\leq\Big(\prod_{i=0}^{t-1}\|\hat{A}(i)\|\Big)\|x_{0}\|+\sum_{j=0}^{t-2}\Big(\prod_{i=j+1}^{t-1}\|\hat{A}(i)\|\Big)\|w(j)\|\\
 & \quad+\|w(t-1)\|.\end{aligned}$$ Here, we have $\|\hat{A}(i)\|=l(i)\|A\|+(1-l(i))\|A+BK\|$, $i\in\mathbb{N}_{0}$. Hence, by letting $$\begin{aligned}
\zeta_{1} & \triangleq\|A\|-\|A+BK\|,\quad\zeta_{0}=\|A+BK\|,\label{eq:zeta-def}\end{aligned}$$ we obtain for $t\in\mathbb{N}$, $$\begin{aligned}
\|x(t)\| & \leq\Big(\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\Big)\|x_{0}\|\\
 & \quad+\sum_{j=0}^{t-2}\Big(\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\Big)\|w(j)\|+\|w(t-1)\|.\end{aligned}$$ By using this inequality, we can also obtain an upper-bound of the Euclidean norm of the state. Specifically, by Corollary 5.4.5 of [@hornmatrixanalysis], there exist $c_{1}>0$ and $c_{2}>c_{1}$ such that $$\begin{aligned}
c_{1}\|y\| & \leq\|y\|_{2}\leq c_{2}\|y\|,\quad y\in\mathbb{R}^{n}.\label{eq:norm-equivalence}\end{aligned}$$ Therefore, we have $$\begin{aligned}
\|x(t)\|_{2} & \leq\frac{c_{2}}{c_{1}}\Bigg(\Big(\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\Big)\|x_{0}\|_{2}\nonumber \\
 & \quad+\sum_{j=0}^{t-2}\Big(\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\Big)\|w(j)\|_{2}+\|w(t-1)\|_{2}\Bigg).\label{eq:two-norm-ineq}\end{aligned}$$ Notice here that the particular values of $c_{1}$ and $c_{2}$ depend on the choice of the vector norm that induces the matrix norm $\|\cdot\|$.

We use (\[eq:two-norm-ineq\]) to provide bounds on the first moment $\mathbb{E}[\|x(t)\|_{2}]$ of the state. First, in the following result, we consider the case where the disturbance is *bounded* and the jamming attacks satisfy Assumption \[SecondAssumption\].

\[TheoremBoundedDisturbance\] Consider the closed-loop networked control system (\[eq:closed-loop-system\]). Suppose that the attacker’s interference power process $\{v(t)\in[0,\infty)\}_{t\in\mathbb{N}_{0}}$ satisfies Assumption \[SecondAssumption\]. Furthermore, suppose that there exists $\overline{w}\geq0$ such that $$\begin{aligned}
\mathbb{P}[\|w(t)\|_{2}\leq\overline{w}] & =1,\quad t\in\mathbb{N}.\label{eq:almost_sure_noise_bound}\end{aligned}$$ If $$\begin{aligned}
 & (1-\hat{p}(\hat{v}))\|A+BK\|+\hat{p}(\hat{v})\|A\|<1,\label{eq:abkcond-vhat}\end{aligned}$$ then there exist $\hat{\mu}\geq0$, $\hat{\theta}\in(0,1)$, and $\hat{d}\geq0$ such that $$\begin{aligned}
\mathbb{E}[\|x(t)\|_{2}] & \leq\hat{\mu}\hat{\theta}^{t}\|x_{0}\|_{2}+\hat{d}\overline{w},\quad t\in\mathbb{N}.\label{eq:noise_to_state_one}\end{aligned}$$

The proof of Theorem \[TheoremBoundedDisturbance\] is given later in the paper. Theorem \[TheoremBoundedDisturbance\] shows that if jamming attacks satisfy Assumption 