Journal of Evolution Equations

, Volume 19, Issue 4, pp 1111–1147 | Cite as

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

  • Birgit Jacob
  • Sven-Ake WegnerEmail author


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.


\({\hbox {C}_{\mathrm{0}}}\)-semigroup Hyperbolic pde Port-Hamiltonian system Well-posedness Pde’s on networks 

Mathematics Subject Classification

Primary 93D15 Secondary 47D06 



The authors would like to thank the referee for her/his very careful review and the insightful feedback which helped to improve the article significantly.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and Natural SciencesUniversity of WuppertalWuppertalGermany
  2. 2.School of Science, Engineering and DesignTeesside UniversityMiddlesbroughUnited Kingdom

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