• Thus <z, (A- jwI)x>=<z,BB*z>=||B*z||2
-- <x, (A- jwI)*z>=<x,C*Cx>=||Cx||2
• so <x,(A-jwI)*z> is real and
•-||Cx||2=< (A-jwI)x,z>=<z,(A-jwI)x>*=||B*z||2
• Therefore B*z = 0 and Cx = 0. So
•(A-jwI)x=0, (A-jwI)*z=0
• Combine the last four equations to
get
•
•
• The
stabilizability of (A,B) gives z=0. Now it is clear that jw is an eigenvalue of H if jw is an unobservable mode
of (C,A).
•(A-BB*X)*X+X(A-BB*X)+XBB*X+C*C=0
• X³0 since A-BB*X is stable.