# Distributed Coordination of Space-Ground Multiresources for Remote Sensing Missions
## 3 Problem Transformation and Decomposition
### Problem Decomposition  
The objective of P2 can be directly separated with respect to \(N\) RSS systems. However, due to the constraints of Equations (12), (14) and (19), the variables \(y(sr^{k}_{n,i,j})(\forall sr^{k}_{n,i,j}\in A_{RL})\) become global variables between the DRS system and RSS systems and thus cannot be classified as local variables of any system. To handle this problem, we introduce the local copies of global variable \(y(sr^{k}_{n,i,j})\) as [25; 26], denoted by \(\hat{y}_{r}(sr^{k}_{n,i,j})\) and \(\hat{y}_{n}(sr^{k}_{n,i,j})\), which belongs to the DRS system and RSS system \(n\), respectively. By substituting the local copies with the global variables into P2, an equivalent form can be obtained  
\[ \begin{array}{l}\text{P3}:\quad\max\sum\limits_{1\leq n\leq N}\sum\limits_{1 \leq i\leq|OM_{n}|}z_{n,i}\\ \text{s.t.}\quad\text{Equations }(\ref{eq:P2})\sim(\ref{eq:P2}),\,(\ref{eq:P2}) \sim(\ref{eq:P2})\text{ and }(\ref{eq:P2})\\ \quad\sum\limits_{j:sr^{k}_{n,i,j}\in A_{RL}}\hat{y}_{n}(sr^{k}_{n,i,j})\leq 1,\quad\forall 1\leq k\leq K,1\leq i\leq|S_{n}|,1\leq n\leq N,\end{array}  \tag{21}\]  
\[ \begin{array}{l}\text{s.t.}\quad\text{Equations }(\ref{eq:P2})\sim(\ref{eq:P2}),\,(\ref{eq:P2})\sim(\ref{eq:P2})\text{ and }(\ref{eq:P2})\\ \quad\sum\limits_{j:sr^{k}_{n,i,j}\in A_{RL}}\hat{y}_{n}(sr^{k}_{n,i,j})\leq 1,\quad\forall 1\leq k\leq K,1\leq j\leq|R|,\end{array}  \tag{22}\]  
\[ \begin{array}{l}\sum\limits_{f\in F_{n}}x(sr^{k}_{n,i,j},f)\leq C(sr^{k}_{n, i,j})\cdot\hat{y}_{n}(sr^{k}_{n,i,j}),\quad\forall sr^{k}_{n,i,j}\in A_{RL} \end{array}  \tag{23}\]  
\[ \begin{array}{l}\hat{y}_{r}(sr^{k}_{n,i,j})=\hat{y}_{n}(sr^{k}_{n,i,j})=y(sr^{ k}_{n,i,j}),\quad\forall sr^{k}_{n,i,j}\in A_{RL}\end{array}  \tag{24}\]  
The consensus constraint Equation (24) imposes that the local copy variables \(\hat{y}_{r}(sr^{k}_{n,i,j})\). \(\hat{y}_{n}(sr^{k}_{n,i,j})\) be consistent with the corresponding global variable \(y(sr^{k}_{n,i,j})\). All variables in problem P3 except the global variables \(\mathbf{y}^{C}=\left\{y(sr^{k}_{n,i,j})\right\}_{sr^{k}_{n,i,j}\in A_{RL}}\) can be divided into individual satellite systems as local variables. Thus, the constraints of P3, except the consistency constraint, can be separated into each RSS system and the DRS system. Note that since DRS systems and RSS systems play different roles in remote sensing missions, their local variables and constraints must be discussed separately. Specifically, the local variables of the RSS system \(n\) are  
\[\left\{ \begin{array}{l}\mathbf{z}_{n}=\left\{z_{n,i}\right\}_{1\leq i\leq|OM_{n} |}\\ \mathbf{x}_{n}=\left(\left\{x(f)\right\}_{f\in F_{n}},\left\{x(a,f)\right\}_{a\in A,f\in F_{n}}\right)\\ \mathbf{y}^{l}_{n}=\left(\left\{y(s^{k}_{n,i,j})\right\}_{0\leq s^{k}_{n,i,j})\in A ^{n}_{O}},\left\{y(s^{k}_{n,i,j})\right\}_{s^{k}_{n,i,j}\in A^{n}_{CL}}\right),\\ \mathbf{y}^{C}_{n}=\left\{\hat{y}_{n}(sr^{k}_{n,i,j})\right\}_{sr^{k}_{n,i,j}\in A ^{n}_{RL}}\end{array} \right. \tag{25}\]where \(A_{O}^{n}\), \(A_{GL}^{n}\), and \(A_{RL}^{n}\) represent the set of observation arcs, ground-satellite link arcs, and intersatellite link arcs belong to the RSS system \(n\), respectively. For example, \(A_{O}^{n}\) is defined as  
\[A_{O}^{n}=\Big{\{}os_{n,i,j}^{k}\Big{|}l^{ck}(o_{n,i})\in R_{O}^{k}(s_{n,j}),1 \leq k\leq K,1\leq i\leq|OM_{n}|,1\leq j\leq|S_{n}|\,\Big{\}}. \tag{26}\]  
The local variable of the DRS system is \(\hat{g}_{r}^{C}=\{\hat{g}_{r}(sr_{n,i,j}^{k})\}_{sr_{n,i,j}^{k}\in A_{RL}}\). Based on the local constraints, the set of feasible solutions for the local variables of RSS system \(n\) is given by  
\[\xi_{n}=\Big{\{}\mathbf{z}_{n},\mathbf{x}_{n},\mathbf{y}_{n}^{L},\hat{g}_{n}^{C}|\text{C1}, \text{C2},\text{C3},\text{C4},\text{C5},\text{C6},\text{C7},\text{C8},\text{ C9},\text{C10},\text{C11}\,\Big{\}} \tag{27}\]  
wherein,  
\[\text{C1}: x(f)=z_{n,i}\cdot b_{n,i},\quad\forall 1\leq i\leq|OM_{n}| \tag{28}\] \[\text{C2}: \sum\limits_{h(\theta)=p}x(a,f)-\sum\limits_{t(\theta)=p}x(a,f)= \left\{