# Attack-Resilient Distributed Convex Optimization of Linear Multi-Agent
Systems Against Malicious Cyber-Attacks over Random Digraphs

Zhi Feng and Guoqiang Hu  This work was supported by Singapore Ministry of
Education Academic Research Fund Tier 1 RG180/17(2017-T1-002-158) and in part
by the Wallenberg-NTU Presidential Postdoctoral Fellow Start-Up Grant. Z. Feng
and G. Hu are with the School of Electrical and Electronic Engineering,
Nanyang Technological University, Singapore 639798 (E-mail:
<EMAIL_ADDRESS>gqhu@ntu.edu.sg).

###### Abstract

This paper addresses a resilient exponential distributed convex optimization
problem for a heterogeneous linear multi-agent system under Denial-of-Service
(DoS) attacks over random digraphs. The random digraphs are caused by
unreliable networks and the DoS attacks, allowed to occur aperiodically, refer
to an interruption of the communication channels carried out by the
intelligent adversaries. In contrast to many existing distributed convex
optimization works over a prefect communication network, the global optimal
solution might not be sought under the adverse influences that result in
performance degradations or even failures of optimization algorithms. The
aforementioned setting poses certain technical challenges to optimization
algorithm design and exponential convergence analysis. In this work, several
resilient algorithms are presented such that a team of agents minimizes a sum
of local non-quadratic cost functions in a safe and reliable manner with
global exponential convergence. Inspired by the preliminary works in [15, 17,
16, 18], an explicit analysis of frequency and duration of attacks is
investigated to guarantee exponential optimal solutions. Numerical simulation
results are presented to demonstrate the effectiveness of the proposed design.

###### Index Terms:

Distributed convex optimization, Linear multi-agent system, Heterogeneous
network, Random digraph, DoS attack.

##  I Introduction

Distributed convex optimization over a multi-agent system has attracted
growing attention over the last decade due to its potential applications
involving source localization in sensor networks [1], resource allocation in
multi-cell networks [2], energy and thermal comfort optimization in smart
building [3], economic dispatch in smart grid [4], etc. The gradient-based
distributed algorithms have been widely provided in the existing works, which
build on either continuous-time or discrete-time agent dynamics to seek an
optimal solution [6, 7, 8, 5, 10, 11, 9, 12, 13]. Each agent needs to
calculate a global optimizer based on the information exchange over a
communication network. The network security plays a fundamental yet important
role in information transmission. Unfortunately, due to malicious attacks,
(e.g., DoS attacks, deception attacks (false data injections, replay attacks),
disclosure attacks (eavesdropping), and Byzantine attacks (faulty agents)),
the secure network environment is hardly guaranteed in practice. The malicious
attacks interrupt, incorrect, or tamper transmitting information so that
efficiency of distributed optimization algorithm is degraded significantly or
even failed. In light of wide applications of distributed optimization
algorithms in cyber-physical systems (safety-critical), and inspired by
studies of security issues in consensus works (e.g., [14, 15, 17, 16, 18, 20,
19]), it is highly desirable to determine how resilient distributed
optimization algorithms are against cyber-attacks. Motivated by the
observation, we aim to address resilient problems of distributed optimization
with certain resilience against DoS attacks over unreliable networks.

###  I-A Related Literature Review

Continuous-Time Distributed Convex Optimization: paralleled with the discrete-
time optimization schemes, the continuous-time distributed optimization
algorithms have attracted much attention due to the well-developed continuous-
time analysis techniques in control and the real-world cyber-physical system.
In particular, the Zero-gradient-sum [5] and Newton-Raphson [6] algorithms are
designed based on the positive and bounded Hessian of the local cost
functions, while the Lagrangian-based algorithm based on the local gradients
is adopted in [7]. The penalty-based optimization strategies are developed in
[5] and [8], while the works investigate the multi-agent system with single-
integrator dynamics only. The authors in [9] further consider distributed
optimization of second-order multi-agent systems. Adaptive schemes are
designed in [10] to achieve distributed optimization of linear agents via
nonsmooth signum functions and gradients that satisfy special structures. All
above designs require continuous communication. To remove this requirement,
the time-based and event-triggered based strategies are presented in [11, 12,
13] over undirected and connected graphs. Besides, these algorithms need known
initial states of each agent, which is difficult to verify in practice.

Resilient Distributed Convex Optimization: the closest related works on this
topic have been recently published in [21, 22, 23, 24] in which the optimal
solutions are obtained for first-order discrete-time multi-agent systems under
Byzantine attacks (faulty nodes). In particular, the authors in [21] present
the resilient optimization algorithm by removing $F$ (maximum amount of
tolerable faults) nodes’ largest and smallest states at each iteration, such
that the optimal solution converges to a convex hull of the set of all non-
faulty nodes’ local minimizers. This algorithm is adopted in [22] to deal with
constrained optimization problems. The local filtering consensus-based
algorithms against $F$ faulty nodes are developed in [23], where the optimal
solution is achieved under the $r$-robust condition. The modified algorithm
named RDO-T is developed in [24], where the number of faulty agents is allowed
to be any large. These aforementioned works assume that the faulty nodes and
the removal of $F$ states are known a prior to the designer. Moreover, these
algorithms rely on the network connectivity of the complete graph or the
undirected and connected graph.

###  I-B Main Contributions

This paper is concerned with a resilient study of distributed optimization
algorithms against cyber-attacks over random digraphs. The malicious DoS
attacks, allowed to occur aperiodically, aim to interrupt information
transmission. In addition to DoS attacks, the random digraphs are induced by
unreliable networked constraints. The main contributions of this paper are
summarized as follows. (1) inspired by our designs in [15, 17, 16, 18], this
is the first to present time-based and event-based distributed optimization
algorithms to achieve resilient distributed optimization of heterogeneous
linear multi-agent networks against DoS attacks over random digraphs. The
proposed algorithms that rely on a consensus-based gradient strategy with a
switching system method to constrain DoS attacks, are capable of exactly
seeking the optimal solutions under attacks over random digraphs. The global
exponential convergence of the proposed algorithm can be ensured, provided
that the frequency and duration of attacks satisfy certain bounded conditions;
(2) these proposed resilient distributed optimization algorithms avoid
continuous-time communication in many existing works (see [6, 7, 8, 5, 9,
10]). The proposed dynamic event-based distributed optimization scheme is
proven to be free of Zeno behavior, and avoid fixed and periodic transmissions
used in the time-based scheme; and (3) in contrast to related optimization
works that require known positive Hessians of the local cost functions [5] and
[6], known gradients satisfying special linear structures [10], known initial
agent states [11, 12, 13], the proposed algorithms in this work remove those
limitations to facilitate practical applications. Moreover, compared to
optimization works in [21, 22, 23, 24] that consider Byzantine faults on nodes
and require the removal of their states to be known a prior, these DoS attacks
on communication networks are time-sequence based and allowed to occur
aperiodically. Another contribution of this paper is that unlike works in [6,
7, 8, 5, 10, 11, 9, 12, 13] and [21, 22, 23, 24] over the undirected graph or
weight-balanced digraph, the studied graphs are directed and time-varying, and
under cyber-attacks, the graphs can be even disconnected or totally paralyzed.

##  II Preliminaries and Problem Formulation

###  II-A Notation

Denote $\mathbb{R}$, $\mathbb{R}^{n}$, and $\mathbb{R}^{n\times m}$ as the
sets of the real numbers, real $n$-dimensional vectors and real $n\times m$
matrices, respectively. Let $\mathbb{N}^{+}$ be the set of positive natural
numbers. Let 0 (1) be the vector with all zeros (ones) with proper dimensions.
Let col$(x_{1},...,x_{n})$ and diag$\\{a_{1},...,a_{n}\\}$ be a column vector
with the entries $x_{i}$ and a diagonal matrix with the entries $a_{i}$,
$i=1,2,\cdots,n$, respectively. $\otimes$ and $\left\|\cdot\right\|$ represent
the Kronecker product and Euclidean norm, respectively. For a real matrix
$M=M^{T}$, let $M>0$ be positive definite. Let $\lambda_{\min}(M)$,
$\lambda_{\max}(M)$ be its minimum and maximum eigenvalues, respectively.
Besides, $\sigma_{\max}(M)$ represents the maximum singular value of a matrix
$M$. For a differentiable function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$,
$\triangledown f$ is the gradient of $f$, and $f$ is strongly convex over a
convex set $\mathbb{R}^{n}$, if $(x-y)^{T}(\triangledown f(x)-\triangledown
f(y))>\iota||x-y||^{2}$ for $\forall x,y\in\mathbb{R}^{n},x\neq y$ and a
scalar $\iota>0$. $f$ is locally Lipschitz at $x\in\mathbb{R}^{n}$ if there
exists a neighbourhood $\mathcal{W}$ and a scalar $l$ so that
$||f(x)-f(y)||\leq l||x-y||$ for $\forall x,y\in\mathcal{W}$; $f$ is locally
Lipschitz on $\mathbb{R}^{n}$ if it is locally Lipschitz at $x$ for $\forall
x\in\mathbb{R}^{n}$.

###  II-B Graph Theory

Static Digraph: let $\mathcal{G}$ $=$
$\left\\{\mathcal{V},\mathcal{E}\right\\}$ be a graph and $\mathcal{V}$ $\in$
$\left\\{1,...,N\right\\}$ be the set of vertices. The set of edges is denoted
as $\mathcal{E}$ $\subseteq$ $\mathcal{V\times V}$. $\mathcal{N}_{i}$ $=$
$\left\\{j\in\mathcal{V\mid}(j,i)\in\mathcal{E}\right\\}$ is the neighborhood
set of vertex $i$. For a directed graph $\mathcal{G}$, $(i,j)\in\mathcal{E}$
means that the information of node $i$ is accessible to node $j$, but not
conversely. A matrix $A$ $=$ $\left[a_{ij}\right]$ is the adjacency matrix of
$\mathcal{G}$, where $a_{ij}>0$ if $(j,i)\in\mathcal{E}$, else $a_{ij}=0$. A
matrix $\mathcal{L}=[l_{ij}]$ is called the Laplacian matrix of $\mathcal{G}$,
where $l_{ii}=\sum^{N}_{j=1}a_{ij}$ and $l_{ij}=-a_{ij}$, $i\neq j$.

Markovian Random Digraph: let
$\mathcal{G}(t)=\left\\{\mathcal{V},\mathcal{E}_{r(t)}\right\\}$ be a time-
varying digraph with $\mathcal{E}_{r(t)}$ being a set of edges, and
$r(t):[0,\infty)\\\ \rightarrow\mathcal{S}=\\{1,2,...,s\\}$ is a piecewise
constant function with $\mathcal{S}$ being an index set of possible digraphs.
The piecewise-constant function $r(t)$ is a Markovian signal. $\mathcal{A}(t)$
$=[a_{ij}^{r(t)}]$ is the adjacency matrix, where $a_{ij}^{r(t)}>0$ if
$(j,i)\in\mathcal{E}_{r(t)}$, else $a_{ij}^{r(t)}=0.$ The neighboring set is
denoted by
$\mathcal{N}_{i}(t)=\left\\{j\in\mathcal{V},(j,i)\in\mathcal{E}_{r(t)}\right\\}$.
Denote $\mathcal{L}(t)=[l^{r(t)}_{ij}]$, where
$l^{r(t)}_{ii}=\sum^{N}_{j=1}a^{r(t)}_{ij}$ and
$l^{r(t)}_{ij}=-a^{r(t)}_{ij}$, $i\neq j$.

