On spectral mappings, higher order circle criteria, and periodically varying systems

For feedback equations of the form
e = x - He
, in which
H
is a linear operator, stability boundaries obtained by the circle criterion method can frequently be improved by finding a function
f(H)
which conformally maps the spectrum of
H
from its initial region in the complex plane into a more suitable region. This idea, which is based on the spectral mapping theorem of Banach algebras, is applied here to a class of equations consisting of a time-invariant part and a periodically time-varying gain. A stability bound is derived which has a minimum whenever the frequency response of the time-invariant part peaks at a frequency near one half that of the time-varying gain.