517.926The paper outlines the methodology underlying the Kalman-Mesarovic realization for dynamic systems with equations of state in the class of linear nonstationary ordinary differential equations. In this sense, the key approaches to solving problems of realization theory in terms of the Rayleigh-Ritz operator are defined.The mathematics for the expansion of matrix-valued operators is developed in the Kalman-Mesarovic realization problem [1, 2] for nonstationary (with coefficients from Lebesgue L p -spaces) multidimensional differential systems in [3,4]. The concepts (categories of the dictionary of mathematical simulation of dynamic systems -D-systems) such as M-operator, M-continuability, the Rayleigh-Ritz operator (a reminiscence of the canonical Rayleigh-Ritz quotient [5]) are defined and formalized. These structures allow approaching, from the general system-theoretic standpoint, the qualitative analysis of the existence properties for the realizations of D-systems in a special class of strong differential ( , )A B -models [6, 7] -an "optimal" mathematical structure of the realization equations for linear continuous controlled D-systems.The purpose of this study is to show that the structure of the Rayleigh-Ritz operator allows a more efficient solution of realization problems in the class of nonstationary ( , )A Â -models (as compared with signal functions (Theorem 1 [6]) and measures (Theorems 2 [6] and 1, 2 [7]) defined in terms of D-systems from Definition 1.8 [1]). FORMULATIONS OF STANDARD REALIZATION PROBLEMS. PRELIMINARIES ON THE RAYLEIGH-RITZ OPERATOR.Let t t T t t 0 1 0 1 < = , : [ , ] be an interval of the number axis R with a Lebesgue measure m, and let P ACL n p m AC T R L T R : ( , ) ( , , ) =´¢ m ; here AC L p , ¢ are standard [8] function spaces of absolutely continuous vectorfunctions and equivalence classes of all vector functions with an L p¢ -norm that are Bochner m-integrable [9] on T; we will use the notation P ACL without an indication that (i) P