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Mathematics > Functional Analysis

arXiv:1804.08505 (math)
[Submitted on 23 Apr 2018]

Title:Standard versus Bounded Real Lemma with infinite-dimensional state space II: The storage function approach

Authors:J.A. Ball, G.J. Groenewald, S. ter Horst
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Abstract:For discrete-time causal linear input/state/output systems, the Bounded Real Lemma explains (under suitable hypotheses) the contractivity of the values of the transfer function over the unit disk for such a system in terms of the existence of a positive-definite solution of a certain Linear Matrix Inequality (the Kalman-Yakubovich-Popov (KYP) inequality). Recent work has extended this result to the setting of infinite-dimensional state space and associated non-rationality of the transfer function, where at least in some cases unbounded solutions of the generalized KYP-inequality are required. This paper is the second installment in a series of papers on the Bounded Real Lemma and the KYP inequality. We adapt Willems' storage-function approach to the infinite-dimensional linear setting, and in this way reprove various results presented in the first installment, where they were obtained as applications of infinite-dimensional State-Space-Similarity theorems, rather than via explicit computation of storage functions.
Comments: 45 pages, to appear in Operator Theory: Advances and Applications volume
Subjects: Functional Analysis (math.FA)
MSC classes: 47A63, 47A48, 93B20, 93C55, 47A56
Cite as: arXiv:1804.08505 [math.FA]
  (or arXiv:1804.08505v1 [math.FA] for this version)
  https://doi.org/10.48550/arXiv.1804.08505

Submission history

From: Sanne ter Horst [view email]
[v1] Mon, 23 Apr 2018 15:27:19 UTC (46 KB)
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