Restatements of input-to-state stability in infinite dimensions : what goes wrong ?

We show by means of counterexamples that many characterizations of input-to-state stability (ISS) known for ODE systems are not valid for general differential equations in Banach spaces. Moreover, these notions or combinations of notions are not equivalent to each other, and can be classified into several groups according to the type and grade of nonuniformity. We introduce the new notion of strong ISS which is equivalent to ISS in the ODE case, but which is strictly weaker than ISS in the infinite-dimensional setting. We characterize strong ISS as a strong asymptotic gain property plus global stability.

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