Suppose that a static-state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-and-hold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space
X
X
and the control space
U
U
are Hilbert spaces, the system is of the form
x
˙
(
t
)
=
A
x
(
t
)
+
B
u
(
t
)
\dot x(t) = Ax(t) + Bu(t)
, where
A
A
is the generator of a strongly continuous semigroup on
X
X
, and the continuous time feedback is
u
(
t
)
=
F
x
(
t
)
u(t) = Fx(t)
. The answer to the above question is known to be “yes” if
X
X
and
U
U
are finite-dimensional spaces. In the infinite-dimensional case, if
F
F
is not compact, then it is easy to find counterexamples. Therefore, we restrict attention to compact feedback. We show that the answer to the above question is “yes”, if
B
B
is a bounded operator from
U
U
into
X
X
. Moreover, if
B
B
is unbounded, we show that the answer “yes” remains correct, provided that the semigroup generated by
A
A
is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.