& <0,\label{eq:almost-sure}\end{aligned}$$ with $\left\Vert \cdot\right\Vert $ denoting the matrix norm induced by the vector norm $\|x\|_{P}\triangleq\sqrt{x^{\mathrm{T}}Px}$ where $P\in\mathbb{R}^{n\times n}$ is a positive-definite matrix. For this matrix norm, (\[eq:abkcond\]) implies (\[eq:almost-sure\]).

Joint Effects of Jamming Interference and Disturbance {#subsec:Joint-Effect-of}
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We next look at the case with disturbance. We observe that in this case, jamming attacks can become considerably more dangerous. Even if the disturbance is very small and the attacker has very limited jamming resources, there still exist attack strategies that can *destabilize* the system while satisfying Assumption \[FirstAssumption\] with very small $\overline{v}$. We illustrate this idea in the following example.

\[ExampleStrategy\] Consider a scalar networked control system (\[eq:system\]), (\[eq:control-input\]) with $x_{0}>0$, $A+BK\in[0,1)$, $A>1$, and constant disturbance $w(t)=w^{*}>0$, $t\in\mathbb{N}_{0}$. Suppose that the conditional probability $p$ of transmission failures is a strictly increasing function (e.g., $p$ given by (\[eq:p\_from\_reference-1\])). For this setup, an attacker can wait for a sufficiently long duration and then attack for a duration with a sufficiently large interference power so that the state norm grows to large values but the average interference power does not go above $\overline{v}$. In particular, for any $\overline{v}>0$, $x_{0}>0$, $z>0$, and $\rho\in(0,1)$, the attack strategy $$\begin{aligned}
v(t) & \triangleq\begin{cases}
v^{*}, & t\in\{\tau_{1},\ldots,\tau_{1}+\tau_{2}-1\},\\
