Robustness of Polynomial Stability with Respect to Sampling

TL;DR

This paper investigates the robustness of polynomial stability in infinite-dimensional systems under sampling, establishing conditions for stability preservation and decay rate estimation with fast sampling and smooth initial states.

Contribution

It provides new conditions ensuring strong stability and decay rate estimates for sampled-data systems with Riesz-spectral generators approaching the imaginary axis.

Findings

Strong stability is preserved under sufficiently fast sampling.

Decay rate of the sampled system can be estimated with smooth initial states.

Conditions are identified for polynomial decay transfer under sampling.

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.