  The bounded real lemma (BRL) is a classical result in systems theory, which
provides a linear matrix inequality criterium for dissipativity, via the
Kalman-Yakubovich-Popov (KYP) inequality. The BRL has many applications, among
others in H-infinity control. Extensions to infinite dimensional systems,
although already present in the work of Yakubovich, have only been studied
systematically in the last few decades. In this context various notions of
stability, observability and controllability exist, and depending on the
hypothesis one may have to allow the KYP-inequality to have unbounded solutions
which forces one to consider the KYP-inequality in a spatial form. In the
present paper we consider the BRL for continuous time, infinite dimensional,
linear well-posed systems. Via an adaptation of Willems' storage function
approach we present a unified way to address both the standard and strict forms
of the BRL. We avoid making use of the Cayley transform and work only in
continuous time. While for the standard bounded real lemma, we obtain analogous
results as there exist for the discrete time case, when treating the strict
case additional conditions are required, at least at this stage. This might be
caused by the fact that the Cayley transform does not preserve exponential
stability, an important property in the strict case, when transferring a
continuous-time system to a discrete-time system.
