# A Stackelberg Game Approach to Model Reference Adaptive Control for Spacecraft Pursuit-Evasion
## 1 Introduction
This assumption is often invalid in practice, as information gathering and decision making involve inherent time lags. A more suitable theoretical basis is the Stackelberg equilibrium, which models sequential decision making [21]. In the PE context, the evader often acts as the \"leader\", making a move first, while the pursuer is the \"follower\", reacting to the leader's action. Therefore, the PE game is more accurately modeled as a dynamic Stackelberg game [22]. While prior works have applied Stackelberg models to surface vehicles [23] and spacecraft using single-step prediction [24], our work distinguishes itself by integrating a Model Reference Adaptive Control architecture. This approach allows the pursuer to dynamically adjust its control gains based on a reference model of the evader's optimal strategy, offering enhanced robustness against unmodeled dynamics and disturbances, which is a primary focus of this paper. Recent approaches to spacecraft pursuit-evasion under \(J_{2}\) perturbations [24] typically employ linearized approximations and fixed-gain control, resulting in suboptimal tracking performance. Our Stackelberg game-based framework introduces three key innovations, as detailed in Table 1. (1) Nonlinear \(J_{2}\) coupling preservation reduces position errors by 62% through exact \(\sin^{2}(kt)\) modeling. (2) Dynamic \(fal(\cdot)\) adaptation enables 22% fuel savings via multi-horizon optimization. (3) Global stability is guaranteed (Theorem 1), overcoming local convergence limitations. While the proposed method operates at 24.8 ms/step compared to 18.7 ms/step in [24], it achieves 6.42% faster execution than the SSE baseline (26.5 ms/step) through parallel computation. The main contributions of this paper are as follows:  
1. In a non-zero-sum framework, a Stackelberg equilibrium-based Model Reference Adaptive Control (MSE) method is proposed for the spacecraft PE game. This method incorporates the dynamic Stackelberg equilibrium game model and uses the Riccati equation to derive the optimal control strategy for the evader. Subsequently, an adaptive control algorithm enables the pursuer to adjust its control gains adaptively. This approach is novel, as it specifically addresses the challenges of \(J_{2}\) perturbations and non-zero-sum game dynamics within the PE context, which has not been thoroughly explored in prior literature. 2. The existence and uniqueness of the solution to the Stackelberg game model are rigorously proven. Furthermore, the proposed MSE algorithm is compared with traditional methods, such as Nash Equilibrium (NE) [18] and Single-step Prediction Stackelberg Equilibrium (SSE) [24]. Numerical simulations demonstrate that the MSE algorithm offers significant advantages in terms of computational efficiency (e.g., an average generation time of 24.79 ms, which is 7.36% less than SSE and 39.11% less than NE), fuel consumption, pursuit success rate, and disturbance rejection capability. 3. While Stackelberg games have been applied to spacecraft PE problems, this work lies in unifying Riccati-based Stackelberg solutions with model reference adaptive control, addressing \(J_{2}\) perturbations and incomplete information--a gap in prior work. The remainder of this article is organized as follows. In Section 2, the spacecraft PE game model is introduced and mapped onto a Stackelberg equilibrium system. In Section 3, the MSE tracking control method is proposed. Subsequently, Section 4 conducts simulation experiments. Finally, Section 5 summarizes the article.