Variants of Two-Measure Input-to-State Stability

Abstract

For attracting closed sets that are not compact we consider different formulations of input-to-state stability properties and related Lyapunov criteria. In the compact case, there exists a well-established hierarchy of such properties and some formulations have been quickly discarded as equivalent to input-to-state stability. In this paper we show for the noncompact case, several new phenomena appear. In particular, input-to-state stability (ISS) does not imply integral input-to-state stability, and ISS is not equivalent to integral-input-to-integral-state stability. The criteria are formulated in terms of measurement functions, which allows a uniform presentation of a number of related results.