On stability and input-to-state stability KL estimates of discrete-time and sampled-data nonlinear systems

We provide an explicit KL stability or input-to-state stability (ISS) estimate for a sampled-data nonlinear system in terms of the KL estimate for the corresponding discrete-time system and a K function describing inter-sample growth. It is straightforward to show that a uniform intersample 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 this fact by means of comparison functions. The formulas we establish are an important analysis tool for several problems which we indicate.