On Asymptotic and Finite-Time Stabilization of Bilinear Systems

TL;DR

This paper investigates stabilization techniques for infinite-dimensional bilinear control systems, covering asymptotic, finite-time, and exponential stabilization, with emphasis on observability and space structure differences.

Contribution

It introduces new stabilization results for bilinear systems in various infinite-dimensional spaces and discusses open problems and applications.

Findings

Analyzed weak, strong, and polynomial stabilization in Hilbert spaces.

Studied exponential stabilization challenges in Banach spaces.

Presented recent results and open problems in finite-time stabilization.

Abstract

This manuscript addresses the analysis and design of feedback laws for the stabilization of bilinear control systems in infinite-dimensional spaces. It first examines weak, strong, and polynomial stabilization within a Hilbert space framework, emphasizing the role of observability conditions. It then studies exponential stabilization in Banach spaces, highlighting the additional challenges arising from the lack of a Hilbertian structure. Finally, it introduces finite-time stabilization, presenting recent results and open problems within the broader context of nonlinear infinite-dimensional control theory. Several applications are also discussed.