A Lock-Evade, Engage or Retreat Game

In this paper we model and analyze a scenario in the two-dimensional plane involving a mobile Attacker and stationary Defender. There are two possibilities for termination: the Attacker can collide with the Defender (engagement) or maneuver to a safe zone away from the Defender (retreat). The Defender is equipped with a directional turret, which it can rotate with a bounded rate. If the turret is aligned with the Attacker’s position, the Defender has a lock on the Attacker and may choose to fire on the Attacker. Thus, whether engaging or retreating, the Attacker has incentive to evade the turret’s line of sight and thereby avoid being locked-on. In the case of retreat, if lock-on occurs the Defender cooperates with the Attacker by withholding fire to allow the Attacker to retreat. However, if the Attacker chooses to engage and lock-on occurs, the Defender will open fire on the Attacker. We model the scenario as a set of differential games with different cost functionals depending on the type of termination. The agents are assumed to have full state information. In the case that the Defender can lock onto the Attacker the pre- and post-lock portions of the game are solved individually and combined together. Otherwise, the Attacker can terminate the game prior to lock-on and the game is comprised of a single stage. The equilibrium strategies are derived in each case, and a partitioning of the state space wherein a particular goal selection is optimal is constructed.