Engage or Retreat Differential Game with N-Targets and Distributed Defensive Assets

A two-player, Engage or Retreat (EoR) differential game is presented in which an attacker must choose whether to capture one of N static targets or retreat across a defined retreat boundary. Throughout the engagement, a defending player is capable of activating defensive assets located at each of the target locations. These defensive assets inflict a cost on the attacker as a function of distance and allow the defender to present a deterrent in an effort to persuade the attacker to elect retreat. The solution of the game is constructed by examining two related subproblems: the Game of Engagement (GoE) and the Optimal Constrained Retreat (OCR). Each subproblem examines a different combination of attacker termination strategies, capture vs retreat, and defender strategies, resist vs cooperate. When solving for the optimal constrained retreat strategy, a value function constraint is imposed on the retreat trajectory in order to ensure that the attacker does not maneuver into an advantageous position. Several solutions are examined to illustrate the types of behaviors found in the equilibrium game solutions.