# An Overview on Denial-of-Service Attacks in Control Systems: Attack Models and Security Analyses
## 3 Deterministic and Game-Theoretical Approaches to Denial-of-Service Attack Modeling
### Game-Theoretic and Optimization-Based Approaches for Attack Modeling
#### 3.2.1 Constraint-Based Models with Binary Actions  
In several studies that deal with DoS attacks, the actions of the attacker and the defender are represented with binary variables corresponding to the choices between attacking and not attacking and similarly between defending and not defending.  
For example, [PERSON] et al. [62] considered the finite-horizon state estimation problem discussed in Section 2.2, where binary variables \(\lambda_{\mathrm{S}}(t)\in\{0,1\}\) and \(\lambda_{\mathrm{A}}(t)\in\{0,1\}\) are used for indicating the decisions of the sensor and the DoS attacker, respectively. If, \(\lambda_{\mathrm{S}}(t)=1\), then the state estimate \(\hat{x}_{\mathrm{S}}(t)\) is attempted to be transmitted over the communication channel at time \(t\). Moreover, the presence of an attack that causes a transmission failure at time \(t\) is represented by \(\lambda_{\mathrm{A}}(t)=1\).  
Notice that, if the attacker and the defender do not have any constraints, it may be ideal for both players to act at every time instant (i.e., \(\lambda_{\mathrm{S}}(t)=\lambda_{\mathrm{A}}(t)=1\) for all \(t\in\mathbb{N}\)). In [62], researchers considered scenarios where both the attacker and the sensor have constraints represented by  
\[\sum_{t=1}^{T}\gamma_{\mathrm{S}}(t)\leq N_{\mathrm{S}},\quad\sum_{t=1}^{T} \lambda_{\mathrm{A}}(t)\leq N_{\mathrm{A}}, \tag{17}\]  
where \(T\in\mathbb{N}\) is the time horizon of the estimation problem, and \(N_{\mathrm{S}}\) and \(N_{\mathrm{A}}\) are positive scalars that are strictly less than \(T\). It is essential for the players to choose the best timing for their actions. To identify the worst-case attack scenarios as well as the best transmission strategies, [PERSON] et al. [62] proposed game-theoretic formulations, where a game is played by the sensor and the attacker. The goal of the defender is to minimize the cost in Equation (9) so as to minimize the estimation error, whereas the attacker wants to maximize Equation (9). For various constrained attack problems, closed-form optimal attack strategies \(\lambda_{\mathrm{A}}(1),\lambda_{\mathrm{A}}(2),\ldots,\lambda_{\mathrm{A}}(T)\) that maximize \(J_{1}(T)\) and \(J_{0}(T)\) are obtained in [63; 64].  
A constraint on the total number of attacks in a given time horizon was considered also by [PERSON] et al. [34] for a networked control problem under DoS attacks. Constraints similar to Equation (17) may characterize energy limitations of a jamming attacker. We note that, besides the formulation with constraints, there are also other formulations where the jamming energy is a part of the cost function of attacker (see, e.g., [76]). The effect of energy-related jamming costs was investigated by [PERSON] et al. [79], who considered a noncooperative game for analyzing the actions of a group of non-malicious nodes together with a malicious node that is capable of jamming and eavesdropping.  
In addition to the energy constraints in Equation (17), the attacker may have additional constraints. In [63], some additional constraints are proposed to explore strategies of an attacker that does not want to get detected by an intruder detection system. It is discussed in [63] that a detection mechanism can check the number of transmission failures in the last \(\tau_{\mathrm{D}}\in\mathbb{N}\) time steps and release an alarm if the failures go above a threshold \(d_{\mathrm{D}}\in\mathbb{N}\). To avoid being detected by such a detection system, the attacker chooses a strategy that satisfies  
\[\sum_{t=k+1}^{k+\tau_{\mathrm{D}}}\lambda_{\mathrm{A}}(t)\leq d_{\mathrm{D}}, \quad k\in\{0,1,\ldots,T-\tau_{\mathrm{D}}\}. \tag{18}\]  
Notice that under Equation (18), the number of attacks in every consecutive \(\tau_{\mathrm{D}}\) time steps does not exceed \(d_{\mathrm{D}}\). An extension of this constraint to infinite-horizon problems resembles the constraints discussed in Section 3.1.