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Diffusive realizations for solutions of some operator equations: The one-dimensional case

Authors: Michel Lenczner, Gérard Montseny and Youssef Yakoubi
Journal: Math. Comp. 81 (2012), 319-344
MSC (2010): Primary 35-xx, 47A62, 01-08, 47G10
Published electronically: July 19, 2011
MathSciNet review: 2833497
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Abstract: In this paper we deal with the derivation of state-realizations of linear operators that are solutions to certain operator linear differential equations in one-dimensional bounded domains. We develop two approaches in the framework of diffusive representations: one with complex diffusive symbols; the other with real diffusive symbols. Then, we illustrate the theories and develop numerical methods for a Lyapunov equation arising from optimal control theory of the heat equation. A practical purpose of this approach is real-time computation on a semi-decentralized architecture with low granularity.

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Additional Information

Michel Lenczner
Affiliation: Femto-St Institute, Time-Frequency 26, Rue de l’Epitaphe, 25030 Besançon, France –and– UTBM, 90010 Belfort Cedex, France

Gérard Montseny
Affiliation: LAAS-CNRS 7, avenue du Colonel Roche 31077 Toulouse Cedex 4, France

Youssef Yakoubi
Affiliation: UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions, F-75005, Paris Cedex, France

Keywords: Integral operators, diffusive representation, operational equation, Lyapunov equation, computational method, real-time computation.
Received by editor(s): September 24, 2009
Received by editor(s) in revised form: September 23, 2010
Published electronically: July 19, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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