# A Zero-Sum Game Framework for Optimal Sensor Placement in Uncertain
Networked Control Systems under Cyber-Attacks

Anh Tung Nguyen1, Sribalaji C. Anand2, and André M. H. Teixeira1 *This work is
supported by the Swedish Research Council under the grants 2018-04396 and
2021-06316 and by the Swedish Foundation for Strategic Research.1 Anh Tung
Nguyen and André M. H. Teixeira are with the Department of Information
Technology, Uppsala University, PO Box 337, SE-75105, Uppsala, Sweden.
{anh.tung.nguyen<EMAIL_ADDRESS>Sribalaji C. Anand is with the
Department of Electrical Engineering, Uppsala University, PO Box 65, SE-75103,
Uppsala, Sweden<EMAIL_ADDRESS>
###### Abstract

This paper proposes a game-theoretic approach to address the problem of
optimal sensor placement against an adversary in uncertain networked control
systems. The problem is formulated as a zero-sum game with two players, namely
a malicious adversary and a detector. Given a protected performance vertex, we
consider a detector, with uncertain system knowledge, that selects another
vertex on which to place a sensor and monitors its output with the aim of
detecting the presence of the adversary. On the other hand, the adversary,
also with uncertain system knowledge, chooses a single vertex and conducts a
cyber-attack on its input. The purpose of the adversary is to drive the attack
vertex as to maximally disrupt the protected performance vertex while
remaining undetected by the detector. As our first contribution, the game
payoff of the above-defined zero-sum game is formulated in terms of the Value-
at-Risk of the adversary’s impact. However, this game payoff corresponds to an
intractable optimization problem. To tackle the problem, we adopt the scenario
approach to approximately compute the game payoff. Then, the optimal monitor
selection is determined by analyzing the equilibrium of the zero-sum game. The
proposed approach is illustrated via a numerical example of a 10-vertex
networked control system.

##  I Introduction

Networked control systems have been playing a crucial role in modeling,
analysis, and operation of real-world large-scale interconnected systems such
as power systems, transportation networks, and water distribution networks.
Those systems consist of multiple interconnected subsystems which generally
communicate with each other via insecure communication channels to share their
information. This insecure protocol may leave the networked control systems
vulnerable to cyber-attacks such as denial-of-service and false-data injection
attacks [1], inflicting serious financial loss and civil damages. Reports on
actual damages such as Stuxnet [2] and Industroyer [3] have described the
catastrophic consequences of such cyber-attacks for an Iranian nuclear program
and a Ukrainian power grid, respectively. Motivated by the above observation,
cyber-physical security has increasingly received much attention from control
society in recent years.

One of the most popular security metrics is the game-theoretic approach that
has been successfully applied to deal with the problem of robustness,
security, and resilience of networked control systems [4]. This approach
affords us to address the robustness and security of networked control systems
within the common well-defined framework of ${\mathcal{H}}_{\infty}$ robust
control design. Further, many other concepts of games considering networked
systems subjected to cyber-attacks such as dynamic [5], stochastic [6],
network monitoring [7, 8], and zero-sum games [9] have been recently studied.
Although the above games were successful in studying control systems subjected
to cyber-attacks such as denial-of-service and stealthy data injection
attacks, the full system model knowledge was assumed to be available to both
the malicious adversary and the detector. This assumption might be restrictive
when it comes to large-scale interconnected systems which can consist of a
huge number of subsystems. This can be explained by a variety of facts such as
$(i)$ limited availability of computational resources for modeling, $(ii)$
limited availability of modeling data, and $(iii)$ modeling errors. Thus, the
adversary and the detector might have limited system knowledge instead of
accurate system parameters, which will be addressed throughout this paper.

In this paper, we deal with the problem of optimal sensor placement against an
adversary in an uncertain networked control system which is represented by
interconnected vertices. Given a protected performance vertex, the detector
monitors the system by selecting a single monitor vertex and placing a sensor
to measure its output with the purpose of detecting cyber-attacks. Meanwhile,
the adversary chooses a single vertex to attack and directly injects attack
signals into its input via the wireless network. The aim of the adversary is
to steer the attack vertex as to maximally disrupt the protected performance
vertex while remaining undetected by the detector. The contributions of this
paper are the following

  1. 1.

The problem of optimal sensor placement against the adversary is formulated as
a zero-sum game between two strategic players, i.e., the adversary and the
detector, with the same uncertain system knowledge.

  2. 2.

Due to the uncertainty, the game payoff of the zero-sum game, which is a min-
max optimization problem, is computationally intractable [10]. To deal with
the problem, we adopt the scenario approach [11] to approximately compute the
above game payoff.

  3. 3.

We show that the existence of a finite solution to the problem is related to
the system-theoretic properties of the dynamical system, namely its invariant
zeros and relative degrees.

  4. 4.

The solutions to the problem of the optimal sensor placement are provided by
investigating the pure and the mixed-strategy equilibrium of the zero-sum game
in a numerical example.

Figure 1: Visualization of a zero-sum game between a detector and an adversary
in a networked control system.

We conclude this section by providing the notations which are used throughout
this paper. The problem description is given in Section II. Thereafter,
Section III formulates the problem of optimal sensor placement as a zero-sum
game with the game payoff based on a risk metric. The evaluation of the game
payoff is carried out in Section IV. Section V presents a numerical example of
the zero-sum game between an adversary and a detector and computes the optimal
monitor selection based on the mixed-strategy Nash equilibrium. Concluding
remarks are provided in Section VI.

Notation: the set of real positive numbers is denoted as ${\mathbb{R}}_{+}$;
${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{n\times m}$ stand for sets of real
$n$-dimensional vectors and $n$-row $m$-column matrices, respectively. Let us
define $e_{i}\in{\mathbb{R}}^{n}$ with all zero elements except the $i$-th
element that is set as $1$. A continuous-time system with the state-space
model $\dot{x}(t)=Ax(t)+Bu(t),\;y(t)=Cx(t)+Du(t)$ is denoted as
$\Sigma\triangleq(A,B,C,D)$. Consider the norm $\left\lVert
x\right\rVert_{{\mathcal{L}}_{2}[0,T]}^{2}\triangleq\int_{0}^{T}\left\lVert
x(t)\right\rVert_{2}^{2}~{}dt$. The space of square-integrable functions is
defined as
${\mathcal{L}}_{2}\triangleq\bigl{\\{}f:{\mathbb{R}}_{+}\rightarrow{\mathbb{R}}~{}|~{}\left\lVert
f\right\rVert_{{\mathcal{L}}_{2}[0,\infty]}<\infty\bigr{\\}}$ and the extended
space is defined as
${\mathcal{L}}_{2e}\triangleq\bigl{\\{}f:{\mathbb{R}}_{+}\rightarrow{\mathbb{R}}~{}|~{}\left\lVert
f\right\rVert_{{\mathcal{L}}_{2}[0,T]}<\infty,~{}\forall~{}0<T<\infty\bigr{\\}}$.
We denote ${\mathbb{I}}_{\mathcal{A}}(x)$ as an indicator function such that
${\mathbb{I}}_{\mathcal{A}}(x)=1$ if $x\in{\mathcal{A}}$, otherwise
${\mathbb{I}}_{\mathcal{A}}(x)=0$. The probability of $X$ is denoted as
${\mathbb{P}}(X)$. For $x\in{\mathbb{R}}$, $\left\lceil{x}\right\rceil$
represents a value rounded to the nearest integer greater than or equal to
$x$. Let ${\mathcal{G}}\triangleq({\mathcal{V}},{\mathcal{E}},A,\Theta)$ be an
undirected weighted digraph with the set of $N$ vertices
${\mathcal{V}}=\\{v_{1},v_{2},...,v_{N}\\}$, the set of edges
${\mathcal{E}}\subseteq{\mathcal{V}}\times{\mathcal{V}}$, the weighted
adjacency matrix $A\triangleq[a_{ij}]$, and the weighted self-loop matrix
$\Theta$. For any $(v_{i},v_{j})\in{\mathcal{E}},~{}i\neq j$, the element of
the weighted matrix $a_{ij}$ is positive, and with
$(v_{i},v_{j})\notin{\mathcal{E}}$ or $i=j$, $a_{ij}=0$. The degree of vertex
$v_{i}$ is denoted as $d_{i}=\sum_{j=1}^{n}a_{ij}$ and the degree matrix of
the graph ${\mathcal{G}}$ is defined as $D={\bf
diag}\big{(}d_{1},d_{2},\dots,d_{N}\big{)}$, where ${\bf diag}$ stands for a
diagonal matrix. For each vertex $v_{i}$, it has a positive weighted self-loop
gain $\theta_{i}>0$. The weighted self-loop matrix of the graph
${\mathcal{G}}$ is defined as $\Theta={\bf
diag}\big{(}\theta_{1},\theta_{2},\dots,\theta_{N}\big{)}$. The Laplacian
matrix, representing the graph ${\mathcal{G}}$, is defined as
$L=[\ell_{ij}]=D-A+\Theta$. Further, ${\mathcal{G}}$ is called an undirected
graph if $A$ is symmetric. An edge of an undirected graph ${\mathcal{G}}$ is
denoted by a pair $(v_{i},v_{j})\in{\mathcal{E}}$. An undirected graph is
connected if for any pair of vertices there exists at least one path between
two vertices. The set of all neighbours of vertex $v_{i}$ is denoted as
${\mathcal{N}}_{i}=\\{v_{j}\in{\mathcal{V}}:(v_{i},v_{j})\in{\mathcal{E}}\\}$.

##  II Problem description

This section firstly presents the description of a networked control system.
Then, we introduce a malicious adversary who with limited system knowledge
conducts a cyber-attack to maliciously affect the system performance.

###  II-A Networked control system description

Consider a networked control system associated with a connected undirected
graph ${\mathcal{G}}\triangleq({\mathcal{V}},{\mathcal{E}},A,\Theta)$ with $N$
vertices, the state-space model of each one-dimensional vertex
$v_{i},~{}i\in\bigl{\\{}1,2,\ldots,N\bigr{\\}}$, is described as

$\displaystyle\dot{x}_{i}^{\Delta}(t)$
$\displaystyle=\sum_{v_{j}\in{\mathcal{N}}_{i}}\ell_{ij}^{\Delta}\big{(}x_{i}^{\Delta}(t)-x_{j}^{\Delta}(t)\big{)}+\tilde{u}_{i}(t),$
(1) $\displaystyle y_{\tau}^{\Delta}(t)$ $\displaystyle=x^{\Delta}_{\tau}(t),$
(2)

where $x_{i}^{\Delta}(t),\;\tilde{u}_{i}(t)\in{\mathbb{R}}$ are the state of
vertex $v_{i}$ and its control input received from its controller over the
wireless network (see Fig. 1), respectively. The performance of the networked
control system (1) is measured via the state of a given vertex
$v_{\tau}\in{\mathcal{V}}$ in (2). The weight parameters
$\ell_{ij}^{\Delta},~{}\forall(v_{i},v_{j})\in{\mathcal{E}},$ are uncertain
and assumed to be structured as
$\ell_{ij}^{\Delta}\triangleq\bar{\ell}_{ij}+\delta_{ij}$, where
$\bar{\ell}_{ij}$ and $\delta_{ij}$ are the nominal value and the bounded
probabilistic uncertainty of $\ell_{ij}^{\Delta}$, respectively.

First, we consider the wireless network healthy, i.e., the absence of cyber-
attacks. Thus, the received control input $\tilde{u}_{i}(t)$ of vertex
$v_{i},~{}i\in\bigl{\\{}1,2,\ldots,N\bigr{\\}}$, is the same as the control
input sent by its controller:

$\displaystyle\tilde{u}_{i}(t)=u_{i}(t)=-\theta_{i}x_{i}^{\Delta}(t),$ (3)

where $u_{i}(t)$ is the control input designed and sent by the controller of
vertex $v_{i}$. $\theta_{i}\in{\mathbb{R}}_{+}$ is an adjustable self-loop
control gain of vertex $v_{i}$.

For convenience, let us denote
$x^{\Delta}(t)\triangleq\big{[}x_{1}^{\Delta}(t),~{}x_{2}^{\Delta}(t),\ldots,~{}x_{N}^{\Delta}(t)\big{]}^{\top}$
as the state of the networked control system. The dynamics of the networked
control system (1) under the control law (3) can be rewritten as

$\displaystyle\dot{x}^{\Delta}(t)$ $\displaystyle=-L^{\Delta}x^{\Delta}(t),$
(4)

where the uncertain matrix $L^{\Delta}$ is defined as:
$L^{\Delta}\triangleq\bar{L}+\Delta$, $\Delta\in\Omega$, where $\Omega$ is a
closed and bounded set, $\bar{L}\triangleq[\bar{\ell}_{ij}]$ and
$\Delta\triangleq[\delta_{ij}]$ are nominal value and bounded uncertainty of
$L^{\Delta}$, respectively. Next, let us make use of the following
assumptions.

###### Assumption II.1

We assume that the healthy networked control system (4) is at its equilibrium
before being attacked.

###### Assumption II.2

The input of the given performance vertex $v_{\tau}$ is protected from any
attacks. Further, its state is unmeasurable.

Then, except for the protected target vertex $v_{\tau}$, a detector monitors
the system by placing a sensor at the output of a single vertex
$v_{m}\in{\mathcal{V}}\setminus\\{v_{\tau}\\}$. On the other hand, the system
is attacked by an adversary, whose detailed descriptions are listed in the
following subsection.

###  II-B Adversary description

This part introduces resources and an attack strategy of the adversary with
limited system knowledge, so-called bounded-rational adversary [10, Def. 2.2].

####  II-B1 System knowledge

The adversary knows the location of the protected target vertex $v_{\tau}$,
the appearance of a detector, the set of $N$ vertices ${\mathcal{V}}$, and the
set of edges ${\mathcal{E}}$. However, the adversary does not know the exact
location of the detector and has limited knowledge about $L^{\Delta}$ in (4).
The adversary only knows $\bar{L}$ and $\Omega$ instead of $L^{\Delta}$.

####  II-B2 Disruption resource

Except for the protected target vertex $v_{\tau}$, the adversary is able to
conduct a cyber-attack on the input of another vertex. The adversary firstly
assumes the location of a monitor vertex $v_{m}$ selected by the detector.
Then, the adversary selects a vertex
$v_{a}\in{\mathcal{V}}\setminus\\{v_{\tau}\\}$ and injects a malicious attack
signal $a(t)\in{\mathbb{R}}$ on its input with the aim of manipulating the
output of the protected target vertex $v_{\tau}$. The control input (3) of
vertex $v_{i},~{}i\in\\{1,2,\ldots,N\\}$, received from its controller over
the attacked wireless network can be described as follows

$\displaystyle\tilde{u}_{i}(t)=u_{i}(t)+\begin{cases}0,~{}&v_{i}\neq v_{a},\\\
a(t),~{}&v_{i}=v_{a}.\end{cases}$ (5)

Thus, the adversary perceives the system model (4) under the control law (5)
with two outputs at the two vertices $v_{\tau}$ and $v_{m}$ as an uncertain
dynamical system described by

$\displaystyle\dot{x}^{\Delta}(t)$
$\displaystyle=-L^{\Delta}x^{\Delta}(t)+e_{a}a(t),$ (6) $\displaystyle
y^{\Delta}_{\tau}(t)$ $\displaystyle=e_{\tau}^{\top}x^{\Delta}(t),$ (7)
$\displaystyle y^{\Delta}_{m}(t)$ $\displaystyle=e_{m}^{\top}x^{\Delta}(t).$
(8)

####  II-B3 Adversary strategy

The goal of the adversary is to maliciously manipulate the output of the
protected target vertex $v_{\tau}$ while remaining stealthy with the detector.
To this end, the adversary conducts the stealthy data injection attack, which
is defined as follows. Consider the above structure of the uncertain
continuous-time system (6)-(8) which is denoted as
$\Sigma^{\Delta}_{\tau,m}\triangleq(-L^{\Delta},e_{a},[e_{\tau},e_{m}]^{\top},0)$,
with target output $y_{\tau}^{\Delta}(t)=e_{\tau}^{\top}x^{\Delta}(t)$ and
monitor output $y^{\Delta}_{m}(t)=e_{m}^{\top}x^{\Delta}(t)$. The input signal
$a(t)$ of the system $\Sigma^{\Delta}_{\tau,m}$ is called the stealthy data
injection attack if the monitor output satisfies $\left\lVert
y^{\Delta}_{m}\right\rVert_{{\mathcal{L}}_{2}[0,T]}^{2}<\sigma$, in which
$\sigma>0$ is called an alarm threshold. Further, the impact of the stealthy
data injection attack is measured via the energy of the target output over the
horizon $[0,T]$, i.e., $\left\lVert
y^{\Delta}_{\tau}\right\rVert_{{\mathcal{L}}_{2}[0,T]}^{2}$.

