Robustness of polynomial stability with respect to sampling^*

Graduate School of System Informatics, Kobe University, Nada, Kobe, Hyogo 657-8501, Japan

^** Corresponding author: wakaiki@ruby.kobe-u.ac.jp

Received: 25 November 2022
Accepted: 28 April 2023

Abstract

We provide a partially affirmative answer to the following question on robustness of polynomial stability with respect to sampling: “Suppose that a continuous-time state-feedback controller achieves the polynomial stability of the infinite-dimensional linear system. We apply an idealized sampler and a zero-order hold to a feedback loop around the controller. Then, is the sampled-data system strongly stable for all sufficiently small sampling periods? Furthermore, is the polynomial decay of the continuous-time system transferred to the sampled-data system under sufficiently fast sampling?” The generator of the open-loop system is assumed to be a Riesz-spectral operator whose eigenvalues are not on the imaginary axis but may approach it asymptotically. We provide conditions for strong stability to be preserved under fast sampling. Moreover, we estimate the decay rate of the state of the sampled-data system with a smooth initial state and a sufficiently small sampling period.