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

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.

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Acknowledgements

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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Correspondence to Sven-Ake Wegner.

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Jacob, B., Wegner, SA. Well-posedness of a class of hyperbolic partial differential equations on the semi-axis. J. Evol. Equ. 19, 1111–1147 (2019). https://doi.org/10.1007/s00028-019-00507-7

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Keywords

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

Mathematics Subject Classification

  • Primary 93D15
  • Secondary 47D06