This paper is concerned with the discrete time, full-information $H_\infty$ control problem. It turns out that, as in the continuous time case, the existence of an internally stabilizing controller which makes the $H_\infty$ norm strictly less than 1 is related to the existence of a stabilizing solution to an algebraic Riccati equation. However the solution of this algebraic Riccati equation has to satisfy an extra condition. Moreover it is interesting to note that in general state feedbacks do not suffice and we have to include the disturbance in our feedback.
Keywords: Discrete time, Algebraic Riccati equation, $H_\infty$ control, Full information, Static feedback.