q**Corollary 12.5:** Suppose (*A,B*) is stabilizable and (*C,A*) is detectable. Then

• has a unique positive semidefinite
solution. Moreover, it is stabilizing.

q**Corollary 12.7**: Suppose *D* has full column rank and denote *R = **D*****D*>0. Let *H* have the form

•

Then *H* Î *dom(Ric)* iff (*A,B*) is stabilizable and has full column rank for all w. Furthermore, *X=Ric(H)*³*0* if *H* Î *dom(Ric)* and Ker(*X*) = 0 if and only if (*D*^**C, A-BR**-1**D*C*) has no stable unobservable modes.

q**Proof:** This is because
has full column rank for all w

Û ((*I-DR**-1**D**)*C, A-BR**-1**D*C*) has no observable modes
on *j*w-axis.