilon\in(0,1-\inf\mathcal{D}(\xi)]$,
$\inf\mathcal{D}(\xi)+\varepsilon$ is a duration-bound of $\xi$. Thus for any
$\varepsilon>0$, $\exists\,\kappa(\varepsilon)\in(0,+\infty)$ such that

$\displaystyle\lvert\Xi(0,t)\rvert\leqslant\kappa(\varepsilon)+(\inf\mathcal{D}(\xi)+\varepsilon){t},\;\forall\,t\geqslant
0.$

The inequality above is also true for $\inf\mathcal{D}(\xi)=1$. Therefore,

$\displaystyle\limsup_{t\rightarrow+\infty}{\frac{\lvert\Xi(0,t)\rvert}{t}}\leqslant\inf\mathcal{D}(\xi)+\varepsilon.$

Due to the arbitrariness of $\varepsilon$, we have

$\displaystyle\limsup_{t\rightarrow+\infty}{\frac{\lvert\Xi(0,t)\rvert}{t}}\leqslant\inf\mathcal{D}(\xi).$

But if the above inequality is replaced by a strict one, then we can find an
$\varepsilon_{0}>0$ that satisfies

$\displaystyle\limsup_{t\rightarrow+\infty}{\frac{\lvert\Xi(0,t)\rvert}{t}}<\inf\mathcal{D}(\xi)-\varepsilon_{0}.$

As a result, there must exist a $T^{\ast}(\varepsilon_{0})\in(0,+\infty)$,
such that

$\displaystyle\frac{\lvert\Xi(0,t)\rvert}{t}$
$\displaystyle<\inf\mathcal{D}(\xi)-\frac{\varepsilon_{0}}{2},\;\forall\,t\geqslant
T^{\ast}.$

Therefore, for any $t\geqslant 0$, we have

$\displaystyle\lvert\Xi(0,t)\rvert\leqslant
T^{\ast}+\left(\inf\mathcal{D}(\xi)-\frac{\varepsilon_{0}}{2}\right)t.$

This, however, means that $\inf\mathcal{D}(\xi)-(\varepsilon_{0}/2)$ is a
duration-bound of $\xi$ and is therefore a contradiction. Thus, (7) is true.

We can prove (8) in a similar way, and the only difference is the case of
$\inf\mathcal{F}(\xi)=+\infty$. In such case, we claim that

$\displaystyle\limsup_{t\rightarrow+\infty}{\frac{n_{\xi}(0,t)}{t}}=+\infty=\inf\mathcal{F}(\xi).$

Otherwise, we can find $\varepsilon_{0}>0$ and $T^{\ast}(\varepsilon_{0})>0$
such that

$\displaystyle\frac{n_{\xi}(0,t)}{t}<\frac{1}{\varepsilon_{0}},\;\forall\,t\geqslant
T^{\ast}.$

Thus, we have

$\displaystyle n_{\xi}(0,t)$ $\displaystyle\leqslant
n_{\xi}(0,T^{\ast})+\frac{t}{\varepsilon_{0}}$
$\displaystyle=\frac{n_{\xi}(0,T^{\ast})}{T^{\ast}}T^{\ast}+\frac{t}{\varepsilon_{0}}$
$\displaystyle<\frac{T^{\ast}}{\varepsilon_{0}}+\frac{t}{\varepsilon_{0}},\;\forall\,t\geqslant
0,$

which means that $\frac{1}{\varepsilon_{0}}$ is a frequency-bound of $\xi$.
This is a contradiction.

If $\inf\mathcal{F}(\xi)<+\infty$, then given any $\varepsilon>0$,
$\inf\mathcal{F}(\xi)+\varepsilon$ must be a frequency-bound of $\xi$. Due to
the arbitrariness of $\varepsilon$, this infers that

$\displaystyle\limsup_{t\rightarrow+\infty}{\frac{n_{\xi}(0,t)}{t}}\leqslant\inf\mathcal{F}(\xi).$

But if the above inequality is replaced by a strict one, then we can find an
$\varepsilon_{0}>0$, such that

$\displaystyle\limsup_{t\rightarrow+\infty}{\frac{n_{\xi}(0,t)}{t}}<\inf\mathcal{F}(\xi)-\varepsilon_{0}.$

This is a contradiction because it infers that
$\inf\mathcal{F}(\xi)-\frac{\varepsilon_{0}}{2}$ is a frequency-bound. Hence,
(8) is true, and the proof is completed.

###  -B Proof of Theorem 1

In view of Assumption 1, $\tau_{n}$ must have a finite upper bound. For this
reason, we have $\inf\mathcal{D}(\xi)=\inf\mathcal{F}(\xi)=0$ when there are
only finite $H_{n}$s in $\xi$. In that case, the conclusion is trivial with
$T=0$ because $\hat{B}_{d}(t)\geqslant\hat{B}_{d}(0)>0$ and
$\hat{B}_{f}(t)\geqslant\hat{B}_{f}(0)>0$.

