Min-max and min-min stackelberg strategies with closed-loop information structure

This paper deals with the min-max and min-min Stackelberg strategies in the case of a closed-loop information structure. Two-player differential one-single stage games are considered with one leader and one follower. We first derive necessary conditions for the existence of the follower to characterize the best response set of the follower and to recast it, under weak assumptions, to an equivalent and more convenient form for expressing the constraints of the leader's optimization problem. Under a standard strict Legendre condition, we then derive optimality necessary conditions for the leader of both min-max and min-min Stackelberg strategies in the general case of nonlinear criteria for finite time horizon games. This leads to an expression of the optimal controls along the associated trajectory. Then, using focal point theory, the necessary conditions are also shown to be sufficient and lead to cheap control. The set of initial states allowing the existence of an optimal trajectory is emphasized. The linear-quadratic case is detailed to illustrate these results.