  In this paper we extend the previously introduced class of boundary
port-Hamiltonian systems to boundary control systems where the variational
derivative of the Hamiltonian functional is replaced by a pair of reciprocal
differential operators. In physical systems modelling, these differential
operators naturally represent the constitutive relations associated with the
implicitly defined energy of the system and obey Maxwell's reciprocity
conditions. On top of the boundary variables associated with the Stokes-Dirac
structure, this leads to additional boundary port variables and to the new
notion of a Stokes-Lagrange subspace. This extended class of boundary
port-Hamiltonian systems is illustrated by a number of examples in the
modelling of elastic rods with local and non-local elasticity relations.
Finally it shown how a Hamiltonian functional on an extended state space can be
associated with the Stokes-Lagrange subspace, and how this leads to an energy
balance equation involving the boundary variables of the Stokes-Dirac structure
as well as of the Stokes-Lagrange subspace.
