---
abstract: 'The problem of wireless networked control under the joint effects of disturbance and jamming attacks is investigated. Specifically, the control input packets are assumed to be transmitted from the controller to a linear plant over an insecure wireless communication channel that faces jamming attacks. The time-varying likelihood of transmission failures on this channel depends on the power of the jamming interference signal emitted by an attacker. We show that jamming attacks and disturbance can jointly prevent stability even if the attacked system without disturbance is stable. We also show that stability under jamming and disturbance can be achieved if the average jamming interference power is restricted in a certain way that we characterize in the paper. We provide a numerical example to illustrate our results.'
author:
- 'Ahmet Cetinkaya, , Hideaki Ishii, , Tomohisa Hayakawa, [^1] [^2] [^3] [^4]'
bibliography:
- 'references.bib'
title: |
    Joint Effects of Jamming Interference and Disturbance in\
    Wireless Networked Control Systems
---

Networked control, cyber-security, wireless networks, jamming interference, disturbance

Introduction
============

As the Internet of Things is gaining popularity, the use of wireless communication channels and the Internet is increasing in remote control applications. These communication technologies are easy to set up and they provide efficiency in the transmission of measurement and control data, but they can create major cyber-security issues in a networked system [@wholejournal2015]. In the framework of cyber-physical systems, researchers have identified a range of potential cyber attacks with different properties [@chong2019tutorial; @dibaji2019systems; @lun2019state]. For instance, an attacker who is knowledgeable about the system dynamics can disrupt control operation by injecting false data into the system or altering measurement and control data [@mo2010false; @fawzi2014secure]. Attackers with limited information can also cause cyber-security issues by means of denial-of-service (DoS) attacks to prevent communication over networks. For example, a jamming attacker can effectively prevent transmission of packets over wireless channels by emitting sufficiently strong interference signals, [@pelechrinis2011]. Jamming attacks can cause performance issues and instability in wireless networked control systems.

The effects of jamming and other Denial-of-Service attacks in control systems have recently been investigated in a few works (see [@cetinkaya-entropy] for an overview). In those works various attack models have been considered. For instance, [@shishehsiam2016] considered a model where the attacker conducts cycles of sleeping and jamming in a repetitive fashion. Moreover, the works [@de2015inputtran; @IFACde2016networked; @cetinkaya2016tac; @feng2017resilient] considered models that allow the timing of attack strategies to be arbitrary as long as the average attack duration and the average frequency of attacks satisfy certain bounds. It was first observed in [@de2015inputtran] that when a networked control system is subject to disturbance, duration and frequency conditions for attacks need to be stronger to guarantee stability in comparison to the case without disturbance.

In this paper, our goal is to investigate the combined effects of disturbance and jamming attacks specifically for *wireless* networked control problems within a probabilistic framework. In particular, we consider the control problem over a wireless channel, where the transmission failure model is characterized through the time-dependent Signal-to-Interference-plus-Noise-Ratio (SINR), which is the ratio of the transmission power of the signal to the jamming attacker’s interference power summed with the channel noise power. A jamming signal with a strong interference power results in a smaller SINR, which ends up increasing the likelihood of a packet loss.

Previously, SINR-based channel models were used by [@IFACli2015jamming; @IFACzhangattack; @IFACli2016sinr] for game-theoretic analysis of remote state estimation problems under jamming attacks. Moreover, in [@IFACzhang2016optimal], a probabilistic channel model was considered in a networked control problem setting and optimal attack policies were explored for the case where the total number of attacks in a fixed interval is bounded. In [@ahmetifacwc2017], we used an SINR-based probabilistic model to investigate a discrete-time networked stabilization problem for scenarios where there is no disturbance, but a jamming attacker can jam the wireless channel at each time instant with a different interference power level that is unknown a priori. Our results in [@ahmetifacwc2017] indicate that stabilization can be achieved if the average interference power is bounded in the long run even if the power can be very large at certain times.

In this paper we consider situations where the jamming attacker can strategically change the interference levels at each time, as in [@ahmetifacwc2017]. However, differently from [@ahmetifacwc2017], we now consider disturbance, and through stochastic analysis, we show that when the dynamics is subject to disturbance, jamming attacks can potentially become more dangerous. Our results indicate that a strategic attacker may take advantage of the disturbance to cause instability even if the attacked system without disturbance is stable. Specifically, the attacker can cause the state norm to grow to arbitrarily large values with arbitrarily high probabilities, while keeping the average jamming interference power below a threshold in the long run. Thus, as in the deterministic case discussed in [@de2015inputtran], a restriction is also needed in this paper. We consider a probabilistic model and the attacker can only partially affect the occurrence probability of a transmission failure. We show that when jamming attacks are restricted so that the wireless channel is not subject to long consecutive emissions of high powered interference signals, then the first moment of the state stays bounded. Interestingly, even under such restrictions, the wireless channel may be attacked at all time instants with small interference powers and thus for any finite interval, there is always a positive probability that all transmission attempts may fail. In this aspect, our setting differs from the deterministic case, where the maximum possible length of a continuous attack duration is required to be bounded to ensure input-to-state stability under disturbance.

As a first step, we investigate the scenarios where the norm of the disturbance is bounded almost surely at each time by a fixed scalar. In such scenarios, the first moment of the state is bounded under attacks from an attacker with sufficiently small resources. Then we explore the more general case where the distribution of the disturbance norm may have infinite support. For this case, we obtain an inequality for the first moment of state that resembles those used for establishing noise-to-state stability in stochastic systems (e.g., [@nunez2014; @zhang2016noise]). In particular, we obtain an upper bound of the first moment of the state by utilizing the second moment of the disturbance. In our analysis, a key technical role is played by a nondecreasing and concave function of the attacker’s interference power that upper-bounds the transmission failure probability. In addition, the use of the first moment of the state in the analysis facilitates the investigation of cross product terms that involve the disturbance and the indicator process for transmission failures through induced matrix norms. A practical consequence is that our results can be used in the scenarios where the transmission failures and the disturbance are statistically dependent. This is for example the case when the disturbance is partially or fully caused by attacker’s actions and the jamming interference in the wireless channel results in packet content errors.

The paper is organized as follows. We explain the wireless networked control problem under jamming attacks in Section \[sec:Networked-Control-Under\]. In Section \[sec:Analysis-of-Networked\], we first discuss the case without disturbance, then we explain the joint effects of jamming interference and disturbance, and moreover, we provide an analysis for the system with disturbance. We present a numerical example in Section \[sec:Illustrative-Numerical-Examples\], and finally we conclude the paper in Section \[sec:Conclusion\]. Our preliminary conference report [@ahmetmtns2018] contains some of the results. In this paper, we provide the proofs, additional detailed discussions, and a new example.

Throughout the paper, we use $\mathbb{N}$ and $\mathbb{N}_{0}$ to denote the sets of positive and nonnegative integers, respectively. Moreover, $\left\Vert \cdot\right\Vert _{2}$ denotes the Euclidean norm, $\mathrm{\mathbb{P}}[\cdot]$ and $\mathbb{E}[\cdot]$ respectively denote the probability and the expectation on a probability space $(\Omega,\mathcal{F},\mathbb{P})$.

Networked Control Under Jamming Attacks  {#sec:Networked-Control-Under}
========================================

We consider the networked control problem of a discrete-time linear plant with a static state feedback controller. As illustrated in Fig. \[Flo:NetworkedOperation\], a wireless communication channel is used for transmission of control input packets from the controller to the plant. This channel is subject to transmission failures at certain times due to interference caused by the jamming signal of an attacker.

![Operation of networked control system under jamming attacks[]{data-label="Flo:NetworkedOperation"}](networkedcontrolsystemdrawing8 "fig:"){width="0.65\columnwidth"} -5pt

-15pt

In the networked control operation, at each time step $t$, the controller computes a control input using the state information and attempts to transmit it on the wireless channel. If the transmission is successful, then the transmitted control input is applied at the plant side. If, on the other hand, there is a transmission failure, then the control input at the plant side is set to $0$. In this setting, the dynamics of the plant is given by $$\begin{aligned}
x(t+1) & =Ax(t)+(1-l(t))Bu_{\mathrm{C}}(t)+w_{\mathrm{P}}(t),\label{eq:system}\end{aligned}$$ where $x(t)\in\mathbb{R}^{n}$ is the state, $u_{\mathrm{C}}(t)\in\mathbb{R}^{m}$ is the control input that is attempted to be transmitted by the controller to the plant at time $t$, $w_{\mathrm{P}}(t)\in\mathbb{R}^{n}$ is the disturbance, and $l(t)\in\{0,1\}$ represents the transmission status (with $l(t)=1$ indicating failure and $l(t)=0$ indicating success). Moreover, $A\in\mathbb{R}^{n\times n}$ is the unstable system matrix and $B^{n\times m}$ is the input matrix.

