Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators

Authors

Abstract

Associated with a skew-symmetric linear operator on the spatial domain \( [a,b] \) we define a Dirac structure which includes the port variables on the boundary of this spatial domain. This Dirac structure is a subspace of a Hilbert space. Naturally, associated with this Dirac structure is an infinite-dimensional system. We parameterize the boundary port variables for which the \( C_{0} \)-semigroup associated with this system is contractive or unitary. Furthermore, this parameterization is used to split the boundary port variables into inputs and outputs. Similarly, we define a linear port controlled Hamiltonian system associated with the previously defined Dirac structure and a symmetric positive operator defining the energy of the system. We illustrate this theory on the example of the Timoshenko beam.

Citation

Journal: SIAM Journal on Control and Optimization
Year: 2005
Volume: 44
Issue: 5
Pages: 1864–1892
Publisher: Society for Industrial & Applied Mathematics (SIAM)
DOI: 10.1137/040611677

BibTeX

@article{Le_Gorrec_2005,
  title={{Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators}},
  volume={44},
  ISSN={1095-7138},
  DOI={10.1137/040611677},
  number={5},
  journal={SIAM Journal on Control and Optimization},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Le Gorrec, Y. and Zwart, H. and Maschke, B.},
  year={2005},
  pages={1864--1892}
}

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References

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