Stokes-Lagrange and Stokes-Dirac representations of $N$-dimensional port-Hamiltonian systems for modelling and control

In this paper, we extend the port-Hamiltonian framework by introducing the concept of Stokes-Lagrange structure, which enables the implicit definition of a Hamiltonian over an $N$-dimensional domain and incorporates energy ports into the system. This new framework parallels the existing Dirac and Stokes-Dirac structures. We propose the Stokes-Lagrange structure as a specific case where the subspace is explicitly defined via differential operators that satisfy an integration by parts formula. By examining various examples through the lens of the Stokes-Lagrange structure, we demonstrate the existence of multiple equivalent system representations. These representations provide significant advantages for both numerical simulation and control design, offering additional tools for the modelling and control of port-Hamiltonian systems.