The likelihood of a transmission failure depends on the power of the jamming interference signal. If the interference power is large, then it is more likely that there is a transmission failure. In particular, with $v(t)\in[0,\infty)$ denoting the jamming interference power at time $t$, the transmission failure indicator $l(t)$ in (\[eq:system\]) is given by $$\begin{aligned}
l(t) & \triangleq{\mathds{1}}[r(t)\leq p(v(t))],\quad t\in\mathbb{N}_{0},\label{eq:ldef}\end{aligned}$$ where, $p\colon[0,\infty)\to[0,1]$ is a Borel-measurable, nondecreasing function, and $r(0),r(1),\ldots$ are independent random variables that are distributed uniformly in $[0,1]$. Furthermore $\{r(t)\in[0,1]\}_{t\in\mathbb{N}_{0}}$ and $\{v(t)\in[0,\infty)\}_{t\in\mathbb{N}_{0}}$ are assumed to be mutually independent processes. Notice that for a fixed scalar $\vartheta$, we represent by $p(\vartheta)$ the conditional probability of a transmission failure given that the jamming interference power is set to $\vartheta$. In particular, (\[eq:ldef\]) implies $$\begin{aligned}
\mathbb{P}[l(t)=1|v(t)=\vartheta] & =\mathbb{P}[r(t)\leq p(\vartheta)|v(t)=\vartheta]\\
 & =\mathbb{P}[r(t)\leq p(\vartheta)]=p(\vartheta).\end{aligned}$$ Observe that, if $v(t)$ is large so that $p(v(t))$ is close to $1$, then it becomes more likely that $r(t)\leq p(v(t))$, and hence by (\[eq:ldef\]), a transmission failure is likely to occur. Note also that transmission failures at different times are *conditionally independent* given the interference powers at those times. Namely, for every $t_{1}<t_{2}<\cdots<t_{k}$, $k\in\mathbb{N}$, $$\begin{aligned}
 & \mathbb{P}[l(t_{1})=1,\ldots,l(t_{k})=1|v(t_{1})=\vartheta_{1},\ldots,v(t_{k})=\vartheta_{k}]\\
 & \quad=\prod_{i=1}^{k}\mathbb{P}[l(t_{i})=1|v(t_{i})=\vartheta_{i}]=\prod_{i=1}^{k}p(\vartheta_{i}).\end{aligned}$$

The characterization in (\[eq:ldef\]) enables us to describe security properties of different wireless channel models by utilizing different $p$ functions. For instance, to describe the additive white Gaussian noise channel with quadrature amplitude modulation scheme considered in [@IFACli2016sinr], $p$ can be selected as $$\begin{aligned}
p(\vartheta)=2Q\Big(\sqrt{c\frac{\xi}{\vartheta+\sigma}}\Big),\label{eq:p_from_reference-1}\end{aligned}$$ where $Q(y)\triangleq\frac{1}{\sqrt{2\pi}}\int_{y}^{\infty}e^{-\frac{s^{2}}{2}}\mathrm{d}s$, $\xi\in(0,\infty)$ and $\sigma\in(0,\infty)$ are constants associated respectively with the transmission power and the power of the channel noise, and $c\in(0,\infty)$ is a constant associated with the parameters of the communication protocol. Notice that the term $\frac{\xi}{\vartheta+\sigma}$ in (\[eq:p\_from\_reference-1\]) corresponds to Signal-to-Interference-plus-Noise-Ratio (SINR), indicating the quality of a wireless channel [@shi2017signal]. Even if there is no attack at time $t$ (i.e., $v(t)=\vartheta=0$), there may still be a transmission failure due to channel noise $\sigma>0$, since $p(0)>0$.

Analysis of Networked Stabilization {#sec:Analysis-of-Networked}
===================================

In this section, we investigate the networked stabilization of the plant (\[eq:system\]) through a state-feedback controller, where the control input transmitted by the controller is given by $$\begin{aligned}
u_{\mathrm{C}}(t) & =Kx(t)+w_{\mathrm{C}}(t),\quad t\in\mathbb{N}_{0},\label{eq:control-input}\end{aligned}$$ where $K\in\mathbb{R}^{m\times n}$ denotes the feedback gain, and $w_{\mathrm{C}}(t)\in\mathbb{R}^{m}$ is used for describing malicious or nonmalicious disturbances on the control input.

With $w(t)\triangleq w_{\mathrm{P}}(t)+(1-l(t))Bw_{\mathrm{C}}(t)$, the closed-loop networked control system (\[eq:system\]), (\[eq:control-input\]) becomes $$\begin{aligned}
x(t+1)=Ax(t)+(1-l(t))BKx(t)+w(t),\,\,t\in\mathbb{N}_{0}.\label{eq:closed-loop-system}\end{aligned}$$ In what follows, we first investigate the stability of (\[eq:closed-loop-system\]) in the disturbance-free case. Then we discuss how a strategic jamming attacker can take advantage of the disturbance to prevent stabilization. Finally, we obtain conditions of stability under disturbance.

Stabilization in the Disturbance-Free Case
------------------------------------------

For the case without disturbance ($w(t)=0$, $t\in\mathbb{N}_{0}$), our previous work [@ahmetifacwc2017] shows that stabilization can be achieved if the long run average jamming interference power is bounded by a sufficiently small scalar. In particular, the characterization in [@ahmetifacwc2017] allows the interference power $v(t)$ to arbitrarily change at each time $t$ as long as it satisfies the following assumption.

\[FirstAssumption\] There exist scalars $\overline{\kappa},\overline{v}\geq0$ such that $$\begin{aligned}
\mathbb{P}\big[\sum_{i=0}^{t-1}v(i)\leq\overline{\kappa}+\overline{v}t\big] & =1,\quad t\in\mathbb{N}.\label{eq:attack_assumption_one}\end{aligned}$$

Here, $\overline{v}\geq0$ is an asymptotic upper-bound on the average interference power (i.e., $\limsup_{k\to\infty}\frac{1}{k}\sum_{t=0}^{k-1}v(t)\leq\overline{v}$). Notice that if $p$ in (\[eq:ldef\]) is a concave function, then $p(\overline{v})$ can be used in the stability analysis as an upper bound on the long run average number of transmission failures. On the other hand, if $p$ is not concave, then a concave function that upper-bounds $p$ can be used for the same purpose. To this end, we utilized in [@ahmetifacwc2017] a continuous, nondecreasing, and concave function $\hat{p}\colon[0,\infty)\to[0,1]$ such that $$\begin{aligned}
\hat{p}(v) & \geq p(v),\quad v\in[0,\infty).\label{eq:p_phat_ineq}\end{aligned}$$ As discussed in [@ahmetifacwc2017], such a $\hat{p}$ always exists, and moreover, $$\begin{aligned}
\limsup_{t\to\infty}\frac{1}{t}\sum_{i=0}^{t-1}l(i) & \leq\hat{p}(\overline{v}),\label{eq:long-run-bound}\end{aligned}$$ almost surely. In other words, the average number of transmission failures is upper bounded in the long run by $\hat{p}(\overline{v})$. The inequality (\[eq:long-run-bound\]) was used in [@ahmetifacwc2017] for establishing stability of the closed-loop system (\[eq:closed-loop-system\]) in the case without disturbance. The analysis in [@ahmetifacwc2017] indicates that if $\overline{v}$ is sufficiently small, then the closed-loop system is asymptotically stable almost surely, implying $\mathbb{P}[\lim_{t\to\infty}\|x(t)\|_{2}=0]=1$.

In addition, moment stability of the networked control system can also be analyzed under Assumption \[FirstAssumption\]. In particular, the following result provides a condition under which the 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 of [@hornmatrixanalysis]). On the other hand, for unstable open-loop dynamics, we have $\|A\|>1$. Notice that the inequality in (\[eq:abkcond\]) holds if the upper bound $\overline{v}$ of the average jamming interference power is sufficiently small so that $\hat{p}(\overline{v})$ is sufficiently close to zero. In such cases, transmission failures happen sufficiently rarely in average, and thus the overall networked control system frequently follows the stable behavior of the closed-loop dynamics and the geometric convergence of the first-moment of the state as in (\[eq:moment-convergence\]) can be guaranteed. As we establish in the proof given in the Appendix, the scalar $\overline{\theta}$ in (\[eq:moment-convergence\]) represents the rate of convergence, and it depends on $\hat{p}(\overline{v})$ as well as the scalars $\|A+BK\|$ and $\|A\|$. In particular, if the bound $\overline{v}$ on the long run average jamming interference power is small, then $\overline{\theta}$ is also small, indicating faster convergence of the first-moment.

