  In the present article, we discuss some aspects of the local stability
analysis for a class of abstract functional differential equations. This is
done under smoothness assumptions which are often satisfied in the presence of
a state-dependent delay. Apart from recapitulating the two classical principles
of linearized stability and instability, we deduce the analogon of the Pliss
reduction principle for the class of differential equations under
consideration. This reduction principle enables to determine the local
stability properties of a solution in the situation where the linearization
does not have any eigenvalues with positive real part but at least one
eigenvalue on the imaginary axis.
