Input-to-State Stability of a Nonlinear Discrete-time System via R-cycles

The input-to-state stability of a particular nonlinear discrete-time system is investigated using a construct to which we refer as an R-cycle. Informally speaking, an R-cycle is a finite subsequence of a state trajectory for which the first and last elements of the subsequence lie in a given set R. We first provide a formal definition of an R-cycle, along with appropriate sufficient conditions to guarantee the existence of R-cycles for the system under investigation. Next, we prove a useful bound on the Euclidean norm of the state for a single R-cycle. By then viewing the full state trajectory as a concatenatation of R-cycles, we are then able to construct a bound on the l_infin norm of the full state trajectory