From now on, we assume that $\xi$ is infinite, and we need the following
auxiliary lemma to proceed.

###### Lemma 2

Under Assumption 1, it holds that

$\displaystyle\limsup_{n\rightarrow\infty}{\frac{\lvert\Xi(0,h_{n}+\tau_{n})\rvert}{h_{n}+\tau_{n}}}=\inf\mathcal{D}(\xi)$
(33)

and

$\displaystyle\limsup_{n\rightarrow\infty}{\frac{n}{h_{n}}}=\inf\mathcal{F}(\xi).$
(34)

Proof. The proof is based on Lemma 1. First, we prove (33) by showing that

$\displaystyle\limsup_{n\rightarrow\infty}{\frac{\lvert\Xi(0,h_{n}+\tau_{n})\rvert}{h_{n}+\tau_{n}}}\leqslant\limsup_{t\rightarrow+\infty}{\frac{\lvert\Xi(0,t)\rvert}{t}}$
(35)

and

$\displaystyle\limsup_{n\rightarrow\infty}{\frac{\lvert\Xi(0,h_{n}+\tau_{n})\rvert}{h_{n}+\tau_{n}}}\geqslant\limsup_{t\rightarrow+\infty}{\frac{\lvert\Xi(0,t)\rvert}{t}}.$
(36)

Recalling Assumption 1 and the fact that $\xi$ is infinite, there must hold
that $h_{n}\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\longrightarrow}}+\infty$. Therefore, (35) is trivial.

To prove (36), we will construct a function $\varphi(t)$ such that

$\displaystyle\varphi(t)\geqslant\frac{\lvert\Xi(0,t)\rvert}{t},$

and show that

$\displaystyle\limsup_{n\rightarrow\infty}{\frac{\lvert\Xi(0,h_{n}+\tau_{n})\rvert}{h_{n}+\tau_{n}}}\geqslant\limsup_{t\rightarrow+\infty}{\varphi(t)}.$

Specifically, we define $\varphi(t)$ as

$\displaystyle\varphi(t)\coloneqq\left\\{\begin{array}[]{ll}0,&t\in[0,h_{1}),\\\
\frac{\lvert\Xi(0,h_{n}+\tau_{n})\rvert}{h_{n}+\tau_{n}},&t\in[h_{n},h_{n+1}).\end{array}\right.$

Then, it is clear that

$\limsup_{n\rightarrow\infty}{\frac{\lvert\Xi(0,h_{n}+\tau_{n})\rvert}{h_{n}+\tau_{n}}}\geqslant\limsup_{t\rightarrow+\infty}{\varphi(t)}.$

On the other hand, note that

${\frac{\lvert\Xi(0,t)\rvert}{t}}=0,\,\forall\,t\in[0,h_{1}),$

and

$\frac{\lvert\Xi(0,h_{n}+\tau_{n})\rvert}{h_{n}+\tau_{n}}\geqslant{\frac{\lvert\Xi(0,t)\rvert}{t}},\,\forall\,t\in[h_{n},h_{n+1}).$

Thus, we can conclude that $\varphi(t)\geqslant\frac{\lvert\Xi(0,t)\rvert}{t}$
and (36) therefore holds. In view of (35), (36), and Lemma 1, then (33) is
true.

The proof of (34) is quite similar. Since $h_{n}\rightarrow+\infty$, we
clearly have

$\displaystyle\limsup_{n\rightarrow\infty}{\frac{n}{h_{n}}}=\limsup_{n\rightarrow\infty}{\frac{n_{\xi}(0,h_{n})}{h_{n}}}\leqslant\limsup_{t\rightarrow+\infty}{\frac{n_{\xi}(0,t)}{t}}.$

On the other hand, if we construct

$\displaystyle\psi(t)\coloneqq\left\\{\begin{array}[]{ll}\frac{1}{h_{1}},&t\in[0,h_{1}),\\\
\frac{n}{h_{n}},&t\in[h_{n},h_{n+1}),\end{array}\right.$

it is easily derived that

$\displaystyle\psi(t)\geqslant\frac{n_{\xi}(0,t)}{t}$

and

$\displaystyle\limsup_{t\rightarrow+\infty}{\psi(t)}=\limsup_{n\rightarrow\infty}{\frac{n}{h_{n}}}.$

Consequently, we have by Lemma 1 that

$\displaystyle\limsup_{n\rightarrow\infty}{\frac{n}{h_{n}}}=\limsup_{t\rightarrow+\infty}\frac{n_{\xi}(0,t)}{t}=\inf\mathcal{F}(\xi),$

and the proof of Lemma 2 is completed. $\square$

Now, we can continue the proof of Theorem 1 when $\xi$ is infinite. Recalling
that $\hat{B}_{d}(h_{n}+\tau_{n})$ and $\hat{B}_{f}(h_{n})$ take the maximum
values from $0$ to $n$, they are clearly non-decreasing in $n$. Hence, it
suffices to find a finite $N_{1}\geqslant\ell$ that makes
$\hat{B}_{d}(h_{N_{1}}+\tau_{N_{1}})$ (resp., $\hat{B}_{f}(h_{N_{1}})$) the
duration-bound (resp., frequency-bound) of $\xi$.

