Formulas relating script K sign ℒ stability estimates of discrete-time and sampled-data nonlinear systems

We provide an explicit script K sign ℒ stability or input-to-state stability (ISS) estimate for a sampled-data nonlinear system in terms of the script K sign ℒ estimate for the corresponding discrete-time system and a script K sign function describing inter-sample growth. It is quite obvious that a uniform inter-sample growth condition, plus an ISS property for the exact discrete-time model of a closed-loop system, implies uniform ISS of the sampled-data nonlinear system. Our results serve to quantify these facts by means of comparison functions. Our results can be used as an alternative to prove and extend results in [1] or extend some results in [4] to a class of nonlinear systems. Finally, the formulas we establish can be used as a tool for some other problems which we indicate.