On the theory of the matrix Riccati equation

Various approaches to the study of the matrix Riccati equation with variable coefficients are investigated. It is proved that the complexification of such an equation determines a flow on the Seigel generalized upper half-plane and on each of the strata forming its boundary. The concept of the matrix cross-ratio of a quadruple of points of the Grassmann manifold $G_n(\mathbf R^{2n})$ is introduced, and applications of it are given. In particular, a criterion is given for the preservation of the isoclinic property for a pair of planes that are displaced by the flow on $G_n(\mathbf R^{2n})$ determined by a matrix Riccati equation with variable coefficients. Bilinear optimal control problems with a quadratic quality criterion are considered. The corresponding extremals are found along with the matrix Riccati equations determined by them.