Wave Propagation in Networks: a System Theoretic Approach

Wave propagation problems on networks arise in many fields of technology. Let us give a brief list of possible applications: •Acoustics: for example the human vocal tract can be considered as a Y-shaped graph; •Mechanics: wave propagation in Timoshenko beam structures. For example, jointed beams and beams with parameter discontinuities can be regarded as network systems; •Electrodynamics: electric transmission networks. In this work we show that a variety of wave propagation problems on networks are solvable (forward in time) and energy passive or conservative. These problems are given in terms of partial differential equations that are interconnected through Kirchhoff type algebraic conditions involving the boundary conditions of these PDEs. The contribution of this work is to show that if all of the single PDEs are solvable and energy passive (or conservative) then also the interconnected PDE-system is solvable and energy passive (respectively, conservative). For related results, see, e.g., [2]. In more technical terms, the subsystems of the network are formulated as distributed parameter boundary control systems (BCS) that are an operator theoretic framework for treating problems concerning PDEs to which control action is inflicted through boundary conditions. In this framework, the interconnections between the subsystems become algebraic conditions in terms of the subsystems’ input and output operators, of which we only require that they are of compatible dimensions. We then show that the interconnections can be constructed so that the composed systems are always solvable and energy passive/conservative.