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Scalar, matrix and operator Riccati equations are considered. Theoretical questions and practical methods of solution of these equations are expounded. The necessary auxiliary facts from algebra, functional analysis and Lie group analysis are given. Theory is illustrated with solutions of numerous examples. The matrix Riccati equations are presented completely. Lie groups on matrices theory have been advanced for an analysis of these equations. Theoretical questions concerning matrix and operator equations are dealt with based on a variety of applied problems from mathematical physics, optimal control of finite dimensional systems and distributed parameters systems.
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