Lie and morse theory for periodic orbits of vector fields and matrix riccati equations, I: General lie-theoretic methods

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The problem of finding periodic solutions of the matrix Riccati equations of linear control theory is interpreted geometrically as a problem of finding periodic orbits of certain one-parameter transformation groups on Grassmann manifolds. For certain control problems the vector fields which generate these groups can be written as a sum of two commuting vector fields, one a gradient vector field, the other a Killing vector field, i.e., an infinitesimal isometry of a metric on the Grassman manifold. For such vector fields, the methods of Morse theory can be adapted to study the periodic orbits. The topological data that is needed to count periodic orbits, i.e., the Poincare polynomial of certain submanifolds of the Grassmann manifold, can be derived from results proved by A. Borel.