## 4 Optimization Design for Satellite Game Mission Deployment Based on Co-Evolution
### Differential Game Model
The first objective is to maintain a safe distance to avoid collisions with the mission satellite, specifically by maximizing the distance from the mission satellite during the engagement process. The second objective is to be as close as possible to the primary satellite. The control variables of the target satellite include the maneuvering times \(t^{O}=\left[t_{1}^{O},t_{2}^{O},\ldots t_{O}^{O}\right]^{T}\), the impulse magnitudes \(u^{O}=\left[u_{1}^{O},u_{2}^{O},\ldots u_{k^{O}}^{O}\right]^{T}\), the angles \(\beta^{O}=\left[\beta_{1}^{O},\beta_{2}^{O}\ldots\beta_{k^{O}}^{O}\right]^{T}\) between the impulses and the \(XOb_{b}Y\) plane, as well as the angles \(\alpha^{O}=\left[\alpha_{1}^{O},\alpha_{2}^{O},\ldots a_{k^{O}}^{O}\right]^{T}\) between their projections in the \(O_{b}-XYZ\) frame and the \(X\)-axis. Here, \(k^{O}\leq k_{o}^{MAX}\), where \(k_{o}^{MAX}\) denotes the maximum number of impulse maneuvers. Similar to the calculation method of the loss function for the mission satellite, for a game-playing duration of \(tr\), the loss function of the target satellite can be expressed as  
\[R=\underset{0\leq n\leq\lfloor\frac{tr}{\gamma}\rfloor+1}{max}\left(max \left(d_{R}^{M}-\|r_{1}^{n}-r_{2}^{n}\|,0\right)-\|r_{2}^{n}\|\right) \tag{8}\]  
Here, \(d_{R}^{M}\) represents the minimum safe distance for the target satellite. During the approach phase, the game-playing duration \(t_{r}\) is equal to the mission duration of the mission satellite. During the sustaining phase, the game-playing duration \(t_{r}\) is the minimum sustainable duration \(T_{Dur}^{MAX}\). The strategy of the target satellite is to optimize parameters such as impulse timings and magnitudes within the maximum flight time to minimize its loss function. The optimization model for the target satellite is as follows:  
\[\underset{u^{O},t^{O},\alpha^{O},\beta^{O}}{min}R\left(u^{A},t^{A},\alpha^{A},\beta^{A};u^{O},t^{O},\alpha^{O},\beta^{O}\right) \tag{9}\]  
\[s.t.\left\{ \begin{array}{l}1\leq k^{O}\leq k_{O}^{MAX}\\ u_{i}^{O}\leq u_{O}^{MAX}\\ t_{i+1}^{O}-t_{i}^{O}\geq\Delta t_{O}^{MIN},1\leq i\leq k^{O}\\ -\tau\leq\alpha_{i}^{A}\leq\pi\\ -\frac{\pi}{2}\leq\beta_{i}^{A}\leq\frac{\pi}{2}\end{array} \right. \tag{10}\]  
Here, \(u^{O},t^{O},\alpha^{O},\beta^{O}\) represent the control strategies of the detected satellite, \(u_{O}^{MAX}\) denotes the upper limit of a single impulse magnitude for the detected satellite, and \(\Delta t_{O}^{MIN}\) represents the minimum time interval between two impulse maneuvers of the target satellite. The co-evolution problem involving a mission satellite and a target satellite requires finding a saddle-point solution that satisfies the following equation under given constraints:  
\[S_{m}^{L}\left(u^{A*},t^{A*},\alpha^{A*},\beta^{A*};u^{O},t^{O},\alpha^{O}, \beta^{O}\right) \leq S_{m}^{L}\left(u^{A},t^{A},\alpha^{A},\beta^{A};u^{O},t^{O}, \alpha^{O},\beta^{O}\right) \tag{11}\]  
\[R\left(u^{A},t^{A},\alpha^{A},\beta^{A};u^{O*},t^{O*},\alpha^{O*},\beta^{O*} \right) \leq R\left(u^{A},t^{A},\alpha^{A},\beta^{A};u^{O},t^{O},\alpha^{O}, \beta^{O}\right) \tag{12}\]  
where the terms \(u^{A*},t^{A*},\alpha^{A*},\beta^{A*}\) represent the optimal control strategy of the task satellite. The terms \(u^{O*},t^{O*},\alpha^{O*},\beta^{O*}\) represent the optimal control strategy of the target satellite. The solutions of Equations (11) and (12) represent saddle-point solutions in the differential game. This indicates that the task satellite and target satellite will reach a stable state during the game process. Their strategies oppose each other to some extent and achieve an optimal balance. Specifically, the task satellite's optimal strategy maximizes its own benefits while considering the target satellite's responses and making reasonable adjustments. The target satellite's optimal strategy maximizes its own objectives while avoiding collisions with the task satellite.