Riccati equations for strongly stabilizable state linear systems

We obtain necessary and sufficient conditions for the existence of strongly stabilizing solutions to Riccati equations for infinite-dimensional systems that are strongly, but not exponentially stabilizable. We illustrate the theory by extending the strict positive-real and bounded-real lemmas to a class of strongly stable systems with colocated actuators and sensors, and apply it to derive a Riccati criterion for the existence of J-spectral factorizations.