Monotonicity of maximal solutions of algebraic Riccati equations (Q1070006)

It is shown that if \(X_+\) is a solution of the Riccati equation (1) \(XBB^*X-A^*X-XA-Q=0,\ldots\), (where the pair (A,B) is stabilizable, and Q is Hermitian) such that Re \(\lambda\) (A-BB\({}^*X_+)\leq 0\), and if \(X_ 1\) is any solution of \(X_ 1B_ 1B^*\!_ 1X_ 1-A^*\!_ 1X_ 1-X_ 1A_ 1-Q_ 1=0\), then \[ \left( \begin{matrix} Q\\ A\end{matrix} \begin{matrix} A^*\\ -BB^*\end{matrix} \right)\geq \left( \begin{matrix} Q_ 1\\ A_ 1\end{matrix} \begin{matrix} A^*\!_ 1\\ -B_ 1B^*\!_ 1\end{matrix} \right) \] implies \(X_+\geq X_ 1\). Based on this monotonicity result it is further shown that a) \(X_+\geq X\) for all other solutions X of (1), b) \(X_+\) is unique, and c) if in addition \(Q\geq 0\), then \(X_+\geq 0\).