Standard versus strict bounded real lemma with infinite-dimensional state space. I. The state-space-similarity approach

Abstract

The bounded real lemma, i.e., the state-space linear matrix inequality characterization (referred to as Kalman--Yakubovich--Popov or KYP-inequality) of when an input/state/output linear system satisfies a dissipation inequality, has recently been studied for infinite-dimensional discrete-time systems in a number of different settings: with or without stability assumptions, with or without controllability/observability assumptions, with or without strict inequalities. In these various settings, sometimes unbounded solutions of the KYP-inequality are required while in other instances bounded solutions suffice. In a series of reports we show how these diverse results can be reconciled and unified. This first instalment focusses on the state-space-similarity approach to the bounded real lemma. We shall show how these results can be seen as corollaries of a new state-space-similarity theorem for infinite-dimensional linear systems