On Hermitian Solutions of the Symmetric Algebraic Riccati Equation
Israel Gohberg, Peter Lancaster, Leiba Rodman · SIAM Journal on Control and Optimization · 1986
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.