  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.
