On Hermitian Solutions of the Symmetric Algebraic Riccati Equation

The structure of the set of hermitian solutions of the matrix quadratic equation XDX - XA - A^ * X - C = 0 is studied under the conditions that C = C^ *, D is positive semidefinite and $(A,D)$ is stabilizable. New features (e.g., nonexistence of the minimal solution) appear in contrast with the known case when $(A,D)$ is controllable.