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

The sets of negative semidefinite solutions T i of two algebraic Riccati equations R i (X)=A ∗ i X+XA i +XB i B ∗ i −C ∗ i C i =(I, X)H i (I, X) ∗ , i=1, 2, are compared under the hypothesis that H1⩽H2. If X+1 and X+2 are the greatest solutions in T 1 and T 2 respectively, then X+1⩽X+2. A more general result will be proved which allows the comparison of other solutions of T 1 and T 2 besides the extremal ones and which in the case of stabilizability leads to a Galois connection between T 1 and T 2. The comparison results are based on one hand on a decomposition of the equations R i (X)=0 into Lyapunov matrix equations and genuine Riccati equations which induce a corresponding decomposition of the solutions in T i, and the other hand on a parametrization of the Riccati components by Ai-invariant subspaces.