Well-posedness and observability of Sturm-Liouville systems on a class of Hilbert spaces

The class of Sturm-Liouville operators on the space of square integrable functions on a finite interval is considered. According to the Riesz-spectral property, the self-adjointness and the positivity of such unbounded linear operators on that space, a class of Hilbert spaces constructed as the domains of the positive (in particular, fractional) powers of any Sturm-Liouville operator is considered. On these spaces, it is shown that any Sturm-Liouville operator is a Riesz-spectral operator that possesses the same eigenvalues as the original ones, associated to rescaled eigenfunctions. This constitutes the first central result of this paper. Properties related to the C_0-semigroup generated by the opposite of such Riesz-spectral operator are also highlighted. In addition as second central result, a characterization of approximate observability by means of point measurement operators is established for such systems. The main results are applied on a diffusion-convection-reaction system in order notably to show that the dynamics operator is the infinitesimal generator of a compact C_0-semigroup on some Sobolev space of integer order, and to establish its observability.