Example 
.
A system that is not controllable but is stabilizable.
Criterion of Stabilizability
i.e. all 
with 
.
Observability:
Detectability: (something weaker)
An unobservable part may exist but it is stable, dual of
stabilizability concept. 
is detectable iff
is stabilizable.
The optimal 
:
and satisfies the Algebraic Ricatti Equation (ARE)
