# Optimal Position and Target Rate for Covert Communication in UAV-Assisted Uplink RSMA Systems
## 4 Optimization Problem  
In this letter, we aim to maximize the ESR by optimizing the deployment of U1 and the target rate of \(\mathbf{s}_{11}\), subject to the covertness constraint and the ECR constraint. The optimization problem is formulated as  
\[\max_{x_{1},y_{1},R_{11}} \overline{R}_{\text{sum}},\] (22a) s.t. \[\overline{\mathbb{P}_{E}^{\dagger}}\geq 1-\delta, \tag{22b}\] \[\overline{R}_{2}=\hat{R}_{2}(1-\mathbb{O}_{11}^{1})\geq\epsilon, \tag{22c}\]  
where (22b) is the covertness constraint, and (22c) is the ECR constraint. To solve the optimization problem (22), we decompose it into two subproblems, as shown in Figure 3. We first discuss the monotonicity of \(\overline{R}_{\text{sum}}\), \(\overline{\mathbb{P}_{E}}\), and \(\overline{R}_{2}\) w.r.t. \(x_{1}\) and \(y_{1}\). We observe that as the distance between U1 and Bob, i.e., \(\|\mathbf{q}_{1}-\mathbf{q}_{b}\|\), decreases, both \(\overline{R}_{\text{sum}}\) and \(\overline{R}_{2}\) increases. On the other hand, \(\overline{\mathbb{P}_{E}}\) decreases since U1 gets farther away from Willie. Then, the first subproblem is to optimize the placement of U1 under the covertness constraint (22b) to minimize the distance between U1 and Bob. The second subproblem is to optimize \(\hat{R}_{11}\) under the ECR constraint (22c) and U1's optimal placement obtained from the first subproblem to maximize the ESR.  
The first subproblem to optimize U1's placement is expressed as  
\[\min_{x_{1},y_{1}} x_{1}^{2}+y_{1}^{2},\] (23a) s.t. \[(x_{1}-x_{w})^{2}+(y_{1}-y_{w})^{2}\leq t, \tag{23b}\] \[t=\frac{\delta g_{0}(P_{1}^{\max}-P_{1}^{\min})}{g_{2w}P_{2} \Big{[}1+W\Big{(}-\frac{\epsilon^{-1/\delta}}{\delta}\Big{)}\Big{]}}-z_{1}^{2}. \tag{23c}\]  
**Lemma 2**.: _The optimal position of U1 is given by \(\mathbf{q}_{1}^{\dagger}=[x_{1}^{\dagger},y_{1}^{\dagger},z_{1}]\) when \(0<t<x_{w}^{2}+y_{w}^{2}\), where \([x_{1}^{\dagger},y_{1}^{\dagger}]=\bigg{(}1-\sqrt{\frac{t}{x_{w}^{2}+y_{w}^{2 }}}\bigg{)}[x_{w},y_{w}]\), otherwise, \(x_{1}^{\dagger}=y_{1}^{\dagger}=0\), when \(t\geq x_{w}^{2}+y_{w}^{2}\)._  
Proof.: See Appendix B.  
From Lemma 2, we notice that the optimal horizontal position of U1 lies on the line between BS and Willie.  
Figure 3: Procedure for solving optimization problem (22).  
The second subproblem to optimize \(\hat{R}_{11}\) is expressed as  
\[\max_{\hat{R}_{11}} \overline{R}_{\text{sum}}(x_{1}^{\dagger},y_{1}^{\dagger}),\] (24a) s.t. \[\mathbb{O}_{11}(x_{1}^{\dagger},y_{1}^{\dagger})\leq 1-\frac{ \epsilon}{\hat{R}_{2}}, \tag{24b}\]  
where \(\overline{R}_{\text{sum}}(x_{1}^{\dagger},y_{1}^{\dagger})\) and \(\mathbb{O}_{11}(x_{1}^{\dagger},y_{1}^{\dagger})\) represent substituting \((x_{1}^{\dagger},y_{1}^{\dagger})\) into \(\overline{R}_{\text{sum}}\) and \(\mathbb{O}_{11}\), respectively. It is shown in (18) that \(\mathbb{O}_{11}\) is a monotonically increasing function w.r.t. \(\hat{R}_{11}\). Thus, the upper limit of \(\hat{R}_{11}\) is expressed as \(\hat{R}_{11}^{\epsilon}\), where \(\hat{R}_{11}^{\epsilon}\) is the solution of \(\mathbb{O}_{11}(x_{1}^{\dagger},y_{1}^{\dagger})=1-\frac{\epsilon}{\hat{R}_{2}}\). Together with Lemma 1, the optimal choice of \(\hat{R}_{11}\) is given by \(\hat{R}_{11}^{\epsilon}=\min\{\hat{R}_{11}^{\epsilon},\hat{R}_{11}^{\dagger}\}\).