We present a "non-standard method" to treat wave equations on networks, leading to a transport process on the doubled directed graph.From the node conditions, we derive a flow governed by a certain adjacency matrix which, in particular, builds the bridge to the theory of difference operators.This approach provides the fundament for a powerful method to examine (boundary-)controllability and to prove stability results for damped and delay-damped networks of wave equations.

We develop a semigroup approach to abstract boundary control problems which allows to characterize the space of all approximately reachable states. We then introduce the "maximal reachability space" giving an upper bound for this space. The abstract results are applied to the flow in a network controlled in a single vertex.

In this paper we examine difference operators with constant coefficients. We show that the type of the generated semigroup is determined by a matrix 𝔹 , originating from the domain of the operator. Moreover, we provide necessary and sufficient conditions for exponential and polynomial stability of the semigroup in terms of the matrix 𝔹 , using results of A. Borichev and Y. Tomilov. We close the paper with an application of our results to flows in networks.