Telegraph systems on networks and port-Hamiltonians. III. Explicit representation and long-term behaviour

TL;DR

This paper derives an explicit formula for the solution semigroup of hyperbolic systems on networks with Kirchhoff boundary conditions and analyzes their long-term behavior using spectral decomposition.

Contribution

It provides a novel explicit representation of the semigroup for hyperbolic network systems and investigates their asymptotic properties.

Findings

Explicit semigroup formula for hyperbolic systems on networks

Analysis of long-term solution behavior

Spectral decomposition as a key tool

Abstract

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.