Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-30T23:53:36.863Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Courant, R. and Hilbert, D.. Methods of Mathematical Physics, Vol. 1 (New York: Interscience, 1953).Google Scholar
2Kato, T.. Notes on some inequalities for linear operators. Math. Ann. US (1952), 208–212.Google Scholar
3Kato, T.. Perturbation Theory for Linear Operators (New York: Springer, 1976).Google Scholar
4Levitan, B. M. and Sargsjan, I. S.. Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators (Providence: American Mathematical Society, 1975).CrossRefGoogle Scholar
5Nambu, T.. Feedback stabilization for distributed parameter systems of parabolic type, II. Arch. Rational Mech. Anal. 79 (1982), 241–259.CrossRefGoogle Scholar
6Nambu, T.. Strong continuity of the solution to the Ljapunov equation XL – BX = C relative to an elliptic operator L. Proc. Japan Acad. 65 (1989), 70–73.Google Scholar
7Nambu, T.. Stiffness of feedback control systems of parabolic type against perturbations to an elliptic operator, submitted.Google Scholar
8Riesz, F. and -Nagy, B. Sz.. Functional Analysis (New York: Frederick Ungar, 1955).Google Scholar
9Sakawa, Y. and Matsushita, T.. Feedback stabilization of a class of distributed systems and construction of a state estimator. IEEE Trans. Automat. Control AC–20 (1975), 748–753.Google Scholar
10Titchmarsh, E. C.. Eigenfunction Expansions: Associated with Second Order Differential Equations, Part 1 (Oxford: Clarendon Press, 1962).Google Scholar
11Wonham, W. M.. On pole assignment in multi-input controllable linear systems. IEEE Trans. Automat. Control. 12 (1967), 660–665.Google Scholar