## 2 Preliminaries and Problem Formulation
### Dual-Mode Security Control Scheme  
In general, the purpose of designing event-based data update strategy is to improve communication utilization. However, the malicious attackers always launch DoS attacks on the S-C channel covertly to destroy information interaction. This will inevitably lead to the accumulation of actual error between the historical update state and the current operation state such that the event-triggered condition is violated for a long time because the state error cannot be reset to zero. These facts motivates us to design a resilience margin for the event-triggered condition (3), which aims to be more robust against DoS attacks. From this, for any aperiodic interval \([\mathcal{T}_{n},\mathcal{T}_{n+1})\), let \(\mathcal{H}_{n}=\{\mathcal{T}_{n}^{\text{on}}\}\cup[\mathcal{T}_{n}^{\text{on }},\mathcal{T}_{n+1})\) be the nth DoS active interval shown in Figure 2, where \(\{\mathcal{T}_{n}^{\text{on}}\}_{n\in\mathbb{N}_{0}}\) represents an injected DoS sequence. Then, for any \(t\in[t_{a},t_{b}]\), we define  
\[\mathcal{D}_{1}(t_{a},t_{b})=\bigcup_{n\in\mathbb{N}_{0}}\mathcal{H}_{n} \bigcap[t_{a},t_{b}] \tag{5}\]  
and  
\[\mathcal{D}_{2}(t_{a},t_{b})=[t_{a},t_{b}]\backslash\mathcal{D}_{1}(t_{a},t_{ b}) \tag{6}\]  
as the union and relative complement of DoS subintervals over interval \([t_{a},t_{b}]\), respectively. Clearly, once the intermittent DoS attacks are successfully launched over a shared communication network, certain packets determined by the event-triggered rule (3) will be lost during the transmission. This implies that these criteria for stability analysis and controller synthesis presented in [24; 25; 27] may no longer be valid. Moreover, these packets that  
Figure 1: Control architecture of MJCPs under DoS attacks. cannot be updated successfully are also considered redundant. In this case, we need to develop an improved event-triggered rule with a resilience margin according to (3). Firstly, the actual state error caused by DoS attacks can be defined as  
\[e_{a}(t)=x(t)-x(t_{k}),\;t\in\left[t_{n}^{\ell},t_{n}^{\ell+1}\right) \tag{7}\]  
where \(\{t_{n}^{\ell}\}_{\ell\in\mathbb{N}_{0}}\subseteq\{t_{k}\}_{k\in\mathbb{N}_{0}}\) denotes a successful transmission sequence during the interval \([\mathcal{T}_{n},\mathcal{T}_{n+1})\). Note that \(e_{a}(t)=e_{s}(t)\) holds if DoS attacks do not exist; otherwise, DoS attacks cause an additional state error that can degrade security performance, i.e., \(e_{a}(t)>e_{s}(t)\) holds. Then, a novel event-triggered rule with a resilience margin can be designed as  
\[ \begin{split}\mathfrak{T}_{*}&=\max\Bigl{\{}t_{k} +\Delta,\inf\Bigl{\{}t\in\mathbb{R}_{>t_{k}}\wedge t_{n}^{\ell}|\varepsilon_{a }^{T}(t)\Theta_{a}e_{a}(t)\\ &\qquad-\sigma x^{T}(t_{k})\Theta_{b}x(t_{k})-(1-\alpha(t)) \mathcal{J}_{d}>0\Bigr{\}}\Bigr{\}},\end{split}  \tag{8}\]  
where \(\mathfrak{T}_{*}\) represents the latest update instant when DoS attacks are no longer injected, \(\Delta\in\mathbb{R}_{>0}\) denotes an unknown constant to be calculated, \(\mathcal{J}_{d}\leq\mathcal{J}_{\max}\) is the performance compensation for condition (3), and \(\mathcal{J}_{\max}\) is the maximum resilience margin. Furthermore, \(\alpha(t)\in\{0,1\}\) is a Dirac measure used to describe DoS on/off properties, and its mathematical expectation can be characterized as  
\[ \begin{cases}\text{Prob}\{\alpha(t)=1\}=\mathbb{E}\{\alpha(t)\}=\alpha\\ \text{Prob}\{\alpha(t)=0\}=1-\alpha\end{cases}  \tag{9}\]  
where \(\alpha\in[0,1]\) is a positive scalar. In this way, we can design a novel dual-mode security controller under the resilient event-triggered rule (8) as follows:  
\[u(t)=\alpha(t)K_{r(t)}^{a}x(t_{k})+(1-\alpha(t))\xi_{r(t)}K_{r(t)}^{b}x(t_{k}) \tag{10}\]  
where \(\xi_{r(t)}\in\mathbb{R}_{>0}\) denotes a given regulated parameters, \(K_{r(t)}^{a}\) and \(K_{r(t)}^{b}\) represent the unknown control gains without or with DoS attacks, respectively. Let \(\tau(t)\triangleq t-t_{n}^{\ell}\) denote the time delay, then the control protocol (10) can be further rewritten as  
\[