From conservation laws to port-Hamiltonian representations of distributed-parameter systems

The treatment of infinite-dimensional Hamiltonian systems in the literature is mostly focussed onsystems with boundary conditions such that the energy exchange through the boundary is zero. On the other hand, in many applications the interaction with the environment (e.g. actuation or measurement) will actually take place through the boundary of the system.In previous work we have developed a framework to represent classes of physical distributed-parameter systems with boundary energy flow as infinite-dimensional port-Hamiltonian systems. Key in this is the notion of a Diracstructure. In the finite-dimensional case Dirac structures can be naturally employed to formalize Hamiltoniansystems with algebraic constraints. In order to allow the inclusion of boundary variables in distributed-parameter systems the concept of (an infinite-dimensional) Dirac structure provides again the right type of generalizationwith respect to the existing framework using Poisson structures. This extends the theory of port-Hamiltonian systems to the distributed-parameter case, and allows to consider mixed lumped- and distributed-parameter systems within the same framework. The aim of this paper is to show how the port-Hamiltonian formulation of distributed-parameter systems is related to, and can be based on, the general framework for describing basic distributed-parameter systems as systems of conservation laws. In this framework each conservation law captures the balance equation associated withsome conserved physical quantity, coupled to a set of closure equations involving the conserved quantities and the fluxes (or generating forces). It turns out that these balance laws define a special type of Dirac structure called the Stokes-Dirac structure, while the closure equations are equivalent to the definition of the Hamiltonian of the system.