Stability via closure relations with applications to dissipative and port-Hamiltonian systems

Authors

Jochen Glück, Birgit Jacob, Annika Meyer, Christian Wyss, Hans Zwart

Abstract

We consider differential operators A that can be represented by means of a so-called closure relation in terms of a simpler operator \( \)A_{{\text {ext}}}\( \) A ext defined on a larger space. We analyse how the spectral properties of A and \( \)A_{{\text {ext}}}\( \) A ext are related and give sufficient conditions for exponential stability of the semigroup generated by A in terms of the semigroup generated by \( \)A_{{\text {ext}}}\( \) A ext . As applications we study the long-term behaviour of a coupled wave–heat system on an interval, parabolic equations on bounded domains that are coupled by matrix-valued potentials, and of linear infinite-dimensional port-Hamiltonian systems with dissipation on an interval.

Keywords

Port-Hamiltonian systems; Closure relations; Exponential stability; \( C_0 \) -semigroups; 93D23; 37K40; 47D06; 34G10

Citation

• Journal: Journal of Evolution Equations
• Year: 2024
• Volume: 24
• Issue: 3
• Pages:
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s00028-024-00992-5

BibTeX

@article{Gl_ck_2024,
  title={{Stability via closure relations with applications to dissipative and port-Hamiltonian systems}},
  volume={24},
  ISSN={1424-3202},
  DOI={10.1007/s00028-024-00992-5},
  number={3},
  journal={Journal of Evolution Equations},
  publisher={Springer Science and Business Media LLC},
  author={Glück, Jochen and Jacob, Birgit and Meyer, Annika and Wyss, Christian and Zwart, Hans},
  year={2024}
}

