Port Hamiltonian formulation of a system of two conservation laws with a moving interface

Authors

Mamadou Diagne, Bernhard Maschke

Abstract

In this paper we consider the port Hamiltonian formulation of systems of two conservation laws defined on two complementary intervals of some interval of the real line and coupled by some moving interface. We recall first how two port Hamiltonian systems coupled by an interface may be expressed as an port Hamiltonian systems augmented with two variables being the characteristic functions of the two spatial domains. Then we consider the case of a moving interface and show that it may be expressed as the previous port Hamiltonian system augmented with an input, being the velocity of the interface and its conjugated output variable. We then discuss the interface relations defining the dynamics of the displacement of the interface and give an illustration with the simple example of two gases coupled by a moving piston.

Keywords

Boundary port Hamiltonian systems; PDE’s; Moving interface; Dirac structure

Citation

• Journal: European Journal of Control
• Year: 2013
• Volume: 19
• Issue: 6
• Pages: 495–504
• Publisher: Elsevier BV
• DOI: 10.1016/j.ejcon.2013.09.001
• Note: Lagrangian and Hamiltonian Methods for Modelling and Control

BibTeX

@article{Diagne_2013,
  title={{Port Hamiltonian formulation of a system of two conservation laws with a moving interface}},
  volume={19},
  ISSN={0947-3580},
  DOI={10.1016/j.ejcon.2013.09.001},
  number={6},
  journal={European Journal of Control},
  publisher={Elsevier BV},
  author={Diagne, Mamadou and Maschke, Bernhard},
  year={2013},
  pages={495--504}
}

Download the bib file

References

• Abgrall, R. & Karni, S. Computations of Compressible Multifluids. Journal of Computational Physics vol. 169 594–623 (2001) – 10.1006/jcph.2000.6685
• Ambroso, A., Boutin, B., Coquel, F., Godlewski, E. & LeFloch, P. G. Coupling Two Scalar Conservation Laws via Dafermos’ Self-Similar Regularization. Numerical Mathematics and Advanced Applications 209–216 doi:10.1007/978-3-540-69777-0_24 – 10.1007/978-3-540-69777-0_24
• Ambroso, A. et al. A Relaxation Method for the Coupling of Systems of Conservation Laws. Hyperbolic Problems: Theory, Numerics, Applications 947–954 (2008) doi:10.1007/978-3-540-75712-2_99 – 10.1007/978-3-540-75712-2_99
• Boutin, Dafermos regularization for interface coupling of conservation laws. Numerics, Applications: Theory (2008)
• Boutin, B., Coquel, F. & Godlewski, E. Dafermos Regularization for Interface Coupling of Conservation Laws. Hyperbolic Problems: Theory, Numerics, Applications 567–575 (2008) doi:10.1007/978-3-540-75712-2_55 – 10.1007/978-3-540-75712-2_55
• 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
• Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• Diagne, M., Dos Santos Martins, V., Couenne, F. & Maschke, B. Well posedness of the model of an extruder in infinite dimension. IEEE Conference on Decision and Control and European Control Conference 1311–1316 (2011) doi:10.1109/cdc.2011.6161288 – 10.1109/cdc.2011.6161288
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• Godlewski, E., Le Thanh, K.-C. & Raviart, P.-A. The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: II. The case of systems. ESAIM: Mathematical Modelling and Numerical Analysis vol. 39 649–692 (2005) – 10.1051/m2an:2005029
• Godlewski, E. & Raviart, P.-A. The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. The scalar case. Numerische Mathematik vol. 97 81–130 (2004) – 10.1007/s00211-002-0438-5
• Hassou, M., Couenne, F., le Gorrec, Y. & Tayakout, M. Modeling and simulation of polymeric nanocapsule formation by emulsion diffusion method. AIChE Journal vol. 55 2094–2105 (2009) – 10.1002/aic.11809
• Jacob, (2012)
• 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
• 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
• Macchelli, A. & Melchiorri, C. Control by interconnection of mixed port Hamiltonian systems. IEEE Transactions on Automatic Control vol. 50 1839–1844 (2005) – 10.1109/tac.2005.858656
• Mader, G., Fösel, G. P. F. & Larsen, L. F. S. Comparison of the transient behavior of microchannel and fin-and-tube evaporators. International Journal of Refrigeration vol. 34 1222–1229 (2011) – 10.1016/j.ijrefrig.2011.01.011
• Maschke, B. & van der Schaft, A. 4 Compositional Modelling of Distributed-Parameter Systems. Lecture Notes in Control and Information Sciences 115–154 (2005) doi:10.1007/11334774_4 – 10.1007/11334774_4
• Olver, (1993)
• Ortega, R., de Rinaldis, A., Spong, M. W., Lee, S. & Nam, K. On Compensation of Wave Reflections in Transmission Lines and Applications to the Overvoltage Problem AC Motor Drives. IEEE Transactions on Automatic Control vol. 49 1757–1762 (2004) – 10.1109/tac.2004.835405
• van der Schaft, The Hamiltonian formulation of energy conserving physical systems with external ports. Archiv für Elektronik und Übertragungstechnik, (1995)
• 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
• Villegas, J. A., Zwart, H., Le Gorrec, Y. & Maschke, B. Exponential Stability of a Class of Boundary Control Systems. IEEE Transactions on Automatic Control vol. 54 142–147 (2009) – 10.1109/tac.2008.2007176
• 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