$H_{\infty}$ constrained control for multi-hierarchies decision players with asymmetric information

In this paper, we consider the asymmetric information problem with multi-hierarchies decision players. The controller A (CA) contains
$n$
players while the external disturbance acts the role of controller B (CB). In the linear control problem, the external disturbance is treated as
$H_{\infty}$
constraint. The controller B shares its observations and the historical control inputs with the controller A, but it cannot obtain the observations of A. Optimal estimators for CA and CB will be presented respectively based on asymmetric observations. As the existence of the asymmetric information, the classical separation principle fails for the reason that the estimation error covariance is coupled. Under assumptions, it is proved that the strategy of CA contains two parts, the common observation and the asymmetric information while the form of CB is related with the common part. Pontryagin's maximum principle is applied to derive the forward and backward stochastic difference equations. A quadratic cost function is minimized by CA and maximized by CB.