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This document is dedicated to a systematic intensive reading of the second paper third.tex. Its task is to truly clarify this paper, enabling readers to understand several key questions by following the paper's logic: what exactly this paper is researching, what impulses are, why impulses can also be seen as a form of artificial intervention, how DoS attacks affect the system in the text, what mechanism the paper relies on for stabilization, and why it leans more towards unified stability analysis rather than complex controller design.

If the first intensive reading emphasized "how the control law is designed and how it acts on the system," then this intensive reading emphasizes "how to establish a unified stability analysis framework when continuous evolution, impulse jumps, attack intervals, and asynchronous behaviors coexist."

Paper Title and Core Theme

The title of this paper is:

Asynchronous Impulsive Stabilization of Large-scale Networks Under Denial-of-Service Attacks

Several keywords can be read from the title itself: Asynchronous, Impulsive Stabilization, Large-scale Networks, Denial-of-Service Attacks. They already clearly indicate the paper's focus. The research object is large-scale networks, the system environment contains DoS attacks, the system behavior is asynchronous, and the most important part of the stabilization means is the impulsive action.

If we translate the title into a core question that is easier to grasp, it can be expressed as:

How can large-scale networks with random disturbances achieve mean-square exponential stability via asynchronous impulsive actions under asynchronous DoS attacks.

The difference between this paper and the first paper begins to show here. The first paper focused more on discrete observation decentralized control design, while the second paper is more concerned with: when continuous feedback, impulse jumps, attack intervals, and asynchronous node behaviors are intertwined, can we use a unified method to determine whether the system is stable?

Therefore, the main focus of the second paper is not just the "control input," but more importantly, impulsive intervention and unified stability analysis.

Why Study Such a Problem

The research background of this paper also stems from several basic facts in real network systems, but its entry point is different from the first paper.

Many Systems Inherently Have Sudden Jumps

In many practical systems, the state does not always change smoothly and continuously. At certain moments, instantaneous corrections, sudden interventions, state resets, instantaneous injections, or impulse stimuli occur. Such behavior would be awkward to describe using traditional continuous system models. Impulsive system models are well-suited to handle this process of "continuous evolution most of the time, with instantaneous jumps at certain moments."

Impulsive Actions Are Powerful, Therefore Worth Studying

Continuous control typically applies adjustments continuously, while impulsive actions occur concentratedly at discrete moments, but each action can be significant. It may shrink the state instantly or amplify it instantly. Precisely because of this, impulses can become part of the stabilization mechanism or part of the destabilization source. Studying impulsive systems essentially means studying how such discrete instantaneous actions change the overall dynamic trend.

Network Attacks Make the Control Environment Less Reliable

As long as the system control links rely on networks, DoS attacks become an unavoidable problem. Attacks can render control inputs ineffective during certain periods or significantly shorten effective control time. Once continuous control is constrained, the system is more likely to deviate from the equilibrium state. In this situation, if the system also has impulsive actions, the stability problem becomes more complex and closer to real networked scenarios.

Asynchronicity Further Complicates the Problem

In reality, impulse moments for different nodes are usually different, and the time intervals during which different nodes are under attack may also differ. Some nodes may be receiving effective control while others are in attack intervals; some nodes have just experienced an impulse jump while others are still evolving according to continuous dynamics. That is, the overall system does not have a neat, unified temporal rhythm. This asynchronicity directly increases the difficulty of analysis.

Putting these backgrounds together, we can see the problem awareness of the second paper. It aims to address:

Whether the exponential stability of a large-scale stochastic network can be uniformly characterized when it simultaneously has continuous evolution, impulse jumps, control failures due to attacks, and node-level asynchronous behaviors.

First, Clarify Some Basic Concepts

To make this intensive reading easier to enter, let's first clarify the core concepts in the second paper.

What is an Impulse

In this paper, an impulse refers to: applying an instantaneous jump or instantaneous correction to the system state at certain discrete moments.

