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

Abstract

In this paper, we consider the following linear system of single input on a separable Hilbert space H: /spl phi//spl dot/(t)=A/spl phi/(t)+bu(t), where the operator A is the generator of a C/sub 0/-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)= /sub H/ for h/spl isin/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.