## 6 Conclusions  
In this paper, we proposed the joint design of a UAV's altitude- and channel statistics-based energy beamforming in order to minimize the overall charging time required for all EHDs by considering the A2G Rician fading channel. To solve the formulated highly nonconvex and nonlinear problem, we first optimized our design for a single EHD by deriving optimal energy beamforming in closed form, thereby developing a low-complexity algorithm to obtain the optimal altitude. Then, considering multiple EHDs, we developed efficient algorithms for the joint design of altitude and energy beamforming based on the dual problem. We further explored two efficient methods with low complexity, yielding a near-optimal solution driven by insights from the design for a single-EHD scenario, as well as a sub-optimal solution by leveraging closed-form weighted-sum energy beamforming. The numerical results demonstrate that compared to conventional methods, the proposed joint design can be used to substantially reduce both the overall charging time as well as the computational complexity. While the overall charging time increases in the Dense Urban environment compared to the Urban environment due to a lower LoS probability, the average execution time remains similar, highlighting the robustness of the proposed methods in terms of complexity reduction.  
Figure 8: Performance comparison of various methods when \(r_{\max}=100\) m in a Dense Urban environment.  
Figure 9: Performance comparison of various methods when \(r_{\max}=150\) m in a Dense Urban environment.  
Although the proposed methods are based on long-term channel statistics, they can be extended to scenarios with perfect CSI by replacing the long-term channel statistics with instantaneous CSI. In addition, the proposed method can be adapted to different channel models, such as Nakagami fading, by redefining statistical expectations and modifying the algorithm with necessary mathematical derivations to incorporate new fading characteristics. Extending the proposed methods to provide a comprehensive analysis of scalability, especially when dealing with large numbers of EHDs in diverse environmental conditions, remains one of our ongoing research topics. Additionally, the joint design of altitude and energy beamforming, taking into account practical factors such as the UAV's energy consumption, is also part of our ongoing work for future research. Moreover, future work will focus on enhancing the proposed model to adapt to dynamic environmental changes that may impact the UAV's ability to maintain optimal altitude, ensuring robustness in real-world scenarios.  
This work was supported in part by the NRF(National Research Foundation of Korea) grant funded by the Korea government (Ministry of Science and ICT) (RS-2023-00214142), and in part by the IITP(Institute of Information & Coummunications Technology Planning & Evaluation)-ICAN(ICT Challenge and Advanced Network of HRD) grant funded by the Korea government (Ministry of Science and ICT) (IITP-2024-RS-2022-00156409).  
**Data Availability Statement:** Data are contained within the article.  
**Conflicts of Interest:** The author declares no conflicts of interest.