•**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 *F**0* such
that *A**0 **:= A- BF**0* is antistable. Let *X**0**=X**0** solve

•* X**0**A**0**+A**0***X**0**-F**0***F**0**+Q=0.*