*First-moment geometric stability* discussed in Proposition \[PropositionMomentConvergence\] is a stronger notion of stochastic stability in comparison to *almost-sure asymptotic stability* explored in [@ahmetifacwc2017]. As expected, first-moment geometric stability condition (\[eq:moment-convergence\]) is more restrictive with respect to the attack parameter $\overline{v}$. Specifically, the almost-sure asymptotic stability condition presented in Theorem 3.5 of [@ahmetifacwc2017] reduces to $$\begin{aligned}
(1-\hat{p}(\overline{v}))\ln\|A+BK\|+\hat{p}(\overline{v})\ln\|A\| & <0,\label{eq:almost-sure}\end{aligned}$$ 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}
-----------------------------------------------------

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 $$\begin{aligned}
v(t) & \triangleq\begin{cases}
v^{*}, & t\in\{\tau_{1},\ldots,\tau_{1}+\tau_{2}-1\},\\
0, & \mathrm{otherwise},
\end{cases}\label{eq:ExampleStrategyV}\end{aligned}$$ with $v^{*}\triangleq p^{-1}(\rho^{\frac{1}{\tau_{2}}})+1$, $\tau_{1}\triangleq\lfloor\frac{\max\{v^{*}-\overline{v},0\}\tau_{2}}{\overline{v}}\rfloor+1$, $\tau_{2}\triangleq\lfloor\max\{\log_{A}(z/w^{*}),0\}\rfloor+1$ guarantees that Assumption \[FirstAssumption\] is satisfied and the state exceeds the value $z$ with probability larger than $\rho$ at time $\tau\triangleq\tau_{1}+\tau_{2}$, i.e., $\mathbb{P}[x(\tau)>z]>\rho.$

To show this, first we define the event $E(\tau_{1},\tau_{2})\in\mathcal{F}$ by $$\begin{aligned}
E(\tau_{1},\tau_{2}) & \triangleq\big\{\omega\in\Omega\colon l(t)=1,t\in\{\tau_{1},\ldots,\tau_{1}+\tau_{2}-1\}\big\}.\end{aligned}$$ This is the event that all packet transmissions during    $t\in\{\tau_{1},\ldots,\tau_{1}+\tau_{2}-1\}$ fail. By (\[eq:ExampleStrategyV\]), we have $\mathbb{P}[E(\tau_{1},\tau_{2})]=p^{\tau_{2}}(v^{*})$. Now, since $x_{0}>0$, $A>1$, and $w^{*}>0$, we obtain $x(t)\geq w^{*},$ $t\in\mathbb{N}$. Therefore, $$\begin{aligned}
 & \mathbb{P}[x(\tau)>z]\geq\mathbb{P}[x(\tau)>z\,|\,E(\tau_{1},\tau_{2})]\mathbb{P}[E(\tau_{1},\tau_{2})]\\
 & \quad\geq\mathbb{P}[A^{\tau_{2}}x(\tau_{1})+\sum_{i=0}^{\tau_{2}-1}A^{i}w^{*}>z\,|\,E(\tau_{1},\tau_{2})]p^{\tau_{2}}(v^{*})\\
 & \quad\geq\mathbb{P}[A^{\tau_{2}}w^{*}>z\,|\,E(\tau_{1},\tau_{2})]p^{\tau_{2}}(v^{*})>1\cdot\rho^{\frac{\tau_{2}}{\tau_{2}}}=\rho.\end{aligned}$$ Furthermore, the attack strategy (\[eq:ExampleStrategyV\]) satisfies Assumption \[FirstAssumption\] with $\overline{\kappa}=0$, because $\tau_{1}\geq\frac{\max\{v^{*}-\overline{v},0\}\tau_{2}}{\overline{v}}\geq\frac{(v^{*}-\overline{v})\tau_{2}}{\overline{v}}$, and thus, $\sum_{i=0}^{\tau-1}v(i)=v^{*}\tau_{2}\leq\overline{v}(\tau_{1}+\tau_{2})=\overline{v}\tau$.

The attack strategy (\[eq:ExampleStrategyV\]) can make the state grow arbitrarily large even if the interference power bound $\overline{v}$ is very small. This attack strategy is effective, because even if the attacker initially waits for a long duration without attacking, the state never reaches a small neighborhood of zero due to the disturbance. Hence, after waiting for a while, the attacker can consecutively attack with high interference powers to cause many transmission failures and make the state norm grow to large values. This is further illustrated in Section \[sec:Illustrative-Numerical-Examples\].

Jamming Interference and Bounded Disturbance
--------------------------------------------

To ensure stability under both disturbance and jamming, the attacks need to be restricted in a way that high jamming interference powers at consecutive times are not allowed. To this end, we consider the following assumption.

\[SecondAssumption\] There exist scalars $\hat{\kappa},\hat{v}\geq0$ such that $$\begin{aligned}
\mathbb{P}\big[\sum_{i=t_{1}}^{t_{2}-1}v(i)\leq\hat{\kappa}+\hat{v}(t_{2}-t_{1})\big] & =1,\label{eq:attack_assumption_two}\end{aligned}$$ for all $t_{1},t_{2}\in\mathbb{N}_{0}$ with $t_{1}<t_{2}$.

Notice that (\[eq:attack\_assumption\_two\]) implies (\[eq:attack\_assumption\_one\]) (with $\overline{\kappa}=\hat{\kappa}$ and $\overline{v}=\hat{v}$), but the converse is not true. Assumption \[SecondAssumption\] is thus more restrictive than Assumption \[FirstAssumption\]. In particular, under Assumption \[SecondAssumption\], the attacker can attack with a jamming interference power $v^{*}>\hat{v}$ consecutively for at most $\lfloor\hat{\kappa}/(v^{*}-\hat{v})\rfloor$ time steps; thus, the destabilizing attacks discussed in Example \[ExampleStrategy\] are avoided.

Assumption \[SecondAssumption\] is related to other characterizations of malicious attacks in the literature. In particular, in the continuous-time deterministic denial-of-service attack characterization of [@de2015inputtran], the number of attacks in a given time frame as well as the total duration of those attacks are bounded by certain ratios of the length of that time frame. Under that characterization, the maximum possible length of a continuous attack duration is bounded, which enables analysis of input-to-state stability under disturbance. The restriction on jamming through Assumption \[SecondAssumption\] is similar, since long consecutive emissions of high powered interference signals are not allowed. We note, however, that Assumption \[SecondAssumption\] allows the scenario where the channel is attacked at all times if the attacker’s interference power for certain times is small. Notice that emission of interference signals in jamming attacks require energy [@pelechrinis2011]. In this respect, Assumption \[SecondAssumption\] can describe the constraints of an attacker with limited energy resources.

In this section, we investigate the networked control system (\[eq:closed-loop-system\]) under bounded disturbance. The analysis is then extended in Section \[subsec:Stabilization-Under-Jamming\] to the case where the disturbance has finite second moments but its norm may not be bounded by a fixed scalar.

In this paper, we consider scenarios where the norm of the disturbance does not approach zero, and hence the state or its moments may not converge to the origin. Instead of exploring asymptotic stability, our goal is to obtain conditions for the first moment of the state to stay bounded. To this end, let $\hat{A}(t)\triangleq l(t)A+(1-l(t))(A+BK)$, $t\in\mathbb{N}_{0}$, and moreover, for every $t_{1},t_{2}\in\mathbb{N}_{0}$ with $t_{1}\leq t_{2},$ let $$\begin{aligned}
F(t_{2},t_{1}) & \triangleq\begin{cases}
\hat{A}(t_{2}), & t_{1}=t_{2},\\
\hat{A}(t_{2})\cdots\hat{A}(t_{1}), & t_{1}<t_{2}.
\end{cases}\end{aligned}$$ For the closed-loop system (\[eq:closed-loop-system\]), we have $x(t)=F(t-1,0)x_{0}+\sum_{j=0}^{t-2}F(t-1,j+1)w(j)+w(t-1)$, for $t\in\mathbb{N}$. Therefore, for any induced norm $\|\cdot\|$, it follows from the triangle inequality and the submultiplicativity property that $$\begin{aligned}
\|x(t)\| & \leq\Big(\prod_{i=0}^{t-1}\|\hat{A}(i)\|\Big)\|x_{0}\|+\sum_{j=0}^{t-2}\Big(\prod_{i=j+1}^{t-1}\|\hat{A}(i)\|\Big)\|w(j)\|\\
 & \quad+\|w(t-1)\|.\end{aligned}$$ 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 \[SecondAssumption\] with a sufficiently small $\hat{v}$ such that (\[eq:abkcond-vhat\]) holds, then the first moment of the state stays bounded. Furthermore, the upper bound given in (\[eq:noise\_to\_state\_one\]) is geometrically decreasing towards the constant $\hat{d}\overline{w}$, where $\overline{w}$ is an upper bound on the Euclidean norm of disturbance $w(t)$. Notice that the condition (\[eq:abkcond-vhat\]) of Theorem \[TheoremBoundedDisturbance\] and the condition (\[eq:abkcond\]) in the disturbance-free case in Proposition \[PropositionMomentConvergence\] are in the same form, but use different scalars $\hat{v}$ and $\overline{v}$ due to the difference of the jamming interference characterizations in Assumptions \[FirstAssumption\] and \[SecondAssumption\]. We remark that for attacks that satisfy both assumptions, we have $\overline{v}\leq\hat{v}$.

