Dirac structures and boundary control systems associated with skew-symmetric differential operators

Y. Le Gorrec, Heiko J. Zwart, B.M. Maschke

Research output: Book/ReportReportOther research output

65 Downloads (Pure)

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 to this Dirac structure is infinite dimensional system. We parameterize the boundary port variables for which the \( C_{0} \)-semigroup associated to 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.
Original languageUndefined
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Publication statusPublished - 2004

Publication series

NameMemorandum / Faculty of Mathematical Sciences
PublisherDepartment of Applied Mathematics, University of Twente
No.1730
ISSN (Print)0169-2690

Keywords

  • MSC-93C25
  • MSC-35M99
  • MSC-70H45
  • IR-65914
  • EWI-3550
  • MSC-47D06
  • MSC-93C20

Cite this

Le Gorrec, Y., Zwart, H. J., & Maschke, B. M. (2004). Dirac structures and boundary control systems associated with skew-symmetric differential operators. (Memorandum / Faculty of Mathematical Sciences; No. 1730). Enschede: University of Twente, Department of Applied Mathematics.
Le Gorrec, Y. ; Zwart, Heiko J. ; Maschke, B.M. / Dirac structures and boundary control systems associated with skew-symmetric differential operators. Enschede : University of Twente, Department of Applied Mathematics, 2004. (Memorandum / Faculty of Mathematical Sciences; 1730).
@book{74e466ea9eae436ba30dae5dd322f412,
title = "Dirac structures and boundary control systems associated with skew-symmetric differential operators",
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 to this Dirac structure is infinite dimensional system. We parameterize the boundary port variables for which the \( C_{0} \)-semigroup associated to 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.",
keywords = "MSC-93C25, MSC-35M99, MSC-70H45, IR-65914, EWI-3550, MSC-47D06, MSC-93C20",
author = "{Le Gorrec}, Y. and Zwart, {Heiko J.} and B.M. Maschke",
note = "Imported from MEMORANDA",
year = "2004",
language = "Undefined",
series = "Memorandum / Faculty of Mathematical Sciences",
publisher = "University of Twente, Department of Applied Mathematics",
number = "1730",

}

Le Gorrec, Y, Zwart, HJ & Maschke, BM 2004, Dirac structures and boundary control systems associated with skew-symmetric differential operators. Memorandum / Faculty of Mathematical Sciences, no. 1730, University of Twente, Department of Applied Mathematics, Enschede.

Dirac structures and boundary control systems associated with skew-symmetric differential operators. / Le Gorrec, Y.; Zwart, Heiko J.; Maschke, B.M.

Enschede : University of Twente, Department of Applied Mathematics, 2004. (Memorandum / Faculty of Mathematical Sciences; No. 1730).

Research output: Book/ReportReportOther research output

TY - BOOK

T1 - Dirac structures and boundary control systems associated with skew-symmetric differential operators

AU - Le Gorrec, Y.

AU - Zwart, Heiko J.

AU - Maschke, B.M.

N1 - Imported from MEMORANDA

PY - 2004

Y1 - 2004

N2 - 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 to this Dirac structure is infinite dimensional system. We parameterize the boundary port variables for which the \( C_{0} \)-semigroup associated to 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.

AB - 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 to this Dirac structure is infinite dimensional system. We parameterize the boundary port variables for which the \( C_{0} \)-semigroup associated to 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.

KW - MSC-93C25

KW - MSC-35M99

KW - MSC-70H45

KW - IR-65914

KW - EWI-3550

KW - MSC-47D06

KW - MSC-93C20

M3 - Report

T3 - Memorandum / Faculty of Mathematical Sciences

BT - Dirac structures and boundary control systems associated with skew-symmetric differential operators

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

Le Gorrec Y, Zwart HJ, Maschke BM. Dirac structures and boundary control systems associated with skew-symmetric differential operators. Enschede: University of Twente, Department of Applied Mathematics, 2004. (Memorandum / Faculty of Mathematical Sciences; 1730).