Linear Closed-Loop Stackelberg Strategies with Two-Step Memory for LQ Games

In this paper, we discuss the closed-loop Stackelberg strategies for linear-quadratic leader-follower games. Different from previous research, our approach adopts a linear closed-loop form with a two-step memory. The primary challenge arises from the leader's encounter with a nonclassical control problem involving constraints. To overcome the difficulty, we first attribute the solvability of the leader-follower games to the forward and backward difference equations by applying the constrained maximum principle. Subsequently, we present explicit linear closed-loop Stackelberg strategies with a two-step memory, based on coupled Riccati equations, achieved by decoupling the forward and backward difference equations. The key technique lies in employing quadratic optimization to determine the gain matrix of the linear closed-loop Stackelberg strategies. Numerical examples substantiate the superiority of our proposed strategies over feedback and one-step memory strategies.