• Define G0:=F0+B*X. Then
• (X0-X)A0+A0*(X0-X)=G0*G0-Q(X)>0
• and X0 > X (by anti-stability of A0).
• Define a non-increasing
sequence of hermitian matrices {Xi}:
• X0 ³ X1 ³ … ³ Xn-1 >X,
• Ai= A- BFi , is
anti-stable, i = 0,…., n-1;
• Fi=-B*Xi-1, i=1, …, n-1;
• XiAi+Ai*Xi=Fi*Fi-Q, i=0,1,…,n-1. (0.8)
• By Induction: We show this
sequence can indeed be defined.
• Introduce Fn=-B*Xn-1 , An= A- BFn.
• We show that An is antistable. Using
(0.8), with i =
n-1, we get
• Xn-1An+An*Xn-1+Q -Fn*Fn-(Fn-Fn-1)*(Fn-Fn-1)=0.