###  II-C Heterogeneous Linear Multi-Agent Model

Consider a multi-agent system consisting of $N$ agents governed by the
following heterogeneous linear dynamics:

$\dot{x}_{i}(t)=A_{i}x_{i}(t)+B_{i}u_{i}(t),\ y_{i}(t)=C_{i}x_{i}(t),\
i\in\mathcal{V},$ (1)

where $x_{i}\in\mathbb{R}^{n_{i}}$ denotes the state of agent $i$,
$u_{i}\in\mathbb{R}^{p_{i}}$ denotes the control input of agent $i$,
$y_{i}\in\mathbb{R}^{q}$ is its output, and $A_{i}\in\mathbb{R}^{n_{i}\times
n_{i}}$, $B_{i}\in\mathbb{R}^{n_{i}\times p_{i}}$,
$C_{i}\in\mathbb{R}^{q\times n_{i}}$ are constant system matrices. Suppose
that the matrix pair $(A_{i},B_{i})$ is controllable and

$\hskip 20.00003pt\text{rank}\left[\begin{array}[]{cc}C_{i}B_{i}&0_{q\times
p_{i}}\\\ -A_{i}B_{i}&B_{i}\\\ \end{array}\right]=n_{i}+q,\ i\in\mathcal{V}.$
(2)

General Distributed Optimization Problem: design a distributed scheme
$u_{i}(t)$ for (1) using local interaction and information over a
communication network so that the output of all agents can reach the optimal
state $\theta^{*}$ that minimize:
$\mathrm{F}(\theta)=\sum_{i=1}^{N}f_{i}(\theta),\ \theta\in\mathbb{R}^{q}$,
where $f_{i}(\theta):\mathbb{R}^{q}\rightarrow\mathbb{R}$ is the private cost
function known to agent $i$ only, and $\theta$ is the global decision variable
to be optimized. From [7], it is equivalent to solve:

$\begin{split}&\min\limits_{y\in\mathbb{R}^{Nq}}\
\tilde{f}(y)=\sum_{i=1}^{N}f_{i}(y_{i}),\ y_{i}\in\mathbb{R}^{q},\\\
\text{subject to}\ &(\ref{Dynamics})\ \text{and}\ y_{i}=y_{j},\ \forall
i,j\in\mathcal{V}=\\{1,2,\cdots,N\\},\end{split}$ (3)

where $y_{i}\in\mathbb{R}^{q}$ is a local estimate on the optimal solution
$\theta^{*}$, and $y=\text{col}(y_{1},\cdots,y_{N})$ is its stack vector of
all estimates.

To solve the problem, two standard assumptions are introduced.

###### Assumption 1

There exists $y^{*}=1_{q}\otimes\theta^{*}$ that minimizes the team cost
function, i.e., $\tilde{f}(y^{*})=\min_{\theta\in\mathbb{R}^{q}}F(\theta)$.

###### Assumption 2

Each function $f_{i}:\mathbb{R}^{q}\rightarrow\mathbb{R}$ is differentiable,
strongly convex, and its gradient is locally Lipschitz on $\mathbb{R}^{q}$.

###### Remark 1

Assumption 1 guarantees the optimal solution set is nonempty. By Assumption 2,
$||\triangledown f_{i}(x_{i})-\triangledown f_{i}(y_{i})||\leq
l_{i}||\triangledown x_{i}-y_{i}||,\ \forall x_{i},y_{i}\in\mathbb{R}^{q}$,
where $\triangledown f_{i}(x_{i})$ and $\triangledown f_{i}(y_{i})$ are the
gradients, and $l_{i}>0$ is the Lipschitz constant.

###  II-D Unreliable Random Communication Network

In large-scale cyber-physical systems, the wireless communication may not be
reliable due to certain physical uncertainties, e.g., failures, quantization
errors, and packet losses in a digital communication [25]. We consider an
unreliable network consisting of $N$ agents whose integrated wireless
communication links are time-varying and failure-prone with certain
probabilities. As considered in [25], a random Markov chain model can be
adopted to capture this situation. This random process describes dynamic
changes of topologies. Let $r(t)$ be a right-continuous homogeneous Markovian
process on the probability space taking values in a finite state space
$\mathcal{S}=\\{1,2,...,s\\}$ with an infinitesimal generator
$\Upsilon=(\gamma_{pq})$, given by $\mathrm{P}_{pq}(t)=\mathrm{\Pr
ob}\\{r(t+h)=q|r(t)=p\\}=\gamma_{pq}h+o(h)$, if $p\neq q$, else,
$1+\gamma_{pp}h+o(h)$, where $\gamma_{pq}\geq 0$ is the transition rate from
the state $p$ to the state $q$, while $\gamma_{pp}=-\sum^{s}_{q=1,p\neq
q}\gamma_{pq},$ and $o(h)$ satisfies: $\lim_{h\rightarrow 0}o(t)/h=0$.

###  II-E Malicious DoS Attack Model

As studied in [18], the DoS attack refers to the interruption of the
communication channels. Without loss of generality, suppose that there exists
an attack sequence $\\{a_{m}\\}_{m\in\mathbb{N}}$ when a DoS attack is lunched
at $a_{m}$, and let the duration of this attack be $\tau_{m}\geq 0$. Then, the
$m$-th DoS attack strategy can be generated with $\mathcal{A}_{m}$ $=$
${a_{m}}\cup[a_{m},a_{m}+\tau_{m})$ with $a_{m+1}>a_{m}+\tau_{m}$ for all
$m\in\mathbb{N}.$ Thus, for given $t\geq t_{0}$, the sets of time instants
where communication is denied (unsuccessful) are described by

$\Xi_{a}(t_{0},t)=\cup\ \mathcal{A}_{m}\cap\ [t_{0},t],\ m\in\mathbb{N},$ (4)

which implies that on the interval $[t_{0},t]$, the sets of time instants
where communication is allowed are:
$\Xi_{s}(t_{0},t)=[t_{0},t]\setminus\Xi_{a}(t_{0},t)$. In words,
$|\Xi_{a}(t_{0},t)|$ and $|\Xi_{s}(t_{0},t)|$ represent the total lengths of
the attacker being active and sleeping over $[t_{0},t]$, respectively.

######  Definition 1 (Attack Frequency)

For any $T_{2}>T_{1}\geq t_{0}$, let $N_{a}(T_{1},T_{2})$ denote the total
number of DoS attacks over $[T_{1},T_{2})$. Then,
$F_{a}(T_{1},T_{2})=\frac{N_{a}(T_{1},T_{2})}{T_{2}-T_{1}}$ denotes the attack
frequency over $[T_{1},T_{2})$ for $\forall T_{2}>T_{1}\geq t_{0}$, where
there exists scalars $N_{0},T_{f}>0$ such that $N_{a}(T_{1},T_{2})\leq
N_{0}+(T_{2}-T_{1})/T_{f}.$

######  Definition 2 (Attack Duration)

For any $T_{2}>T_{1}\geq t_{0},$ let $|\Xi_{a}(T_{1},T_{2})|$ be the total
time interval under attacks over $[T_{1},T_{2}).$ The attack duration over
$[T_{1},T_{2})$ is defined as: there exist scalars $T_{0}\geq 0$, $T_{a}>1$ so
that $|\Xi_{a}(T_{1},T_{2})|\leq T_{0}+(T_{2}-T_{1})/T_{a}.$

###### Remark 2

As mentioned in the preliminary works [15, 17, 16, 18] for multi-agent systems
under malicious cyber-attacks, Definitions 1 and 2 are firstly introduced in
[15] to specify DoS attack signals in terms of the frequency and time-ratio
constraints.

###  II-F Main Objective

The main objective of the paper is to solve a resilient distributed
optimization issue of heterogeneous linear multi-agent systems.

###### Problem 1

(Resilient Distributed Convex Optimization against DoS Attacks over Random
Digraphs)
Develop a resilient distributed algorithm $u_{i}$ so that the output of each
agent cooperatively seeks $\theta^{*}$ under DoS attacks over random digraphs.
The problem in (3) is reformulated as

$\begin{split}&\min\limits_{y\in\mathbb{R}^{Nq}}\
\tilde{f}(y)=\sum_{i=1}^{N}f_{i}(y_{i}),\ y_{i}\in\mathbb{R}^{q},\\\
\text{subject to}\
&\dot{x}_{i}(t)=A_{i}x_{i}(t)+B_{i}u_{i}(t),y_{i}(t)=C_{i}x_{i}(t),\\\
\text{and}\ &(\mathcal{L}(t)\otimes I_{q})y=\textbf{0},\
t\in\Xi_{s}(t_{0},t)\cup\Xi_{a}(t_{0},t).\end{split}$ (5)

###### Remark 3

Solving resilient exponential distributed optimization in Problem 1 is much
more challenging in fourfold: (1) Agent dynamics: we investigate general
linear multi-agent systems with nonidentical dynamics and dimensions; (2)
Communication: the unreliable communication network leads to time-varying
directed graphs and moreover, the interruptions of communication channels
caused by DoS attacks make existing distributed optimization algorithms
degraded or totally failed; (3) Assumptions: the local gradient is nonlinear
without needing to satisfy special structures in many existing works and only
the local Lipschitz property is required; and 4) Design requirements: propose
the initialization-free resilient distributed optimization algorithms with the
discrete-time communication nature to provide global exponential convergence
and resilience features against DoS attacks over random digraphs. The existing
designs in [6, 7, 8, 5, 10, 11, 9, 12, 13] cannot be directly applied.

##  III Resilient Exponential Distributed Optimization Against Dos ATTACKS
over Random Networks

Due to unreliable networks described in the subsection II-D, the underlying
topologies are time-varying and random. As stated in [26], define that a
sequence of Laplacian matrices $\\{\mathcal{L}(t)\\}$ admits a common
stationary distribution $\pi>0$ if $\mathcal{L}(t)\pi=\textbf{0}$. Let
$\mathcal{L}_{s}(t)$ be a mirror of $\mathcal{L}_{un}(t)$, i.e.,
$\mathcal{L}_{s}(t)=(\mathcal{L}_{un}+\mathcal{L}_{un}^{T})/2$, where
$\mathcal{L}_{un}=\sum_{p=1}^{N}L_{p}(t)$ is Laplacian matrix of a union of
digraphs. Define the minimum cut of $\\{\mathcal{L}(t)\\}$ as
$l_{c}(t)=\min_{\mathcal{S}\subset\mathcal{V},\mathcal{S}\neq\emptyset}\sum_{i\in\mathcal{S},j\in\bar{\mathcal{S}}}\mathcal{L}_{s}(t)$,
where $\bar{\mathcal{S}}$ is the complement of $\mathcal{S}$. Then, we say
that this sequence of $\\{\mathcal{L}(t)\\}$ has a minimum cut $c$, if
$l_{c}(t)\geq c>0$.

###### Assumption 3

Assume that the sequence of Laplacian matrices $\\{\mathcal{L}(t)\\}$ has a
stationary distribution $\pi$ with a minimum cut.

###### Lemma 1

By Assumption 3, let $\mathcal{L}(t)$ be the Laplacian matrix with a
stationary distribution $\pi>0$. Then, there exists a weighted matrix
$Q(t)=\mathcal{L}(t)\Pi+\Pi\mathcal{L}^{T}(t)$, $\Pi=\text{diag}\\{\pi\\}$, so
that for $\pi_{min}=\min_{p\in\mathcal{V}}\\{\pi_{p}\\}$ and a vector
$\xi\in\mathbb{R}^{N}$ satisfying $\pi^{T}\xi=0$, we have

$\xi^{T}Q(t)\xi=\sum_{i=1}^{N}\sum_{j\in\mathcal{N}(t)}Q_{ij}(t)(\xi_{i}-\xi_{j})^{2}\geq\frac{\pi_{min}c}{N^{2}}||\xi||^{2}.$
(6)

###### Proof:

without loss of generality, we suppose that $||\xi||\neq 0$, where it
satisfies $\xi_{1}\leq\xi_{2}\leq\cdots\leq\xi_{N}$. Since $||\xi||\neq 0$ and
$\pi^{T}\xi=0$, we have $\xi_{1}<0$ and $\xi_{N}>0$. Define
$i_{*}\in\text{argmax}_{i\geq 2}|\xi_{i}-\xi_{i-1}|$ and
$\varepsilon=|\xi_{i_{*}}-\xi_{i_{*}-1}|$. Then, we obtain that

$\displaystyle\xi^{T}Q(t)\xi$
$\displaystyle=\sum_{i=1}^{N}\sum_{i<j}Q_{ij}(t)(\xi_{i}-\xi_{j})^{2}\geq\sum_{i\leq
i_{*}}\sum_{i_{*}<j}Q_{ij}(t)\varepsilon^{2}$
$\displaystyle\geq\pi_{min}l_{c}(t)\varepsilon^{2}\geq\frac{\pi_{min}c}{N^{2}}||\xi||^{2},$
(7)

where the fact $|\xi_{i}|\leq N\varepsilon$ is used to get the last
inequality. ∎

###  III-A Distributed Time-Based Strategy Against DoS Attacks

Based on the DoS attack model in the subsection II-E, without loss of
generality, suppose that there exists an infinite sequence of uniformly
bounded non-overlapping time intervals $[t_{k},t_{k+1}),k=0,1,2\cdots$. Each
agent $i$ thus updates its controller $u_{i}(t)$ over time intervals
$[t_{k},t_{k+1})$. Then, in the presence of the DoS attacks, we denote the
attack-induced new time sequence as
$0=t_{0}=a_{0}<t_{1}<a_{1}<a_{1}+\tau_{1}\leq
t_{2}<\cdots<t_{k}<a_{k}<a_{k}+\tau_{k}\leq t_{k+1}<\cdots$.

Next, we propose the following distributed algorithm to analyze the effect of
DoS attacks over random digraphs.

