The structural property of a class of vector-valued hyperbolic equations and applications

In this paper, we study the structural properties of a class of vector-valued hyperbolic equations with appropriate boundary conditions, including the spectrum determined growth condition. We prove that the equations associate with a C 0 semigroup. By the structural analysis, we obtained a sufficient and necessary condition for being at least an eigenvalue on the imaginary axis. In particular, using the asymptotic analysis technique we prove that the spectrum of the operator determined by the equations is distributed in a strip parallel to the imaginary axis and is union of finitely many separable sets. Furthermore, we prove that the root vectors of the operator are complete and there is a sequence of root vectors that forms a Riesz basis with parentheses for the Hilbert state space. As applications of our results, we give some concrete examples in controlled complex network of Euler–Bernoulli beams.