# Finite rank, relatively bounded perturbations of semigroups generators - Part II: Spectrum and riesz basis assignment with applications to feedback systems

## Abstract

This paper is motivated by, and ultimately directed to, boundary feedback partial differential equations of both parabolic and hyperbolic type, defined on a bounded domain. It is written, however, in abstract form. It centers on the (feedback) operator AF=A+P; A the infinitesimal generator of a s.c. semigroup on H; P an Abounded, one dimensional range operator (typically nondissipative), so that P=(A·, a)b, for a, b ∈ H. While Part I studied the question of generation of a s.c. semigroup on H by AF and lack thereof, the present Part II focuses on the following topics: (i) spectrum assignment of AF, given A and a ∈ H, via a suitable vector b ∈ H; alternatively, given A, via a suitable pair of vectors a, b ∈ H; (ii) spectrality of AF-and lack thereof-when A is assumed spectral (constructive counterexamples include the case where P is bounded but the eigenvalues of A have zero gap, as well as the case where P is genuinely Abounded). The main result gives a set of sufficient conditions on the eigenvalues {λn} of A, the given vector a ∈ H and a given suitable sequence {εn} of nonzero complex numbers, which guarantee the existence of a suitable vector b ∈ H such that AF possesses the following two desirable properties: (i) the eigenvalues of AF are precisely equal to λn+εn; (ii) the corresponding eigenvectors of AF form a Riesz basis (a fortiori, AF is spectral). While finitely many εn′s can be preassigned arbitrarily, it must be however that εn → 0 « sufficiently fast ». Applications include various types of boundary feedback stabilization problems for both parabolic and hyperbolic partial differential equations. An illustration to the damped wave equation is also included. © 1986 Nicola Zanichelli Editore.

## Publication Title

Annali di Matematica Pura ed Applicata

## Recommended Citation

Lasiecka, I., & Triggiani, R.
(1986). Finite rank, relatively bounded perturbations of semigroups generators - Part II: Spectrum and riesz basis assignment with applications to feedback systems.* Annali di Matematica Pura ed Applicata**, 143* (1), 47-100.
https://doi.org/10.1007/BF01769210