This paper presents a novel approach to address the problem of intercepting non-cooperative targets with multiple satellites in Earth orbit. The multi-satellite interception problem is formulated as a multi-player pursuit–evasion game that explicitly accounts for stochastic disturbances and control constraints. By combining differential game theory with stochastic optimization techniques, the paper derives optimal interception trajectories that ensure safety and performance under modeling uncertainties. A linear exponential quadratic cost functional is established, and corresponding Nash equilibrium strategies are obtained to determine the optimal control laws. Numerical simulations validate the effectiveness and robustness of the proposed approach in achieving reliable interception performance.

Gaussian-enhanced reinforcement learning for scalable evasion strategies in multi-agent pursuit-evasion games

In this paper, we unify and extend the linear quadratic pursuit–evasion games to dynamic equations on time scales. A mixed strategy for a single pursuer and evader is studied in two settings. In the open-loop setting, the corresponding controls are expressed in terms of a zero-input difference. In the closed-loop setting, the corresponding controls require a mixing feedback term when rewriting the system in extended state form. Finally, we offer a numerical simulation.

Cops and Robber is a well-studied two-player pursuit-evasion game played on a graph, where a group of cops tries to capture the robber. The cop number of a graph is the minimum number of cops required to capture the robber. Gavenčiak et al. [Eur. J. of Comb. 72, 45-69 (2018)] studied the game on intersection graphs and established that the cop number for the class of string graphs is at most 15, and asked as an open question to improve this bound for string graphs and subclasses of string graphs. We address this question and establish that the cop number of a string graph is at most 13. To this end, we develop a novel guarding technique. We further establish that this technique can be useful for other Cops and Robber games on graphs admitting a representation. In particular, we show that four cops have a winning strategy for a variant of Cops and Robber, named Fully Active Cops and Robber, on planar graphs, addressing an open question of Gromovikov et al [Austr. J. Comb. 76(2), 248-265 (2020)]. In passing, we also improve the known bounds on the cop number of boxicity 2 graphs. Finally, as a corollary of our result on the cop number of string graphs, we establish that the chromatic number of string graphs with girth at least 5 is at most 14.

Optimal control problems under incomplete information, particularly in pursuit-evasion scenarios, present significant mathematical challenges. This study extends a basic time-optimal pursuit-evasion game by introducing a time-dependent observer parameter, $\lambda(t)$, which enhances the model's realism without altering the fundamental control strategy. We derive an optimal control law for the pursuer, based on current observations, and explicitly calculate the minimum capture time for a piecewise constant $\lambda(t)$. This work provides an analytical framework for managing uncertainty in dynamic environments, with direct applications in robotics, autonomous navigation, and search-and-rescue operations.

Dynamic Historical Data-Based Reinforcement Learning for Pursuit–Evasion Games of Nonholonomic Vehicles With Input Saturation

Abstract With the increasing complexity of the space environment and continuous advancements in satellite maneuverability, orbital game theory has become a crucial research direction in aerospace engineering. To address the need for prior target information acquisition in missions such as space debris removal and on-orbit servicing, a scenario of observation and counter-observation similar to the pursuit-evasion game is proposed. Leveraging the Multi-objective optimization algorithm, we systematically investigate the observation and counter-observation game (OCOG) problem under continuous-thrust conditions for two satellites. First, a sequential decision-making model for continuous-thrust maneuvers is established, transforming the OCOG problem into a bilateral optimization problem incorporating multiple constraints, including maneuverability, illumination conditions, relative distance, line-of-sight (LOS) angular velocity, and relative velocity. Second, a switching-type payoff function characterizing effective imaging duration is constructed, and a hyperbolic tangent function is introduced to achieve continuous smoothing of discrete switching characteristics, effectively resolving differentiation challenges at constraint boundaries. Furthermore, a clustering multi-objective optimization genetic algorithm (CMOGA) is proposed, which employs non-dominated sorting and crowding distance computation to effectively maintain the Pareto front distribution of the solution set. By introducing a clustering mechanism, the algorithm ensures solution diversity while significantly reducing computational complexity. Comprehensive simulations and comparative analyses demonstrate the proposed method’s significant advantages in collision avoidance, robustness, and optimality. This research provides an effective maneuvering strategy for future space orbital games.

An approximate analytical method for solving the coplanar spacecraft pursuit-evasion game barrier

Abstract In this manuscript, we study the orbit pursuit-evasion game problem in the Cartesian model under J2 perturbations, give the dynamic equation of both spacecraft and derive the costate dynamics. This paper proposes a dimensionality reduction technology based on the transition problem and the saddle-point solution is obtained through the Costate Mapping Hybrid Optimal Method. Unlike the traditional direct method for solving costate, this method obtains the pursuer’s costate in the one-side interception problem by costate mapping and then considers the maneuver of the evader to solve the costate’s optimal value of the two-point boundary value problem. In numerical results, this method is consistent with other methods, and the computation time is about 2-4 seconds. The simulations show that this technology reduce the dimensionality of the problem and the sensitivity of the initial costate, the proposed method can obtain the saddle-point solution quickly and does not have the convergence problem. This work also compares the differences between the optimal solutions of the minimum-time interception problem and saddle-point solutions of the orbital game problem under different thrust configurations, then discusses the effects of the evader thrust and initial relative distance on the optimal control law of the pursuer.