First, we show the existence of a finite integer $N_{2}$, such that
$\hat{B}_{d}(h_{N_{2}}+\tau_{N_{2}})$ is a duration-bound of $\xi$. Assume by
contradiction, that is, for any $n\geqslant\ell$ and any $\tilde{\kappa}>0$,
there exists a $t^{\ast}(n,\tilde{\kappa})\geqslant 0$ satisfying

$\displaystyle\lvert\Xi(0,t^{\ast})\rvert$
$\displaystyle>\tilde{\kappa}+\hat{B}_{d}(h_{n}+\tau_{n})\,{t^{\ast}}$
$\displaystyle\geqslant\tilde{\kappa}+\left(\mbox{max}_{\ell\leqslant
i\leqslant n}\\{\theta B_{d}(i)+(1-\theta)\\}\right)t^{\ast}$
$\displaystyle\geqslant\tilde{\kappa}+\left(\theta
B_{d}(n)+(1-\theta)\right){t^{\ast}}.$ (39)

In view of Lemma 2, there exists a subsequence
$\\{n_{k}\\}_{k\in\mathbb{N}_{+}}\subseteq\\{n\\}_{n\in\mathbb{N}_{+}}$, such
that

$\displaystyle\lim_{k\rightarrow\infty}{B_{d}(n_{k})}=\inf\mathcal{D}(\xi).$

Thus, for
$\varepsilon_{1}\coloneqq\frac{\left(1-\inf\mathcal{D}(\xi)\right)(1-\theta)}{2}>0$,
we can pick an $n_{0}(\varepsilon_{1})\in\\{n_{k}\\}_{k\in\mathbb{N}_{+}}$ to
satisfy

$\displaystyle\theta B_{d}(n_{0})+(1-\theta)$
$\displaystyle>\theta\inf\mathcal{D}(\xi)+(1-\theta)-\varepsilon_{1}$
$\displaystyle=\inf\mathcal{D}(\xi)+\varepsilon_{1}.$

Hence, it follows from (-B) that, for any $\tilde{\kappa}>0$ and $n=n_{0}$,
there exists a $t^{\ast}(n_{0},\tilde{\kappa})\geqslant 0$ which satisfies

$\displaystyle\lvert\Xi(0,t^{\ast})\rvert>\tilde{\kappa}+\left(\inf\mathcal{D}(\xi)+\varepsilon_{1}\right){t^{\ast}}.$

This is a contradiction because $\inf\mathcal{D}(\xi)+\varepsilon_{1}$ is a
duration-bound of $\xi$. The existence of $N_{2}$ is therefore proved.

Analogously, we can find an $N_{3}\geqslant\ell$ such that
$\hat{B}_{f}(h_{N_{3}})$ is a frequency-bound of $\xi$. In fact, by taking
$\varepsilon_{2}\coloneqq\frac{(1-\theta)\inf\mathcal{F}(\xi)}{2\theta}>0$, we
can use Lemma 2 to find an $n_{0}$ that satisfies

$\displaystyle\frac{n_{0}}{\theta
h_{n_{0}}}>\frac{\inf\mathcal{F}(\xi)}{\theta}-\varepsilon_{2}=\inf\mathcal{F}(\xi)+\varepsilon_{2}.$

If the desired $N_{3}$ does not exist, it will be similarly derived for any
$\tilde{\Lambda}>0$ that there exists a
$t^{\ast}(n_{0},\tilde{\Lambda})\geqslant 0$ satisfying

$\displaystyle
n_{\xi}(0,t^{\ast})>\tilde{\Lambda}+\left(\inf\mathcal{F}(\xi)+\varepsilon_{2}\right){t^{\ast}}.$

This is a contradiction.

Taking $N_{1}=\mbox{max}\\{N_{2},N_{3}\\}$ and $T=h_{N_{1}}+\tau_{N_{1}}$
(which must be finite because of Assumption 1), the proof of Theorem 1 is
completed.

###  -C Proof of Theorem 2

The proof will be divided into two parts. In the first part we show that the
system (18) achieves consensus if $\Delta_{k}$ stays within a certain range
after some finite instant. In the second part, we further show that
$\Delta_{k}$ designed in (24) and (III-B) converges into the required range
within finite time and remains there forever.

The following lemma corresponds to the first part.