The system evolves according to continuous differential equations most of the time, but at certain specific moments, the state is immediately altered once. This alteration could be a reduction or an amplification. Mathematically, it is usually written as some jump mapping.

Why Impulses Can Also Be Understood as Artificial Intervention

Because although impulses are not written as traditional continuous control input $u(t)$, they essentially apply a rule-based correction to the state directly at certain moments. You can think of it as scheduled correction, instantaneous reset, discrete correction, or instantaneous intervention. From a control philosophy perspective, this is still a regulation mechanism, so impulsive control can fully be regarded as a control method.

What Are Stabilizing Impulses and Destabilizing Impulses

If after an impulsive action, the state magnitude decreases, then it is beneficial for stability; such impulses are called stabilizing impulses. If after an impulsive action, the state magnitude increases, then it is detrimental to stability; such impulses are called destabilizing impulses.

A very important point of the second paper is that it does not only study "always beneficial impulses," but incorporates both types of impulses into the same theoretical framework.

What is Unified Stability Analysis

Unified stability analysis can be understood as: not cutting various situations into many independent small pieces for separate proofs, but using a common analysis framework to unify them into the same stability criterion.

For this paper, what needs to be uniformly handled includes impulse and non-impulse intervals, attack and non-attack intervals, stabilizing and destabilizing impulses, and asynchronous behaviors of different nodes. This unified perspective is the theoretical focus of the second paper.

What Problem Does the Second Paper Truly Aim to Solve

If we state the problem of this paper more completely, what it truly aims to answer is:

Under asynchronous DoS attacks, when a large-scale stochastic network is simultaneously affected by continuous feedback control, asynchronous impulsive intervention, and random disturbances, can a unified mean-square exponential stability criterion be established?

There are several layers of meaning that must be grasped simultaneously.

First, the system is not purely continuous; it has both continuous evolution and discrete jumps. Second, the direction of impulsive actions is not fixed; some impulses favor stability, some destroy stability. Third, DoS attacks cause continuous feedback control to fail during certain intervals. Fourth, impulse moments and attack moments for different nodes may all be asynchronous.

Therefore, this paper discusses a very typical stability problem of asynchronous hybrid systems. Its focus is not on studying a single factor in isolation, but on studying how the overall system stability can be uniformly characterized when these factors are superimposed.

What is the System State

In the paper, the state of the $a$-th node is denoted as $\eta_a(t) \in \mathbb{R}^l$

Here, $\eta_a(t)$ represents the state vector of the $a$-th node at time $t$. The total state of the entire large-scale network can be written as $\eta(t)=\big(\eta_1^\top(t),\eta_2^\top(t),\dots,\eta_L^\top(t)\big)^\top$

This indicates that the "system stability" discussed in the second paper ultimately still asks: Can the overall state $\eta(t)$ converge to zero?

This point is essentially consistent with the first paper. Although the model structures of the two papers differ—one leaning towards discrete observation control, the other towards impulsive hybrid systems—both ultimately need to aggregate local node behaviors into a global stability conclusion for the entire network state.

How is the System Dynamics Constituted

At non-impulse moments, the system can be summarized as $\mathrm{d}\eta_a(t)=\Big[F_a(t,\eta_a(t))+\sum_{b=1}^{L}d_{ab}\eta_b(t)+\alpha_a(t)\rho_a\eta_a(t)\Big]\mathrm{d}t+G_a(t,\eta_a(t))\mathrm{d}B(t)$

In this equation, each term corresponds to a mechanism of action in the system.

The Self-Dynamics Term Describes How the Node Itself Evolves

$F_a(t,\eta_a(t))$ represents the natural evolution law of node $a$. It tells us how the node state itself changes over time without coupling and control influences.

The Coupling Term Reflects the Network Structure

$\sum_{b=1}^{L}d_{ab}\eta_b(t)$ indicates that node $a$ is influenced by the states of other nodes. Here, $d_{ab}$ characterizes the coupling relationship between nodes. This term shows that the system does not operate with nodes independently; local fluctuations propagate through the network topology.

