qTheorem 12.1: Suppose H Î dom(Ric) and X=Ric(H). Then
• (i) X is real symmetric
• (ii) X satisfies the algebraic
Riccati equation
•A*X+XA+XRX+Q=0
• (iii) A+RX is stable.
qProof: (i) Let
. We show X1*X2 is symmetric.
q Note that there exists a stable matrix H_ in Rn´n such that
•
• Pre-multiply this
equation by
•
• to get
•
•