TITLE: Riccati & Lyapunov equations
QUESTION [5 upvotes]: We know that
Lyapunov equation: $A^TP + PA + Q = 0$
Algebraic Riccati equation: $A^TP + PA + Q + PBR^{-1}B^TP= 0$
It seems that the difference between the two lies in $B = 0$ (zero input) in Lyapunov Eq
and both are infinite horizon in the case above. 
Is there any other engineering-sense difference (not mathematics) between the two Eqs?
Thanks!

REPLY [3 votes]: Replace $A$ by $A-BK$ with $K=R^{-1}B^\top P$ in the Lyapunov equation yields the algebraic Riccati equation. Hope this helps.