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


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.

This is a preview of subscription content, access via your institution.

Fig. 1


  1. 1.

    B. Augner and B. Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evol. Equ. Control Theory, 3(2):207–229, 2014.

    MathSciNet  Article  Google Scholar 

  2. 2.

    H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York, 2011.

    Google Scholar 

  3. 3.

    K.-J. Engel. Generator property and stability for generalized difference operators. J. Evol. Equ., 13(2):311–334, 2013.

    MathSciNet  Article  Google Scholar 

  4. 4.

    K.-J. Engel and R. Nagel. One-Parameter Semigroups for Linear Evolution Equations. Springer-Verlag, New York, 2000.

    Google Scholar 

  5. 5.

    B. Farkas and S.-A. Wegner. Variations on Barbălat’s Lemma. Am. Math. Monthly 128(8):825–830 (2016).

    Article  Google Scholar 

  6. 6.

    G. B. Folland. Introduction to Partial Differential Equations. second ed. Princeton University Press, Princeton, NJ, 1995.

    Google Scholar 

  7. 7.

    R. A. Horn and C. R. Johnson. Matrix Analysis, second ed. Cambridge University Press, Cambridge, 2013.

    Google Scholar 

  8. 8.

    B. Jacob and J. T. Kaiser. Well-posedness of networks for 1-D hyperbolic partial differential equations. Journal of Evolution Equations, 19(1):91–109, 2019.

    MathSciNet  Article  Google Scholar 

  9. 9.

    B. Jacob, K. Morris, and H. Zwart. $\text{ C }_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain. Journal of Evolution Equations, 15:493–502, 2015 .

    MathSciNet  Article  Google Scholar 

  10. 10.

    B. Jacob and H. J. Zwart. Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Operator Theory: Advances and Applications, vol. 223. Birkhäuser/Springer Basel AG, Basel, 2012, Linear Operators and Linear Systems.

  11. 11.

    T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics. Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

  12. 12.

    Y. Le Gorrec, H. J. Zwart, and B. Maschke. Dirac structures and boundary control systems associated with skew-symmetric differential operators. SIAM J. Control Optim., 44(5):1864–1892, 2005.

    MathSciNet  Article  Google Scholar 

  13. 13.

    G. Leoni, A First Course in Sobolev Spaces, first ed., Graduate Studies in Mathematics, vol. 105. American Mathematical Society, Providence, 2009.

  14. 14.

    D. Mugnolo. Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems. Springer, 2014.

  15. 15.

    C. Schubert, C. Seifert, J. Voigt, and M. Waurick, Boundary systems and (skew-)self-adjoint operators on infinite metric graphs. Math. Nachr., 288(14-15):1776–1785, 2015.

    MathSciNet  Article  Google Scholar 

  16. 16.

    O. Staffans, Well-Posed Linear Systems, Encyclopedia of Mathematics and its Applications, vol. 103. Cambridge University Press, Cambridge, 2005.

    Google Scholar 

  17. 17.

    G. Tao. A simple alternative to the Barbălat Lemma. IEEE Trans. Automat. Control, 42(8):698, 2017.

    MATH  Google Scholar 

  18. 18.

    A. E. Taylor. General Theory of Functions and Integration, second ed. Dover Publications, Inc., New York, 1985.

    Google Scholar 

  19. 19.

    G. Todorova and B. Yordanov. Weighted $L^2$-estimates of dissipative wave equations with variable coefficients. J. Differential Equations, 246(12):4497-4518, 2009.

    MathSciNet  Article  Google Scholar 

  20. 20.

    M. Tucsnak and G. Weiss. Well-posed systems—the LTI case and beyond. Automatica J. IFAC, 50(7), 1757–1779, 2014.

    MathSciNet  Article  Google Scholar 

  21. 21.

    A. J. van der Schaft and B. M. Maschke. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. J. Geom. Phys., 42(1-2), 166–194, 2002.

    MathSciNet  Article  Google Scholar 

  22. 22.

    J. A. Villegas, A port-Hamiltonian approach to distributed parameter systems. Ph.D. dissertation, Department of Applied Mathematics, University of Twente,, Enschede, The Netherlands, 2007.

  23. 23.

    J. A. Villegas, H. Zwart, Y. Le Gorrec, and B. Maschke. Exponential stability of a class of boundary control systems. IEEE Trans. Automat. Control, 54(1), 142–147, 2009.

    MathSciNet  Article  Google Scholar 

  24. 24.

    S.-A. Wegner. Boundary triplets for skew-symmetric operators and the generation of strongly continuous semigroups. Analysis Mathematica, 43(4):657–686, 2017.

    MathSciNet  Article  Google Scholar 

  25. 25.

    G. Weiss. Regular linear systems with feedback. Math. Control Signals Systems, 7(1), 23–57, 1994.

    MathSciNet  Article  Google Scholar 

  26. 26.

    H. Zwart. Transfer functions for infinite-dimensional systems. Systems Control Lett., 52(3-4):247–255, 2004.

    MathSciNet  Article  Google Scholar 

  27. 27.

    H. Zwart, Y. Le Gorrec, B. Maschke, and J. Villegas. Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM Control Optim. Calc. Var., 16(4):1077–1093, 2010.

    MathSciNet  Article  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Sven-Ake Wegner.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


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

Mathematics Subject Classification

  • Primary 93D15
  • Secondary 47D06