# Multi-Satellite Task Parallelism via Priority-Aware Decomposition and Dynamic Resource Mapping
## 4 Methodology
### Dynamic Scheduling and Task Dependency Constraint Handling Based on Temporal Logic
Resource limitations for each satellite \(s\) are imposed by the linear constraint  
\[\sum_{T_{ij}\in\mathcal{M}}R(T_{ij})\;x_{ijs}\leq R_{s},\quad\forall s, \tag{37}\]  
where \(R(T_{ij})\) denotes the resource requirement of task \(T_{ij}\), and \(R_{s}\) is the resource capacity of satellite \(s\). Thus, the overall MINLP formulation is  
\[ \begin{split}\min_{x_{ij}(t),t_{ij}}&\quad\max_{T_{ij} \in\mathcal{M}}\bigl{(}t_{ij}+d_{ij}\bigr{)},\\ \text{s.t.}&\quad t_{ij}=\sum_{t=1}^{T}t\,x_{ij}(t), \quad\forall T_{ij}\in\mathcal{M},\\ &\quad\sum_{t=1}^{T}x_{ij}(t)=1,\quad x_{ij}(t)\in\{0,1\},\quad \forall T_{ij}\in\mathcal{M},\\ &\quad t_{ij}\geq t_{ik}+\tau_{ik}-M\,(1-z_{ijk}),\quad\forall(T _{ik},T_{ij}),\\ &\quad\sum_{t=1}^{T}x_{ik}(t)\leq T\,z_{ijk},\quad\sum_{t=1}^{T}x _{ik}(t)\geq z_{ijk},\quad\forall(T_{ik},T_{ij}),\\ &\quad\sum_{T_{ij}\in\mathcal{M}}R(T_{ij})\,x_{ijs}\leq R_{s}, \quad\forall s,\end{split}  \tag{38}\]  
Since all LTL formulas are systematically transformed into the linear inequalities Equations (35) and (36) via auxiliary variables and big-M constraints, and the resource constraints (37) are inherently linear, the MINLP in Equation (38) comprises a finite set of linear inequalities. Therefore, if there exists an assignment of the binary variables \(\{x_{ij}(t),z_{ijk}\}\) satisfying Equation (38), a feasible schedule exists. Consequently, under the condition that all temporal logic and resource constraints are met, the dynamic scheduling problem admits a feasible solution. Based on the concept above, we proposed a scheduling algorithm that employs MINLP to dynamically adjust task scheduling in real time based on the system state and task dependencies. The main idea of the proposed TL-MIS is shown in Algorithm 3. ```
Input: Task set \(M\), satellite set \(S\), resource constraints \(R\), dependency constraints \(\phi_{ij}\) Output: Optimal task scheduling \(x_{ij}(t)\) Initialize: \(T_{s}\leftarrow\) task states; \(R_{s}\leftarrow\) available resources; while\(M\
eq\mathcal{O}\)do

Construct scheduling problem with current task set and constraints; foreach\(T_{i}\in M\)do

iftask dependencies \(\phi_{ij}\) are satisfiedthen

Assign task \(T_{i}\) to satellite \(s\in S\);

Determine start time \(t_{i}\) and duration \(\Delta t_{i}\);

Update resource usage \(R_{s}\gets R_{s}-R(T_{i})\);

Update task states \(T_{s}\);

Remove completed tasks from \(M\);

Return:\(x_{ij}(t)\);
```  
**Algorithm 3**Dynamic Scheduling Using Temporal Logic and MINLP (TL-MIS)  
This algorithm starts by initializing the task states and available resources (Lines 2-3). In each iteration, it constructs a scheduling problem based on the current set of tasks and constraints (Line 5). Then, for every task \(T_{i}\) in the set \(M\), if the task dependencies \(\phi_{ij}\) (as specified by temporal logic) are satisfied, the algorithm assigns \(T_{i}\) to a satellite from \(S\), determines its start time \(t_{i}\) and duration \(\Delta t_{i}\) (Lines 6-9), and updates the available resourcesby subtracting the resources required for \(T_{i}\) (Line 10). After processing all tasks in the current iteration, the task states are updated (Line 11), and completed tasks are removed from \(M\) (Line 12). Finally, the algorithm returns the optimal task scheduling \(x_{ij}(t)\), which specifies the execution schedule that meets all the temporal and resource constraints. The proposed dynamic scheduling approach leverages temporal logic to encode task dependencies directly into the scheduling model, allowing for a flexible and robust adaptation to the dynamic nature of multi-satellite cooperative computing. The use of MINLP ensures that the solution satisfies all the constraints, providing a feasible and efficient scheduling strategy.