## 1 Introduction  
Radio frequency (RF)-based wireless energy transfer (WET) is a key technology enabling the development of intelligent and self-sustaining Internet of Everything (IoE) networks in the 6G era [1, 2, 3, 4, 5, 6, 7]. It provides the continuous supply of power to wireless devices over the air, such as wearable electronic devices, extended reality devices, and robotics, without the need for frequent battery replacements or wired power lines. Moreover, compared to traditional power systems, RF-based WET can significantly enhance the quality of service for powering devices by adapting to different physical conditions and service requirements while also improving throughput and robustness [1, 2, 3, 4, 5, 6, 7]. In recent years, unmanned aerial vehicles (UAVs) have gained significant attention in various scenarios due to their deployment flexibility, mobility, and cost-effectiveness, leading to their widespread adoption across various applications, including military operations, cargo delivery, disaster management, and communication platforms [8, 9, 10, 11, 12, 13, 14, 15]. They provide greater flexibility in system design and operation in wireless networks as the deployment position and path of a UAV can be adjusted, yielding significant advantages such as coverage enhancements [8, 9, 10, 11, 12, 13, 14, 15]. Due to its advantages, UAV-enabled wireless energy transfer (WET) has received great attention for providing ground devices with more efficient and stable power comparedto conventional WET systems, which use fixed-location energy transmitters [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. In particular, UAVs are able to move flexibly in three-dimensional (3D) space, utilizing favorable line-of-sight (LoS) channels with ground devices. As such, UAVs can serve as a new type of aerial energy transmitter, reducing the transmission distances for powering devices while avoiding obstacles and shadow fading, even in remote areas where conventional fixed-location energy transmitters are not available. Accordingly, UAV-enabled WET is able to overcome energy bottlenecks as well as meet urgent energy demands, extending the operational lifespan of energy-constrained devices, especially in dynamic or hard-to-reach environments such as smart cities, wireless sensor networks, and maritime networks, among others [14; 15; 16; 17; 18]. However, effective UAV trajectory design is required to reap potential gains from UAV-enabled WET. In practical UAV-enabled WET scenarios, an UAV may move closer to a device in order to reduce the transmission distance for improved power efficiency. This may result in the UAV moving farther away from another device, thereby decreasing the overall energy transfer efficiency. To address this challenge, UAV trajectory design has been extensively studied to improve the energy transfer performance for multiple devices [14; 15; 16; 17; 18; 19; 20; 21]. Specifically, in a single UAV-enabled WET system, an optimal one-dimensional (1D) trajectory design at a fixed altitude was proposed to maximize the minimum received energy among all devices during a given charging period [16]. For a more general context, the design of a two-dimensional (2D) UAV trajectory at a fixed altitude was studied to optimize the energy transferred to all devices during a given charging period [17], with further investigations in a more practical scenario [18]. Moreover, the design of a 3D UAV trajectory within a specified altitude range was explored to maximize the total received energy across all devices for a given charging period [19]. The joint design of 2D UAV trajectory and orientation of 1D directional antenna array at the UAV with a fixed altitude was also studied [20] and further extended to a structure that includes a 3D directional antenna array at the UAV [21]. Moreover, a multi-UAV-enabled WET system capable of covering a large area was proposed, and effective trajectory designs for multiple UAVs were studied to enhance energy transfer performance across various scenarios [15; 22; 23; 24].