Stability of a controlled strings network with a circuit

A strings network with a circuit is considered under the assumptions that the network is connected and are fixed at the exterior nodes and that the controllers are placed on the inner nodes of the network. By the semigroup theory and the spectral analysis, it is shown that the system satisfies the Riesz basis property and the spectrum determined growth condition. Thus, by the spectral analysis in imaginary axis, some sufficient conditions are provided for the asymptotical stability of the network.