On Casimir functionals for field theories in Port-Hamiltonian description for control purposes

Abstract

We consider infinite dimensional Port-Hamiltonian systems in an evolutionary formulation. Based on this system representation conditions for Casimir densities (functionals) will be derived where in this context the variational derivative plays an extraordinary role. Furthermore the coupling of finite and infinite dimensional systems will be analyzed in the spirit of the control by interconnection problem. Our Hamiltonian representation differs significantly from the well-established one using Stokes-Dirac structures that are based on skew-adjoint differential operators and the use of energy variables. We mainly base our considerations on a bundle structure with regard to dependent and independent coordinates as well as on differential-geometric objects induced by that structure.