## 2 Problem Formulation and Preliminaries
### AUV System I with Time-Varying Delay  
Motivated by [19], we consider the class of continuous-time linear uncertain time-varying delay systems of the form:  
\[ \begin{array}{rcl}\dot{x}(t)&=&(A+\Delta A)x(t)+(A_{d}+\Delta A_{d})x(t-\tau (t))+Bu(t)+B_{w}w(t),\\ y(t)&=&Cx(t),\\ x(t)&=&\phi(t),\ t\in[-\tau_{M},0],\end{array}  \tag{1}\]  
where \(x(t)\in\mathbb{R}^{n}\) is the state at time \(t\), \(u(t)\in\mathbb{R}^{m}\) is the control input, \(w(t)\in\mathbb{R}^{p}\) is the external disturbance that belongs to \(\mathbb{L}_{2}[0,+\infty)\), \(y(t)\in\mathbb{R}^{q}\) is the measured output, and \(\phi(t)\) is the initial state condition. \(A\), \(A_{d}\), \(B\), \(B_{w}\), and \(C\) are known matrices of appropriate dimensions. \(\tau(t)\) is a nonnegative time-varying delay affecting the system states, assumed to be bounded as \(0\leq\tau_{m}\leq\tau(t)\leq\tau_{M}\) and \(\dot{\tau}(t)\leq\mu<\infty\), where \(\mu\in\mathbb{R}^{+}\). The state feedback controller can be designed and formulated as follows:  
\[u(t)=Kx(t) \tag{2}\]  
where \(K\) is the feedback control gain. According to [19; 20], \(\Delta A\) and \(\Delta A_{d}\) represent the UBB uncertainties in system (1) satisfying:  
\[\Delta A=M_{1}\Delta(t)N_{1},\ \ \Delta A_{d}=M_{2}\Delta(t)N_{2} \tag{3}\]  
where \(M_{1}\), \(N_{1}\), \(M_{2}\), and \(N_{2}\) are known as constant matrices with appropriate dimensions and \(\Delta(t)\) is an unknown matrix with Lebesgue measurable elements and satisfies \(\Delta(t)^{\top}\Delta(t)\leq I\). Note that the behavior of the system can be affected by the presence of complexities, such as random uncertainties that occur during the process of system modeling. The appearance of parameter uncertainties often exhibits a random nature, due to unpredictable changes. As a result, the ROU is usually controlled by Bernoulli-distributed stochastic variables. In recent years, the concept of randomly occurring uncertainties has been investigated [17; 18]. In our paper, the randomly occurring UBB uncertainties are assumed to follow a Bernoulli distribution, which is expressed as follows:  
\[Prob\{\alpha(t)=1\}=\bar{\alpha},\ Prob\{\alpha(t)=0\}=1-\bar{\alpha},\]  
where \(\bar{\alpha}\in[0,1]\) is a known constant. The stochastic variable \(\alpha(t)\in\mathbb{R}\) is introduced to characterize the randomly occurring uncertainties. Now, considering the randomly occurring uncertainties [17] and the sensor and actuator attacks [24] in system (1), we have the following formulation:  
\[ \begin{array}{rcl}\dot{x}(t)&=&(A+\alpha(t)\Delta A)x(t)+(A_{d}+\alpha(t) \Delta A_{d})x(t-\tau(t))\\ &&+B[u(t)+\beta(t)\chi_{a}(t)]+B_{w}w(t),\\ y(t)&=&Cx(t)+\gamma(t)\chi_{s}(t),\\ x(t)&=&\phi(t),\ t\in[-\tau_{M},0],\end{array}  \tag{4}\]  
where \(x(t)\in\mathbb{R}^{n}\) denotes the state vector affected by the actuator attack \(\chi_{a}(t)\), \(y(t)\in\mathbb{R}^{q}\) is the measurable output affected by the sensor attack \(\chi_{s}(t)\), and nonlinear functions \(\chi_{a}(t)\)and \(\chi_{s}(t)\) are used to model the cyber signals inserted by attackers. The probability of the event that the system's actuator or sensor has been affected by a cyber-attack is denoted by \(\tilde{\chi}_{a}\in[0,1]\) (or \(\tilde{\chi}_{s}\in[0,1]\)). According to [25], both \(\beta(t)\) and \(\gamma(t)\) are Markovian stochastic processes with the binary state (0 or 1), which satisfy the following probability:  
\[ \begin{array}{l}E\{\beta(t)\}=Prob\{\beta(t)=1\}=\tilde{\beta},\\ E\{\gamma(t)\}=Prob\{\gamma(t)=1\}=\tilde{\gamma}.\end{array}  \tag{5}\]  
where an actuator cyber-attack \(\chi_{a}(t)\) (or a sensor cyber-attack \(\chi_{s}(t)\)) happens when event \(\beta(t)=1\) (or \(\gamma(t)=1\)), which indicates that the actuator (or the sensor) of the system is the target of a cyber-attack. On the other hand, event \(\beta(t)=0\) (or \(\gamma(t)=0\)) indicates that there is no cyber-attack on the actuator (or on the sensor).