• Since JH is symmetric Þ :
• (-X1*X2+X2*X1) H_=H_*(-X1*X2+X2*X1 )*= -H_*(-X1*X2+X2*X1 )
• This is Lyapunov
equation. Since H_
is stable, the unique solution is
•-X1*X2+X2*X1
=0.
• i.e., X1*X2 is symmetric.
Þ X=(X1-1)*(X1*X2)X1-1 is symmetric.
• (ii) Start with the equation
•
• and post-multiply by X1-1 to get
•
• now pre-multiply by [X -I]:
q
• This is precisely the
Riccati equation.
• (iii)
• Thus A + RX is stable because H_ is.