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Oscillations in Systems with Periodic Coefficients and Sector-restricted Nonlinearities

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Differential Operators and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 117))

Abstract

One of the fields of Applied Mathematics that grew up from the pioneering papers of A.M. Liapunov and M.G. Krein is the theory of dynamical systems with periodic coefficients. The results of M.G. Krein and V.A. Yakubovich concerning stability of linear periodic Hamiltonian systems turned out to have applications in the so-called linear-quadratic theory of the controlled systems. The present paper deals with a “subset” of this theory: existence of nonlinear oscillations (periodic and almost periodic solutions) in systems with sector-restricted nonlinearities (the so-called absolutely stable systems). Since almost all results obtained for differential equations have their discrete-time (more or less) counterpart, both continuous time and discrete time periodic cases are presented here. The existence conditions are expressed in terms of an associated periodic Hamiltonian system that is required to be dichotomic and strongly disconjugate. This property may be checked in terms of the properties of an associated matrix Riccati equation or of some Linear Matrix Inequalities.

Realized during author’s stage at Weizmann Institute of Science, Dept. of Theoretical Mathematics, as Meyerhoff Visiting Professor.

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Author information

Authors and Affiliations

  1. Dept. of Mathematics, Bucharest University, Romania

    A. Halanay

  2. Dept. of Automatic Control, Craiova University, A.I. Cuza Str., no.13, RO-1100, Craiova, Romania

    V. L. Răsvan

Authors
  1. A. Halanay
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  2. V. L. Răsvan
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Editor information

Editors and Affiliations

  1. Department of Theoretical Physics, University of Odessa, 270026, Odessa, Ukraine

    V. M. Adamyan

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

    I. Gohberg

  3. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    M. Gorbachuk  & V. Gorbachuk  & 

  4. Department of Mathematics, Vrije Universiteit, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands

    M. A. Kaashoek

  5. Department of Mathematics, Technical University of Vienna, Wiedner Hauptstrasse 8-10/1411, 1040, Vienna, Austria

    H. Langer

  6. Institute of Mathematics, Economics and Mechanics, Odessa State University, 270057, Dvoryanskaya str. 2, Odessa, Ukraine

    G. Popov

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Halanay, A., Răsvan, V.L. (2000). Oscillations in Systems with Periodic Coefficients and Sector-restricted Nonlinearities. In: Adamyan, V.M., et al. Differential Operators and Related Topics. Operator Theory: Advances and Applications, vol 117. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8403-7_12

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  • DOI: https://doi.org/10.1007/978-3-0348-8403-7_12

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9552-1

  • Online ISBN: 978-3-0348-8403-7

  • eBook Packages: Springer Book Archive

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