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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.
Journal of Evolution Equations – Springer Journals
Published: May 1, 2008
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