  We introduce the notions of semi-uniform input-to-state stability and its
subclass, polynomial input-to-state stability, for infinite-dimensional
systems. We establish a characterization of semi-uniform input-to-state
stability based on attractivity properties as in the uniform case. Sufficient
conditions for linear systems to be polynomially input-to-state stable are
provided, which restrict the range of the input operator depending on the rate
of polynomial decay of the product of the semigroup and the resolvent of its
generator. We also show that a class of bilinear systems are polynomially
integral input-to-state stable under a certain smoothness assumption on
nonlinear operators.
