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 H-1 less-than-or-equal-to H-2. If X1+ and X2+ are the greatest solutions in T1 and T2 respectively, then X1+ less-than-or-equal-to X2+. A more general result will be proved which allows the comparison of other solutions of T1 and T2 besides the extremal ones and which in the case of stabilizability leads to a Galois connection between T1 and T2. 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 A(i)-invariant subspaces.