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

We introduce 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 an input operator depending of the polynomial decay rate of a $C_0$-semigroup. We also show that bilinear systems are polynomially integral input-to-state stable under a certain smoothing assumption on nonlinear operators.