Download the bib file

References

• Arendt W (2002) Chapter 1 Semigroups and evolution equations: Functional calculus, regularity and kernel estimates. In: Handbook of Differential Equations: Evolutionary Equations. Elsevier, pp 1–85 – 10.1016/s1874-5717(04)80003-3
• B. Augner. Stabilisation of Infinite-dimensional Port-Hamiltonian System. PhD thesis, University of Wuppertal, 2016.
• Augner B, Jacob B (2014) Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. EECT 3(2):207–229. https://doi.org/10.3934/eect.2014.3.207 – 10.3934/eect.2014.3.207
• Batty C, Paunonen L, Seifert D (2016) Optimal energy decay in a one-dimensional coupled wave–heat system. J Evol Equ 16(3):649–664. https://doi.org/10.1007/s00028-015-0316-0 – 10.1007/s00028-015-0316-0
• Batty C, Paunonen L, Seifert D (2019) Optimal Energy Decay for the Wave-Heat System on a Rectangular Domain. SIAM J Math Anal 51(2):808–819. https://doi.org/10.1137/18m1195796 – 10.1137/18m1195796
• Cardoso-Ribeiro FL, Matignon D, Lefèvre L (2021) Dissipative Shallow Water Equations: a port-Hamiltonian formulation. IFAC-PapersOnLine 54(19):167–172. https://doi.org/10.1016/j.ifacol.2021.11.073 – 10.1016/j.ifacol.2021.11.073
• Daners D (2000) Robin boundary value problems on arbitrary domains. Trans Amer Math Soc 352(9):4207–4236. https://doi.org/10.1090/s0002-9947-00-02444-2 – 10.1090/s0002-9947-00-02444-2
• Daners D, Glück J, Kennedy JB (2016) Eventually and asymptotically positive semigroups on Banach lattices. Journal of Differential Equations 261(5):2607–2649. https://doi.org/10.1016/j.jde.2016.05.007 – 10.1016/j.jde.2016.05.007
• R. Dautray and J.-L. Lions. Mathematical analysis and numerical methods for science and technology. Vol. 5: Evolution problems I. With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon. Transl. from the French by Alan Craig. Translation editor: Ian N. Sneddon. Berlin: Springer, 2nd printing edition, 2000.
• Dobrick A, Glück J (2023) Convergence to equilibrium for linear parabolic systems coupled by matrix‐valued potentials. Mathematische Nachrichten 297(2):577–594. https://doi.org/10.1002/mana.202300087 – 10.1002/mana.202300087
• V. Duindam, A. Macchelli, S. Stramigioli, and H. Bruyninckx, editors. Modeling and Control of Complex Physical Systems. Springer-Verlag, Berlin, 2009.
• Eberard D, Maschke BM, van der Schaft AJ (2007) An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on Mathematical Physics 60(2):175–198. https://doi.org/10.1016/s0034-4877(07)00024-9 – 10.1016/s0034-4877(07)00024-9
• K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Grad. Texts Math. Berlin: Springer, 2000.
• L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010.
• Guidotti P, Merino S (1997) Hopf Bifurcation in a Scalar Reaction Diffusion Equation. Journal of Differential Equations 140(1):209–222. https://doi.org/10.1006/jdeq.1997.3307 – 10.1006/jdeq.1997.3307
• Humaloja J-P, Paunonen L (2018) Robust Regulation of Infinite-Dimensional Port-Hamiltonian Systems. IEEE Trans Automat Contr 63(5):1480–1486. https://doi.org/10.1109/tac.2017.2748055 – 10.1109/tac.2017.2748055
• Jacob B, Kaiser JT (2019) On Exact Controllability of Infinite-Dimensional Linear Port-Hamiltonian Systems. IEEE Control Syst Lett 3(3):661–666. https://doi.org/10.1109/lcsys.2019.2916814 – 10.1109/lcsys.2019.2916814
• Jacob B, Skrepek N (2021) Stability of the multidimensional wave equation in port-Hamiltonian modelling. In: 2021 60th IEEE Conference on Decision and Control (CDC). IEEE, pp 6188–6193 – 10.1109/cdc45484.2021.9683501
• Jacob B, Zwart H (2018) An operator theoretic approach to infinite‐dimensional control systems. GAMM-Mitteilungen 41(4):e201800010. https://doi.org/10.1002/gamm.201800010 – 10.1002/gamm.201800010
• Jacob B, Zwart HJ (2012) Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces. Springer Basel, Basel – 10.1007/978-3-0348-0399-1
• Jäschke J, Ehrhardt M, Günther M, Jacob B (2022) A port-Hamiltonian formulation of coupled heat transfer. Mathematical and Computer Modelling of Dynamical Systems 28(1):78–94. https://doi.org/10.1080/13873954.2022.2038637 – 10.1080/13873954.2022.2038637
• T. Kato. Perturbation theory for linear operators. Corr. printing of the 2nd ed, volume 132 of Grundlehren Math. Wiss. Springer, Cham, 1980.
• 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
• R. Nagel, editor. One-parameter semigroups of positive operators, volume 1184 of Lect. Notes Math. Springer, Cham, 1986.
• Ng ACS (2020) Optimal Energy Decay in a One-Dimensional Wave-Heat-Wave System. In: Springer Proceedings in Mathematics & Statistics. Springer International Publishing, Cham, pp 293–314 – 10.1007/978-3-030-46079-2_17
• Ramirez H, Le Gorrec Y, Macchelli A, Zwart H (2014) Exponential Stabilization of Boundary Controlled Port-Hamiltonian Systems With Dynamic Feedback. IEEE Trans Automat Contr 59(10):2849–2855. https://doi.org/10.1109/tac.2014.2315754 – 10.1109/tac.2014.2315754
• Ramirez H, Zwart H, Le Gorrec Y (2017) Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control. Automatica 85:61–69. https://doi.org/10.1016/j.automatica.2017.07.045 – 10.1016/j.automatica.2017.07.045
• 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
• Schmid J (2022) Stabilization of port-Hamiltonian systems with discontinuous energy densities. EECT 11(5):1775. https://doi.org/10.3934/eect.2021063 – 10.3934/eect.2021063
• J. Schmid and H. Zwart. Stabilization of port-hamiltonian systems by nonlinear boundary control in the presence of disturbances. ESAIM: Control, Optimisation & Calculus of Variations, 26:1 – 37, 2021.
• Schwenninger FL, Zwart H (2014) Generators with a closure relation. Operators and Matrices (1):157–165. https://doi.org/10.7153/oam-08-08 – 10.7153/oam-08-08
• Skrepek N (2021) Well-posedness of linear first order port-Hamiltonian Systems on multidimensional spatial domains. EECT 10(4):965–1006. https://doi.org/10.3934/eect.2020098 – 10.3934/eect.2020098
• S. Trostorff and M. Waurick. Characterisation for exponential stability of port-Hamiltonian systems, 2022.
• van der Schaft A (2007) Port-Hamiltonian systems: an introductory survey. In: Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006. EMS Press, pp 1339–1365 – 10.4171/022-3/65
• J. Villegas. A Port-Hamiltonian Approach to Distributed Parameter Systems. PhD thesis, University of Twente, Netherlands, 2007.
• Villegas JA, Zwart H, Le Gorrec Y, Maschke B (2009) Exponential Stability of a Class of Boundary Control Systems. IEEE Trans Automat Contr 54(1):142–147. https://doi.org/10.1109/tac.2008.2007176 – 10.1109/tac.2008.2007176
• M. Waurick and H. Zwart. Asymptotic stability of port-Hamiltonian systems. 2022. Preprint, available at arXiv:2210.11775.
• Zhou W, Hamroun B, Gorrec YL, Couenne F (2015) Infinite Dimensional Port Hamiltonian Representation of reaction diffusion processes. IFAC-PapersOnLine 48(1):476–481. https://doi.org/10.1016/j.ifacol.2015.05.119 – 10.1016/j.ifacol.2015.05.119
• Zwart H, Le Gorrec Y, Maschke B (2016) Building systems from simple hyperbolic ones. Systems & Control Letters 91:1–6. https://doi.org/10.1016/j.sysconle.2016.02.002 – 10.1016/j.sysconle.2016.02.002
• Zwart H, Le Gorrec Y, Maschke B, Villegas J (2009) Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: COCV 16(4):1077–1093. https://doi.org/10.1051/cocv/2009036 – 10.1051/cocv/2009036