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” - Orlov, Yury (IEEE transactions on automatic control, 2017)
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.