###### Lemma 3

Consider the sampled data MAS (18)–(21) subject to a DoS sequence $\xi$
satisfying Assumption 1. The system achieves consensus if there exist finite
constants $\check{t}$, $\underline{\Delta}$, and $\bar{\Delta}$ such that,

$\displaystyle\left\\{\begin{array}[]{lcl}0<\underline{\Delta}\leqslant\Delta_{k}<+\infty,&\forall\,t_{k}\geqslant
0,\\\
\Delta_{k}\leqslant\bar{\Delta},&\forall\,t_{k}\geqslant\check{t}.\end{array}\right.$
(42)

Here, $\bar{\Delta}$ should satisfy

$\displaystyle\lvert 1-\bar{\Delta}\lambda_{N}(\mathscr{L})\rvert<1$ (43)

and

$\displaystyle B_{d}+B_{f}\bar{\Delta}<1,$ (44)

where $B_{d}$ and $B_{f}$ are, respectively, the duration & frequency bounds
of $\xi$.

Proof. In order to recast the system (18) into a compact form, we introduce
some notations first. We denote by
$\bar{x}\coloneqq\frac{1}{N}\sum_{i=1}^{N}x_{i}$ the mean value of the $N$
agents, and by
$e\coloneqq(x_{1}-\bar{x},x_{2}-\bar{x},\cdots,x_{N}-\bar{x})^{T}$ the
consensus error vector. Notably, we have
$\bar{x}=\frac{1}{N}\sum_{i=1}^{N}x_{i0}$ because $\dot{\bar{x}}=0$. In view
of (18) and (21), we can derive that

$\displaystyle e(t_{k+1})=\left\\{\begin{array}[]{lcl}e(t_{k}),&\text{if
}t_{k}\in\Xi(0,+\infty)\\\ (I-\Delta_{k}\mathscr{L})e(t_{k}),&\text{if
}t_{k}\in\Theta(0,+\infty)\end{array}\right.$

It is easy to verify that $e\perp\mathbf{1}_{N}$, and therefore

$\displaystyle\quad\lvert(I-\Delta_{k}\mathscr{L})e(t_{k})\rvert$
$\displaystyle\leqslant\mbox{max}\left\\{\lvert
1-\Delta_{k}\lambda_{2}(\mathscr{L})\rvert,\,\lvert
1-\Delta_{k}\lambda_{N}(\mathscr{L})\rvert\right\\}\cdot\lvert e(t_{k})\rvert$
$\displaystyle\leqslant\mbox{max}\left\\{\lvert
1-\underline{\Delta}\lambda_{2}(\mathscr{L})\rvert,\,\lvert
1-\bar{\Delta}\lambda_{N}(\mathscr{L})\rvert\right\\}\cdot\lvert
e(t_{k})\rvert$ $\displaystyle\eqqcolon\gamma_{2}\lvert e(t_{k})\rvert<\lvert
e(t_{k})\rvert,\quad\forall\,t_{k}\geqslant\check{t}.$ (46)

Here, the second inequality follows from (42), and the last inequality follows
from (43). If we denote by $t_{k(\check{t})}$ the first transmission attempt
no earlier than $\check{t}$, and by $\alpha(\tau,s)$ the number of successful
transmissions on $[\tau,s]$, it can be derived from (-C) that

$\displaystyle\lvert e(t_{m})\rvert$
$\displaystyle=\gamma_{2}^{\alpha(\check{t},t_{m})-1}\lvert
e(t_{k(\check{t})})\rvert,\,\forall\,m>k(\check{t}).$

Since $\check{t}$ is finite and
$0<\underline{\Delta}\leqslant\Delta_{k}<+\infty$, we have finite
$t_{k(\check{t})}$ and $\lvert e(t_{k(\check{t})})\rvert$. On the other hand,
when $\Delta_{k}\leqslant\bar{\Delta}$ and $B_{d}+B_{f}\bar{\Delta}<1$, it
follows from lemma 3 in [6] that $\alpha(\check{t},t_{m})\rightarrow+\infty$
as $t_{m}\rightarrow+\infty$. Together with $\gamma_{2}<1$, this yields that
$\lvert e(t_{m})\rvert\stackrel{{\scriptstyle
t_{m}\rightarrow+\infty}}{{\longrightarrow}}0$ and consequently
$e(t)\stackrel{{\scriptstyle t\rightarrow+\infty}}{{\longrightarrow}}0$.
Hence, consensus is achieved. $\square$

Next, we present another lemma to show that $\Delta_{k}$ designed in (24) and
(III-B) converges into the range determined by (42)–(44) within finite time.

###### Lemma 4

Under Assumption 1, there exists a finite $\check{t}\geqslant 0$, such that
$\Delta_{k}$ generated by (24) and (III-B) satisfies (42)–(44).

