Using H² norm to bound H^∞ norm from above on real rational modules

Various optimal control strategies exist in the literature. Prominent approaches are Robust Control and Linear Quadratic Regulators, the first one being related to the H^∞ norm of a system, the second one to the H² norm. In 1994, F. De Bruyne et al [1] showed that assuming knowledge of the poles of a transfer function one can derive upper bounds on the H^∞ norm as a constant multiple of its H² norm. We strengthen these results by providing tight upper bounds also for the case where the transfer functions are restricted to those having a real valued impulse response. Moreover the results are extended by studying spaces consisting of transfer functions with a common denominator polynomial. These spaces, called rational modules, have the feature that their analytic properties, captured in the integral kernel reproducing them, are accessible by means of purely algebraic techniques.