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”

Abstract

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.