Optimal Linear Closed-Loop Stackelberg Strategy with Asymmetric Information

In this paper, we consider the optimal linear closed-loop Stackelberg strategy for stochastic system where the state is unknown but observed though linear stochastic measurement systems. Particularly, the information structure is asymmetric where the leader can obtain the follower's measurement information and its historical control inputs, while the follower can only obtain its own measurement information and historical inputs, which have led to major challenge since the 1970s. It is well known that the closed-loop Stackelberg optimal control in general is a hard problem and no explicit solution can be derived. Instead, this paper aims to present a linear closed-loop solution to the problem. However, how to give a rational definition for the linear closed-loop Stackelberg controller is a difficulty. By using innovation, this paper defines the linear closed-loop Stackelberg controller with specified given gain matrices. Then the analytical solutions for the leader and the follower are given, respectively, which are the feedback form of the Kalman filtering and the optimal gain matrices for the leader and follower are obtained by decoupled solving two Riccati equations. Finally, by using the forward iteration algorithm, the coupled forward and backward Riccati equations are solved in infinite horizon, which overcomes the obstacles that the separation principle becomes invalid.