Phase Portrait of the Matrix Riccati Equation

The matrix Riccati equation which arises from optimal control and filtering problems is a quadratic differential equation on the space of real symmetric n × n matrices. It is closely related, via compactification of the phase space, to the differential equations on the Grassmann manifold and the Lagrange–Grassmann manifold whose flows are generated by the action of one-parameter subgroups of the general linear group and of the symplectic group respectively. We. determine the complete phase portraits of the Riccati equations on all three spaces. The asymptotic behavior of every solution is described. The phase portraits are characterized topologically as well as set-theoretically. Although the Riccati equation is not generally a Morse–Smale vector field, we are able to show that it possesses suitable generalizations of many of the important properties of Morse–Smale vector fields. In particular, the Riccati equation satisfies a generalized version of the Morse inequalities for a Morse–Smale dynamical system. In fact, for the Riccati equation, the inequalities are actually equalities.