$C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain

TL;DR

This paper provides a simple boundary condition test to determine when hyperbolic PDEs on a 1D domain generate $C_0$-semigroups, aiding analysis of systems like beams, waves, and transmission lines.

Contribution

It introduces a straightforward criterion for $C_0$-semigroup generation based on boundary conditions for hyperbolic PDEs on a 1D domain.

Findings

Boundary condition test for $C_0$-semigroup generation

Application to models of beams, waves, and transmission lines

Illustrated with several examples

Abstract

Hyperbolic partial differential equations on a one-dimensional spatial domain are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of non-homogeneous transmission lines. The main result of this paper is a simple test for $C_0$-semigroup generation in terms of the boundary conditions. The result is illustrated with several examples.