On the theory of the matrix Riccati equation

M. I. Zelikin

Sbornik. Mathematics, Tome 73 (1992) no. 2, pp. 341-354

Résumé

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.