Well-posedness of a class of infinite-dimensional port-Hamiltonian systems with boundary control and observation

Authors

Bouchra Elghazi, Birgit Jacob, Hans Zwart

Abstract

We characterize the well-posedness of a class of infinite-dimensional port-Hamiltonian systems with boundary control and observation. This class includes in particular the Euler-Bernoulli beam equations and more generally 1D linear infinite-dimensional port-Hamiltonian systems with boundary control and observation as well as coupled systems. It is known, that for the Timoshenko beam models internal well-posedness implies well-posedness of the overall system. By means of an example we show that this is not true for the Euler-Bernoulli beam models. An easy verifiable equivalent condition for well-posedness of the overall system will be presented. We will conclude the paper by applying the obtained results to several Euler-Bernoulli beam models.

Keywords

well-posed distributed parameter systems; port-Hamiltonian systems; impedance passive system; Euler-Bernoulli beam equations

Citation

• Journal: IFAC-PapersOnLine
• Year: 2025
• Volume: 59
• Issue: 8
• Pages: 102–107
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2025.08.074
• Note: 5th IFAC Workshop on Control of Systems Governed by Partial Differential Equations - CPDE 2025- Beijing, China, June 18 - 20, 2025

BibTeX

@article{Elghazi_2025,
  title={{Well-posedness of a class of infinite-dimensional port-Hamiltonian systems with boundary control and observation}},
  volume={59},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2025.08.074},
  number={8},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Elghazi, Bouchra and Jacob, Birgit and Zwart, Hans},
  year={2025},
  pages={102--107}
}

Download the bib file

References

• Augner, B. Well-Posedness and Stability of Infinite-Dimensional Linear Port-Hamiltonian Systems with Nonlinear Boundary Feedback. SIAM J. Control Optim. 57, 1818–1844 (2019) – 10.1137/15m1024901
• Augner, (2016)
• Augner, B. & Jacob, B. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory 3, 207–229 (2014) – 10.3934/eect.2014.3.207
• Curtain, (2020)
• Bao-Zhu Guo & Jun-Min Wang. The well-posedness and stability of a beam equation with conjugate variables assigned at the same boundary point. IEEE Trans. Automat. Contr. 50, 2087–2093 (2005) – 10.1109/tac.2005.860275
• Humaloja, J.-P. & Paunonen, L. Robust Regulation of Infinite-Dimensional Port-Hamiltonian Systems. IEEE Trans. Automat. Contr. 63, 1480–1486 (2018) – 10.1109/tac.2017.2748055
• Humaloja, J.-P., Paunonen, L. & Pohjolainen, S. Robust regulation for first-order port-hamiltonian systems. 2016 European Control Conference (ECC) 2203–2208 (2016) doi:10.1109/ecc.2016.7810618 – 10.1109/ecc.2016.7810618
• Jacob, (2012)
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM J. Control Optim. 44, 1864–1892 (2005) – 10.1137/040611677
• Staffans, Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems. Math. Control Signals Syst. (2002)
• Staffans, (2005)
• van der Schaft, Port-Hamiltonian systems: an introductory survey. Proceedings of the International Congress of Mathematicians (2006)
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. FnT in Systems and Control 1, 173–378 (2014) – 10.1561/2600000002
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42, 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Villegas, (2007)
• Weiss, G. Regular linear systems with feedback. Math. Control Signal Systems 7, 23–57 (1994) – 10.1007/bf01211484
• Zwart, H., Le Gorrec, Y., Maschke, B. & Villegas, J. Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: COCV 16, 1077–1093 (2009) – 10.1051/cocv/2009036