Due to limited system knowledge, the uncertain system dynamics (6)-(8) are not
explicitly available to the adversary. Such an issue causes a difficulty for
the adversary in designing of the attack strategy. To deal with the issue, the
next section adopts a risk metric to evaluate the attack impact over the
probabilistic uncertainty set, which can be evaluated by the adversary to
select an attack vertex.

##  III Problem formulation

We consider that both the adversary and the detector have the same bounded
uncertainty about the system knowledge. Based on this assumption, for a given
uncertain parameter and attack and monitor vertices, the attack impact is
characterized via an optimal control problem. Then, we aggregate the attack
impact over the probabilistic uncertainty set by means of a risk metric.
Finally, the problem of optimal selection of attack and monitor vertices is
formulated as a zero-sum game between two strategic players, the adversary and
the detector, where the game payoff corresponds to the risk of the attack
impact evaluated over the probabilistic uncertainty.

###  III-A Stealthy data injection attack policy

Due to the presence of uncertainty in the system model (6)-(8), the attack
impact $J_{\tau}(v_{a},v_{m};\Delta,a)$ on the target vertex $v_{\tau}$ by the
attack vector $a\in{\mathcal{L}}_{2e}$ becomes a function of the random
variable $\Delta\in\Omega$

$\displaystyle J_{\tau}(v_{a},v_{m};\Delta,a)\triangleq\left\lVert
y^{\Delta}_{\tau}\right\rVert_{{\mathcal{L}}_{2}}^{2}{\mathbb{I}}_{{\mathcal{A}}}(a),$
(9) $\displaystyle{\mathcal{A}}\triangleq\\{a|~{}\left\lVert
y^{\Delta}_{m}\right\rVert_{{\mathcal{L}}_{2}}^{2}\leq\sigma,\eqref{sys:x_unc},\eqref{sys:ym},~{}x(0)=0\\},$
(10)

where $y^{\Delta}_{\tau}(t)$ and $y^{\Delta}_{m}(t)$ are the output of the
target vertex $v_{\tau}$ and the output of the monitor vertex $v_{m}$,
respectively. From (9), the worst-case attack impact on the target vertex
$v_{\tau}$ with the random variable $\Delta\in\Omega$ can be formulated as
follows

$\displaystyle\sup_{a\in{\mathcal{L}}_{2e}}J_{\tau}(v_{a},v_{m};\Delta,a).$
(11)

It is worth noting that (11) is introduced to evaluate the worst-case attack
impact for each pair of $v_{a}$ and $v_{m}$, thus allowing one to compare the
impact for different pairs of attack and monitor vertices. Further, the worst-
case attack impact (11) is proportional to the alarm threshold $\sigma$ for
all possible pairs of $v_{a}$ and $v_{m}$. Therefore, without loss of
generality, let us set the alarm threshold $\sigma=1$ in the remainder of this
paper.

###### Remark 1

Due to the random variable $\Delta\in\Omega$, the worst-case impact (11)
becomes a random variable. Thus, in order to compare the worst-case impacts
made by pairs of $v_{a}$ and $v_{m}$ over the uncertainty set $\Omega$, we
need to employ a risk metric which will be introduced in the rest of this
subsection.

After investigating the worst-case attack impact (11) on the target vertex
$v_{\tau}$ with all the possible pairs of attack $v_{a}$ and monitor vertices
$v_{m}$, the adversary firstly chooses the attack vertex $v_{a}$ such that the
corresponding risk (defined in Definition III.1) is maximized [10]. Then, the
adversary directly injects the stealthy data injection attack on the input of
the selected attack vertex $v_{a}$. To this end, the adversary deals with the
following optimization problem:

$\displaystyle\max_{v_{a}\neq v_{\tau}\in{\mathcal{V}}}$
$\displaystyle~{}{\mathcal{J}}_{\tau}(v_{a},v_{m}),$ (12)
$\displaystyle{\mathcal{J}}_{\tau}(v_{a},v_{m})={\mathcal{R}}_{\Delta\in\Omega}$
$\displaystyle\Big{[}~{}\sup_{a\in{\mathcal{L}}_{2e}}J_{\tau}(v_{a},v_{m};\Delta,a)\Big{]},$
(13)

where ${\mathcal{R}}_{\Delta\in\Omega}$ is a risk metric evaluated over the
probabilistic uncertainty set. In this paper, we use the well-known Value-at-
Risk [12] as our risk metric, which is defined below.

###### Definition III.1

(Value-at-Risk (VaR)): Given a random variable $X$ and $\beta\in(0,1)$, the
VaR is defined as

$\displaystyle\text{VaR}_{\beta}(X)\triangleq\operatornamewithlimits{inf\vphantom{p}}\big{\\{}x|{\mathbb{P}}\big{[}X\leq
x\big{]}\geq 1-\beta\big{\\}}.$ (14)

With a specified level $\beta\in(0,1)$, VaRβ is the lowest amount of $x$ such
that with probability $1-\beta$, the random variable $X$ does not exceed $x$.
$\triangleleft$

In order to counter the adversary, the detector adopts the game-theoretic
approach to design its detection strategy, which will be introduced in the
next part.

###  III-B Game-theoretic approach to sensor placement

The detector chooses a vertex $v_{m}\in{\mathcal{V}}\setminus\\{v_{\tau}\\}$
and monitors its output with the purpose of minimizing the risk (13). Hence,
the detector addresses the following problem.

###### Problem 1

(Optimal monitor selection) Given a target vertex $v_{\tau}$ and an arbitrary
attack vertex $v_{a}$, select a monitor vertex that minimizes the risk
corresponding to the worst-case attack impact
${\mathcal{J}}_{\tau}(v_{a},v_{m})$ defined in (13).

The above detector objective is converted into the following optimization
problem:

$\displaystyle\min_{v_{m}\neq
v_{\tau}\in{\mathcal{V}}}{\mathcal{J}}_{\tau}(v_{a},v_{m}).$ (15)

From the scenario of a single-adversary-single-detector we are considering,
the adversary objective (12), and the detector objective (15), we formulate
Problem 1 as a zero-sum game with the game payoff (13) between two players,
i.e., the adversary and the detector, as follows:

$\displaystyle\min_{v_{m}\neq v_{\tau}\in{\mathcal{V}}}~{}\max_{v_{a}\neq
v_{\tau}\in{\mathcal{V}}}~{}{\mathcal{J}}_{\tau}(v_{a},v_{m}).$ (16)

The min-max optimization problem (16) admits a saddle-point equilibrium
$(v_{a}^{\star},v_{m}^{\star})$ [4] if and only if it satisfies

$\displaystyle-\infty<{\mathcal{J}}_{\tau}(v_{a},v_{m}^{\star})\leq$
$\displaystyle{\mathcal{J}}_{\tau}(v_{a}^{\star},v_{m}^{\star})\leq{\mathcal{J}}_{\tau}(v_{a}^{\star},v_{m})<\infty,~{}$
(17) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
v_{a},v_{m}\in{\mathcal{V}}\setminus\\{v_{\tau}\\}.$ (18)

The game payoff of the saddle-point equilibrium
${\mathcal{J}}_{\tau}(v_{a}^{\star},v_{m}^{\star})$ implies that a deviation
of the attack vertex $\forall
v_{a}\in{\mathcal{V}}\setminus\\{v_{\tau},v_{a}^{\star}\\}$ does not gain the
game payoff and a deviation of the monitor vertex $\forall
v_{m}\in{\mathcal{V}}\setminus\\{v_{\tau},v_{m}^{\star}\\}$ does not decrease
the game payoff.

###### Remark 2

Since the zero-sum game (16) determined by discrete decisions of the adversary
and the detector might be solved via linear programming [13, Ch. 5], we need
to evaluate the game payoff defined in (13) for all the possible pairs of
$v_{a}$ and $v_{m}$. However, computing (13) requires us not only to address
the non-convexity of the worst-case impact (11) but also to devise a
computationally efficient approximation of (13) over a continuous uncertainty
set.

The next section will give us an efficient method to approximately compute the
game payoff (13) for each selected pair of $v_{a}$ and $v_{m}$.

##  IV Evaluating the game payoff

There are two difficulties in solving the zero-sum game (16). The first
difficulty is that: for any given pair of $v_{a},v_{m}$, and uncertainty
$\Delta\in\Omega$, the function
$\sup_{a\in{\mathcal{L}}_{2e}}J_{\tau}(v_{a},v_{m};\Delta,a)$ is a non-convex
optimization problem. Secondly, since the set $\Omega$ is continuous, the
problem of assessing the game payoff (13) is computationally intractable.
Thus, in this section, we aim to address both difficulties by invoking the
scenario approach [11] that discretizes the uncertainty set $\Omega$.

###  IV-A Worst-case attack impact for a sampled uncertainty point

We begin by considering the case of a sampled uncertainty realization
$\Delta_{i}\in\Omega$. Let us denote the value of the corresponding uncertain
Laplacian matrix in (6) as $L^{\Delta_{i}}$ and the uncertain system (6)-(8)
as
$\Sigma^{\Delta_{i}}_{\tau,m}\triangleq\big{(}-L^{\Delta_{i}},e_{a},[e_{\tau},e_{m}]^{\top},0\big{)}$
with the attack input at vertex $v_{a}$, the target output at vertex
$v_{\tau}$, and the monitor output at vertex $v_{m}$. For such an isolated
uncertainty, the worst-case attack impact can be written as

$\sup_{a\in{\mathcal{L}}_{2e}}J_{\tau}(v_{a},v_{m};\Delta_{i},a)$ (19)

Following the details in [14, Prop. 1], the optimal control problem (19) can
be equivalently rewritten as the following convex SDP

$\displaystyle\gamma_{i}^{\star}\triangleq\underset{\gamma_{i}\in{\mathbb{R}}_{+},P_{i}=P_{i}^{\top}\geq
0}{\min}$ $\displaystyle~{}~{}~{}~{}\gamma_{i}$ (20) s.t. $\displaystyle
R\big{(}\Sigma^{\Delta_{i}}_{\tau,m},P_{i},\gamma_{i}\big{)}\leq 0,$

where

$\displaystyle
R\big{(}\Sigma^{\Delta_{i}}_{\tau,m},P_{i},\gamma_{i}\big{)}\triangleq$
$\displaystyle\left[\begin{array}[]{cc}-L^{\Delta_{i}}P_{i}-P_{i}L^{\Delta_{i}}{}&~{}P_{i}e_{a}\\\
e_{a}^{\top}P_{i}{}&~{}0\\\ \end{array}\right]$ (23)
$\displaystyle-\left[\begin{array}[]{cc}\gamma_{i}e_{m}e^{\top}_{m}+e_{\tau}e^{\top}_{\tau}{}&~{}0\\\
0{}&~{}0\\\ \end{array}\right].$ (26)

Next, we tackle the game payoff evaluation over a continuous set of
uncertainties $\Omega$ by first approximating the continuous uncertainty set
$\Omega$ with a discrete set $\Omega_{M_{1}}$ of sampled uncertainty
realizations, with cardinality $M_{1}$, and then using the point-wise
evaluation of the worst-case attack impact described in (20).

###  IV-B Approximate game payoff function

The game payoff (13) is difficult to determine since the risk metric operates
over a continuous set $\Omega$. To this end, we adopt the scenario approach
[11] to approximate the continuous uncertainty set $\Omega$, and consequently
determine the approximate game payoff (13). Before this, we rewrite (13) for a
given $\beta\in(0,1)$ as (27).

$\displaystyle{\mathcal{J}}_{\tau}(v_{a},v_{m})=~{}~{}$
$\displaystyle\operatornamewithlimits{inf\vphantom{p}}\gamma$ (27)
$\displaystyle~{}\text{s.t.}~{}~{}\mathbb{P}_{\Omega}[X\leq\gamma]\geq
1-\beta$ (28)

where
$X=\sup_{a\in{\mathcal{L}}_{2e}}J_{\tau}(v_{a},v_{m};\Delta,a),~{}\Delta\in\Omega$,
and the subscript to the probability operator denotes that it operates over
the set $\Omega$. Next, we apply the scenario approach to determine the
approximate value of the optimization problem (27) in the following theorem.

###### Theorem IV.1

Let $\epsilon_{1}\in(0,1)$ represent the accuracy with which the probability
operator $\mathbb{P}_{\Omega}$ in (27) is approximated. Let
$\beta_{1}\in(0,1)$ represent the confidence with which the accuracy
$\epsilon_{1}$ is guaranteed, i.e.,

$\mathbb{P}\\{|\mathbb{P}_{\Omega}(X\leq\gamma)-\hat{\mathbb{P}}_{M_{1}}|\geq\epsilon_{1}\\}\leq\beta_{1}.$
(29)

Here $\hat{\mathbb{P}}_{M_{1}}$ represents the approximation of the
probability operator $\mathbb{P}_{\Omega}$ in (27) defined as

$\hat{\mathbb{P}}_{M_{1}}\triangleq\frac{1}{M_{1}}\sum_{i=1}^{M_{1}}\mathbb{I}\left(X\leq\gamma\right),\;\text{where}\;M_{1}\geq\frac{1}{2\epsilon_{1}^{2}}\text{log}\frac{2}{\beta_{1}}.$
(30)

Then, the $\text{VaR}_{\beta}$ defined in (13) can be obtained with an
accuracy $\epsilon_{1}$ and confidence $\beta_{1}$ by solving

$\hat{\gamma}=\left\\{\begin{aligned} \min&\quad\gamma\\\
\textrm{s.t.}&\quad\frac{1}{M_{1}}\sum_{i=1}^{M_{1}}\mathbb{I}\left(\gamma_{i}^{\star}\leq\gamma\right)\geq
1-\beta_{1}\end{aligned}\right\\},$ (31)

where $\hat{\gamma}$ represents the $\text{VaR}_{\beta}$ with an accuracy
$\epsilon_{1}$. The value of $\gamma^{\star}_{i},i\in\\{1,2,\dots,M_{1}\\}$,
is obtained by solving (20). $\triangleleft$

###### Proof:

The proof follows directly from our previous results in [10, Th. 4.4]. ∎

###### Remark 3

Solving (31) with the risk metric defined in Definition III.1 gives us a
measure of risk for a corresponding pair of $v_{a}$ and $v_{m}$ that has been
evaluated over the explicit probabilistic uncertainty set $\Omega$. This risk
measure is different from the worst-case impact (11), which is a function of a
random variable $\Delta\in\Omega$.

Theorem IV.1 provides a method to compute the approximate value of game payoff
(13) which was difficult to compute previously. In order to evaluate the
result of Theorem IV.1, the next subsection will address the feasibility of
the optimization problem (31).

###  IV-C Feasibility analysis

For $M_{1}$ sampled uncertainty $\Delta_{i}\in\Omega_{M_{1}}$, the following
lemma gives us the necessary and sufficient condition to ensure that the
problem (31) is feasible and therefore admits a finite upper bound.

