Zero-sum turret defense differential game with singular surfaces

We examine a two-player, zero-sum, differential turret defense game, which consists of two players, a mobile Attacker and a stationary Defender. The Attacker is modeled as a mobile agent moving with a constant speed about a infinite, two-dimensional plane, and it is capable of making instantaneous changes in direction. The Defender is modeled as a stationary target located at the origin, and it is capable of steering a direction of focus toward the Attacker. The game terminates when the Attacker intercepts the Defender. Over the course of the game, the Attacker incurs an integral cost. The cost functional consists of a constant time penalty and a Defender generated cost based on the relative agents positions. The goal of the Attacker is to terminate the game with minimal cost, while the Defender strives to maximize the Attacker's cost. We show that the global equilibrium solution contains three singular surfaces that divide the state space into regions with qualitatively different equilibrium control strategies for the two players.