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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