Equilibrium points of the Riccati equation: Geometric structure

Given the algebraic Riccati equation (ARE) -A´K - KA + KBB´K - Q = 0 and its unique maximal symmetric solution K+, J. C. Willems [5] proved that the set of real symmetric solutions is in one-to-one correspondence with the set of invariant subspaces of A- BB´K+. We prove that this bijection is actually a homeomorphism. This enables us to apply several theorems of Shayman [3], [4] on the variety of invariant subspaces of a finite-dimensional linear operator. We obtain a detailed description of the solution set of the ARE. We give a necessary and sufficient condition for the set to be finite. We compute the number of connected components, and show that the connected components need not be manifolds. However, they are always unions of manifolds, and we give a formula for their dimension.