As we establish in the proof of Theorem \[TheoremBoundedDisturbance\], the first moment upper bound in (\[eq:noise\_to\_state\_one\]) depends on attack parameters $\hat{\kappa}$ and $\hat{v}$. In particular, the values of $\hat{\mu}$ and $\hat{d}$ are large, if $\hat{k}$ and $\hat{v}$ take large values. Moreover, the scalar $\hat{\theta}$ is directly related to the term $(1-\hat{p}(\hat{v}))\|A+BK\|+\hat{p}(\hat{v})\|A\|$ on the left-hand side of (\[eq:abkcond-vhat\]). If this term is close to zero, then $\hat{\theta}$ is close to zero, which indicates faster convergence of the bound in (\[eq:noise\_to\_state\_one\]) towards the constant $\hat{d}\overline{w}$. We note that $(1-\hat{p}(\hat{v}))\|A+BK\|+\hat{p}(\hat{v})\|A\|$ represents the behavior of the overall networked control system and it is composed of the convex combination of terms $\|A+BK\|$ and $\|A\|$ weighted respectively with the lower bound $(1-\hat{p}(\hat{v}))$ of the long-term ratio of successful transmissions and upper bound $\hat{p}(\hat{v})$ of the long-term ratio of failed transmissions.

Our analysis approach differs from the more classical approaches utilized in the cases where the transmission failure indicator process $\{l(t)\in\{0,1\}\}_{t\in\mathbb{N}_{0}}$ is a Bernoulli process or a Markov chain. In those cases, stability analysis can rely on the probability of failures $\mathbb{P}[l(t)=1]$ and conditional failure probabilities $\mathbb{P}[l(t)=q|l(t-1)=r]$, $q,r\in\{0,1\}$, (see [@quevedo2011packetized; @costa2004discrete]). We remark that in our case, precise information of such probability terms is not available due to the uncertainty in the generation of attacks. Specifically, the interference power $v(t)$ at a given time $t$ is part of attacker’s strategy and cannot be known with certainty. As a result, the transmission failure probability at that time is also uncertain and cannot be used in the analysis.

A crucial role in our analysis is played by the following lemma, where we investigate the products of affine functions that involve the transmission failure indicator $l(\cdot)$ and obtain upper bounds for their expected values. As shown later in the proof of Theorem \[TheoremBoundedDisturbance\], such upper bounds allow us to conduct stability analysis without relying on transmission failure probabilities for each time step. In the derivation of these bounds, an essential step is to exploit the concavity of the upper-bounding function $\hat{p}$ given in (\[eq:p\_phat\_ineq\]).

\[LemmaDOne\] Suppose that the attacker’s interference power process $\{v(t)\in[0,\infty)\}_{t\in\mathbb{N}_{0}}$ satisfies Assumption \[SecondAssumption\]. Then for every $\alpha_{1}\geq0$, $\alpha_{0}\geq0$ that satisfy $$\begin{aligned}
\alpha_{1}\hat{p}(\hat{v})+\alpha_{0} & <1,\label{eq:alpha_ineq}\end{aligned}$$ there exist scalars $\mu\geq0$ and $\theta\in(0,1)$ such that $$\begin{aligned}
\mathbb{E}[\prod_{i=t_{1}}^{t_{2}-1}(\alpha_{1}l(i)+\alpha_{0})] & \leq\mu\theta^{(t_{2}-t_{1})},\label{eq:geometric-lemma-result}\end{aligned}$$ for $t_{1},t_{2}\in\mathbb{N}_{0}$ with $t_{1}<t_{2}$.

For the case where $\alpha_{1}+\alpha_{0}=0$, (\[eq:geometric-lemma-result\]) holds for any $\mu\geq0$ and $\theta\in(0,1)$. In the following, we consider the case where $\alpha_{1}+\alpha_{0}>0$. First, by Lemma 2.1 of [@ahmetifacwc2017], $$\mathbb{E}[\prod_{i=t_{1}}^{t_{2}-1}(\alpha_{1}l(i)+\alpha_{0})]=\mathbb{E}[\prod_{i=t_{1}}^{t_{2}-1}(\alpha_{1}p(v(i))+\alpha_{0})].\label{eq:first-eq}$$ Next, by (\[eq:first-eq\]), $\alpha_{1}\geq0$, and $p(v)\leq\hat{p}(v)$, $v\in[0,\infty)$, we get $$\begin{aligned}
\mathbb{E}\big[\prod_{i=t_{1}}^{t_{2}-1}(\alpha_{1}l(i)+\alpha_{0})\big] & \leq\mathbb{E}\big[\prod_{i=t_{1}}^{t_{2}-1}h(v(i))\big],\label{eq:expxhineq-1-1}\end{aligned}$$ where $h(v)\triangleq\alpha_{1}\hat{p}(v)+\alpha_{0}$. We note that $h(\cdot)$ is nondecreasing, concave, and continuous, as $\hat{p}(\cdot)$ also has such properties and $\alpha_{1}\geq0$. To obtain an upper-bound for $\mathbb{E}[\prod_{i=t_{1}}^{t_{2}-1}h(v(i))]$ in (\[eq:expxhineq-1-1\]), we first show $$\begin{aligned}
\prod_{i=t_{1}}^{t_{2}-1}h(v(i)) & \leq h^{(t_{2}-t_{1})}(\frac{1}{t_{2}-t_{1}}\sum_{i=t_{1}}^{t_{2}-1}v(i)).\label{eq:powerbound}\end{aligned}$$ We note that (\[eq:powerbound\]) holds if $h(v(i))=0$ for some $i\in\{t_{1},\ldots,t_{2}-1\}$. Now, consider the case where $h(v(i))>0$ for all $i\in\{t_{1},\ldots,t_{2}-1\}$. For this case, we have $$\begin{aligned}
\ln\prod_{i=t_{1}}^{t_{2}-1}h(v(i)) & =(t_{2}-t_{1})\Big(\frac{1}{t_{2}-t_{1}}\sum_{i=t_{1}}^{t_{2}-1}\ln h(v(i))\Big).\label{eq:firstln}\end{aligned}$$ Here, $\ln h(\cdot)$ is concave, since it is the composition of a nondecreasing concave function $\ln(\cdot)$ and a concave function $h(\cdot)$ (see Proposition 2.16 in [@avriel2010generalized] and Section 3.2.4 in [@IFACboyd2004convex]). Thus, by (\[eq:firstln\]), $$\begin{aligned}
\ln\prod_{i=t_{1}}^{t_{2}-1}h(v(i)) & \leq(t_{2}-t_{1})\ln h(\frac{1}{t_{2}-t_{1}}\sum_{i=t_{1}}^{t_{2}-1}v(i)),\label{eq:lnineq2}\end{aligned}$$ which implies (\[eq:powerbound\]). The interference power process $v(\cdot)$ satisfies (\[eq:attack\_assumption\_two\]) in Assumption \[SecondAssumption\], and hence, $\frac{1}{t_{2}-t_{1}}\sum_{i=t_{1}}^{t_{2}-1}v(i)\leq\frac{\hat{\kappa}}{t_{2}-t_{1}}+\hat{v}$, almost surely. Thus, noting that $h(\cdot)$ is a nondecreasing function, by (\[eq:powerbound\]), we obtain $\prod_{i=t_{1}}^{t_{2}-1}h(v(i))\leq h^{t_{2}-t_{1}}(\frac{\hat{\kappa}}{t_{2}-t_{1}}+\hat{v})$, almost surely. Consequently, we have $$\begin{aligned}
\mathbb{E}\big[\prod_{i=t_{1}}^{t_{2}-1}h(v(i))\big] & \leq h^{t_{2}-t_{1}}(\frac{\hat{\kappa}}{t_{2}-t_{1}}+\hat{v}).\label{eq:hfineq-1-1}\end{aligned}$$ Now, it follows from (\[eq:alpha\_ineq\]) that $h(\hat{v})<1$. Therefore, by the continuity of $h(\cdot)$, there exists $\delta>0$ such that $h(\delta+\hat{v})<1$. As a result, for sufficiently large values of $t_{2}-t_{1}$, we have $h(\frac{\hat{\kappa}}{t_{2}-t_{1}}+\hat{v})<1$.

Let $T^{*}$ be a positive integer such that $h(\frac{\hat{\kappa}}{T^{*}}+\hat{v})<1$ and let $$\begin{aligned}
\theta & \triangleq h(\frac{\hat{\kappa}}{T^{*}}+\hat{v}).\label{eq:ThetaDef}\end{aligned}$$ It follows from (\[eq:hfineq-1-1\]) that $$\begin{aligned}
\mathbb{E}\big[\prod_{i=t_{1}}^{t_{2}-1}h(v(i))\big] & \leq\theta^{t_{2}-t_{1}},\label{eq:hjt_part1}\end{aligned}$$ for all $t_{1},t_{2}\in\mathbb{N}_{0}$ such that $t_{2}-t_{1}\geq T^{*}$. If $T^{*}=1$, then (\[eq:geometric-lemma-result\]) holds, by (\[eq:hjt\_part1\]). If, on the other hand, $T^{*}>1$, then by using $h(v(t))\leq\alpha_{1}+\alpha_{0}$, $t\in\mathbb{N}_{0}$, we obtain $$\begin{aligned}
\mathbb{E}\big[\prod_{i=t_{1}}^{t_{2}-1}h(v(i))\big] & \leq(\alpha_{1}+\alpha_{0})^{t_{2}-t_{1}}\leq(\alpha_{1}+\alpha_{0})^{T^{*}-1}\label{eq:hjt_part2}\end{aligned}$$ for all $t_{1},t_{2}\in\mathbb{N}_{0}$ such that $0<t_{2}-t_{1}<T^{*}$. Letting $$\begin{aligned}
\mu & \triangleq(\alpha_{1}+\alpha_{0})^{T^{*}-1}\theta^{-(T^{*}-1)},\label{eq:MuDef}\end{aligned}$$ we obtain (\[eq:geometric-lemma-result\]), by (\[eq:hjt\_part1\]) and (\[eq:hjt\_part2\]).