######  Lemma IV.2 (Boundedness)

Consider $M_{1}$ i.i.d. realizations of uncertainty
$\Delta_{i}\in\Omega_{M_{1}}$. The optimal solution of (31) with these $M_{1}$
realizations of uncertainty is bounded if and only if the optimal value of
(20) is bounded for at least $\left\lceil{M_{1}(1-\beta_{1})}\right\rceil$
system realizations. $\triangleleft$

###### Proof:

The proof follows directly from our previous results in [10, Lem. 4.5]. ∎

Then, we investigate the feasibility of the optimization problem (20) for a
system realization corresponding to a given sampled uncertainty
$\Delta_{i}\in\Omega_{M_{1}}$. Let us denote the following systems
$\Sigma^{\Delta_{i}}_{\tau}\triangleq(-L^{\Delta_{i}},e_{a},e^{\top}_{\tau},0)$
and $\Sigma^{\Delta_{i}}_{m}\triangleq(-L^{\Delta_{i}},e_{a},e^{\top}_{m},0)$.
Inspired by [15, Th. 2], the feasibility of the optimization problem (20) is
related to the invariant zeros of $\Sigma^{\Delta_{i}}_{\tau}$ and
$\Sigma^{\Delta_{i}}_{m}$, which are defined as follows.

###### Definition IV.1

(Invariant zeros) Consider the strictly proper system
$\Sigma\triangleq(A,B,C,0)$ with $A,B,$ and $C$ are real matrices with
appropriate dimensions. A tuple
$(\lambda,\bar{x},g)\in{\mathbb{C}}\times{\mathbb{C}}^{N}\times{\mathbb{C}}$
is a zero dynamics of $\Sigma$ if it satisfies

$\displaystyle\left[\begin{array}[]{cc}\lambda I-A{}{}{}{}&-B\\\ C&0\\\
\end{array}\right]\left[\begin{array}[]{c}\bar{x}\\\ g\\\
\end{array}\right]=\left[\begin{array}[]{c}0\\\ 0\\\
\end{array}\right],~{}~{}~{}\bar{x}\neq 0.$ (38)

In this case, a finite $\lambda$ is called a finite invariant zero of
$\Sigma$. Further, the strictly proper system $\Sigma$ always has at least one
invariant zero at infinity [16, Ch. 3]. $\triangleleft$

More specifically, let us state the following lemma.

###### Lemma IV.3

[15, Th. 2] Consider the two following continuous time systems
$\Sigma^{\Delta_{i}}_{\tau}\triangleq(-L^{\Delta_{i}},e_{a},e^{\top}_{\tau},0)$
and $\Sigma^{\Delta_{i}}_{m}\triangleq(-L^{\Delta_{i}},e_{a},e^{\top}_{m},0)$.
The optimization problem (20) is feasible if and only if the unstable
invariant zeros of $\Sigma^{\Delta_{i}}_{m}$ are also invariant zeros of
$\Sigma^{\Delta_{i}}_{\tau}$. $\triangleleft$

Inspired by Lemma IV.3, we will investigate both finite and infinite invariant
zeros of the two systems $\Sigma^{\Delta_{i}}_{m}$ and
$\Sigma^{\Delta_{i}}_{\tau}$.

#### Finite invariant zeros

Let us state the following lemma that considers the finite invariant zeros of
$\Sigma^{\Delta_{i}}_{m}$.

###### Lemma IV.4

Consider a networked control system associated with a connected undirected
graph ${\mathcal{G}}\triangleq({\mathcal{V}},{\mathcal{E}},A,\Theta)$, whose
closed-loop dynamics is described in (6)-(8) for a given sampled uncertainty
$\Delta_{i}\in\Omega_{M_{1}}$. Suppose that the networked control system is
driven by the stealthy data injection attack at a single attack vertex
$v_{a}$, and observed by a single monitor vertex $v_{m}$, resulting in the
state-space model
$\Sigma^{\Delta_{i}}_{m}\triangleq(-L^{\Delta_{i}},e_{a},e^{\top}_{m},0)$.
Then, there exist self-loop control gains
$\theta_{i},~{}i\in\\{1,2,\ldots,N\\},$ in (3) such that the networked control
system $\Sigma^{\Delta_{i}}_{m}$ has no finite unstable invariant zero.
$\triangleleft$

###### Proof:

We postpone the proof to Appendix A. ∎

The constructive proof of Lemma IV.4 (see Appendix A) gives us a design
procedure to ensure that the system $\Sigma^{\Delta_{i}}_{m}$ has no finite
unstable zero.

#### Infinite invariant zeros

We now investigate the infinite invariant zeros of the systems
$\Sigma^{\Delta_{i}}_{m}$ and $\Sigma^{\Delta_{i}}_{\tau}$. In the
investigation, we make use of known results connecting infinite invariant
zeros mentioned in Definition IV.1 and the relative degree of a linear system,
which is defined below.

###### Definition IV.2

(Relative degree) [17, Ch. 13] Consider the strictly proper system
$\Sigma\triangleq(A,B,C,0)$ with $A\in{\mathbb{R}}^{n\times n}$, $B$, and $C$
are real matrices with appropriate dimensions. The system $\Sigma$ is said to
have relative degree $r~{}(1\leq r\leq n)$ if the following conditions satisfy

$\displaystyle CA^{k}B=0,~{}~{}0\leq k<r-1,$ $\displaystyle CA^{r-1}B\neq 0.$
(39)

Based on Definition IV.2, let us denote $r_{\tau a}$ and $r_{ma}$ as the
relative degrees of $\Sigma^{\Delta_{i}}_{\tau}$ and
$\Sigma^{\Delta_{i}}_{m}$, respectively. In the scope of this study, we have
assumed that the cyber-attack (5) has no direct impact on the outputs (7) and
(8), resulting in strictly proper systems $\Sigma^{\Delta_{i}}_{\tau}$ and
$\Sigma^{\Delta_{i}}_{m},~{}\forall\Delta_{i}\in\Omega_{M_{1}}$. This implies
that the relative degrees $r_{\tau a}$ and $r_{ma}$ of
$\Sigma^{\Delta_{i}}_{\tau}$ and $\Sigma^{\Delta_{i}}_{m}$ are positive,
yielding their infinite zeros. By following our existing result related to
those infinite zeros [9, Th. 7] the infinite zeros of
$\Sigma_{m}^{\Delta_{i}}$ are also the infinite zeros of
$\Sigma_{\tau}^{\Delta_{i}}$ if and only if the following condition holds

$\displaystyle r_{ma}\leq r_{\tau a}.$ (40)

#### Boundedness of solutions

After analyzing both finite and infinite zeros of the two systems
$\Sigma_{m}^{\Delta_{i}}$ and $\Sigma_{\tau}^{\Delta_{i}}$, the following
theorem gives us a sufficient condition to ensure the feasibility of the
optimization problem (20), and thus of the existence of a finite upper bound
on the corresponding optimal value.

###### Theorem IV.5

Consider the strictly proper systems
$\Sigma^{\Delta_{i}}_{\tau}\triangleq(-L^{\Delta_{i}},e_{a},e^{\top}_{\tau},0)$
and $\Sigma^{\Delta_{i}}_{m}\triangleq(-L^{\Delta_{i}},e_{a},e^{\top}_{m},0)$,
in which the two systems have the same stealthy data injection attack input at
a single attack vertex $v_{a}$ but different output vertices, i.e., $v_{\tau}$
for $\Sigma^{\Delta_{i}}_{\tau}$ and $v_{m}$ for $\Sigma^{\Delta_{i}}_{m}$.
Suppose the systems $\Sigma^{\Delta_{i}}_{\tau}$ and $\Sigma^{\Delta_{i}}_{m}$
have relative degrees $r_{\tau a}$ and $r_{ma}$, respectively. Then, the
problem (20) admits a finite solution if

  1. 1.

the self-loop control gains $\theta_{i},~{}i\in\\{1,2,\ldots,N\\}$, in (3) are
chosen such that the system $\Sigma^{\Delta_{i}}_{m}$ has no finite unstable
zeros; and

  2. 2.

the condition (40) holds. $\triangleleft$

###### Proof:

The proof is postponed to Appendix B. ∎

The sufficient condition (40) will be verified by computing the approximate
game payoffs (31) in Theorem IV.1 and the equilibrium of the zero-sum game
(16) will be analyzed via a numerical example in the next section.

##  V Numerical examples

To validate the obtained results, through a numerical example, this section
$i)$ applies (31) with two different values of $\beta$ to the example with the
aim of verifying (40); $ii)$ examines the saddle-point equilibrium of the
zero-sum game (16) with the two different values of $\beta$; $iii)$ computes
the mixed-strategy Nash equilibrium of the zero-sum game in case there is no
saddle-point equilibrium. Let us take an example of a 10-vertex networked
control system depicted in Fig. 3. The simulation parameters are chosen as
follows:

$\displaystyle L^{\Delta}$
$\displaystyle\triangleq[\ell_{ij}^{\Delta}]=[\bar{\ell}_{ij}]+[\delta_{ij}]+\Theta,$
(41) $\displaystyle\bar{\ell}_{ij}$
$\displaystyle=-10,~{}\delta_{ij}\in[-0.5,0.5],~{}\forall(v_{i},v_{j})\in{\mathcal{E}},i\neq
j,$ (42) $\displaystyle\bar{\ell}_{ij}$
$\displaystyle=\delta_{ij}=0,~{}\forall(v_{i},v_{j})\notin{\mathcal{E}},$ (43)
$\displaystyle\ell_{ii}^{\Delta}$
$\displaystyle=-\sum_{v_{j}\in{\mathcal{N}}_{i}}\big{(}\bar{\ell}_{ij}+\delta_{ij}\big{)},~{}\theta_{0}=0.5.$
(44)

###  V-A Computing the approximate game payoff

To compute (31), let us choose $\epsilon_{1}=0.06,\beta_{1}=0.08$, and
$M_{1}=450$, which satisfy (30). For any sampled uncertainty
$\Delta_{i}\in\Omega_{M_{1}}$, the chosen uniform offset self-loop control
gain $\theta_{0}$ (see Appendix A) ensures that $\Sigma^{\Delta_{i}}_{m}$ has
no finite unstable zero, which validates Lemma IV.4. We will present two cases
by selecting two values of the specified level $\beta$ in (27), i.e.,
$\beta_{a}=0.08$ and $\beta_{b}=0.15$. Suppose that $v_{5}$ is the protected
target vertex (see Assumption II.2 and Fig. 3). There are two possible monitor
vertices $v_{2}$ and $v_{6}$, which satisfy the necessary and sufficient
condition (40) for any $v_{a}\in{\mathcal{V}}\setminus\\{v_{5}\\}$ (see Fig.
3). For more clarity, we compute the approximate game payoff (31) w.r.t. the
target vertex $v_{5}$ for each pair of
$v_{a}\in{\mathcal{V}}\setminus\\{v_{5}\\}$ and $v_{m}\in\\{v_{2},v_{6}\\}$ in
the cases $\beta=\beta_{a}$ and $\beta=\beta_{b}$, which gives us

$\displaystyle{\mathcal{J}}_{5}(v_{a},v_{m=2};\beta_{a})\leq
1.5848,~{}{\mathcal{J}}_{5}(v_{a},v_{m=6};\beta_{a})\leq 1.5055,$ (45)
$\displaystyle{\mathcal{J}}_{5}(v_{a},v_{m=2};\beta_{b})\leq
1.5550,~{}{\mathcal{J}}_{5}(v_{a},v_{m=6};\beta_{b})\leq 1.4803.$ (46)

Otherwise, there exits at least an attack vertex
$v_{a}\in{\mathcal{V}}\setminus\\{v_{5}\\}$ pairing with an arbitrary monitor
$v_{m}\in{\mathcal{V}}\setminus\\{v_{2},v_{5},v_{6}\\}$ to yield infinite game
payoffs, e.g., ${\mathcal{J}}_{5}(v_{a=3},v_{m=1};\beta_{a})=\infty$,
${\mathcal{J}}_{5}(v_{a=3},v_{m=1};\beta_{b})=\infty$,
${\mathcal{J}}_{5}(v_{a=10},v_{m=3};\beta_{a})=\infty$, and
${\mathcal{J}}_{5}(v_{a=10},v_{m=3};\beta_{b})=\infty$. In order to explain
those infinite values, we verify the condition (40) by checking the relative
degrees among those vertices via Fig. 3, i.e., $r_{a=3,m=5}=2<r_{a=3,m=1}=3$
and $r_{a=10,m=5}=2<r_{a=10,m=3}=3$, which violate the necessary and
sufficient condition (40).

Figure 2: 10-vertex networked control system with target vertex $v_{5}$.

Figure 3: Approximate game payoff (31) with $\beta=\beta_{a}=0.08$ and
$\beta=\beta_{b}=0.15$ in case the detector selects $v_{m=2}$ or $v_{m=6}$ and
the adversary attacks $v_{a=1}$ or $v_{a=10}$. The other game payoffs yielded
by the other choices of $v_{a}$ and $v_{m}$ are removed due to the
ineffectiveness.

###  V-B Examining the saddle-point equilibrium

Next, we will investigate the equilibrium of the zero-sum game in the cases
$\beta=\beta_{a}$ and $\beta=\beta_{b}$. Fig. 3 illustrates the game payoffs
for $v_{a}\in\\{v_{1},v_{10}\\}$ and $v_{m}\in\\{v_{2},v_{6}\\}$ corresponding
to $\Delta_{i}\in\Omega_{M_{1}}$. In both cases $\beta=\beta_{a}$ and
$\beta=\beta_{b}$, since those game payoffs dominate the values of the other
choices of $v_{a}\in{\mathcal{V}}\setminus\\{v_{1},v_{5},v_{10}\\}$ and
$v_{m}\in\\{v_{2},v_{6}\\}$, we only show four marked-lines in Fig. 3.

#### In the first case $\beta=\beta_{a}=0.08$

the crossing points of the green dotted-line and marked-lines are the
approximate game payoffs with $\beta=\beta_{a}$ for the corresponding pairs of
attack and monitor vertices (see Box A in Fig. 3). By observing those
approximate game payoffs in Fig. 3, one has

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\mathcal{J}}_{5}(\forall
v_{a}\in{\mathcal{V}}\setminus\\{v_{5},v_{10}\\},v_{m=6};\beta_{a})$ (47)
$\displaystyle<{\mathcal{J}}_{5}(v_{a=10},v_{m=6};\beta_{a})<{\mathcal{J}}_{5}(v_{a=10},v_{m=2};\beta_{a}).$
(48)

According to the definition of the saddle-point equilibrium in (18), the
inequalities (48) imply that the example admits a saddle-point equilibrium
$(v_{a}^{\star}=v_{10},v_{m}^{\star}=v_{6})$ with $\beta=\beta_{a}$.

#### In the second case $\beta=\beta_{b}=0.15$

the approximate game payoffs are the crossing points of the marked-lines and
the blue dashed-line (see Box B in Fig. 3). Those crossing points give us

$\displaystyle{\mathcal{J}}_{5}(v_{a=1},v_{m=2};\beta_{b})=1.4603,$ (49)
$\displaystyle{\mathcal{J}}_{5}(v_{a=10},v_{m=6};\beta_{b})=1.4803,$ (50)
$\displaystyle{\mathcal{J}}_{5}(v_{a=1},v_{m=6};\beta_{b})=1.4856,$ (51)
$\displaystyle{\mathcal{J}}_{5}(v_{a=10},v_{m=2};\beta_{b})=1.5550.$ (52)

From (52), we will examine whether a saddle-point equilibrium exists. If the
detector monitors $v_{m=2}$, the adversary simply attacks $v_{a=10}$ to
maximize the risk. But, in the case of $v_{a=10}$, the detector can move to
$v_{m=6}$ to reduce the risk since
${\mathcal{J}}_{5}(v_{a=10},v_{m=6};\beta_{b})<{\mathcal{J}}_{5}(v_{a=10},v_{m=2};\beta_{b})$.
Then, the adversary can obtain a higher risk by attacking $v_{a=1}$ instead of
$v_{a=10}$, i.e.,
${\mathcal{J}}_{5}(v_{a=1},v_{m=6};\beta_{b})>{\mathcal{J}}_{5}(v_{a=10},v_{m=6};\beta_{b})$.
Monitoring $v_{m=2}$ yields a lower risk for the detector, i.e.,
${\mathcal{J}}_{5}(v_{a=1},v_{m=2};\beta_{b})<{\mathcal{J}}_{5}(v_{a=1},v_{m=6};\beta_{b})$.
The story comes back to the beginning since the adversary simply attacks
$v_{a=10}$ to maximize the risk. The above observation implies that the
example with $\beta=\beta_{b}$ does not admit a saddle-point equilibrium
defined in (18). However, the game always admits a mixed-strategy Nash
equilibrium [4], which will be computed in the next subsection.

