A Galois correspondence between sets of semidefinite solutions of continuous-time algebraic Riccati equations (Q1329965)

The paper is devoted to comparison of solutions of Riccati equations. Let \[ {\mathcal R}_ i(X)= A^*_ i X+ X A_ i+ X B_ i B^*_ i X- C_ i^*C_ i= 0\tag{1} \] be an algebraic Riccati equation, where \(A_ i\), \(B_ i\), \(C_ i\) are complex matrices of sizes \(n\times n\), \(n\times p\), \(q\times n\) respectively. Let \({\mathcal T}_ i= \{X\mid X\leq 0,\;{\mathcal R}_ i(X)= 0\}\) denote the set of negative semidefinite solutions of (1) and \[ {\mathcal H}_ i= \left[\begin{matrix} -C^*_ i C_ i & A^*_ i\\ A_ i & B_ i B^*_ i\end{matrix}\right]. \] Under the assumption that \({\mathcal H}_ 1\leq {\mathcal H}_ 2\), the paper compares solutions in \({\mathcal T}_ 1\) and \({\mathcal T}_ 2\). In particular, the author shows that if \(X_{i^ +}\) is the greatest solution of (1) then \(X_{1^ +}\leq X_{2^ +}\). A similar inequality for the least element is also true. The paper also establishes a Galois correspondence between other elements in \({\mathcal T}_ 1\) and \({\mathcal T}_ 2\), rather than the greatest and the least elements.