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}
}
References
- Beattie, C., Mehrmann, V., Xu, H. & Zwart, H. Linear port-Hamiltonian descriptor systems. Mathematics of Control, Signals, and Systems vol. 30 (2018) – 10.1007/s00498-018-0223-3
- Castaños, F., Gromov, D., Hayward, V. & Michalska, H. Implicit and explicit representations of continuous-time port-Hamiltonian systems. Systems & Control Letters vol. 62 324–330 (2013) – 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.
- Rashad, R., Califano, F., van der Schaft, A. J. & Stramigioli, S. Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information vol. 37 1400–1422 (2020) – 10.1093/imamci/dnaa018
- van der Schaft, A. J. Implicit Hamiltonian systems with symmetry. Reports on Mathematical Physics vol. 41 203–221 (1998) – 10.1016/s0034-4877(98)80176-6
- van der Schaft, A. J. Port-Hamiltonian Differential-Algebraic Systems. Surveys in Differential-Algebraic Equations I 173–226 (2013) doi:10.1007/978-3-642-34928-7_5 – 10.1007/978-3-642-34928-7_5
- van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
- Altmann, R., Mehrmann, V. & Unger, B. Port-Hamiltonian formulations of poroelastic network models. Mathematical and Computer Modelling of Dynamical Systems vol. 27 429–452 (2021) – 10.1080/13873954.2021.1975137
- Altmann, R. & Schulze, P. A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows. Systems & Control Letters vol. 100 51–55 (2017) – 10.1016/j.sysconle.2016.12.005
- Califano, F., Rashad, R., Schuller, F. P. & Stramigioli, S. Energetic decomposition of distributed systems with moving material domains: The port-Hamiltonian model of fluid-structure interaction. Journal of Geometry and Physics vol. 175 104477 (2022) – 10.1016/j.geomphys.2022.104477
- Cardoso-Ribeiro, F. L., Matignon, D. & Pommier-Budinger, V. Port-Hamiltonian model of two-dimensional shallow water equations in moving containers. IMA Journal of Mathematical Control and Information vol. 37 1348–1366 (2020) – 10.1093/imamci/dnaa016
- Cardoso-Ribeiro, F. L., Matignon, D. & Pommier-Budinger, V. A port-Hamiltonian model of liquid sloshing in moving containers and application to a fluid-structure system. Journal of Fluids and Structures vol. 69 402–427 (2017) – 10.1016/j.jfluidstructs.2016.12.007
- Gernandt, H., Haller, F. E., Reis, T. & Schaft, A. J. van der. Port-Hamiltonian formulation of nonlinear electrical circuits. Journal of Geometry and Physics vol. 159 103959 (2021) – 10.1016/j.geomphys.2020.103959
- Macchelli, A., van der Schaft, A. J. & Melchiorri, C. 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 (2004) doi:10.1109/cdc.2004.1429324 – 10.1109/cdc.2004.1429324
- Maschke, B. M. & van der Schaft, A. J. PORT-CONTROLLED HAMILTONIAN SYSTEMS: MODELLING ORIGINS AND SYSTEMTHEORETIC PROPERTIES. Nonlinear Control Systems Design 1992 359–365 (1993) doi:10.1016/b978-0-08-041901-5.50064-6 – 10.1016/b978-0-08-041901-5.50064-6
- Serhani, A., Haine, G. & Matignon, D. Anisotropic heterogeneous n-D heat equation with boundary control and observation: I. Modeling as port-Hamiltonian system. IFAC-PapersOnLine vol. 52 51–56 (2019) – 10.1016/j.ifacol.2019.07.009
- Vu, N. M. T., Lefèvre, L. & Maschke, B. A structured control model for the thermo-magneto-hydrodynamics of plasmas in tokamaks. Mathematical and Computer Modelling of Dynamical Systems vol. 22 181–206 (2016) – 10.1080/13873954.2016.1154874
- Zhou, W., Hamroun, B., Couenne, F. & Le Gorrec, Y. Distributed port-Hamiltonian modelling for irreversible processes. Mathematical and Computer Modelling of Dynamical Systems vol. 23 3–22 (2016) – 10.1080/13873954.2016.1237970
- Macchelli, A., Gorrec, Y. L., Ramírez, H., Zwart, H. & Califano, F. Control Design for Linear Port-Hamiltonian Boundary Control Systems: An Overview. SEMA SIMAI Springer Series 57–72 (2021) doi:10.1007/978-3-030-61742-4_4 – 10.1007/978-3-030-61742-4_4
- Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM Journal on Control and Optimization vol. 43 743–767 (2004) – 10.1137/s0363012903429530
- Toledo, J., Wu, Y., Ramírez, H. & Le Gorrec, Y. Observer-based boundary control of distributed port-Hamiltonian systems. Automatica vol. 120 109130 (2020) – 10.1016/j.automatica.2020.109130
- Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates. Applied Mathematical Modelling vol. 75 940–960 (2019) – 10.1016/j.apm.2019.04.035
- Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian formulation and symplectic discretization of plate models Part II: Kirchhoff model for thin plates. Applied Mathematical Modelling vol. 75 961–981 (2019) – 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, A. J. & Baños, A. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica vol. 43 212–225 (2007) – 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. Dirac structures and their composition on Hilbert spaces. Journal of Mathematical Analysis and Applications vol. 372 402–422 (2010) – 10.1016/j.jmaa.2010.07.004
- Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 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. First-order Hamiltonian field theory and mechanics. Mathematical and Computer Modelling of Dynamical Systems vol. 17 105–121 (2011) – 10.1080/13873954.2010.537526
- Schöberl, M. & Siuka, A. Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators. Automatica vol. 50 607–613 (2014) – 10.1016/j.automatica.2013.11.035
- Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 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.
