## 1 Introduction  
Autonomous underwater vehicles (AUVs) have become indispensable robotic devices for ocean exploration that carry out various military and civilian applications, including ocean floor surveying, oceanographic data gathering, minefield surveying, pipeline inspection, and anti-submarine warfare [1, 2, 3]. Research in underwater vehicles has been very active in the past few decades with the advancement of control, sensing, communication, and computing technologies. In practice, controlling the stability of AUV is a challenging task during underwater operations and time-varying disturbances [4], i.e., the payload, mass, dynamics, and buoyancy will change when performing different tasks. In addition, AUV suffers from marine environmental external disturbances, i.e., wakes, ocean currents [5, 6], and unpredictable uncertainties [7, 8, 9]. In recent years, the security of autonomous marine vehicles has attracted much attention because attackers can access a set of sensors and actuator devices, modify their software or environment, and carry out coordinated attacks on the system design. In cyber-security, control technology has been widely applied, and it is well known that control performance depends on the quality of control input and sensor measurement signals [10]. Recently, sensor and actuator attacks were presented in adaptive neural dissipative control for Markovian jump cyber-physical systems [11], resilient control of cyber-physical systems [12], and cyber-attacks in industrial control systems [13]. A secure tracking control was developed in [14] to guarantee the prescribed security in systems with sensor and actuator attacks. Actuator saturation and probabilistic cyber-attacks were discussed in uncertain networked control systems [15]. The hybrid-driven-based resilient control for T-S fuzzy systems with time-delay and cyber-attack was investigated in [16]. The behavior of the system can be affected by complexities such as random uncertainties that occur during system modeling. These complexities are subject to random changes in complicated circumstances, exceptional environmental disturbances, random failures, modeling errors, varying geometry and material properties, repairs of system components, and so forth, which may occur in a probabilistic way. The appearance of parameter uncertainties often exhibits a random nature, because of unpredictable changes. As a result, the so-called randomly occurring uncertainty (ROU) is usually controlled by Bernoulli-distributed stochastic variables. In recent years, the concept of randomly occurring uncertainties has been investigated in [17; 18]. Time-varying delay frequently occurs in AUV controls, and it may cause undesirable transient system responses, diminish system performance, and even lead to instability. References [19; 20] discussed unknown uncertainties and time-varying delays in AUV systems. In recent years, the concept of time-varying delay and delays in vehicle systems has been investigated [21; 22; 23]. In [19; 20], robustness against unpredictable uncertainties was investigated by considering unknown but norm-bounded (UBB) uncertainties. However, note that UBB random uncertainties were not handled in [19; 20]. Inspired by [19; 20], our paper addresses robustness against unpredictable random uncertainties by considering UBB random uncertainties. Moreover, we handle time-varying delay in AUV controls. However, to the best of our knowledge, cyber-attacks were not handled in the control of AUV systems. To the best of our knowledge, no papers are available on the stabilization of AUV systems with time-varying delay, randomly occurring uncertainties, and cyber-attacks. In our paper, we design a robust \(H_{\infty}\) control for autonomous underwater vehicle systems under time-varying delay, uncertain random models, and cyber-attacks.