###  V-C Computing mixed-strategy Nash equilibrium

We compute the mixed-strategy Nash equilibrium for the example with the cases
$\beta=\beta_{a}$ and $\beta=\beta_{b}$. Let us denote
${\mathbb{P}}(v_{a};\beta)$ and ${\mathbb{P}}(v_{m};\beta)$,
$\beta\in\\{\beta_{a}=0.08,\beta_{b}=0.15\\}$ as the probabilities for attack
$v_{a}$ and monitor vertices $v_{m}$, respectively. For convenience, we denote
$\bar{{\mathbb{P}}}(v_{a};\beta)=\left[{\mathbb{P}}(v_{a=1};\beta),\ldots,{\mathbb{P}}(v_{a=10};\beta)\right]^{\top},(v_{a}\neq
v_{5})$ and
$\bar{{\mathbb{P}}}(v_{m};\beta)=\left[{\mathbb{P}}(v_{m=1};\beta),\ldots,{\mathbb{P}}(v_{m=10};\beta)\right]^{\top},(v_{m}\neq
v_{5})$. The expected game payoff of the example w.r.t. the target vertex
$v_{5}$ for attack vertex $v_{a}$ and monitor vertex $v_{m}$ is

$\displaystyle
Q_{5}(v_{a},v_{m};\beta)=\bar{{\mathbb{P}}}(v_{a};\beta)^{\top}\bar{{\mathcal{J}}}_{5}\bar{{\mathbb{P}}}(v_{m};\beta),$
(53)

where
$\bar{{\mathcal{J}}}_{5}=\big{[}{\mathcal{J}}_{5}(v_{i},v_{j};\beta)_{ij}\big{]}$
is a $9\times 9$-game matrix, whose $ij$-entry is filled by
${\mathcal{J}}_{5}(v_{a=i},v_{m=j};\beta)$. Similarly to (18), there exits a
saddle point $(v_{a}^{\star},v_{m}^{\star})$ if it satisfies

$\displaystyle Q_{5}(v_{a},v_{m}^{\star};\beta)\leq$ $\displaystyle
Q_{5}(v_{a}^{\star},v_{m}^{\star};\beta)\leq
Q_{5}(v_{a}^{\star},v_{m};\beta),$ (54)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
v_{a},v_{m}\in{\mathcal{V}}\setminus\\{v_{\tau}\\}.$ (55)

The saddle point $(v_{a}^{\star},v_{m}^{\star})$ in (55) indicates that a
deviation of selecting $v_{a}(v_{m})$ does not increase(decrease) the optimal
expected game payoff $Q_{5}(v_{a}^{\star},v_{m}^{\star};\beta)$. Further,
since the possible choices of the detector are restricted to
$\\{v_{2},v_{6}\\}$, we simply obtain ${\mathbb{P}}(\forall
v_{m}\in{\mathcal{V}}\setminus\\{v_{2},v_{5},v_{6}\\};\beta)=0$. More
specifically, by using linear programming [13, Ch. 5] to compute (55), we
receive the following optimal solution

#### In the first case $\beta=\beta_{a}=0.08$

$\displaystyle{\mathbb{P}}^{\star}(v_{m=6};\beta_{a})=100\%,~{}{\mathbb{P}}^{\star}(v_{m=2};\beta_{a})=0\%,$
(56)
$\displaystyle{\mathbb{P}}^{\star}(v_{a=10};\beta_{a})=100\%,~{}{\mathbb{P}}^{\star}(\forall
v_{a}\in{\mathcal{V}}\setminus\\{5,10\\};\beta_{a})=0\%.$ (57)

The above optimal solution once again confirms that a pair
$(v_{a}^{\star}=v_{10},v_{m}^{\star}=v_{6})$ is the pure saddle-point
equilibrium (18) of the example with $\beta=\beta_{a}$, which was also
verified in (48).

#### In the second case $\beta=\beta_{b}=0.15$

we obtain the following optimal solution

$\displaystyle{\mathbb{P}}^{\star}(v_{m=6};\beta_{b})\approx
94.72\%,~{}{\mathbb{P}}^{\star}(v_{m=2};\beta_{b})\approx 5.28\%,$ (58)
$\displaystyle{\mathbb{P}}^{\star}(v_{a=10};\beta_{b})\approx
25.29\%,~{}{\mathbb{P}}^{\star}(v_{a=1};\beta_{b})\approx 74.71\%,$ (59)
$\displaystyle{\mathbb{P}}^{\star}(\forall
v_{a}\in{\mathcal{V}}\setminus\\{1,5,10\\};\beta_{b})=0\%.$ (60)

The above optimal solution clearly show that the example does not admit a pure
saddle-point equilibrium (18) with $\beta=\beta_{b}$, which was discussed at
the end of the previous subsection.

##  VI Conclusion

In this paper, we studied a continuous-time networked control system attacked
by an adversary with uncertain system knowledge. The purpose of the adversary
was to manipulate the output of a protected target vertex by directly
conducting the stealthy data injection attack on another vertex. Meanwhile, an
optimal sensor placement problem was formulated such that a detector with the
same uncertain system knowledge places a sensor at a vertex in order to unmask
the adversary. We developed a risk-based game-theoretic framework to describe
the interactions between the two players, the adversary and the detector, in
the presence on probabilistic parameter uncertainty. In particular, we
formulate the optimal decisions as a zero-sum game, where the game payoff is
taken as a risk metric evaluated over the probabilistic uncertainty set. Due
to the continuous nature of the uncertainty set, the zero-sum game could not
be solved directly. Thus, we employed the scenario approach to approximately
compute the game payoff over a number of samples of uncertain parameters.
After approximately evaluating the game payoff for each pair of monitor and
attack vertices, the mixed-strategy Nash equilibrium of the zero-sum game was
also computed by linear programming. In future works, our game will be
expanded to consider multiple attack and monitor vertices. Characterizing an
analytical solution to the equilibrium of the game between the adversary and
the detector would also be a promising topic.

## Appendix

###  VI-A Proof of Lemma IV.4

Let us denote a tuple
$(\lambda^{\Delta_{i}}_{m},\bar{x}^{\Delta_{i}}_{m},g^{\Delta_{i}}_{m})\in{\mathbb{C}}\times{\mathbb{C}}^{N}\times{\mathbb{C}}$
as a zero dynamics of $\Sigma_{m}^{\Delta_{i}}$, where a finite
$\lambda_{m}^{\Delta_{i}}$ is called a finite invariant zero of
$\Sigma_{m}^{\Delta_{i}}$. From Definition IV.1, one has that the tuple
$(\lambda^{\Delta_{i}}_{m},\bar{x}^{\Delta_{i}}_{m},g^{\Delta_{i}}_{m})$
satisfies

$\displaystyle\left[\begin{array}[]{cc}\lambda^{\Delta_{i}}_{m}I+L^{\Delta_{i}}&-e_{a}\\\
e_{m}^{\top}&0\\\
\end{array}\right]\left[\begin{array}[]{c}\bar{x}^{\Delta_{i}}_{m}\\\
g^{\Delta_{i}}_{m}\\\ \end{array}\right]=\left[\begin{array}[]{c}0\\\ 0\\\
\end{array}\right].$ (67)

The above equation is rewritten as

$\displaystyle\left[\begin{array}[]{cc}(\lambda^{\Delta_{i}}_{m}-\theta_{0})I+L^{\Delta_{i}}+\theta_{0}I&-e_{a}\\\
e_{m}^{\top}&0\\\
\end{array}\right]\left[\begin{array}[]{c}\bar{x}^{\Delta_{i}}_{m}\\\ g_{m}\\\
\end{array}\right]=\left[\begin{array}[]{c}0\\\ 0\\\ \end{array}\right],$ (74)

where $\theta_{0}\in{\mathbb{R}}_{+}$ is a uniform offset self-loop control
gain. From (LABEL:pen_mtr_lam_m1), the finite value
$(\lambda^{\Delta_{i}}_{m}-\theta_{0})\in{\mathbb{C}}$ is an invariant zero of
a new state-space model
$\Sigma_{0m}^{\Delta_{i}}\triangleq(-L^{\Delta_{i}}-\theta_{0}I,e_{a},e^{\top}_{m},0)$.
For all $\lambda_{m}^{\Delta_{i}}\in{\mathbb{C}}$ satisfies
(LABEL:pen_mtr_lam_m1), the control gain $\theta_{0}$ can be adjusted such
that $\theta_{0}>$ Re$(\lambda_{m}^{\Delta_{i}})$, resulting in that
$\Sigma_{0m}^{\Delta_{i}}$ has no finite unstable zero. Then, the self-loop
control gains $\theta_{i},~{}i\in\\{1,2,\ldots,N\\},$ in (3) are tuned with
$\theta_{0}$ such that the system $\Sigma_{m}^{\Delta_{i}}$ is identical with
$\Sigma_{0m}^{\Delta_{i}}$. By this tuning procedure, the system
$\Sigma^{\Delta_{i}}_{m}$ also has no finite unstable invariant zero.
$\blacksquare$

###  VI-B Proof of Theorem IV.5

Based on Lemma IV.3, the optimization problem (20) is feasible if and only if
$\Sigma^{\Delta_{i}}_{m}$ has unstable invariant zeros that are also invariant
zeros of $\Sigma^{\Delta_{i}}_{\tau}$. By applying the control design
procedure in the proof of Lemma IV.4 (see Appendix A), we ensure that
$\Sigma^{\Delta_{i}}_{m}$ has no finite unstable invariant zeros, which leaves
us to analyze infinite zeros of those systems. Recall the equivalence between
the relative degree of a SISO system and the degree of its infinite zero.
Hence, a necessary condition to guarantee the feasibility of the optimization
(20) is that the number of infinite invariant zeros of
$\Sigma^{\Delta_{i}}_{m}$ is not greater than that of
$\Sigma^{\Delta_{i}}_{\tau}$. This implies $r_{ma}\leq r_{\tau a}$. For
sufficiency, it remains to show that if $r_{ma}\leq r_{\tau a}$, any infinite
zeros of $\Sigma^{\Delta_{i}}_{m}$ are also infinite zeros of
$\Sigma^{\Delta_{i}}_{\tau}$. The proof directly follows our previous results
[9, Th. 7]. $\blacksquare$

## References

  * [1] A. Teixeira, I. Shames, H. Sandberg, and K. H. Johansson, “A secure control framework for resource-limited adversaries,” Automatica, vol. 51, pp. 135–148, 2015.
  * [2] N. Falliere, L. O. Murchu, and E. Chien, “W32. stuxnet dossier,” White paper, Symantec Corp., Security Response, vol. 5, no. 6, p. 29, 2011.
  * [3] N. Kshetri and J. Voas, “Hacking power grids: A current problem,” Computer, vol. 50, no. 12, pp. 91–95, 2017.
  * [4] Q. Zhu and T. Basar, “Game-theoretic methods for robustness, security, and resilience of cyberphysical control systems: games-in-games principle for optimal cross-layer resilient control systems,” IEEE Control Systems Magazine, vol. 35, no. 1, pp. 46–65, 2015.
  * [5] A. Gupta, C. Langbort, and T. Başar, “Dynamic games with asymmetric information and resource constrained players with applications to security of cyberphysical systems,” IEEE Transactions on Control of Network Systems, vol. 4, no. 1, pp. 71–81, 2016.
  * [6] F. Miao, Q. Zhu, M. Pajic, and G. J. Pappas, “A hybrid stochastic game for secure control of cyber-physical systems,” Automatica, vol. 93, pp. 55–63, 2018.
  * [7] J. Milošević, M. Dahan, S. Amin, and H. Sandberg, “A network monitoring game with heterogeneous component criticality levels,” in 2019 IEEE 58th Conference on Decision and Control (CDC), pp. 4379–4384, IEEE, 2019.
  * [8] M. Pirani, E. Nekouei, H. Sandberg, and K. H. Johansson, “A game-theoretic framework for the security-aware sensor placement problem in networked control systems,” IEEE Transactions on Automatic Control, 2021.
  * [9] A. T. Nguyen, A. M. H. Teixeira, and A. Medvedev, “A single-adversary-single-detector zero-sum game in networked control systems,” arXiv preprint arXiv:2205.14001, 2022.
  * [10] S. C. Anand, A. M. H. Teixeira, and A. Ahlén, “Risk assessment of stealthy attacks on uncertain control systems,” arXiv preprint arXiv:2106.07071v1, 2021.
  * [11] G. C. Calafiore and F. Dabbene, “Probabilistic robust control,” in 2007 American Control Conference, pp. 147–158, IEEE, 2007.
  * [12] D. Duffie and J. Pan, “An overview of value at risk,” Journal of derivatives, vol. 4, no. 3, pp. 7–49, 1997.
  * [13] S. Boyd, S. P. Boyd, and L. Vandenberghe, Convex optimization.  Cambridge university press, 2004.
  * [14] A. M. H. Teixeira, “Security metrics for control systems,” in Safety, Security and Privacy for Cyber-Physical Systems, pp. 99–121, Springer, 2021\.
  * [15] A. Teixeira, H. Sandberg, and K. H. Johansson, “Strategic stealthy attacks: the output-to-output $\ell_{2}$-gain,” in 2015 54th IEEE Conference on Decision and Control (CDC), pp. 2582–2587, IEEE, 2015.
  * [16] G. F. Franklin, J. D. Powell, A. Emami-Naeini, and J. D. Powell, Feedback control of dynamic systems, vol. 4.  Prentice hall Upper Saddle River, NJ, 2002.
  * [17] H. K. Khalil, “Nonlinear systems third edition,” Patience Hall, vol. 115, 2002.


# Dynamical Properties and Some Classes of Non-porous Subsets of Lebesgue
Spaces

Stefan Ivković  Mathematical Institute of the Serbian Academy of Sciences and
Arts, p.p. 367, Kneza Mihaila 36, 11000 Beograd, Serbia.
<EMAIL_ADDRESS>,  Serap Öztop  Department of Mathematics, Faculty of
Science, Istanbul University, Istanbul, Turkey<EMAIL_ADDRESS>and
Seyyed Mohammad Tabatabaie∗ Department of Mathematics, University of Qom, Qom,
Iran<EMAIL_ADDRESS>

###### Abstract.

In this paper, we introduce several classes of non-$\sigma$-porous subsets of
a general Lebesgue space. Also, we study some linear dynamics of operators and
show that the set of all non-hypercyclic vectors of a sequences of weighted
translation operators on $L^{p}$-spaces is not $\sigma$-porous.

