Stabilizability, uncertainty and the choice of sampling rate

We shall in this contribution study a class of first-order sampled-data control systems with unknown nonlinear structure and with sampling rate not necessary fast enough, aiming at understanding how stabilizability depends quantitatively upon the choice of the sampling rate and the “size” of the uncertainty. We shall show that if the unknown nonlinear function has a linear growth rate with its “slope” (denoted by L) being a measure of the “size” of uncertainty, then the sampling rate should not exceed 1/L multiplied by a constant (≈7.53) for the system to be globally stabilizable. If, however, the unknown nonlinear function has a growth rate faster than linear, and if the system is disturbed by noises modeled as the standard Brownian motion, then an example is given, showing that the corresponding sampled-data system is not stabilizable in general, no matter how fast the sampling rate is