Optimal Constrained Retreat within the Turret Defense Differential Game

We extend the Turret Defense Differential Game by considering the possibility for the mobile Attacker to retreat to a particular region of the state space in lieu of engaging the Turret. In this case, the Turret cooperates with the Attacker by escorting the Attacker to the retreat surface, thereby minimizing its cost. Along the retreat trajectory, the cost associated with the Value of the Game of Engagement must be greater than the cost associated with retreating. We refer to this scenario as Optimal Constrained Retreat, wherein the aforementioned requirement is imposed as a path constraint. The optimality conditions are developed herein. We specify a boundary value problem with fixed initial state and compute the optimal trajectory numerically via backwards shooting. The corresponding trajectory contains a constrained arc wherein the path constraint associated with the Game of Engagement becomes active.