## 4 Optimization Design for Satellite Game Mission Deployment Based on Co-Evolution  
### Differential Game Model  
When both the mission satellite and the target satellite possess maneuvering capabilities, their trajectories are not solely determined by unilateral strategies but are jointly shaped by their mutual game-playing and interactive behaviors. In response to this scenario, this section constructs a model based on differential game theory and solves it using a cooperative game algorithm, aiming to optimize the best control strategies between the detecting satellite and the target satellite. The entire mission process of the task satellite can be described as first approaching the target satellite, then maintaining operation for a certain duration while meeting mission requirements. Therefore, the complete mission can be divided into two phases: the approach phase and the sustained phase. For the approach phase, the mission satellite's strategy is to optimize the control variables to minimize the loss function corresponding to its mission scenario within the maximum flight time \(T_{f}\). The control parameters of the mission satellite include the total mission time \(t_{f}\), the maneuvering times \(t^{A}=\left[t_{1}^{A},t_{2}^{A},\ldots t_{k^{A}}^{A}\right]^{T}\), the impulse magnitudes \(u^{A}=\left[u_{1}^{A},u_{2}^{A},\ldots u_{k^{A}}^{A}\right]^{T}\), the angles \(\beta^{A}=\left[\beta_{1}^{A},\beta_{2}^{A},\ldots\beta_{k^{A}}^{A}\right]^{T}\) between the impulses and the \(XO_{b}Y\) plane, as well as the angles \(\alpha^{A}=\left[\alpha_{1}^{A},\alpha_{2}^{A},\ldots\alpha_{k^{A}}^{A}\right] ^{T}\) between their projections in the \(O_{b}-XYZ\) frame and the \(X\)-axis. Here, \(k^{A}\leq k_{A}^{MAX}\), where \(k_{A}^{MAX}\) denotes the maximum number of impulse maneuvers. The optimization model of the mission satellite can be expressed in the following form:  
\[\underset{u^{A},t^{A},t^{A},\alpha^{A},\beta^{A}}{min}S_{m}^{Appr}\left(u^{A},t^{A},\alpha^{A},\beta^{A};u^{O},t^{O},\alpha^{O},\beta^{O}\right) \tag{4}\]  
\[s.t.\left\{ \begin{array}{l}t_{f}^{A}\leq T_{f}\\ 1\leq k^{A}\leq k_{A}^{MAX}\\ u_{i}^{A}\leq u_{A}^{MAX}\\ t_{i}^{A}\leq t_{i}^{MAX}\\ -\pi\leq\alpha_{i}^{A}\leq\pi\\ -\frac{\pi}{2}\leq\beta_{i}^{A}\leq\frac{\pi}{2}\end{array} \right.,1\leq i\leq k ^{A} \tag{5}\]  
\(u^{O},t^{O},\alpha^{O},\beta^{O}\) represent the control strategies of the target satellite, which influence the optimal strategies of the mission satellite. \(t_{f}^{A}\) is the total mission time for the mission satellite to execute the approach phase, and \(u_{A}^{MAX}\) is the maximum value of a single impulse for the mission satellite. The constraints ensure that the control variables of the mission satellite remain within the physically achievable and mission-compliant ranges, thereby guaranteeing the practical feasibility of the optimization problem. At the end of the approach phase, the mission satellite has already met all requirements except for the duration. Therefore, during the sustaining phase, the mission satellite does not need to optimize the mission time. The optimization model for the mission satellite is as follows:  
\[\underset{u^{A},t^{A},\alpha^{A},\beta^{A}}{min}S_{m}^{Dur}\left(u^{A},t^{A}, \alpha^{A},\beta^{A};u^{O},t^{O},\alpha^{O},\beta^{O}\right) \tag{6}\]  
\[s.t.\left\{ \begin{array}{l}1\leq k^{A}\leq k_{A}^{MAX}\\ u_{i}^{A}\leq u_{i}^{MAX}\\ t_{i}^{A}\leq t_{i}^{MAX}\\ -\pi\leq\alpha_{i}^{A}\leq\pi\\ -\frac{\pi}{2}\leq\beta_{i}^{A}\leq\frac{\pi}{2}\end{array} \right.,1\leq i \leq k^{A} \tag{7}\]The loss function of the target satellite is independent of the mission phase, and its optimization objective does not necessarily solely involve evading the mission satellite. In practice, different objectives can be set for the target satellite according to varying requirements to make it more intelligent. This article outlines two objectives for the target satellite design.