Equivalent types of ISS Lyapunov functions for discontinuous discrete-time systems

In this note, we study input-to-state stability (ISS) of discontinuous discrete-time systems via ISS Lyapunov functions (ISS LFs). For continuous discrete-time systems it is well-known that the existence of a dissipation-form ISS LF is equivalent to the existence of an implication-form ISS LF. For discontinuous discrete-time systems it was recently shown that this equivalence is no longer satisfied, and a stronger definition of an implication-form ISS LF is introduced. Moreover, we consider max-form ISS LFs. Here, we give a sufficient and necessary condition under which the existence of all these three forms of ISS LFs are equivalent in the sense that the existence of each of these forms implies the existence of both the other forms. Most importantly, this condition, called global K-boundedness, is shown to be also a necessary condition for ISS. To give a complete characterization we consider the case of ISS with respect to two measurement functions, which includes ISS with respect to a single measure and classic ISS as special cases.