## 2 System Model  
Consider a SatCom system comprising a single ES serving \(K\) single-antenna user equipments (UEs), a single multibeam satellite, and \(K\) UAVs with individual RISs (i.e., \(K\) RISs) equipped with \(N\) reflecting elements. In our system model, a UAV equipped with an RIS operates as a mobile reflector between the satellite and the UEs (even though UAV-based communications may have flight-time issues caused by battery constraints, the proposed scheme can also apply to an RIS attached to manned aircraft or building surfaces, which can be relatively free from battery constraints), as shown in Figure 1. The satellite is equipped with a reflective array antenna consisting of \(K\) feeds (i.e., a single feed per beam) in which a transparent satellite payload is considered. The data for the \(K\) UEs are multiplexed in frequency at the ES and transmitted to the satellite through the feeder link operating in the optical band. Subsequently, at the satellite side, the received signal is demultiplexed, amplified, translated to RF streams, and then transmitted to the UEs via the corresponding UAVs through multiple beams. Each UAV can passively reflect the incident signal from the satellite using an RIS, and each reflecting element induces a phase shift of the incident wave (i.e., passive beamforming). We assume that the ES allocates a suitable RIS to each UE (i.e., one RIS per UE) and sends a control signal, including the reflection phase shift information, to each RIS through a dedicated wireless control link. In Section 3, we consider the unicast system, i.e., each RIS serves each corresponding UE. Subsequently, in Section 4, the unicast system is extended to the multicast system, i.e., each RIS serves multiple UEs in a corresponding multicast group. Note that multibeam SatCom systems can use the advantages of beamspace isolation and resource allocation for improved spectral efficiency [19]. Depending on the location of the UAV, the UAVs located far away from each other will not suffer from interbeam interference due to the beamspace isolation, whereas the UAVs with similar locations can  
Figure 1: Illustrative system model of an RIS-assisted SatCom system. suffer from interbeam interference, which can degrade the performance. To tackle this challenge, we make the assumption that the ES allocates orthogonal frequencies among UAVs/UEs located in similar positions, thereby supporting dynamic spectrum and power allocation [20], as shown in Figure 1. Consequently, multibeam satellite systems can be an interbeam interference-free environment. The received signal at UE \(k\) is then obtained as  
\[y_{k}=\sqrt{\rho_{k}}f_{k}^{\mathrm{H}}\mathbf{\Phi}_{k}\mathbf{g}_{k}x_{k}+n_{k}, \tag{1}\]  
where \(\rho_{k}\) is the signal-to-noise ratio (SNR) for UE \(k\) and \(\mathbf{f}_{k}\in\mathbb{C}^{N\times 1}\) and \(\mathbf{g}_{k}\in\mathbb{C}^{N\times 1}\) are, respectively, the fading channels between UE \(k\) and RIS \(k\) and between the satellite and RIS \(k\). Moreover, \(x_{k}\) and \(n_{k}\) are the transmit signal and additive complex Gaussian noise with zero mean and unit variance, respectively, and \(\mathbf{\Phi}_{k}\in\mathbb{C}^{N\times N}\) is a diagonal reflection matrix functioning as a passive beamformer. The reflection matrix is determined from the control signal transmitted by the ES, and \(\mathbf{\Phi}_{k}\) can be obtained as follows:  
\[\mathbf{\Phi}_{k}=\mathrm{diag}\Big{(}\sqrt{\alpha_{k,k}}\mathrm{e}^{i\theta_{k,1 }},\sqrt{\alpha_{k,2}}\mathrm{e}^{i\theta_{k,2}},\cdots,\sqrt{\alpha_{k,N}} \mathrm{e}^{i\theta_{k,N}}\Big{)}, \tag{2}\]  
where \(\sqrt{\alpha_{k,n}}\in[0,1]\) and \(\theta_{k,n}\in[0,2\pi)\) are, respectively, the amplitude and phase shifting response of the reflection element \(n\) in RIS \(k\).