On the extraction of the boundary conditions and the boundary ports in second-order field theories

In this paper, we consider second-order field theories in a variational setting. From the variational principle, the Euler-Lagrange equations follow in an unambiguous way, but it is well known that this is not true for the Cartan form. This also has consequences on the derivation of the boundary conditions when non-trivial variations are allowed on the boundary. By posing extra conditions on the set of possible boundary terms, we exploit the degree of freedom in the Cartan form to extract physical meaningful boundary expressions. The same mathematical machinery will be applied to derive the boundary ports in a Hamiltonian representation of the partial differential equations which is crucial for energy based control approaches. Our results will be visualized for mechanical systems such as beam and plate models.