Sturm - Liouville Systems Are Riesz - Spectral Systems

where x, u and y are the system state, input and output, respectively, A is a densely defined differential linear operator on an (infinite-dimensional) Hilbert space (e.g., L(a, b), a, b ∈ R), which generates a C0-semigroup, and B, C and D are bounded linear operators. Moreover, if A is a Riesz-spectral operator, it possesses several interesting properties, regarding in particular observability and controllability. In many physical systems (e.g., vibration problems in mechanics, diffusion problems), A, or −A, is a SturmLiouville operator (see, e.g., Renardy and Rogers, 1993, Naylor and Sell, 1982, Ray, 1981, p. 157). This is also the case for chemical reactor models with axial dispersion (see, e.g., Winkin et al., 2000, Laabissi et al., 2001). In order to encompass all these applications in one single unifying framework, it is natural to define the class of Sturm-Liouville systems. This is accomplished in Section 2. Many theoretical results regarding Sturm-Liouville (S-L) operators or S-L problems are available in the scientific literature (see, e.g., Sagan, 1961; Birkhoff, 1962; Young, 1972; Renardy and Rogers, 1993). In Section 3 we deduce from these properties that any S-L system is a Riesz-spectral system on L(a, b). To the authors’ knowledge, the concept of the Sturm-Liouville system is new and so is the result concerning its connection with Riesz spectral systems, under this form. The authors would like to stress the fact that this result is obtained by gathering a number of properties that are dispersed in the literature, and expressed in a form that can be useful for system theory and control, by emphasizing the concept of SturmLiouville systems. Such an application to systems analysis is given in Section 4.