Lemma \[LemmaDOne\] shows that under Assumption \[SecondAssumption\], the expectation term $\mathbb{E}[\prod_{i=t_{1}}^{t_{2}-1}(\alpha_{1}l(i)+\alpha_{0})]$ with $\alpha_{1}\geq0$, $\alpha_{0}\geq0$ satisfying (\[eq:alpha\_ineq\]), converges to zero at a geometric rate. By using this lemma, we obtain the following result.

\[LemmaSum\] Suppose that the attacker’s interference power process $\{v(t)\in[0,\infty)\}_{t\in\mathbb{N}_{0}}$ satisfies Assumption \[SecondAssumption\]. Then for every $\alpha_{1}\geq0$, $\alpha_{0}\geq0$ that satisfy (\[eq:alpha\_ineq\]), there exists a scalar $d\geq0$ such that $$\begin{aligned}
\sum_{j=0}^{t-2}\mathbb{E}\Big[\prod_{i=j+1}^{t-1}(\alpha_{1}l(i)+\alpha_{0})\Big] & \leq d,\quad t\in\{2,3,\ldots\}.\label{eq:sum_result}\end{aligned}$$

Since (\[eq:alpha\_ineq\]) holds, it follows from Lemma \[LemmaDOne\] that $\mathbb{E}[\prod_{i=j+1}^{t-1}(\alpha_{1}l(i)+\alpha_{0})]\leq\mu\theta^{(t-j-1)}$, where $\mu\geq0$ and $\theta\in(0,1)$ are scalars that depend on $\alpha_{1}$ and $\alpha_{0}$. Letting $$\begin{aligned}
d & \triangleq\mu/(1-\theta),\label{eq:DDef}\end{aligned}$$ we obtain$\sum_{j=0}^{t-2}\mathbb{E}\Big[\prod_{i=j+1}^{t-1}(\alpha_{1}l(i)+\alpha_{0})\Big]\leq\sum_{j=0}^{t-2}\mu\theta^{(t-j-1)}=\mu\sum_{i=1}^{t-1}\theta^{i}\leq\mu\sum_{i=0}^{\infty}\theta^{i}=d$, which completes the proof.

In Lemmas \[LemmaDOne\] and \[LemmaSum\], we obtained the upper-bounding inequalities (\[eq:geometric-lemma-result\]) and (\[eq:sum\_result\]) concerning the transmission failure indicator process $\{l(t)\in\{0,1\}\}_{t\in\mathbb{N}_{0}}$. In our proof of Theorem \[TheoremBoundedDisturbance\] given below, we utilize these inequalities.

*Proof of Theorem \[TheoremBoundedDisturbance\]*: By (\[eq:two-norm-ineq\]) and (\[eq:almost\_sure\_noise\_bound\]), $$\begin{aligned}
\|x(t)\|_{2} & \leq\frac{c_{2}}{c_{1}}\left(\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\right)\|x_{0}\|_{2}\\
 & \quad+\frac{c_{2}}{c_{1}}\Big(\sum_{j=0}^{t-2}\big(\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big)+1\Big)\overline{w},\end{aligned}$$ almost surely, and hence, for $t\in\mathbb{N}$, $$\begin{aligned}
\mathbb{E}[\|x(t)\|_{2}] & \leq\frac{c_{2}}{c_{1}}\mathbb{E}\big[\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]\|x_{0}\|_{2}\nonumber \\
 & \quad+\frac{c_{2}}{c_{1}}\Big(\sum_{j=0}^{t-2}\mathbb{E}\big[\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]+1\Big)\overline{w}.\label{eq:expectation_bounded_noise_case}\end{aligned}$$

Next, we apply Lemmas \[LemmaDOne\] and \[LemmaSum\] to obtain upper-bounds for the expectation terms on the right-hand side of (\[eq:expectation\_bounded\_noise\_case\]). First, since $\|A\|>1$, (\[eq:abkcond-vhat\]) implies $\|A+BK\|\in[0,1)$. Thus, we have $\zeta_{1}>0$ and $\zeta_{0}\in[0,1)$. By letting $\alpha_{1}=\zeta_{1}$ and $\alpha_{0}=\zeta_{0}$, (\[eq:abkcond-vhat\]) implies (\[eq:alpha\_ineq\]). Therefore, by Lemmas \[LemmaDOne\] and \[LemmaSum\], we have $\mathbb{E}\big[\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]\leq\mu\theta^{t}$ and $\sum_{j=0}^{t-2}\mathbb{E}\big[\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]\leq d$, where $\mu\geq0$, $\theta\in(0,1)$, and $d\geq0$ are scalars that depend on $\zeta_{1}$ and $\zeta_{0}$. Hence, with $$\begin{aligned}
 & \hat{\theta}\triangleq\theta,\quad\hat{\mu}\triangleq\mu c_{2}/c_{1},\quad\hat{d}\triangleq(d+1)c_{2}/c_{1},\label{eq:thetamudhat}\end{aligned}$$ the inequality (\[eq:noise\_to\_state\_one\]) follows from (\[eq:expectation\_bounded\_noise\_case\]). $\blacksquare$

\[RemarkFirstHatValues\] By using (\[eq:thetamudhat\]) together with (\[eq:ThetaDef\]), (\[eq:MuDef\]), and (\[eq:DDef\]), we can obtain the values of $\hat{\theta}$, $\hat{\mu}$, and $\hat{d}$ as $$\begin{aligned}
 & \hat{\theta}=(1-\hat{p}(\frac{\hat{\kappa}}{T^{*}}+\hat{v}))\|A+BK\|+\hat{p}(\frac{\hat{\kappa}}{T^{*}}+\hat{v})\|A\|,\label{eq:ThetaHatValue}\\
 & \hat{\mu}=\frac{c_{2}}{c_{1}}\|A\|{}^{T^{*}-1}\hat{\theta}^{-(T^{*}-1)},\label{eq:MuHatValue}\\
 & \hat{d}=\frac{c_{2}}{c_{1}}\Big(\frac{\|A\|{}^{T^{*}-1}\hat{\theta}^{-(T^{*}-1)}}{1-\hat{\theta}}+1\Big),\label{eq:DHatValue}\end{aligned}$$ where $T^{*}$ is a positive integer that satisfies $$\begin{aligned}
 & (1-\hat{p}(\frac{\hat{\kappa}}{T^{*}}+\hat{v}))\|A+BK\|+\hat{p}(\frac{\hat{\kappa}}{T^{*}}+\hat{v})\|A\|<1,\end{aligned}$$ and $c_{1},c_{2}>0$ are scalars that satisfy (\[eq:norm-equivalence\]). The attacker with large resources can cause the state norm to grow large. This is also indicated in the upper bound for the first moment given in (\[eq:noise\_to\_state\_one\]). In particular, if $\hat{\kappa}$ in Assumption \[SecondAssumption\] is large, then $T^{*}$ is large, which makes $\hat{\mu}$ large, as $\hat{\mu}$ is an increasing function of $T^{*}$. Further, since $$\begin{aligned}
 & \frac{c_{2}}{c_{1}}\Big(\frac{\hat{\mu}}{1-(1-\hat{p}(\hat{v}))\|A+BK\|-\hat{p}(\hat{v})\|A\|}+1\Big)\leq\hat{d},\end{aligned}$$ we observe that $\hat{d}$ is large for large values of $\hat{\mu}$ and $\hat{v}$. On the other hand, for large values of $T^{*}$, $\hat{\theta}$ is close to $(1-\hat{p}(\hat{v}))\|A+BK\|+\hat{p}(\hat{v})\|A\|$. If the upper bound $\hat{v}$ of average interference powers is large, then $\hat{\mu}\hat{\theta}^{t}$ in (\[eq:noise\_to\_state\_one\]) converges slowly, since $\hat{\theta}$ is close to $1$.

