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Exponential bases in Sobolev spaces in control and observation problems for the wave equation

Published online by Cambridge University Press:  11 July 2007

S. A. Avdonin ,
S. A. Ivanov  and
D. L. Russell
S. A. Avdonin
Affiliation:
Department of Applied Mathematics and Control, St Petersburg State University, Bibliotechnaya sq. 2, 198904 St Petersburg, Russia Department of Mathematics and Statistics, The Flinders University of South Australia, GPO Box 2100, Adelaide SA 5001, Australia (avdonin@ist.flinders.edu.au)
S. A. Ivanov
Affiliation:
Russian Centre of Laser Physics, St Petersburg State University, Ul'yanovskaya 1, 198904 St Petersburg, Russia (sergei.ivanov@pobox.spbu.ru)
D. L. Russell
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA (russell@calvin.math.vt.edu)
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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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 130 , Issue 5 , October 2000 , pp. 947 - 970
Copyright
Copyright © Royal Society of Edinburgh 2000

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