Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint

Authors

Andrea Brugnoli, Ghislain Haine, Denis Matignon

Abstract

In this paper, we prove that a large class of linear evolution partial differential equations defines a Stokes-Dirac structure over Hilbert spaces. To do so, the theory of boundary control system is employed. This definition encompasses problems from mechanics that cannot be handled by the geometric setting given in the seminal paper by van der Schaft and Maschke in 2002. Many worked-out examples stemming from continuum mechanics and physics are presented in detail, and a particular focus is given to the functional spaces in duality at the boundary of the geometrical domain. For each example, the connection between the differential operators and the associated Hilbert complexes is illustrated.

Citation

• Journal: Communications in Analysis and Mechanics
• Year: 2023
• Volume: 15
• Issue: 3
• Pages: 362–387
• Publisher: American Institute of Mathematical Sciences (AIMS)
• DOI: 10.3934/cam.2023018

BibTeX

@article{Brugnoli_2023,
  title={{Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint}},
  volume={15},
  ISSN={2836-3310},
  DOI={10.3934/cam.2023018},
  number={3},
  journal={Communications in Analysis and Mechanics},
  publisher={American Institute of Mathematical Sciences (AIMS)},
  author={Brugnoli, Andrea and Haine, Ghislain and Matignon, Denis},
  year={2023},
  pages={362--387}
}