$\displaystyle u_{i}$
$\displaystyle=-K_{i}x_{i}-(U_{i}-K_{i}X_{i})\varrho_{i}+W_{i}\vartheta_{i},$
(8a) $\displaystyle\dot{\varrho}_{i}$
$\displaystyle=\vartheta_{i}=-\triangledown f_{i}(y_{i})-\beta e_{\varrho
zi}-\alpha\beta e_{yi},$ (8b) $\displaystyle\dot{z}_{i}$
$\displaystyle=\alpha\beta e_{yi},t\in[t_{k},t_{k+1}),\ i\in\mathcal{V},$ (8c)

where $\alpha,\beta\in\mathbb{R}$ are positive constant gains, $\vartheta_{i}$
is the intermediate variable, and $e_{\varrho zi}$, $e_{yi}$ are consensus
errors under DoS attacks

$e_{\varrho
zi}=\left\\{\begin{array}[]{c}\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\varrho_{i}-\varrho_{j}+z_{i}-z_{j}),\text{
if}\ t\in[t_{k},a_{k}),\\\ \textbf{0},\text{ if}\ t\in[a_{k},t_{k+1}),\
k=0,1,2\cdots,\end{array}\right.$ (9)
$e_{yi}=\left\\{\begin{array}[]{c}\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(y_{i}-y_{j}),\text{
if}\ t\in[t_{k},a_{k}),\\\ \textbf{0},\text{ if}\ t\in[a_{k},t_{k+1}),\
k=0,1,2\cdots,\end{array}\right.$ (10)

where $\varrho_{i},z_{i}\in\mathbb{R}^{q}$ denote the auxiliary states. In
(8), $K_{i}\in\mathbb{R}^{p_{i}\times n_{i}}$ and $(U_{i},W_{i},X_{i})$ is the
solution to the following equations:

$B_{i}U_{i}=A_{i}X_{i},\ B_{i}W_{i}=X_{i},\ C_{i}X_{i}=I_{q},\
i\in\mathcal{V}.$ (11)

###### Remark 4

The proposed algorithm in (8)-(11) has properties: 1) it is distributed as
each agent communicates with its neighbors only; 2) each agent transmits
exchange information over random digraphs, while in the presence of DoS
attacks, the information is interrupted; and 3)
$\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(z_{i}-z_{j})$ in $e_{\varrho zi}$ is
relative internal state information, which is used to avoid zero-sum initial
requirements. Besides, the solution of (11) is ensured by (2).

Define $\tilde{\mathcal{L}}(t)=\mathcal{L}(t)\otimes I_{q}$. Substituting
(8)-(10) into (1) gives the following closed-loop system under attacks and
random digraphs.

$\dot{\chi}=\left\\{\begin{array}[]{l}\left[\begin{array}[]{ccc}(A-BK)x+BW\vartheta-B(U-KX)\varrho\\\
-\triangledown\tilde{f}(y)-\alpha\beta\tilde{\mathcal{L}}(t)y-\beta\tilde{\mathcal{L}}(t)(\varrho+z)\\\
\alpha\beta\tilde{\mathcal{L}}(t)y\end{array}\right],\\\ \text{ if}\
t\in[t_{k},a_{k}),k=0,1,2\cdots\\\
\left[\begin{array}[]{ccc}(A-BK)x+BW\vartheta-B(U-KX)\varrho\\\
-\triangledown\tilde{f}(y)\\\ \textbf{0}\end{array}\right],\\\ \text{ if}\
t\in[a_{k},t_{k+1}),k=0,1,2\cdots\end{array}\right.$ (12)

where $\chi=\text{col}(x,\varrho,z)$ and
$x,\varrho,z,\vartheta,\triangledown\tilde{f}(y)$ are stacked vectors of
$x_{i}$, $\varrho_{i}$, $z_{i}$, $\vartheta_{i}$, $\triangledown
f_{i}(y_{i})$, and $A,B,C,K,U,W,X$ are stacked block diagonal matrices of
$A_{i}$, $B_{i}$, $C_{i}$, $K_{i}$, $U_{i}$, $W_{i}$, $X_{i}$, respectively.

Next, we present the exponential resilient distributed optimization result
against DoS attacks and random digraphs.

###### Theorem 1

Given Assumptions 1-3, Problem 1 is solvable for any $x_{i}(0)$,
$\varrho_{i}(0)$, and $z_{i}(0)$ under the proposed resilient distributed
optimization algorithm in (8)-(10), provided that $K_{i}$ is chosen so that
$A_{i}-B_{i}K_{i}$ is Hurwitz, $U_{i},W_{i},X_{i}$ are solutions to (11), and
for scalars $\lambda_{a},\lambda_{b},u>0$ to be determined later, the
following two arrack-related conditions are satisfied:

(1). There exists constants $\eta^{\ast}\in(0,\lambda_{a})$ and $\mu\geq 1$ so
that $T_{f}$ in the attack frequency Definition 1 satisfies the condition:

$T_{f}>T^{*}_{f}=\ln(\mu)/\eta^{\ast},$ (13)

(2). There exist constants $\lambda_{a}$, $\lambda_{b}>0$ such that $T_{a}$ in
the attack duration Definition 2 satisfies the condition:

$\hskip
20.00003ptT_{a}>T^{*}_{a}=(\lambda_{a}+\lambda_{b})/(\lambda_{a}-\eta^{\ast}).$
(14)

Moreover, the estimated states converge exponentially, i.e.,

$\|\tilde{\chi}(t)\|^{2}\leq\varsigma
e^{-\eta(t-t_{0})}||\tilde{\chi}(t_{0})||^{2},\ \forall t_{0}\geq 0,$ (15)

where
$\tilde{\chi}=\text{col}(\tilde{x}-X\tilde{\varrho},\tilde{\varrho}+\tilde{z},\tilde{y})$
with $\tilde{x}=x-\bar{x}$, $\tilde{\varrho}=\varrho-\bar{\varrho}$, and
$\tilde{z}=z-\bar{z}$ representing the state transformations, $\varsigma$ is a
positive scalar to be determined later, and
$\eta=\lambda_{a}-(\lambda_{a}+\lambda_{b})/T_{a}-\eta^{*}>0$.

###### Proof:

the idea is to show the convergence of $(x,\varrho,z)$ to the equilibrium
point, which will include four steps below:
i) consider the case that the communication is not subject to DoS attacks,
i.e., $t\in[t_{k},a_{k})$. We first show that the output $y_{i}$ at the
equilibrium point is an optimal solution of (5).

In the absence of DoS attacks, (12) can be rewritten as

$\displaystyle\dot{x}$ $\displaystyle=(A-BK)x+BW\vartheta-B(U-KX)\varrho,$
(16a) $\displaystyle\dot{\varrho}$
$\displaystyle=\vartheta=-\triangledown\tilde{f}(y)-\alpha\beta\tilde{\mathcal{L}}(t)y-\beta\tilde{\mathcal{L}}(t)(\varrho+z),$
(16b) $\displaystyle\dot{z}$
$\displaystyle=\alpha\beta\tilde{\mathcal{L}}(t)y.$ (16c)

By (11), we obtain the equilibrium point $(\bar{x},\bar{\varrho},\bar{z})$
from

$\displaystyle\dot{\bar{x}}$ $\displaystyle=\textbf{0}\ \Longrightarrow\
(A-BK)(\bar{x}-X\bar{\varrho})+X\dot{\bar{\varrho}}=\textbf{0},$ (17a)
$\displaystyle\dot{\bar{\varrho}}$ $\displaystyle=\textbf{0}\ \Longrightarrow\
-\triangledown\tilde{f}(y)-\alpha\beta\tilde{\mathcal{L}}(t)\bar{y}-\beta\tilde{\mathcal{L}}(t)(\bar{\varrho}+\bar{z})=\textbf{0},$
(17b) $\displaystyle\dot{\bar{z}}$ $\displaystyle=\textbf{0}\ \Longrightarrow\
\alpha\beta\tilde{\mathcal{L}}(t)\bar{y}=\alpha\beta\tilde{\mathcal{L}}(t)C\bar{x}=\textbf{0}.$
(17c)

In the sequel, we show that the equilibrium point is the solution. Deducing
from (17), the equilibrium point satisfies

$(A-BK)(\bar{x}-X\bar{\varrho})=\textbf{0},\
\sum_{i=1}^{N}\triangledown\tilde{f}_{i}(\bar{y}_{i})=\textbf{0},\
\bar{y}_{i}=C_{i}\bar{x}_{i}=y^{*},$ (18)

where $y^{*}\in\mathbb{R}^{q}$, and since $\tilde{f}$ is strongly convex,
$\sum_{i=1}^{N}\triangledown\tilde{f}_{i}(y^{*})=\textbf{0}$ implies that
$y^{*}$ is the optimal solution. Since $K$ is selected such that $A-BK$ is
Hurwitz, it follows from (11) and (18) that

$\bar{x}=X\bar{\varrho}\ \ \text{and}\ \
\bar{y}=\bar{\varrho}=\textbf{1}\otimes y^{*}.$ (19)

Thus,
$(\bar{x},\bar{\varrho},\bar{z})=(X\bar{\varrho},\bar{\varrho},\bar{z})$, and
$y_{i}$ at the equilibrium point is the optimal solution of (5) in the absence
of attacks.

Let
$F=\triangledown\tilde{f}(\tilde{y}+\bar{y})-\triangledown\tilde{f}(\bar{y})$
and $\tilde{y}=C\tilde{x}$. Since $\tilde{x}=x-\bar{x}$,
$\tilde{\varrho}=\varrho-\bar{\varrho}$, $\tilde{z}=z-\bar{z}$, we further
have the following system

$\displaystyle\dot{\tilde{x}}$
$\displaystyle=(A-BK)(\tilde{x}-X\tilde{\varrho})+X\dot{\tilde{\varrho}},$
(20a) $\displaystyle\dot{\tilde{\varrho}}$
$\displaystyle=-F-\alpha\beta\tilde{\mathcal{L}}(t)\tilde{y}-\beta\tilde{\mathcal{L}}(t)(\tilde{\varrho}+\tilde{z}),$
(20b) $\displaystyle\dot{\tilde{z}}$
$\displaystyle=\alpha\beta\tilde{\mathcal{L}}(t)\tilde{y}.$ (20c)

Choose the following nonnegative Lyapunov function candidate
$V_{a}=V_{1a}+V_{2a}+V_{3a}$, where

$\displaystyle V_{1a}$
$\displaystyle=(\tilde{x}-X\tilde{\varrho})^{T}P_{a}(\tilde{x}-X\tilde{\varrho}),$
(21a) $\displaystyle V_{2a}$
$\displaystyle=\frac{1}{2}(\tilde{\varrho}+\tilde{z})^{T}(\Pi\otimes
I_{q})(\tilde{\varrho}+\tilde{z}),$ (21b) $\displaystyle V_{3a}$
$\displaystyle=\frac{1}{2}\tilde{y}^{T}(\Pi\otimes I_{q})\tilde{y},$ (21c)

where $P_{a}>0$ and $\Pi=\text{diag}\\{\pi\\}$ is defined in Lemma 1.

Then, the time derivatives of (21a) along (20) is described by

$\displaystyle\dot{V}_{1a}$
$\displaystyle=(\tilde{x}-X\tilde{\varrho})^{T}(P_{a}\tilde{A}+\tilde{A}^{T}P_{a})(\tilde{x}-X\tilde{\varrho})$
(22)
$\displaystyle=-(\tilde{x}-X\tilde{\varrho})^{T}O(\tilde{x}-X\tilde{\varrho})\leq-\epsilon_{0}(\tilde{x}-X\tilde{\varrho})^{T}(\tilde{x}-X\tilde{\varrho}),$

where $\tilde{A}=A-BK$ is Hurwitz so that there is a positive define matrix
$O$ satisfying $P_{a}\tilde{A}+\tilde{A}^{T}P_{a}=-O$ and
$\epsilon_{0}=\lambda_{min}(O)>0$. Then, it follows from (21) and (20b)-(20c)
that

$\displaystyle\dot{V}_{2a}$
$\displaystyle=(\tilde{\varrho}+\tilde{z})^{T}(\Pi\otimes
I_{q})(\dot{\tilde{\varrho}}+\alpha\beta\tilde{\mathcal{L}}(t)\tilde{y}),$
$\displaystyle=(\tilde{\varrho}+\tilde{z})^{T}(\Pi\otimes
I_{q})(-F-\beta\tilde{\mathcal{L}}(t)(\tilde{\varrho}+\tilde{z}))$
$\displaystyle=-(\tilde{\varrho}+\tilde{z})^{T}(\Pi\otimes
I_{q})F-\beta(\tilde{\varrho}+\tilde{z})^{T}Q(t)(\tilde{\varrho}+\tilde{z}).$
(23)

The time derivative of (21c) is expressed as

$\displaystyle\dot{V}_{3a}$ $\displaystyle=\tilde{y}^{T}(\Pi\otimes
I_{q})C\dot{\tilde{x}}=\tilde{y}^{T}(\Pi\otimes
I_{q})C(\tilde{A}(\tilde{x}-X\tilde{\varrho})+X\dot{\tilde{\varrho}})$
$\displaystyle=\tilde{y}^{T}(\Pi\otimes
I_{q})C\tilde{A}(\tilde{x}-X\tilde{\varrho})-\tilde{y}^{T}(\Pi\otimes I_{q})F$
$\displaystyle\ \ \ +\tilde{y}^{T}(\Pi\otimes
I_{q})CX(-\alpha\beta\tilde{\mathcal{L}}(t)\tilde{y}-\beta\tilde{\mathcal{L}}(t)(\tilde{\varrho}+\tilde{z}))$
$\displaystyle\leq\epsilon_{1}||\Pi
C||^{2}\tilde{y}^{T}\tilde{y}+\frac{||\tilde{A}||^{2}}{4\epsilon_{1}}(\tilde{x}-X\tilde{\varrho})^{T}(\tilde{x}-X\tilde{\varrho})$
$\displaystyle\ \ \ +\tilde{y}^{T}(\Pi\otimes
I_{q})(-F-\alpha\beta\tilde{\mathcal{L}}(t)\tilde{y}-\beta\tilde{\mathcal{L}}(t)(\tilde{\varrho}+\tilde{z})),$
(24)

where $C_{i}X_{i}=I_{q}$ in (11) and the Young inequality are used to derive
the last inequality, and $\epsilon_{1}$ is a positive constant.

