Port representations of the telegrapher's equations

This article studies the telegrapher's equations with boundary port variables. Firstly, a link is made between the telegrapher's equations and a skew-symmetric linear operator on a spatial domain. Associated to this linear operator is a Dirac structure which includes the port variables on the boundary of this spatial domain. Secondly, we present all partitions of the port variables into inputs and outputs for which the state dynamics is dissipative. Particularly, we recognize the possible input-outputs for which the system is impedance energy-preserving, i.e., $\frac{1}{2}\frac{d}{dt}\|x(t)\|^2=u(t)^Ty(t)$, as well as scattering energy-preserving, i.e.,$\frac{1}{2}\frac{d}{dt}\|x(t)\|^2=\|u(t)\|^2 -\|y(t)\|^2$. Additionally, we show how to represent the corresponding system as an abstract infinite-dimensional system, i.e., $\dot{x}(t) =Ax(t) +Bu(t)$ and $y(t) = Cx(t)+Du(t)$.