A Structural Observation on Port-Hamiltonian Systems

Authors

Rainer H. Picard, Sascha Trostorff, Bruce Watson, Marcus Waurick

Abstract

We study port-Hamiltonian systems on a familiy of intervals and characterise all boundary conditions leading to \( m \)-accretive realisations of the port-Hamiltonian operator and thus to generators of contractive semigroups. The proofs are based on a structural observation that the port-Hamiltonian operator can be transformed to the derivative on a familiy of reference intervals by suitable congruence relations allowing for studying the simpler case of a transport equation. Moreover, we provide well-posedness results for associated control problems without assuming any additional regularity of the operators involved.

Citation

• Journal: SIAM Journal on Control and Optimization
• Year: 2023
• Volume: 61
• Issue: 2
• Pages: 511–535
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/21m1441365

BibTeX

@article{Picard_2023,
  title={{A Structural Observation on Port-Hamiltonian Systems}},
  volume={61},
  ISSN={1095-7138},
  DOI={10.1137/21m1441365},
  number={2},
  journal={SIAM Journal on Control and Optimization},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Picard, Rainer H. and Trostorff, Sascha and Watson, Bruce and Waurick, Marcus},
  year={2023},
  pages={511--535}
}

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References

• Engel, K.-J. Generator property and stability for generalized difference operators. Journal of Evolution Equations vol. 13 311–334 (2013) – 10.1007/s00028-013-0179-1
• Engel K.-J.. One-Parameter Semigroups for Linear Evolution Equations (2000)
• Heidari, H. & Zwart, H. Port-Hamiltonian modelling of nonlocal longitudinal vibrations in a viscoelastic nanorod. Mathematical and Computer Modelling of Dynamical Systems vol. 25 447–462 (2019) – 10.1080/13873954.2019.1659374
• Jacob, B. & Kaiser, J. T. Well-posedness of systems of 1-D hyperbolic partial differential equations. Journal of Evolution Equations vol. 19 91–109 (2018) – 10.1007/s00028-018-0470-2
• Jacob, B., Morris, K. & Zwart, H. C 0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain. Journal of Evolution Equations vol. 15 493–502 (2015) – 10.1007/s00028-014-0271-1
• Jacob, B. & Wegner, S.-A. Well-posedness of a class of hyperbolic partial differential equations on the semi-axis. Journal of Evolution Equations vol. 19 1111–1147 (2019) – 10.1007/s00028-019-00507-7
• Jacob, B. & Zwart, H. An operator theoretic approach to infinite‐dimensional control systems. GAMM-Mitteilungen vol. 41 (2018) – 10.1002/gamm.201800010
• Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• Picard, R. A structural observation for linear material laws in classical mathematical physics. Mathematical Methods in the Applied Sciences vol. 32 1768–1803 (2009) – 10.1002/mma.1110
• Picard, R. Mother operators and their descendants. Journal of Mathematical Analysis and Applications vol. 403 54–62 (2013) – 10.1016/j.jmaa.2013.02.004
• Picard, R. & McGhee, D. Partial Differential Equations. (2011) doi:10.1515/9783110250275 – 10.1515/9783110250275
• Picard, R., Trostorff, S. & Waurick, M. Well-posedness viaMonotonicity – an Overview. Operator Theory: Advances and Applications 397–452 (2015) doi:10.1007/978-3-319-18494-4_25 – 10.1007/978-3-319-18494-4_25
• Picard, R., Trostorff, S. & Waurick, M. On a comprehensive class of linear control problems. IMA Journal of Mathematical Control and Information vol. 33 257–291 (2014) – 10.1093/imamci/dnu035
• Seifert, C., Trostorff, S. & Waurick, M. Evolutionary Equations. Operator Theory: Advances and Applications (Springer International Publishing, 2022). doi:10.1007/978-3-030-89397-2 – 10.1007/978-3-030-89397-2
• Showalter R. E.. Monotone Operators in Banach Space and Nonlinear Partial Differential Equations (1997)
• Trostorff, S. A characterization of boundary conditions yielding maximal monotone operators. Journal of Functional Analysis vol. 267 2787–2822 (2014) – 10.1016/j.jfa.2014.08.009
• Trostorff, S. Semigroups and evolutionary equations. Semigroup Forum vol. 103 661–699 (2021) – 10.1007/s00233-021-10208-8
• van der Schaft, A., Maschke, B. & Ortega, R. Network modelling of physical systems: a geometric approach. Lecture Notes in Control and Information Sciences 253–276 (2001) doi:10.1007/bfb0110387 – 10.1007/bfb0110387
• Waurick, M. & Wegner, S.-A. Dissipative extensions and port-Hamiltonian operators on networks. Journal of Differential Equations vol. 269 6830–6874 (2020) – 10.1016/j.jde.2020.05.014
• 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: Control, Optimisation and Calculus of Variations vol. 16 1077–1093 (2009) – 10.1051/cocv/2009036