Intelligent strategy resolution methods and mechanism analysis in two-on-one impulsive orbital pursuit–evasion games

In recent years, with the development of micro-satellite clusters and large-scale satellite constellations, the likelihood of multiple spacecraft engaging in orbital pursuit–evasion games has increased. This paper establishes a novel interception domain theory for planar orbital multi-player “encirclement-capture” differential games, and it proves the partitioned structure and classification properties of Nash equilibrium solutions. The main contributions of our study are the following: (1) Proposing the first rigorous definition of interception domains in orbital pursuit–evasion games, proving their convexity, developing computational methods for domain intersections, and establishing a complete classification of equilibrium for planar multi-pursuer interception games, which establishes a theoretical foundation for analyzing multi-spacecraft orbital pursuit–evasion games. (2) Analyzing Nash equilibrium properties for “encirclement-capture” differential games with two, three, or more pursuers, classifying degenerate/non-degenerate scenarios via spatial inclusion relationships. The equilibrium results indicate that as the number of pursuers increases, the game tends toward a degenerate scenario where the likelihood of redundant pursuers (whose actions do not affect the game outcome) rises.

Prescribed-Time Nash Equilibrium Seeking for Pursuit–Evasion Game Under Intermittent Control With Undirected/Directed Graph

Computing the Pursuing Control in Proximate Orbital Pursuit-Evasion Game by Polynomial Approximation

This study investigates emergent behaviors in multi-agent pursuit-evasion games within a bounded 2D grid world, where both pursuers and evaders employ multi-agent reinforcement learning (MARL) algorithms to develop adaptive strategies. We define six fundamental pursuit actions—flank, engage, ambush, drive, chase, and intercept—which combine to form 21 types of composite actions during two-pursuer coordination. After training with MARL algorithms, pursuers achieved a 99.9% success rate in 1,000 randomized pursuit-evasion trials, demonstrating the effectiveness of the learned strategies. To systematically identify and measure emergent behaviors, we propose a K-means-based clustering methodology that analyzes the trajectory evolution of both pursuers and evaders. By treating the full set of game trajectories as statistical samples, this approach enables the detection of distinct behavioral patterns and cooperative strategies. Through analysis, we uncover emergent behaviors such as lazy pursuit, where one pursuer minimizes effort while complementing the other’s actions, and serpentine movement, characterized by alternating drive and intercept actions. We identify four key cooperative pursuit strategies, statistically analyzing their occurrence frequency and corresponding trajectory characteristics: serpentine corner encirclement, stepwise corner approach, same-side edge confinement, and pincer flank attack. These findings provide significant insights into the mechanisms of behavioral emergence and the optimization of cooperative strategies in multi-agent games.

The tethered space net robot offers an effective solution for active space debris removal due to its large capture envelope. However, most existing studies overlook the evasive behavior of non-cooperative targets. To address this, we model an orbital pursuit–evasion game involving a tethered net and propose a game theory-based leader–follower tracking control strategy. In this framework, a virtual leader—defined as the geometric center of four followers—engages in a zero-sum game with the evader. An adaptive dynamic programming method is employed to handle input saturation and compute the Nash Equilibrium strategy. In the follower formation tracking phase, a synchronous distributed model predictive control approach is proposed to update all followers’ control simultaneously, ensuring accurate tracking while meeting safety constraints. The feasibility and stability of the proposed method are theoretically analyzed. Additionally, a body-fixed reference frame is introduced to reduce the capture angle. Simulation results show that the proposed strategy successfully captures the target and outperforms existing methods in both formation keeping and control efficiency.

Approximate Analytical Approach for Spacecraft Pursuit–Evasion Game With Reachability Analysis

Closed-Loop Strategy Synthesis for Real-Time Spacecraft Pursuit–Evasion Games in Elliptical Orbits

Safety-Aware Pursuit-Evasion Game Based on Control Barrier Function and Reinforcement Learning

Pursuit-Evasion Game for Spacecraft With Incomplete Information Under <i>J</i>₂ Perturbation

Abstract In this article, an iterative learning nonzero‐sum game (NSG) control approach is investigated for incomplete‐information pursuit‐evasion (PE) systems subject to uncertainty. Due to the non‐shared quality of the cost functions, the NSG frame is established for the incomplete‐information PE issue. To deal with the uncertainty, the upper bounds of the cost functions are derived for both pursuer and evader. Then, the suboptimal game control strategies are suggested for the uncertain complete‐information PE game. For the incomplete‐information case, an adaptive learning gain estimator is devised to estimate the control gain of the evader. In the light of this estimation, the incomplete‐information pursuit control policy is proposed to guarantee the asymptotic stability of the whole system including the PE system and the gain estimation error system. Meanwhile, a matrix iterative learning algorithm is proposed to solve the modified Riccati equation. Finally, the simulated outcomes are presented to validate the efficacy of the designed incomplete‐information PE control approaches.