  The stability of functional differential equations under delayed feedback is
investigated near a Hopf bifurcation. Necessary and sufficient conditions are
derived for the stability of the equilibrium solution using averaging theory.
The results are used to compare delayed versus undelayed feedback, as well as
discrete versus distributed delays. Conditions are obtained for which delayed
feedback with partial state information can yield stability where undelayed
feedback is ineffective. Furthermore, it is shown that if the feedback is
stabilizing (respectively, destabilizing), then a discrete delay is locally the
most stabilizing (resp., destabilizing) one among delay distributions having
the same mean. The result also holds globally if one considers delays that are
symmetrically distributed about their mean.
