Boundary Integrability of Multiple Stokes--Dirac Structures

Authors

Gou Nishida, Bernhard Maschke, Ryojun Ikeura

Abstract

A distributed port-Hamiltonian (DPH) system is a framework of boundary controls based on energy for partial differential equations (PDEs). A DPH system is defined in terms of a Stokes–Dirac structure that implies boundary integrability in the sense of Stokes’ theorem. Because of the boundary integrability, the energy flows of the systems distributed on a domain can be transformed into energy flows across the boundary of the domain. In general, there might exist boundary nonintegrable energy flows in PDEs. Such energy flows can be modeled by an extended representation of DPH systems, called a distributed energy flow. This paper shows that DPH systems with distributed energy flows can be transformed into standard DPH systems without distributed energy flows by adding extra DPH systems. As a result, we can apply boundary controls and boundary interconnections based on energy to a larger class of PDEs. We call this transformation the boundary completion of DPH systems. This paper derives the necessary and sufficient conditions for such boundary completions.

Citation

• Journal: SIAM Journal on Control and Optimization
• Year: 2015
• Volume: 53
• Issue: 2
• Pages: 800–815
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/110856058

BibTeX

@article{Nishida_2015,
  title={{Boundary Integrability of Multiple Stokes--Dirac Structures}},
  volume={53},
  ISSN={1095-7138},
  DOI={10.1137/110856058},
  number={2},
  journal={SIAM Journal on Control and Optimization},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Nishida, Gou and Maschke, Bernhard and Ikeura, Ryojun},
  year={2015},
  pages={800--815}
}

Download the bib file

References

• van der Schaft A (2000) L2 - Gain and Passivity Techniques in Nonlinear Control. Springer London, London – 10.1007/978-1-4471-0507-7
• Duindam V, Macchelli A, Stramigioli S, Bruyninckx H (2009) Modeling and Control of Complex Physical Systems. Springer Berlin Heidelberg, Berlin, Heidelberg – 10.1007/978-3-642-03196-0
• 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
• Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations , John Wiley, Chichester, UK, 1993.
• 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
• 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
• Eberard, D., Maschke, B. M. & van der Schaft, A. J. An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on Mathematical Physics vol. 60 175–198 (2007) – 10.1016/s0034-4877(07)00024-9
• Voss, T. & Scherpen, J. M. A. Structure Preserving Spatial Discretization of a 1-D Piezoelectric Timoshenko Beam. Multiscale Modeling & Simulation vol. 9 129–154 (2011) – 10.1137/100789038
• Nishida G., Berlin (2007)
• Jeltsema, D. & Schaft, A. van der. Pseudo-gradient and Lagrangian boundary control system formulation of electromagnetic fields. Journal of Physics A: Mathematical and Theoretical vol. 40 11627–11643 (2007) – 10.1088/1751-8113/40/38/013
• Jeltsema, D. & Van Der Schaft, A. J. Lagrangian and Hamiltonian formulation of transmission line systems with boundary energy flow. Reports on Mathematical Physics vol. 63 55–74 (2009) – 10.1016/s0034-4877(09)00009-3
• Nishida G., NJ (2008)
• D. Karnopp, D. Margolis, and R. Rosenberg, System Dynamics, Modeling and Simulation of Mechatronic Systems , 4th ed., Wiley, New York, 2006.
• Olver PJ (1993) Applications of Lie Groups to Differential Equations. Springer New York, New York, NY – 10.1007/978-1-4612-4350-2
• H. Flanders, Differential Forms with Applications to the Physical Sciences , Academic, New York, 1963.
• Morita S (2001) Geometry of Differential Forms. American Mathematical Society, Providence, Rhode Island – 10.1090/mmono/201
• D.G.B. Edelen, Applied Exterior Calculus , John Wiley, New York, 1985.
• P. Hammond, Energy Methods in Electromagnetism , Clarendon, Oxford, 1981.
• G. Nishida, M. Yamakita, and Z. Luo, Global boundary connection of Stokes-Dirac structures for Morse-Smale flows on compact manifolds , in Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto University, Kyoto, 2006.
• Nishida G., IFAC, Pretoria (2007)
• Franco A.A., Vienna (2006)