Weak solutions of port-Hamiltonian systems

Authors

Abstract

We consider port-Hamiltonian systems from a geometric perspective, where the quantities involved such as state, flows, and efforts evolve in (possibly infinite-dimensional) Banach spaces. The main contribution of this article is the introduction of a weak solution concept. In this framework we show that a suitable part of the state is weakly differentiable in a space that, for ordinary evolution equations, plays the role of an extrapolation space. Through examples, we demonstrate that this approach is consistent with the weak solution framework commonly used for partial differential equations.

Citation

Journal: Mathematical Control and Related Fields
Year: 2026
Volume: 0
Issue: 0
Pages: 0–0
Publisher: American Institute of Mathematical Sciences (AIMS)
DOI: 10.3934/mcrf.2026036

BibTeX

@article{Reis_2026,
  title={{Weak solutions of port-Hamiltonian systems}},
  volume={0},
  ISSN={2156-8499},
  DOI={10.3934/mcrf.2026036},
  number={0},
  journal={Mathematical Control and Related Fields},
  publisher={American Institute of Mathematical Sciences (AIMS)},
  author={Reis, Timo},
  year={2026},
  pages={0--0}
}

Download the bib file

References

R. A. Adams and J. J. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics, $2^{nd}$ edition, Academic Press, Amsterdam, San Diego, Oxford, London, 2003.
H. W. Alt, Linear Functional Analysis, Universitext, Springer-Verlag, London, 2016.
Arnold VI (1989) Mathematical Methods of Classical Mechanics. Springer New York – 10.1007/978-1-4757-2063-1
Brugnoli A, Haine G, Matignon D (2023) Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint. CAM 15(3):362–387. https://doi.org/10.3934/cam.2023018 – 10.3934/cam.2023018
Kurula M, Zwart H, van der Schaft A, Behrndt J (2010) Dirac structures and their composition on Hilbert spaces. Journal of Mathematical Analysis and Applications 372(2):402–422. https://doi.org/10.1016/j.jmaa.2010.07.004 – 10.1016/j.jmaa.2010.07.004
Cervera J, van der Schaft AJ, Baños A (2007) Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica 43(2):212–225. https://doi.org/10.1016/j.automatica.2006.08.014 – 10.1016/j.automatica.2006.08.014
Ciarlet PG (2013) Linear and Nonlinear Functional Analysis with Applications. Society for Industrial and Applied Mathematics – 10.1137/1.9781611972597
Diestel J, Uhl J Jr (1977) Vector Measures. Mathematical Surveys and Monographs – 10.1090/surv/015
Duindam V, Macchelli A, Stramigioli S, Bruyninckx H (2009) Modeling and Control of Complex Physical Systems. Springer Berlin Heidelberg – 10.1007/978-3-642-03196-0
L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, American Mathematical Society, Providence, R.I., 2010.
Gernandt H, Haller FE, Reis T (2021) A Linear Relation Approach to Port-Hamiltonian Differential-Algebraic Equations. SIAM J Matrix Anal Appl 42(2):1011–1044. https://doi.org/10.1137/20m1371166 – 10.1137/20m1371166
Gernandt H, Haller FE, Reis T, Schaft AJ van der (2021) Port-Hamiltonian formulation of nonlinear electrical circuits. Journal of Geometry and Physics 159:103959. https://doi.org/10.1016/j.geomphys.2020.103959 – 10.1016/j.geomphys.2020.103959
Jacob B, Zwart HJ (2012) Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces. Springer Basel – 10.1007/978-3-0348-0399-1
van der Schaft A, Jeltsema D (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control 1(2–3):173–378. https://doi.org/10.1561/2600000002 – 10.1561/2600000002
Le Gorrec Y, Zwart H, Maschke B (2005) Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM J Control Optim 44(5):1864–1892. https://doi.org/10.1137/040611677 – 10.1137/040611677
van der Schaft AJ, Maschke BM (2002) Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42(1–2):166–194. https://doi.org/10.1016/s0393-0440(01)00083-3 – 10.1016/s0393-0440(01)00083-3
van der Schaft A, Maschke B (2020) Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems. Vietnam J Math 48(4):929–939. https://doi.org/10.1007/s10013-020-00419-x – 10.1007/s10013-020-00419-x
Macchelli A, van der Schaft AJ, Melchiorri C (2004) Port Hamiltonian formulation of infinite dimensional systems I. Modeling. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 3762-3767 Vol.4 – 10.1109/cdc.2004.1429324
Macchelli A, van der Schaft AJ, Melchiorri C (2004) Port Hamiltonian formulation of infinite dimensional systems II. Boundary control by interconnection. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 3768-3773 Vol.4 – 10.1109/cdc.2004.1429325
Reis T (2021) Some notes on port-Hamiltonian systems on Banach spaces. IFAC-PapersOnLine 54(19):223–229. https://doi.org/10.1016/j.ifacol.2021.11.082 – 10.1016/j.ifacol.2021.11.082
W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.
O. Staffans, Well-posed Linear Systems, vol. 103 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2005.
L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, vol. 3 of Lecture Notes of the Unione Matematica Italiana, Springer, Berlin, Heidelberg, 2007.
Tucsnak M, Weiss G (2009) Observation and Control for Operator Semigroups. Birkhäuser Basel – 10.1007/978-3-7643-8994-9
van der Schaft AJ (2013) Port-Hamiltonian Differential-Algebraic Systems. Surveys in Differential-Algebraic Equations I 173–226 – 10.1007/978-3-642-34928-7_5
J. A. Villegas, A Port-Hamiltonian Approach to Distributed Parameter Systems, PhD thesis, University of Twente, 2007.