Stokes-Lagrange and Stokes-Dirac representations of N-dimensional port-Hamiltonian systems for modeling and control

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Abstract

In this paper, we extended the port-Hamiltonian framework by introducing the concept of Stokes-Lagrange structure, which enables the implicit definition of a Hamiltonian over an N-dimensional domain and incorporates energy ports into the system. This new framework parallels the existing Dirac and Stokes-Dirac structures. We proposed the Stokes-Lagrange structure as a specific case where the subspace is explicitly defined via differential operators that satisfy an integration-by-parts formula. By examining various examples through the lens of the Stokes-Lagrange structure, we demonstrated the existence of multiple equivalent system representations. These representations provide significant advantages for both numerical simulation and control design, offering additional tools for the modeling and control of port-Hamiltonian systems.

Keywords

port-Hamiltonian systems / Stokes-Dirac structure / Stokes-Lagrange structure / Reissner-Mindlin model / Kirchhoff-Love model / Maxwell’s equations / passivity-based control

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References

[1] A. van der Schaft, D. Jeltsema, Port-Hamiltonian systems theory: An introductory overview, Found. Trends Syst. Control., 1 (2014), 173-378. https://doi.org/10.1561/2600000002

[2] V. Duindam, A. Macchelli, S. Stramigioli, H. Bruyninckx, Modeling and control of com- plex physical systems: the port-Hamiltonian approach, Springer Berlin, Heidelberg, 2009. https://doi.org/10.1007/978-3-642-03196-0

[3] F. Gay-Balmaz, H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: Continuum systems, J. Geom. Phys., 111 (2017), 194-212. https://doi.org/10.1016/j.geomphys.2016.08.019

[4] A. van der Schaft, B. Maschke, Hamiltonian formulation of distributed-parameter systems with boundary energy flow, J. Geom. Phys., 42 (2002), 166-194. https://doi.org/10.1016/S0393- 0440(01)00083-3

[5] B. Jacob, H. J. Zwart, Linear port-Hamiltonian systems on infinite-dimensional spaces, Springer, 2012. https://doi.org/10.1007/978-3-0348-0399-1

[6] A. Brugnoli, G. Haine, D. Matignon, Stokes-Dirac structures for distributed parameter port- Hamiltonian systems: An analytical viewpoint, Commun. Anal. Mech., 15 (2023), 362-387. https://doi.org/10.3934/cam.2023018

[7] R. Rashad, F. Califano, A. J. van der Schaft, S. Stramigioli, Twenty years of distributed port- Hamiltonian systems: A literature review, IMA J. Math. Control Inf., 37 (2020), 1400-1422. https://doi.org/10.1093/imamci/dnaa018

[8] C. Beattie, V. Mehrmann, H. Xu, H. Zwart, Linear port-Hamiltonian descriptor systems, Math. Control Signals Syst., 30 (2018), 1-27. https://doi.org/10.1007/s00498-018-0223-3

[9] H. Zwart, V. Mehrmann, Abstract dissipative Hamiltonian differential-dlgebraic equations are everywhere, DAE Panel, 2 (2014). https://doi.org/10.52825/dae-p.v2i.957

[10] V. Mehrmann, B. Unger, Control of port-Hamiltonian differential-algebraic systems and applica- tions, Acta Numer., 32 (2023), 395-515. https://doi.org/10.1017/S0962492922000083

[11] A. van der Schaft, B. Maschke Generalized port-Hamiltonian DAE systems, Systems Control Lett., 121 (2018), 31-37. https://doi.org/10.1016/j.sysconle.2018.09.008

[12] A. van der Schaft, B. Maschke, Dirac and Lagrange algebraic constraints in nonlinear port- Hamiltonian systems, Vietnam J. Math., 48 (2020), 929-939. https://doi.org/10.1007/s10013-020- 00419-x

[13] M. Yaghi, F. Couenne, A. Galfré, L. Lefèvre, B. Maschke, Port-Hamiltonian formulation of the solidification process for a pure substance: A phase field approach, IFAC-PapersOnLine, 55 (2022), 93-98. https://doi.org/10.1016/j.ifacol.2022.08.036

[14] B. Jacob, K. Morris, On solvability of dissipative partial differential-algebraic equations, IEEE Control Syst. Lett, 6 (2022), 3188-3193. https://doi.org/10.1109/LCSYS.2022.3183479

[15] H. Heidari, H. Zwart, Port-Hamiltonian modelling of nonlocal longitudinal vibra- tions in a viscoelastic nanorod, Math. Comp. Model. Dyn., 25 (2019), 447-462. https://doi.org/10.1080/13873954.2019.1659374

[16] H. Heidari, H. Zwart, Nonlocal longitudinal vibration in a nanorod, a system theoretic analysis, Math. Model. Nat. Phenom., 17 (2022), 24. https://doi.org/10.1051/mmnp/2022028

[17] A. Bendimerad-Hohl, G. Haine, L. Lefèvre, D. Matignon, Implicit port-Hamiltonian systems: structure-preserving discretization for the nonlocal vibrations in a viscoelastic nanorod, and for a seepage model, IFAC-PapersOnLine, 56 (2023), 6789-6795.