Download the bib file

References

• Beattie C, Mehrmann V, Xu H, Zwart H (2018) Linear port-Hamiltonian descriptor systems. Math Control Signals Syst 30(4):17. https://doi.org/10.1007/s00498-018-0223-3 – 10.1007/s00498-018-0223-3
• Castaños F, Gromov D, Hayward V, Michalska H (2013) Implicit and explicit representations of continuous-time port-Hamiltonian systems. Systems & Control Letters 62(4):324–330. https://doi.org/10.1016/j.sysconle.2013.01.007 – 10.1016/j.sysconle.2013.01.007
• V. Duindam, A. Macchelli, S. Stramigioli, H. Bruyninckx, Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach, Springer-Verlag, Berlin Heidelberg, 2009. https://doi.org/10.1007/978-3-642-03196-0
• Rashad R, Califano F, van der Schaft AJ, Stramigioli S (2020) Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information 37(4):1400–1422. https://doi.org/10.1093/imamci/dnaa018 – 10.1093/imamci/dnaa018
• van der Schaft AJ (1998) Implicit Hamiltonian systems with symmetry. Reports on Mathematical Physics 41(2):203–221. https://doi.org/10.1016/s0034-4877(98)80176-6 – 10.1016/s0034-4877(98)80176-6
• van der Schaft AJ (2013) Port-Hamiltonian Differential-Algebraic Systems. In: Surveys in Differential-Algebraic Equations I. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 173–226 – 10.1007/978-3-642-34928-7_5
• 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
• Altmann R, Mehrmann V, Unger B (2021) Port-Hamiltonian formulations of poroelastic network models. Mathematical and Computer Modelling of Dynamical Systems 27(1):429–452. https://doi.org/10.1080/13873954.2021.1975137 – 10.1080/13873954.2021.1975137
• Altmann R, Schulze P (2017) A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows. Systems & Control Letters 100:51–55. https://doi.org/10.1016/j.sysconle.2016.12.005 – 10.1016/j.sysconle.2016.12.005
• Califano F, Rashad R, Schuller FP, Stramigioli S (2022) Energetic decomposition of distributed systems with moving material domains: The port-Hamiltonian model of fluid-structure interaction. Journal of Geometry and Physics 175:104477. https://doi.org/10.1016/j.geomphys.2022.104477 – 10.1016/j.geomphys.2022.104477
• Cardoso-Ribeiro FL, Matignon D, Pommier-Budinger V (2020) Port-Hamiltonian model of two-dimensional shallow water equations in moving containers. IMA Journal of Mathematical Control and Information 37(4):1348–1366. https://doi.org/10.1093/imamci/dnaa016 – 10.1093/imamci/dnaa016
• Cardoso-Ribeiro FL, Matignon D, Pommier-Budinger V (2017) A port-Hamiltonian model of liquid sloshing in moving containers and application to a fluid-structure system. Journal of Fluids and Structures 69:402–427. https://doi.org/10.1016/j.jfluidstructs.2016.12.007 – 10.1016/j.jfluidstructs.2016.12.007
• 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
• Macchelli A, van der Schaft AJ, Melchiorri C (2004) Port Hamiltonian formulation of infinite dimensional systems I. Modeling. In: 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601). IEEE, pp 3762–3767 Vol.4 – 10.1109/cdc.2004.1429324
• Maschke BM, van der Schaft AJ (1993) PORT-CONTROLLED HAMILTONIAN SYSTEMS: MODELLING ORIGINS AND SYSTEMTHEORETIC PROPERTIES. In: Nonlinear Control Systems Design 1992. Elsevier, pp 359–365 – 10.1016/b978-0-08-041901-5.50064-6
• Serhani A, Haine G, Matignon D (2019) Anisotropic heterogeneous n-D heat equation with boundary control and observation: I. Modeling as port-Hamiltonian system. IFAC-PapersOnLine 52(7):51–56. https://doi.org/10.1016/j.ifacol.2019.07.009 – 10.1016/j.ifacol.2019.07.009
• Vu NMT, Lefèvre L, Maschke B (2016) A structured control model for the thermo-magneto-hydrodynamics of plasmas in tokamaks. Mathematical and Computer Modelling of Dynamical Systems 22(3):181–206. https://doi.org/10.1080/13873954.2016.1154874 – 10.1080/13873954.2016.1154874
• Zhou W, Hamroun B, Couenne F, Le Gorrec Y (2016) Distributed port-Hamiltonian modelling for irreversible processes. Mathematical and Computer Modelling of Dynamical Systems 23(1):3–22. https://doi.org/10.1080/13873954.2016.1237970 – 10.1080/13873954.2016.1237970
• Macchelli A, Gorrec YL, Ramírez H, Zwart H, Califano F (2021) Control Design for Linear Port-Hamiltonian Boundary Control Systems: An Overview. In: SEMA SIMAI Springer Series. Springer International Publishing, Cham, pp 57–72 – 10.1007/978-3-030-61742-4_4
