Banasiak, JacekBloch, Adam2023-04-042023-04-042022-12Jacek Banasiak, Adam Błoch. Telegraph systems on networks and port-Hamiltonians. Ⅲ. Explicit representation and long-term behaviour. Evolution Equations and Control Theory, 2022, 11(6): 2165-2181. doi: 10.3934/eect.2022016.2163-2480 (online)10.3934/eect.2022016http://hdl.handle.net/2263/90330The research was completed while A. B. was a doctoral candidate in the Interdisciplinary Doctoral School at Łodź University of Technology, Poland.In this paper we present an explicit formula for the semigroup governing the solution to hyperbolic systems on a metric graph, satisfying general linear Kirchhoff's type boundary conditions. Further, we use this representation to establish the long term behaviour of the solutions. The crucial role is played by the spectral decomposition of the boundary matrix.en© 2023 American Institute of Mathematical SciencesSpectral projections of matricesPerron–Frobenius theoryLong-term behaviourSemigroups of operatorsHyperbolic systems on networksPort-HamiltoniansPostprint Article