first-moment of the state ($\mathbb{E}[\|x(t)\|_{2}]$) converges to zero at a geometric rate. In presentation of this result, we use induced matrix norms (see Section 5.6 in [@hornmatrixanalysis]). Specifically, for a given matrix $M\in\mathbb{R}^{n\times n}$, let $\|M\|$ denote the induced matrix norm defined by $\|M\|\triangleq\sup_{x\in\mathbb{R}^{n}\setminus\{0\}}\frac{\|Mx\|}{\|x\|}$, where $\|\cdot\|$ on the right-hand side denotes a vector norm on $\mathbb{R}^{n}$.

\[PropositionMomentConvergence\]Consider the closed-loop networked control system (\[eq:system\]), (\[eq:control-input\]) for the case where $w(t)=0$, $t\in\mathbb{N}_{0}$. Suppose that the attacker’s interference power process $\{v(t)\in[0,\infty)\}_{t\in\mathbb{N}_{0}}$ satisfies Assumption \[FirstAssumption\]. Moreover, assume $$\begin{aligned}
 & (1-\hat{p}(\overline{v}))\|A+BK\|+\hat{p}(\overline{v})\|A\|<1.\label{eq:abkcond}\end{aligned}$$ Then the closed-loop system (\[eq:system\]), (\[eq:control-input\]) is first-moment geometrically stable, that is, there exist $\overline{\mu}\geq0$ and $\overline{\theta}\in(0,1)$ such that $$\begin{aligned}
\mathbb{E}[\|x(t)\|_{2}] & \leq\overline{\mu}\overline{\theta}^{t}\|x_{0}\|_{2},\quad t\in\mathbb{N}.\label{eq:moment-convergence}\end{aligned}$$

See the Appendix.

Proposition \[PropositionMomentConvergence\] provides a method to check the first-moment geometric stability of the system (\[eq:system\]), (\[eq:control-input\]) under jamming attacks that satisfy Assumption \[FirstAssumption\]. The scalars $\|A+BK\|$ and $\|A\|$ in condition (\[eq:abkcond\]) respectively represent the behavior of the closed-loop dynamics under successful transmissions and the open-loop dynamics under failed transmissions. Here, we select the matrix norm $\|\cdot\|$ to ensure $\|A+BK\|<1$. This is possible since the feedback gain $K$ is designed to make $A+BK$ a Schur matrix, for which such a matrix norm can be constructed (see Corollary 9.3.4 