• Macchelli A, Melchiorri C (2004) Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM J Control Optim 43(2):743–767. https://doi.org/10.1137/s0363012903429530 – 10.1137/s0363012903429530
• Toledo J, Wu Y, Ramírez H, Le Gorrec Y (2020) Observer-based boundary control of distributed port-Hamiltonian systems. Automatica 120:109130. https://doi.org/10.1016/j.automatica.2020.109130 – 10.1016/j.automatica.2020.109130
• Brugnoli A, Alazard D, Pommier-Budinger V, Matignon D (2019) Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates. Applied Mathematical Modelling 75:940–960. https://doi.org/10.1016/j.apm.2019.04.035 – 10.1016/j.apm.2019.04.035
• Brugnoli A, Alazard D, Pommier-Budinger V, Matignon D (2019) Port-Hamiltonian formulation and symplectic discretization of plate models Part II: Kirchhoff model for thin plates. Applied Mathematical Modelling 75:961–981. https://doi.org/10.1016/j.apm.2019.04.036 – 10.1016/j.apm.2019.04.036
• A. Brugnoli, A port-Hamiltonian formulation of flexible structures. Modelling and structure-preserving finite element discretization, PhD thesis, Université de Toulouse, ISAE-SUPAERO, 2020.
• A. Serhani, Systèmes couplés d’EDPs, vus comme des systèmes Hamiltoniens à ports avec dissipation : Analyse théorique et simulation numérique, PhD thesis, Université de Toulouse, ISAE-SUPAERO, 2020.
• 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
• G. Haine, D. Matignon, F. Monteghetti, Long-time behavior of a coupled heat-wave system using a structure-preserving finite element method, Math. Rep., 24 (2022), 187–215.
• 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
• Ortega R, van der Schaft A, Maschke B, Escobar G (2002) Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38(4):585–596. https://doi.org/10.1016/s0005-1098(01)00278-3 – 10.1016/s0005-1098(01)00278-3
• A. J. van der Schaft, Interconnection and geometry, in The Mathematics of Systems and Control: from Intelligent Control to Behavioral Systems, University of Groningen, 1999,203–218.
• Schöberl M, Schlacher K (2011) First-order Hamiltonian field theory and mechanics. Mathematical and Computer Modelling of Dynamical Systems 17(1):105–121. https://doi.org/10.1080/13873954.2010.537526 – 10.1080/13873954.2010.537526
• Schöberl M, Siuka A (2014) Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators. Automatica 50(2):607–613. https://doi.org/10.1016/j.automatica.2013.11.035 – 10.1016/j.automatica.2013.11.035
• Courant TJ (1990) Dirac manifolds. Trans Amer Math Soc 319(2):631–661. https://doi.org/10.1090/s0002-9947-1990-0998124-1 – 10.1090/s0002-9947-1990-0998124-1
• G. Nishida, M. Yamakita, A higher order Stokes-Dirac structure for distributed-parameter port-Hamiltonian systems, in Proceedings of the 2004 American Control Conference (ACC), vol. 6, 2004, 5004–5009. https://doi.org/10.23919/ACC.2004.1384643
• Yoshimura H, Marsden JE (2006) Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems. Journal of Geometry and Physics 57(1):133–156. https://doi.org/10.1016/j.geomphys.2006.02.009 – 10.1016/j.geomphys.2006.02.009
• Jiménez F, Yoshimura H (2015) Dirac structures in vakonomic mechanics. Journal of Geometry and Physics 94:158–178. https://doi.org/10.1016/j.geomphys.2014.11.002 – 10.1016/j.geomphys.2014.11.002
• Schöberl M, Schlacher K (2018) On the extraction of the boundary conditions and the boundary ports in second-order field theories. Journal of Mathematical Physics 59(10):102902. https://doi.org/10.1063/1.5024847 – 10.1063/1.5024847
• van der Schaft AJ, Maschke BM (2013) Port-Hamiltonian Systems on Graphs. SIAM J Control Optim 51(2):906–937. https://doi.org/10.1137/110840091 – 10.1137/110840091
• 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
• 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
• B. Jacob, H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Operator Theory: Advances and Applications, Birkhäuser Basel, 2012. https://doi.org/10.1007/978-3-0348-0399-1
• 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
• Gay-Balmaz F, Yoshimura H (2017) A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems. Journal of Geometry and Physics 111:169–193. https://doi.org/10.1016/j.geomphys.2016.08.018 – 10.1016/j.geomphys.2016.08.018
