An opinion is sometimes heard that the Onsagerian description of heat and mass exchange is not compatible with the commonly used engineering approach based on the hydrodynamic theory of boundary layer, with transfer coefficients and driving forces. However, in this work we prove the compatibility of both descriptions for the same driving forces. Applying a general approach based on the minimum of entropy production to a lumped relaxing system with simultaneous heat and mass transfer between two subsystems, we find the canonical (Hamilton's) structure of the nonequilibrium dynamics, and show a general self-consistent way of derivation of these dynamics. The relaxation dynamics is derived from a 'thermodynamic Lagrangian' L-sigma which uses two dissipation functions: a rate dependent one Phi*; and a state dependent one Psi. Two possible variational approaches are compared: the first, which uses dependent variables of state connected by constraints stemming from conservation laws (DVA); and the second, in which the constraints are eliminated in advance, so that the model contains only independent variables (IVA). The first approach is novel, the second is basically an integral version of Onsager's approach in a two-phase context. Both approaches are analyzed via methods of optimal control theory, and make use of ideas based on the Hamilton-Jacobi-Bellman equation (H-J-B theory). It is shown that the DVA has a number of virtues with respect to the IVA: it can deal with the non-truncated thermodynamic entropy and with absolute values of thermodynamic adjoints (T-1, T-1 mu(i)), and it gives a complementary relaxation picture for these adjoints, as governed by the relaxation matrix K-T, the transpose of the state relaxation matrix K. Other new physical results elucidate a general optimal-control scheme to construct a non-equilibrium thermodynamic entropy S as the principal function which satisfies an autonomous H-J-B equation and the related Hamilton-Jacobi theory under the constraint of a vanishing thermodynamic Hamiltonian H-sigma = Phi-Psi, necessary for S to be a state function. The properties of this entropy are herein analyzed, and it is concluded that the proper non-equilibrium S is that evaluated additively over the homogeneous subsystems. This result, along with the proven Lagrangian or Hamiltonian dynamics of the heat and mass exchange, enhances our confidence to the Onsager's theory.