Proof. First, it is clear that $\Delta_{k}\leqslant\Delta_{0}<+\infty$,
$\forall\,t_{k}\geqslant 0$. Recalling the definition of $\Delta_{0}$ below
(24), we have (43) holds with $\bar{\Delta}=\Delta_{0}$.

When $\xi$ is finite, $\inf\mathcal{D}(\xi)=\inf\mathcal{F}(\xi)=0$, and hence
any $B_{d}\in(0,1)$ & $B_{f}>0$ must be the duration & frequency bounds of
$\xi$. By taking them to be sufficiently small, (44) can be satisfied with
$\bar{\Delta}=\Delta_{0}$. On the other hand, because $\xi$ is finite, the set
$\\{\Delta_{k}\\}_{k\in\mathbb{N}_{+}}$ only contains finite positive elements
and thus has a positive lower-bound $\underline{\Delta}$. Then by taking
$\check{t}=0$ and $\bar{\Delta}=\Delta_{0}$, (42)–(44) are all satisfied.

When $\xi$ is infinite, it is easy to derive from Lemma 2 that

$\displaystyle\lim_{n\rightarrow\infty}{\hat{B}_{d}(h_{n}+\tau_{n})}<1$

and

$\displaystyle\lim_{n\rightarrow\infty}{\hat{B}_{f}(h_{n})}<+\infty.$

Here, the inequalities are true because $\ell\geqslant 2$ and Assumption 1
holds. Since $\hat{B}_{d}(h_{n}+\tau_{n})$ and $\hat{B}_{f}(h_{n})$ are non-
decreasing in $n$, $\Delta_{k}$ generated by (III-B) must be non-increasing
and lower-bounded by

$\displaystyle\underline{\Delta}$
$\displaystyle=\min\left\\{{\Delta_{0},\lim_{n\rightarrow\infty}\Bigg{[}\frac{1}{\gamma_{1}\hat{B}_{f}(h_{n})}\left(1-\hat{B}_{d}(h_{n}+\tau_{n})\right)}\Bigg{]}\right\\}$
$\displaystyle>0.$

Now it remains to find a $\bar{\Delta}$ satisfying (43)–(44) and a finite
$\check{t}$, such that $\Delta_{k}\leqslant\bar{\Delta}$ holds for any
$t_{k}\geqslant\check{t}$. According to Theorem 1, there exists a finite $T$
satisfying $\hat{B}_{d}(T)\in\mathcal{D}(\xi)$ and
$\hat{B}_{f}(T)\in\mathcal{F}(\xi)$. Then, (III-B) yields for any
$t_{k}\geqslant\check{t}\coloneqq T$ that

$\Delta_{k}\leqslant\Delta_{k(\check{t})}$

and

$\displaystyle\hat{B}_{d}(T)+\Delta_{k(\check{t})}\hat{B}_{f}(T)$
$\displaystyle\leqslant\hat{B}_{d}(T)+\frac{1}{\gamma_{1}}\left(1-\hat{B}_{d}(T)\right)$
$\displaystyle<1.$

By letting $\bar{\Delta}=\min\\{\Delta_{0},\Delta_{k(\check{t})}\\}$, we have
(42)–(44) be satisfied with $\check{t}=T$, $B_{d}=\hat{B}_{d}(T)$ and
$B_{f}=\hat{B}_{f}(T)$. The proof is completed. $\square$

With Lemmas 3 and 4, the conclusion of Theorem 2 directly follows.

###  -D Proof of Theorem 3

According to Corollary 3.1.1 in [29], if Assumption 3 is satisfied, then we
have

$\displaystyle V(t,X(t))\leqslant V(0,X_{0})\mu^{\alpha(0,t)}\mbox{exp}(\beta
t).$ (47)

Here, $\alpha(\tau,s)$ denotes the number of successful control instants on
$[\tau,s]$. For further analysis, it is necessary to estimate the value of
$\alpha(0,t)$, which is addressed by the following lemma.

###### Lemma 5

Consider a DoS sequence satisfying Assumption 1 and a sequence of control
instants $\\{t_{k}\\}_{k\in\mathbb{N_{+}}}$ satisfying $\Delta_{k}\coloneqq
t_{k+1}-t_{k}<+\infty$. If for some finite instant $\check{t}\geqslant 0$,
there exists a $\bar{\Delta}>0$ such that
$\Delta_{k}\leqslant\bar{\Delta},\,\forall\,t_{k}\geqslant\check{t}$, then, it
holds that

$\displaystyle\alpha(0,t)\geqslant\frac{t}{\bar{\Delta}}\left(1-B_{d}-\bar{\Delta}B_{f}\right)-C_{1},\,\forall\,t\geqslant
t_{k(\check{t})}.$ (48)

Here, $t_{k(\check{t})}<+\infty$ represents the first control instant no
earlier than $\check{t}$, $B_{d}$ and $B_{f}$ are the duration & frequency
bounds of $\xi$,
$C_{1}\coloneqq\frac{t_{k(\check{t})}+\kappa}{\bar{\Delta}}+1+\Lambda$, and
$\kappa$ and $\Lambda$ are given in Definitions 1 and 2.