###### Key words and phrases:

non-$\sigma$-porous sets, Lebesgue spaces, $\sigma$-porous operators, locally
compact groups, locally compact hypergroups, hypercyclic vectors

###### 2010 Mathematics Subject Classification:

47A16, 28A05, 43A15, 43A62.
∗Corresponding author

##  1\. Introduction

$\sigma$-porous sets, as a collection of very thin subsets of metric spaces,
were introduced and studied first time in [8] through a research on boundary
behavior of functions, and then were applied in differentiation and Banach
spaces theories in [3, 14]. The concepts related to porosity have been active
topics in recent decades because they can be adapted for many known notions in
several kind of metric spaces; see the monograph [21]. $\sigma$-porous subsets
of $\mathbb{R}$ are null and of first category, while in every complete metric
space without any isolated points these two categories are different [20]. On
the other hand, linear dynamics including hypercyclicity in operator theory
received attention during the last years; see books [2, 11] and for instance
[6, 16, 17]. Recently, F. Bayart in [1] through study of hypercyclic shifts
(which was previously studied in [15]; see also [10]) proved that the set of
non-hypercyclic vectors of some classes of weighted shift operators on
$\ell^{2}(\mathbb{Z})$ is a non-$\sigma$-porous set. This would be a new
example of a first category set which is not $\sigma$-porous. In this work, by
some idea from the proof of [1, Theorem1] first we introduce a class of
non-$\sigma$-porous subsets of general Lebesgue spaces, and then we develop
the main result of [1] to sequences of weighted translation operators on
general Lebesgue spaces in the context of discrete groups and hypergroups. In
particular, we prove that if $p\geq 1$, $K$ is a discrete hypergroup,
$(a_{n})$ is a sequence with distinct terms in $K$, and
$w:K\rightarrow(0,\infty)$ is a bounded measurable function such that

$\sum_{n\in\mathbb{N}}\frac{1}{w(a_{0})w(a_{1})\ldots
w(a_{n})}\chi_{\\{a_{n+1}\\}}\in L^{p}(K),$

then the set of all non-hypercyclic vectors of the sequence
$(\Lambda_{n})_{n}$ is not $\sigma$-porous, where the operators $\Lambda_{n}$
are given in Definition 3.8. Also, we study non-$\sigma$-porosity of non-
hypercyclic vectors of weighted composition operators on $L^{\infty}(\Omega)$
for a general measure space $\Omega$ equipped with a nonnegative Radon measure
and on $L^{p}(\mathbb{R},\tau)$, where $\tau$ is the Lebesgue measure on
$\mathbb{R}$. We show that if $G$ is a locally compact group, $\mu$ is a left
Haar measure on $G$, $a\in G$, and $w:G\to(0,\infty)$ be a weight such that

$\big{(}\frac{1}{w(a)w(a^{2})\ldots w(a^{n})}\big{)}_{n}\in
L^{\infty}(G,\mu),$

then the set of all non-hypercyclic vectors of the weighted translation
operator $T_{a,w,\infty}$ on $L^{\infty}(G,\mu)$ is not $\sigma$-porous.

##  2\. Non-$\sigma$-porous subsets of Lebesgue spaces

In this section, we will introduce some classes of non-$\sigma$-porous subsets
of Lebesgue spaces related to a fixed function. First, we recall the
definition of the main notion of this paper.

######  Definition 2.1.

Let $0<\lambda<1$. A subset $E$ of a metric space $X$ is called _$\lambda$
-porous_ at $x\in E$ if for each $\delta>0$ there is an element $y\in
B(x;\delta)\setminus\\{x\\}$ such that

$B(y;\lambda\,d(x,y))\cap E=\varnothing.$

$E$ is called _$\lambda$ -porous_ if it is $\lambda$-porous at every element
of $E$. Also, $E$ is called _$\sigma$ -$\lambda$-porous_ if it is a countable
union of $\lambda$-porous subsets of $X$.

The following lemma plays a key role in the proof of main results of this
section. This fact is a special case of [19, Lemma2]; see also [1, Lemma2].

######  Lemma 2.2.

Let $\mathcal{F}$ be a non-empty family of non-empty closed subsets of a
complete metric space $X$ such that for each $F\in\mathcal{F}$ and each $x\in
X$ and $r>0$ with $B(x;r)\cap F\neq\varnothing$, there exists an element
$J\in\mathcal{F}$ such that

$\varnothing\neq J\cap B(x;r)\subseteq F\cap B(x;r)$

and $F\cap B(x;r)$ is not $\lambda$-porous at all elements of $J\cap B(x;r)$.
Then, every set in $\mathcal{F}$ is not $\sigma$-$\lambda$-porous.

The next result is a development of of [1, Theorem1]. Same as [1], the proof
of this theorem is based on Lemma 2.2.

######  Theorem 2.3.

Let $p\geq 1$, $\Omega$ be a locally compact Hausdorff space, $\mu$ be a
nonnegative Radon measure on $\Omega$, and $A\subseteq\Omega$ be a Borel set
such that

$|f|\chi_{A}\leq\|f\|_{p}\text{ a.e. }\qquad(f\in L^{p}(\Omega,\mu)).$ (2.1)

Then, for each measurable function $g$ on $\Omega$ with $g\chi_{A}\in
L^{p}(\Omega,\mu)$, the set

$\Gamma_{g}:=\big{\\{}f\in L^{p}(\Omega,\mu):\,|f|\geq|g|\chi_{A}\,\,\,{\rm
a.e.}\big{\\}}$

is not $\sigma$-porous in $L^{p}(\Omega,\mu)$.

###### Proof.

Fix an arbitrary number $0<\lambda\leq\frac{1}{2}$, and pick
$0<\beta<\lambda$. Denote

$\mathcal{F}:=\big{\\{}\Gamma_{g}:\,\,g\chi_{A}\in
L^{p}(\Omega,\mu)\big{\\}}.$

We will show that the collection $\mathcal{F}$ satisfies the conditions of
Lemma 2.2. Let $g\in L^{p}(\Omega,\mu)$. Without lossing the generality, we
can assume that $g$ is a nonnegative function. Trivially,
$\Gamma_{g}\neq\varnothing$. Let $(f_{n})$ be a sequence in $\Gamma_{g}$ and
$f_{n}\rightarrow f$ in $L^{p}(\Omega,\mu)$. Then, by (2.1), $|f|\geq
g\chi_{A}$ a.e., and so $f\in\Gamma_{g}$. Therefore, every element of the
collection $\mathcal{F}$ is a closed subset of $L^{p}(\Omega,\mu)$. Now,
assume that $f\in L^{p}(\Omega,\mu)$ and $r>0$ with
$B(f;r)\cap\Gamma_{g}\neq\varnothing$. We find a measurable function $h$ with
$0\leq h\chi_{A}\in L^{p}(\Omega,\mu)$ such that

$\varnothing\neq B(f;r)\cap\Gamma_{h}\subseteq B(f;r)\cap\Gamma_{g},$

and $B(f;r)\cap\Gamma_{g}$ is not $\lambda$-porous at elements of
$B(f;r)\cap\Gamma_{h}$.

Since $\big{(}|f|+\beta^{-1}g\chi_{A}\big{)}^{p}\in L^{1}(\Omega,\mu)$ and
$\mu$ is a Radon measure, the mapping $\nu$ defined by

$\nu(B):=\int_{B}\big{(}|f|+\beta^{-1}g\chi_{A}\big{)}^{p}\,d\mu\qquad(\text{for
every Borel set }B\subseteq\Omega)$

is a Radon measure [9]. Hence, there are some $0<\epsilon<1$, a function $k\in
B(f;r)\cap\Gamma_{g}$ and a compact subset $D$ of $\Omega$ with $\mu(D)>0$
such that

$\|k-f\|_{p}<\epsilon^{1/p}\,r.$

and

$\int_{D^{c}}\big{(}|f|+\beta^{-1}g\chi_{A}\big{)}^{p}\,d\mu<(1-\epsilon)\,r^{p}.$
(2.2)

Pick some $\alpha$ with

$\|k-f\|_{p}<\alpha<\epsilon^{1/p}\,r,$

and denote

$\delta:=\frac{\epsilon^{1/p}\,r-\alpha}{2\mu(D)^{\frac{1}{p}}}.$

Now, we define two functions $h,\xi:\Omega\rightarrow\mathbb{C}$ by

$h:=(g\chi_{A}+\delta)\chi_{D}+\beta^{-1}g\chi_{A}\,\chi_{\Omega\setminus
D}\qquad\text{and}\qquad\xi:=(|k|+\delta)\eta\,\chi_{D}+h\chi_{\Omega\setminus
D},$

where

$\eta(x):=\left\\{\begin{array}[]{ll}\frac{k(x)}{|k(x)|},&\mbox{ if }k(x)\neq
0\\\ \\\ 1,&\mbox{ if }k(x)=0\end{array}\right.$

for all $x\in\Omega$. Since $D$ is compact, we have $h\chi_{A}\in
L^{p}(\Omega,\mu)$. Also, for each $x\in D$,

$\displaystyle|k(x)-\xi(x)|$
$\displaystyle=\big{|}k(x)-\big{(}|k(x)|+\delta\big{)}\,\eta(x)\big{|}$
$\displaystyle=\big{|}k(x)-k(x)-\delta\,\eta(x)\big{|}$
$\displaystyle=\delta,$

and therefore

$\displaystyle\|(\xi-k)\,\chi_{D}\|_{p}=\delta\,\mu(D)^{\frac{1}{p}}=\frac{\epsilon^{1/p}\,r-\alpha}{2}.$

This implies that

$\displaystyle\|(\xi-f)\,\chi_{D}\|_{p}$
$\displaystyle\leq\|(\xi-k)\,\chi_{D}\|_{p}+\|(k-f)\,\chi_{D}\|_{p}$
$\displaystyle\leq\frac{\epsilon^{1/p}\,r-\alpha}{2}+\alpha<\epsilon^{1/p}\,r.$

Hence,

$\displaystyle\|\xi-f\|_{p}^{p}$
$\displaystyle=\int_{D}|\xi-f|^{p}\,d\mu+\int_{\Omega\setminus
D}|\xi-f|^{p}\,d\mu$ $\displaystyle<\epsilon\,r^{p}+\int_{\Omega\setminus
D}|\beta^{-1}g\chi_{A}-f|^{p}\,d\mu$
$\displaystyle\leq\epsilon\,r^{p}+\int_{\Omega\setminus
D}(\beta^{-1}g\chi_{A}+|f|)^{p}\,d\mu$
$\displaystyle<\epsilon\,r^{p}+(1-\epsilon)\,r^{p}=r^{p},$

and so, $\xi\in B(f;r)$. Moreover,

$|\xi(x)|=|k(x)|+\delta\geq g(x)+\delta=h(x)\quad\text{a.e. on }D\cap A,$

and for each $x\in(\Omega\setminus D)\cap A$ we have $|\xi(x)|=h(x)$. This
shows that $\xi\in\Gamma_{h}$, and so

$\varnothing\neq B(f;r)\cap\Gamma_{h}\subseteq B(f;r)\cap\Gamma_{g}$

because $h\geq g$. Now, let $u\in B(f;r)\cap\Gamma_{h}$ and put
$r^{\prime}:=\min\\{\delta,\lambda\,(r-\|f-u\|_{p})\\}$. Let $v\in
B(u;r^{\prime})$. We define the function $\gamma:\Omega\rightarrow\mathbb{C}$
by

$\gamma(x):=\left\\{\begin{array}[]{ll}v(x),&\mbox{ if }x\in D\\\ \\\
\Big{(}|v(x)|+\beta|u(x)-v(x)|\Big{)}\,\theta(x),&\mbox{ if
}x\in\Omega\setminus D\end{array}\right.$

where

$\theta(x):=\left\\{\begin{array}[]{ll}\frac{v(x)}{|v(x)|},&\mbox{ if
}v(x)\neq 0\\\ \\\ 1,&\mbox{ if }v(x)=0.\end{array}\right.$

Therefore, for each $x\in\Omega\setminus D$ we have

$|\gamma(x)-v(x)|=\beta\,|u(x)-v(x)|\qquad\text{and}\qquad|\gamma(x)|\geq\beta\,|u(x)|.$

Easily,

$\displaystyle\|\gamma-v\|_{p}^{p}$
$\displaystyle=\|(\gamma-v)\,\chi_{D}\|_{p}^{p}+\|(\gamma-v)\,\chi_{\Omega\setminus
D}\|_{p}^{p}$ $\displaystyle=\|(\gamma-v)\,\chi_{\Omega\setminus D}\|_{p}^{p}$
$\displaystyle=\beta^{p}\,\|(u-v)\,\chi_{\Omega\setminus D}\|_{p}^{p}$
$\displaystyle\leq\beta^{p}\,\|u-v\|_{p}^{p}<\lambda^{p}\,\|u-v\|_{p}^{p},$

and hence,

$\gamma\in B\big{(}v;\lambda\,\|u-v\|_{p}\big{)}\subseteq B(f;r).$

In addition,

$|\gamma(x)|\geq\beta\,|u(x)|\geq\beta\,h(x)=g(x)\quad\text{for a.e.
}x\in(\Omega\setminus D)\cap A$

and

$|\gamma(x)|=|v(x)|\geq|u(x)|-\delta\geq g(x)\quad\text{for a.e. }x\in D\cap
A,$

because $\|u-v\|_{p}\leq\delta$ and also $|u|\geq h$. Therefore,

$B\big{(}v;\lambda\,\|u-v\|_{p}\big{)}\cap
B(f;r)\cap\Gamma_{g}\neq\varnothing.$

and this competes the proof. ∎

######  Remark 2.4.

Note that, in general, the condition (2.1) in the statement of Theorem 2.3
does not implies that $\Omega$ is a discrete space. In particular, if in the
condition (2.1) we set $A:=\Omega$, then it implies that
$L^{p}(\Omega,\mu)\subseteq L^{\infty}(\Omega,\mu)$, and this inclusion is
equivalent to

$\alpha:=\inf\\{\mu(E):\,\mu(E)>0\\}>0,$ (2.3)

and equivalently, for each $q>p$, $L^{p}(\Omega,\mu)\subseteq
L^{q}(\Omega,\mu)$; see [18]. If in addition, ${\rm supp}\mu=\Omega$, then the
condition (2.3) implies that for each $x\in\Omega$,

$\mu(\\{x\\})=\inf\\{\mu(F):\,F\text{ is a compact neighborhood of }x\\}>0.$

Specially, if $\Omega$ is a locally compact group (or hypergroup) and $\mu$ is
a left Haar measure of it, then the condition (2.1) implies that $\Omega$ is a
discrete topological space.

The next result is a direct conclusion of Theorem 2.3.

######  Corollary 2.5.

Let $\Omega$ be a discrete topological space and
$\varphi:=(\varphi_{j})_{j\in\Omega}\subseteq[1,\infty)$ such that for each
$j$, $\varphi_{j}\geq 1$. Put
$\mu_{\varphi}:=\sum_{j\in\Omega}\varphi_{j}\,\delta_{j}$, where $\delta_{j}$
is the point-mass measure at $j$. Then, for each $g\in
L^{p}(\Omega,\mu_{\varphi})$, the set

$\Gamma_{g}:=\big{\\{}f\in L^{p}(\Omega,\mu_{\varphi}):\,|f|\geq|g|\big{\\}}$

is not $\sigma$-porous in $L^{p}(\Omega,\mu_{\varphi})$.

###### Proof.

Just note that for each $k\in\Omega$ and $f\in L^{p}(\Omega,\mu_{\varphi})$,

$\|f\|_{p}^{p}=\sum_{j\in\Omega}|f(j)|^{p}\,\mu_{\varphi}(\\{j\\})\geq|f(k)|\,\varphi_{k}\geq|f(k)|^{p}.$

∎

In particular, if a set is endowed with the counting measure, we get the fact.

######  Corollary 2.6.

Let $p\geq 1$ and $A$ be a non-empty set. Then, for each $g\in\ell^{p}(A)$,
the set

$\Gamma_{g}:=\big{\\{}f\in\ell^{p}(A):\,|f|\geq|g|\big{\\}}$

is not $\sigma$-porous in $\ell^{p}(A)$.

