  A stability analysis of out of equilibrium and boundary driven systems is
presented. It is performed in the framework of the hydrodynamic macroscopic
fluctuation theory and assuming the additivity principle whose interpretation
is discussed with the help of a Hamiltonian description. An extension of Le
Chatelier principle for out of equilibrium situations is presented which allows
to formulate the conditions of validity of the additivity principle. Examples
of application of these results in the realm of classical and quantum systems
are provided.
