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.

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Annali di Matematica Pura ed Applicata

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