# The singular minimum entropy $H_\infty$ control problem

A.A. Stoorvogel

In this paper we search for controllers which minimize an entropy function of the closed loop transfer matrix under the constraint of internal stability and under the constraint that the closed loop transfer matrix has $H_\infty$ norm less than some a priori given bound $\gamma$. We find an explicit expression for the infimum. Moreover, we give a charaterization when the infimum is attained (contrary to the regular case, for the singular minimum entropy $H_\infty$ control problem the infimum is not always attained). Keywords: $H_\infty$ control, algebraic Riccati equation, quadratic matrix inequality, minimum entropy.