On decomposing of infinite-dimensional Sylvester equations

In this paper we study certain infinite-dimensional Sylvester equations. The equations are closely related to robust output regulation of infinite-dimensional systems. If the signal generator is finite-dimensional or has discrete spectrum and a complete set of orthonormal eigenvectors, there are some known sufficient conditions for the decomposing of these Sylvester equations. In this paper we generalize these conditions to the case where the signal generator has discrete spectrum and a complete set of orthonormal generalized eigenvectors. We also study how these conditions are related to an infinite-dimensional version of the internal model of finite-dimensional control theory. We show that under certain assumptions on the spectra of the closed-loop system and the signal generator these conditions are equivalent to the concept of an internal model.