## 3 Problem Formulation
### Mathematical Formulation  
Assume that at time slot \(t\), the computational task generated by a mobile device is divided into two parts based on the offloading ratio: one part is processed locally on the device, and the other part is offloaded to the UAV server for processing. These two parts are executed in parallel. This implies that, during a specific time slot \(t\), the task processing delay consists of two components: the delay for local processing and the delay for offloading the task to the UAV edge server and processing it on the server. In the task processing workflow, the energy consumption of UAVs is divided into two major components: mobility energy consumption and computational energy consumption. Mobility energy consumption is primarily influenced by the UAV's movement behavior, while computational energy consumption is mainly determined by the task volume and the UAV's CPU frequency. Assume that each UAV \(u\) serves only one mobile device at each time slot \(t\). Let \(\eta_{l,u}(t)\) indicate whether mobile device \(l\) is served by UAV \(u\) at time slot \(t\). The selection of mobile devices, UAV mobility, task offloading, and the UAV's CPU frequency adjustment collectively impact both task processing delay and UAV energy consumption. Therefore, this study aims to jointly optimize mobile device selection, multi-UAV mobility control, UAV CPU frequency adjustment, and task offloading allocation, with the objective of minimizing the weighted sum of task processing delay and UAV energy consumption across all time slots while satisfying the total computational task requirements and service period constraints. The optimization problem is formulated as follows:  
\[min_{\eta_{l,u}(t),z_{u}(t+1),\alpha_{l,u}(t),k(u)}\sum_{t=1}^{T}\sum_{l=1}^{N }\eta_{l,u}(t)[w_{0}t^{\mu\epsilon}(t)+w_{1}e_{to}(t)] \tag{23}\]  
\[C_{1}:x_{u}(t),x_{l}(t)\in[0,L],\forall t,\forall l,\forall u, \tag{24}\]  
\[C_{2}:y_{u}(t),y_{l}(t)\in[0,W],\forall t,\forall l,\forall u, \tag{25}\]  
\[C_{3}:\eta_{l,u}(t)\in 0,1,\forall t,\forall l,\forall u, \tag{26}\]  
\[C_{4}:\sum_{t=1}^{N}\eta_{l,u}(t)=M,\forall t,\forall u, \tag{27}\]  
\[C_{5}:w_{0}+w_{1}=1,w_{0}\in[0,1],w_{1}\in[0,1], \tag{28}\]  
\[C_{6}:||z_{u}(t)-z_{v}(t)||^{2}\
eq 0,\forall u\
eq v, \tag{29}\]  
\[C_{7}:b_{l,u}\in 0,1,\forall u,\forall l,\forall t, \tag{30}\]\[C_{8}:\sum_{t=1}^{T}c_{u}^{lo}(t)\leq E_{max},\forall u,\] (31) \[C_{9}:0\leq\alpha_{l,u}(t)\leq 1,\forall t,\forall l,\forall u,\] (32) \[C_{1}:x_{u}(t),x_{l}(t)\in[0,L],\forall t,\forall l,\forall u, \tag{24}\]0 (33) The constraints are defined as follows:  
* Constraints \(C_{1}\) and \(C_{2}\) specify that the horizontal and vertical coordinates of UAVs and mobile devices must not exceed the predefined geographical boundaries, ensuring that all devices operate within the designated area. * Constraint \(C_{3}\) indicates whether mobile device \(l\) is served by UAV \(u\) at time slot \(t\), where \(\eta_{l,u}(t)=0\) denotes that mobile device \(l\) is not served by UAV \(u\), and \(\eta_{l,u}(t)=1\) denotes that mobile device \(l\) is served by UAV \(u\). * Constraint \(C_{4}\) ensures that each UAV serves exactly one mobile device at any given time slot \(t\). * Constraint \(C_{5}\) defines \(w_{0}\) as the weight factor for task processing delay and \(w_{1}\) as the weight factor for UAV energy consumption, with \(w_{0}+w_{1}=1\). This constraint explicitly captures the tradeoff between processing delay and energy consumption. * Constraint \(C_{6}\) enforces that the positions of UAVs must not overlap, preventing potential collisions and crashes. * Constraint \(C_{7}\) describes the communication link obstruction between mobile device \(l\) and UAV \(u\), where \(b_{l,u}(t)=0\) indicates a line-of-sight (LoS) link, and \(b_{l,u}(t)=1\) indicates a non-line-of-sight (NLoS) link. * Constraint \(C_{8}\) ensures that the total energy consumption of all UAVs does not exceed their maximum battery capacity \(E_{max}\) over the entire service period.