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The Bounded Real Lemma, i.e., the state-space linear matrix inequalitycharacterization (referred to as Kalman-Yakubovich-Popov or KYP inequality) ofwhen an input/state/output linear system satisfies a dissipation inequality,has recently been studied for infinite-dimensional discrete-time systems in anumber of different settings: with or without stability assumptions, with orwithout controllability/observability assumptions, with or without strictinequalities. In these various settings, sometimes unbounded solutions of theKYP inequality are required while in other instances bounded solutions suffice.In a series of reports we show how these diverse results can be reconciled andunified. This first instalment focusses on the state-space-similarity approachto the bounded real lemma. We shall show how these results can be seen ascorollaries of a new State-Space-Similarity theorem for infinite-dimensionallinear systems.

Standard versus Strict Bounded Real Lemma with infinite-dimensional state space I: The State-Space-Similarity Approach