Eigenvalues and Eigenvectors of Semigroup Generators Obtained from Diagonal Generators by Feedback

We study infinite-dimensional well-posed linear systems with output feedback such that the closed-loop system is well-posed. The generator A of the open-loop system is assumed to be diagonal, i.e., the state space X (a Hilbert space) has a Riesz basis consisting of eigenvectors of A. We investigate when the closed-loop generator A is Riesz spectral, i.e, its generalized eigenvectors form a Riesz basis in X. We construct a new Riesz basis in X using the sequence of eigenvectors of A and the control operator B. If this new basis is, in a certain sense, close to a subset of the generalized eigenvectors of A , then we conclude that A is Riesz spectral. This approach leads to several results on Riesz spectralness of A where the closed-loop eigenvectors need not be computed. We illustrate the usefulness of our results through several examples concerning the stabilization of systems described by partial differential equations in one space dimension. For the systems in the examples we show that the closed-loop generator is Riesz spectral. Our method allows us to simplify long computations which were necessary otherwise.