Dominance Regions of Pursuit-evasion Games in Non-anticipative Information Patterns

TL;DR

The paper addresses how to characterize the evader's dominance region under a non-anticipative information pattern in pursuit–evasion games, including environments with obstacles. It uses geometric tools—shortest-path distance $d_L$, Cartesian ovals, and Apollonius circles—to link the evader's dominance region to its open-loop reachable set and to design pursuer strategies. Key contributions include proving the dominance region equals the evader's open-loop reachable set in obstacle-free settings, establishing the existence of a PELIDR non-anticipative strategy without obstacles, presenting a counterexample with obstacles plus a necessary condition, giving a sufficient condition for a single corner obstacle, and applying these insights to static target defense scenarios. The results offer principled geometric guidelines for constructing non-anticipative strategies in obstacle-rich pursuit–evasion problems and inform practical defense planning.

Abstract

The evader's dominance region is an important concept and the foundation of geometric methods for pursuit-evasion games. This article mainly reveals the relevant properties of the evader's dominance region, especially in non-anticipative information patterns. We can use these properties to research pursuit-evasion games in non-anticipative information patterns. The core problem is under what condition the pursuer has a non-anticipative strategy to prevent the evader leaving its initial dominance region before being captured regardless of the evader's strategy. We first define the evader's dominance region by the shortest path distance, and we rigorously prove for the first time that the initial dominance region of the evader is the reachable region of the evader in the open-loop sense. Subsequently, we prove that there exists a non-anticipative strategy by which the pursuer can capture the evader before the evader leaves its initial dominance region's closure in the absence of obstacles. For cases with obstacles, we provide a counter example to illustrate that such a non-anticipative strategy does not always exist, and provide a necessary condition for the existence of such strategy. Finally, we consider a scenario with a single corner obstacle and provide a sufficient condition for the existence of such a non-anticipative strategy. At the end of this article, we discuss the application of the evader's dominance region in target defense games. This article has important reference significance for the design of non-anticipative strategies in pursuit-evasion games with obstacles.

Figures (18)

• Figure 1: Cartesian oval of the first type.
• Figure 2: Apollonius circle.
• Figure 3: Cartesian oval of the second type.
• Figure 4: The non-anticipative strategy $\delta_{t_0,\mathbf{x}_0}^*$ is generated by $\gamma$ like \ref{['nonanti_stra2']} with initial time $t_0$ and initial state $\mathbf{x}_0=(x_{p0},x_{e0})$.
• Figure 5: $\chi(x)$ and $\psi(x)$ in the proof of \ref{['lemma1']}.