Modeling and Stability Analysis of a Wave Network with Parallel Edges and Loops

In this paper we model the dynamics of a wave network consisting of three parallel edges and two loops by determining the number of the controllers and their locations. By frequency approach, we prove the asymptotic stability of the controlled system equipped with four controllers. We then deduce a simple form of the characteristic equation of the controlled system and prove that it is not exponentially stable. These methods and results are valid for other networks containing parallel edges and loops.