Standard ARE

q**Theorem 12.4:** Suppose *H* has the form

•

Then *H* Î *dom(Ric)* iff (*A,B*) is stabilizable and (*C,A*) has no unobservable modes on the
imaginary axis. Furthermore, *X=Ric(H)*³*0.* And *X* > 0 if and only if (*C,A*) has no stable
unobservable modes.

q**Proof: **Only need to show that, assuming (*A,B*) is stabilizable, *H* has no *j*w eigenvalues iff (*C,A*) has no unobservable
modes on the imaginary axis.

Suppose that *j*w is an eigenvalue and is a corresponding
eigenvector. Then

• Re-arrange: (*A- j**w**I*)*x=BB*z, -*(*A-j**w**I*)**z=C*Cx*