Reach-Avoid Games With Two Cooperative Attackers: Value Function and Singular Surfaces

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.