Well-posedness of a class of hyperbolic partial differential equations on the semi-axis

TL;DR

This paper investigates the well-posedness of a class of hyperbolic PDEs on the semi-axis, characterizing boundary conditions for semigroup generation and establishing boundary control systems, with applications to transport networks and vibrating strings.

Contribution

It provides a comprehensive characterization of boundary conditions for hyperbolic PDEs to generate semigroups and develops a framework for boundary control and observation in infinite domains.

Findings

Boundary conditions for semigroup generation identified.

Boundary control systems shown to be well-posed.

Applications to transport networks and vibrating strings demonstrated.

Abstract

In this article we study a class of hyperbolic partial differential equations of order one on the semi-axis. The so-called port-Hamiltonian systems cover for instance the wave equation and the transport equation, but also networks of the aforementioned equations fit into this framework. Our main results firstly characterize the boundary conditions which turn the corresponding linear operator into the generator of a strongly continuous semigroup. Secondly, we equip the equation with inputs (control) and outputs (observation) at the boundary and prove that this leads to a well-posed boundary control system. We illustrate our results via an example of coupled transport equations on a network, that allows to model transport from and to infinity. Moreover, we study a vibrating string of infinite length with one endpoint. Here, we show that our results allow to treat cases where the physical constants of the string tend to zero at infinity.