The Infinite-Dimensional Standard and Strict Bounded Real Lemmas in Continuous Time: The storage function approach.

The bounded real lemma is a classical result in systems theory, which provides a linear matrix inequality criterium for dissipativity, via the Kalman-Yakubovich-Popov (KYP) inequality, that has many applications, among others in H-inftinity 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 bounded real lemma 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 bounded real lemma. We avoid making use of the Cayley transform and work only in the continuous-time case. 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 condition in the strict bounded real case) when transferring a continuous-time system to a discrete-time system.