• Gay-Balmaz F, Yoshimura H (2017) A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: Continuum systems. Journal of Geometry and Physics 111:194–212. https://doi.org/10.1016/j.geomphys.2016.08.019 – 10.1016/j.geomphys.2016.08.019
• R. F. Curtain, G. Weiss, Well-posedness of triples of operators (in the sense of linear systems theory), in Control and Estimation of Distributed Parameter Systems (Vorau, 1988), Birkhäuser Basel, 1989, 41–59. https://doi.org/10.1090/conm/097/1021030
• M. Kurula, H. Zwart, Linear wave systems on n-D spatial domains, International Journal of Control, 88 (2015), 1063–1077.
• Salamon D (1987) Infinite Dimensional Linear Systems With Unbounded Control and Observation: A Functional Analytic Approach. Transactions of the American Mathematical Society 300(2):383. https://doi.org/10.2307/2000351 – 10.2307/2000351
• O. J. Staffans, Well-posed linear systems, vol. 103 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2005.
• M. Tucsnak, G. Weiss, Observation and control for operator semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. https://doi.org/10.1007/978-3-7643-8994-9
• Tucsnak M, Weiss G (2014) Well-posed systems—The LTI case and beyond. Automatica 50(7):1757–1779. https://doi.org/10.1016/j.automatica.2014.04.016 – 10.1016/j.automatica.2014.04.016
• G. Weiss, O. J. Staffans and M. Tucsnak, Well-posed linear systems - a survey with emphasis on conservative systems, Int. J. Ap. Mat. Com-Pol., 11 (2001), 7–33.
• Arnold DN, Falk RS, Winther R (2006) Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15:1–155. https://doi.org/10.1017/s0962492906210018 – 10.1017/s0962492906210018
• Olver PJ (1993) Applications of Lie Groups to Differential Equations. Springer New York, New York, NY – 10.1007/978-1-4612-4350-2
• M. Renardy, R. C. Rogers, An introduction to partial differential equations, vol. 13 of Texts in Applied Mathematics, Springer Science & Business Media, 2006.
• R. Rashad, A. Brugnoli, F. Califano, E. Luesink, S. Stramigioli, Intrinsic nonlinear elasticity: An exterior calculus formulation, arXiv preprint arXiv: 2303.06082.
• Wegner S-A (2017) Boundary triplets for skew-symmetric operators and the generation of strongly continuous semigroups. Anal Math 43(4):657–686. https://doi.org/10.1007/s10476-017-0509-6 – 10.1007/s10476-017-0509-6
• Monk P (2003) Finite Element Methods for Maxwell’s Equations. Oxford University Press – 10.1093/acprof:oso/9780198508885.001.0001
• Arnold DN, Hu K (2021) Complexes from Complexes. Found Comput Math 21(6):1739–1774. https://doi.org/10.1007/s10208-021-09498-9 – 10.1007/s10208-021-09498-9
• Pauly D, Zulehner W (2018) The divDiv-complex and applications to biharmonic equations. Applicable Analysis 99(9):1579–1630. https://doi.org/10.1080/00036811.2018.1542685 – 10.1080/00036811.2018.1542685
• Pauly D, Zulehner W (2022) The elasticity complex: compact embeddings and regular decompositions. Applicable Analysis 102(16):4393–4421. https://doi.org/10.1080/00036811.2022.2117497 – 10.1080/00036811.2022.2117497
• Amara M, Capatina-Papaghiuc D, Chatti A (2002) Bending Moment Mixed Method for the Kirchhoff–Love Plate Model. SIAM J Numer Anal 40(5):1632–1649. https://doi.org/10.1137/s0036142900379680 – 10.1137/s0036142900379680
• Weiss G, Staffans OJ (2013) Maxwell’s Equations as a Scattering Passive Linear System. SIAM J Control Optim 51(5):3722–3756. https://doi.org/10.1137/120869444 – 10.1137/120869444
• Cardoso-Ribeiro FL, Matignon D, Lefèvre L (2020) A partitioned finite element method for power-preserving discretization of open systems of conservation laws. IMA Journal of Mathematical Control and Information 38(2):493–533. https://doi.org/10.1093/imamci/dnaa038 – 10.1093/imamci/dnaa038
• Haine G, Matignon D, Serhani A (2022) Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation as a Port-Hamiltonian System. International Journal of Numerical Analysis and Modeling 20(1):92–133. https://doi.org/10.4208/ijnam2023-1005 – 10.4208/ijnam2023-1005
• Brugnoli A, Rashad R, Stramigioli S (2022) Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus. Journal of Computational Physics 471:111601. https://doi.org/10.1016/j.jcp.2022.111601 – 10.1016/j.jcp.2022.111601