The situation for $L^{\infty}$-spaces is different.

######  Theorem 2.7.

Let $\Omega$ be a locally compact Hausdorff space and $\mu$ be a nonnegative
Radon measure on $\Omega$. Then, for each $g\in L^{\infty}(\Omega,\mu)$, the
set

$\Gamma_{g}:=\big{\\{}f\in L^{\infty}(\Omega,\mu):\,|f|\geq|g|\,\,\,{\rm
a.e.}\big{\\}}$

is not $\sigma$-porous in $L^{\infty}(\Omega,\mu)$.

###### Proof.

Same as the proof of Theorem 2.3 fix $0<\lambda\leq\frac{1}{2}$, and set

$\mathcal{F}:=\big{\\{}\Gamma_{g}:\,\,g\in L^{\infty}(\Omega,\mu)\big{\\}}.$

This collection satisfies the conditions of Lemma 2.2. Trivially, $\Gamma_{g}$
is a closed subset of $L^{\infty}(\Omega,\mu)$ for all $g\in
L^{\infty}(\Omega,\mu)$. Let Assume that $0\leq g\in L^{\infty}(\Omega,\mu)$,
and let $f\in L^{\infty}(\Omega,\mu)$ and $r>0$. If
$B(f;r)\cap\Gamma_{g}\neq\varnothing$, we choose some $k\in
B(f;r)\cap\Gamma_{g}$ and we find some $\varepsilon\in(0,1)$ such that
$\|k-f\|_{\infty}<\varepsilon r$. Pick some $\delta\in(0,(1-\varepsilon)r)$,
and set

$h:=g+\delta\qquad\text{and}\qquad\xi:=(|k|+\delta)\eta,$

where $\eta$ is as in the proof of Theorem 2.3. Then, we get

$\|\xi-f\|_{\infty}\leq\|\xi-k\|_{\infty}+\|k-f\|_{\infty}\leq\delta+\varepsilon
r<(1-\varepsilon)r+\varepsilon r=r,$

so $\xi\in B(f;r)$, and $\xi\in\Gamma_{h}$ since $|k|\geq g$. Next, let $u\in
B(f;r)\cap\Gamma_{h}$, and set
$r^{\prime}:=\min\\{\delta,\lambda\,(r-\|f-u\|_{\infty})\\}$. Pick some $v\in
B(u;r^{\prime})$. Then, $\|u-v\|_{\infty}<r^{\prime}\leq\delta$, so
$|v|\geq|u|-\delta\geq h-\delta=g$ a.e., so $v\in\Gamma_{g}$. Thus,

$v\in B(v;\lambda\|u-v\|_{\infty})\cap B(f;r)\cap\Gamma_{g}.$

This completes the proof. ∎

######  Remark 2.8.

The main Theorem 2.3 is valid also for the sequence space $c_{0}$, because the
sequences with finitely many non-zero coefficients approximate sequences in
$c_{0}$.

At the end of this section, we give a class of non-$\sigma$-porous subsets of
the $L^{p}$-space on real line. In the proof of this result, which is also
based on Lemma 2.2, we apply some functions defined in the proof of Theorem
2.3.

######  Theorem 2.9.

Let $p\geq 1$, and $\tau$ be the Lebesgue measure on $\mathbb{R}$. For each
$g\in L^{p}(\mathbb{R},\tau)$ put

$\Theta_{g}:=\big{\\{}f\in
L^{p}(\mathbb{R},\tau):\,\|f\chi_{[m,m+1]}\|_{p}\geq\|g\chi_{[m,m+1]}\|_{p}\,\text{
for all }m\in\mathbb{Z}\big{\\}}.$

Then, $\Theta_{g}$ is not $\sigma$-porous in $L^{p}(\mathbb{R},\tau)$.

###### Proof.

Let $0<\lambda\leq\frac{1}{2}$, and $0<\beta<\lambda$. Denote

$\mathcal{F}:=\big{\\{}\Theta_{g}:\,\,g\in L^{p}(\mathbb{R},\tau)\big{\\}}.$

We prove that the collection $\mathcal{F}$ satisfies the conditions of Lemma
2.2. Let $0\leq g\in L^{p}(\mathbb{R},\tau)$. Then, easily
$\Theta_{g}\neq\varnothing$ and it is closed in $L^{p}(\mathbb{R},\tau)$. Now,
assume that $f\in L^{p}(\mathbb{R},\tau)$ and $r>0$ with
$B(f;r)\cap\Theta_{g}\neq\varnothing$. Then, there exists a large enough
number $N\in\mathbb{N}$, some $0<\epsilon<1$ and a function $k\in
B(f;r)\cap\Theta_{g}$ such that

$\|k-f\|_{p}<\epsilon^{\frac{1}{p}}\,r\qquad\text{and}\qquad\int_{[-N,N]^{c}}\big{(}|f|+\beta^{-1}g\big{)}^{p}\,d\tau<(1-\epsilon)\,r^{p}.$

Pick some $\alpha$ with $\|k-f\|_{p}<\alpha<\epsilon^{\frac{1}{p}}\,r,$ and
denote $\delta:=\frac{\epsilon^{\frac{1}{p}}\,r-\alpha}{2(2N)^{\frac{1}{p}}}$.
Put

$A_{1}:=\\{m\in[N]:\,g=0\,\text{ a.e. on }[m,m+1]\\},\quad A_{2}:=[N]\setminus
A_{1}$

and

$B_{1}:=\\{m\in[N]:\,k=0\,\text{ a.e. on }[m,m+1]\\},\quad B_{2}:=[N]\setminus
B_{1},$

where $[N]:=\\{-N,\ldots,N-1\\}$, and then define

$\rho:=\sum_{m\in A_{1}}\chi_{[m,m+1]}+\sum_{m\in
A_{2}}\frac{g\chi_{[m,m+1]}}{\|g\chi_{[m,m+1]}\|_{p}},$

and

$\eta:=\sum_{m\in B_{1}}\chi_{[m,m+1]}+\sum_{m\in
B_{2}}\frac{k\chi_{[m,m+1]}}{\|k\chi_{[m,m+1]}\|_{p}}.$

Now, we define $h,\xi:\mathbb{R}\rightarrow\mathbb{C}$ by

$h:=g\chi_{[-N,N]}+\delta\rho+\beta^{-1}g\,\chi_{[-N,N]^{c}}$

and

$\xi:=|k|\,\chi_{[-N,N]}+\delta\eta+h\chi_{[-N,N]^{c}}.$

Clearly, $h\in L^{p}(\mathbb{R},\tau)$. For each $x\in[-N,N]$ we have
$|k(x)-\xi(x)|=\delta\,|\eta(x)|$, and so

$\displaystyle\|(k-\xi)\chi_{[-N,N]}\|_{p}^{p}$
$\displaystyle=\delta^{p}\,\|\eta\chi_{[-N,N]}\|_{p}^{p}$
$\displaystyle=\delta^{p}\,\sum_{m\in[N]}\|\eta\chi_{[m,m+1]}\|_{p}^{p}$
$\displaystyle=\delta^{p}\,2N.$

Hence, $\|(k-\xi)\chi_{[-N,N]}\|_{p}=\delta\,(2N)^{\frac{1}{p}}$. Now, similar
to the proof of Theorem 2.3 we have $\xi\in B(f;r)$. Moreover,

$\|\xi\chi_{[m,m+1]}\|_{p}=\|k\chi_{[m,m+1]}\|_{p}+\delta\geq\|g\chi_{[m,m+1]}\|_{p}+\delta=\|h\chi_{[m,m+1]}\|_{p}$

for all $m\in[N]$. And also for each $m\notin[N]$,

$\|\xi\chi_{[m,m+1]}\|_{p}=\|h\chi_{[m,m+1]}\|_{p}\geq\|g\chi_{[m,m+1]}\|_{p}.$

So,

$\xi\in B(f;r)\cap\Theta_{h}\subseteq B(f;r)\cap\Theta_{g}.$

Now, let $u\in B(f;r)\cap\Theta_{h}$ and put
$r^{\prime}:=\min\\{\delta,\lambda\,(r-\|f-u\|_{p})\\}$. Let $v\in
B(u;r^{\prime})$. We define the function
$\gamma:\mathbb{R}\rightarrow\mathbb{C}$ by

$\gamma(x):=\left\\{\begin{array}[]{ll}v(x),&\mbox{ if }x\in[-N,N]\\\ \\\
\Big{(}|v(x)|+\beta|u(x)-v(x)|\Big{)}\,\theta(x),&\mbox{ if
}x\in[-N,N]^{c}\end{array}\right.$

where

$\theta(x):=\left\\{\begin{array}[]{ll}\frac{v(x)}{|v(x)|},&\mbox{ if
}v(x)\neq 0\\\ \\\ 1,&\mbox{ if }v(x)=0.\end{array}\right.$

Similar to the proof of Theorem 2.3, we have $\gamma\in
B\big{(}v;\lambda\,\|u-v\|_{p}\big{)}$. Now, for each $m\notin[N]$,

$|\gamma|\chi_{(m,m+1)}=(|v|+\beta|u-v|)\chi_{(m,m+1)}\geq\beta|u|\chi_{(m,m+1)}.$

Hence,

$\|\gamma\chi_{[m,m+1]}\|_{p}\geq\beta\,\|u\chi_{[m,m+1]}\|_{p}\geq\beta\,\|h\chi_{[m,m+1]}\|_{p}$

since $u\in B(f;r)\cap\Theta_{h}$. However, in this case we have
$(m,m+1)\in{[-N,N]}^{c}$, so $h\chi_{(m,m+1)}=\beta^{-1}g\chi_{(m,m+1)}$.
Thus, $\beta\|h\chi_{[m,m+1]}\|_{p}=\|g\chi_{[m,m+1]}\|_{p}$. If $m\in[N]$, we
have $\gamma\chi_{[m,m+1]}=v\chi_{[m,m+1]}$ because
$\gamma\chi_{[-N,N]}=v\chi_{[-N,N]}$ and $[m,m+1]\subseteq{[-N,N]}$. We get

$\left|\,\|u\chi_{[m,m+1]}\|_{p}-\|v\chi_{[m,m+1]}\|_{p}\right|\leq||(u-v)\chi_{[m,m+1]}\|_{p}\leq\|u-v\|_{p}<\delta$

because $v\in B(u;r^{\prime})$, hence

$\displaystyle\|\gamma\chi_{[m,m+1]}\|_{p}$
$\displaystyle=\|v\chi_{[m,m+1]}\|_{p}$
$\displaystyle\geq\|u\chi_{[m,m+1]}\|_{p}-\delta$
$\displaystyle\geq\|h\chi_{[m,m+1]}\|_{p}-\delta$
$\displaystyle=\|g\chi_{[m,m+1]}\|_{p}.$

Therefore,

$\gamma\in B\big{(}v;\lambda\,\|u-v\|_{p}\big{)}\cap B(f;r)\cap\Theta_{g},$

and the proof is complete. ∎

##  3\. Applications

In this section, we will apply the results of the previous section, to prove
that the set of all non-hypercyclic vectors of some sequences of weighted
translation operators is non-$\sigma$-porous.

######  Definition 3.1.

Let $\mathcal{X}$ be a Banach space. A sequence $(T_{n})_{n\in\mathbb{N}_{0}}$
of operators in $B(\mathcal{X})$ is called hypercyclic if there is an element
$x\in\mathcal{X}$ (called _hypercyclic vector_) such that the orbit
$\\{T_{n}(x):\,n\in\mathbb{N}_{0}\\}$ is dense in $\mathcal{X}$. The set of
all hypercyclic vectors of a sequence $(T_{n})_{n\in\mathbb{N}_{0}}$ is
denoted by $HC((T_{n})_{n\in\mathbb{N}_{0}})$. An operator $T\in
B(\mathcal{X})$ is called _hypercyclic_ if the sequence
$(T^{n})_{n\in\mathbb{N}_{0}}$ is hypercyclic.

Let $G$ be a locally compact group and $a\in G$. Then, for each function
$f:G\to\mathbb{C}$ we define $L_{a}f:G\rightarrow\mathbb{C}$ by
$L_{a}f(x):=f(a^{-1}x)$ for all $x\in G$. Note that if $p\geq 1$, then the
left translation operator

$L_{a}:L^{p}(G)\rightarrow L^{p}(G),\quad f\mapsto L_{a}f$

is not hypercyclic because $\|L_{a}\|\leq 1$. Hypercyclicity of _weigted_
translation operators on $L^{p}(G)$ and regarding an aperiodic element $a$ was
studied in [5] (an element $a\in G$ is called _aperiodic_ if the closed
subgroup of $G$ generated by $a$ is not compact).

######  Definition 3.2.

Let $G$ be a locally compact group with a left Haar measure $\mu$. Fix $p\geq
1$. We denote $L^{p}(G):=L^{p}(G,\mu)$. Assume that $w:G\rightarrow(0,\infty)$
is a bounded measurable function (called a _weight_) and $a\in G$. Then, the
weighted translation operator $T_{a,w,p}:L^{p}(G)\rightarrow L^{p}(G)$ is
defined by

$T_{a,w,p}(f):=w\,L_{a}f,\qquad(f\in L^{p}(G)).$

For each $n\in\mathbb{N}$ we denote
$\varphi_{n}:=w\,L_{a}w\,\ldots\,L_{a^{n-1}}w,$ where $a^{0}:=e$, the identity
element of $G$.

######  Theorem 3.3.

Let $p\geq 1$, $G$ be a discrete group and $a\in G$. Let $\mu$ be a left Haar
measure on $G$ with $\mu(\\{e\\})\geq 1$ and $(\gamma_{n})_{n}$ be an
unbounded sequence of non-negative integers. Let $w:G\rightarrow(0,\infty)$ be
a bounded function such that for some finite nonempty set $F\subseteq G$ and
some $N>0$ we have

$a^{\gamma_{n}}F\cap F=\varnothing\qquad(n\geq N),$

and

$\beta:=\inf\left\\{\prod_{k=1}^{\gamma_{n}}w(a^{k}t):\,n\geq N,\,t\in
F\right\\}>0.$

Then, the set

$\Lambda:=\big{\\{}f\in
L^{p}(G,\mu):\,\|T_{a,w,p}^{\gamma_{n}}f-\chi_{F}\|_{p}\geq\mu(F)^{\frac{1}{p}}\,\text{
for all }n\geq N\big{\\}}$

is non-$\sigma$-porous.

###### Proof.

Let $\Gamma:=\\{f\in L^{p}(G,\mu):\,|f|\geq\frac{1}{\beta}\chi_{F}\\}$. Then,
$\Gamma$ is not $\sigma$-porous in $L^{p}(G,\mu)$ thanks to Theorem 2.7. Also,
for each $f\in\Gamma$ and $n\geq N$ we have

$\displaystyle\|T_{a,w,p}^{\gamma_{n}}f-\chi_{F}\|_{p}^{p}$
$\displaystyle=\int_{G}|\prod_{k=1}^{n}w(a^{-\gamma_{n}+k}x)\,f(a^{-\gamma_{n}}x)-\chi_{F}(x)|^{p}\,d\mu(x)$
$\displaystyle=\int_{G}|\prod_{k=1}^{\gamma_{n}}w(a^{k}x)\,f(x)-\chi_{F}(a^{\gamma_{n}}x)|^{p}\,d\mu(x)$
$\displaystyle=\int_{G}|\prod_{k=1}^{\gamma_{n}}w(a^{k}x)\,f(x)-\chi_{a^{-\gamma_{n}}F}(x)|^{p}\,d\mu(x)$
$\displaystyle\geq\int_{F}|\prod_{k=1}^{\gamma_{n}}w(a^{k}x)\,f(x)-\chi_{a^{-\gamma_{n}}F}(x)|^{p}\,d\mu(x)$
$\displaystyle=\int_{F}|\prod_{k=1}^{\gamma_{n}}w(a^{k}x)\,f(x)|^{p}\,d\mu(x)$
$\displaystyle\geq\int_{F}|\beta\,\frac{1}{\beta}|^{p}\,d\mu(x)$
$\displaystyle=\mu(F).$

This completes the proof. ∎

######  Example 3.4.

Let $G$ be the additive group $\mathbb{Z}$ with the counting measure. Let $F$
be a finite non-empty subset of $\mathbb{Z}$. Put $N:=\max\\{|j|:j\in F\\}$.
If $w:=(w_{n})_{n\in\mathbb{Z}}\subseteq(0,\infty)$ is a bounded sequence with
$w_{n}\geq 1$ for all $n\geq N$. Then the required conditions in the previous
theorem hold with respect to $F$ and $a:=1$.

