## 4 Distribution Method for Observation Scheduling in the AEOS Constellation
### The Decentralized Framework of PI
According to whether \(q_{c}\)*(\(\mathbf{a}_{c}\oplus g\)) and \(q_{s}\)*(\(\mathbf{a}_{s}\odot g\)) is greater or less than zero, there are four cases: Case 1. \(q_{s}(\mathbf{a}_{c}\odot g)<0\) and \(q_{c}\)*(\(\mathbf{a}_{c}\odot g\)) > 0. Naturally (18) holds, then \(F_{s}\) and \(F_{c}\) are both increased. In this way, the global fitness value Y(\(\mathbf{a}^{\prime}\)) is increased. Case 2. \(q_{s}(\mathbf{a}_{c}\odot g)\geq 0\) and \(q_{c}\)*(\(\mathbf{a}_{c}\oplus g\)) > 0. Then, the satisfaction of (18) implies that the reduction of \(F_{s}\) is less that the increase of \(F_{c}\). Hence, Y(\(\mathbf{a}^{\prime}\)) is increased. Case 3. \(q_{s}(\mathbf{a}_{c}\odot g)\geq 0\) and \(q_{c}\)*(\(\mathbf{a}_{c}\oplus g\)) \(\leq 0\). Then, (18) never holds. Case 4. \(q_{s}(\mathbf{a}_{c}\odot g)<0\) and \(q_{c}\)*(\(\mathbf{a}_{c}\oplus g\)) \(\leq 0\). Then, we have \(\left|\,q_{c}\)*(\(\mathbf{a}_{c}\odot g\)) \(\right|<1\,q_{s}(\mathbf{a}_{c}\odot g)\,\). That means, the increase of \(F_{s}\) is greater than the decrease of \(F_{c}\), and hence Y(\(\mathbf{a}^{\prime}\)) is increased correspondingly. The significance value of a target can be broadcasted among AEOSs by using ISLs. According to Proposition 1, based on the received significance value, an AEOS can determine whether to remove or insert a target from or into its current target sequence so as to increase the global fitness value. However, the sparse ISL communication topology makes it difficult to deliver a target's significance value to all satellites, leading to the possibility of the system getting trapped in a local optimum [34]. For example, consider a set of AEOSs \(S=\{s_{1},s_{2},s_{3}\}\) and four targets \(G=\{g_{1},g_{2},g_{3},g_{4}\}\). There are ISLs between \(s_{1}\) and \(s_{2}\) as well as between \(s_{2}\) and \(s_{3}\). As illustrated in Figure 4, the current schedule solution is \(\mathbf{a}_{1}=\{g_{1}\}\), \(\mathbf{a}_{2}=\{g_{2},g_{3}\}\), and \(\mathbf{a}_{3}=\{g_{4}\}\). We now assume that i) \(q_{s2}\)*(\(\mathbf{a}_{s2}\odot g_{1}\)) \(\leq q_{s1}\)*(\(\mathbf{a}_{s1}\odot g_{1}\)), and ii) the global fitness will be increased if target \(g_{1}\) is removed from \(\mathbf{a}_{1}\) to \(\mathbf{a}_{3}\). However, since \(q_{s2}\)*(\(\mathbf{a}_{s2}\odot g_{1}\)) \(\leq q_{s1}\)(\(\mathbf{a}_{s1}\odot g_{1}\)), \(g_{1}\) cannot be added into \(\mathbf{a}_{s2}\) by Proposition 1, and further \(g_{1}\) cannot be added into \(\mathbf{a}_{s3}\), leading to a local optimum. To deliver the target significance value to all satellites, PI algorithm framework introduces three message lists \(\mathbf{Z}_{s}\), \(\mathbf{Q}_{s}\), and \(\mathbf{u}_{s}\) stored on each satellite \(s\), which are defined as follows:  
Figure 4: An example of local optimum where \(\mathbf{a}_{s1}=\{g_{1}\}\), \(\mathbf{a}_{s2}=\{g_{2},g_{3}\}\), and \(\mathbf{a}_{s2}=\{g_{4}\}\). \(\mathbf{Z}_{s}=[Z_{s1},\,Z_{s2},\,\ldots,\,Z_{sn}]^{T}\) is a vector that keeps track of which target is assigned to which satellite. The entry \(Z_{sg}=c\) implies that \(s\) thinks that target \(g\) is scheduled to satellite \(c\), and if \(s\) deems \(g\) to be unassigned, \(Z_{sg}\) towards infinity, i.e., \(Z_{sg}\rightarrow\infty\). \(\mathbf{Q}_{s}=[Q_{s1},\,Q_{s2},\,\ldots,\,Q_{sn}]^{T}\) is a vector recording the significance values of all targets. If \(Z_{sg}\rightarrow\infty\), we set \(Q_{sg}\rightarrow\infty\). \(\mathbf{u}_{s}=[u_{s1},\,u_{s2}\,\ldots,\,u_{sn}]^{T}\) is a vector where the entry \(u_{sc}\) records the time when (\(s\) thinks that) satellite \(c\) has received the latest message from other satellites. Once a message is passed, the timestamp \(u_{sc}\) is updated as follows:  
\[u_{sc}=\left\{ \begin{array}{ll}\tau_{r},&\text{ if }H(s,c)=1\\ \max\limits_{h\in S:H(s,h)=1}u_{hc},&\text{ \ \ \ otherwise}\end{array} \right. \tag{19}\]  
where \(\tau_{r}\) is the time that \(s\) receives the message from \(c\).