This paper completely solves the discrete time minimum entropy control problem. It is shown that in discrete time the central controller has the additional interpretation as the controller which minimizes a minimum entropy criterion. This is completely analogous to the continuous time. However, although the $H_\infty$ control problems for discrete and continuous time can be connected via the bilinear transform, it is shown that this is not the case for the corresponding minimum entropy control problems and hence the bilinear transform does not connect the central controllers in continuous and discrete time.
Keywords: $H_\infty$ control, algebraic Riccati equation, discrete time systems, minimum entropy, bilinear transform.