[18] B. Maschke, A. van der Schaft, Linear boundary port-Hamiltonian systems with implicitly defined energy, arXiv preprint arXiv:2305.13772. https://doi.org/10.48550/arXiv.2305.13772

[19] K. Krhač, B. Maschke, A. van der Schaft, Port-Hamiltonian systems with energy and power ports, IFAC-PapersOnLine, 58 (2024), 280-285. https://doi.org/10.1016/j.ifacol.2024.08.294

[20] A. van der Schaft, B. Maschke, Boundary control port-Hamiltonian systems with power and energy ports, in 26th International Symposium on Mathematical Theory of Networks and Systems, Cambridge, UK, 2024, 120-122.

[21] M. Schöberl, K. Schlacher, First-order Hamiltonian field theory and mechanics, Math. Comp. Model. Dyn., 17 (2011), 105-121. https://doi.org/10.1080/13873954.2010.537526

[22] M. Schöberl, A. Siuka, Analysis and comparison of port-Hamiltonian formulations for field theories - demonstrated by means of the Mindlin plate, in 2013 European Control Conference (ECC), 2013, 548-553. https://doi.org/10.23919/ECC.2013.6669137

[23] M. Schöberl, A. Siuka, Jet bundle formulation of infinite-dimensional port- Hamiltonian systems using differential operators, Automatica, 50 (2014), 607-613. https://doi.org/10.1016/j.automatica.2013.11.035

[24] A. Bendimerad-Hohl, D. Matignon, G. Haine, L. Lefèvre, On Stokes-Lagrange and Stokes-Dirac representations for 1D distributed port-Hamiltonian systems, IFAC-PapersOnLine, 58 (2024), 238-243. https://doi.org/10.1016/j.ifacol.2024.10.174

[25] M. Kurula, H. Zwart, A. van der Schaft, J. Behrndt, Dirac structures and their composition on Hilbert spaces, J. Math. Anal. Appl., 372 (2010), 402-422. https://doi.org/10.1016/j.jmaa.2010.07.004

[26] G. Haine, D. Matignon, A. Serhani, Numerical analysis of a structure-preserving space- discretization for an anisotropic and heterogeneous boundary controlled N-dimensional wave equation as a port-Hamiltonian system , Int. J. Numer. Anal. Model., 20 (2023), 92-133. https://doi.org/10.4208/ijnam2023-1005

[27] P. J. Olver, Applications of Lie groups to differential equations, New York: Springer-Verlag, 1993. https://doi.org/10.1007/978-1-4612-4350-2

[28] A. C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., 54 (1983), 4703-4710. https://doi.org/10.1063/1.332803

[29] G. Romano, R. Barretta, M. Diaco, F. M. de Sciarra, Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, Int. J. Mech. Sci., 121 (2017), 151-156. https://doi.org/10.1016/j.ijmecsci.2016.10.036

[30] W. Rudin, Functional analysis, 2nd edition, International series in pure and applied mathematics, McGraw-Hill, 1991.

[31] T. Preuster, B. Maschke, M. Schaller, Jet space extensions of infinite-dimensional Hamiltonian systems, IFAC-PapersOnLine, 58 (2024), 298-303. https://doi.org/10.1016/j.ifacol.2024.08.297

[32] M. Schöberl, K. Schlacher, Lagrangian and port-Hamiltonian formulation for distributed-parameter systems, IFAC-PapersOnLine, 48 (2015), 610-615. https://doi.org/10.1016/j.ifacol.2015.05.025

[33] A. van der Schaft, B. Maschke, Differential operator Dirac structures, IFAC-PapersOnLine, 54 (2021), 198-203. https://doi.org/10.1016/j.ifacol.2021.11.078

[34] A. F. Gangi, Constitutive equations and reciprocity, Geophys. J. Int., 143 (2020), 311-318. https://doi.org/10.1046/j.1365-246X.2000.01259.x

[35] H. Gernandt, F. M. Philipp, T. Preuster, M. Schaller, On the equivalence of geometric and descriptor representations of linear port-Hamiltonian systems, in Systems Theory and PDEs , Springer, 2022, 149-165. https://doi.org/10.1007/978-3-031-64991-2 6

[36] V. Mehrmann, A. van der Schaft, Differential-algebraic systems with dissipative Hamiltonian structure, Math. Control Signals Syst., 35 (2023), 541-584. https://doi.org/10.1007/s00498-023- 00349-2

[37] J. Bognár, Indefinite inner product spaces , Springer Berlin, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-65567-8