- Yoshimura, H. & Marsden, J. E. Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems. Journal of Geometry and Physics vol. 57 133–156 (2006) – 10.1016/j.geomphys.2006.02.009
- Jiménez, F. & Yoshimura, H. Dirac structures in vakonomic mechanics. Journal of Geometry and Physics vol. 94 158–178 (2015) – 10.1016/j.geomphys.2014.11.002
- Schöberl, M. & Schlacher, K. On the extraction of the boundary conditions and the boundary ports in second-order field theories. Journal of Mathematical Physics vol. 59 (2018) – 10.1063/1.5024847
- van der Schaft, A. J. & Maschke, B. M. Port-Hamiltonian Systems on Graphs. SIAM Journal on Control and Optimization vol. 51 906–937 (2013) – 10.1137/110840091
- van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
- 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
- B. Jacob, H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Operator Theory: Advances and Applications, Birkhäuser Basel, 2012.
- Skrepek, N. Well-posedness of linear first order port-Hamiltonian Systems on multidimensional spatial domains. Evolution Equations & Control Theory vol. 10 965 (2021) – 10.3934/eect.2020098
- Gay-Balmaz, F. & Yoshimura, H. A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems. Journal of Geometry and Physics vol. 111 169–193 (2017) – 10.1016/j.geomphys.2016.08.018
- Gay-Balmaz, F. & Yoshimura, H. A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: Continuum systems. Journal of Geometry and Physics vol. 111 194–212 (2017) – 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.
- M. Kurula, H. Zwart, Linear wave systems on n-D spatial domains, International Journal of Control, 88 (2015), 1063–1077.
- Salamon, D. Infinite Dimensional Linear Systems With Unbounded Control and Observation: A Functional Analytic Approach. Transactions of the American Mathematical Society vol. 300 383 (1987) – 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.
- Tucsnak, M. & Weiss, G. Well-posed systems—The LTI case and beyond. Automatica vol. 50 1757–1779 (2014) – 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, D. N., Falk, R. S. & Winther, R. Finite element exterior calculus, homological techniques, and applications. Acta Numerica vol. 15 1–155 (2006) – 10.1017/s0962492906210018
- Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1993). doi:10.1007/978-1-4612-4350-2 – 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. Boundary triplets for skew-symmetric operators and the generation of strongly continuous semigroups. Analysis Mathematica vol. 43 657–686 (2017) – 10.1007/s10476-017-0509-6
- Monk, P. Finite Element Methods for Maxwell’s Equations. (2003) doi:10.1093/acprof:oso/9780198508885.001.0001 – 10.1093/acprof:oso/9780198508885.001.0001
- Arnold, D. N. & Hu, K. Complexes from Complexes. Foundations of Computational Mathematics vol. 21 1739–1774 (2021) – 10.1007/s10208-021-09498-9
- Pauly, D. & Zulehner, W. The divDiv-complex and applications to biharmonic equations. Applicable Analysis vol. 99 1579–1630 (2018) – 10.1080/00036811.2018.1542685
- Pauly, D. & Zulehner, W. The elasticity complex: compact embeddings and regular decompositions. Applicable Analysis vol. 102 4393–4421 (2022) – 10.1080/00036811.2022.2117497
- Amara, M., Capatina-Papaghiuc, D. & Chatti, A. Bending Moment Mixed Method for the Kirchhoff–Love Plate Model. SIAM Journal on Numerical Analysis vol. 40 1632–1649 (2002) – 10.1137/s0036142900379680
- Weiss, G. & Staffans, O. J. Maxwell’s Equations as a Scattering Passive Linear System. SIAM Journal on Control and Optimization vol. 51 3722–3756 (2013) – 10.1137/120869444
- Cardoso-Ribeiro, F. L., Matignon, D. & Lefèvre, L. A partitioned finite element method for power-preserving discretization of open systems of conservation laws. IMA Journal of Mathematical Control and Information vol. 38 493–533 (2020) – 10.1093/imamci/dnaa038
- Ghislain Haine, G. H., Denis Matignon, D. M. & Anass Serhani, A. S. 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 vol. 20 92–133 (2023) – 10.4208/ijnam2023-1005
- Brugnoli, A., Rashad, R. & Stramigioli, S. Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus. Journal of Computational Physics vol. 471 111601 (2022) – 10.1016/j.jcp.2022.111601