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A Kernel Representation of Dirac Structures forInfinite-dimensional Systems

Published online by Cambridge University Press:  19 August 2014

O.V. Iftime ,
M. Roman  and
A. Sandovici
O.V. Iftime
Affiliation:
Department of Economics, Econometrics and Finance, University of Groningen Nettelbosje 2, 9747 AE, Groningen, The Netherlands
M. Roman
Affiliation:
Department of Mathematics, “Gheorghe Asachi” Technical University B-dul Carol I, nr. 11, 700506, Iaşi, Romania
A. Sandovici*
Affiliation:
Department of Mathematics, “Gheorghe Asachi” Technical University B-dul Carol I, nr. 11, 700506, Iaşi, Romania
*
⋆Corresponding author. E-mail: [email protected]
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Abstract

Dirac structures are used as the underlying structure to mathematically formalizeport-Hamiltonian systems. This note approaches the Dirac structures forinfinite-dimensional systems using the theory of linear relations on Hilbert spaces.First, a kernel representation for a Dirac structure is proposed. The one-to-onecorrespondence between Dirac structures and unitary operators is revisited. Further, theproposed kernel representation and a scattering representation are constructively related.Several illustrative examples are also presented in the paper.

Keywords

Dirac structurelinear relationHilbert spaceinfinite–dimensional system
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 295 - 308
Copyright
© EDP Sciences, 2014

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References

Arens, R.. Operational calculus of linear relations. Pacific J. Math. Vol., 11 (1961), no. 1, 9–23. CrossRefGoogle Scholar
T.Ya. Azizov, I.S. Iokhvidov. Linear operators in spaces with an indefinite metric. John Wiley and Sons, Chichester, 1989.
J. Bognar. Indefinite inner product spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1974.
Cervera, J., van der Schaft, A.J., Baños, A.. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica, 43 (2007), 212–225. CrossRefGoogle Scholar
Courant, T.. Dirac manifolds. Tr. Am. Math. Soc., 319 (1990), 631–661. CrossRefGoogle Scholar
Derkach, V., Hassi, S., Malamud, M., de Snoo, H.. Boundary Relations and Their Weyl Families. Tr. of Amer. Math. Soc., 358 (2006), 5351–5400. CrossRefGoogle Scholar
Dorfman, I.. Dirac structures of integrable evolution equations. Phys. Lett. A, 125 (1987), 240–246. CrossRefGoogle Scholar
Foias, C., Frazho, A.E.. Redheffer products and the lifting of contractions on Hilbert space. Journal of Operator Theory, 11 (1984), 193–196. Google Scholar
G. Golo. Interconnection Structures in Port-Based Modeling: Tools for Analysis and Simulation. Ph.D. Thesis, University of Twente, September 2002.
G. Golo, O.V. Iftime, A. van der Schaft. Interconnection Structures in Physical Systems: a Mathematical Formulation. In Proceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems, Notre Dame University, USA, Editors D.S. Gilliam and J. Rosenthal, 2002.
G. Golo, O.V. Iftime, H. Zwart, A. van der Schaft. Tools for analysis of Dirac structures on Hilbert spaces. Memorandum no. 1729, Faculty of Applied Mathematics, University of Twente, July 2004.
Y. Gorrec Le, H.J. Zwart, B. Maschke. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM J. Control and Optimization, 2005, 1864–1892.
O.V. Iftime, A. Sandovici, G. Golo. Tools for analysis of Dirac Structures on Banach Spaces. In Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Sevilla, Spain, pp. 3856–3861, December 12–15, 2005.
O.V. Iftime, A. Sandovici. Interconnection of Dirac structures via kernel/image representation. In Proceedings of the American Control Conference, San Francisco, California, USA, pp. 3571-3576, June 29 - July 1, 2011.
Iftime, O.V., Sandovici, A.. On different types of representations of Dirac structures on Hilbert spaces. ROMAI J., 7 (2011), no. 2, 79–88.Google Scholar
M. Kurula, A. van der Schaft, H. Zwart. Composition of Infinite-Dimensional Linear Dirac-Type Structures. Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, 2006.
Kurula, M., Zwart, H.J., van der Schaft, A., Behrndt, J.. Dirac structure and their composition on Hilbert spaces. Journal of Mathematical Analysis and Applications, 372 (2010), 402–422. CrossRefGoogle Scholar
Polyuga, R.V., van der Schaft, A.. Effort- and flow-constraint reduction methods for structure preserving model reduction of port-Hamiltonian systems. Systems and Control Letters, 61 (2012), 412-421. CrossRefGoogle Scholar
Redheffer, R.. On Certain linear fractional transformations. Journal of Mathematics and Physics, 39 (1960), 269–286. CrossRefGoogle Scholar
A. van der Schaft, B. Maschke. Modeling and Control of Mechanical Systems. Imperial College Press, Ch. Interconnected Mechanical Systems, part I: geometry of interconnection and implicit Hamiltonian systems, (1997), 1–15.
A. van der Schaft. The Mathematics of Systems and Control: from Intelligent Control to Behavioral Systems. University of Groningen, Ch. Interconnection and Geometry, (1999), 203–217 .
van der Schaft, A., Maschke, B.. Hamiltonian formulation of distributed parameter systems with boundary energy flow. Journal of Geometry and Physics, 42 (2002), 166–194. CrossRefGoogle Scholar
J. Villegas. A Port-Hamiltonian Approach to Distributed Parameter Systems. PhD Thesis, University of Twente, 2007.
H. Yoshimura, H. Jacobs, J.E. Marsden. Interconnection of Dirac Structures and Lagrange-Dirac Dynamical Systems. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, Budapest, Hungary, (2010), 5–9.