  In this work, the relation between input-to-state stability and integral
input-to-state stability is studied for linear infinite-dimensional systems
with an unbounded control operator. Although a special focus is laid on the
case $L^{\infty}$, general function spaces are considered for the inputs. We
show that integral input-to-state stability can be characterized in terms of
input-to-state stability with respect to Orlicz spaces. Since we consider
linear systems, the results can also be formulated in terms of admissibility.
For parabolic diagonal systems with scalar inputs, both stability notions with
respect to $L^\infty$ are equivalent.