Proof. Consider any connected component of $\Theta(t_{k(\check{t})},$
$+\infty)$, which must be in the form of $[s_{a},s_{b})$, $s_{b}\geqslant
s_{a}$. Here, $\Theta(\tau,s)$ is defined in (3). Since
$[s_{a},s_{b})\subseteq[t_{k(\check{t})},+\infty)$, it contains no less than
$\left(({s_{b}-s_{a}})/{\bar{\Delta}}\right)-1$ successful control instants.
Therefore, $\alpha(t_{k(\check{t})},t)$ must be no less than
$\lvert\Theta(t_{k(\check{t})},t)\rvert/\bar{\Delta}$ minus the number of
connected components of $\Theta(t_{k(\check{t})},t)$. Combining this with
$B_{d}\in\mathcal{D}(\xi)$ and $B_{f}\in\mathcal{F}(\xi)$, we obtain that

$\displaystyle\quad\alpha(t_{k(\check{t})},t)$
$\displaystyle\geqslant\frac{\lvert\Theta(t_{k(\check{t})},t)\rvert}{\bar{\Delta}}-\left(n_{\xi}(t_{k(\check{t})},t)+1\right)$
$\displaystyle\geqslant\frac{\lvert\Theta(0,t)\rvert-
t_{k(\check{t})}}{\bar{\Delta}}-\left(n_{\xi}(0,t)+1\right)$
$\displaystyle\geqslant\frac{1}{\bar{\Delta}}\left(\bigg{(}1-B_{d}\bigg{)}t-t_{k(\check{t})}-\kappa\right)-\left(n_{\xi}(0,t)+1\right)$
$\displaystyle\geqslant\frac{t}{\bar{\Delta}}\left(1-B_{d}-\bar{\Delta}B_{f}\right)-C_{1}.$

Since $\alpha(0,t)\geqslant\alpha(t_{k(\check{t})},t)$, (48) directly follows.
$\square$

In order to use (48), we need to show that $\Delta_{k}$ generated by (31) and
(IV-B) satisfy the conditions in Lemma 5. In addition, the upper-bound
$\bar{\Delta}$ in (48) should satisfy

$\displaystyle\bar{\Delta}\left(\beta-
B_{f}{\chi}\right)<-\chi\left(1-B_{d}\right)$ (49)

so that $\mu^{\alpha(0,t)}\mbox{exp}(\beta t)\rightarrow 0$ as
$t\rightarrow+\infty$. To this end, we introduce the following lemma.

###### Lemma 6

Under Assumption 1, if $1>B_{d}\in\mathcal{D}(\xi)$ and
$+\infty>B_{f}\in\mathcal{F}(\xi)$, then $\Delta_{k}$ generated by (31) and
(IV-B) satisfy the following items.

  1. 1.

$\Delta_{k}<+\infty$.

  2. 2.

There exists a $\underline{\Delta}>0$, such that
$\Delta_{k}\geqslant\underline{\Delta}$, $\forall\,k\in\mathbb{N}_{+}$.

  3. 3.

There exist a constant $\bar{\Delta}>0$ satisfying (49) and a finite
$\check{t}\geqslant 0$, such that
$\Delta_{k}\leqslant\bar{\Delta},\,\forall\,t_{k}\geqslant\check{t}$.

Proof. The item 1) is trivial. In fact, we have
$\Delta_{k}\leqslant\Delta_{0}$ according to (31) and (IV-B).

When $\xi$ is finite, the set $\\{\Delta_{k}\\}_{k\in\mathbb{N}_{+}}$ contains
finite positive elements, so $\Delta_{k}$ must have a positive lower-bound.
Hence, the item 2) is satisfied. In addition, we have
$\inf\mathcal{D}(\xi)=\inf\mathcal{F}(\xi)=0$ due to Assumption 1. Then, any
finite $B_{d}\in(0,1)$ and $B_{f}>0$ are the duration & frequency bounds of
$\xi$, respectively. By choosing them to be sufficiently small, (49) is
satisfied with $\bar{\Delta}=\Delta_{0}$. Thus, the item 3) is satisfied with
$\bar{\Delta}=\Delta_{0}$ and $\check{t}=0$.

