Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: an analytical viewpoint

TL;DR

This paper establishes that a broad class of linear PDEs in distributed parameter systems can be characterized as Stokes-Dirac structures using boundary control theory, expanding the geometric framework to include more mechanics problems.

Contribution

It introduces a new analytical approach to define Stokes-Dirac structures for linear PDEs, including those outside the scope of previous geometric formulations, with detailed examples from mechanics and physics.

Findings

Defines Stokes-Dirac structures for linear PDEs using boundary control theory

Includes examples from continuum mechanics and physics

Connects differential operators with Hilbert complexes at boundaries

Abstract

In this paper we prove that a large class of linear evolution PDEs defines a Stokes-Dirac structure over Hilbert spaces. To do so, the theory of boundary control system is employed. This definition encompasses problems from mechanics, that cannot be handled by the seminal geometric setting given in [van der Schaft and Maschke, Hamiltonian formulation of distributed-parameter systems with boundary energy flow, 2002 ]. Many worked-out examples stemming from continuum mechanics and physics are presented in detail, and a particular focus is given on the functional spaces in duality at the boundary of the geometrical domain. For each example, the connection between the differential operators and the associated Hilbert complexes is illustrated.