## 5 Stability Analysis  
From the developments in Section 4.2, it is clear that the overall system involves interactions between continuous- and discrete-time dynamics. Hence, it would be realistic to model the closed-loop system as a hybrid dynamical system. In view of (8)-(11) and since ZOH is employed, we obtain the following impulsive model:  
\[ \begin{array}{l}\dot{x}(t)=\mathcal{A}_{1}x(t)+\mathcal{B}_{1}e(t)+ \mathcal{E}_{1}w(t)\\ \dot{e}(t)=\mathcal{A}_{2}x(t)+\mathcal{B}_{2}e(t)+\mathcal{E}_{2}w(t)\\ u(t)=-Kx(t_{k})\\ \dot{e}(t_{k})=0\end{array} \qquad\qquad\qquad\qquad\forall t\in[t_{k},t_{k+1}). \tag{19}\]  
Additionally, a time variable with the following dynamics needs to be included as an auxiliary variable  
\[\dot{\tau}(t)=1\quad\forall t\in[t_{k},t_{k+1}),\qquad\tau(t_{k}^{+})=0. \tag{20}\]The variable \(\tau\) is used to track the elapsed time from the transmission instant \(t_{k},k\in\mathbb{N}\) until the next sampling time \(t_{k+1}\), where \(\tau\) is reset to zero, as described in (20). We employ the framework [40] to define the closed-loop system as a hybrid dynamical system, where the _jump dynamics_ refer to the discrete-time changes and the _flow dynamics_ refer to continuous-time behavior. Additionally, the _flow set_ and the _jump set_ are used to indicate whether the system is operating in continuous or discrete time, respectively. The reader may refer to [40] for more information on these terms. Let \(\xi:=(x,e,\tau)\in\mathbb{R}^{n_{x}}\times\mathbb{R}^{n_{x}}\times\mathbb{R}\). Then, the hybrid dynamical system is  
\[\xi(t) \in \left[ \begin{array}{ll}\mathcal{A}_{1}x(t)+\mathcal{B}_{1}e(t) +\mathcal{E}_{1}w(t)\\ \mathcal{A}_{2}x(t)+\mathcal{B}_{2}e(t)+\mathcal{E}_{2}w(t)\\ 1\end{array} \right],\quad\xi(t)\in\mathcal{C} \tag{21}\] \[\xi(t_{k}^{+}) \in \left[ \begin{array}{ll}x\\ 0\\ 0\end{array} \right],\qquad\qquad\qquad\qquad\qquad\xi(t)\in\mathcal{D},\]  
where the flow set \(\mathcal{C}\) and the jump set \(\mathcal{D}\) are constructed based on the ETC (12) as follows:  
\[\mathcal{C} = \big{\{}\xi(t):|e(t)|\leq\sigma|x(t)|\;\;\text{or}\;\;\tau\in[0, T]\big{\}} \tag{22}\] \[\mathcal{D} = \big{\{}\xi(t):|e(t)|\geq\sigma|x(t)|\;\;\text{and}\;\;\tau\geq T \big{\}}.\]  
Note that the flow and jump sets \(C,D\) are closed and the flow and jump maps in (21) are continuous, which ensures that the hybrid model (21) is well-posed; see [40] for more details. Now, we present the main results. **Theorem 1**. : _Consider systems (21) and (22) with Assumption 1. Take \(\sigma=\frac{\sqrt{\xi}}{\gamma}\) and \(T\in(0,\mathcal{T}(\gamma,L))\) in (22). Then, systems (21), (22) are \(\mathcal{L}_{2}\)-gain-stable from \(w\) to \(x\) with an \(\mathcal{L}_{2}\)-gain that is less than or equal to \(\sqrt{\eta}\)._  
Proof of Theorem 1.: Recall that \(\xi=(x,e,\tau)\), as defined in (21). For all \(\xi\in C\cup D\), we define \(R(\xi):=V(x)+\max\{0,\tilde{\tau}\phi(\tau)W^{2}(e)\}\) with \(V(x)\) and \(W(e)\), as defined in (15) and (14). We study the dynamics of \(R(\xi)\) during the flow and jump phases. Let \(\xi\in D\), we obtain, in view of (21), and the fact that \(W(0)=0\)  
\[R(G(\xi)) = V(x)+\max\{0,\tilde{\gamma}\phi(0)W^{2}(0)\}\] \[\leq R(\xi),\]  
where \(G(\xi):=(x,0,0)\). Let \(\xi\in C\), and suppose that \(\phi(\tau)<0\), which implies that \(\tau>T\). Hence, \(\gamma^{2}W^{2}(e)\leq\varepsilon|x|^{2}\) in view of (22) since \(\xi\in C\).