Let $\iota_{min}=\min_{i\in\mathcal{V}}\\{\iota_{i}\\}$,
$l_{max}=\max_{i\in\mathcal{V}}\\{l_{i}\\}$ and $\epsilon_{2}>0$ is a scalar.
Due to the fact that
$F=\triangledown\tilde{f}(\tilde{y}+\bar{y})-\triangledown\tilde{f}(\bar{y})$
and according to Assumption 2, we can further obtain that

$\displaystyle\tilde{y}^{T}(\Pi\otimes
I_{q})F\geq\iota_{min}\pi_{min}||\tilde{y}||^{2},\ ||F||\leq
l_{max}||\tilde{y}||,$ (25a)
$\displaystyle(\tilde{\varrho}+\tilde{z})^{T}(\Pi\otimes
I_{q})F\leq\frac{||(\Pi\otimes
I_{q})F||^{2}}{4\epsilon_{2}}+\epsilon_{2}||\tilde{\varrho}+\tilde{z}||^{2}.$
(25b)

Thus, combining (22)-(25) gives rise to

$\displaystyle\dot{V}_{a}$
$\displaystyle\leq[\frac{||\tilde{A}||^{2}}{4\epsilon_{1}}-\epsilon_{0}](\tilde{x}-X\tilde{\varrho})^{T}(\tilde{x}-X\tilde{\varrho})+\epsilon_{1}||(\Pi\otimes
I_{q})C||^{2}\tilde{y}^{T}\tilde{y}$
$\displaystyle-\iota_{min}\pi_{min}\tilde{y}^{T}\tilde{y}-\alpha\beta\tilde{y}^{T}Q(t)\tilde{y}-\beta\tilde{y}^{T}(\Pi\otimes
I_{q})\tilde{\mathcal{L}}(t)(\tilde{\varrho}+\tilde{z}))$
$\displaystyle+\frac{l^{2}_{max}\pi^{2}_{max}}{4\epsilon_{2}}||\tilde{y}||^{2}+\epsilon_{2}||\tilde{\varrho}+\tilde{z}||^{2}-\beta(\tilde{\varrho}+\tilde{z})^{T}Q(t)(\tilde{\varrho}+\tilde{z})$
$\displaystyle\leq-\left[\begin{array}[]{ccc}\tilde{x}-X\tilde{\varrho}\\\
\tilde{\varrho}+\tilde{z}\\\
\tilde{y}\end{array}\right]^{T}\varOmega_{a}\left[\begin{array}[]{ccc}\tilde{x}-X\tilde{\varrho}\\\
\tilde{\varrho}+\tilde{z}\\\
\tilde{y}\end{array}\right]\leq-\lambda_{a}V_{a},$ (32)

where $\lambda_{a}=\lambda_{min}(\varOmega_{a})/\lambda_{max}(\varGamma_{a})$,
$\varGamma_{a}=\frac{1}{2}\text{diag}\\{2P_{a},\Pi\otimes I_{q},\Pi\otimes\\\
I_{q}\\}$,
$\varOmega_{a}=\left[\begin{array}[]{ccc}\varOmega^{1}_{a}&\textbf{0}&\textbf{0}\\\
&\varOmega^{2}_{a}&\varOmega^{3}_{a}\\\
&*&\varOmega^{4}_{a}\end{array}\right]>0$,
$\varOmega^{1}_{a}=(\epsilon_{0}-\frac{||\tilde{A}||^{2}}{4\epsilon_{1}})\textbf{
I}$,
$\varOmega^{2}_{a}=(\beta\frac{\pi_{min}c}{N^{2}}-\epsilon_{2})\textbf{I}$,
$\varOmega^{3}_{a}=\frac{\beta}{2}(\Pi\otimes I_{q})\tilde{\mathcal{L}}$,
$\varOmega^{4}_{a}=(\alpha\beta\frac{\pi_{min}c}{N^{2}}+l_{min}\pi_{min}-\epsilon_{1}||(\Pi\otimes
I_{q})C||^{2}-\frac{l^{2}_{max}\pi^{2}_{max}}{4\epsilon_{2}})\textbf{I}$,
where $\pi_{max}=\max_{i\in\mathcal{V}}\\{\pi_{i}\\}$ and I is the identify
matrix with proper dimensions.

Hence, denote
$\tilde{\chi}=\text{col}(\tilde{x}-X\tilde{\varrho},\tilde{\varrho}+\tilde{z},\tilde{y})$
and there exists scalars $\epsilon_{i}>0$, $i=0,1,2$ and matrices
$\varGamma_{a}=\text{diag}\\{P_{a},\Pi\otimes I_{q},\Pi\otimes I_{q}\\}$,
$P_{a}>0$, $\Pi=\text{diag}\\{\pi\\}$, such that we have

$V_{a}=\frac{1}{2}\tilde{\chi}^{T}\varGamma_{a}\tilde{\chi}\ \Rightarrow\
\dot{V}_{a}\leq-\lambda_{a}V_{a},\ t\in[t_{k},a_{k}).$ (33)

ii) consider the case that in the presence of DoS attacks, i.e.,
$t\in[a_{k},t_{k+1})$, we choose the Lyapunov function candidate as

$V_{b}=\frac{1}{2}\tilde{\chi}^{T}\varGamma_{b}\tilde{\chi},\
\varGamma_{b}=\left[\begin{array}[]{ccc}2P_{b}&\textbf{0}&\textbf{0}\\\
&S\otimes I_{q}&\textbf{0}\\\ &*&S\otimes I_{q}\end{array}\right],$ (34)

where $P_{b}>0$, and $S>0$ is a diagonal matrix.

Then, the time derivative of $V_{b}$ along (12) is given by

$\displaystyle\dot{V}_{b}$
$\displaystyle=(\tilde{x}-X\tilde{\varrho})^{T}(P_{b}\tilde{A}+\tilde{A}^{T}P_{b})(\tilde{x}-X\tilde{\varrho})-(\tilde{\varrho}+\tilde{z})^{T}(S\otimes
I_{q})F$ $\displaystyle\ \ \ +\tilde{y}^{T}(S\otimes
I_{q})C\tilde{A}(\tilde{x}-X\tilde{\varrho})-\tilde{y}^{T}(S\otimes I_{q})F$
$\displaystyle\leq[\frac{||\tilde{A}||^{2}}{4\tilde{\epsilon}_{1}}-\tilde{\epsilon}_{0}](\tilde{x}-X\tilde{\varrho})^{T}(\tilde{x}-X\tilde{\varrho})+\tilde{\epsilon}_{1}||(S\otimes
I_{q})C||^{2}\tilde{y}^{T}\tilde{y}$ $\displaystyle\ \ \
-\iota_{min}\lambda_{min}(S)\tilde{y}^{T}\tilde{y}+\frac{l^{2}_{max}}{4\tilde{\epsilon}_{2}}\tilde{y}^{T}\tilde{y}+\tilde{\epsilon}_{2}(\tilde{\varrho}+\tilde{z})^{T}(\tilde{\varrho}+\tilde{z})$
$\displaystyle\leq\tilde{\chi}^{T}\varOmega_{b}\tilde{\chi}\leq\lambda_{b}V_{b},$
(35)

where $\lambda_{b}=\sigma_{max}(\varOmega_{b})>0$, and $\varOmega_{b}$ can be
described by

$\varOmega_{b}=\left[\begin{array}[]{ccc}(\frac{||\tilde{A}||^{2}}{4\tilde{\epsilon}_{1}}-\tilde{\epsilon}_{0})\textbf{
I}&\textbf{0}&\textbf{0}\\\ &\tilde{\epsilon}_{2}\textbf{I}&\textbf{0}\\\
&*&\tilde{\epsilon}_{3}\textbf{I}\end{array}\right],$ (36)

where $\tilde{\epsilon}_{0}=\lambda_{min}(\tilde{O})$ with
$P_{b}\tilde{A}+\tilde{A}^{T}P_{b}=-\tilde{O}$,
$\tilde{\epsilon}_{3}=\tilde{\epsilon}_{1}||(S\otimes I_{q})\\\
C||^{2}+\frac{l^{2}_{max}}{4\tilde{\epsilon}_{2}}-\iota_{min}\lambda_{min}(S)$,
and $\tilde{\epsilon}_{1},\tilde{\epsilon}_{2},\tilde{\epsilon}_{3}>0$.

iii) we analyze the exponential convergence of the closed-loop system from a
hybrid perspective in [15, 17, 16, 18].

Denote $\delta(t)\in\\{a,b\\}$ as a switching signal. Then, we can select a
piecewise Lyapunov function candidate: $V(t)=V_{a}(\tilde{\chi})$, if
$t\in[t_{k},a_{k})$, and $V(t)=V_{b}(\tilde{\chi})$, if $t\in[a_{k},t_{k+1}),$
where $V_{a}(\tilde{\chi})$ and $V_{b}(\tilde{\chi})$ are defined in (33) and
(34), respectively.

We suppose that $V_{a}$ is activated in $[t_{k},a_{k})$, while $V_{b}$ is
activated in $[a_{k},t_{k+1})$. Then, it follows from (33) and (35) that we
get

$V(t)=\left\\{\begin{array}[]{c}e^{-\lambda_{a}(t-t_{k})}V_{a}(t_{k}),\
\text{if}\ t\in[t_{k},a_{k}),\\\ e^{\lambda_{b}(t-a_{k})}V_{b}(a_{k}),\ \text{
if}\ t\in[a_{k},t_{k+1}).\end{array}\right.$ (37)

Note that a closed-loop system is switched at $t=t^{+}_{k}$ or $t=a^{+}_{k}$.
Let
$\mu=\max\\{\lambda_{max}(\varGamma_{a})/\lambda_{min}(\varGamma_{b}),\lambda_{max}(\varGamma_{b})/\lambda_{min}(\varGamma_{a})\\}\geq
1$, and next, we discuss the switching situation in two cases:

Case a): if $t\in[t_{k},a_{k})$, it follows from (37) that

$\displaystyle V(t)$ $\displaystyle\leq
e^{-\lambda_{a}(t-t_{k})}V_{a}(t_{k})\leq\mu
e^{-\lambda_{a}(t-t_{k})}V_{b}(t^{-}_{k})$ $\displaystyle\leq\mu
e^{-\lambda_{a}(t-t_{k})}[e^{\lambda_{b}(t_{k}-t_{k-1})}V_{b}(t_{k-1})]$
$\displaystyle\leq\mu
e^{-\lambda_{a}(t-t_{k})}e^{\lambda_{b}(t_{k}-t_{k-1})}[\mu
V_{a}(t^{-}_{k-1})]$
$\displaystyle=\mu^{2}e^{-\lambda_{a}(t-t_{k})}e^{\lambda_{b}(t_{k}-t_{k-1})}V_{a}(t^{-}_{k-1})$
$\displaystyle\leq\mu^{2}e^{-\lambda_{a}(t-t_{k})}e^{\lambda_{b}(t_{k}-t_{k-1})}[e^{-\lambda_{a}(t_{k-1}-t_{k-2})}V_{a}(t_{k-2})]$
$\displaystyle\leq\cdots\leq\mu^{k+1}e^{-\lambda_{a}|\Sigma_{s}(t_{0},t)|}e^{\lambda_{b}|\Sigma_{a}(t_{0},t)|}V_{a}(t_{0}).$
(38)

