then the pair
F 0 G
(3-18)
BH A [ )
is stabilizable. If in addition, the word "stabilizable" in (B.1) is replaced by "controllable" then the pair of (3-18) is controllable.
The next theorem shows that when K1 and K2 are selected to stabilize (3-17) the pair (A, K2) is observable. This is the precise
condition needed for Theorem 2.1 and the final condition required for our discussion.
Theorem 3.4: If all eigenvalues of A are in the closed right half-plane
and the system LCT described by (3-17) is asymptotically stable then the pair (A, K2) is observable.
Proof: We use contradiction. Suppose that the system LCT is asymptotically stable but (A, K2) is not observable. This implies that there exists a vector v such that
xI-A
V = 0 (3-19)
K2
for some X which is an eigenvalue of A. Consequently, we can write
XI - F + GK1 GK2 [ 0 (3-20)
BH XI - A] v