Interconnection of Dirac structures via kernel/image representation

Dirac structures are used to mathematically formalize the power-conserving interconnection structure of physical systems. For finite-dimensional systems several representations are available and it is known that the composition (or interconnection) of two Dirac structures is again a Dirac structure. It is also known that for infinite-dimensional systems the composition of two Dirac structures may not be a Dirac structure. In this paper, the theory of linear relations is used in the first instance to provide different representations of infinite dimensional Dirac structures (on Hilbert spaces): an orthogonal decomposition, a scattering representation, a constructive kernel representation and an image representation. Some links between scattering and kernel/image representations of Dirac structures are also discussed. The Hilbert space setting is large enough from the point of view of the applications. Further, necessary and sufficient conditions (in terms of the scattering representation and in terms of kernel/image representations) for preserving the Dirac structure on Hilbert spaces under the composition (interconnection) are also presented. Complete proofs and illustrative example(s) will be included in a follow up paper.