Jamming and Disturbance with Finite Second Moment {#subsec:Stabilization-Under-Jamming}
-------------------------------------------------

In Theorem \[TheoremBoundedDisturbance\], we explored the case where the disturbance norm is bounded at each time almost surely. Next, we investigate scenarios where the disturbance may not be bounded. We obtain a relation between the state and the disturbance similar to those used for noise-to-state stability analysis of stochastic systems (e.g., [@nunez2014; @zhang2016noise]). Specifically, in the next result, we provide an upper bound for the first moment of state by utilizing the second moment of disturbance.

\[TheoremDependentDisturbance\] 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 $\mathbb{E}[\|w(t)\|_{2}^{2}]<\infty$, $t\in\mathbb{N}_{0}$. If $$\begin{aligned}
 & (1-\hat{p}(\hat{v}))\|A+BK\|^{2}+\hat{p}(\hat{v})\|A\|^{2}<1,\label{eq:abkcond-difficult}\end{aligned}$$ then there exist $\hat{\mu},\hat{f}\geq0$, and $\hat{\theta}\in(0,1)$ such that for $t\in\mathbb{N}$, $$\begin{aligned}
\mathbb{E}[\|x(t)\|_{2}]\leq\hat{\mu}\hat{\theta}^{t}\|x_{0}\|_{2}+\hat{f}\max_{i\in\{0,\ldots,t-1\}}(\mathbb{E}[\|w(i)\|_{2}^{2}])^{\frac{1}{2}}.\label{eq:noise_to_state_three}\end{aligned}$$

The proof of this result relies on the following lemma.

\[LemmaSumTwo\] Suppose that the attacker’s interference power process $\{v(t)\in[0,\infty)\}_{t\in\mathbb{N}_{0}}$ satisfies Assumption \[SecondAssumption\]. Then for every $\gamma_{1}\geq0$, $\gamma_{0}\geq0$ that satisfy $$\begin{aligned}
(\gamma_{1}^{2}+2\gamma_{1}\gamma_{0})\hat{p}(\hat{v})+\gamma_{0}^{2} & <1,\label{eq:gamma_cond}\end{aligned}$$ there exists a scalar $f\geq0$ such that $$\begin{aligned}
\sum_{j=0}^{t-2}\Big(\mathbb{E}\Big[\big(\prod_{i=j+1}^{t-1}(\gamma_{1}l(i)+\gamma_{0})\big)^{2}\Big]\Big)^{\frac{1}{2}} & \leq f,\quad t\in\{2,3,\ldots\}.\label{eq:l_d1_ineq-1}\end{aligned}$$

For every $j\in\{0,1,\ldots,t-2\}$, $t\in\{2,3,\ldots\}$, we have $(\prod_{i=j+1}^{t-1}(\gamma_{1}l(i)+\gamma_{0})){}^{2}=\prod_{i=j+1}^{t-1}(\gamma_{1}l(i)+\gamma_{0})^{2}=\prod_{i=j+1}^{t-1}(\gamma_{1}^{2}l^{2}(i)+2\gamma_{1}\gamma_{0}l(i)+\gamma_{0}^{2}).$ Let $\alpha_{1}\triangleq\gamma_{1}^{2}+2\gamma_{1}\gamma_{0}$ and $\alpha_{0}\triangleq\gamma_{0}^{2}$. Since $l^{2}(i)=l(i)$, it follows that $$\mathbb{E}\Big[\big(\prod_{i=j+1}^{t-1}(\gamma_{1}l(i)+\gamma_{0})\big)^{2}\Big]=\mathbb{E}\Big[\prod_{i=j+1}^{t-1}(\alpha_{1}l(i)+\alpha_{0})\Big].\label{eq:gamma-alpha-eq}$$ By (\[eq:gamma\_cond\]), (\[eq:alpha\_ineq\]) holds. Therefore, it follows from Lemma \[LemmaDOne\] that $\mathbb{E}\Big[\prod_{i=j+1}^{t-1}(\alpha_{1}l(i)+\alpha_{0})\Big]\leq\mu\theta^{(t-j-1)}$, where $\mu\geq0$ and $\theta\in(0,1)$ are scalars that depend on $\alpha_{1}$ and $\alpha_{0}$. Letting $$\begin{aligned}
f & \triangleq\mu^{\frac{1}{2}}/(1-\theta^{\frac{1}{2}}),\label{eq:FDef}\end{aligned}$$ we obtain $$\begin{aligned}
 & \sum_{j=0}^{t-2}\Big(\mathbb{E}\Big[\big(\prod_{i=j+1}^{t-1}(\gamma_{1}l(i)+\gamma_{0})\big)^{2}\Big]\Big)^{\frac{1}{2}}\leq\sum_{j=0}^{t-2}\mu^{\frac{1}{2}}\theta^{\frac{1}{2}(t-j-1)}\nonumber \\
 & \quad=\mu^{\frac{1}{2}}\sum_{i=1}^{t-1}\theta^{\frac{1}{2}i}\leq\mu^{\frac{1}{2}}\sum_{i=0}^{\infty}\theta^{\frac{1}{2}i}=f,\label{eq:expineq-1}\end{aligned}$$ which completes the proof.

Lemma \[LemmaSumTwo\] enables us to deal with quadratic terms that involve the failure indicator $l(\cdot)$. We are now ready to prove Theorem \[TheoremDependentDisturbance\].

*Proof of Theorem \[TheoremDependentDisturbance\]*: By (\[eq:two-norm-ineq\]), $$\begin{aligned}
\mathbb{E}[\|x(t)\|_{2}] & \leq\frac{c_{2}}{c_{1}}\mathbb{E}\big[\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]\|x_{0}\|_{2}\nonumber \\
 & \quad+\frac{c_{2}}{c_{1}}\sum_{j=0}^{t-2}\mathbb{E}\Big[\Big(\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\alpha_{0})\Big)\|w(j)\|_{2}\Big]\nonumber \\
 & \quad+\frac{c_{2}}{c_{1}}\mathbb{E}[\|w(t-1)\|_{2}],\quad t\in\mathbb{N}.\label{eq:expectation_two-norm}\end{aligned}$$ To show (\[eq:noise\_to\_state\_three\]), we obtain upper bounds for the expectation terms on the right-hand side of (\[eq:expectation\_two-norm\]) by using Schwarz’s and Jensen’s inequalities. First, by Schwarz’s inequality (see Section 6.8 of [@williams2010probability]), $$\begin{aligned}
 & \mathbb{E}\Big[\Big(\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\Big)\|w(j)\|_{2}\Big]\nonumber \\
 & \quad\leq\Big(\mathbb{E}\big[\big(\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big)^{2}\big]\Big)^{\frac{1}{2}}\big(\mathbb{E}[\|w(j)\|_{2}^{2}]\big)^{\frac{1}{2}}.\label{eq:schwarz-result}\end{aligned}$$ Furthermore, by Jensen’s inequality (see Section 6.6 of [@williams2010probability]), $$\begin{aligned}
\mathbb{E}[\|w(t-1)\|_{2}] & \leq\big(\mathbb{E}[\|w(t-1)\|_{2}^{2}]\big)^{\frac{1}{2}}.\label{eq:jensen-result}\end{aligned}$$ As $\Big(\mathbb{E}\big[\big(\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big)^{2}\big]\Big)^{\frac{1}{2}}\geq0$, we obtain from (\[eq:expectation\_two-norm\])–(\[eq:jensen-result\]), $$\begin{aligned}
 & \mathbb{E}[\|x(t)\|_{2}]\leq\frac{c_{2}}{c_{1}}\mathbb{E}\big[\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]\|x_{0}\|_{2}\nonumber \\
 & \,\,\quad+\frac{c_{2}}{c_{1}}\Bigg(\sum_{j=0}^{t-2}\Big(\mathbb{E}\Big[\big(\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big)^{2}\Big]\Big)^{\frac{1}{2}}+1\Bigg)\nonumber \\
 & \,\,\qquad\cdot\max_{i\in\{0,\ldots,t-1\}}(\mathbb{E}[\|w(i)\|_{2}^{2}])^{\frac{1}{2}},\quad t\in\mathbb{N}.\label{eq:exp_ineq_in_noise_three}\end{aligned}$$ First, we apply Lemma \[LemmaDOne\] to find an upper bound of the term $\mathbb{E}\big[\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]$ on the right-hand side of (\[eq:exp\_ineq\_in\_noise\_three\]). To this end let $\alpha_{1}\triangleq\zeta_{1}$ and $\alpha_{2}\triangleq\zeta_{2}$. Since $\|A\|>1$, it follows from (\[eq:abkcond-difficult\]) that $\|A+BK\|<1$. As a result, $\alpha_{1}=\zeta_{1}>0$ and $\alpha_{0}=\zeta_{0}\in[0,1)$. Furthermore, (\[eq:abkcond-difficult\]) implies $(\zeta_{1}^{2}+2\zeta_{1}\zeta_{0})\hat{p}(\hat{v})+\zeta_{0}^{2}<1$. Using this inequality together with $\hat{p}(\hat{v})\leq1$, we obtain $$\begin{aligned}
(\zeta_{1}\hat{p}(\hat{v})+\zeta_{0})^{2} & =\zeta_{1}^{2}\hat{p}^{2}(\hat{v})+2\zeta_{1}\zeta_{0}\hat{p}(\hat{v})+\zeta_{0}^{2}\\
 & \leq(\zeta_{1}^{2}+2\zeta_{1}\zeta_{0})\hat{p}(\hat{v})+\zeta_{0}^{2}<1,\end{aligned}$$ which implies (\[eq:alpha\_ineq\]). It then follows from Lemma \[LemmaDOne\] that $$\begin{aligned}
\mathbb{E}\big[\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big] & \leq\mu\theta^{t},\label{eq:up1}\end{aligned}$$ where $\mu\geq0$ and $\theta\in(0,1)$ depend on $\zeta_{1}$ and $\zeta_{0}$. Next, we apply Lemma \[LemmaSumTwo\] to find an upper bound of the summation term $\sum_{j=0}^{t-2}\Big(\mathbb{E}\Big[\big(\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big)^{2}\Big]\Big)^{\frac{1}{2}}$. Specifically, let $\gamma_{1}\triangleq\zeta_{1}$ and $\gamma_{0}\triangleq\zeta_{0}$. By (\[eq:abkcond-difficult\]), we have (\[eq:gamma\_cond\]). Noting that $\gamma_{1}>0$ and $\gamma_{0}\in[0,1)$, we obtain by Lemma \[LemmaSumTwo\] that $$\begin{aligned}
\sum_{j=0}^{t-2}\Big(\mathbb{E}\Big[\big(\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big)^{2}\Big]\Big)^{\frac{1}{2}} & \leq f,\label{eq:up2}\end{aligned}$$ where $f\geq0$ depends on $\zeta_{1}$ and $\zeta_{0}$. Now, by letting $$\begin{aligned}
\hat{\theta}\triangleq\theta,\quad\hat{\mu} & \triangleq\mu c_{2}/c_{1},\quad\hat{f}\triangleq(f+1)c_{2}/c_{1},\label{eq:SecondThetaMuFHatDef}\end{aligned}$$ we obtain (\[eq:noise\_to\_state\_three\]) from (\[eq:exp\_ineq\_in\_noise\_three\])–(\[eq:up2\]). $\blacksquare$

