Port-representation of bi-Hamiltonian structure for infinite-dimensional symmetry
Authors
null Gou Nishida, null Masaki Yamakita, null Zhi-wei Luo
Abstract
In this paper, the port-representation of conservation laws is extended to a wider class of symmetries, the infinite-dimensional symmetry expressed by the bi-Hamiltonian system. It is known from Noether’s theorem that a conservation law is associated with an invariant property called a symmetry. In certain cases, the symmetry appears in a system as a hidden infinite-dimensional structure. Such a structure can be defined by using a recursive operator consisting of a Hamiltonian pair and is called a bi-Hamiltonian structure. The bi-Hamiltonian structure induces a hierarchical set of conservation laws. This concept can be used for reducing a system possessing a bi-Hamiltonian structure to simpler port-representations of the conservation laws. Finally, a boundary observer for symmetry destruction is shown.
Citation
- Journal: 2007 46th IEEE Conference on Decision and Control
- Year: 2007
- Volume:
- Issue:
- Pages: 5588–5593
- Publisher: IEEE
- DOI: 10.1109/cdc.2007.4434262
BibTeX
@inproceedings{Gou_Nishida_2007,
title={{Port-representation of bi-Hamiltonian structure for infinite-dimensional symmetry}},
DOI={10.1109/cdc.2007.4434262},
booktitle={{2007 46th IEEE Conference on Decision and Control}},
publisher={IEEE},
author={Gou Nishida and Masaki Yamakita and Zhi-wei Luo},
year={2007},
pages={5588--5593}
}
References
- van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
- Doran, C. & Lasenby, A. Geometric Algebra for Physicists. (2003) doi:10.1017/cbo9780511807497 – 10.1017/cbo9780511807497
- dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations (1993)
- morita, Geometry of Differential Forms A (0)
- Preiswerk, F., Arnold, P., Fasel, B. & Cattin, P. C. Robust tumour tracking from 2D imaging using a population-based statistical motion model. 2012 IEEE Workshop on Mathematical Methods in Biomedical Image Analysis 209–214 (2012) doi:10.1109/mmbia.2012.6164749 – 10.1109/mmbia.2012.6164749
- 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
- Eberard, D., Lefevre, L. & Maschke, B. M. Multiscale coupling in heterogeneous diffusion processes : a port-based approach. Proceedings. 2005 International Conference Physics and Control, 2005. 543–547 doi:10.1109/phycon.2005.1514042 – 10.1109/phycon.2005.1514042
- villegas, boundary control for a class of dissipative differential operators including diffusion systems. Proc 7th Int Symp on Mathematical Theory of Networks and Systems (2006)
- van der Schaft, A. J. Implicit Hamiltonian systems with symmetry. Reports on Mathematical Physics 41, 203–221 (1998) – 10.1016/s0034-4877(98)80176-6
- Blankenstein, G. & van der Schaft, A. J. Symmetry and reduction in implicit generalized Hamiltonian systems. Reports on Mathematical Physics 47, 57–100 (2001) – 10.1016/s0034-4877(01)90006-0
- Grizzle, J. & Marcus, S. The structure of nonlinear control systems possessing symmetries. IEEE Trans. Automat. Contr. 30, 248–258 (1985) – 10.1109/tac.1985.1103927
- Nishida, G. & Yamakita, M. A higher order Stokes-Dirac structure for distributed-parameter port-Hamiltonian systems. Proceedings of the 2004 American Control Conference 5004–5009 vol.6 (2004) doi:10.23919/acc.2004.1384643 – 10.23919/acc.2004.1384643
- Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM J. Control Optim. 43, 743–767 (2004) – 10.1137/s0363012903429530
- eberard, conservative systems with ports on contact manifolds. Proc IFAC World Cong (2005)
- maschke, from conservation laws to port-hamiltonian representations of distributed-parameter systems. Proc IFAC World Cong (2005)
- van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42, 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
- Nishida, G., Yamakita, M. & Luo, Z. Field Port-Lagrangian Representation of Conservation Laws for Variational Symmetries. Proceedings of the 45th IEEE Conference on Decision and Control 5875–5881 (2006) doi:10.1109/cdc.2006.376974 – 10.1109/cdc.2006.376974
- Gou Nishida & Yamakita, M. Formal Distributed Port-Hamiltonian Representation of Field Equations. Proceedings of the 44th IEEE Conference on Decision and Control 6009–6015 doi:10.1109/cdc.2005.1583123 – 10.1109/cdc.2005.1583123