The Continuous Feedback Term Provides Daily Stabilization

$\alpha_a(t)\rho_a\eta_a(t)$ is the continuous feedback control term. Usually $\rho_a<0$, so it corresponds to negative feedback. This negative feedback continuously suppresses state growth when the control link is available.

The Random Disturbance Term Reflects the Uncertain Environment

$G_a(t,\eta_a(t))\mathrm{d}B(t)$ indicates that the system is continuously affected by random noise, so stability analysis must be conducted in a stochastic sense.

If we only look at this continuous part, the system already includes self-dynamics, network coupling, continuous feedback, and random disturbances. After superimposing the subsequent impulse jumps, the entire model becomes a typical stochastic hybrid system.

How Do DoS Attacks Enter the Model

The second paper handles DoS attacks in a very direct way, using a switching signal $\alpha_a(t)$ to describe whether the control channel is available:

$\alpha_a(t)= \begin{cases} 1, & \text{when control is available} \\ 0, & \text{when DoS attack causes control failure} \end{cases}$

This means that when $\alpha_a(t)=1$, the continuous feedback control for node $a$ is effective; when $\alpha_a(t)=0$, the DoS attack temporarily renders this part of the control ineffective.

The advantage of this modeling approach is obvious. The impact of attacks on system stability is written very directly into the dynamic equation. The system has continuous control support during some time periods, while during others, it can only rely on its own dynamics, coupling relationships, and possibly existing impulsive actions to continue evolving.

Therefore, DoS attacks in the second paper are not peripheral background but enter the system equation directly through $\alpha_a(t)$, substantially altering the system's contraction capability.

How Are Impulses Represented Mathematically

At impulse moments, the state undergoes an instantaneous jump. A typical form can be written as $\eta_a(t_k^a)=\gamma_a\eta_a(t_k^{a-})$, or more generally as $\eta_a(t_k^a)=\gamma_a^k\eta_a(t_k^{a-})$

Here:

• $t_k^a$ denotes the $k$-th impulse moment of the $a$-th node
• $\eta_a(t_k^{a-})$ denotes the state just before the impulse occurs
• $\eta_a(t_k^a)$ denotes the state just after the impulse occurs
• $\gamma_a$ or $\gamma_a^k$ denotes the impulse strength

The meaning of this expression is very intuitive. The system has almost no continuous transition before and after the impulse; instead, at that moment, it directly completes a state correction according to some proportion or mapping. Therefore, impulsive control studies how this instantaneous correction at discrete moments affects the long-term stability trend of the entire system.

Why Are Impulses Also a Form of Artificial Intervention

This is a key point in understanding the second paper. If one only looks at the equation form, many might think impulses are different from control inputs because they are not written as $u(t)$ in continuous time. But from the essence of control, impulses can fully be understood as an intervention mechanism.

The reason is simple. Impulses essentially apply an externally defined mapping to the system state at certain moments. This mapping may correspond to scheduled correction, discrete reset, instantaneous correction, fault recovery, or some planned intervention. It's just that its action does not occur continuously but is concentrated at discrete moments.

Therefore, in a control sense, the second paper can be understood as having two types of regulation methods simultaneously:

• One is the continuous feedback control that exists persistently during normal times
• One is the impulsive intervention that occurs at specific moments

This also explains why the second paper resembles a hybrid control system more. It does not rely solely on daily continuous suppression but also allows instantaneous correction of the state at critical moments.

What Do Stabilizing and Destabilizing Impulses Really Mean

Whether an impulse favors stability depends on how it instantaneously changes the state.

If $|\gamma_a|<1$, then after the impulse, the state magnitude shrinks immediately. For example, the original state magnitude might be $10$, and after the impulse, it becomes $8$. Such impulses enhance the system's contractility and are therefore called stabilizing impulses.

If $|\gamma_a|>1$, then after the impulse, the state magnitude amplifies immediately. For example, the original state magnitude might be $10$, and after the impulse, it becomes $12$. Such impulses weaken the system's stability and are therefore called destabilizing impulses.

