## 2 System Model and Problem Formulation
### Wireless Power Transfer with Aerial Channel Model  
We denote the channel vector between the UAV and the \(u\)-th EHD on the ground as \(\mathbf{h}_{u}\in\mathbb{C}^{N\times 1}\); for WPT operation, the harvested power of the \(u\)-th EHD is given by [1; 2; 3; 4]. \[P_{u}^{\text{WPT}}=\zeta_{u}\operatorname{Tr}\left(\mathbf{h}_{u}\mathbf{h}_{u }^{H}\mathbf{X}\right),\;\forall u\in\mathcal{U}, \tag{7}\]  
where \(\zeta_{u}\) (\(\in(0,1]\)) is the energy-harvesting efficiency of the \(u\)-th EHD, which is determined by the RF circuit. Without loss of generality, we assume that \(\zeta_{1}=\cdots=\zeta_{U}=1\) for simplicity. We note that any constant value of \(\zeta_{u}\) can be incorporated into our design problem. For the UAV-enabled WPT to the EHDs on the ground, we adopt an air-to-ground (A2G) Rician fading channel to capture the unique characteristics of the UAV aerial channel in practical scenarios [8; 9; 10; 11; 18]. This channel type comprises both a deterministic line-of-sight (LoS) component and a Rayleigh-distributed non-line-of-sight (NLoS) component. In addition, given that the propagation characteristics of the A2G channel, such as obstacle density and link quality, vary with the UAV altitude, the probability of having an LoS link, as well as the path loss exponent and the Rician factor, is modeled as a function of the UAV altitude. Consequently, the A2G Rician fading channel model effectively describes the aerial channel by accounting for both the dominant LoS path and the combined effects of large-scale and small-scale fading. The A2G Rician fading channel vector between the UAV and the \(u\)-th EHD is expressed as [11; 18; 27]  
\[\mathbf{h}_{u}=\sqrt{\beta_{0}d_{u}^{-\alpha_{u}}}\Bigg{(}\sqrt{\frac{\mathcal{ K}_{u}}{\mathcal{K}_{u}+1}}\mathbf{a}(\phi_{u})+\sqrt{\frac{1}{\mathcal{K}_{u}+1 }}\mathbf{h}_{u}^{\text{NLoS}}\Bigg{)}, \tag{8}\]  
where \(\beta_{0}\) denotes the path loss at the reference distance and \(d_{u}\) is the distance between the UAV and the \(u\)-th EHD, which is given by  
\[d_{u}=\sqrt{r_{u}^{2}+h^{2}}. \tag{9}\]  
In addition, \(\alpha_{u}\) represents the path loss exponent between the UAV and the \(u\)-th EHD. The deterministic LoS component is denoted by \(\mathbf{a}(\phi_{u})\), where \(\phi_{u}\) indicates the \(u\)-th EHD's angle of departure (AoD), and is constructed based on (4) [18; 27]. In addition, \(\mathcal{K}_{u}\) denotes the Rician factor, which represents the power ratio between the LoS and NLoS components, where the Rayleigh-distributed NLoS component is expressed as follows: \(\mathbf{h}_{u}^{\text{NLoS}}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{N})\in \mathbb{C}^{N\times 1}\)[18; 27]. Details of each parameter are reported below. The probability of having an LoS link is modeled as a function of the UAV altitude [5; 6; 7; 8; 9; 10; 11; 18] as follows:  
\[P_{\text{LoS}}(\psi_{u})=\frac{1}{1+a_{1}\exp\Big{(}-b_{1}\big{(}\frac{180}{ \pi}\psi_{u}-a_{1}\big{)}\Big{)}}, \tag{10}\]  
where \(a_{1}>0\) and \(b_{1}>0\) are the environment-dependent (i.e., suburban, urban, dense urban, and high-rise urban) parameters. In addition, \(\psi_{u}\) denotes the elevation angle (in radians) between the UAV and the \(u\)-th EHD, which is expressed as  
\[\psi_{u}=\arctan\Big{(}\frac{h}{r_{u}}\Big{)}. \tag{11}\]  
In addition, the path loss exponent for the \(u\)-th EHD is modeled as a function of \(P_{\text{LoS}}(\psi_{u})\)[8; 9; 10; 11; 18]; therefore, it is expressed as follows:  
\[\alpha_{u}=a_{2}P_{\text{LoS}}(\psi_{u})+b_{2}, \tag{12}\]where coefficients \(a_{2}\) and \(b_{2}\) are expressed as  
\[a_{2} =\big{(}\alpha_{\frac{\pi}{2}}-\alpha_{0}\big{)}\frac{1+a_{1}\exp(a_ {1}b_{1})}{a_{1}\exp(a_{1}b_{1})},\] \[b_{2} =\alpha_{0}-\frac{a_{2}}{1+a_{1}\exp(a_{1}b_{1})}.\]  
respectively, where \(\alpha_{0}\) and \(\alpha_{\frac{\pi}{2}}\) denote the path loss exponents of the ground links and aerial links, respectively [7; 8; 9].