A new approach to the study of controllability and observabihty by means of analysis of the controhabihty and observabihty matrices and gramians was proposed. It rehes on the lemma of expansion of the matrix integrals proved by expanding the resolvents of the matrices A _m and A and the properties of convolution of the matrix function in the complex domain. Analysis of the structural properties of gramians demonstrated that the coefficients of the characteristic equations and the Faddeev matrices in the decomposition of the resolvent of the system dynamics matrix play the leading part in their formation. An example illustrating the properties of gramians was given.

Consideration was given to the methods for solution of the differential and algebraic Lyapunov and Sylvester equations in the time and frequency domains. Their solutions are represented as various finite and infinite grammians. The proposed approach to calculation of the grammians lies in expanding them as the sums of the matrix bilinear or quadratic forms generated with the use of the Faddeev matrices and representing each the solution of the linear matrix algebraic equation corresponding to an individual matrix eigenvalue. A lemma was proved representing explicitly the finite and infinite grammians as the matrix exponents depending on the combined spectrum of the original matrices. This result is generalized to the cases where the spectrum of one matrix contains an eigenvalue of the multiplicity two. Examples illustrating calculation of the finite and infinite grammians were discussed.

Spectral decompositions for the solutions of Lyapunov equation obtained earlier are generalized to a more general class of solutions of Krein matrix equations including as a special case the standard Sylvester equation. Eigen parts of these decompositions are calculated using residues of matrix resolvents and their derivatives. In particular, spectral decompositions for the solutions of algebraic and discrete Lyapunov equations are obtained in a more general formulation. The practical significance of the obtained spectral expansions is that they allow one to characterize the contribution of individual eigen-components or their pairwise combinations into the asymptotic dynamics of the system perturbation energy.