Anisotropic heterogeneous n-D heat equation with boundary control and observation : I. Modeling as port-Hamiltonian system

Abstract

The aim of this paper is to recast the heat equation with boundary control and observation in the port-Hamiltonian formalism. The anisotropic and heteregenous case in an n-D geometrical domain is systematically developped. Three different points of view are presented. The first two are thermodynamically founded, taking either entropy or energy as Hamiltonian functional. With the choice of entropy, the second principle can be recovered. With the choice of energy, following Zhou et al. (2017), extra physical variables are introduced allowing to recover the first principle. The third formulation is classical from a mathematical perspective, although less meaningful physically speaking; however the Hamiltonian proves to be a Lyapunov functional, which is useful for boundary control purposes. Moreover, all these three formulations can be discretized with a structure-preserving scheme, as presented in the companion paper Serhani et al. (2019a).