The following fact is a direct conclusion of the previous theorem.

######  Corollary 3.5.

Let $p\geq 1$, $G$ be a discrete group and $a\in G$ with infinite order. Let
$\mu$ be the counting measure on $G$ and $(\gamma_{n})_{n}$ be an unbounded
sequence of non-negative integers. Let $w:G\rightarrow(0,\infty)$ be a bounded
function such that for some $t\in G$,

$\inf\left\\{\prod_{k=1}^{\gamma_{n}}w(a^{k}t):\,n\in\mathbb{N}\right\\}>0.$

Then, the set

$\big{\\{}f\in L^{p}(G,\mu):\,\|T_{a,w}^{\gamma_{n}}f-\chi_{\\{t\\}}\|_{p}\geq
1\,\text{ for all }n\big{\\}}$

is non-$\sigma$-porous.

######  Theorem 3.6.

Let $p\geq 1$, $G$ be a discrete group, and $a\in G$. Let $\mu$ be a left Haar
measure on $G$ with $\mu(\\{e\\})\geq 1$. Let $(\gamma_{n})_{n}$ be an
unbounded sequence of non-negative integers and let $w:G\rightarrow(0,\infty)$
be a bounded function such that

$\inf_{n\in\mathbb{N}}\prod_{k=1}^{\gamma_{n}}w(a^{k})>0.$

Then, the set

$\Gamma:=\big{\\{}f\in
L^{p}(G,\mu):\,|f(e)|\,\inf_{n\in\mathbb{N}}\prod_{k=1}^{\gamma_{n}}w(a^{k})\geq
1\big{\\}}$

is non-$\sigma$-porous. In particular, setting $T_{n}:=T_{a,w,p}^{\gamma_{n}}$
for all $n$, the set of all non-hypercyclic vectors of the sequence
$(T_{n})_{n}$ is not $\sigma$-porous in $L^{p}(G,\mu)$.

###### Proof.

Since $\mu(\\{e\\})\geq 1$, applying Theorem 2.3 the set $\Gamma$ is
non-$\sigma$-porous, because

$[\inf_{n\in\mathbb{N}}\prod_{k=1}^{\gamma_{n}}w(a^{k})]^{-1}\,\chi_{\\{e\\}}\in
L^{p}(G,\mu).$

Let $f\in\Gamma$. If $n$ is a nonnegative integer, then for every $x$ in $G$
we have

$\|T_{n}f\|_{p}\geq\big{|}\varphi_{\gamma_{n}}(x)\,L_{a^{\gamma_{n}}}f(x)\big{|},$

and so setting $x=a^{{\gamma_{n}}}$ we have

$\displaystyle\|T_{n}f\|_{p}$
$\displaystyle\geq\big{|}\varphi_{n}(a^{\gamma_{n}})\,L_{a^{\gamma_{n}}}f(a^{\gamma_{n}})\big{|}$
$\displaystyle=\Big{[}\prod_{k=1}^{\gamma_{n}}w(a^{k})\Big{]}\,|f(e)|$
$\displaystyle\geq|f(e)|\,\inf_{m\in\mathbb{N}}\prod_{k=1}^{\gamma_{m}}w(a^{k})\geq
1.$

This implies that the set $\\{T_{n}f:\,n\in\mathbb{N}\\}$ is not dense in
$L^{p}(G,\mu)$, and so $\Gamma$ is a subset of the set of all non-hypercyclic
vectors of $T$. This completes the proof. ∎

Now, we recall the definition of hypergroups which are generalizations of
locally compact groups; see the monograph [4] and the basic paper [12] for
more details. In locally compact hypergroups the convolution of two Dirac
measures is not necessarily a Dirac measure. Let $K$ be a locally compact
Hausdorff space. We denote by $\mathbb{M}(K)$ the space of all regular complex
Borel measures on $K$, and by $\delta_{x}$ the Dirac measure at the point $x$.
The support of a measure $\mu\in\mathbb{M}(K)$ is denoted by
$\text{supp}(\mu)$.

######  Definition 3.7.

Suppose that $K$ is a locally compact Hausdorff space,
$(\mu,\nu)\mapsto\mu\ast\nu$ is a bilinear positive-continuous mapping from
$\mathbb{M}(K)\times\mathbb{M}(K)$ into $\mathbb{M}(K)$ (called convolution),
and $x\mapsto x^{-}$ is an involutive homeomorphism on $K$ (called involution)
with the following properties:

  1. (i)

$\mathbb{M}(K)$ with $\ast$ is a complex associative algebra;

  2. (ii)

if $x,y\in K$, then $\delta_{x}\ast\delta_{y}$ is a probability measure with
compact support;

  3. (iii)

the mapping $(x,y)\mapsto\text{supp}(\delta_{x}\ast\delta_{y})$ from $K\times
K$ into $\textbf{C}(K)$ is continuous, where $\textbf{C}(K)$ is the set of all
non-empty compact subsets of $K$ equipped with Michael topology;

  4. (iv)

there exists a (necessarily unique) element $e\in K$ (called identity) such
that for all $x\in K$,
$\delta_{x}\ast\delta_{e}=\delta_{e}\ast\delta_{x}=\delta_{x}$;

  5. (v)

for all $x,y\in K$, $e\in\text{supp}(\delta_{x}\ast\delta_{y})$ if and only if
$x=y^{-}$;

Then, $K\equiv(K,\ast,^{-},e)$ is called a locally compact hypergroup.

A nonzero nonnegative regular Borel measure $m$ on $K$ is called the (left)
Haar measure if for each $x\in K$, $\delta_{x}\ast m=m$. For each $x,y\in K$
and measurable function $f:K\rightarrow\mathbb{C}$ we denote

$f(x*y):=\int_{K}f\,d(\delta_{x}*\delta_{y}),$

while this integral exists.

######  Definition 3.8.

Suppose that $a:=(a_{n})_{n\in\mathbb{N}_{0}}$ is a sequence in a hypergroup
$K$, and $w$ is a weight function on $K$. For each $n\in\mathbb{N}_{0}$ we
define the bounded linear operator $\Lambda_{n+1}$ on $L^{p}(K)$ by

$\Lambda_{n+1}f(x):=w(a_{0}\ast x)\,w(a_{1}\ast x)\ldots w(a_{n}\ast
x)\,f(a_{n+1}\ast x)\qquad(f\in L^{p}(K))$

for all $x\in K$. Also, we assume that $\Lambda_{0}$ is the identity operator
on $L^{p}(K)$.

Some linear dynamical properties of this sequence of operators were studied in
[13]. The sequence $\\{\Lambda_{n}\\}_{n}$ is a generalization of the usual
powers of a single weighted translation operator on $L^{p}(G)$, where $G$ is a
locally compact group. In fact, any locally compact group $G$ with the mapping

$\mu\ast\nu\mapsto\int_{G}\int_{G}\delta_{xy}d\mu(x)d\nu(y)\qquad(\mu,\nu\in\mathbb{M}(G))$

as convolution, and $x\mapsto x^{-1}$ from $G$ onto $G$ as involution is a
locally compact hypergroup. Let $\eta:=(a_{n})_{n\in\mathbb{N}_{0}}$ be a
sequence in $G$, and $w$ be a weight on $G$. Then for each $f\in L^{p}(G)$,
$n\in\mathbb{N}_{0}$ and $x\in G$, we have

$\Lambda_{n+1}f(x)=w(a_{0}x)\,w(a_{1}x)\ldots w(a_{n}x)\,f(a_{n+1}x).$

In particular, let $a\in G$ and for each $n\in\mathbb{N}_{0}$, put
$a_{n}:=a^{-n}$. Then, $\Lambda_{n}=T_{a,w,p}^{n}$ for all $n\in\mathbb{N}$.
In this case, the operator $T_{a,w,p}$ is hypercyclic if and only if the
sequence $(\Lambda_{n})_{n}$ is hypercyclic.

Let $K$ be a discrete hypergroup with the convolution $\ast$ between Radon
measures of $K$ and the involution $\cdot^{-}:K\rightarrow K$. Then, by [12,
Theorem7.1A], the measure $\mu$ on $K$ given by

$\mu(\\{x\\}):=\frac{1}{\delta_{x}\ast\delta_{x^{-}}(\\{e\\})},\qquad(x\in K)$
(3.1)

is a left Haar measure on $K$.

######  Proposition 3.9.

Let $K$ be a discrete hypergroup, $\mu$ be the Haar measure (3.1), and $p\geq
1$. Then for each $g\in L^{p}(K,\mu)$, the set

$\big{\\{}f\in L^{p}(K,\mu):\,|f|\geq|g|\big{\\}}$

is not $\sigma$-porous in $L^{p}(K,\mu)$.

###### Proof.

Just note that for each $x\in K$ we have $\mu(\\{x\\})\geq 1$ because

$1=\delta_{x}\ast\delta_{x^{-}}(K)\geq\delta_{x}\ast\delta_{x^{-}}(\\{e\\}).$

Hence, the measure space $(K,\mu)$ satisfies the condition of Corollary 2.5. ∎

Let $a:=(a_{n})_{n\in\mathbb{N}}$ be a sequence in a discrete hypergroup $K$
such that $a_{n}\neq a_{m}$ for each $m\neq n$, and let
$w:K\rightarrow(0,\infty)$ be bounded. We define
$h_{a,w}:K\rightarrow\mathbb{C}$ by

$h_{a,w}:=\sum_{n\in\mathbb{N}_{0}}\frac{1}{w(a_{0})w(a_{1})\ldots
w(a_{n})}\chi_{\\{a_{n+1}\\}}.$

######  Theorem 3.10.

Let $p\geq 1$, and $K$ be a discrete hypergroup endowed with the left Haar
measure (3.1). Let $a:=(a_{n})_{n\in\mathbb{N}_{0}}\subseteq K$ with distinct
terms, and $w$ be a weight on $K$ such that $h_{a,w}\in L^{p}(K)$. Then, the
set of all non-hypercyclic vectors of the sequence $(\Lambda_{n})_{n}$ is not
$\sigma$-porous.

###### Proof.

First, thanks to Proposition 3.9, the set

$E:=\Big{\\{}f\in L^{p}(K):\,|f(a_{n+1})|\geq\frac{1}{w(a_{0})w(a_{1})\ldots
w(a_{n})}\,\text{ for all }n\Big{\\}}$

is not $\sigma$-porous because it equals to the set $\Big{\\{}f\in
L^{p}(K):\,|f|\geq h_{a,w}\Big{\\}}.$ Now, for each $f\in E$,

$\displaystyle\|\Lambda_{n+1}f\|_{p}$ $\displaystyle\geq\sup_{x\in
K}w(a_{0}\ast x)\,w(a_{1}\ast x)\ldots w(a_{n}\ast x)\,|f(a_{n+1}\ast x)|$
$\displaystyle\geq w(a_{0})\,w(a_{1})\ldots w(a_{n})\,|f(a_{n+1})|\geq 1$

for all $n\in\mathbb{N}_{0}$. This implies that $0$ does not belong to the
closure of $\\{\Lambda_{n}f:\,n\in\mathbb{N}\\}$ in $L^{p}(K)$, and so
$E\subseteq[HC((\Lambda_{n})_{n})]^{c}$. This completes the proof. ∎

Since any group is a hypergroup, we can give the fact below.

######  Corollary 3.11.

Let $p\geq 1$, and $G$ be a discrete group. Let $a\in G$ be of infinite order,
$(\gamma_{n})_{n\in\mathbb{N}_{0}}\subseteq\mathbb{N}$ be with distinct terms
and $w:G\rightarrow(0,\infty)$ be a weight such that

$\big{(}\frac{1}{w(a^{\gamma_{0}})w(a^{\gamma_{1}})\ldots
w(a^{\gamma_{n}})}\big{)}_{n}\in\ell^{p}(G).$

Then, the set of all non-hypercyclic vectors of the sequence
$(T_{a,w,p}^{\gamma_{n}})_{n}$ is not $\sigma$-porous in $\ell^{p}(G)$.

Now, we can write the next corollary which is a generalization of [1, Theorem
1].

######  Corollary 3.12.

Let $p\geq 1$, $(\gamma_{n})_{n}\subseteq\mathbb{N}$ be strictly increasing
and $(w_{n})_{n\in\mathbb{Z}}$ be a bounded sequence in $(0,\infty)$ such that

$\big{(}\frac{1}{w_{\gamma_{0}}w_{\gamma_{1}}w_{\gamma_{2}}\ldots
w_{\gamma_{n}}}\big{)}_{n}\in\ell^{p}(\mathbb{Z}).$

Then, the set of all non-hypercyclic vectors of the sequence $(T_{n})_{n}$ is
not $\sigma$-porous, where

$(T_{n+1}a)_{k}:=w_{\gamma_{0}}w_{\gamma_{1}}w_{\gamma_{2}}\ldots
w_{\gamma_{n}}a_{k+\gamma_{n+1}}\qquad(k\in\mathbb{N}_{0})$

for all $a:=(a_{j})_{j}\in\ell^{p}(\mathbb{Z})$.

Applying Theorem 2.7 we can speak regarding some more general situation in the
case of $p=\infty$. Let $\Omega$ be a locally compact Hausdorff space endowed
with a nonnegative Radon measure $\mu$. Let $w:\Omega\rightarrow(0,\infty)$ be
a bounded measurable function, and $\alpha:\Omega\rightarrow\Omega$ be a bi-
measurable mapping such that $\|f\circ\alpha^{\pm
1}\|_{\infty}=\|f\|_{\infty}$ for all $f\in L^{\infty}(\Omega,\mu)$. Then, we
define $T_{\alpha,w,\infty}:L^{\infty}(\Omega,\mu)\rightarrow
L^{\infty}(\Omega,\mu)$ by

$T_{\alpha,w,\infty}(f):=w\,(f\circ\alpha)\qquad(f\in
L^{\infty}(\Omega,\mu)).$

If $\Omega$ be a locally compact group and $a\in\Omega$, setting
$\alpha_{a}(x):=ax$ for all $x\in\Omega$, we denote
$T_{a,w,\infty}:=T_{\alpha_{a},w,\infty}$. Note that $\alpha^{-1}$ means the
inverse function of $\alpha$, and for each $k\in\mathbb{N}$,
$\alpha^{-k}:=(\alpha^{-1})^{k}$.

######  Theorem 3.13.

Let $T_{\alpha,w,\infty}$ be the weighted composition operator defined as
above and let $\\{\gamma_{n}\\}_{n}\subseteq\mathbb{N}$ be a fixed unbounded
sequence. Suppose that there exists a sequence $\\{A_{n}\\}_{n}$ of disjoint
subsets of $\Omega$ with $\mu(A_{n})>0$ for all $n$ such that

$j_{\alpha,w}:=\sum_{n\in\mathbb{N}}\frac{1}{(w\circ\alpha^{-\gamma_{n}})\,(w\circ\alpha^{-\gamma_{n}+1})\ldots(w\circ\alpha^{-1})}\chi_{A_{n}}\in
L^{\infty}(\Omega,\mu).$

Then, the set $\\{f\in
L^{\infty}(\Omega,\mu):\,\|T^{\gamma_{n}}_{\alpha,w,\infty}(f)\|_{\infty}\geq
1\,\,\text{for all }n\\}$ is not $\sigma$-porous. In particular, the set of
all non-hypercyclic vectors of the sequence
$\\{T^{\gamma_{n}}_{\alpha,w,\infty}\\}_{n}$ is not $\sigma$-porous.

