The Algebraic Riccati Equation without Complete Controllability
Hȧrald K. Wimmer · SIAM Journal on Algebraic and Discrete Methods · 1982
The equation $XDX + XA + A^*X - C = 0$ is studied under the assumptions that D is positive semidefinite and to pure imaginary eigenvalues of the Hamiltonian matrix $\begin{pmatrix} A & D \\ C & { - A^* } \end{pmatrix}$ correspond only elementary divisors of even degree. A necessary and sufficient condition on the existence and uniqueness of hermitian solutions is given. The approach is based on symplectic transformations of Hamiltonian matrices.