On General Properties of Eigenvalues and Eigenfunctions of a Sturm–Liouville Operator: Comments on “ISS With Respect to Boundary Disturbances for 1-D Parabolic PDEs”

In the paper "ISS with respect to boundary disturbances for 1-D parabolic PDEs" (IEEE Transactions on Automatic Control, vol. 61, pp. 3712-3724, 2016), input-to-state stability properties are established for 1-D spatially varying parabolic partial differential equations (PDEs) under certai...

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Bibliographic Details
Published in:IEEE transactions on automatic control Vol. 62; no. 11; pp. 5970 - 5973
Main Author: Orlov, Yury
Format: Journal Article
Language:English
Published: IEEE 01.11.2017
Subjects:
Autonomous systems
Boundary conditions
Convergence
distributed parameter systems
Eigenvalues and eigenfunctions
Force
Fourier series
Heating systems
linear systems
Sturm–Liouville problem
ISSN:0018-9286, 1558-2523
Online Access:Get full text
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Summary:In the paper "ISS with respect to boundary disturbances for 1-D parabolic PDEs" (IEEE Transactions on Automatic Control, vol. 61, pp. 3712-3724, 2016), input-to-state stability properties are established for 1-D spatially varying parabolic partial differential equations (PDEs) under certain assumptions, imposed on eigenvalues and eigenfunctions of an associated Sturm-Liouville operator. A key assumption on the absolute convergence of an associated Fourier series, composed of the normalized eigenfunctions and inverse eigenvalues of the Sturm-Liouville operator, is analyzed in the present note. General properties of the Sturm-Liouville operator are carried out to demonstrate that such a key assumption becomes redundant for the underlying PDEs with sign-definite sufficiently smooth coefficients.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2017.2694425