A new approach to the controllability and observability properties, based on controllability and observability matrices links, is considered. The approach is based on matrices A, Am integrals expansion Lemma and properties of the matrices function convolution in complex plane. Coefficients of characteristic equations and Faddeev's matrices in the resolvent expansion of the system dynamic matrix play a dominant role in matrices integrals expansion forming. The gramians expansion on dynamic system matrix spectrum provide a new notion for the solutions of the Lyapunov and Sylvester equations. The results are illustrated by the example for two-zones furnace control.