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The Ljapunov equation and an application to stabilisation of one-dimensional diffusion equations*

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

A Ljapunov equation XL − BX = C appears in stabilisation studies of linear systems. Here, the operators L, B, and C are given linear operators working in infinite-dimensional Hilbert spaces, which are derived from a specific control system. We have so far considered the case where L is a general elliptic operator of order 2 in a bounded domain of an Euclidean space. When L is instead a self-adjoint elliptic operator working in an interval of ℝ1, we derive here a stronger geometrical character of the solution X to the Ljapunov equation. The result is applied to stabilisation of one-dimensional diffusion equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 1-2 , 1986 , pp. 39 - 52
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Agmon, S.. Lectures on Elliptic Boundary Value Problems(Princeton: Van Nostrand, 1965).Google Scholar
2Fujiwara, D.. Concrete characterization of the domain of fractional powers of some elliptic differential operators of the second order. Proc. Japan Acad. Ser. A Math. Sci. 43 (1967), 82–86.CrossRefGoogle Scholar
3Levinson, N.. Gap and Density Theorems (New York: Amer. Math. Soc. Colloquium Publications, 1940).CrossRefGoogle Scholar
4Nambu, T.. On the stabilization of diffusion equations: boundary observation and feetiback. J. Differential Equations 52 (1984), 204–233.CrossRefGoogle Scholar
5Nambu, T.. On stabilization of partial differential equations of parabolic type: boundary observation and feedback. Funkcial. Ekvac. 28 (1985), 267–298.Google Scholar
6Titchmarsh, E. C.. The Theory of Functions (Oxford: Clarendon Press, 1939).Google Scholar