Analyticity and Riesz basis property of semigroups associated to damped vibrations

Abstract

Second order equations of the formz(t) + A 0 z(t) + Dż(t) = 0 are considered. Such equations are often used as a model for transverse motions of thin beams in the presence of damping. We derive various properties of the operator matrix A = 0 I −A 0 −D associated with the second order problem above. We develop sufficient conditions for analyticity of the associated semigroup and for the existence of a Riesz basis consisting of eigenvectors and associated vectors of A in the phase space.

A structurally damped plate equation with Dirichlet–Neumann boundary conditions

“…Such systems have been studied in detail also from the viewpoint of dynamical systems and control theory, see e.g. [5,20,23,29] and the references therein. In the L 2 case, the basic generation results were already obtained in [6].…”

Locally Definitizable Operators: The Local Structure of the Spectrum

“…[15,21,22,25,31,34,38,39,41,66,93], KreinFeller operators [46], λ-dependent boundary value problems, see e.g. [24,30,42,62,63,82], operator polynomials [68,69,71,72,74,75,76,87], second order systems [50,89,90] and in the study of problems of Klein-Gordon type [83].…”

Спектральная Теория Операторных Матриц, Встречающихся В Моделях Механики

“…По-кажем, что условие σ ess (A −1 0 D) = {0} эквивалентно условию дефинизируемости для A . Оказывается, что в случае, когда A имеет компактную резольвенту, в от-личие от ситуации, рассмотренной в [23] и [24], оператор A перестает быть дефини-зируемым.…”