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Robustness of a feedback control scheme for one-dimensional diffusion equations: perturbation to the Sturm-Liouville operator

Published online by Cambridge University Press:  14 November 2011

Takao Nambu
Takao Nambu
Affiliation:
Department of Mathematics, Faculty of Engineering, Kumamoto University, Kumamoto 860, Japan
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Synopsis

We study the stabilisation of a one-dimensional diffusion equation by means of static feedback. The equation contains the so-called Sturm-Liouville operator (S-L operator). A perturbation, often interpreted as an error in modelling physical systems, enters the principal part and the boundary condition of the S-L operator. Since the perturbation is not subordinate to the operator, the classical perturbation theory is no longer available. We show, however, that the feedback stabilisation scheme for the unperturbed equation is effective also for the perturbed equation as long as the perturbation is small in an adequate topology. The key idea is to show the strong continuity of the eigenfunctions for the S-L operator relative to the coefficients of the operator.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 121 , Issue 3-4 , 1992 , pp. 349 - 359
Copyright
Copyright © Royal Society of Edinburgh 1992

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