On spectrum and Riesz basis assignment of infinite dimensional linear systems by bounded linear feedbacks

In this paper, we consider the following linear system of single input on a separable Hilbert space H: φ˙(t)=Aφ(t)+bu(t), where the operator A is the generator of a C₀-semigroup on R and the vector b is not necessarily in H (for the case of boundary controls). We assume that the operator A has compact resolvents, the spectrum of A is discrete and simple and the eigenvectors of A form a Riesz basis in H. We study the spectrum assignability of the system by bounded linear feedbacks of the form: u(t)=<h,φ(t)>_H for h∈H. Under some conditions on the distribution of the spectrum of A and the relative largeness of b we prove the necessary and sufficient condition of Sun (1981) for a given set of points to be assigned to the system by a bounded linear feedback. Given an assignable spectrum set we present explicitly the linear feedback law which realizes it and prove that the eigenvectors of the resulted feedback system form also a Riesz basis in H