Generalized solutions of Riccati equalities and inequalities
  The Riccati inequality and equality are studied for infinite dimensional linear discrete time stationary systems with respect to the scattering supply rate. The results obtained are an addition to and based on our earlier work on the Kalman-Yakubovich-Popov inequality in [6]. The main theorems are closely related to the results of Yu. M. Arlinski\u{\i} in [3]. The main difference is that we do not assume the original system to be a passive scattering system, and we allow the solutions of the Riccati inequality and equality to satisfy weaker conditions. 
D. Z. Arov, M. A. Kaashoek, D. R. Pik
math.FA