Case b): if $t\in[a_{k},t_{k+1})$, it follows from (37) that

$\displaystyle V(t)$ $\displaystyle\leq
e^{\lambda_{b}(t-a_{k})}V_{b}(a_{k})\leq\mu
e^{\lambda_{b}(t-a_{k})}V_{b}(a^{-}_{k})$ $\displaystyle\leq\mu
e^{\lambda_{b}(t-a_{k})}e^{-\lambda_{a}(a_{k}-a_{k-1})}[\mu
V_{b}(a^{-}_{k-1})]$
$\displaystyle\leq\mu^{2}e^{\lambda_{a}(t-a_{k})}e^{-\lambda_{a}(a_{k}-a_{k-1})}[e^{\lambda_{b}(a_{k-1}-a_{k-2})}V_{b}(a_{k-2})]$
$\displaystyle\leq\cdots$
$\displaystyle\leq\mu^{k+1}e^{-\lambda_{a}|\Sigma_{s}(t_{0},t)|}e^{\lambda_{b}|\Sigma_{a}(t_{0},t)|}V_{a}(t_{0}).$
(39)

iv) bounds on DoS attack frequency and duration

Notice that $N_{a}(t_{0},t)=k+1$ for $t\in[t_{k},a_{k})$ or
$t\in[a_{k},t_{k+1})$. Thus, for $\forall t\geq t_{0},$ it follows from (38)
and (39) that

$V(t)\leq\mu^{N_{a}(t_{0},t)}e^{-\lambda_{a}|\Xi_{s}(t_{0},t)|}e^{\lambda_{b}|\Xi_{a}(t_{0},t)|}V(t_{0}).$
(40)

Notice that for all $t\geq t_{0},$
$|\Xi_{s}(t_{0},t)|=t-t_{0}-|\Xi_{a}(t_{0},t)|$ and $|\Xi_{a}(t_{0},t)|\leq
T_{0}+(t-t_{0})/T_{a}$ by Definition 2. Thus, we have

$\displaystyle-\lambda_{a}(t-t_{0}-|\Xi_{a}(t_{0},t)|)+\lambda_{b}|\Xi_{a}(t_{0},t)|$
$\displaystyle=-\lambda_{a}(t-t_{0})+(\lambda_{a}+\lambda_{b})|\Xi_{a}(t_{0},t)|$
$\displaystyle\leq-\lambda_{a}(t-t_{0})+(\lambda_{a}+\lambda_{b})[T_{0}+(t-t_{0})/T_{a}].$
(41)

Substituting (41) into (40) yields

$\displaystyle V(t)$
$\displaystyle\leq\mu^{N_{a}(t_{0},t)}e^{-\lambda_{a}(t-t_{0}-|\Xi_{a}(t_{0},t)|)}e^{\lambda_{b}|\Xi_{a}(t_{0},t)|}V(t_{0})$
$\displaystyle\leq
e^{(\lambda_{a}+\lambda_{b})T_{0}}e^{-\lambda_{a}(t-t_{0})}e^{\frac{(\lambda_{a}+\lambda_{b})}{T_{a}}(t-t_{0})}$
$\displaystyle\ \ \ \times e^{\ln(\mu)N_{a}(t_{0},t)}V(t_{0}).$ (42)

By exploiting the attack condition in (13), we can obtain that

$\ln(\mu)N_{a}(t_{0},t)\leq\ln(\mu)N_{0}+\eta^{*}(t-t_{0}).$ (43)

Let $\eta=\lambda_{\alpha}-(\lambda_{a}+\lambda_{b})/T_{a}-\eta^{\ast}>0$.
Based on another attack condition in (14), and using (43), we further rewrite
(42) as

$V(t)\leq e^{(\lambda_{a}+\lambda_{b})T_{0}+\ln(\mu)N_{0}}\
e^{-\eta(t-t_{0})}V(t_{0}).$ (44)

Further, it follows from (33), (34), (37) and (44) that

$||\tilde{\chi}(t)||^{2}\leq\varsigma
e^{-\eta(t-t_{0})}||\tilde{\chi}(t_{0})||^{2},$ (45)

where
$\varsigma=e^{(\lambda_{a}+\lambda_{b})T_{0}+2\ln(\mu)}\varsigma_{a}/\varsigma_{b}$,
$\varsigma_{a}=\max\\{\lambda_{max}(\varGamma_{a}),\lambda_{max}(\\\
\varGamma_{b})\\}$, and
$\varsigma_{b}=\min\\{\lambda_{min}(\varGamma_{a}),\lambda_{min}(\varGamma_{b})\\}$.

Thus, it follows from (45) that $\tilde{x}-X\tilde{\varrho}$,
$\tilde{\varrho}+\tilde{z}$, and $\tilde{y}$ are bounded, and converge to zero
exponentially. That is,
$\lim_{t\rightarrow\infty}(\tilde{x}-X\tilde{\varrho})=\textbf{0}$,
$\lim_{t\rightarrow\infty}(\tilde{\varrho}+\tilde{z})=\textbf{0}$, and
$\lim_{t\rightarrow\infty}\tilde{y}=\textbf{0}$. It follows from
$C_{i}X_{i}=I_{q}$ in (11) that
$\lim_{t\rightarrow\infty}C(\tilde{x}-X\tilde{\varrho})=\lim_{t\rightarrow\infty}(\tilde{y}-\tilde{\varrho})=\textbf{0}$,
i.e., $\lim_{t\rightarrow\infty}\tilde{\varrho}=\textbf{0}$. Thus,
$\lim_{t\rightarrow\infty}\tilde{z}=\textbf{0}$. Then, based on
$\lim_{t\rightarrow\infty}\tilde{y}=\textbf{0}$, the problem in (5) is
solvable. Hence, it can be concluded that $y_{i}$ converges to $y^{*}$
exponentially for heterogeneous linear multi-agent systems under DoS attacks
over random digraphs. ∎

In the absence of attacks, Theorem 1 is reduced as follows.

Corollary 1: In the absence of attacks, the optimal solution is achieved under
the following distributed algorithm, provided that $A_{i}-B_{i}K_{i}$ is
Hurwitz and $(U_{i},W_{i},X_{i})$ is a solution to (11).

$\displaystyle u_{i}$
$\displaystyle=-K_{i}x_{i}-(U_{i}-K_{i}X_{i})\varrho_{i}+W_{i}\vartheta_{i},$
(46a) $\displaystyle\dot{\varrho}_{i}$
$\displaystyle=\vartheta_{i}=-\beta\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\varrho_{i}-\varrho_{j}+z_{i}-z_{j})$
$\displaystyle\ \ \ \ \ \ -\triangledown
f_{i}(y_{i})-\alpha\beta\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(y_{i}-y_{j}),$
(46b) $\displaystyle\dot{z}_{i}$
$\displaystyle=\alpha\beta\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(y_{i}-y_{j}),i\in\mathcal{V}.$
(46c)

###### Proof:

the proof is similar to Step (i), and is omitted. ∎

###  III-B Distributed Event-Triggered Strategy Against DoS Attacks

The time-based scheme in the subsection A is implemented by fixed and periodic
samplings, which require all agents to exchange information synchronously. Let
$\\{t^{i}_{k}\\}$ denote a sequence of some transmission time instants with
$0=t^{i}_{0}<t^{i}_{1}<t^{i}_{1}\cdots<t^{i}_{k}<\cdots$. Unlike known time-
based transmission time instants $t_{k}$, an event-triggered transmission
attempt $t_{k}^{i}$ will be scheduled to be resilient against DoS attacks. If
$t_{k}^{i}\in\Xi_{a}(t_{0},t)$, an information transmission attempt suffers
from the DoS attack, and thereby is unsuccessful. If
$t_{k}^{i}\in\Xi_{s}(t_{0},t)$, the information can be successfully
transmitted. It is thus desirable to develop a new way to update the
distributed optimization algorithm and determine the next triggering instants
for information transmissions over insecure communication. The main challenges
include twofolds: i) how to design a distributed dynamic event-triggered
condition without Zeno behavior, and ii) how to illustrate the validity of
distributed optimization algorithm to guarantee global exponential
convergence.

Now, propose the following distributed optimization algorithm to handle the
effect of DoS attacks over random digraphs.

$\displaystyle u_{i}$
$\displaystyle=-K_{i}x_{i}-(U_{i}-K_{i}X_{i})\varrho_{i}+W_{i}\vartheta_{i},$
(47a) $\displaystyle\dot{\varrho}_{i}$
$\displaystyle=\vartheta_{i}=-\triangledown f_{i}(y_{i})-\beta\bar{e}_{\varrho
zi}-\alpha\beta\bar{e}_{yi},$ (47b) $\displaystyle\dot{z}_{i}$
$\displaystyle=\alpha\beta\bar{e}_{yi},t\in[t^{i}_{k},t^{i}_{k+1}),\
i\in\mathcal{V},$ (47c)

where the consensus errors under DoS attacks are described by

$\bar{e}_{\varrho zi}=\left\\{\begin{array}[]{c}e_{\varrho zi},\ \text{ if}\
t\in[t^{i}_{k},t^{i}_{k+1})\ \cap\ t=t^{i}_{k}\in\Xi_{s}(t_{0},t),\\\
\textbf{0},\ \ \text{ if}\ t=t^{i}_{k}\in\Xi_{a}(t_{0},t),\end{array}\right.$
(48) $\bar{e}_{yi}=\left\\{\begin{array}[]{c}e_{yi},\ \ \ \text{ if}\
t\in[t^{i}_{k},t^{i}_{k+1})\ \cap\ t=t^{i}_{k}\in\Xi_{s}(t_{0},t),\\\
\textbf{0},\ \ \ \text{ if}\
t=t^{i}_{k}\in\Xi_{a}(t_{0},t),\end{array}\right.$ (49)

where $e_{\varrho
zi}=\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{\varrho}_{i}-\hat{\varrho}_{j}+\hat{z}_{i}-\hat{z}_{j})$,
$\hat{\varrho}_{i}=\varrho_{i}(t^{i}_{k})$, $\hat{z}_{i}=z_{i}(t^{i}_{k})$,
and $e_{yi}=\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{y}_{i}-\hat{y}_{j})$,
$\hat{y}_{i}=y_{i}(t^{i}_{k})$.

To specify event time instants, denote the measurement errors

$\tilde{e}_{yi}=\hat{y}_{i}(t)-y_{i}(t),\ \tilde{e}_{\varrho
zi}=\hat{\varrho}_{i}(t)+\hat{z}_{i}(t)-(\varrho_{i}(t)+z_{i}(t)).$ (50)

Then, a dynamic event-triggering scheme is developed as

$t^{i}_{k+1}=\left\\{\begin{array}[]{c}\underset{t>t^{i}_{k}}{\inf}\\{t-t_{k}^{i}|\sigma_{gi}g_{i}(t)\leq\eta_{gi}(t)\
\text{or}\ \sigma_{hi}h_{i}(t)\leq\eta_{hi}(t)\\},\\\ \text{ if}\
t^{i}_{k}\in\Xi_{s}(t_{0},t),\\\ t_{k}^{i}+\kappa^{i}_{k},\ \ \ \ \ \ \ \ \
\text{ if}\ t^{i}_{k}\in\Xi_{a}(t_{0},t),\end{array}\right.$ (51)

where $\kappa^{i}_{k}>0$ is a dwell time to be determined, $\sigma_{gi}>0$,
$\sigma_{hi}>0$, and the triggering functions $g_{i}(t)$, $h_{i}(t)$ are given
by

$g_{i}(t)=\tilde{e}_{yi}^{2}-\theta_{gi}e_{yi}^{2},\
h_{i}(t)=\tilde{e}^{2}_{\varrho zi}-\theta_{hi}e^{2}_{\varrho zi},$ (52)

where $\theta_{gi}$, $\theta_{hi}\in[0,1)$, and $\eta_{gi}(t)$, $\eta_{hi}(t)$
are auxiliary variables satisfying for $\eta_{gi}(t_{0})$, $\eta_{hi}(t_{0})$,
$k_{gi},k_{hi}>0$, and $\delta_{gi},\delta_{hi}\in[0,1)$,

$\dot{\eta}_{gi}=\left\\{\begin{array}[]{c}-k_{gi}\eta_{gi}-\delta_{gi}g_{i}(t),\
\ \text{ if}\ t^{i}_{k}\in\Xi_{s}(t_{0},t),\\\ \ \ \ \ \ \ \ 0,\ \ \ \ \ \ \ \
\ \ \ \ \ \ \text{ if}\ t^{i}_{k}\in\Xi_{a}(t_{0},t),\end{array}\right.$ (53)
$\dot{\eta}_{hi}=\left\\{\begin{array}[]{c}-k_{hi}\eta_{hi}-\delta_{hi}h_{i}(t),\
\ \text{ if}\ t^{i}_{k}\in\Xi_{s}(t_{0},t),\\\ \ \ \ \ \ \ \ 0,\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \text{ if}\ t^{i}_{k}\in\Xi_{a}(t_{0},t).\end{array}\right.$
(54)

Next, we firstly introduce the following lemma.

