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

TL;DR

This paper explores explicit and implicit mathematical representations of 1D distributed port-Hamiltonian systems, deriving power balance equations and transformations, with applications to models like beams and underground water seepage.

Contribution

It introduces implicit Stokes-Lagrange subspace formulations for 1D port-Hamiltonian systems and establishes bijective transformations between explicit and implicit forms.

Findings

Derived power balance equations for new implicit representations.

Proposed bijective transformations between explicit and implicit models.

Validated transformations commute with flow-constraint operators.

Abstract

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.