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

TL;DR

This paper extends the bounded real lemma to infinite-dimensional continuous-time systems using a storage function approach, providing a unified framework for standard and strict cases without relying on the Cayley transform.

Contribution

It introduces a continuous-time, infinite-dimensional bounded real lemma framework that unifies standard and strict cases through a storage function approach, avoiding the Cayley transform.

Findings

Unified approach for standard and strict BRL in continuous time

Results analogous to discrete time BRL for standard case

Additional conditions needed for strict case due to stability issues

Abstract

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.