###### Lemma 2

Under the dynamical event-triggered scheme (51), the term $\eta_{i}(t)$ below
is always positive for $\forall\eta_{gi}(t_{0}),\eta_{hi}(t_{0})>0$,

$\eta_{i}(t)=\eta_{gi}(t)+\eta_{hi}(t)>0,\ \forall i\in\mathcal{V}.$ (55)

###### Proof:

under the proposed event-triggered scheme (51)-(54), the transmission attempts
$\\{t^{i}_{k}\\}_{k\in\mathrm{N}^{+}}=\\{t^{i}_{k_{s}}\\}_{s\in\mathrm{N}^{+}}\bigcup\
\\{t^{i}_{k_{a}}\\}_{a\in\mathrm{N}^{+}}$ can be generated, where
$\\{t^{i}_{k_{s}}\\}_{s\in\mathrm{N}^{+}}$ denotes the successful transmission
attempts while $\\{t^{i}_{k_{a}}\\}_{a\in\mathrm{N}^{+}}$ denotes the
unsuccessful transmission attempts. Next, two cases are considered:

Case i): choose any $t^{i}_{k_{s_{0}}}\in\Xi_{s}(t_{0},\infty)$ and let
$a_{1}=\min_{a}\\{t^{i}_{k_{a}}>t^{i}_{k_{s_{0}}}\\}$. Then, when
$t\in[t^{i}_{k},t^{i}_{k+1})$,
$\tilde{e}^{2}_{yi}(t)-\theta_{gi}e^{2}_{yi}\leq\eta_{gi}(t)/\sigma_{gi}.$
Hence, we can obtain from (53) that

$\dot{\eta}_{gi}(t)=-k_{gi}\eta_{gi}(t)-\delta_{gi}g_{i}(t)\geq-(k_{gi}+\delta_{gi}/\sigma_{gi})\eta_{gi}(t),$
(56)

which implies that based on the mathematical induction, we have that for
$t\in[t^{i}_{k},t^{i}_{k+1})\in[t^{i}_{k_{s_{0}}},t^{i}_{k_{a_{1}}})$,

$\displaystyle\eta_{gi}(t)$
$\displaystyle\geq\eta_{gi}(t^{i}_{k})e^{-(k_{gi}+\delta_{gi}/\sigma_{gi})(t-t^{i}_{k})}\geq\cdots$
$\displaystyle\geq\eta_{gi}(t^{i}_{k_{s_{0}}})e^{-(k_{gi}+\delta_{gi}/\sigma_{gi})(t-t^{i}_{k_{s_{0}}})}>0.$
(57)

Case ii): denote $s_{1}=\min_{s}\\{t^{i}_{k_{s}}>t^{i}_{k_{a_{1}}}\\}$, and we
have that for
$t\in[t^{i}_{k},t^{i}_{k+1})\in[t^{i}_{k_{a_{1}}},t^{i}_{k_{s_{1}}})$, it
follows from (53) that $\dot{\eta}_{gi}(t)=0$. Thus, $\eta_{gi}(t)$ is time-
invariant and since $\eta_{gi}(t_{0})>0$, we have

$\eta_{gi}(t)=\eta_{gi}(t^{i}_{k})=\cdots=\eta_{gi}(t^{i}_{k_{a_{1}}})>0.$
(58)

Then, we obtain from both cases that $\eta_{gi}(t)>0,\forall i\in\mathcal{V}$.

Similarly, it can be derived from (54) that $\eta_{hi}(t)>0,\forall
i\in\mathcal{V}$ for both cases by following the same procedure. ∎

Substituting (47)-(49) into (1) yields the following system

$\dot{\chi}=\left\\{\begin{array}[]{l}\left[\begin{array}[]{ccc}(A-BK)x+BW\vartheta-B(U-KX)\varrho\\\
-\triangledown\tilde{f}(y)-\alpha\beta\tilde{\mathcal{L}}(t)\hat{y}-\beta\tilde{\mathcal{L}}(t)(\hat{\varrho}+\hat{z})\\\
\alpha\beta\tilde{\mathcal{L}}(t)\hat{y}\end{array}\right],\\\ \text{ if}\
t\in[t^{i}_{k},t^{i}_{k+1})\cap t=t^{i}_{k}\in\Xi_{s}(t_{0},t),\\\
\left[\begin{array}[]{ccc}(A-BK)x+BW\vartheta-B(U-KX)\varrho\\\
-\triangledown\tilde{f}(y)\\\ \textbf{0}\end{array}\right],\\\ \text{ if}\
t=t^{i}_{k}\in\Xi_{a}(t_{0},t),k=0,1,2\cdots,\end{array}\right.$ (59)

where $\hat{y},\hat{\varrho},\hat{z}$ are the stack vectors of
$\hat{y}_{i},\hat{\varrho}_{i},\hat{z}_{i}$, respectively, and

$\hat{y}_{i}(t)=y_{i}+\tilde{e}_{yi},\
\hat{\varrho}_{i}(t)=\varrho_{i}+\tilde{e}_{\varrho i},\
\hat{z}_{i}(t)=z_{i}+\tilde{e}_{zi}.$ (60)

###### Theorem 2

Given Assumptions 1-3, Problem 1 is solvable for any $x_{i}(0)$,
$\varrho_{i}(0)$ and $z_{i}(0)$ under the proposed resilient distributed
optimization algorithm in (47)-(49) together with the distributed event-
triggered scheme in (51)-(54), provided that $A_{i}-B_{i}K_{i}$ is Hurwitz,
$U_{i},W_{i},X_{i}$ are solutions to (11), and

$T_{f}>(\ln(\tilde{\mu})+(\tilde{\lambda}_{a}+\tilde{\lambda}_{b})\kappa_{*})/\tilde{\eta}^{\ast},\
T_{a}>(\tilde{\lambda}_{a}+\tilde{\lambda}_{b})/(\tilde{\lambda}_{a}-\tilde{\eta}^{\ast}),$
(61)

where $\tilde{\lambda}_{a},\tilde{\lambda}_{b},\tilde{u},\kappa_{*}$ are
positive scalars to be determined later.

###### Proof:

the proof includes the following steps:

Step 1 (two intervals classification): two time intervals where (51)-(54)
holds and does not hold are characterized. Consider sequences:
$\\{t_{k}^{i}\\}_{k\in\mathbb{N}^{+}}$ and $\\{a_{k}\\}_{k\in\mathbb{N}^{+}}.$
The following $\mathcal{F}:=\\{(i,k)\in\mathcal{V\times}\mathbb{N}^{+}\text{
}|t_{k}^{i}\in\cup_{k\in\mathbb{N}^{+}}\mathcal{A}_{k}\\}$ is a set of
integers related to an updating attempt occurring under attacks. Due to the
finite sampling rate, a time interval will necessarily elapse from
$t^{i}_{k}+a_{k}$ to the time when agents successfully sample and transmit. It
is upper bounded by $\sup_{(i,k)\in\mathcal{F}}\kappa^{i}_{k}\leq\kappa_{*}$.
Hence, a DoS free interval of a length greater than $\kappa_{*}$ ensures that
each agent can sample and transmit. The $m$-th time interval where (51)-(54)
do not need to hold is

$\mathfrak{A}_{m}=[a_{m},a_{m}+\tau_{m}+\kappa_{*}).$ (62)

Thus, the time interval $[\tau,t)$ consists of the following two union of sub-
intervals: $[\tau,t)=\tilde{\Xi}_{s}(\tau,t)\cup\tilde{\Xi}_{a}(\tau,t)$ with

$\displaystyle\hskip 11.99998pt\tilde{\Xi}_{a}(\tau,t):=\cup\
\mathfrak{A}_{m}\cap[\tau,t],\
\tilde{\Xi}_{s}(\tau,t):=[\tau,t]\backslash\tilde{\Xi}_{a}(\tau,t).$ (63)

Step 2 (Lyapunov stability analysis)
i) consider the time interval $\tilde{\Xi}_{s}(\tau,t)$ over which (51)-(54)
hold. Denote $\tilde{\varGamma}_{a}=\text{diag}\\{2\tilde{P}_{a},\Pi\otimes
I_{q},\Pi\otimes I_{q}\\}$ and then select

$W_{a}=V_{a}+V_{\eta}=\frac{1}{2}\tilde{\chi}^{T}\tilde{\varGamma}_{a}\tilde{\chi}+\sum_{i=1}^{N}\eta_{i}(t).$
(64)

Then, the time derivative of $W_{a}$ along (59)-(60) is given by

$\displaystyle\dot{W}_{a}$
$\displaystyle=\tilde{y}^{T}\tilde{\Pi}C\tilde{A}(\tilde{x}-X\tilde{\varrho})-(\tilde{y}+\tilde{\varrho}+\tilde{z})^{T}\tilde{\Pi}F+\tilde{y}^{T}\tilde{\Pi}(-\alpha\beta\tilde{\mathcal{L}}(t)\tilde{y}$
$\displaystyle-\beta\tilde{\mathcal{L}}(t)(\tilde{\varrho}+\tilde{z}))-(\tilde{x}-X\tilde{\varrho})^{T}O(\tilde{x}-X\tilde{\varrho})-\beta\tilde{y}^{T}\tilde{\Pi}\tilde{\mathcal{L}}(t)$
$\displaystyle\times(\alpha\tilde{e}_{y}+\tilde{e}_{\varrho
z})-\beta(\tilde{\varrho}+\tilde{z})^{T}[Q(t)(\tilde{\varrho}+\tilde{z})+\tilde{\Pi}\tilde{\mathcal{L}}(t)\tilde{e}_{\varrho
z}]+\dot{V}_{\eta}$
$\displaystyle\leq-[\epsilon_{0}-\frac{||\tilde{A}||^{2}}{4\epsilon_{1}}](\tilde{x}-X\tilde{\varrho})^{T}(\tilde{x}-X\tilde{\varrho})+\epsilon_{1}||\tilde{\Pi}C||^{2}\tilde{y}^{T}\tilde{y}$
$\displaystyle-\iota_{min}\pi_{min}\tilde{y}^{T}\tilde{y}-\alpha\beta\tilde{y}^{T}Q(t)\tilde{y}-\beta\tilde{y}^{T}\tilde{\Pi}\tilde{\mathcal{L}}(t)(\tilde{\varrho}+\tilde{z})$
$\displaystyle+\frac{l^{2}_{max}\pi^{2}_{max}}{4\epsilon_{2}}||\tilde{y}||^{2}+\epsilon_{2}||\tilde{\varrho}+\tilde{z}||^{2}-\beta(\tilde{\varrho}+\tilde{z})^{T}Q(t)(\tilde{\varrho}+\tilde{z})$
$\displaystyle+\frac{1}{4c_{1}}||\alpha\beta\tilde{\Pi}\tilde{\mathcal{L}}(t)||^{2}||\tilde{y}||^{2}+c_{1}||\tilde{e}_{y}||^{2}+\frac{1}{4c_{2}}||\beta\tilde{\Pi}\tilde{\mathcal{L}}(t)||^{2}||\tilde{y}||^{2}$
$\displaystyle+c_{2}||\tilde{e}_{\varrho
z}||^{2}+\frac{1}{4c_{2}}||\beta\tilde{\Pi}\tilde{\mathcal{L}}(t)||^{2}||\tilde{\varrho}+\tilde{z}||^{2}+c_{2}||\tilde{e}_{\varrho
z}||^{2}$
$\displaystyle-\sum_{i=1}^{N}k_{gi}\eta_{gi}-\sum_{i=1}^{N}\delta_{gi}(\tilde{e}^{2}_{yi}-\theta_{gi}e^{2}_{yi})+\sum_{i=1}^{N}\theta_{gi}(e^{2}_{yi}-e^{2}_{yi})$
$\displaystyle-\sum_{i=1}^{N}k_{hi}\eta_{hi}-\sum_{i=1}^{N}\delta_{hi}(\tilde{e}_{\varrho
zi}^{2}-\theta_{hi}e_{\varrho zi}^{2})+\sum_{i=1}^{N}\theta_{hi}(e_{\varrho
zi}^{2}-e_{\varrho zi}^{2})$
$\displaystyle\leq-\left[\begin{array}[]{ccc}\tilde{x}-X\tilde{\varrho}\\\
\tilde{\varrho}+\tilde{z}\\\
\tilde{y}\end{array}\right]^{T}\tilde{\varOmega}_{a}(t)\left[\begin{array}[]{ccc}\tilde{x}-X\tilde{\varrho}\\\
\tilde{\varrho}+\tilde{z}\\\
\tilde{y}\end{array}\right]-\sum_{i=1}^{N}k_{gi}\eta_{gi}$ (71)
$\displaystyle\ \ \
+\sum_{i=1}^{N}(1-\delta_{gi})(\tilde{e}^{2}_{yi}-\theta_{gi}e^{2}_{yi})-\sum_{i=1}^{N}k_{hi}\eta_{hi}$
$\displaystyle\ \ \ +\sum_{i=1}^{N}(1-\delta_{hi})[\tilde{e}_{\varrho
zi}^{2}-\theta_{hi}e_{\varrho zi}^{2}]$
$\displaystyle\leq-\tilde{\chi}^{T}\tilde{\varOmega}_{a}(t)\tilde{\chi}-\sum_{i=1}^{N}k_{ci}\eta_{i}\leq-\tilde{\lambda}_{a}W_{a},$
(72)

where $\tilde{\Pi}=\Pi\otimes I_{q}$,
$\tilde{\lambda}_{a}=\min\\{\lambda_{min}(\tilde{\varOmega}_{a}),\underline{k}_{c}\\}$,
$\underline{k}_{c}=\min\\{k_{ci}\\}$,
$k_{ci}=\min\\{k_{gi}-\frac{1-\delta_{gi}}{\sigma_{gi}},k_{hi}-\frac{1-\delta_{hi}}{\sigma_{hi}}\\}>0$,
$c_{1}=1-c_{3}\overline{\theta}_{g}\|\tilde{L}\|^{2}>0$,