The reason the second paper particularly emphasizes this distinction is that many existing works often only study the case where "impulses are always beneficial" or separate the two types of impulses into two completely different sets of conclusions. What this paper aims to do more is: incorporate both stabilizing and destabilizing impulses into the same analysis framework and see whether the system as a whole can still be stable.

This step is very important because instantaneous interventions in real systems do not always bring ideal effects. Sometimes they help the system return to the equilibrium point; sometimes they introduce additional disturbances. Putting both possibilities into the theory makes the conclusions more general.

What Mechanism Does the Second Paper Rely on to Stabilize the System

If asked "what does the system rely on to stabilize?" the answer cannot be just "impulses" or just "continuous feedback." A more accurate understanding is:

It relies on continuous feedback control, stabilizing impulsive actions, and the net contraction mechanism within the overall analysis framework to jointly achieve stability.

Continuous Feedback Provides Daily Suppression

During non-attack intervals, the continuous feedback term, $\alpha_a(t)\rho_a\eta_a(t)$ continuously takes effect. Since $\rho_a<0$, it continuously suppresses state growth. This is the system's daily stabilization mechanism.

Stabilizing Impulses Provide Instantaneous Correction

At impulse moments, if the impulse is stabilizing, the state is instantaneously shrunk. This action is like performing a quick correction at critical moments. It cannot replace continuous feedback but can significantly improve the system's contraction trend.

Attacks and Destabilizing Impulses Disrupt This Contraction

DoS attacks cause continuous feedback to fail during certain intervals, and destabilizing impulses instantly amplify the state at certain moments. That is, both favorable and unfavorable factors exist simultaneously on the system's timeline.

Final Stability Depends on the Net Effect

What the paper actually proves is that under certain parameter and timing conditions, the cumulative favorable influences are sufficient to outweigh the unfavorable ones, so the system as a whole remains mean-square exponentially stable.

This also explains why the second paper emphasizes "unified stability analysis" more. What it truly aims to answer is not whether a single control component is effective, but whether the overall effect accumulated over the entire timeline ultimately shows contraction or growth.

Why the System May Still Be Stable When DoS Exists

If we only look at continuous feedback control, then when $\alpha_a(t)=0$, the control term indeed fails, and the system loses part of its stabilization support. But the system structure in the second paper is more complex than the simple dichotomy of "controlled or uncontrolled."

Because besides continuous feedback, the system may also receive additional corrections at impulse moments. If some impulses are stabilizing, they pull the state smaller at discrete moments. Thus, even if continuous control is limited by attacks during some intervals, the system may not immediately lose stability. The key question lies in how long the attack intervals last, whether stabilizing impulses appear timely enough, to what extent destabilizing factors accumulate, and how the overall balance of these factors is.

Therefore, the difficulty of the second paper lies precisely here: it studies not "the system becomes completely unsolvable once attacks come," but "whether the system can maintain convergence relying on the overall mechanism when attacks exist and impulsive actions are complex."

What Does "Stability" Specifically Refer to in the Second Paper

The second paper also discusses mean-square exponential stability. A typical form can be written as $\mathbb{E}\lVert \eta(t)\rVert^2 \le C e^{-\omega (t-t_0)}$

Where:

• $\eta(t)$ represents the overall system state
• $C>0$ is a constant
• $\omega>0$ is the exponential decay rate

This conclusion means that the expectation of the squared system state decreases exponentially over time. That is, despite the presence of random noise, network attacks, and impulse jumps in the system, the overall state can still converge to zero at a definite rate.

This definition is important because it reminds us that the paper pursues not a weak sense of "boundedness" or "eventually not exploding," but a strong sense of exponential convergence conclusion.

What is Unified Stability Analysis, and Why is it the Main Thread of the Second Paper

Unified stability analysis is the most important theoretical keyword to grasp in the second paper. Its core idea is:

**Using a common analysis