Theorem \[TheoremDependentDisturbance\] is applicable to scenarios where the condition (\[eq:almost\_sure\_noise\_bound\]) of Theorem \[TheoremBoundedDisturbance\] may fail to hold. In particular, if the entries of the disturbance vector has distributions with infinite support, then (\[eq:almost\_sure\_noise\_bound\]) does not hold (e.g., $w(t)\sim\mathcal{N}(m,\Sigma)$ where $m\in\mathbb{R}^{n}$ and $\Sigma\in\mathbb{R}^{n\times n}$ is a positive-definite matrix). In such cases, Theorem \[TheoremDependentDisturbance\] can be utilized. If $\mathbb{E}[\|w(t)\|_{2}^{2}]\leq\tilde{w}$ holds for all $t\in\mathbb{N}_{0}$ with a scalar $\tilde{w}\geq0$, then it follows from (\[eq:noise\_to\_state\_three\]) that $\limsup_{t\to\infty}\mathbb{E}[\|x(t)\|_{2}]\leq\hat{f}\tilde{w}^{\frac{1}{2}}$, indicating the long-run boundedness of expected state norm.

Although Theorem \[TheoremDependentDisturbance\] is applicable to a wider range of scenarios in terms of the disturbance, the condition (\[eq:abkcond-difficult\]) concerning the average jamming attack interference power is more restrictive than the condition (\[eq:abkcond-vhat\]) of Theorem \[TheoremBoundedDisturbance\]. In particular, we have $\big((1-\hat{p}(\hat{v}))\|A+BK\|+\hat{p}(\hat{v})\|A\|\big)^{2}<(1-\hat{p}(\hat{v}))\|A+BK\|^{2}+\hat{p}(\hat{v})\|A\|^{2}$ for $\hat{p}(\hat{v})\in(0,1)$ indicating that (\[eq:abkcond-difficult\]) implies (\[eq:abkcond-vhat\]), but not vice versa.

It is interesting that both Theorems \[TheoremBoundedDisturbance\] and \[TheoremDependentDisturbance\] can be used for assessing stability in the scenarios where the transmission failure indicator process $\{l(t)\}_{t\in\mathbb{N}_{0}}$ and the disturbance process $\{w(t)\}_{t\in\mathbb{N}_{0}}$ are dependent. This is the case, e.g., when the state measurements received by the controller are subject to noise. Further, the processes $\{l(t)\}_{t\in\mathbb{N}_{0}}$ and $\{w(t)\}_{t\in\mathbb{N}_{0}}$ would also be dependent if the disturbance is partially/fully caused by the attacker, and the jamming interference results in errors in the control input packet content.

Numerical Example {#sec:Illustrative-Numerical-Examples}
=================

Consider the networked control system (\[eq:closed-loop-system\]) with $$\begin{aligned}
A=\left[\begin{array}{cc}
0.1 & -1\\
1.1 & 1.8
\end{array}\right],\, & B=\left[\begin{array}{c}
0\\
1
\end{array}\right],\,K=[-0.9277\,-1.2615],\end{aligned}$$ and the wireless channel whose transmission failure probability function $p$ is given by (\[eq:p\_from\_reference-1\]) where $c=1$, $\xi=3$, and $\sigma=0.4$.

In [@ahmetifacwc2017], we investigated almost sure asymptotic stability of this system for the disturbance-free case ($w(t)=0$, $t\in\mathbb{N}_{0}$). There, we used the concave, continuous, and nondecreasing function $\hat{p}(\vartheta)\triangleq p(\vartheta+\psi)$ with $\psi\triangleq(c\xi-3\sigma)/3$, which upper-bounds $p$ according to (\[eq:p\_phat\_ineq\]). By Theorem 3.5 of [@ahmetifacwc2017], the closed-loop system is almost surely asymptotically stable under any jamming attacks that satisfy Assumption \[FirstAssumption\] with $\overline{v}\leq3.5$. The analysis in [@ahmetifacwc2017] is Lyapunov-based, and for the case with $\overline{v}=3.5$, it utilizes the Lyapunov-like function $V(x)\triangleq x^{\mathrm{T}}Px$ with the positive-definite matrix $$\begin{aligned}
P & =\left[\begin{array}{cc}
0.7728 & 0.8554\\
0.8554 & 3.2649
\end{array}\right].\label{eq:example-p-matrix}\end{aligned}$$ The matrix $P$ is also useful for the first-moment stability analysis of this system. In particular, we use the matrix norm $\|\cdot\|$ induced by the vector norm $\|x\|_{P}\triangleq\sqrt{x^{\mathrm{T}}Px}$. By using this matrix norm, the condition (\[eq:abkcond\]) is satisfied for $\overline{v}\leq1.29$. From Proposition \[PropositionMomentConvergence\], the networked control system (\[eq:closed-loop-system\]) without disturbance is first-moment geometrically stable under jamming attacks that satisfy Assumption \[FirstAssumption\] with $\overline{v}\leq1.29$. Hence, in the disturbance-free case $\mathbb{E}[\|x(t)\|_{2}]$ converges to zero with a geometric rate.

![Effect of jamming attacks following (\[eq:ExampleStrategyV\]) with and without disturbance (Top: $\tau_{1}=960$, $\tau_{2}=40$, $v^{*}=32$; Bottom: $\tau_{1}=1440$, $\tau_{2}=60$, $v^{*}=32$). Approximate first moments $\mathbb{E}[\|x(t)\|_{2}]$ are obtained through $500$ simulations with the initial state $x_{0}=[1,\,1]^{\mathrm{T}}$. []{data-label="Flo:Example1"}](example1twofigures500samplepaths "fig:"){width="0.91\columnwidth"} -14pt

-16pt

### Disturbance-free scenario

As discussed in Section \[subsec:Joint-Effect-of\], Assumption \[FirstAssumption\] allows the attacker to jam the channel with very large interference powers after waiting without attacking for sufficiently long durations. For instance, for the attack strategy considered in (\[eq:ExampleStrategyV\]) with $\tau_{1}=960$, $\tau_{2}=40$, and $v^{*}=32$, Assumption \[FirstAssumption\] is satisfied with $\overline{\kappa}=0$ and $\overline{v}=1.28$. In the disturbance-free case, this attack strategy does not create a problem for stability since $\overline{v}$ is sufficiently small. In particular, after the long duration $\tau_{1}$ without attacks, the state norm gets very close to zero, and as a result, the state norm after the attack period of $\tau_{2}$ time steps is also small.

### Scenarios with disturbance

By contrast, in the case with disturbance, the attack strategy (\[eq:ExampleStrategyV\]) makes the state norm grow at time $\tau_{1}+\tau_{2}$. This is because, even after the long attack-free duration, the state norm cannot get close to zero due to the disturbance. This is shown in the top part of Fig. \[Flo:Example1\] where the disturbance $w(t)$ at each time $t$ is uniformly distributed in $[-0.5,0.5]$. Under disturbance, the length $\tau_{2}$ of the attack period directly affects the growth of the state norm. The attacker can increase the waiting time $\tau_{1}$ to attack with a longer duration $\tau_{2}$ with the same high interference power $v^{*}$ to make the state norm grow, while still satisfying Assumption \[FirstAssumption\].

