On implicit and explicit representations for 1D distributed port-Hamiltonian systems

First, two examples of 1D distributed port-Hamiltonian systems with dissipation, given in explicit (descriptor) form, are considered: the Dzekster model for the seepage of underground water and a nanorod model with non-local viscous damping. Implicit representations in Stokes-Lagrange subspaces are formulated. These formulations lead to modified Hamiltonian functions with spatial differential operators. The associated power balance equations are derived, together with the new boundary ports. Second, the port-Hamiltonian formulations for the Timoshenko and the Euler-Bernoulli beams are recalled, the latter being a flow-constrained version of the former. Implicit representations of these models in Stokes-Lagrange subspaces and corresponding power balance equations are derived. Bijective transformations are proposed between the explicit and implicit representations. It is proven these transformations commute with the flow-constraint projection operator.