Stackelberg strategy with closed-loop information structure for linear-quadratic games

This paper deals with the Stackelberg strategy in the case of a closed-loop information structure. Two players differential games are considered with one leader and one follower. The Stackelberg controls in this case are hard to obtain since the necessary conditions to be satisfied by both players cannot be easily defined. The main difficulty is due to the presence of the partial derivative of the leader's control with respect to state in the necessary condition for the follower. We first derive necessary conditions for the Stackelberg equilibrium in the general case of nonlinear criteria for finite time horizon games. 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. An extension to infinite time horizon games is proposed. The Linear Quadratic case is detailed to illustrate these results.

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