$c_{2}=(1-c_{3}\overline{\theta}_{h}\|\tilde{L}\|^{2})/2>0$,
$\overline{\theta}_{g}=\max\\{\theta_{gi}\\}$,
$\overline{\theta}_{h}=\max\\{\theta_{hi}\\}$, $\tilde{\varOmega}_{a}$ is
similar to $\varOmega_{a}$ with
$\tilde{\varOmega}^{2}_{a}=\varOmega^{2}_{a}-(\frac{1}{4c_{2}}\|\beta\tilde{\Pi}\tilde{L}\|^{2}+\frac{1}{4c_{3}}\|\tilde{L}\|^{2})\textbf{I}$,
$\tilde{\varOmega}^{3}_{a}=\varOmega^{3}_{a}-((\frac{1}{4c_{1}}+\frac{1}{4c_{2}})\|\alpha\beta\tilde{\Pi}\tilde{L}\|^{2}+\frac{1}{4c_{3}}\|\tilde{L}\|^{2})\textbf{I}$.

ii) consider the time interval $\tilde{\Xi}_{a}(\tau,t)$ over which (51)-(54)
do not necessarily hold. Let $\tilde{P}_{b}>0$, $\tilde{S}=\bar{S}\otimes
I_{q}$ with $\bar{S}>$ being a diagonal matrix and choose the Lyapunov
function candidate as

$W_{b}=\frac{1}{2}\tilde{\chi}^{T}\tilde{\varGamma}_{b}\tilde{\chi}+\sum_{i=1}^{N}\eta_{i}(t),\
\tilde{\varGamma}_{b}=\left[\begin{array}[]{ccc}2\tilde{P}_{b}&\textbf{0}&\textbf{0}\\\
&\tilde{S}&\textbf{0}\\\ &*&\tilde{S}\end{array}\right].$ (73)

Similar to (35), the time derivative of $W_{b}$ along (59) is

$\dot{W}_{b}\leq\tilde{\chi}^{T}\tilde{\varOmega}_{b}\tilde{\chi}\leq\tilde{\lambda}_{b}W_{b},$
(74)

where $\lambda_{b}=\max\\{\sigma_{max}(\tilde{\varOmega}_{b}),1\\}>0$, and
$\tilde{\varOmega}_{b}$ is similarly defined as (36) with
$\tilde{\epsilon}_{3}=\tilde{\epsilon}_{1}||(\tilde{S}\otimes
I_{q})C||^{2}+\frac{l^{2}_{max}\pi^{2}_{max}}{4\tilde{\epsilon}_{2}}-\iota_{min}\lambda_{min}(\tilde{S})$.

iii) convergence analysis of a piecewise Lyapunov function.

Similar to (37)-(39), let $W(t)=W_{\tilde{\delta}(t)}$ and
$\tilde{\delta}(t)\in\\{a,b\\}$, where $W_{a}$ and $W_{b}$ are defined in (64)
and (73), respectively. Suppose that $W_{a}$ is activated in
$[a_{m-1}+\tau_{m-1},a_{m})$ and $W_{b}$ is activated in
$[a_{m},a_{m}+\tau_{m}+\kappa_{*})$. Then, by the Comparison lemma, we get

$W(t)=\left\\{\begin{array}[]{c}e^{-\tilde{\lambda}_{a}(t-t_{m-1}-\tau_{m-1})}W_{a}(t_{m-1}+\tau_{m-1}),\\\
e^{\tilde{\lambda}_{b}(t-a_{m})}W_{b}(a_{m}).\end{array}\right.$ (75)

Denote
$\tilde{\mu}=\max\\{\lambda_{max}(\tilde{\varGamma}_{a})/\lambda_{min}(\tilde{\varGamma}_{b}),\lambda_{max}(\tilde{\varGamma}_{b})/\lambda_{min}(\tilde{\varGamma}_{a})\\}\geq
1$. Next, we discuss the situation in two cases:

Case a): if $t\in[a_{m-1}+\tau_{m-1},a_{m})$, it follows from (75) that

$\displaystyle W(t)$ $\displaystyle\leq
e^{-\tilde{\lambda}_{a}(t-a_{m-1}-\tau_{m-1})}W_{a}(a_{m-1}+\tau_{m-1})$
$\displaystyle\leq\tilde{\mu}e^{-\tilde{\lambda}_{a}(t-a_{m-1}-\tau_{m-1})}W_{b}(a^{-}_{m-1}+\tau^{-}_{m-1})\leq\cdots$
$\displaystyle\leq\tilde{\mu}^{m}e^{-\tilde{\lambda}_{a}|\tilde{\Sigma}_{s}(t_{0},t)|}e^{\tilde{\lambda}_{b}|\tilde{\Sigma}_{a}(t_{0},t)|}W_{a}(t_{0}).$
(76)

Case b): if $t\in[a_{m},a_{m}+\tau_{m}+\kappa_{*})$, similarly

$\displaystyle W(t)$ $\displaystyle\leq
e^{\tilde{\lambda}_{b}(t-a_{m})}W_{b}(a_{m})\leq\tilde{\mu}e^{\tilde{\lambda}_{b}(t-a_{m})}W_{a}(a^{-}_{m})$
$\displaystyle\leq
e^{\tilde{\lambda}_{b}(t-a_{m})}[e^{-\tilde{\lambda}_{a}(a_{m}-a_{m-1}-\tau_{m-1})}W_{a}(a_{m-1}+\tau_{m-1})]$
$\displaystyle\leq\cdots\leq\tilde{\mu}^{m+1}e^{-\tilde{\lambda}_{a}|\tilde{\Sigma}_{s}(t_{0},t)|}e^{\tilde{\lambda}_{b}|\tilde{\Sigma}_{a}(t_{0},t)|}W_{a}(t_{0}).$
(77)

iv) bounds on the attack frequency and duration.

From Definition 1, $N_{a}(t_{0},t)=m$ for $t\in[a_{m-1}+\tau_{m-1},a_{m})$ and
$m+1$ for $t\in[a_{m},a_{m}+\tau_{m}+\kappa_{*})$. Thus, for $\forall t\geq
t_{0}$,

$W(t)\leq\mu^{N_{a}(t_{0},t)}e^{-\tilde{\lambda}_{a}|\tilde{\Xi}_{s}(t_{0},t)|}e^{\tilde{\lambda}_{b}|\tilde{\Xi}_{a}(t_{0},t)|}W(t_{0}).$
(78)

Notice that for all $t\geq t_{0},$
$|\tilde{\Xi}_{s}(t_{0},t)|=t-t_{0}-|\tilde{\Xi}_{a}(t_{0},t)|$ and
$|\tilde{\Xi}_{a}(t_{0},t)|\leq|\Xi_{a}(t_{0},t)|+(1+N_{a}(t_{0},t))\kappa_{*}$.
By Definition 2, we have:
$-\tilde{\lambda}_{a}(t-t_{0}-|\tilde{\Xi}_{a}(t_{0},t)|)+\tilde{\lambda}_{b}|\tilde{\Xi}_{a}(t_{0},t)|=-\tilde{\lambda}_{a}(t-t_{0})+(\tilde{\lambda}_{a}+\tilde{\lambda}_{b})[T_{0}+(t-t_{0})/T_{a}]$.
Thus, it follows from (78) that

$\displaystyle W(t)$ $\displaystyle\leq
e^{(\tilde{\lambda}_{a}+\tilde{\lambda}_{b})T_{0}}e^{-\tilde{\lambda}_{a}(t-t_{0})}e^{\frac{(\tilde{\lambda}_{a}+\tilde{\lambda}_{b})}{\tau_{a}}(t-t_{0})}$
$\displaystyle\ \ \ \times
e^{[\ln(\tilde{\mu})+(\tilde{\lambda}_{a}+\tilde{\lambda}_{b})N_{a}(t_{0},t)]}W(t_{0}).$
(79)

From (61), let
$\tilde{\eta}=\tilde{\lambda}_{a}-(\tilde{\lambda}_{a}+\tilde{\lambda}_{b})/\tau_{a}-\tilde{\eta}^{\ast}>0$
so that

$W(t)\leq
e^{(\tilde{\lambda}_{a}+\tilde{\lambda}_{b})(T_{0}+\kappa_{*})+\tilde{u}N_{0}}e^{-\tilde{\eta}(t-t_{0})}W(t_{0}).$
(80)

Thus, for certain scalar $\tilde{\varsigma}>0$, we can have

$\left\|\left[\begin{array}[]{cc}\tilde{\chi}(t)\\\
\eta(t)\end{array}\right]\right\|^{2}\leq\tilde{\varsigma}e^{-\tilde{\eta}(t-t_{0})}\left\|\left[\begin{array}[]{cc}\tilde{\chi}(t_{0})\\\
\eta(t_{0})\end{array}\right]\right\|^{2}.$ (81)

Hence, we obtain that $y_{i}$ converges to $y^{*}$ exponentially. ∎

###### Theorem 3

The event-triggering instants determined by (51)-(54) guarantee that no agents
will exhibit Zeno behavior.

###### Proof:

suppose that there exists Zeno behavior. That is, there exists an agent so
that at certain time $\mathcal{T}_{0}$,
$\lim_{t\rightarrow\infty}t^{i}_{k}=\mathcal{T}_{0}$. From the property of
limits, we have that for certain $\varepsilon_{0}>0$, there exists a positive
integer $\mathcal{N}(\varepsilon_{0})$ such that

$t^{i}_{k}\in[\mathcal{T}_{0}-\varepsilon_{0},\mathcal{T}_{0}+\varepsilon_{0}],\
\forall k\geq\mathcal{N}(\varepsilon_{0}),$ (82)

which implies that
$t^{i}_{\mathcal{N}(\varepsilon_{0})+1}-t^{i}_{\mathcal{N}(\varepsilon_{0})}<2\varepsilon_{0}$.

