## 4 Theoretical Results
By analyzing the case of \(T_{k}-t_{k}\geq\tau_{u}^{s}\) or \(T_{k}-t_{k}<\tau_{u}^{s}\), one has \(\int_{t_{k}}^{t}\left\|e_{u_{s}}\right\|^{2}\mathrm{d}\omega\leq\sigma_{u}^{s }\left\|\Pi_{u}^{s}\bar{K}\right\|^{2}\int_{t_{k}}^{t}\left\|X_{c}\right\|^{2} \mathrm{d}\omega\), \(t\in[t_{k},T_{k}]\). This means that Label (15) holds for \(t\in[t_{k},T_{k}]\). If \(t_{s}^{s}<t_{k}\), there must exist \(0\leq\bar{k}<k\) such that \(t_{s}^{s}=t_{\bar{k}}\). Then, one can prove Label (15) for \(t\in[t_{k},T_{k}]\) via the parallel process for the case of \(t_{s}^{s}=t_{k}\), where \(t_{k}\) is replaced by \(t_{\bar{k}}\). As a result, \(\int_{t_{0}}^{t}E^{\mathrm{T}}\mathrm{E}\mathrm{d}\omega\leq\theta\rho_{0}^{2 }\int_{t_{0}}^{t}X_{c}^{\mathrm{T}}X_{c}\mathrm{d}\omega\) can be proved for \(t\in[t_{k},t_{k+1})\) by the contradiction with \(T_{k}\). Then, by induction, one has \(\int_{t_{0}}^{t}E^{\mathrm{T}}\mathrm{E}\mathrm{d}\omega\leq\theta\rho_{0}^{2 }\int_{t_{0}}^{t}X_{c}^{\mathrm{T}}X_{c}\mathrm{d}\omega\) for \(t\in[t_{0},\infty)\). Thereby, the stability can be proved. Moreover, the stability implies the uniform continuity of \(\left\|X_{c}\right\|^{2}\). The limit of the monotone decreasing function \(-(1-\theta)\epsilon_{2}\rho_{0}^{2}\int_{t_{0}}^{t}X_{c}^{\mathrm{T}}X_{c} \mathrm{d}\omega\) exists, since the function is lower bounded by \(-V(0)\). Then, from [PERSON]'s Lemma (see [29] for more details), \(\left\|X_{c}\right\|^{2}\) converges to zero. Therefore, the closed-loop system is asymptotically stable and the proof is completed. **Remark 2**. : _The proof of Theorem 1 implies that, to guarantee the stability, one can also adopt a time-triggered control where the sampling period of sensor and actuator nodes are \(\tau_{u}^{s},s=1,\ldots,p\) and \(\tau_{u}^{l},l=1,\ldots,r\), respectively. Hence, for stability, the event conditions are not necessary. However, the time-triggered control is not preferable to save communication resources. As shown in the Simulation section, by introducing event-triggered control, quite a larger average inter-event time can be obtained._  
For given \(K\), \(L\) and \(\rho_{0}\), the ellipsoid \(\Gamma(P_{1},P_{2})\) describes the admissible region for initial states such that the closed-loop system is asymptotically stable. Thereby, one may expect to find the largest one among them. To define the \"largest\", a measure that can reflect the geometrical size of \(\Gamma(P_{1},P_{2})\) is required. Generally, the measure is often considered as volume. However, Ref. [30] pointed out that the volume optimization can lead ellipsoids to be \"flat\" in some directions. On the contrary, the trace optimization yields the ellipsoids that tend to be homogeneous in all directions. Referring to [18; 30], we consider the following optimization problem to find the largest ellipsoid for given \(K\), \(L\), \(\rho_{0}\) and some \(\epsilon_{1},\epsilon_{2}>0\):  
\[ \begin{split}&\min-tr(P_{1}^{-1}+P_{2}^{-1})\\ &\text{subject to (\ref{eq:1}) and (\ref{eq:2}),}\end{split}  \tag{18}\]  
where the optimization variables are \(\{P_{1},P_{2},H_{1},H_{2}\}\). However, the optimization problem (18) is nonlinear subject to \(P_{1}^{-1}\) and \(P_{2}^{-2}\). Hence, to transfer the above problem into a linear form, we propose the following theorem, where the LMI optimization problem can be solved by the standard LMI toolbox in MATLAB (8.4.0.150421, The MathWorks, Natick, Massachusetts, United States, 2014). Theorem 2: _The solutions to Label (18) can be obtained from the following LMI optimization problem:_  
\[\min-tr(Q_{1}+Q_{2})\] subject to \[\left( \begin{array}{cccccccc}\bar{\varphi}_{11}&BD_{j}^{-}\bar{H}_{2}&BD_{ j}^{+}&0&BD_{j}^{+}K&\varepsilon_{2}\rho_{0}Q_{1}&0&0\\ *&\bar{\varphi}_{22}&0&L&0&0&\varepsilon_{2}\rho_{0}Q_{2}&\varepsilon_{1}Q_{2 }\\ *&*&-\varepsilon_{2}I&0&0&0&0\\ *&*&*&-\varepsilon_{2}I&0&0&0&0\\ *&*&*&*&-\varepsilon_{1}I&0&0&0\\ *&*&*&*&*&-\varepsilon_{2}I&0&0\\ *&*&*&*&*&*&-\varepsilon_{1}I\\ \end{array} \right)<0,j\in M, \tag{19}\] \[\left(