Analyticity and Riesz basis property of semigroups associated to damped vibrations

Abstract.

Second order equations of the form $\ddot{z}(t) + A_0z(t) + D\dot{z}(t) = 0$ 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 $\mathcal{A} = \left[ {\begin{array}{*{20}c} 0 & I\\ { -A_0 } & { -D}\\ \end{array}} \right]$ 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 ${\mathcal{A}}$ in the phase space.