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The Extended Field of Operator Theory pp 37–72Cite as

The Infinite-dimensional Continuous Time Kalman-Yakubovich-Popov Inequality

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 171))

Abstract

We study the set M Σ of all generalized positive self-adjoint solutions (that may be unbounded and have an unbounded inverse) of the KYP (Kalman-Yakubovich-Popov) inequality for a infinite-dimensional linear time-invariant system Σ in continuous time with scattering supply rate. It is shown that if M Σ is nonempty, then the transfer function of Σ coincides with a Schur class function in some right half-plane. For a minimal system Σ the converse is also true. In this case the set of all H ∈ M Σ with the property that the system is still minimal when the original norm in the state space is replaced by the norm induced by H is shown to have a minimal and a maximal solution, which correspond to the available storage and the required supply, respectively. The notions of strong H-stability, H-*-stability and H-bistability are introduced and discussed. We show by an example that the various versions of H-stability depend crucially on the particular choice of H ∈ M Σ. In this example, depending on the choice of the original realization, some or all H ∈ M Σ will be unbounded and/or have an unbounded inverse.

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Author information

Authors and Affiliations

  1. Division of Mathematical Analysis Institute of Physics and Mathematics, South-Ukrainian Pedagogical University, 65020, Odessa, Ukraine

    Damir Z. Arov

  2. Department of Mathematics, Åbo Akademi University, FIN-20500, Åbo, Finland

    Olof J. Staffans

Authors
  1. Damir Z. Arov
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  2. Olof J. Staffans
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Editor information

Editors and Affiliations

  1. School of Mathematics and Statistics, University of Newcastle, Newcastle upon Tyne, NE1 7RU, UK

    Michael A. Dritschel

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© 2006 Birkhäuser Verlag Basel/Switzerland

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Arov, D.Z., Staffans, O.J. (2006). The Infinite-dimensional Continuous Time Kalman-Yakubovich-Popov Inequality. In: Dritschel, M.A. (eds) The Extended Field of Operator Theory. Operator Theory: Advances and Applications, vol 171. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7980-3_3

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