Analyticity and Riesz basis property of semigroups associated to damped vibrations

Abstract

Second order equations of the form $z^{\prime \prime }+A_{0}z+D{z}^{\prime }=0$ in an abstract Hilbert space 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 $A$ 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.