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Linear Operators and Matrices pp 223–233Cite as

Birkhäuser

Lyapunov Functions and Solutions of the Lyapunov Matrix Equation for Marginally Stable Systems

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 130))

Abstract

We consider linear systems of differential equations \(I\ddot x + B\dot x + Cx = 0\) where I is the identity matrix and B and C are general complex n x n matrices. Our main interest is to determine conditions for complete marginal stability of these systems. To this end we find solutions of the Lyapunov matrix equation and characterize the set of matrices (B,C) which guarantees marginal stability. The theory is applied to gyroscopic systems, to indefinite damped systems, and to circulatory systems, showing how to choose certain parameter matrices to get sufficient conditions for marginal stability. Comparison is made with some known results for equations with real system matrices. Moreover more general cases are investigated and several examples are given.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Technical University of Denmark, Matematiktorvet, Bldg. 303, Kgs. Lyngby, DK-2800, Denmark

    Wolfhard Kliem & Christian Pommer

Authors
  1. Wolfhard Kliem
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  2. Christian Pommer
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Editor information

Editors and Affiliations

  1. School of Mathematical Sciences Raymond and Beverly Sackler, Faculty of Exact Sciences Tel Aviv University, Ramat Aviv, 69978, Israel

    I. Gohberg

  2. Mathematik, Technische Universität Wien, Wiedner Hauptstrasse 8–10/1411, Wien, 1040, Austria

    H. Langer

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© 2002 Springer Basel AG

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Kliem, W., Pommer, C. (2002). Lyapunov Functions and Solutions of the Lyapunov Matrix Equation for Marginally Stable Systems. In: Gohberg, I., Langer, H. (eds) Linear Operators and Matrices. Operator Theory: Advances and Applications, vol 130. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8181-4_17

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  • DOI: https://doi.org/10.1007/978-3-0348-8181-4_17

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9467-8

  • Online ISBN: 978-3-0348-8181-4

  • eBook Packages: Springer Book Archive

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