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Exponential Bases in Sobolev Spaces in Control and Observation Problems

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Optimal Control of Partial Differential Equations

Part of the book series: ISNM International Series of Numerical Mathematics ((ISNM,volume 133))

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Abstract

The Fourier method in control systems reduces the study of controllability/observability to the study of related exponential families. In this paper we present examples of such systems, specifically those for which we can prove that the related exponential families form a Riesz basis in corresponding appropriately defined Sobolev spaces. This makes it possible to choose ‘natural’ pairs of spaces: the state space / observability space and the control space / state space, depending on whether an observation or a control problem is studied, respectively, so that the observation and control operators are isomorphisms.

This work was partially supported by the US National Science Foundation (grant #DMS-9501036) and by the Russian Fundamental Research Foundation (grant # 97-01-01115).

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References

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Control, St. Petersburg State University, Bibliotechnaya sq. 2, 198904, St. Petersburg, Russia

    Sergei A. Avdonin & David L. Russell

  2. Department Mathematics and Statistics, The Flinders University of South Australia, GPO Box 2100, Adelaide, SA, 5001, Australia

    Sergei A. Avdonin

  3. Russian Center of Laser Physics, St. Petersburg State University, Ul’yanovskaya 1, 198904, St. Petersburg, Russia

    Sergei A. Ivanov

  4. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

    David L. Russell

Authors
  1. Sergei A. Avdonin
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  2. Sergei A. Ivanov
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  3. David L. Russell
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Editor information

Editors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, Arcisstraße 21, 80333, München, Germany

    Karl-Heinz Hoffmann

  2. Mathematisches Institut, Universität Bayreuth, 95440, Bayreuth, Germany

    Günter Leugering

  3. Fakultät für Mathematik, TU Chemnitz, 09107, Chemnitz, Germany

    Fredi Tröltzsch

  4. 53111, Bonn, Germany

    Stiftung Caesar

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© 1999 Springer Basel AG

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Avdonin, S.A., Ivanov, S.A., Russell, D.L. (1999). Exponential Bases in Sobolev Spaces in Control and Observation Problems. In: Hoffmann, KH., Leugering, G., Tröltzsch, F., Caesar, S. (eds) Optimal Control of Partial Differential Equations. ISNM International Series of Numerical Mathematics, vol 133. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8691-8_3

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  • DOI: https://doi.org/10.1007/978-3-0348-8691-8_3

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9731-0

  • Online ISBN: 978-3-0348-8691-8

  • eBook Packages: Springer Book Archive

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