STABILITY ANALYSIS OF SOLUTIONS OF A LINEAR MATRIX DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS

The article presents the method for analyzing the stability of linear matrix differential equations with constant coefficients. One of the classical typesof such equations is the class of linear matrix differential equations, which includes the Lyapunov equation as a particular case. Matrix differentialequations arise in problems of stability theory, practical stability, optimal control theory, and state estimation of systems under uncertainty. Therefore,it is necessary to compute and analyze the qualitative properties of solutions to matrix differential equations. This involves addressing problems of existence, uniqueness, continuation, and analysis of stability conditions for various types of such mathematical equations. The method proposed in thearticle is based on algebraic properties of eigenvalues, Jordan forms of matrices, and characteristics of polynomial roots. A theorem is established regardingthe conditions for stability, asymptotic stability, and instability of solutions to linear matrix differential equations with constant coefficients.The developed approach includes the computation of the maximal real parts of eigenvalues and the analysis of the Jordan form structure of the systemmatrices. As a consequence, corresponding stability conditions for the Lyapunov matrix equation are also obtained. An algorithm is proposed for computingthe maximal real part of the roots of a polynomial, as well as for finding all roots. The approach relies on the Routh – Hurwitz theorem. The article also presents results of computational experiments.