In the bottom part of Fig. \[Flo:Example1\], we see that for the same level of disturbance but with $\tau_{2}=60$, the state is driven to larger values. Notice that with $\tau_{1}=1440$, $\tau_{2}=60$, and $v^{*}=32$, Assumption \[FirstAssumption\] is also satisfied with $\overline{\kappa}=0$ and $\overline{v}=1.28$. Although after the time $\tau_{1}+\tau_{2}$, the effect of the attack diminishes, the attacker can repeat cycles of sleeping and jamming, and the state norm may grow if the attacker uses higher interference powers for longer durations.

To guarantee a predetermined bound on the expected state norm, interference power levels need to be restricted. This is achieved by Assumption \[SecondAssumption\]. Under Assumption \[SecondAssumption\], the attacker can attack with a jamming interference power $v^{*}>\hat{v}$ consecutively for at most $\lfloor\hat{\kappa}/(v^{*}-\hat{v})\rfloor$ time steps. For instance, with $\hat{\kappa}=1228.8$ and $\hat{v}=1.28$, the jamming attacks in the top part of Fig. \[Flo:Example1\] satisfy Assumption \[SecondAssumption\]. However, the jamming attacks in the bottom part of the figure do not satisfy Assumption \[SecondAssumption\] with the same $\hat{\kappa}$ and $\hat{v}$ due to the longer attack duration. For a duration of $60$ time steps, the maximum allowed interference power is $v^{*}=21.76$. We remark that the parameters $\hat{\kappa}$ and $\hat{v}$ can be selected to reflect the capabilities of the attacker.

If the jamming strategy satisfies Assumption \[SecondAssumption\] with $\hat{v}\leq1.29$, then by Theorem \[TheoremBoundedDisturbance\], the first moment of the state satisfies the bound in (\[eq:noise\_to\_state\_one\]) for any bounded disturbance. If the disturbance is not bounded, then Theorem \[TheoremDependentDisturbance\] can be applied; by Theorem \[TheoremDependentDisturbance\], the bound in (\[eq:noise\_to\_state\_three\]) holds if the attacker is less powerful with $\hat{v}\leq0.345$.

![Approximate expected values of total state norm (top) and total transmission power (bottom) for different countermeasure parameters compared to the case with no countermeasures from Fig. \[Flo:Example1\].[]{data-label="Flo:Countermeasures"}](countermeasurefigure "fig:"){width="0.91\columnwidth"} -14pt

-18pt

### Countermeasures against jamming

The damaging effects of the jamming attacks can be reduced by adjusting the transmission power ($\xi$ in (\[eq:p\_from\_reference-1\])). In particular, the controller can improve the overall performance by increasing the transmission power when there are many consecutive failures. This countermeasure against jamming can be described as follows. If $l(t-i)=1$ for each $i\in\{1,\ldots,N_{\mathrm{C}}\}$ (representing $N_{\mathrm{C}}$ total consecutive failures), then at time $t$, the transmission power $\xi(t)$ is set to a value $\xi_{\mathrm{C}}$ (larger than the nominal value $3$ used above) for a duration of $T_{\mathrm{C}}$ time steps. Thus, at those time steps, failures become less likely. After $T_{\mathrm{C}}$ time steps, the transmission power is set back to its nominal (lower) value and the countermeasure system restarts counting consecutive failures. We explore the effectiveness of this countermeasure against the attacks that are illustrated in the bottom part of Fig. \[Flo:Example1\]. Specifically, Fig. \[Flo:Countermeasures\] shows the expected total state norm ($\sum_{i=0}^{t-1}\mathbb{E}[\|x(i)\|_{2}]$) and expected total transmission power ($\sum_{i=0}^{t-1}\mathbb{E}[\xi(i)]$) approximated through 500 simulations for different parameter values $\xi_{\mathrm{C}}\in\{6,12\},N_{\mathrm{C}}\in\{2,4\},T_{\mathrm{C}}\in\{4,8\}$. The results indicate that the effects of jamming can be mitigated by temporarily increasing transmission powers, and the performance gets better with larger total transmission power use.

Conclusion {#sec:Conclusion}
==========

We explored the networked control problem under jamming attacks with time-varying interference power. Specifically, we investigated the joint effects of jamming attacks and disturbance, and obtained conditions under which the first moment of the state stays bounded. Our results indicate that if the disturbance is known to be bounded, larger average jamming interference powers can be allowed.

One of our future research directions is to increase robustness properties of the overall system by utilizing predictive control approaches proposed previously in [@feng2017resilient]. Another future work is to provide an analysis of the networked control system under time-varying transmission powers. In this line of research, for a wireless networked control problem without attacks, [@varma2017stochastic] recently explored stability and energy-efficiency under time-varying transmission powers.

Proof of Proposition \[PropositionMomentConvergence\]
=====================================================

The proof of Proposition \[PropositionMomentConvergence\] is based on the following result.

\[LemmaDTwo\] Suppose that the attacker’s interference power process $\{v(t)\in[0,\infty)\}_{t\in\mathbb{N}_{0}}$ satisfies Assumption \[FirstAssumption\]. Then for every $\alpha_{1}\geq0$, $\alpha_{0}\geq0$ that satisfy $$\begin{aligned}
\alpha_{1}\hat{p}(\overline{v})+\alpha_{0} & <1,\label{eq:alpha_ineq-1}\end{aligned}$$ there exist scalars $\mu\geq0$ and $\theta\in(0,1)$ such that $$\begin{aligned}
\mathbb{E}[\prod_{i=0}^{t-1}(\alpha_{1}l(i)+\alpha_{0})] & \leq\mu\theta^{t},\quad t\in\mathbb{N}.\label{eq:geometric-lemma-result-1}\end{aligned}$$

The proof is similar to that of Lemma \[LemmaDOne\]. In particular, we have (\[eq:expxhineq-1-1\]) and (\[eq:powerbound\]) with $t_{1}=0$, $t_{2}=t$, $h(v)\triangleq\alpha_{1}\hat{p}(v)+\alpha_{0}$, and hence, $\mathbb{E}[\prod_{i=0}^{t-1}(\alpha_{1}l(i)+\alpha_{0})]\leq\mathbb{E}[h^{t}(\frac{1}{t}\sum_{i=0}^{t-1}v(i))]$. By Assumption \[FirstAssumption\], we then obtain $\mathbb{E}[h^{t}(\frac{1}{t}\sum_{i=0}^{t-1}v(i))]\leq h^{t}(\frac{\overline{\kappa}}{t}+\overline{v})$. Therefore, by (\[eq:alpha\_ineq-1\]), after letting $T^{*}$ be a positive integer such that $h(\frac{\overline{\kappa}}{T^{*}}+\overline{v})<1$ and defining $$\begin{aligned}
\theta\triangleq h(\frac{\overline{\kappa}}{T^{*}}+\overline{v}),\quad & \mu\triangleq(\alpha_{1}+\alpha_{0})^{T^{*}-1}\theta^{-(T^{*}-1)},\label{eq:thetamuappendix}\end{aligned}$$ we obtain (\[eq:geometric-lemma-result-1\]).

*Proof of Proposition \[PropositionMomentConvergence\]*: By (\[eq:two-norm-ineq\]) with $w(t)=0$, $t\in\mathbb{N}_{0}$, we have $$\begin{aligned}
\mathbb{E}[\|x(t)\|_{2}] & \leq\frac{c_{2}}{c_{1}}\mathbb{E}\Big[\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\Big]\|x_{0}\|_{2}.\label{eq:two-norm-ineq-1-1}\end{aligned}$$ Next, we apply Lemma \[LemmaDTwo\]. First, since $\|A\|>1$, (\[eq:abkcond\]) implies $\|A+BK\|\in[0,1)$, and thus, $\zeta_{1}>0$, $\zeta_{0}\in[0,1)$. With $\alpha_{1}=\zeta_{1}$ and $\alpha_{0}=\zeta_{0}$, (\[eq:abkcond\]) implies (\[eq:alpha\_ineq-1\]). Therefore, by Lemma \[LemmaDTwo\], we have $\mathbb{E}\big[\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]\leq\mu\theta^{t}$, where $\mu\geq0$ and $\theta\in(0,1)$. Hence, by (\[eq:two-norm-ineq-1-1\]), the inequality (\[eq:moment-convergence\]) holds with $\overline{\mu}\triangleq\frac{c_{2}}{c_{1}}\mu$ and $\overline{\theta}=\theta$. $\blacksquare$

[^1]: This work is supported by JST ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), by JST CREST Grant No. JPMJCR15K3, and by JSPS under Grant-in-Aid for Scientific Research Grant No. 18H01460.

[^2]: Ahmet Cetinkaya is with the National Institute of Informatics, Tokyo, 101-8430, Japan. (e-mail: [cetinkaya@nii.ac.jp]{}).

[^3]: Hideaki Ishii is with the Department of Computer Science, Tokyo Institute of Technology, Yokohama 226-8502, Japan. (e-mail: [ishii@c.titech.ac.jp]{}).

[^4]: Tomohisa Hayakawa is with the Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan. (e-mail: [hayakawa@sc.e.titech.ac.jp]{}).
