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We introduce the notions of semi-uniform input-to-state stability and itssubclass, polynomial input-to-state stability, for infinite-dimensionalsystems. We establish a characterization of semi-uniform input-to-statestability based on attractivity properties as in the uniform case. Sufficientconditions for linear systems to be polynomially input-to-state stable areprovided, which restrict the range of the input operator depending on the rateof polynomial decay of the product of the semigroup and the resolvent of itsgenerator. We also show that a class of bilinear systems are polynomiallyintegral input-to-state stable under a certain smoothness assumption onnonlinear operators.

Semi-uniform Input-to-state Stability of Infinite-dimensional Systems