# A Stackelberg Game Approach to Model Reference Adaptive Control for Spacecraft Pursuit-Evasion
## 1 Introduction  
The rapid development of space technology has increased the importance of spacecraft in many fields such as space exploration, satellite communication, and military reconnaissance [1, 2, 3]. At the same time, the problem of Pursuit-Evasion (PE) games between spacecraft has become increasingly significant [4]. In mission scenarios such as satellite formation flight, space debris cleanup, and space confrontation, the development of strategies to achieve efficient pursuit has become a key technical challenge [5, 6, 7]. For example, in satellite formation flight, each satellite must maintain a specific relative position relationship. Once there is a deviation, it is necessary to return the satellite to a predetermined orbit with the help of reasonable control strategies. This involves a decision-making process similar to that of evasion tracking [8]. In space debris cleanup missions, the pursuer has to develop a strategy to approach and capture debris, which can be regarded as a target with the intention of evasion [9]. It is these practical application requirements that drive the in-depth investigation of the spacecraft PE game problem. When modeling the PE game problem, traditional control methods usually construct it as a special structure, a competitive scenario where the total of the pursuer's and evader's cost functions equals zero, essentially representing a zero-sum dynamic [10]. In recent years, the study of zero-sum games has attracted extensive attention from many scholars. Within this framework, the ultimate equilibrium state of the PE game converges at a saddle point, with the determination of this saddle point balance achieved through the solution of the Hamilton-Jacobi-Isaacs (HJI) equation [11]. In the literature [12], the scholar crafted an integral reinforcement learning algorithm tailored for a category ofzero-sum differential game issues, characterized by entirely unknown dynamic attributes of the linear system. Reference [13] broadens the investigative horizon by applying the zero-sum game concept to nonlinear systems. It innovatively integrates neural network methodologies into adaptive dynamic programming, thereby effectively deriving the saddle-point equilibrium solution within such nonlinear frameworks. The study in [14] also focuses on the nonlinear zero-sum differential game problem and proposes a model-free control algorithm based on neural networks, which provides new ideas and methods for solving such problems. However, in practical situations, the goals of pursuers and evaders are not completely opposite, which leads to overly aggressive strategy design, neglects the reasonable allocation of safety and resources, and limits the flexibility of strategy design. In some cases, the zero-sum framework can lead to moral and ethical dilemmas as it encourages absolute victory for one party without considering the loss of the other [15]. While zero-sum games simplify analysis, they often fail to capture real-world scenarios where objectives are not strictly adversarial. In contrast, non-zero-sum frameworks accommodate partial competition and cooperation, enabling strategies that balance safety, fuel efficiency, and mission success; a non-zero-sum framework more accurately models the complex interplay of competition and potential cooperation between the two parties. It allows for more flexible strategies that can adapt to different mission requirements and consider long-term stability [16; 17]. This has led to the development of methods like Q-learning for non-zero-sum games with incomplete information [18] and optimal continuous thrust strategies within a differential game framework [19]. A significant challenge in PE games is the inherent asymmetry of information and the sequential nature of decision making. Most traditional game-theoretic solutions are based on the Nash Equilibrium (NE), which assumes that players act simultaneously with full knowledge of each other's strategies [20].