Journal of Operator Theory

Volume 80, Issue 1, Summer 2018 pp. 225-253.

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

Authors: Joseph A. Ball (1), Gilbert J. Groenewald (2), and Sanne Ter Horst (3)
Author institution: (1) Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, U.S.A.
(2) Department of Mathematics, Unit for BMI, North-West University, Potchefstroom 2531, South Africa
(3) Department of Mathematics, Unit for BMI, North-West University, Potchefstroom 2531, South Africa

Summary: 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.

DOI: http://dx.doi.org/10.7900/jot.2017sep28.2175
Keywords: KYP-inequality, state-space-similarity theorem, bounded real lemma, infinite dimensional linear system, minimal system

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