## 2 Problem Description
\begin{table}
\begin{tabular}{c c c c} \hline
**Notations** & **Description** & **Notations** & **Description** \\ \hline \(I\) & Set of observation tasks & \(A_{I}\) & Set of task index \\ \(J\) & Set of involved AEOSs & \(A_{r}\) & Set of AEOS index \\ \(TW_{ij}\) & Set of VTWs for task \(i\) in AEOS \(j\) & \(S_{t}\) & Vector of task state at time step \(t\) \\ \(n_{tuji}\) & Number of VTWs for task \(i\) in AEOS \(j\) & \(s_{i}^{t}\) & Vector of state element of task \(i\). \\ \(tw_{ijk}\) & Start time of \(k\) th VTW for task \(i\) in AEOS \(j\) & \(x_{i}\) & Vector of static state elements of task \(i\). \\ \(twe_{ijk}\) & End time of \(k\) th VTW for task \(i\) in AEOS \(j\) & \(d_{i}^{t}\) & Vector of dynamic state elements of task \(i\). \\ \(rs_{i}\) & Start time of required OTW of task \(i\) & \(X\) & Matrix of static elements \\ \(re_{i}\) & End time of required OTW of task \(i\) & \(D_{t}\) & Matrix of dynamic elements \\ \(ts_{ij,}\) & Start time of observation for task \(i\) in AEOS \(j\) & \(P_{i}\) & Observation profit of task \(i\) \\ \(te_{ij}\) & End time of observation for task \(i\) in AEOS \(j\) & \(st_{i}^{t}\) & Storage occupied by task \(i\). \\ \(\theta_{pij}\) & Pitch attitude angle to task \(i\) by AEOS \(j\) & \(rc_{i}^{t}\) & Remaining storage in all satellites for task \(i\). \\ \(\theta_{rij}\) & Roll attitude angle to task \(i\) by AEOS \(j\), & \(s_{i}\) & Required storage of task \(i\) \\ \(v_{p}\) & Angular velocity of the pitch maneuver & \(C_{j}\) & Memory capacity of AEOS \(j\) \\ \(v_{r}\) & Angular velocity of the roll maneuver & \(n_{tuji}\) & Number of VTWs for task \(i\) on AEOS \(j\) \\ \(d_{i}\) & Observation duration of task \(i\) & \(ftw_{i}^{t}\) & FTWs for task \(i\) at time step \(t\). \\ \(ttw_{i}^{t}\) & Total length of FTWs in all satellites for task \(i\) at time step \(t\). & Earliest end time of task \(i\) in all satellites at time step \(t\) \\  & If task \(i\) is observed by AEOS \(j\), & If task \(i\) is observed by AEOS \(j\) and start \\ \(x_{ij}\) & and \(x_{ij}=0\) otherwise. & \(y_{ij\tau}\) & If task \(i\) is \(\tau\), and \(y_{ij\tau}=0\) otherwise. \\ \(tr_{mnj}\) & Attitude transition time between two adjacent tasks \(m\) and \(n\) & \(sch_{i}^{t}\) & If task \(i\) has been scheduled at time step \(t\), \\ \hline \end{tabular}
\end{table} 
Table 1: Symbols and notations. 1. Objective function: The objective function of the MAEOSSP in this study is to maximize the total observation profit of all scheduled tasks. The objective function is defined as follows: \[\text{Maximize}:\ f=\sum_{j\in J}\sum_{i\in I}x_{ij}P_{i},\] (1) where \(x_{ij}=1\) if task \(i\) is observed by satellite \(j\), and \(x_{ij}=0\) otherwise. 2. Uniqueness constraint: All tasks are one-time tasks. That is, each task can be observed no more than once over the entire time horizon. \[\sum_{j=1}^{N}\sum_{k=1}^{n_{traij}}\sum_{\tau=tus_{ijk}}^{twe_{ijk}}y_{ij\tau} \tau\leq 1,\ i\in I,\] (2) where \(y_{ij\tau}=1\) if task \(i\) is observed by satellite \(j\) and the start time of task \(i\) is \(\tau\), and \(y_{ij\tau}=0\) otherwise. 3. Memory capacity constraint: The total storage consumed by observed tasks for each satellite cannot exceed the memory capacity of the satellite. \[\sum_{i\in I}x_{ij}s_{i}\leq C_{j},\ j\in J,\] (3) 4. Observation time request constraint: Task \(i\) can be observed by any satellite, but the observation should start and be completed within the required observation time by users. \[rs_{i}\leq ts_{i}<te_{i}\leq re_{i},\ i\in I\] (4) 5. VTW constraint: If task \(i\) is observed by satellite \(j\), the observation should start and be completed within VTW \(tw_{ijk}\), where \(tw_{ijk}\in TW_{ij}\). \[tw_{ijk}\leq ts_{i}<te_{i}\leq twe_{ijk},\ i\in I,\ x_{ij}=1\] (5) 6.