• Let Gn:=Fn+B*X. Then
• (Xn-1-X)An+An*(Xn-1-X)=Gn*Gn-Q(X )+(Fn-Fn-1)*(Fn-Fn-1)>0
• which
implies that An is antistable by Lyapunov
stability theorem since Xn-1- X>0.
• Let Xn be the unique solution of
• Xn An +An*Xn = Fn*Fn –Q. (0.9)
• Then Xn is hermitian. Next, we have
• (Xn-X)An+An*(Xn-X)=Gn*Gn-Q(X )>0
• (Xn-1-Xn)An+An*(Xn-1-Xn)=(Fn-Fn-1)*(Fn-Fn-1)³0
• Since An is antistable, we have Xn-1³ Xn >X.
• Therefore, we have a
non-increasing sequence {Xi}.
•