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

Authors:
Y. Le Gorrec;H. Zwart;B. Maschke
Affiliations:
-;-;-
Venue:
SIAM Journal on Control and Optimization
Year:
2005

Citing 0
Cited 2

Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws

Journal of Computational Physics
Stability and passivity preserving Petrov-Galerkin approximation of linear infinite-dimensional systems

Automatica (Journal of IFAC)

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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.