## 2 System Description and Preliminaries
### Network Control Based on Dynamic Event-Triggered Scheme
\tag{17}\]  
where \(\kappa_{0}\geq 0\), \(\varsigma\) is a function of class \(K_{\infty}\) with Lipschitz continuous [24]. Compared with static event triggering scheme, dynamic event triggering scheme is less conservative that do not have to satisfy \(\sigma x^{T}(t)Wx(t)-e^{T}(t)We(t)\geq 0\). **Remark 1**. : _Notably, several factors that need to be carefully chosen for the communication strategy may be found in the event-triggered mechanism (16). Firstly, a bigger \(\sigma\) may accept a larger measured error \(e(t)\), which results in fewer data packets being triggered for the controller update. \(\sigma\) describes how tight the triggering process is. The internal dynamic variable \(\kappa(t)\) participation is then described by the parameter \(\rho\)._  
**Remark 2**. : _The internal variable \(\kappa(t)\) under the starting \(\kappa_{0}\geq 0\) meets the following criterion for the dynamic event-triggered scheme (16)_  
\[\kappa(t)\geq 0,\quad\rho>0.\]  
For the rest of this paper, we will consider the flexible manipulator subject to the semi-Markov jump process in the network environment, through the design of the corresponding event-trigger controller to make the system state asymptotically stable and meet the \(\mathcal{H}_{\infty}\) performance of parameter \(\gamma\). The event-trigger controller is selected as follows:  
\[u(t)=K_{i}(x(t-\tau(t))-e(t)) \tag{18}\]  
**Remark 3**. : _Compared with the traditional static ETM, there are two advantages on the proposed DETM. First, the DETM can effectively avoid the transmission of redundant data and reduce the number of triggering compared with static ETM. Second, the event-triggered parameter \(\varsigma(\kappa(t))\) can be dynamically selected adaptively, which provides greater potential for optimization of the event-trigger mechanism. In this paper, the parameter \(\varsigma(\kappa(t))\) is assumed to be a direct proportional function which belongs to a function of class \(K_{\infty}\) with Lipschitz continuous. Furthermore, more options for the event-trigger parameter can be seen in existing results [31; 32]._  
In the development of network technology, some hackers seek personal gains by launching network attacks, which represents both an immoral and illegal act. In this article, we consider a common form of network attack: DoS attacks. The Bernoulli process is used to describe the phenomena, the following is suggested for the affected controller:  
\[\bar{u}(t)=h(t)u(t), \tag{19}\]  
where the Bernoulli distributed variables \(h(t)\) display the frequency of DoS assaults and \(\mathbb{E}\{h(t)=h_{1}\}\). Introduce into switch matrix \(L\) to describe DoS attacks:  
\[L=\left\{ \begin{array}{cc}1&\text{No attack},\\ 0&\text{DoS attack},\end{array} \right.\]  
In summary, the manipulator's closed-loop control system is stated as  
\[\left\{ \begin{array}{l}\dot{x}(t)=A_{i}x(t)+B_{1i}\omega(t)+B_{2}LK_{i}x(t- \tau(t))-B_{2}LK_{i}e(t)\\ z(t)=Cx(t),\quad t\in\left[t_{k}h+\tau_{k_{i}}t_{k+1}h+\tau_{k_{i+1}}\right] \\ x(t)=\vartheta(t),\quad t\in\left[-\tau_{2},0\right]\end{array} \right. \tag{20}\]  
where \(\vartheta(t)\) is the initial function of \(x(t)\). Some necessary lemmas are presented to complete the following theorem. **Problem Statement:**_For a given semi-Markov flexible robotic arm system suffering from DoS attacks in (5), the dynamic event-triggered controller is designed in (19), so that the resulting system in (20) is asymptotically stable with a \(\mathcal{H}_{\infty}\) performance._  
**Lemma 2**. : _[_33_]_ _For matrix \(R>0\), scalar \(a,b\), satisfy \(a<b\). The function \(x(s):[a,b]\rightarrow\mathbb{R}^{n}\) is continuously differentiable, and the following integral inequality is true_  
\[-(b-a)\int_{a}^{b}\dot{x}^{T}(s)R\dot{x}(s)ds\leq-\left[\int_{a}^{b}\dot{x}(s )ds\right]^{T}R\left[\int_{a}^{b}\dot{x}(s)ds\right] \tag{21}\]  
**Lemma 3**.