•Lemma ARE: [Ran
abd Vreugdenhil, 1988] Suppose (A,B) is controllable and there is an X = X* such that
• Q(X):=XA+A*X+XBB*X+Q<0.
•Then there exists a solution X+>X to the Riccati equation
• X+A+A*X++X+BB*X++Q=0 (0.7)
•such that A+BB*X+ is antistable
•Proof. Let X be such that Q(X) < 0. Choose F0 such
that A0 := A- BF0 is antistable. Let X0=X0* solve
• X0A0+A0*X0-F0*F0+Q=0.