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

TL;DR

This paper develops a method to extract meaningful boundary conditions and ports in second-order field theories using variational principles, aiding energy-based control in mechanical systems like beams and plates.

Contribution

It introduces a novel approach to derive boundary conditions and ports from the Cartan form in second-order field theories, enhancing control applications.

Findings

Derived boundary conditions for mechanical systems

Applied the method to beam and plate models

Facilitated energy-based control design

Abstract

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 has also 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.