###### Proof.

Let $E:=\\{f\in L^{\infty}(\Omega,\mu):\,|f|\geq j_{\alpha,w}\\}$. Then, $E$
is not $\sigma$-porous thanks to Theorem 2.7. For each $f\in E$ and
$n\in\mathbb{N}$ we have

$\displaystyle\|T^{\gamma_{n}}_{\alpha,w,\infty}(f)\|_{\infty}$
$\displaystyle=\|\prod_{k=1}^{\gamma_{n}}(w\circ\alpha^{\gamma_{n}-k})\,(f\circ\alpha^{\gamma_{n}})\|_{\infty}$
$\displaystyle=\|\prod_{k=1}^{\gamma_{n}}(w\circ\alpha^{-k})\,f\|_{\infty}$
$\displaystyle\geq\|\prod_{k=1}^{\gamma_{n}}(w\circ\alpha^{-k})\,\chi_{A_{n}}\,f\|_{\infty}$
$\displaystyle\geq\|\prod_{k=1}^{\gamma_{n}}(w\circ\alpha^{-k})\,\chi_{A_{n}}\,j_{\alpha,w}\|_{\infty}$
$\displaystyle=1.$

This completes the proof. ∎

######  Corollary 3.14.

Let $G$ be a locally compact group and $\mu$ be a left Haar measure on $G$.
Let $a\in G$ and $w:G\to(0,\infty)$ be a bounded measurable function. Then, if

$\big{(}\frac{1}{w(a)w(a^{2})\ldots w(a^{n})}\big{)}_{n}\in
L^{\infty}(G,\mu),$

then the set of all non-hypercyclic vectors of the operator $T_{a,w,\infty}$
on $L^{\infty}(G,\mu)$ is not $\sigma$-porous.

######  Corollary 3.15.

If $(w_{n})_{n\in\mathbb{Z}}$ is a bounded sequence such that

$\big{(}\frac{1}{w_{1}\ldots w_{n}}\big{)}_{n}\in\ell^{\infty},$

then the set of all non-hypercyclic vectors of the sequence
$(T_{\gamma_{n},w})_{n}$ is not $\sigma$-porous in $\ell^{\infty}$.

######  Theorem 3.16.

Let $T_{\alpha,w,\infty}$ be the weighted composition operator on
$L^{\infty}(\Omega,\mu)$ and let $F\subseteq\Omega$ be a Borel set with
$0<\mu(F)<\infty$. Let there exists a constant $N>0$ such that for all $n\geq
N$,

$\alpha^{n}(F)\cap F=\varnothing,$ (3.2)

and

$\beta:=\inf\\{\prod_{k=1}^{n}(w\circ\alpha^{-k})(t):\,n\geq N,\,t\in F\\}\neq
0.$

Then, the set

$\\{f\in
L^{\infty}(\Omega,\mu):\,\|T^{n}_{\alpha,w,\infty}f-\chi_{F}\|_{\infty}\geq
1\,\text{for all }n\geq N\\}$

is not $\sigma$-porous in $L^{\infty}(\Omega,\mu)$.

###### Proof.

Let $\Gamma:=\\{f\in
L^{\infty}(\Omega,\mu):\,|f|\geq\frac{1}{\beta}\chi_{F}\\}$. Then by Theorem
2.7, $\Gamma$ is not $\sigma$-porous in $L^{\infty}(\Omega,\mu)$. Also, for
each $f\in\Gamma$ we have

$\displaystyle\|T^{n}_{\alpha,w,\infty}f-\chi_{F}\|_{\infty}$
$\displaystyle=\|\prod_{k=1}^{n}(w\circ\alpha^{n-k})\,(f\circ\alpha^{n})-\chi_{F}\|_{\infty}$
$\displaystyle=\|\prod_{k=1}^{n}(w\circ\alpha^{-k})\,f-\chi_{F}\circ\alpha^{n}\|_{\infty}$
$\displaystyle=\|\prod_{k=1}^{n}(w\circ\alpha^{-k})\,f-\chi_{\alpha^{n}(F)}\|_{\infty}$
$\displaystyle\geq\|\prod_{k=1}^{n}(w\circ\alpha^{-k})\,f\chi_{F}-\chi_{\alpha^{n}(F)}\chi_{F}\|_{\infty}$
$\displaystyle=\|\prod_{k=1}^{n}(w\circ\alpha^{-k})\,f\chi_{F}\|_{\infty}$
$\displaystyle\geq\beta\|f\chi_{F}\|_{\infty}\geq\beta\,\frac{1}{\beta}\,\|\chi_{F}\|_{\infty}=1.$

This completes the proof. ∎

######  Example 3.17.

Let $\Omega:=\mathbb{R}$ and $\mu$ be the Lebesgue measure. Put
$\alpha(t):=t-1$ for all $t\in\mathbb{R}$ and $F:=[0,1]$. If $w\in
C_{b}(\mathbb{R})$ such that $|w(t)|\geq 1$ for all $t\geq k>0$ and
$\inf\\{|w(t)|:\,t\in[0,1]\\}>0$, then the required conditions in the previous
theorem hold with respect to $F$.

With some similar proof, one can prove the next fact without the condition
(3.2).

######  Theorem 3.18.

Let $T_{\alpha,w,\infty}$ be the weighted composition operator on
$L^{\infty}(\Omega,\mu)$ and let $F\subseteq\Omega$ be a Borel set with
$0<\mu(F)<\infty$ such that

$\inf\\{\prod_{k=1}^{n}(w\circ\alpha^{-k})(t):\,n\geq N,\,t\in F\\}\neq 0.$

Then, the set

$\\{f\in L^{\infty}(\Omega,\mu):\,\|T^{n}_{\alpha,w,\infty}f\|_{\infty}\geq
1\,\text{for all }n\geq N\\}$

is not $\sigma$-porous in $L^{\infty}(\Omega,\mu)$. In particular, the set of
all non-hypercyclic vectors of the operator $T_{\alpha,w,\infty}$ is not
$\sigma$-porous.

In sequel, we find some application for Theorem 2.9 regarding hypercyclicity
of shift operators on $L^{p}(\mathbb{R},\tau)$.

######  Theorem 3.19.

Consider the weighted translation operator $T_{\alpha,w}$ on
$L^{p}(\mathbb{R},\tau)$ given by $T_{\alpha,w}f:=w\cdot(f\circ\alpha)$, where
$0<w,w^{-1}\in C_{b}(\mathbb{R})$ and $\alpha(t)=t+1$. For each
$n\in\mathbb{N}$ put $A_{n}:=[n,n+1]=\alpha^{n}([0,1])$. Set

$y_{\alpha,w}:=\sum_{n\in\mathbb{N}}\frac{1}{\inf_{t\in
A_{n}}\prod_{k=1}^{n}(w\circ\alpha^{-k})(t)}\chi_{A_{n}}$

and assume that $y_{\alpha,w}\in L^{p}(\mathbb{R},\tau)$ (in particular
$\inf_{t\in A_{n}}\prod_{k=1}^{n}(w\circ\alpha^{-k})(t)>0$ for all
$n\in\mathbb{N}$). Then, the set

$\\{f\in L^{p}(\mathbb{R},\tau):\,\|T_{\alpha,w}^{n}(f)\|_{p}\geq 1\,\text{for
all }n\in\mathbb{N}\\}$

is not $\sigma$-porous.

###### Proof.

By Theorem 2.9, the set

$E:=\\{f\in
L^{p}(\mathbb{R},\tau):\,\|f\chi_{A_{n}}\|_{p}\geq\|y_{\alpha,w}\chi_{A_{n}}\|_{p}\,\text{
for all }n\in\mathbb{N}\\}$

is not $\sigma$-porous, because it equals to

$\\{f\in
L^{p}(\mathbb{R},\tau):\,\|f\chi_{[m,m+1]}\|_{p}\geq\|y_{\alpha,w}\chi_{[m,m+1]}\|_{p}\,\text{for
all }m\in\mathbb{Z}\\},$

as $y_{\alpha,w}\chi_{[m,m+1]}=0$ for all $m\in\mathbb{Z}$ with $m\leq 0$.
Now, note that for each $f\in E$ and $n\in\mathbb{N}$,

$\displaystyle\|T_{\alpha,w}^{n}(f)\|_{p}^{p}$
$\displaystyle=\int_{\mathbb{R}}\left[\prod_{k=1}^{n}(w\circ\alpha^{n-k})(t)\right]^{p}\,|(f\circ\alpha^{n})(t)|^{p}\,d\tau$
$\displaystyle=\int_{\mathbb{R}}\left[\prod_{k=1}^{n}(w\circ\alpha^{-k})(t)\right]^{p}\,|f(t)|^{p}\,d\tau$
$\displaystyle\geq\int_{A_{n}}\left[\prod_{k=1}^{n}(w\circ\alpha^{-k})(t)\right]^{p}\,|f(t)|^{p}\,d\tau$
$\displaystyle\geq\inf_{t\in
A_{n}}\left[\prod_{k=1}^{n}(w\circ\alpha^{-k})(t)\right]^{p}\|y_{\alpha,w}\chi_{A_{n}}\|_{p}^{p}$
$\displaystyle=\inf_{t\in
A_{n}}\left[\prod_{k=1}^{n}(w\circ\alpha^{-k})(t)\right]^{p}\,\frac{1}{\inf_{t\in
A_{n}}\left[\prod_{k=1}^{n}(w\circ\alpha^{-k})(t)\right]^{p}}\,\tau(A_{n})=1.$

∎

Assume now that there exists some $l\in\mathbb{Z}$ such that

$\beta:=\inf\\{\prod_{k=1}^{n}(w\circ\alpha^{-k})(t):\,t\in[l,l+1],\,n\in\mathbb{N}\\}>0.$

Put

$F:=\\{f\in
L^{p}(\mathbb{R},\tau):\,\|f\chi_{[m,m+1]}\|_{p}\geq\|\frac{1}{\beta}\chi_{[l,l+1]}\chi_{[m,m+1]}\|_{p}\,\text{
for all }m\in\mathbb{Z}\\}.$

So by Theorem 2.9, $F$ is not $\sigma$-porous. For every $f\in F$ we have

$\displaystyle\|T_{\alpha,w}^{n}(f)\|_{p}^{p}$
$\displaystyle=\int_{\mathbb{R}}\left[\prod_{k=1}^{n}(w\circ\alpha^{n-k})(t)\right]^{p}\,|(f\circ\alpha^{n})(t)|^{p}\,d\tau$
$\displaystyle=\int_{\mathbb{R}}\left[\prod_{k=1}^{n}(w\circ\alpha^{-k})(t)\right]^{p}\,|f(t)|^{p}\,d\tau$
$\displaystyle\geq\int_{[l,l+1]}\left[\prod_{k=1}^{n}(w\circ\alpha^{-k})(t)\right]^{p}\,|f(t)|^{p}\,d\tau$
$\displaystyle\geq 1.$

Hence, the set

$\\{f\in L^{p}(\mathbb{R},\tau):\,\|T_{\alpha,w}^{n}(f)\|_{p}\geq 1\,\text{for
all }n\in\mathbb{N}\\}$

is not $\sigma$-porous.

Next, suppose that $\alpha$ is an aperiodic function on $\mathbb{R}$ (this
means that for each compact set $C\subset\mathbb{R}$, there exists a constant
$N>0$ such that $\alpha^{n}(C)\cap C=\varnothing$ for all $n\geq N$) and
$\beta>0$, where $\beta$ is as above. Then, the set

$\\{f\in
L^{p}(\mathbb{R},\tau):\,\|T_{\alpha,w}^{n}(f)-\chi_{[l,l+1]}\|_{p}\geq
1\,\text{ for all }n\geq N\\}$

is not $\sigma$-porous. Indeed, for all $f\in F$, and $n\geq N$ we have

$\|T_{\alpha,w}^{n}(f)-\chi_{[l,l+1]}\|_{p}\geq\|\prod_{k=1}^{n}\,(w\circ\alpha^{-k})\,f\chi_{[l,l+1]}\|_{p}$

by the similar calculations as in the proof of Theorem 3.16. However,

$\|\prod_{k=1}^{n}\,(w\circ\alpha^{-k})\,f\chi_{[l,l+1]}\|_{p}\geq\beta\,\|f\chi_{[l,l+1]}\|_{p}\geq
1.$

## References

  * [1] F. Bayart, _Porosity and hypercyclic operators_ , Proc. Amer. Math. Soc. 133(11) (2005) 3309-3316.
  * [2] F. Bayart and É. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Math. 179, Cambridge University Press, Cambridge, 2009.
  * [3] C. L. Belna, M. J. Evans and P.D.Humke, _Symmetric and ordinary differentiation_ , Proc. Amer. Math. Soc. 72(2) (1978) 261-267.
  * [4] W.R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, De Kruyter, Berlin, 1995.
  * [5] C-C. Chen and C-H. Chu, Hypercyclic weighted translations on groups, Proc. Amer. Math. Soc. 139 (2011) 2839-2846.
  * [6] C-C. Chen, S. Öztop and S. M. Tabatabaie, Disjoint dynamics on weighted Orlicz spaces, _Disjoint dynamics on weighted Orlicz spaces_ , Complex Anal. Oper. Theory, 14(72) (2020). https://doi.org/10.1007/s11785-020-01034-x
  * [7] C.-C. Chen and S. M. Tabatabaie, _Chaotic operators on hypergroups_ , Oper. Mat. 12(1) (2018) 143-156.
  * [8] E. P. Dolženko, _Boundary properties of arbitrary functions_ , Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967) 3-14.
  * [9] G.B. Folland, _Real Analysis, Modern Techniques and Their Applications; Second Edition,_ John Wiley and Sons, Inc. New York (1999).
  * [10] K-G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math. 139 (2000) 47-68.
  * [11] K-G. Grosse-Erdmann and A. Peris, Linear Chaos, Universitext, Springer, 2011.
  * [12] R. I. Jewett, _Spaces with an abstract convolution of measures_ , Adv. Math., 18 (1975) 1-101.
  * [13] V. Kumar and S. M. Tabatabaie, _Hypercyclic sequences of weighted translations on hypergroups_ , Semigroup Forum, 103 (2021) 916–934.
  * [14] D. Preiss and L. Zajíček, _Fréchet differentiation of convex functions in a Banach space with a separable dual_ , Proc. Amer. Math. Soc. 91(2) (1984) 202–204.
  * [15] H. Salas, _Hypercyclic weighted shifts_ , Trans. Amer. Math. Soc. 347 (1995) 993-1004.
  * [16] Y. Sawano, S. M. Tabatabaie and F. Shahhoseini, _Disjoint Dynamics of Weighted Translations on Solid Spaces_ , Topology Appl. 298, 107709, 14 pp. (2021) DOI:10.1016/J.TOPOL.2021.107709
  * [17] S. M. Tabatabaie and S. Ivkovic, _Linear dynamics of cosine operator functions on solid Banach function spaces_ , Positivity, 25 (2021) 1437–1448.
  * [18] A. Villani, Another note on the inclusion $L^{p}(\mu)\subset L^{q}(\mu)$, Amer. Math. Monthly, 92 (1985) 485–487.
  * [19] L. Zájiček, _Porosity and $\sigma$-porous_, Real Anal. Exchange 13 (1987/1988) 314-350.
  * [20] L. Zajíček, _Small non- $\sigma$-porous sets in topologically complete metric spaces_, Colloq. Math. 77(2) (1998) 293-304.
  * [21] L. Zájiček, _On $\sigma$-porous sets in abstract spaces_, Abstr. Appl. Anal. 5 (2005) 509-534.