Conditions for the 2-D characteristic polynomial of a matrix to be very strict Hurwitz

Conditions for the bi-variate characteristic polynomial of a matrix to be very strict Hurwitz are presented. These conditions are based on two different formulations of the 2-D continuous Lyapunov equation. The necessary and sufficient conditions for the existence of positive definite solutions for the first formulation with constant coefficients are presented. It is shown that such an existence is only sufficient but not necessary for the characteristic polynomial to be very strict Hurwitz. Further, the testing of zeros at infinite distant points requires the use of a class of very strict positive real functions. An alternative formulation of the Lyapunov equation with frequency dependent coefficients is also presented based on a new condition for the very strict Hurwitz property. This second formulation of the Lyapunov equation leads to necessary and sufficient conditions for the very strict Hurwitz property.>