TY - JOUR

T1 - Spectral properties of infinite-dimensional closed-loop systems

AU - Weiss, G.

AU - Xu, C. Z.

PY - 2005/8

Y1 - 2005/8

N2 - We consider feedback systems obtained from infinite-dimensional well-posed linear systems by output feedback. Thus, our framework allows for unbounded control and observation operators. Our aim is to investigate the relationship between the open-loop system, the feedback operator K and the spectrum (in particular, the eigenvalues and eigenvectors) of the closed-loop generator AK. We give a useful characterization of that part of the spectrum σ (A K) which lies in the resolvent set of A, the open-loop generator, via the "characteristic equation" involving the open-loop transfer function. We show that certain points of σ (A) cannot be eigenvalues of AK if the input and output are scalar (so that K is a number) and K ≠ 0. We devote special attention to the case when the output feedback operator K is compact. It is relatively easy to prove that in this case, σ e (A), the essential spectrum of A, is invariant, that is, it is equal to σe(AK). A related but much harder problem is to determine the largest subset of σ(A) which remains invariant under compact output feedback. This set, which we call the immovable spectrum of A, strictly contains σ e(A). We give an explicit characterization of the immovable spectrum and we investigate the consequences of our results for a certain class of well-posed systems obtained by interconnecting an infinite chain of identical systems.

AB - We consider feedback systems obtained from infinite-dimensional well-posed linear systems by output feedback. Thus, our framework allows for unbounded control and observation operators. Our aim is to investigate the relationship between the open-loop system, the feedback operator K and the spectrum (in particular, the eigenvalues and eigenvectors) of the closed-loop generator AK. We give a useful characterization of that part of the spectrum σ (A K) which lies in the resolvent set of A, the open-loop generator, via the "characteristic equation" involving the open-loop transfer function. We show that certain points of σ (A) cannot be eigenvalues of AK if the input and output are scalar (so that K is a number) and K ≠ 0. We devote special attention to the case when the output feedback operator K is compact. It is relatively easy to prove that in this case, σ e (A), the essential spectrum of A, is invariant, that is, it is equal to σe(AK). A related but much harder problem is to determine the largest subset of σ(A) which remains invariant under compact output feedback. This set, which we call the immovable spectrum of A, strictly contains σ e(A). We give an explicit characterization of the immovable spectrum and we investigate the consequences of our results for a certain class of well-posed systems obtained by interconnecting an infinite chain of identical systems.

UR - http://www.scopus.com/inward/record.url?scp=23044502321&partnerID=8YFLogxK

U2 - 10.1007/s00498-005-0154-7

DO - 10.1007/s00498-005-0154-7

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AN - SCOPUS:23044502321

SN - 0932-4194

VL - 17

SP - 153

EP - 172

JO - Mathematics of Control, Signals, and Systems

JF - Mathematics of Control, Signals, and Systems

IS - 3

ER -