Rapid stabilizability of infinite-dimensional control systems with time delays

TL;DR

This paper proves that linear infinite-dimensional control systems with constant delays can be rapidly stabilized using static feedback, and the delay does not affect this property.

Contribution

It demonstrates that delay terms do not hinder rapid stabilizability and that static feedback suffices for stabilization in such systems.

Findings

Delay does not affect rapid stabilizability.

Static feedback achieves stabilization.

Results apply to systems with compact semigroup generators.

Abstract

In this paper, we investigate the rapid stabilizability of linear infinite-dimensional control systems with constant delays. Under the assumptions that the state operator generates an immediately compact semigroup and that the delay coefficient is constant, we establish two main results: (i) the presence of a time-delay term does not affect the rapid stabilizability of the control system, that is, this property depends only on the state and control operators; (ii) static feedback is sufficient to achieve rapid stabilization of the system. Applications are also presented.