Search

This article studies a zero-sum reach-avoid game in the 2-D plane with one defender and two cooperative attackers. The attackers try to enter the target region protected by the defender who tries to capture both of them. This work aims to obtain the value function and equilibrium strategies, for which the analysis of singular surfaces is needed. To this end, we investigate the game problem in two steps. First, a constrained game scenario is considered in which the defender needs to capture the attackers in a predefined order. The value function is constructed using geometric tools and proved using the viscosity solution method. The attackers can cooperate with each other by adopting the equilibrium strategies corresponding to the value function. Second, on the basis of the solution to the first game scenario, we analyze the general game problem in which there is no capture order restriction for the defender. The singular surfaces are discussed and the local properties of the value function in the vicinity of the singular surfaces are given. It is shown that the defender can perform better than to capture the attackers in any predefined order. Moreover, the construction of singular surfaces is illustrated with numerical results.

This article studies a planar Homicidal Chauffeur reach-avoid differential game, where the pursuer is a Dubins car and the evader has simple motion. The pursuer aims to protect a goal region from the evader. The game is solved in an analytical approach instead of solving Hamilton–Jacobi–Isaacs equations numerically. First, an evasion region is introduced, based on which a pursuit strategy guaranteeing the winning of a simple-motion pursuer under specific conditions is proposed. Motivated by the simple-motion pursuer, a strategy for a Dubins-car pursuer is proposed when the pursuer–evader configuration satisfies separation condition (SC) and interception orientation (IO). The necessary and sufficient condition on capture radius, minimum turning radius, and speed ratio to guarantee the pursuit winning is derived. When the IO is deviated (Non-IO), a heading adjustment pursuit strategy is proposed, and the condition to achieve IO within a finite time is given. Based on it, a two-step pursuit strategy is proposed for the SC and Non-IO case. A nonconvex optimization problem is introduced to give a condition guaranteeing the winning of the pursuer. A polynomial equation gives a lower bound of the nonconvex problem, providing a sufficient and efficient pursuit winning condition. Finally, we extend to multiplayer games by collecting pairwise outcomes for pursuer–evader matchings. Simulations are provided to illustrate the theoretical results.

This article considers a planar multiple-pursuer single-evader reach-avoid game played in a constant flow field. Such a game problem arises in practical applications, such as maritime escort, aircraft confrontation in airflow fields, and defense of moving targets. The dominance boundary, which serves as an analogy to the Apollonius circle and the Cartesian oval, is investigated. The existence of semipermeable surfaces in the game is shown. Under reasonable assumptions, there exists a semipermeable surface, which is part of the barrier surface separating the winning regions for the evader and pursuers. The winning strategies are designed for both sides of players in their corresponding winning regions. It is also shown that at most two pursuers are necessary for successful protection, as in the situation where there is no flow field. Finally, numerical results are given to illustrate the conclusions.