On the application of Pontryagin's criterion to distributed feedback systems

Let
h(x, t)
be a polynomial with complex coefficients in the two complex variables
z
and
t
. Pontryagin[1] has developed necessary and sufficient conditions that the function
H(z) = h(z, e^{z})
have zeros with only negative real parts. The usual case in a stability analysis of a feedback system is to be given a ratio of such polynomials for the open-loop case, and to find right-half plane (rhp) zeros of the denominator of the closed-loop gain. In this letter an attempt is made to extend the Pontryagin criterion to the open-loop closed-loop type of analysis. While some interesting results are obtained, it is concluded that this is not a fruitful area of research.