When $\xi$ is infinite, it is easy to derive from Lemma 2 that

$\displaystyle{\lim_{n\rightarrow\infty}{\hat{B}_{d}(h_{n}+\tau_{n})}<1}$

and

$\displaystyle{\lim_{n\rightarrow\infty}{\hat{B}_{f}(h_{n})}<+\infty.}$

Recalling the monotonicity of $\hat{B}_{d}(h_{n}+\tau_{n})$ and
$\hat{B}_{f}(h_{n})$, we have $\Delta_{k}$ generated by (IV-B) non-increasing.
Consequently, $\Delta_{k}$ is lower-bounded by

$\displaystyle\underline{\Delta}=\lim_{n\rightarrow\infty}\frac{\chi\left(1-\hat{B}_{d}(h_{n}+\tau_{n})\right)}{\gamma_{3}\left(\hat{B}_{f}(h_{n}){\chi}-\beta\right)}>0.$

Hence, the item 2) is satisfied. As for 3), it is also satisfied if we take
$\check{t}\coloneqq T$ and $\bar{\Delta}=\Delta_{k(\check{t})}$ where the
finite $T$ is determined by Theorem 1. In fact, it follows from (IV-B) that

$\displaystyle\qquad\bar{\Delta}\left(\beta-\hat{B}_{f}(T){\chi}\right)$
$\displaystyle\leqslant\frac{\chi\left(1-\hat{B}_{d}(T)\right)}{\gamma_{3}\left(\hat{B}_{f}(T){\chi}-\beta\right)}\left(\beta-\hat{B}_{f}(T){\chi}\right)$
$\displaystyle<-\chi\left(1-\hat{B}_{d}(T)\right),$

where the second inequality is because $\gamma_{3}>1$, $\beta>0$ and
$\chi=\ln\mu<0$. Hence, $\bar{\Delta}$ satisfies (49) as $\hat{B}_{d}(T)$ &
$\hat{B}_{f}(T)$ are the duration & frequency bounds of $\xi$. Additionally,
the monotonicity of $\Delta_{k}$ indicates that
$\Delta_{k}\leqslant\bar{\Delta}$ for any $t_{k}\geqslant\check{t}$. The proof
is completed. $\square$

With Lemmas 5 and 6, we can substitute (48) into (47) and obtain the following
inequality

$\displaystyle V(t,X(t))$ $\displaystyle\leqslant V(0,X_{0})e^{\beta
t}\mu^{\frac{t}{\bar{\Delta}}\left(1-B_{d}-\bar{\Delta}B_{f}\right)-C_{1}}$
$\displaystyle\leqslant V(0,X_{0})\mu^{-C_{1}}e^{\beta
t+\chi\left(\frac{t}{\bar{\Delta}}\left(1-B_{d}-{\bar{\Delta}}B_{f}\right)\right)}$
$\displaystyle=V(0,X_{0})\mu^{-C_{1}}e^{\left(\bar{\Delta}\left(\beta-
B_{f}{\chi}\right)+\chi\left(1-B_{d}\right)\right)\frac{t}{\bar{\Delta}}},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\forall\,t\geqslant
t_{k(\check{t})}.$

Since $\Delta_{k}<+\infty$, $t_{k(\check{t})}$ must be finite. And due to 3)
in Assumption 3, $V(t,X(t))$ cannot escape on $[0,t_{k(\check{t})}]$. In view
of (49), $V(t,X(t))$ exponentially converges to $0$, and thus the system (28)
is exponentially stabilized by the impulsive controller (29)–(30) (see Theorem
3.2.1 in [29]). In addition, $\Delta_{k}$ is lower-bounded by
$\underline{\Delta}>0$ according to Lemma 6.