[38] R. Ortega, A. van der Schaft, F. Castanos, A. Astolfi, Control by interconnection and standard passivity-based control of port-Hamiltonian systems, IEEE Trans. Autom. Control., 53 (2008), 2527-2542. https://doi.org/10.1109/TAC.2008.2006930

[39] M. Erbay, B. Jacob, K. Morris, On the Weierstraß form of infinite-dimensional differential algebraic equations, J. Evol. Equ., 24 (2024), 73. https://doi.org/10.1007/s00028-024-01003-3

[40] E. Hairer, C. Lubich, G. Wanner, Geometric numerical integration: structure-preserving algo- rithms for ordinary differential equations: structure-preserving algorithms for ordinary differential equations , Springer Berlin, Heidelberg, 2006. https://doi.org/10.1007/3-540-30666-8

[41] R. Altmann, P. Schulze, A novel energy-based modeling framework, Math. Control Signals Syst., 2025. https://doi.org/10.1007/s00498-024-00405-5

[42] P. Destuynder, M. Salaün, Mathematical analysis of thin plate models , Springer Berlin, Heidelberg, 2013. https://doi.org/10.1007/978-3-642-51761-7

[43] R. D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, J. Appl. Mech., 18 (1951), 31-38. https://doi.org/10.1115/1.4010217

[44] A. Brugnoli, D. Alazard, V. Pommier-Budinger, D. Matignon, Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates, Appl. Math. Model., 75 (2019), 940-960. https://doi.org/10.1016/j.apm.2019.04.035

[45] C. Ponce, Y. Wu, Y. Le Gorrec, H. Ramirez, A systematic methodology for port-Hamiltonian modeling of multidimensional flexible linear mechanical systems, Appl. Math. Model., 134 (2024), 434-451. https://doi.org/10.1016/j.apm.2024.05.040

[46] A. Brugnoli, D. Alazard, V. Pommier-Budinger, D. Matignon, Port-Hamiltonian formulation and symplectic discretization of plate models Part II: Kirchhoff model for thin plates, Appl. Math. Model., 75 (2019), 961-981. https://doi.org/10.1016/j.apm.2019.04.036

[47] A. Kovetz, Electromagnetic theory , Oxford University Press, 2020. https://doi.org/10.1119/1.1371014

[48] D. N. Arnold, Finite element exterior calculus , SIAM, 2018. https://doi.org/10.1137/1.9781611975543

[49] A. Buffa, H. Ammari, J. C. Nédélec, A justification of eddy currents model for the Maxwell equa- tions, SIAM J. Appl. Math., 60 (2020), 1805-1823. https://doi.org/10.1137/S0036139998348979

[50] T. Reis, T. Stykel, Passivity, port-Hamiltonian formulation and solution estimates for a cou- pled magneto-quasistatic system, Evol. Equations Control Theory, 12 (2023), 1208-1232. https://doi.org/10.3934/eect.2023008

[51] F. Assous, P. Ciarlet, S. Labrunie, Mathematical foundations of computational electromagnetism , Springer, 2018 https://doi.org/10.1007/978-3-319-70842-3

[52] F. L. Cardoso-Ribeiro, G. Haine, Y. Le Gorrec, D. Matignon, H. Ramirez, Port-Hamiltonian formulations for the modeling, simulation and control of fluids, Comput. Fluids., 283 (2024), 106407. https://doi.org/10.1016/j.compfluid.2024.106407

[53] G. Haine, D. Matignon Incompressible Navier-Stokes Equation as port-Hamiltonian system: velocity formulation versus vorticity formulation, IFAC-PapersOnLine, 54 (2021), 161-166. https://doi.org/10.1016/j.ifacol.2021.11.072

[54] E. S. Dzektser, Generalization of the equation of motion of ground waters with free surface, Dokl. Akad. Nauk SSSR, 202 (1972), 1031-1033.

[55] P. Borja, J. Ferguson, A. van der Schaft, Interconnection schemes in modeling and control, IEEE Control Syst. Lett, 7 (2023), 2287-2292. https://doi.org/10.1109/LCSYS.2023.3286124

[56] A. Macchelli, A. van der Schaft, C. Melchiorri, Multi-variable port-Hamiltonian model of piezo- electric material, in 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566), 2004, 897-- 902. https://doi.org/10.1109/IROS.2004.1389466

[57] A. Macchelli, Energy shaping of distributed parameter port-Hamiltonian systems based on finite element approximation, Systems Control Lett., 60 (2011), 579-589. https://doi.org/10.1016/j.sysconle.2011.04.016

[58] H. Gernandt, F. Philipp, T. Preuster, M. Schaller, Extension theory via boundary triplets for infinite-dimensional implicit port-Hamiltonian systems, arXiv preprint arXiv:2503.17874.