Field Port-Lagrangian Representation of Conservation Laws for Variational Symmetries

This paper studies the relation between conservation laws and symmetries from the point of view of port-representation. First, the formal structure of conservation laws is defined by the field port-Lagrangian system for such symmetries. Next, it is shown that a variational complex notation naturally defines the system decomposition that extracts a self-adjoint subsystem from a non-self-adjoint system. Then, the above discussion is extended to one of non-trivial symmetries, a variational symmetry. Since the necessary and sufficient condition of the variational symmetry is a self-adjointness, we apply the decomposition given by the previous section to non-self-adjoint systems. Thirdly, we propose the procedure to realize the boundary observer for a destruction of symmetries. Finally, we show a shaping of conservation laws in the sense of generalized energy shaping