## References

  * [1] C. De Persis and P. Tesi, “Input-to-state stabilizing control under denial-of-service,” IEEE Transactions on Automatic Control, vol. 60, no. 11, pp. 2930–2944, 2015.
  * [2] A. Wood and J. Stankovic, “Denial of service in sensor networks,” Computer, vol. 35, no. 10, pp. 54–62, 2002.
  * [3] S. Amin, A. A. Cárdenas, and S. Shankar Sastry, “Safe and secure networked control systems under denial-of-service attacks,” in Hybrid Systems: Computation and Control, pp. 31–45, Berlin, Heidelberg: Springer, 2009.
  * [4] S. Amin, G. A. Schwartz, and S. Shankar Sastry, “Security of interdependent and identical networked control systems,” Automatica, vol. 49, no. 1, pp. 186–192, 2013.
  * [5] Y. Zhu and W. X. Zheng, “Observer-based control for cyber-physical systems with periodic DoS attacks via a cyclic switching strategy,” IEEE Transactions on Automatic Control, vol. 65, no. 8, pp. 3714–3721, 2020.
  * [6] C. De Persis and P. Tesi, “Networked control of nonlinear systems under denial-of-service,” Systems & Control Letters, vol. 96, pp. 124–131, 2016.
  * [7] S. Feng and P. Tesi, “Networked control systems under denial-of-service: Co-located vs. remote architectures,” Systems & Control Letters, vol. 108, pp. 40–47, 2017.
  * [8] M. Wakaiki, A. Cetinkaya, and H. Ishii, “Stabilization of networked control systems under DoS attacks and output quantization,” IEEE Transactions on Automatic Control, vol. 65, no. 8, pp. 3560–3575, 2020.
  * [9] S. Feng and H. Ishii, “Dynamic quantized consensus of general linear multiagent systems under denial-of-service attacks,” IEEE Transactions on Control of Network Systems, vol. 9, no. 2, pp. 562–574, 2022.
  * [10] W. Xu, G. Hu, D. W. C. Ho, and Z. Feng, “Distributed secure cooperative control under denial-of-service attacks from multiple adversaries,” IEEE Transactions on Cybernetics, vol. 50, no. 8, pp. 3458–3467, 2020.
  * [11] V. S. Dolk, P. Tesi, C. De Persis, and W. P. M. H. Heemels, “Event-triggered control systems under denial-of-service attacks,” IEEE Transactions on Control of Network Systems, vol. 4, no. 1, pp. 93–105, 2017.
  * [12] A.-Y. Lu and G.-H. Yang, “Input-to-state stabilizing control for cyber-physical systems with multiple transmission channels under denial of service,” IEEE Transactions on Automatic Control, vol. 63, no. 6, pp. 1813–1820, 2018.
  * [13] R. Kato, A. Cetinkaya, and H. Ishii, “Linearization-based quantized stabilization of nonlinear systems under DoS attacks,” IEEE Transactions on Automatic Control, vol. 67, no. 12, pp. 6826–6833, 2022.
  * [14] S. Feng, A. Cetinkaya, H. Ishii, P. Tesi, and C. De Persis, “Resilient quantized control under denial-of-service: Variable bit rate quantization,” Automatica, vol. 141, p. 110302, 2022.
  * [15] D. Zhang, C. Deng, and G. Feng, “Resilient cooperative output regulation for nonlinear multiagent systems under DoS attacks,” IEEE Transactions on Automatic Control, vol. 68, no. 4, pp. 2521–2528, 2023.
  * [16] T. Li, B. Chen, L. Yu, and W.-A. Zhang, “Active security control approach against DoS attacks in cyber-physical systems,” IEEE Transactions on Automatic Control, vol. 66, no. 9, pp. 4303–4310, 2021.
  * [17] Z. Abdollahi Biron, S. Dey, and P. Pisu, “Real-time detection and estimation of denial of service attack in connected vehicle systems,” IEEE Transactions on Intelligent Transportation Systems, vol. 19, no. 12, pp. 3893–3902, 2018.
  * [18] Y. Li, L. Shi, P. Cheng, J. Chen, and D. E. Quevedo, “Jamming attacks on remote state estimation in cyber-physical systems: A game-theoretic approach,” IEEE Transactions on Automatic Control, vol. 60, no. 10, pp. 2831–2836, 2015.
  * [19] C. D. Aliprantis and O. Burkinshaw, Principles of Real Analysis.  Academic, 3rd ed., 1998.
  * [20] W. Rudin, Principles of Mathematical Analysis.  New York: McGraw Hill, 1976.
  * [21] T. Yang, “Impulsive control,” IEEE Transactions on Automatic Control, vol. 44, no. 5, pp. 1081–1083, 1999.
  * [22] R. Saber and R. Murray, “Consensus protocols for networks of dynamic agents,” in Proceedings of the 2003 American Control Conference, 2003., vol. 2, pp. 951–956, 2003.
  * [23] J. Berneburg and C. Nowzari, “Robust dynamic event-triggered coordination with a designable minimum interevent time,” IEEE Transactions on Automatic Control, vol. 66, no. 8, pp. 3417–3428, 2021.
  * [24] G. Xie, H. Liu, L. Wang, and Y. Jia, “Consensus in networked multi-agent systems via sampled control: Fixed topology case,” in 2009 American Control Conference, pp. 3902–3907, 2009.
  * [25] H. Su and S. Miao, “Consensus on directed matrix-weighted networks,” IEEE Transactions on Automatic Control, vol. 68, no. 4, pp. 2529–2535, 2023.
  * [26] K. You and L. Xie, “Network topology and communication data rate for consensusability of discrete-time multi-agent systems,” IEEE Transactions on Automatic Control, vol. 56, no. 10, pp. 2262–2275, 2011.
  * [27] X. Li, D. Peng, and J. Cao, “Lyapunov stability for impulsive systems via event-triggered impulsive control,” IEEE Transactions on Automatic Control, vol. 65, no. 11, pp. 4908–4913, 2020.
  * [28] Z. Li, C. Wen, and Y. Soh, “Analysis and design of impulsive control systems,” IEEE Transactions on Automatic Control, vol. 46, no. 6, pp. 894–897, 2001.
  * [29] V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations.  World Scientific, 1989.
  * [30] H. Khalil, Nonlinear Systems.  London: Pearson, 2001.