Based on (1), (47) and (60), the upper right-hand Dini derivative of
$\|\tilde{e}_{yi}\|$ and $\|\tilde{e}_{\varrho zi}\|$ can be derived as

$\displaystyle D^{+}\|\tilde{e}_{yi}\|$
$\displaystyle=\|-\dot{y}_{i}\|\leq\|C_{i}\tilde{A}_{i}(x_{i}-X_{i}\varrho_{i})+\dot{\varrho}_{i}\|,$
(83a) $\displaystyle D^{+}\|\tilde{e}_{\varrho zi}\|$
$\displaystyle=\|-\dot{\varrho}_{i}-\dot{z}_{i}\|\leq\|\dot{\varrho}_{i}+\dot{z}_{i}\|,$
(83b) $\displaystyle\dot{\varrho}_{i}$ $\displaystyle=-\triangledown
f_{i}(y_{i})-\beta\bar{e}_{\varrho zi}-\alpha\beta\bar{e}_{yi},$ (83c)
$\displaystyle\dot{z}_{i}$ $\displaystyle=\alpha\beta\bar{e}_{yi},\
t\in[t^{i}_{k},t^{i}_{k+1}).$ (83d)

Case a): for $t\in[t^{i}_{k},t^{i}_{k+1})\cap t=t^{i}_{k}\in\Xi_{s}(t_{0},t)$,
we get

$D^{+}\|\tilde{e}_{yi}\|\leq\|C_{i}\tilde{A}_{i}(x_{i}-X_{i}\varrho_{i})-\triangledown
f_{i}(y_{i})-\beta e_{\varrho zi}-\alpha\beta e_{yi}\|.$ (84)

From Theorem 2, we have that all the states $\tilde{x}-X\tilde{\varrho}$,
$\tilde{\varrho}+\tilde{z}$, $\tilde{y}$ are bounded. Thus, for certain scalar
$C_{0}>0$,

$D^{+}\|\tilde{e}_{yi}\|\leq C_{0}.$ (85)

Since an event is triggered if (51) is satisfied and
$||\tilde{e}_{yi}(t^{i}_{k})||=0$, we have
$||\tilde{e}_{yi}(t)||^{2}\geq\theta_{gi}e^{2}_{yi}+\eta_{gi}(t)/\sigma_{gi}\geq\eta_{gi}(t)/\sigma_{gi}$
. Thus,

$||\tilde{e}_{yi}(t)||\geq\sqrt{\eta_{gi}(t)/\sigma_{gi}}\geq\sqrt{\eta_{gi}(0)/\sigma_{gi}}e^{-\frac{k_{gi}+\delta_{gi}/\sigma_{gi}}{2}t}.$
(86)

Hence, it follows from (85) and (86) that

$t^{i}_{\mathcal{N}(\varepsilon_{0})+1}-t^{i}_{\mathcal{N}(\varepsilon_{0})}\geq\frac{1}{C_{0}}\sqrt{\eta_{gi}(0)/\sigma_{gi}}e^{-\frac{k_{gi}+\delta_{gi}/\sigma_{gi}}{2}t^{i}_{\mathcal{N}(\varepsilon_{0})+1}}.$

Select $\varepsilon_{0}$ as the solution of
$\frac{1}{C_{0}}\sqrt{\eta_{gi}(0)/\sigma_{gi}}e^{-\frac{k_{gi}+\delta_{gi}/\sigma_{gi}}{2}\mathcal{T}_{0}}=2\varepsilon_{0}e^{\frac{k_{gi}+\delta_{gi}/\sigma_{gi}}{2}\varepsilon_{0}}$.
Then, we obtain that

$t^{i}_{\mathcal{N}(\varepsilon_{0})+1}-t^{i}_{\mathcal{N}(\varepsilon_{0})}\geq\frac{1}{C_{0}}\sqrt{\eta_{gi}(0)/\sigma_{gi}}e^{-\frac{k_{gi}+\delta_{gi}/\sigma_{gi}}{2}(\mathcal{T}_{0}+\varepsilon_{0})}=2\varepsilon_{0},$

which contracts (82). Thus, no Zero behavior exists for Case a).

Case b): for $t\in[t^{i}_{k},t^{i}_{k+1})\cap t=t^{i}_{k}\in\Xi_{s}(t_{0},t)$,
$D^{+}\|\tilde{e}_{\varrho zi}\|\leq\|-\triangledown f_{i}(y_{i})-\beta
e_{\varrho zi}\|$. Following a similar procedure, we can also show that no
Zero behavior exists for $\tilde{e}_{\varrho zi}$.

Overall, Zero behavior can be avoided for agent $i$. ∎

###### Remark 5

Unlike traditional time-triggered control schemes in (8), an event-based
transmission strategy is presented in (47) to save limited communication
resources and to be resilient against attacks. The defined $\\{t^{i}_{k}\\}$
is an aperiodic transmission sequence determined by (51) for agent i. This
means that each agent has its own triggering rule, that is, they transmit data
in an asynchronous manner. In addition, the triggering process of each agent
does not affect each other, that is, an agent and its neighbors are considered
independent from the perspective of activating the triggering rules. Each
agent needs to measure its own information and has access to its neighboring
information, which essentially reflects the basic requirement of distributed
control strategy.

###### Remark 6

Inspired by [27] and [28], a new dynamical event-triggered communication
scheme in (51)-(54) is designed over an insecure and unreliable network. For
linear multi-agent systems, the proposed scheme in (51)-(54) generates
transmission updates under DoS attacks. General speaking, there exist several
methods to exclude Zeno behavior, e.g., a pre-defined dwell time design in
[18] or combination with sample-data schemes in [29]. In contrast to the above
literature, the dynamical event-triggered communication scheme in (51)-(54) is
developed to avoid the continuous communication and involvement of global
graph information. In addition, its effectiveness has been verified even in
the presence of DoS attacks over random digraphs.

##  IV Numerical Simulations

In this section, numerical simulation examples are presented to verify the
effectiveness of the proposed designs.

Consider a group of agents described by the following general heterogeneous
linear dynamics with different dimensions:

$\displaystyle\dot{x}_{i}(t)=A_{i}x_{i}(t)+B_{i}u_{i}(t),\
y_{i}(t)=C_{i}x_{i}(t),\ \text{where},$ (87) $\displaystyle A_{i}$
$\displaystyle=\left[\begin{array}[]{cc}0&1\\\
0&0\end{array}\right],B_{i}=\left[\begin{array}[]{cc}0&1\\\
1&-2\end{array}\right],C^{T}_{i}=\left[\begin{array}[]{c}1\\\
1\end{array}\right],\ i=1,$ (94) $\displaystyle A_{i}$
$\displaystyle=\left[\begin{array}[]{cc}0&-1\\\
1&-2\end{array}\right],B_{i}=\left[\begin{array}[]{cc}1&0\\\
3&-1\end{array}\right],C^{T}_{i}=\left[\begin{array}[]{c}-1\\\
1\end{array}\right],\ i=2,$ (101) $\displaystyle A_{i}$
$\displaystyle=\left[\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
0.5&1&-2\end{array}\right],B_{i}=\left[\begin{array}[]{cc}1&0\\\ 0&1\\\
1&0\end{array}\right],C_{i}^{T}=\left[\begin{array}[]{c}1\\\ -1\\\
1\end{array}\right],i=3.$ (111)

In order to cooperatively solve the original optimization problem:
$\mathrm{F}(\theta)=\sum_{i=1}^{N}f_{i}(\theta),\ i=1,2,3$, the local
objective functions for each agent are described by
$f_{1}(\theta)=-2e^{-0.5\theta}+0.5e^{0.3\theta}$,
$f_{2}(\theta)=\theta^{4}+2\theta^{2}+2$, and
$f_{3}(\theta)=0.5\theta^{2}\ln(1+\theta^{2})+\theta^{2}$ [11].

The random digraphs for a team of three agents are shown in Fig. 1, where the
random process is captured by the Markov chain with the state space being
$\mathcal{S}=\\{1,2,3\\}$, and its transition rate matrix is
$\Upsilon=\left(\begin{array}[]{ccc}-0.1&0.02&0.08\\\ 0.3&-0.5&0.2\\\
0.1&0.1&-0.2\end{array}\right)$, whose row summation is zero and all off-
diagonal elements are nonnegative. The initial distribution is given by
$[0.5882,0.1500,0.3235].$ As can be seen, each graph is disconnected, while
the union of graphs contains a directed spanning tree to satisfy Assumption 3.

Next, executions of resilient distributed optimization algorithms with respect
to three different cases are presented. The values of the initial states
$x_{i}(0)$, $\varrho_{i}(0)$ and $z_{i}(0)$ are randomly chosen in an interval
$[-10,10]$. The controller gain matrix $K_{i}$ is chosen so that
$A_{i}-B_{i}K_{i}$ is Hurwitz, which are provided as: $K_{1}=[3,5;1.5,1]$,
$K_{2}=[0.75,-1;1.25,-4]$, $K_{3}=[2.167,1,0.333;0,3,1]$. Thus, the gain
matrices $U_{i}$, $W_{i}$, $X_{i}$ (solution to (11)) can be determined by:
$U_{1}=[1;0.5]$, $U_{2}=[-0.5;0]$, $U_{3}=[-1;0]$, $W_{1}=[1.5;0.5]$,
$W_{2}=[-0.5;-2]$, $W_{3}=[0;-1]$, $X_{1}=[0.5;0.5]$, $X_{2}=[-0.5;\\\ 0.5]$,
and $X_{3}=[0;-1;0]$, respectively.

Figure 1: The communication digraphs for a team of three agents. Figure 2: The
signal $r(t)$ with respect to random digraphs in Fig. 1. Figure 3: Execution
of algorithm (46) over random digraphs under $\alpha=2$, $\beta=1$: (a) the
states $y_{i}$; (b) optimal errors $\ln||y_{i}-y_{*}||$, $i=1,2,3$. Figure 4:
The sum of local objective functions $\sum^{3}_{i=1}f_{i}(y_{i})$.

Case 1: execution of reliable distributed optimization algorithm (46) over
random digraphs and without DoS attacks

In this simulation, the resilient distributed optimization algorithm (46) is
performed under only unreliable random digraphs in the absence of DoS attacks.
Figs. 2-4 depict the simulation results for the execution of (46). The signal
$r(t)$ in Fig. 2 describes the random process of unreliable digraphs in Fig.
1. Fig. 3 shows the trajectories of both output states $y_{i}$ and optimal
errors $\ln||y_{i}-y_{*}||$, respectively, while Fig. 4 illustrates the
evolution of the sum of local objective functions
$\sum^{3}_{i=1}f_{i}(y_{i})$. It can be seen that these outputs reach
consensus and converge to the optimal solution $y_{*}$.

To illustrate the effect of the parameter $\beta$, Fig. 5 illustrate the
simulation results with different $\beta$. As we can see, the larger value of
$\beta$ results in a faster convergence as discussed.

---
Figure 5: Execution of the proposed algorithm (46) over random digraphs under
(a) $\alpha=2$, $\beta=0.5$; and (b) $\alpha=2$, $\beta=1.5$.

Case 2: execution of time-based resilient distributed optimization algorithm
(8) under DoS attacks over random digraphs

In this simulation, the resilient distributed optimization algorithm (8) with
time-based communication is performed under DoS attacks over random digraphs.
The simulation result is shown in Fig. 6. The DoS attack signal is simulated
in Fig. 6(a) with $\tau_{a}=3s$. Based on (13) and (14), the two conditions
are satisfied with $F_{a}(t_{0},t)=\frac{N_{a}(t_{0},t)}{t-t_{0}}\leq 0.01$
and $T_{a}>(\lambda_{a}+\lambda_{b})/(\lambda_{a}-\eta^{\ast})=2$. That is,
the attack cannot occur more than $0.01$ times during a unit of time. Figs.
6(b) and 6(c) show the outputs and the errors, while Fig. 6(d) shows the sum
of local cost functions. As can be seen, these outputs reach consensus and
converge to the solution $y_{*}$ under DoS attacks over random digraphs.

Figure 6: Execution of the proposed algorithm (8) under DoS attacks over
random digraphs for $\alpha=2$, $\beta=1$: (a) Dos attack signals; (b) states
$y_{i}$; (c) optimal errors $\ln||y_{i}-y_{*}||$, (d) sum of local objective
functions $\sum^{3}_{i=1}f_{i}(y_{i})$, $i=1,2,3$.

Case 3: execution of event-based resilient distributed optimization algorithm
(47) under DoS attacks over random digraphs

In this part, the event-triggered resilient distributed optimization algorithm
(47) has been performed under attacks over random digraphs. The simulation
environments including DoS attacks and control parameters are set the same as
those in Case 2. Then, the simulation results are depicted in Fig. 7, from
which it can be seen that these outputs reach consensus and can converge to
$y_{*}$. Thus, the optimization problem is solvable by the proposed
distributed algorithm (47) with an event-based communication strategy under
DoS attacks over random digraphs. Fig. 8 depicts the event time instants of
agents, and there exists no Zeno-behavior.

Figure 7: Execution of the proposed algorithm (47) under DoS attacks over
random digraphs for $\alpha=2$, $\beta=1$: (a) Dos attack signals; (b) states
$y_{i}$; (c) optimal errors $\ln||y_{i}-y_{*}||$, (d) sum of local objective
functions $\sum^{3}_{i=1}f_{i}(y_{i})$, $i=1,2,3$. Figure 8: The event time
instants of all agents.

##  V Conclusion

In this paper, resilient exponential distributed convex optimization problems
were studied for linear multi-agent systems under DoS attacks over random
digraphs. The two types of time-based and event-based resilient distributed
optimization algorithms were proposed to solve these problems, respectively.
Under both algorithms, the global minimizer was achieved exponentially,
provided that an explicit analysis of the frequency and duration of attacks
was established. In addition, it was proved that there were no Zeno behavior
occurring under the dynamic event-triggered condition. Future work will
investigate resilient constrained optimization.

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